gam_terms/

term_builder.rs

//! Term construction: bridge from parsed formula terms to `TermCollectionSpec`.
//!
//! This module takes the AST produced by `inference::formula_dsl` and a loaded
//! dataset, resolves column references, infers knot counts and center strategies,
//! and produces a `TermCollectionSpec` ready for `build_term_collection_design`.

use std::collections::{BTreeMap, BTreeSet, HashMap};
use std::path::PathBuf;

use ndarray::{Array2, ArrayView1};

use crate::basis::{
    BSplineBasisSpec, BSplineBoundaryConditions, BSplineEndpointBoundaryCondition,
    BSplineIdentifiability, BSplineKnotSpec, CenterCountRequest, CenterStrategy,
    ConstantCurvatureBasisSpec, ConstantCurvatureIdentifiability, DuchonBasisSpec,
    DuchonNullspaceOrder, DuchonOperatorPenaltySpec, DuchonSpectralBasis, MaternBasisSpec,
    MaternIdentifiability, MaternLengthScale, MaternNu, MeasureJetBasisSpec,
    MeasureJetIdentifiability, OneDimensionalBoundary, SpatialIdentifiability, SphereMethod,
    SphereWahbaKernel, SphericalSplineBasisSpec, SphericalSplineIdentifiability,
    ThinPlateBasisSpec, auto_spatial_center_strategy, count_unique_coordinate_rows,
    default_num_centers, default_spatial_center_strategy, default_spherical_harmonic_degree,
    plan_spatial_basis, select_r_uniform_subsample_centers, thin_plate_penalty_order,
};
use crate::inference::formula_dsl::{
    ParsedTerm, SmoothKind, option_bool, option_f64, option_f64_strict, option_usize,
    option_usize_any, option_usize_any_strict, option_usize_strict, strip_quotes,
};
use crate::smooth::{
    BySmoothKind, ByVarKind, ByVariableSpec, FactorSmoothFlavour, FactorSmoothSpec,
    LinearCoefficientGeometry, LinearTermSpec, RandomEffectTermSpec, ShapeConstraint,
    SmoothBasisSpec, SmoothTermSpec, TensorBSplineIdentifiability,
    TensorBSplinePenaltyDecomposition, TensorBSplineSpec, TermCollectionSpec,
};
use gam_data::{ColumnKindTag, DataError, EncodedDataset as Dataset};
use gam_problem::types::ColIdx;
use gam_runtime::resource::ResourcePolicy;

/// Default B-spline degree when a smooth's `degree=` option is absent. Cubic
/// (degree 3) is the standard GAM convention: C² continuity with a low knot
/// count.
const DEFAULT_BSPLINE_DEGREE: usize = 3;

/// Default difference-penalty order when a smooth's `penalty_order=` (alias
/// `m=`) option is absent. Second-order (curvature) is the standard P-spline
/// convention.
const DEFAULT_PENALTY_ORDER: usize = 2;

/// Admissible `lmax=` for the truncated Wahba sphere kernels, matching the
/// documented range on [`SphereWahbaKernel::SobolevTruncated`]. The lower end
/// keeps at least a few degrees of resolution; the upper end is the bound the
/// device kernel bakes in as a compile-time `#define`.
const SPHERE_TRUNCATION_LMAX_RANGE: std::ops::RangeInclusive<usize> = 5..=200;

/// Default basis dimension for one-dimensional cyclic cubic P-splines.
///
/// Periodic smooths spend no coefficients on free endpoints, so they should not
/// inherit the larger open B-spline knot ceiling by default.  This is still only
/// a default: callers can request a richer periodic space with `k=`.
const CYCLIC_DEFAULT_BASIS_DIM: usize = 12;

/// Default shared-marginal basis dimension for `bs="fs"`/`bs="sz"` factor smooths,
/// matching mgcv's factor-smooth default `k=10`. A factor smooth shares one
/// marginal across all levels; a modest basis recovers the shared signal without
/// over-fitting each group's within-group noise (gam#903). Overridden by an
/// explicit `k`/`basis_dim`.
const FACTOR_SMOOTH_DEFAULT_BASIS_DIM: usize = 10;

/// Default row-chunk size for the out-of-core PCA-basis smooth when the
/// `chunk_size=` option is absent. Streams the design in row blocks to bound
/// peak memory independent of the dataset row count.
const DEFAULT_PCA_CHUNK_SIZE: usize = 4096;

// ---------------------------------------------------------------------------
// Typed errors
// ---------------------------------------------------------------------------

/// Typed errors emitted by term-builder helpers. `Display` reproduces the exact
/// pre-refactor `format!(...)` text byte-for-byte, so callers that string-match
/// on the message (tests, log assertions) keep working unchanged. Public-API
/// functions still return `Result<_, String>` and use `.to_string()` shims at
/// their boundary to stay compatible with callers in protected modules.
#[derive(Clone, Debug)]
pub enum TermBuilderError {
    /// Column-resolution / column-kind lookup failures whose context is purely
    /// internal (column-kind table out-of-sync, alias map missing an entry,
    /// etc.). User-facing "this formula references a column that doesn't
    /// exist" diagnostics use the dedicated `ColumnNotFound` variant so the
    /// FFI boundary can lift the structured payload into a Python
    /// `ColumnNotFoundError` without parsing prose.
    MissingColumn { reason: String },
    /// A formula referenced a column that is not present in the input data.
    /// Mirrors `DataError::ColumnNotFound` field-for-field so the conversion
    /// across module boundaries is a pure data move (no re-derivation, no
    /// string re-parsing). Public callers see byte-identical `Display`
    /// output to the legacy `missing_column_message` text.
    ColumnNotFound {
        name: String,
        role: Option<String>,
        available: Vec<String>,
        similar: Vec<String>,
        tsv_hint: bool,
    },
    /// User-specified configuration is internally inconsistent (e.g. too few
    /// variables for a smooth type, conflicting size options, requested basis
    /// dimension below the polynomial nullspace).
    IncompatibleConfig { reason: String },
    /// Option parsing failure: malformed numeric expression, unknown option
    /// key, out-of-range integer, list-length mismatch, etc.
    InvalidOption { reason: String },
    /// User requested a feature that is intentionally not supported (unknown
    /// smooth type / method / kernel / identifiability, non-zero anchor,
    /// internal-only token, etc.).
    UnsupportedFeature { reason: String },
    /// Input data is degenerate for the requested term (constant column,
    /// non-finite categorical entries, ...).
    DegenerateData { reason: String },
    /// Term-collection-stage formula error — a node that the caller was
    /// supposed to resolve upstream reached the builder.
    MalformedFormula { reason: String },
}

impl std::fmt::Display for TermBuilderError {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        match self {
            TermBuilderError::MissingColumn { reason }
            | TermBuilderError::IncompatibleConfig { reason }
            | TermBuilderError::InvalidOption { reason }
            | TermBuilderError::UnsupportedFeature { reason }
            | TermBuilderError::DegenerateData { reason }
            | TermBuilderError::MalformedFormula { reason } => f.write_str(reason),
            // Delegate to the canonical `DataError::ColumnNotFound` formatter
            // so a single source of truth defines the human text. The
            // intermediate `DataError` constructed here owns its strings only
            // for the duration of the Display call — no allocation cost
            // beyond the original payload that this variant already holds.
            TermBuilderError::ColumnNotFound {
                name,
                role,
                available,
                similar,
                tsv_hint,
            } => {
                let canonical = DataError::ColumnNotFound {
                    name: name.clone(),
                    role: role.clone(),
                    available: available.clone(),
                    similar: similar.clone(),
                    tsv_hint: *tsv_hint,
                };
                std::fmt::Display::fmt(&canonical, f)
            }
        }
    }
}

impl From<TermBuilderError> for String {
    fn from(err: TermBuilderError) -> String {
        err.to_string()
    }
}

/// Catchall lift for the term-builder's internal `Result<_, String>` helpers
/// (numeric expression parsing, option lookup, boundary-condition parsing,
/// ...) that flow into `build_termspec` via `?`. Maps to
/// `IncompatibleConfig`, which is the most appropriate generic bucket for
/// option/config-style failures — leaf sites that emit structured payloads
/// (`From<DataError>` for column-not-found) bypass this fallback.
impl From<String> for TermBuilderError {
    fn from(reason: String) -> Self {
        Self::IncompatibleConfig { reason }
    }
}

/// Typed lift from data-layer errors. `DataError::ColumnNotFound` becomes
/// `TermBuilderError::ColumnNotFound` field-for-field — no stringification,
/// no information loss — so the FFI boundary downstream can dispatch on
/// the typed variant. Other `DataError` variants degrade into
/// `MissingColumn` since they describe column-resolution-time failures
/// without a dedicated structured destination.
impl From<DataError> for TermBuilderError {
    fn from(err: DataError) -> Self {
        match err {
            DataError::ColumnNotFound {
                name,
                role,
                available,
                similar,
                tsv_hint,
            } => Self::ColumnNotFound {
                name,
                role,
                available,
                similar,
                tsv_hint,
            },
            DataError::SchemaMismatch { reason }
            | DataError::ParseError { reason }
            | DataError::EncodingFailure { reason }
            | DataError::EmptyInput { reason }
            | DataError::InvalidValue { reason } => Self::MissingColumn { reason },
        }
    }
}

// Constructor helpers — keep error-site code compact and consistent.
impl TermBuilderError {
    #[inline]
    fn missing_column(reason: impl Into<String>) -> Self {
        TermBuilderError::MissingColumn {
            reason: reason.into(),
        }
    }
    #[inline]
    fn incompatible_config(reason: impl Into<String>) -> Self {
        TermBuilderError::IncompatibleConfig {
            reason: reason.into(),
        }
    }
    #[inline]
    fn invalid_option(reason: impl Into<String>) -> Self {
        TermBuilderError::InvalidOption {
            reason: reason.into(),
        }
    }
    #[inline]
    fn unsupported_feature(reason: impl Into<String>) -> Self {
        TermBuilderError::UnsupportedFeature {
            reason: reason.into(),
        }
    }
    #[inline]
    fn degenerate_data(reason: impl Into<String>) -> Self {
        TermBuilderError::DegenerateData {
            reason: reason.into(),
        }
    }
    #[inline]
    fn malformed_formula(reason: impl Into<String>) -> Self {
        TermBuilderError::MalformedFormula {
            reason: reason.into(),
        }
    }
}

// ---------------------------------------------------------------------------
// Column resolution
// ---------------------------------------------------------------------------

/// Resolve a bare column name to its index, returning a typed
/// `DataError::ColumnNotFound` on miss so the FFI boundary can surface a
/// structured `gamfit.ColumnNotFoundError(column=…, available=…)` rather
/// than rely on string-classification of human prose. Internal callers that
/// still flow `Result<_, String>` get byte-identical text via
/// `From<DataError> for String`.
pub fn resolve_col(col_map: &HashMap<String, usize>, name: &str) -> Result<usize, DataError> {
    col_map
        .get(name)
        .copied()
        .ok_or_else(|| DataError::column_not_found(col_map, name, None))
}

/// Like `resolve_col` but tags the missing-column payload with a role label
/// (`"response"`, `"entry"`, `"exit"`, `"event"`, `"z"`, `"id"`, …) so the
/// boundary-side Python exception can disambiguate which formula slot held
/// the bad reference.
pub fn resolve_role_col(
    col_map: &HashMap<String, usize>,
    name: &str,
    role: &str,
) -> Result<usize, DataError> {
    col_map
        .get(name)
        .copied()
        .ok_or_else(|| DataError::column_not_found(col_map, name, Some(role)))
}

fn encoded_levels_for_column(ds: &Dataset, col: ColIdx) -> Vec<(u64, String)> {
    let mut seen = BTreeSet::<u64>::new();
    for value in ds.values.column(col.get()) {
        if value.is_finite() {
            seen.insert(gam_data::canonical_level_bits(*value));
        }
    }
    let schema_levels = ds
        .schema
        .columns
        .get(col.get())
        .map(|column| column.levels.as_slice())
        .unwrap_or(&[]);
    seen.into_iter()
        .enumerate()
        .map(|(idx, bits)| {
            let fallback = format!("level{}", idx + 1);
            let label = schema_levels.get(idx).cloned().unwrap_or(fallback);
            (bits, label)
        })
        .collect()
}

/// Internal option key carrying the row count that n-scaling BASIS DEFAULTS
/// (radial center counts, spatial plans) must size from. A factor-by smooth
/// expands into per-level blocks that each see ONLY their level's rows, so
/// sizing the default from the pooled row count over-provisions every level —
/// measured on the #1561 by-group location-scale fixture: `s(x, bs='tp',
/// by=group)` at n=200 (100/group) got ~50 centers PER LEVEL, an
/// ill-conditioned 100-column mean block whose truth-recovery floor (0.111)
/// no λ could beat, while the same smooth sized for the level's own 100 rows
/// recovers to ~0.036. Explicit user `centers=`/`k=` bypass the default and
/// are unaffected. Stripped at the top of [`build_smooth_basis`] like
/// `__by_col`, so per-kind option allow-lists never see it.
const DEFAULT_SIZING_ROWS_OPTION: &str = "__default_sizing_rows";

/// The smallest per-level row count of a categorical by-column: the effective
/// sample size each by-level smooth block actually fits. `None` when the
/// column has no finite rows (callers fall back to the pooled count).
fn min_categorical_by_level_rows(ds: &Dataset, by_col: usize) -> Option<usize> {
    let mut counts: BTreeMap<u64, usize> = BTreeMap::new();
    for value in ds.values.column(by_col) {
        if value.is_finite() {
            *counts
                .entry(gam_data::canonical_level_bits(*value))
                .or_insert(0) += 1;
        }
    }
    counts.values().copied().min()
}

/// Insert [`DEFAULT_SIZING_ROWS_OPTION`] into `inner_options` when the by
/// column is categorical (numeric-by smooths keep one shared block over all
/// rows, so pooled sizing stays correct there).
fn inject_by_level_sizing_rows(
    inner_options: &mut BTreeMap<String, String>,
    ds: &Dataset,
    by_col: usize,
) {
    if matches!(
        ds.column_kinds.get(by_col).copied(),
        Some(ColumnKindTag::Categorical)
    ) && let Some(min_rows) = min_categorical_by_level_rows(ds, by_col)
    {
        inner_options.insert(DEFAULT_SIZING_ROWS_OPTION.to_string(), min_rows.to_string());
    }
}

pub fn column_map_with_alias(
    col_map: &HashMap<String, usize>,
    alias: &str,
    target_column: &str,
) -> HashMap<String, usize> {
    let mut aliased = col_map.clone();
    if let Some(idx) = col_map.get(target_column).copied() {
        aliased.entry(alias.to_string()).or_insert(idx);
    }
    aliased
}

/// The canonical marginal-slope alias: `z` in a formula binds to the column
/// named by `z_column`.
pub const MARGINAL_SLOPE_Z_ALIAS: &str = "z";

/// Whether writing `z` in a formula would resolve to `z_column` for this frame.
///
/// `column_map_with_alias` inserts with `or_insert`, so a frame that carries its
/// own real `z` column keeps it and the alias is inert — writing `z` there means
/// that column, which is legitimate. The alias is live only when `z_column`
/// exists and `z` does not, and only then does `z` silently denote the score.
pub fn marginal_slope_z_alias_is_live(col_map: &HashMap<String, usize>, z_column: &str) -> bool {
    col_map.contains_key(z_column) && !col_map.contains_key(MARGINAL_SLOPE_Z_ALIAS)
}

// ---------------------------------------------------------------------------
// ParsedTerm[] + Dataset → TermCollectionSpec
// ---------------------------------------------------------------------------

pub fn build_termspec(
    terms: &[ParsedTerm],
    ds: &Dataset,
    col_map: &HashMap<String, usize>,
    inference_notes: &mut Vec<String>,
    policy: &ResourcePolicy,
) -> Result<TermCollectionSpec, TermBuilderError> {
    let mut linear_terms = Vec::<LinearTermSpec>::new();
    let mut random_terms = Vec::<RandomEffectTermSpec>::new();
    let mut smooth_terms = Vec::<SmoothTermSpec>::new();
    let smooth_coordinate_count = terms
        .iter()
        .map(|term| match term {
            ParsedTerm::Smooth { vars, .. } => vars.len(),
            _ => 0,
        })
        .sum::<usize>();

    for t in terms {
        match t {
            ParsedTerm::Linear {
                name,
                explicit,
                double_penalty,
                coefficient_min,
                coefficient_max,
            } => {
                let col = resolve_col(col_map, name)?;
                let auto_kind = ds.column_kinds.get(col).copied().ok_or_else(|| {
                    TermBuilderError::missing_column(format!(
                        "internal column-kind lookup failed for '{name}'"
                    ))
                    .to_string()
                })?;
                if *explicit {
                    linear_terms.push(LinearTermSpec {
                        name: name.clone(),
                        feature_col: col,
                        feature_cols: vec![col],
                        categorical_levels: vec![],
                        // Parametric terms are unpenalized/MLE by default.
                        // `double_penalty=true` is an explicit shrinkage choice
                        // carried by the parsed term.
                        double_penalty: *double_penalty,
                        coefficient_geometry: LinearCoefficientGeometry::Unconstrained,
                        coefficient_min: *coefficient_min,
                        coefficient_max: *coefficient_max,
                        frozen_function_mass: None,
                    });
                } else {
                    match auto_kind {
                        ColumnKindTag::Continuous | ColumnKindTag::Binary => {
                            linear_terms.push(LinearTermSpec {
                                name: name.clone(),
                                feature_col: col,
                                feature_cols: vec![col],
                                categorical_levels: vec![],
                                // Preserve the parser's explicit opt-in. Bare
                                // numeric terms arrive as `false`.
                                double_penalty: *double_penalty,
                                coefficient_geometry: LinearCoefficientGeometry::Unconstrained,
                                coefficient_min: *coefficient_min,
                                coefficient_max: *coefficient_max,
                                frozen_function_mass: None,
                            });
                        }
                        ColumnKindTag::Categorical => {
                            if coefficient_min.is_some() || coefficient_max.is_some() {
                                return Err(TermBuilderError::incompatible_config(format!(
                                    "coefficient constraints are not supported for categorical auto-random-effect term '{name}'; use group({name}) or an unconstrained numeric term"
                                )));
                            }
                            random_terms.push(RandomEffectTermSpec {
                                name: name.clone(),
                                feature_col: col,
                                drop_first_level: false,
                                penalized: true,
                                frozen_levels: None,
                                // A BARE categorical main effect (`+ g`) is a FIXED
                                // parametric factor. Although it is auto-promoted to
                                // a penalized random block above, an *unseen* level
                                // at predict must raise a schema mismatch rather than
                                // be mapped to the factor's centering point (#2102).
                                lenient_unseen: false,
                            });
                        }
                    }
                }
            }
            ParsedTerm::BoundedLinear {
                name,
                min,
                max,
                prior,
                double_penalty,
            } => {
                let col = resolve_col(col_map, name)?;
                let auto_kind = ds.column_kinds.get(col).copied().ok_or_else(|| {
                    TermBuilderError::missing_column(format!(
                        "internal column-kind lookup failed for '{name}'"
                    ))
                    .to_string()
                })?;
                if !matches!(auto_kind, ColumnKindTag::Continuous | ColumnKindTag::Binary) {
                    return Err(TermBuilderError::incompatible_config(format!(
                        "bounded() currently supports only numeric columns, got categorical '{name}'"
                    )));
                }
                linear_terms.push(LinearTermSpec {
                    name: name.clone(),
                    feature_col: col,
                    feature_cols: vec![col],
                    categorical_levels: vec![],
                    double_penalty: *double_penalty,
                    coefficient_geometry: LinearCoefficientGeometry::Bounded {
                        min: *min,
                        max: *max,
                        prior: prior.clone(),
                    },
                    coefficient_min: None,
                    coefficient_max: None,
                    frozen_function_mass: None,
                });
            }
            ParsedTerm::RandomEffect {
                name,
                lenient_unseen,
            } => {
                let col = resolve_col(col_map, name)?;
                random_terms.push(RandomEffectTermSpec {
                    name: name.clone(),
                    feature_col: col,
                    drop_first_level: false,
                    penalized: true,
                    frozen_levels: None,
                    // Unseen-level policy is fixed by the wrapper the user wrote
                    // (`formula_dsl`): a genuine random effect
                    // (`group(g)`/`re(g)`/`s(g, bs="re")`) shrinks a held-out
                    // group to the population mean and so tolerates unseen
                    // levels; a fixed `factor(g)`, like a bare `+ g` categorical
                    // main effect, must reject an unseen level rather than
                    // collapse onto the centering point (#2137/#2102).
                    lenient_unseen: *lenient_unseen,
                });
            }
            ParsedTerm::Smooth {
                label,
                vars,
                kind,
                options,
            } => {
                let smooth_vars = vars.clone();
                let by_name = options.get("by").cloned();
                // `bs="sz"` (sum-to-zero), like `bs="fs"`/`bs="re"`, is a
                // factor-smooth family handled natively by `build_smooth_basis`'s
                // fs/sz/re path: it detects the categorical factor among the
                // variables and emits a `SmoothBasisSpec::FactorSmooth { Sz }`
                // with the correct single-penalty marginal and modest default
                // basis. Route sz straight through `build_smooth_basis` rather
                // than intercepting it into a legacy `FactorSumToZero` envelope
                // here (which left `sz(fac, x)` mis-typed as `FactorSumToZero`
                // instead of the expected `FactorSmooth { Sz }`).
                let cols = smooth_vars
                    .iter()
                    .map(|v| resolve_col(col_map, v))
                    .collect::<Result<Vec<_>, _>>()?;
                let mut inner_options = options.clone();
                inner_options.remove("by");
                // `ordered=` is consumed here (ByVarKind::Factor routing) and
                // must not propagate to the inner basis builder, which has no
                // allow-list entry for it and would reject it as an unknown option.
                inner_options.remove("ordered");
                // Pop the shape constraint before `build_smooth_basis` runs so
                // it never reaches the per-kind `validate_known_options`
                // allow-lists (the constraint is a property of the smooth term,
                // not of any one basis kind). Basis-incompatible requests still
                // fail loudly downstream via `shape_supports_basis`.
                let shape = match inner_options.remove("shape") {
                    None => ShapeConstraint::None,
                    Some(raw) => crate::smooth::parse_shape_constraint(&raw)
                        .map_err(TermBuilderError::invalid_option)?,
                };
                // A categorical by= expands into per-level blocks below; size
                // the inner basis's n-scaling defaults from the smallest
                // level's rows, not the pooled count (see
                // `DEFAULT_SIZING_ROWS_OPTION`).
                if let Some(by_name) = by_name.as_deref() {
                    let by_col = resolve_col(col_map, by_name)?;
                    inject_by_level_sizing_rows(&mut inner_options, ds, by_col);
                }
                let inner_basis = build_smooth_basis(
                    *kind,
                    &smooth_vars,
                    &cols,
                    &inner_options,
                    ds,
                    inference_notes,
                    policy,
                    smooth_coordinate_count,
                )?;
                // `bs="sz"` deliberately stays typed as `SmoothBasisSpec::FactorSmooth
                // { Sz }` (#1403, owner-confirmed in #1887): the `FactorSumToZero`
                // envelope is the *legacy, mis-typed* representation. `build_factor_smooth`
                // reuses the sum-to-zero construction internally as its single source of
                // truth for the zero-sum geometry (term_specs.rs) while keeping the
                // freeze-consistent `FactorSmooth` metadata shape shared by fs/sz/re, so
                // there is no reason to re-wrap the spec into the legacy envelope here —
                // doing so (#1981) mis-typed `sz(fac, x)` back to `FactorSumToZero` and
                // broke the refit/predict freeze path's `(FactorSmooth, …)` metadata match.
                if let Some(by_name) = by_name {
                    let by_col = resolve_col(col_map, &by_name)?;
                    match ds.column_kinds.get(by_col).copied().ok_or_else(|| {
                        format!("internal column-kind lookup failed for by variable '{by_name}'")
                    })? {
                        ColumnKindTag::Categorical => {
                            let levels = encoded_levels_for_column(ds, ColIdx::new(by_col));
                            // A penalized random block for this factor already
                            // owns its full level offsets when EITHER an explicit
                            // `group(factor)` appears, OR a *bare* categorical
                            // `+ factor` does — the latter is auto-promoted to a
                            // penalized random-effect block (see the
                            // `ParsedTerm::Linear` / `ColumnKindTag::Categorical`
                            // arm above, `penalized: true`). Both representations
                            // carry the same per-level offsets, so #1457: the
                            // `by=` branch must NOT additionally add its own
                            // unpenalized treatment-coded main effect, which would
                            // double-represent the factor (two `g` design blocks +
                            // a spurious extra smoothing parameter).
                            let penalized_group_owner_present =
                                terms.iter().any(|other| match other {
                                    ParsedTerm::RandomEffect { name, .. } => name == &by_name,
                                    ParsedTerm::Linear {
                                        name,
                                        explicit: false,
                                        ..
                                    } if name == &by_name => col_map
                                        .get(name)
                                        .and_then(|c| ds.column_kinds.get(*c).copied())
                                        .map(|kind| matches!(kind, ColumnKindTag::Categorical))
                                        .unwrap_or(false),
                                    _ => false,
                                });
                            // Add an unpenalized treatment-coded fixed main
                            // effect for a standalone factor-by smooth, unless
                            // the same factor already has an explicit
                            // `group(factor)` term OR a bare categorical `+
                            // factor` that was auto-promoted to a penalized
                            // random block (#1457).  In those mixed-model forms
                            // the penalized random intercept is the coherent
                            // owner of level offsets; adding a no-pooling fixed
                            // factor effect would bypass random-effect
                            // shrinkage and degrade BLUP-style predictions.
                            if !random_terms.iter().any(|rt| rt.name == by_name)
                                && !penalized_group_owner_present
                            {
                                random_terms.push(RandomEffectTermSpec {
                                    name: by_name.clone(),
                                    feature_col: by_col,
                                    drop_first_level: true,
                                    penalized: false,
                                    frozen_levels: None,
                                    // Unpenalized treatment-coded FIXED factor main
                                    // effect for a factor-by smooth: an unseen level
                                    // is out of contract and must raise, not center
                                    // (#2102).
                                    lenient_unseen: false,
                                });
                            }
                            // Unordered factor-by smooths are independent
                            // level-specific smooths. Preserve that
                            // term-spec structure explicitly so later
                            // hierarchy/identifiability passes can see the
                            // per-level ownership rather than a generic
                            // BySmooth envelope.
                            for (level_bits, level_label) in levels {
                                smooth_terms.push(SmoothTermSpec {
                                    frozen_parametric_residualization: None,
                                    name: format!("{label}:by={by_name}[{level_label}]"),
                                    basis: SmoothBasisSpec::ByVariable {
                                        inner: Box::new(inner_basis.clone()),
                                        by_col,
                                        kind: BySmoothKind::Level { level_bits },
                                        by: ByVariableSpec::Level {
                                            value_bits: level_bits,
                                            label: level_label,
                                        },
                                    },
                                    shape: shape.clone(),
                                    joint_null_rotation: None,
                                });
                            }
                        }
                        ColumnKindTag::Binary | ColumnKindTag::Continuous => {
                            smooth_terms.push(SmoothTermSpec {
                                frozen_parametric_residualization: None,
                                name: label.clone(),
                                basis: SmoothBasisSpec::ByVariable {
                                    inner: Box::new(inner_basis),
                                    by_col,
                                    kind: BySmoothKind::Numeric,
                                    by: ByVariableSpec::Numeric,
                                },
                                shape,
                                joint_null_rotation: None,
                            });
                        }
                    }
                } else {
                    smooth_terms.push(SmoothTermSpec {
                        frozen_parametric_residualization: None,
                        name: label.clone(),
                        basis: inner_basis,
                        shape,
                        joint_null_rotation: None,
                    });
                }
            }
            ParsedTerm::LinkWiggle { .. }
            | ParsedTerm::TimeWiggle { .. }
            | ParsedTerm::LinkConfig { .. }
            | ParsedTerm::SurvivalConfig { .. } => {
                // Consumed at formula level, not design terms.
            }
            ParsedTerm::LogSlopeSurface { .. } => {
                return Err(TermBuilderError::malformed_formula(
                    "logslope(...) declarations must be resolved by the marginal-slope formula path before building a term spec",
                ));
            }
            ParsedTerm::Interaction {
                vars,
                double_penalty,
            } => {
                // A linear `:` interaction realizes one design column equal to
                // the elementwise product of its operands. Numeric (continuous/
                // binary) operands multiply directly; a categorical operand is
                // a factor, so the product is expanded factor-aware: one design
                // column per surviving cell of the factor(s), each an indicator
                // `1[factor == level]` gating the numeric product.
                //
                // Coding is MARGINALITY-AWARE (gam#1158, gam#1159). A categorical
                // operand `g` is treatment-coded (its lexicographically first
                // reference level dropped) ONLY when the lower-order term obtained
                // by removing `g` from this interaction is also present in the
                // model — that lower-order term is what makes the dropped level
                // identifiable, exactly mgcv's marginality rule. When that parent
                // is ABSENT (the interaction-only form), dropping the reference
                // level instead pins a group to the reference fit (a rank-deficient
                // design), so we keep ALL levels (full dummy coding) and rely on a
                // single intercept cell-drop below for identifiability:
                //   * `y ~ x:g` with no `x` main effect → "common intercept,
                //     separate slopes": every group keeps its own x-slope.
                //   * `y ~ g:h` with no `g`/`h` main effects → the saturated
                //     cell-means model: full cross of all levels minus one
                //     reference cell absorbed by the intercept.
                // When the parents ARE present (`x + x:g`, or `g*h` = `g + h +
                // g:h`), the historical treatment coding is preserved so those
                // forms stay correct.
                //
                // A main effect for var V is a `Linear`/`BoundedLinear`/
                // `RandomEffect` ParsedTerm whose referenced name is V (an
                // auto-detected categorical `Linear` becomes a RandomEffect main
                // effect; either spelling counts). We only treat such standalone
                // main-effect terms as parents — not V appearing inside another
                // interaction.
                let main_effect_present = |target: &str| -> bool {
                    terms.iter().any(|other| match other {
                        ParsedTerm::Linear { name, .. }
                        | ParsedTerm::BoundedLinear { name, .. }
                        | ParsedTerm::RandomEffect { name, .. } => name == target,
                        _ => false,
                    })
                };
                // The lower-order parent of dropping operand `drop_var` from this
                // interaction is present iff EVERY other operand is a main effect.
                // For the two cases we care about (`x:g`, `g:h`) the interaction
                // has two operands, so this reduces to "is the single remaining
                // operand a main effect"; the general form handles any arity.
                let parent_present = |drop_var: &str| -> bool {
                    vars.iter()
                        .filter(|v| v.as_str() != drop_var)
                        .all(|v| main_effect_present(v))
                };

                let mut numeric_cols = Vec::<usize>::new();
                // Per categorical operand: (var name, col, kept levels, was the
                // reference level dropped / treatment-coded?).
                let mut categorical_factors =
                    Vec::<(String, usize, Vec<(u64, String)>, bool)>::new();
                for var in vars {
                    let col = resolve_col(col_map, var)?;
                    let kind = ds.column_kinds.get(col).copied().ok_or_else(|| {
                        TermBuilderError::missing_column(format!(
                            "internal column-kind lookup failed for '{var}'"
                        ))
                        .to_string()
                    })?;
                    match kind {
                        ColumnKindTag::Continuous | ColumnKindTag::Binary => numeric_cols.push(col),
                        ColumnKindTag::Categorical => {
                            let mut levels = encoded_levels_for_column(ds, ColIdx::new(col));
                            // Treatment-code (drop the reference level) only when
                            // the marginal parent that identifies it is present;
                            // otherwise keep every level (full dummy coding).
                            let treatment_coded = parent_present(var);
                            if treatment_coded && levels.len() > 1 {
                                levels.remove(0);
                            }
                            if levels.is_empty() {
                                return Err(TermBuilderError::incompatible_config(format!(
                                    "interaction `{}` references categorical column `{var}` with no usable levels",
                                    vars.join(":")
                                )));
                            }
                            categorical_factors.push((var.clone(), col, levels, treatment_coded));
                        }
                    }
                }

                let label = vars.join(":");

                if categorical_factors.is_empty() {
                    // Pure numeric `:` interaction — single product column,
                    // identical to the historical behaviour.
                    linear_terms.push(LinearTermSpec {
                        name: label,
                        feature_col: numeric_cols[0],
                        feature_cols: numeric_cols,
                        categorical_levels: vec![],
                        // Interactions are recoverable as zero by default.
                        double_penalty: *double_penalty,
                        coefficient_geometry: LinearCoefficientGeometry::Unconstrained,
                        coefficient_min: None,
                        coefficient_max: None,
                        frozen_function_mass: None,
                    });
                    inference_notes.push(format!(
                        "wired linear interaction `{}` as product of numeric columns",
                        vars.join(":")
                    ));
                } else {
                    // Factor-aware expansion: cartesian product over the kept
                    // levels of every categorical operand. Each cell yields one
                    // column gating the numeric product (or, with no numeric
                    // operand, a pure cell indicator).
                    let mut cells: Vec<Vec<(usize, u64, String)>> = vec![Vec::new()];
                    for (_var, col, levels, _treatment_coded) in &categorical_factors {
                        let mut next = Vec::with_capacity(cells.len() * levels.len());
                        for cell in &cells {
                            for (bits, level_label) in levels {
                                let mut extended = cell.clone();
                                extended.push((*col, *bits, level_label.clone()));
                                next.push(extended);
                            }
                        }
                        cells = next;
                    }

                    // Intercept-identifiability cell drop. When the cells are PURE
                    // INDICATORS (no numeric operand) and at least one factor was
                    // dummy-coded (kept all its levels), the full set of cell
                    // columns sums to the all-ones intercept and is rank-deficient
                    // against it. Drop exactly ONE reference cell — the cell where
                    // every factor sits at its reference (lexicographically first)
                    // level — so the remaining saturated cells are identifiable
                    // (rank n_g*n_h - 1 cells + intercept). With a numeric operand
                    // the cells gate `x` and sum to `x`, not the intercept, so no
                    // cell is dropped (the collinearity there is with the absent
                    // `x` main effect, which is exactly why full coding is right).
                    let any_dummy_coded = categorical_factors
                        .iter()
                        .any(|(_, _, _, treatment_coded)| !*treatment_coded);
                    if numeric_cols.is_empty() && any_dummy_coded {
                        // The reference cell pairs each factor's column with the
                        // bits of its lexicographically-first (index 0) level.
                        let reference_cell: Vec<(usize, u64)> = categorical_factors
                            .iter()
                            .map(|(_, col, _, _)| {
                                let levels = encoded_levels_for_column(ds, ColIdx::new(*col));
                                (*col, levels[0].0)
                            })
                            .collect();
                        cells.retain(|cell| {
                            !reference_cell.iter().all(|(rcol, rbits)| {
                                cell.iter()
                                    .any(|(col, bits, _)| col == rcol && bits == rbits)
                            })
                        });
                    }

                    let n_cells = cells.len();
                    for cell in cells {
                        let cell_suffix = cell
                            .iter()
                            .map(|(_, _, level_label)| level_label.as_str())
                            .collect::<Vec<_>>()
                            .join(":");
                        let categorical_levels =
                            cell.iter().map(|(col, bits, _)| (*col, *bits)).collect();
                        // `feature_col` is required to point at a real column;
                        // use the first numeric operand when present, otherwise
                        // the first categorical column (its raw value is never
                        // multiplied — `realized_design_column` starts from ones
                        // and only gates by the level indicators).
                        let feature_col = numeric_cols
                            .first()
                            .copied()
                            .unwrap_or(categorical_factors[0].1);
                        linear_terms.push(LinearTermSpec {
                            name: format!("{label}:{cell_suffix}"),
                            feature_col,
                            feature_cols: numeric_cols.clone(),
                            categorical_levels,
                            double_penalty: *double_penalty,
                            coefficient_geometry: LinearCoefficientGeometry::Unconstrained,
                            coefficient_min: None,
                            coefficient_max: None,
                            frozen_function_mass: None,
                        });
                    }
                    let all_treatment_coded = !any_dummy_coded;
                    let coding = if all_treatment_coded {
                        "treatment-coded"
                    } else {
                        "marginality-aware (full dummy / saturated)"
                    };
                    inference_notes.push(format!(
                        "wired factor-aware linear interaction `{}` as {} {} cell column(s)",
                        vars.join(":"),
                        n_cells,
                        coding
                    ));
                }
            }
        }
    }

    Ok(TermCollectionSpec {
        linear_terms,
        random_effect_terms: random_terms,
        smooth_terms,
    })
}

fn split_list_option(raw: &str) -> Vec<String> {
    let t = raw.trim();
    // Accept the Python/JSON list form `[a, b]` AND mgcv's R-vector forms
    // `c(a, b)` / `(a, b)` as bracketed wrappers around a comma-separated body.
    // mgcv-style formulas pass per-margin numeric options as `k=c(5,5)` /
    // `period=c(2*pi, pi)`; without R-vector peeling here those entries were
    // split into `["c(5", "5)"]` and the downstream numeric parser then
    // misreported the leading garbage as the invalid digit.
    let inner = t
        .strip_prefix('[')
        .and_then(|u| u.strip_suffix(']'))
        .or_else(|| {
            t.strip_prefix("c(")
                .or_else(|| t.strip_prefix("C("))
                .or_else(|| t.strip_prefix('('))
                .and_then(|u| u.strip_suffix(')'))
        })
        .unwrap_or(t);
    inner
        .split(',')
        .map(|v| v.trim().to_string())
        .filter(|v| !v.is_empty())
        .collect()
}

fn parse_numeric_expr(raw: &str) -> Result<f64, String> {
    let mut acc = 1.0f64;
    let normalized = raw.replace(' ', "");
    if normalized.eq_ignore_ascii_case("none") {
        return Err("None is not numeric".to_string());
    }
    for factor in normalized.split('*') {
        if factor.is_empty() {
            return Err(format!("invalid numeric expression '{raw}'"));
        }
        let value = if factor.eq_ignore_ascii_case("pi") || factor == "π" {
            std::f64::consts::PI
        } else if factor.eq_ignore_ascii_case("tau") || factor == "τ" {
            std::f64::consts::TAU
        } else if let Some(prefix) = factor
            .strip_suffix("pi")
            .or_else(|| factor.strip_suffix("π"))
        {
            let coefficient = if prefix.is_empty() {
                1.0
            } else {
                prefix
                    .parse::<f64>()
                    .map_err(|err| format!("invalid numeric expression '{raw}': {err}"))?
            };
            coefficient * std::f64::consts::PI
        } else if let Some(prefix) = factor
            .strip_suffix("tau")
            .or_else(|| factor.strip_suffix("τ"))
        {
            let coefficient = if prefix.is_empty() {
                1.0
            } else {
                prefix
                    .parse::<f64>()
                    .map_err(|err| format!("invalid numeric expression '{raw}': {err}"))?
            };
            coefficient * std::f64::consts::TAU
        } else {
            factor
                .parse::<f64>()
                .map_err(|err| format!("invalid numeric expression '{raw}': {err}"))?
        };
        acc *= value;
    }
    Ok(acc)
}

/// Read an endpoint/period option as a numeric *expression* (`2*pi`, `tau`,
/// `0.5*tau`, `6.283185307179586`, ...) — the same grammar that `period=` and
/// `origin=` already accept via [`parse_numeric_expr`].
///
/// Returns `Ok(None)` when the key is absent, `Ok(Some(v))` when it parses, and
/// a hard `Err` when the key is *present but unparseable*. The crucial contrast
/// is with the lenient [`option_f64`], which collapses an unparseable value to
/// `None` and lets the caller silently substitute the data range — wrapping a
/// cyclic smooth at the wrong period with no diagnostic (the #815 failure mode).
fn option_numeric_expr(
    options: &BTreeMap<String, String>,
    key: &str,
) -> Result<Option<f64>, String> {
    match options.get(key) {
        None => Ok(None),
        Some(raw) => parse_numeric_expr(raw)
            .map(Some)
            .map_err(|err| format!("option `{key}={raw}` is not a valid numeric value: {err}")),
    }
}

fn parse_periods_option(
    options: &BTreeMap<String, String>,
    dim: usize,
) -> Result<Option<Vec<Option<f64>>>, String> {
    let Some(raw) = options.get("period") else {
        return Ok(None);
    };
    let values = split_list_option(raw);
    let mut periods = vec![None; dim];
    if values.len() == 1 && dim == 1 {
        periods[0] = Some(parse_numeric_expr(&values[0])?);
    } else {
        if values.len() != dim {
            return Err(format!(
                "period list length {} must match smooth dimension {}",
                values.len(),
                dim
            ));
        }
        for (i, v) in values.iter().enumerate() {
            if v.eq_ignore_ascii_case("none") {
                continue;
            }
            periods[i] = Some(parse_numeric_expr(v)?);
        }
    }
    Ok(Some(periods))
}

fn parse_periodic_axes_option(
    options: &BTreeMap<String, String>,
    dim: usize,
) -> Result<Option<Vec<Option<f64>>>, String> {
    let Some(raw_axes) = options.get("periodic") else {
        return Ok(None);
    };
    let mut periods = parse_periods_option(options, dim)?.unwrap_or_else(|| vec![None; dim]);
    // Scalar boolean form (`periodic=true` / `false`, `yes` / `no`) applies to
    // every axis — the documented per-axis-flag broadcast (see the doc on
    // `parse_periodic_axes`, the tensor sibling that already accepts it). A
    // 1-D `duchon(x, periodic=true)` lands here: the cyclic *domain* is then
    // resolved from the data range by `parse_cyclic_boundary` (the 1-D builder
    // consults `boundary` first), so a finite explicit period is NOT required —
    // we only need to NOT mis-read "true" as an axis index (#1074). `false`
    // means no axis is periodic.
    let lowered = raw_axes.trim().to_ascii_lowercase();
    if matches!(lowered.as_str(), "true" | "yes" | "y") {
        return Ok(Some(periods));
    }
    // `false` means NO axis is periodic. Return `None` — NOT
    // `Some(vec![None; dim])` — because the radial 1-D consumer treats a
    // `Some([None])` as "periodicity requested, derive the wrap period from
    // the data range" (see the Duchon builder arm below, which back-fills
    // `axes[0] = data_span` for a lone `None`) and the 1-D builder routes on
    // `spec.periodic.is_some()`. Emitting `Some([None])` here therefore
    // silently produced a *periodic* smooth for an explicit `periodic=false`
    // — the exact regression this branch now avoids, matching the bracketed
    // `[false]` form handled by the per-axis boolean block below.
    if matches!(lowered.as_str(), "false" | "no" | "n") {
        return Ok(None);
    }
    let axes = split_list_option(raw_axes);
    if axes.is_empty() {
        return Ok(Some(periods));
    }

    // Boolean forms `periodic=true` / `periodic=[true, false, ...]`, mirroring
    // `parse_tensor_periodic_axes`. The radial 1-D builders (`duchon`/`tps`/
    // `matern`) intentionally DERIVE the wrap period from the closed center
    // lattice when none is supplied (`prepare_periodic_duchon_centers_1d_with_period`,
    // gam#580: `None => span`), so a boolean-selected periodic axis legitimately
    // omits `period`. Without this branch, `duchon(x, periodic=true)`-style
    // radial formulas failed with the misleading "invalid periodic axis 'true'".
    let is_bool = |t: &str| {
        matches!(
            t.to_ascii_lowercase().as_str(),
            "true" | "yes" | "y" | "false" | "no" | "n"
        )
    };
    let is_truthy = |t: &str| matches!(t.to_ascii_lowercase().as_str(), "true" | "yes" | "y");

    // Scalar boolean: `periodic=true` / `periodic=false`.
    if axes.len() == 1 && is_bool(&axes[0]) {
        if !is_truthy(&axes[0]) {
            // Non-periodic: return None so the 1-D builder (which routes on
            // `spec.periodic.is_some()`) does NOT take the periodic path.
            return Ok(None);
        }
        // Every axis periodic; honor any explicit per-axis period, else leave
        // `None` for the caller (formula arm) / builder to derive the span.
        return Ok(Some(periods));
    }

    // Per-axis boolean list: `periodic=[true, false, ...]` (length must match dim).
    if axes.iter().all(|a| is_bool(a)) {
        if axes.len() != dim {
            return Err(format!(
                "periodic flag list length {} must match smooth dimension {dim}",
                axes.len()
            ));
        }
        if !axes.iter().any(|a| is_truthy(a)) {
            return Ok(None);
        }
        for (i, a) in axes.iter().enumerate() {
            if !is_truthy(a) {
                periods[i] = None;
            }
        }
        return Ok(Some(periods));
    }

    // Index-list form: `periodic=[0, 2]`. Each listed axis must carry an
    // explicit finite period — an index gives no per-axis span-derive hint.
    for a in &axes {
        let axis = a
            .parse::<usize>()
            .map_err(|err| format!("invalid periodic axis '{a}': {err}"))?;
        if axis >= dim {
            return Err(format!(
                "periodic axis {axis} out of range for {dim}D smooth"
            ));
        }
        if periods[axis].is_none() {
            return Err(format!(
                "periodic axis {axis} requires period[{axis}] to be finite"
            ));
        }
    }
    // Axes not listed are non-periodic even if period list has a finite placeholder.
    let listed: std::collections::BTreeSet<usize> = axes
        .iter()
        .filter_map(|a| a.parse::<usize>().ok())
        .collect();
    for i in 0..dim {
        if !listed.contains(&i) {
            periods[i] = None;
        }
    }
    Ok(Some(periods))
}

// ---------------------------------------------------------------------------
// Smooth basis spec construction
// ---------------------------------------------------------------------------

fn parse_option_list(raw: &str) -> Vec<String> {
    let trimmed = raw.trim();
    // Accept both the Python/JSON list form `[a, b]` and mgcv's R vector form
    // `c(a, b)` (and a bare `(a, b)`) as the bracketed wrapper around a
    // comma-separated option list. mgcv writes per-margin options as
    // `bs=c('tp','tp')` / `m=c(2,2)`, so the `c(...)` form must round-trip
    // through the same splitter the `[...]` form uses.
    let inner = trimmed
        .strip_prefix('[')
        .and_then(|v| v.strip_suffix(']'))
        .or_else(|| {
            trimmed
                .strip_prefix("c(")
                .or_else(|| trimmed.strip_prefix("C("))
                .or_else(|| trimmed.strip_prefix('('))
                .and_then(|v| v.strip_suffix(')'))
        })
        .unwrap_or(trimmed);
    inner
        .split(',')
        .map(|v| {
            v.trim()
                .trim_matches('"')
                .trim_matches('\'')
                .to_ascii_lowercase()
        })
        .filter(|v| !v.is_empty())
        .collect()
}

fn parse_periodic_axes(
    options: &BTreeMap<String, String>,
    dim: usize,
) -> Result<Vec<bool>, String> {
    let mut axes = vec![false; dim];
    if let Some(raw) = options.get("periodic").or_else(|| options.get("cyclic")) {
        let lowered = raw.trim().to_ascii_lowercase();
        if matches!(lowered.as_str(), "true" | "yes" | "y") {
            axes.fill(true);
            return Ok(axes);
        }
        // `false` leaves every axis non-periodic, which `axes` already is.
        if matches!(lowered.as_str(), "false" | "no" | "n") {
            return Ok(axes);
        }
        for axis_raw in parse_option_list(raw) {
            let axis = axis_raw
                .parse::<usize>()
                .map_err(|err| format!("invalid periodic axis '{axis_raw}': {err}"))?;
            if axis >= dim {
                return Err(format!(
                    "periodic axis {axis} out of range for {dim}D smooth"
                ));
            }
            axes[axis] = true;
        }
    }
    if let Some(raw) = options.get("boundary").or_else(|| options.get("bc")) {
        let boundary = parse_option_list(raw);
        if boundary.len() == dim {
            for (axis, value) in boundary.iter().enumerate() {
                if matches!(value.as_str(), "periodic" | "cyclic" | "cc") {
                    axes[axis] = true;
                }
            }
        } else if dim == 1
            && matches!(
                boundary.first().map(String::as_str),
                Some("periodic" | "cyclic" | "cc")
            )
        {
            axes[0] = true;
        }
    }
    Ok(axes)
}

fn parse_optional_numeric_list(
    options: &BTreeMap<String, String>,
    keys: &[&str],
    dim: usize,
) -> Result<Vec<Option<f64>>, String> {
    let Some(raw) = keys.iter().find_map(|key| options.get(*key)) else {
        return Ok(vec![None; dim]);
    };
    let values = split_list_option(raw);
    let mut out = vec![None; dim];
    if values.len() == 1 && dim == 1 {
        if !values[0].eq_ignore_ascii_case("none") {
            out[0] = Some(parse_numeric_expr(&values[0])?);
        }
        return Ok(out);
    }
    if values.len() != dim {
        return Err(format!(
            "numeric option list length {} must match smooth dimension {}",
            values.len(),
            dim
        ));
    }
    for (i, value) in values.iter().enumerate() {
        if !value.eq_ignore_ascii_case("none") {
            out[i] = Some(parse_numeric_expr(value)?);
        }
    }
    Ok(out)
}

fn parse_periods(
    options: &BTreeMap<String, String>,
    periodic_axes: &[bool],
) -> Result<Vec<Option<f64>>, String> {
    let dim = periodic_axes.len();
    // Broadcast a single-element `period=[v]` onto the lone periodic axis
    // of a multi-axis smooth (e.g. `te(th, h, bc=['periodic','natural'],
    // period=[2*pi])`): with only one periodic margin, the value can only
    // belong there.
    let lone_periodic_broadcast = options
        .get("period")
        .or_else(|| options.get("periods"))
        .and_then(|raw| {
            let values = split_list_option(raw);
            if values.len() != 1 || dim <= 1 {
                return None;
            }
            let mut iter = periodic_axes.iter().enumerate().filter(|(_, p)| **p);
            let first = iter.next()?;
            if iter.next().is_some() {
                return None;
            }
            Some((first.0, values.into_iter().next()?))
        });
    let periods = if let Some((axis, value)) = lone_periodic_broadcast {
        let mut out = vec![None; dim];
        if !value.eq_ignore_ascii_case("none") {
            out[axis] = Some(parse_numeric_expr(&value)?);
        }
        out
    } else {
        parse_optional_numeric_list(options, &["period", "periods"], dim)?
    };
    for (axis, (periodic, period)) in periodic_axes.iter().zip(periods.iter()).enumerate() {
        if *periodic
            && let Some(value) = period
            && (!value.is_finite() || *value <= 0.0)
        {
            return Err(format!(
                "period for periodic axis {axis} must be finite and positive, got {value}"
            ));
        }
    }
    Ok(periods)
}

fn parse_period_origins(
    options: &BTreeMap<String, String>,
    periodic_axes: &[bool],
) -> Result<Vec<Option<f64>>, String> {
    parse_optional_numeric_list(
        options,
        &[
            "origin",
            "origins",
            "period_origin",
            "period-origin",
            "domain_origin",
        ],
        periodic_axes.len(),
    )
}

/// Parse a per-axis periodic flag list for tensor smooths. Accepts three forms:
/// - `periodic=true` / `periodic=false` (scalar applied to every axis),
/// - `periodic=[true, false, ...]` (one flag per axis, length `dim`),
/// - `periodic=c(1, 1)` / `c(0, 0)` (a length-`dim` 0/1 mask, mgcv's
///   per-margin spelling — distinguished from an axis-index list by the
///   repeated 0/1 value), and
/// - `periodic=[0, 2, ...]` (axis indices that are periodic; others are not).
///
/// `boundary=[..., "periodic"/"cyclic"/"cc", ...]` may also flip individual
/// axes on; non-matching tokens leave the existing flag unchanged.
fn parse_tensor_periodic_axes(
    options: &BTreeMap<String, String>,
    dim: usize,
) -> Result<Vec<bool>, String> {
    let mut axes = vec![false; dim];
    if let Some(raw) = options.get("periodic").or_else(|| options.get("cyclic")) {
        let lowered = raw.trim().to_ascii_lowercase();
        match lowered.as_str() {
            "true" | "yes" | "y" => {
                axes.fill(true);
            }
            "false" | "no" | "n" => {
                // Already false; allow `boundary=` below to flip axes if set.
            }
            _ => {
                let entries = parse_option_list(raw);
                let all_bool = !entries.is_empty()
                    && entries.iter().all(|v| {
                        matches!(
                            v.as_str(),
                            "true" | "yes" | "y" | "false" | "no" | "n" | "none"
                        )
                    });
                // mgcv writes per-margin flag vectors as `periodic=c(1,1)` /
                // `periodic=c(0,0)` — a length-`dim` mask where each entry is a
                // 0/1 flag for THAT margin, not an axis index. A bare axis-index
                // list (`periodic=[0,1]`, `periodic=[0]`) lists DISTINCT margin
                // indices to turn on. The two collide only when the list is all
                // 0/1 of length `dim`; disambiguate by the repeated-value
                // signature `c(1,1)`/`c(0,0)` (a valid axis-index set never
                // repeats an index), which is the canonical mask spelling. This
                // is what makes the leading tensor margin honor its periodic flag
                // (#1751: `periodic=c(1,1)` previously parsed `1,1` as axis
                // indices, marking only axis 1 and dropping axis 0).
                let all_zero_one =
                    !entries.is_empty() && entries.iter().all(|v| v == "0" || v == "1");
                let has_repeat = {
                    let mut seen = std::collections::BTreeSet::new();
                    !entries.iter().all(|v| seen.insert(v.clone()))
                };
                let numeric_mask = all_zero_one && entries.len() == dim && has_repeat;
                if all_bool || numeric_mask {
                    if entries.len() != dim {
                        return Err(format!(
                            "periodic list length {} must match smooth dimension {}",
                            entries.len(),
                            dim
                        ));
                    }
                    for (i, v) in entries.iter().enumerate() {
                        axes[i] = matches!(v.as_str(), "true" | "yes" | "y" | "1");
                    }
                } else {
                    for axis_raw in entries {
                        let axis = axis_raw
                            .parse::<usize>()
                            .map_err(|err| format!("invalid periodic axis '{axis_raw}': {err}"))?;
                        if axis >= dim {
                            return Err(format!(
                                "periodic axis {axis} out of range for {dim}D smooth"
                            ));
                        }
                        axes[axis] = true;
                    }
                }
            }
        }
    }
    if let Some(raw) = options.get("boundary").or_else(|| options.get("bc")) {
        let boundary = parse_option_list(raw);
        if boundary.len() == dim {
            for (axis, value) in boundary.iter().enumerate() {
                if matches!(value.as_str(), "periodic" | "cyclic" | "cc") {
                    axes[axis] = true;
                }
            }
        }
    }
    // A per-margin basis vector (`bs=c('cc','ps')` / `type=[...]`) declares each
    // margin's basis family, and a cyclic family (`cc`/`cp`/`cyclic`) makes THAT
    // margin periodic — exactly as the 1-D `s(x, bs='cc')` smooth wraps its lone
    // axis. Without this, the per-margin `cc` token was validated but discarded:
    // every `bs=c(...)` spelling collapsed to the same open B-spline tensor
    // (#1752). Only honor the vector form here; a scalar `bs='cc'` on a tensor is
    // ambiguous about which margins wrap, so it does not flip any axis on.
    if let Some(raw) = options.get("bs").or_else(|| options.get("type"))
        && bs_selector_is_vector(raw)
    {
        let per_margin = parse_option_list(raw);
        if per_margin.len() == dim {
            for (axis, margin_bs) in per_margin.iter().enumerate() {
                if matches!(canonicalize_smooth_type(margin_bs), "cc" | "cp" | "cyclic") {
                    axes[axis] = true;
                }
            }
        }
    }
    Ok(axes)
}

/// Validate the per-margin `boundary=`/`bc=` tokens on a tensor-product smooth.
///
/// The tensor `boundary`/`bc` list selects, per margin, whether the margin
/// *wraps* (a `periodic`/`cyclic`/`cc` token, consumed by
/// [`parse_tensor_periodic_axes`]) or is an ordinary non-periodic margin. In the
/// tensor DSL a *non-periodic* margin is spelled `clamped` — in the B-spline
/// sense of a **clamped knot vector**, i.e. the standard open spline that is
/// free at its two ends and does not wrap (exactly how the callers document it:
/// "non-periodic / clamped … free at the two ends, no wrap"). It is therefore an
/// inert marker here, not a zero-derivative endpoint reparameterization: a
/// cylinder `te(theta, z, boundary=['periodic','clamped'], …)` is a cyclic θ
/// margin tensor-producted with an ordinary open z margin, the direct analog of
/// mgcv `te(bs=c("cc","ps"))` / `te(bs=c("cc","cr"))`.
///
/// The periodic selectors and the inert non-periodic markers
/// (`clamped`/`open`/`natural`/`free`/`none`/empty) are accepted; anything else
/// (e.g. a genuine `anchored` zero-value endpoint constraint, which has no
/// ordinary-margin meaning in a tensor) is surfaced as a clean
/// unsupported-feature error rather than silently dropped. Previously `clamped`
/// itself was rejected, so the cylinder/torus mixed-boundary tensors — the exact
/// construction the manifold quality suite builds — could not be fit at all.
fn validate_tensor_boundary_tokens(
    options: &BTreeMap<String, String>,
    dim: usize,
) -> Result<(), String> {
    let Some(raw) = options.get("boundary").or_else(|| options.get("bc")) else {
        return Ok(());
    };
    let entries = parse_option_list(raw);
    for (axis, value) in entries.iter().enumerate() {
        let inert = matches!(
            value.trim().to_ascii_lowercase().as_str(),
            "clamped" | "open" | "natural" | "free" | "none" | "" | "periodic" | "cyclic" | "cc"
        );
        if !inert {
            return Err(TermBuilderError::unsupported_feature(format!(
                "tensor smooth margin {axis} boundary token '{value}' is not supported \
                 (got bc/boundary={raw:?} on a {dim}-D tensor); tensor margins accept the periodic \
                 selectors (periodic/cyclic/cc) or the non-periodic markers (clamped/open/natural/free). \
                 Apply anchored/zero-value endpoint constraints with a 1-D s(x, bc=...) term instead."
            ))
            .to_string());
        }
    }
    Ok(())
}

fn tensor_k_axis_option_axis(
    key: &str,
    cols: &[usize],
    ds: &Dataset,
) -> Result<Option<usize>, String> {
    let Some(suffix) = key.strip_prefix("k_") else {
        return Ok(None);
    };
    if suffix.is_empty() {
        return Err("tensor k axis option must be named k_<axis> or k_<variable>".to_string());
    }
    if let Ok(axis) = suffix.parse::<usize>() {
        return if axis < cols.len() {
            Ok(Some(axis))
        } else {
            Err(format!(
                "tensor k axis option `{key}` references axis {axis}, but the smooth has {} margins",
                cols.len()
            ))
        };
    }

    let mut matches = cols
        .iter()
        .enumerate()
        .filter(|(_, col)| ds.headers.get(**col).is_some_and(|name| name == suffix))
        .map(|(axis, _)| axis);
    let first = matches.next();
    if matches.next().is_some() {
        return Err(format!(
            "tensor k axis option `{key}` matches more than one margin named `{suffix}`"
        ));
    }
    first.map(Some).ok_or_else(|| {
        let margin_names = cols
            .iter()
            .enumerate()
            .map(|(axis, col)| {
                let name = ds
                    .headers
                    .get(*col)
                    .map(String::as_str)
                    .unwrap_or("<unnamed>");
                format!("{axis}:{name}")
            })
            .collect::<Vec<_>>()
            .join(", ");
        format!(
            "tensor k axis option `{key}` does not match a margin index or name; tensor margins are [{margin_names}]"
        )
    })
}

fn is_tensor_k_axis_option_key(key: &str) -> bool {
    key.strip_prefix("k_")
        .is_some_and(|suffix| !suffix.is_empty())
}

/// Parse a per-margin basis dimension list (`k=<scalar>`, `k=[k0, k1, ...]`,
/// or axis aliases like `k_x=...` / `k_0=...`). A scalar is broadcast across
/// all axes; `None` returns the heuristic from the data column.
fn parse_tensor_k_list(
    options: &BTreeMap<String, String>,
    cols: &[usize],
    ds: &Dataset,
) -> Result<(Vec<usize>, bool), String> {
    let mut axis_values = vec![None; cols.len()];
    let mut saw_axis_alias = false;
    for (key, value) in options {
        let Some(axis) = tensor_k_axis_option_axis(key, cols, ds)? else {
            continue;
        };
        saw_axis_alias = true;
        if axis_values[axis].is_some() {
            return Err(format!("tensor k axis {axis} is specified more than once"));
        }
        let k: usize = value
            .parse()
            .map_err(|err| format!("invalid tensor k option `{key}={value}`: {err}"))?;
        axis_values[axis] = Some(k);
    }

    let raw = options
        .get("k")
        .or_else(|| options.get("basis_dim"))
        .or_else(|| options.get("basis-dim"))
        .or_else(|| options.get("basisdim"));
    if saw_axis_alias {
        if raw.is_some() {
            return Err(
                "tensor k axis aliases cannot be combined with k= or basis_dim=".to_string(),
            );
        }
        if let Some(missing_axis) = axis_values.iter().position(Option::is_none) {
            let margin_name = cols
                .get(missing_axis)
                .and_then(|col| ds.headers.get(*col))
                .map(String::as_str)
                .unwrap_or("<unnamed>");
            return Err(format!(
                "tensor k axis aliases must specify every margin; missing axis {missing_axis} ({margin_name})"
            ));
        }
        return Ok((
            axis_values
                .into_iter()
                .map(|k| k.expect("missing axis values rejected above"))
                .collect(),
            false,
        ));
    }
    let Some(raw) = raw else {
        let inferred = heuristic_tensor_margin_knots(cols, ds);
        return Ok((inferred, true));
    };
    let entries = split_list_option(raw);
    if entries.len() == 1 {
        let k: usize = entries[0]
            .parse()
            .map_err(|err| format!("invalid tensor k '{}': {err}", entries[0]))?;
        return Ok((vec![k; cols.len()], false));
    }
    if entries.len() != cols.len() {
        return Err(format!(
            "tensor k list length {} must match smooth dimension {}",
            entries.len(),
            cols.len()
        ));
    }
    let mut out = Vec::with_capacity(entries.len());
    for entry in entries {
        let k: usize = entry
            .parse()
            .map_err(|err| format!("invalid tensor k '{entry}': {err}"))?;
        out.push(k);
    }
    Ok((out, false))
}

/// Parse the `identifiability=` option for tensor-product smooths. Mirrors the
/// vocabulary of the Matern/Duchon parsers so the formula DSL is consistent.
///
/// `kind` selects the default identifiability when no explicit
/// `identifiability=` option is supplied: `te(...)` ([`SmoothKind::Te`]) keeps
/// the full-tensor sum-to-zero default, while `ti(...)` ([`SmoothKind::Ti`])
/// defaults to per-margin sum-to-zero so the marginal main effects are excluded
/// (the mgcv tensor-interaction semantics). An explicit option always wins.
fn parse_tensor_identifiability(
    options: &BTreeMap<String, String>,
    kind: SmoothKind,
) -> Result<TensorBSplineIdentifiability, String> {
    let Some(raw) = options.get("identifiability").map(String::as_str) else {
        return Ok(match kind {
            SmoothKind::Ti => TensorBSplineIdentifiability::MarginalSumToZero,
            _ => TensorBSplineIdentifiability::default(),
        });
    };
    match raw.trim().to_ascii_lowercase().as_str() {
        "none" => Ok(TensorBSplineIdentifiability::None),
        "sum_tozero" | "sum-to-zero" | "center_sum_tozero" | "center-sum-to-zero" | "centered"
        | "sumtozero" => Ok(TensorBSplineIdentifiability::SumToZero),
        "marginal_sum_tozero" | "marginal-sum-to-zero" | "marginal_sumtozero"
        | "marginalsumtozero" | "interaction" => {
            Ok(TensorBSplineIdentifiability::MarginalSumToZero)
        }
        other => Err(TermBuilderError::unsupported_feature(format!(
            "invalid tensor identifiability '{other}'; expected one of: none, sum_tozero, marginal_sum_tozero"
        ))
        .to_string()),
    }
}

fn bspline_boundary_declares_periodic_axis(options: &BTreeMap<String, String>) -> bool {
    options
        .get("boundary")
        .or_else(|| options.get("bc"))
        .map(|raw| {
            parse_option_list(raw)
                .into_iter()
                .any(|value| matches!(value.as_str(), "periodic" | "cyclic" | "cc"))
        })
        .unwrap_or(false)
}

/// Canonical-name lookup for the `bs=`/`type=` smooth selector.
///
/// User-facing names — including mgcv-compatible spellings whose semantics
/// match an existing gamfit smooth exactly — collapse to the engine-internal
/// canonical names used by the dispatch in [`build_smooth_basis`]. Adding a
/// new exactly-equivalent alias is a one-line entry here; the match arms
/// below remain the single dispatch site.
///
/// Aliases listed here MUST be true semantic equivalents of the canonical
/// target, not approximations. mgcv names whose semantics differ from any
/// gamfit smooth (e.g. `bs="ts"` shrinkage thin-plate, `bs="ad"` adaptive)
/// are intentionally NOT mapped here — they should reach the unsupported-type
/// path so users get a real diagnostic instead of a silent semantic
/// substitution. mgcv's `bs="cr"`/`"cs"` (cubic regression and its shrinkage
/// twin) are handled directly in the [`build_smooth_basis`] dispatch — they
/// are not aliased here because the `cr`/`cs` distinction controls a default
/// (`double_penalty`) that the canonical-name layer cannot see.
///
/// Unrecognised inputs pass through unchanged so the dispatch can produce its
/// usual "unsupported smooth type" error, preserving the existing diagnostic
/// surface for genuine typos.
pub(crate) fn canonicalize_smooth_type(raw: &str) -> &str {
    match raw {
        // Thin-plate spline. mgcv `bs="tp"` is the default thin-plate
        // regression spline — exact semantic equivalent of gamfit's `"tps"`.
        "tp" => "tps",
        // Gaussian process / Matérn. mgcv `bs="gp"` defaults to a Matérn
        // covariance kernel with REML smoothing parameter selection, which
        // matches gamfit's `"matern"` exactly (same kernel-Gram identity,
        // same REML route).
        "gp" => "matern",
        // Constant-curvature (M_κ) geodesic-kernel smooth (#944). All aliases
        // collapse to one canonical type so `bs="curv"`/`bs="mkappa"` cannot
        // diverge from `curv(...)`.
        "curv" | "constant_curvature" | "mkappa" => "curvature",
        // Measure-jet spline: multiscale local-jet-residual energy of the
        // empirical measure. No mgcv equivalent (mgcv has no measure-learned
        // geometry smooth), so no mgcv alias is mapped.
        "mjs" | "measure_jet" | "web" => "measurejet",
        other => other,
    }
}

/// Is `margin_bs` a per-margin basis name that the tensor builder realizes as a
/// penalized 1-D B-spline margin?
///
/// gam's tensor product is built from penalized B-spline marginals. mgcv's
/// thin-plate (`tp`/`tps`), P-spline (`ps`), B-spline (`bs`), cubic-regression
/// (`cr`/`cs`), and cyclic (`cc`/`cp`/`cyclic`) marginals are all penalized
/// splines spanning the same per-axis smoothing space, so a B-spline margin
/// reproduces the same tensor smoothing class. Margin kinds with fundamentally
/// different structure (adaptive, random-effect, sphere) are NOT accepted as
/// tensor margins.
pub(crate) fn tensor_margin_bs_is_supported(margin_bs: &str) -> bool {
    matches!(
        canonicalize_smooth_type(margin_bs),
        "tps" | "ps" | "bs" | "bspline" | "cr" | "cs" | "cc" | "cp" | "cyclic"
    )
}

/// Does the smooth request a periodic/cyclic axis via its options?
///
/// Mirrors the boundary-condition reading used by the periodic-aware dispatch
/// branches. Factored out so the type resolver and `build_smooth_basis` agree
/// on a single notion of "periodic requested".
pub(crate) fn smooth_options_declare_periodic(options: &BTreeMap<String, String>) -> bool {
    options.contains_key("periodic")
        || options.contains_key("cyclic")
        || options
            .get("boundary")
            .or_else(|| options.get("bc"))
            .map(|boundary| {
                boundary.to_ascii_lowercase().contains("periodic")
                    || boundary.to_ascii_lowercase().contains("cyclic")
            })
            .unwrap_or(false)
}

/// Resolve the canonical engine-internal smooth-type name for a term.
///
/// Reads the user-facing `type=`/`bs=` selector and collapses mgcv-compatible
/// aliases (`tp`→`tps`, `gp`→`matern`) via [`canonicalize_smooth_type`], or
/// derives the default from the smooth kind/arity when no selector is given.
/// This is the single source of truth for the dispatch in
/// [`build_smooth_basis`]; other call sites (e.g. predictor-specific basis
/// policy) use it so the classification never drifts from the dispatch.
/// Is the raw `bs=`/`type=` selector a vector literal (`c('tp','tp')`,
/// `['tp','tp']`, `(tp, tp)`) rather than a scalar smooth-type name?
///
/// mgcv's tensor smooths take a *per-margin* basis vector
/// (`te(x1, x2, bs=c('tp','tp'))`). Such a value is not a scalar canonical
/// type and must not be fed through [`canonicalize_smooth_type`] — it has to be
/// recognized as a tensor request and split into per-margin types. A scalar
/// selector (`bs="tp"`) is left untouched.
pub(crate) fn bs_selector_is_vector(raw: &str) -> bool {
    let trimmed = raw.trim();
    let bracketed = (trimmed.starts_with('[') && trimmed.ends_with(']'))
        || (trimmed.starts_with("c(") || trimmed.starts_with("C(")) && trimmed.ends_with(')')
        || (trimmed.starts_with('(') && trimmed.ends_with(')'));
    bracketed && !parse_option_list(trimmed).is_empty()
}

pub fn resolve_smooth_type_name(
    kind: SmoothKind,
    n_cols: usize,
    options: &BTreeMap<String, String>,
) -> String {
    let selector = options.get("type").or_else(|| options.get("bs"));
    // A per-margin basis vector is a tensor request, never a scalar type. Route
    // it to the tensor builder, which reads the per-margin types out of the
    // same `bs=` option. (A vector on a non-tensor smooth is ill-formed and
    // falls through to the scalar path below so the existing diagnostic fires.)
    if let Some(raw) = selector
        && bs_selector_is_vector(raw)
        && matches!(kind, SmoothKind::Te | SmoothKind::Ti | SmoothKind::T2)
    {
        return "tensor".to_string();
    }
    selector
        .map(|s| canonicalize_smooth_type(&s.to_ascii_lowercase()).to_string())
        .unwrap_or_else(|| match kind {
            SmoothKind::Te | SmoothKind::Ti | SmoothKind::T2 => "tensor".to_string(),
            SmoothKind::S if n_cols == 1 => "bspline".to_string(),
            // Mixed periodic Euclidean radial kernels are not separable on the
            // cylinder. Use a tensor product with a cyclic margin so s(theta,h)
            // honors seam continuity while preserving the formula-level s(...).
            SmoothKind::S if smooth_options_declare_periodic(options) => "tensor".to_string(),
            SmoothKind::S => "tps".to_string(),
        })
}

/// Does this canonical smooth type size its basis through the generous spatial
/// center heuristic ([`crate::basis::default_num_centers`])?
///
/// Only the radial spatial bases (thin-plate, Matérn/GP, Duchon) route their
/// default basis dimension through `plan_spatial_basis(.., Default, ..)`. The
/// B-spline, cyclic, tensor, and factor-smooth bases use their own modest
/// knot-based defaults, so they are unaffected by — and must not be perturbed
/// by — secondary-predictor basis-parsimony adjustments (#501).
pub fn smooth_type_uses_spatial_center_heuristic(canonical_type: &str) -> bool {
    matches!(canonical_type, "tps" | "matern" | "duchon")
}

pub fn build_smooth_basis(
    kind: SmoothKind,
    vars: &[String],
    cols: &[usize],
    options: &BTreeMap<String, String>,
    ds: &Dataset,
    inference_notes: &mut Vec<String>,
    policy: &ResourcePolicy,
    smooth_coordinate_count: usize,
) -> Result<SmoothBasisSpec, String> {
    // Strip the internal by-level sizing carrier before any per-kind option
    // allow-list runs (the `__by_col` pattern): `sizing_rows` feeds every
    // n-scaling BASIS DEFAULT below; explicit user counts are untouched.
    let stripped_sizing_options;
    let (options, sizing_rows) = match options.get(DEFAULT_SIZING_ROWS_OPTION) {
        Some(raw) => {
            let rows = raw.parse::<usize>().map_err(|_| {
                format!("internal by-level sizing rows carrier is not a count: '{raw}'")
            })?;
            let mut cleaned = options.clone();
            cleaned.remove(DEFAULT_SIZING_ROWS_OPTION);
            stripped_sizing_options = cleaned;
            (&stripped_sizing_options, rows)
        }
        None => (options, ds.values.nrows()),
    };
    // Fail fast on degenerate input: a smooth whose (non-categorical) coordinate
    // columns collapse to a SINGLE distinct point can only ever fit the response
    // mean — its design matrix is rank-1. For a UNIVARIATE smooth this is exactly
    // "the one column is constant": `smooth(x)`/`matern(x)` on constant `x` would
    // otherwise silently fit the mean of `y` with no visible cue (Duchon already
    // errors loudly via the basis layer; this makes the diagnosis explicit and
    // uniform). For a general MULTIVARIATE Euclidean smooth (tensor, tps,
    // matern, ...) a single constant coordinate is NOT degenerate — the basis
    // still varies along the other coordinate(s) and the penalty absorbs the
    // rank-deficient direction (a constant-`x2` slice of `tps(x1, x2)` is a
    // well-posed 1-D function of `x1`). Such a term is degenerate only when
    // EVERY coordinate is constant at once, i.e. the joint input is a single
    // point. Test the JOINT cardinality, not each column independently, so the
    // loud diagnosis still fires for the genuinely rank-1 case without rejecting
    // well-posed lower-dimensional slices.
    //
    // The SPHERE/SOS term is the exception (handled separately just below): its
    // spherical-harmonic / Wahba basis is intrinsically a function of BOTH
    // angular coordinates, so a constant latitude or longitude is not an honest
    // lower-D slice but an unidentifiable axis (every point on a single meridian
    // or parallel) — that case is rejected per-coordinate.
    let coord_cols: Vec<(&String, usize)> = vars
        .iter()
        .zip(cols.iter().copied())
        .filter(|(_, col)| !matches!(ds.column_kinds.get(*col), Some(ColumnKindTag::Categorical)))
        .collect();
    if !coord_cols.is_empty() {
        let views: Vec<ArrayView1<'_, f64>> = coord_cols
            .iter()
            .map(|(_, col)| ds.values.column(*col))
            .collect();
        let n_rows = views[0].len();
        let mut distinct_points = std::collections::HashSet::<Vec<u64>>::new();
        for r in 0..n_rows {
            let key: Vec<u64> = views
                .iter()
                .map(|v| gam_data::canonical_level_bits(v[r]))
                .collect();
            distinct_points.insert(key);
            if distinct_points.len() > 1 {
                break;
            }
        }
        if distinct_points.len() <= 1 {
            return Err(TermBuilderError::degenerate_data(if coord_cols.len() == 1 {
                let var = coord_cols[0].0;
                format!(
                    "smooth term over '{var}' has only one unique value in the training data \
                     — a smooth on a constant column is degenerate and would only fit the response mean. \
                     Remove `{var}` from the smooth, drop the term, or check the data."
                )
            } else {
                let names = coord_cols
                    .iter()
                    .map(|(v, _)| v.as_str())
                    .collect::<Vec<_>>()
                    .join(", ");
                format!(
                    "smooth term over ({names}) has only one unique joint coordinate in the training \
                     data — every coordinate is constant, so the smooth is degenerate and would only \
                     fit the response mean. Drop the term or check the data."
                )
            })
            .to_string());
        }

        // Sphere/SOS exception: the S² smooth is intrinsically a function of
        // BOTH angular coordinates, so a single constant axis is unidentifiable
        // (every point on one meridian or one parallel), not an honest 1-D
        // slice. Reject it per-coordinate at fit-time with a coordinate-named
        // error. This runs ONLY during term construction (build_smooth_basis);
        // predict rebuilds the design from the frozen resolvedspec and never
        // re-enters this path, so a constant predict grid (e.g. a single query
        // point on a fixed meridian) is never re-validated (#frozen-mass).
        if matches!(
            resolve_smooth_type_name(kind, cols.len(), options).as_str(),
            "sphere" | "s2" | "sos"
        ) {
            for (axis, (var, col)) in coord_cols.iter().enumerate() {
                let column = ds.values.column(*col);
                let mut distinct = std::collections::HashSet::<u64>::new();
                for &value in column.iter() {
                    distinct.insert(gam_data::canonical_level_bits(value));
                    if distinct.len() > 1 {
                        break;
                    }
                }
                if distinct.len() <= 1 {
                    // Axis 0 is latitude, axis 1 longitude (formula order
                    // `sphere(lat, lon)`); name the collapsed slice accordingly.
                    let slice = if axis == 0 {
                        "a single parallel (constant latitude)"
                    } else {
                        "a single meridian (constant longitude)"
                    };
                    return Err(TermBuilderError::degenerate_data(format!(
                        "sphere smooth has a constant '{var}' column — every point lies on \
                         {slice}, so the 2-sphere term is degenerate and unidentifiable along \
                         that axis. A spherical smooth needs genuine variation in BOTH latitude \
                         and longitude; vary '{var}', drop the term, or fit a 1-D smooth on the \
                         varying coordinate."
                    ))
                    .to_string());
                }
            }
        }
    }
    if let Some(by_name) = options.get("by").cloned() {
        let by_col = options
            .get("__by_col")
            .and_then(|raw| raw.parse::<usize>().ok())
            .or_else(|| vars.iter().position(|v| v == &by_name).map(|idx| cols[idx]))
            .ok_or_else(|| format!("unknown by= column '{by_name}'"))?;
        let mut inner_options = options.clone();
        inner_options.remove("by");
        inner_options.remove("__by_col");
        inner_options.remove("id");
        // Size the inner basis's n-scaling defaults from the smallest
        // by-level's rows (see `DEFAULT_SIZING_ROWS_OPTION`); numeric-by
        // smooths keep pooled sizing.
        inject_by_level_sizing_rows(&mut inner_options, ds, by_col);
        let inner = build_smooth_basis(
            kind,
            vars,
            cols,
            &inner_options,
            ds,
            inference_notes,
            policy,
            smooth_coordinate_count,
        )?;
        let by_kind = match ds.column_kinds.get(by_col).copied() {
            Some(ColumnKindTag::Categorical) => ByVarKind::Factor {
                feature_col: by_col,
                ordered: option_bool(options, "ordered").unwrap_or(false),
                frozen_levels: None,
            },
            Some(ColumnKindTag::Continuous | ColumnKindTag::Binary) => ByVarKind::Numeric {
                feature_col: by_col,
            },
            None => {
                return Err(format!(
                    "internal column-kind lookup failed for by='{by_name}'"
                ));
            }
        };
        return Ok(SmoothBasisSpec::BySmooth {
            smooth: Box::new(inner),
            by_kind,
        });
    }

    let smooth_double_penalty = option_bool(options, "double_penalty").unwrap_or(true);
    let type_opt = resolve_smooth_type_name(kind, cols.len(), options);

    if matches!(type_opt.as_str(), "fs" | "sz" | "re") {
        validate_known_options(
            type_opt.as_str(),
            options,
            &[
                "type",
                "bs",
                "k",
                "basis_dim",
                "basis-dim",
                "basisdim",
                "knots",
                "knot_placement",
                "knot-placement",
                "knotplacement",
                "degree",
                "penalty_order",
                "m",
                "double_penalty",
                "ordered",
            ],
        )?;
        if cols.len() != 2 {
            return Err(format!(
                "{} factor-smooth currently expects exactly two variables (one numeric, one categorical)",
                type_opt
            ));
        }
        let kinds = cols
            .iter()
            .map(|&c| ds.column_kinds.get(c).copied())
            .collect::<Vec<_>>();
        let (cont_idx, group_idx) = if type_opt == "re" {
            // mgcv random-slope examples are often s(g, x, bs="re").
            match (kinds[0], kinds[1]) {
                (Some(ColumnKindTag::Categorical), _) => (1usize, 0usize),
                (_, Some(ColumnKindTag::Categorical)) => (0usize, 1usize),
                _ => (1usize, 0usize),
            }
        } else {
            match (kinds[0], kinds[1]) {
                (_, Some(ColumnKindTag::Categorical)) => (0usize, 1usize),
                (Some(ColumnKindTag::Categorical), _) => (1usize, 0usize),
                _ => {
                    return Err(format!(
                        "{} factor-smooth requires one categorical factor variable",
                        type_opt
                    ));
                }
            }
        };
        let c = cols[cont_idx];
        let (minv, maxv) = col_minmax(ds.values.column(c))?;
        let degree = if type_opt == "re" {
            1
        } else {
            option_usize(options, "degree").unwrap_or(DEFAULT_BSPLINE_DEGREE)
        };
        // For a factor smooth every group's curve is fit from THAT group's rows
        // alone, so the marginal's flexibility must respect the least-resolved
        // group, not the pooled column. The pooled heuristic can hand the marginal
        // a basis that saturates (or exceeds) a small group's sample — e.g. the
        // sleepstudy panel has 8 training days per subject, and a default cubic
        // basis of 8 functions interpolates each subject's 8 points, leaving no
        // room for the wiggliness penalty to collapse the curve toward the
        // per-subject line. The factor smooth then fits within-group noise and
        // extrapolates badly (held-out forecast worse than the population mean).
        //
        // Cap the marginal basis below the minimum per-group covariate resolution
        // so the penalty always retains residual degrees of freedom to shrink each
        // group's curvature toward its linear null space (the random-slope
        // estimand). This small-group cap composes with a separate upper bound at
        // mgcv's factor-smooth default k=10 (FACTOR_SMOOTH_DEFAULT_BASIS_DIM,
        // applied below), so even ample-data groups get the modest SHARED marginal
        // a factor smooth wants rather than the full pooled basis. The explicit
        // `re` random-effect form takes neither cap: it is a raw linear `[1, x]`
        // random effect (0 internal knots), handled in the branch above.
        let pooled_internal = heuristic_knots_for_column(ds.values.column(c));
        let default_internal = if type_opt == "re" {
            // `bs="re"` is a PARAMETRIC random effect, not a smooth of the
            // covariate: `s(x, g, bs="re")` is the mgcv random intercept+slope
            // `(1 + x | g)`, i.e. a per-group line `[1, x]`, penalized by an iid
            // ridge. A degree-1 marginal with ZERO internal knots spans exactly
            // that linear space (2 coefficients per group). Using the pooled
            // knot heuristic here instead turned the marginal into a
            // piecewise-linear B-spline (e.g. 6 functions/group on sleepstudy),
            // i.e. a *smooth* with kinks rather than a random slope — many extra
            // collinear-across-levels coefficients that ill-condition the joint
            // Newton/REML solve (minutes-long fits, and a singular block when
            // combined with a separate random intercept `s(g, bs="re")`). The
            // raw linear basis is both the correct `re` semantics and fast.
            0
        } else {
            let min_group_resolution =
                min_per_group_unique_count(ds.values.column(c), ds.values.column(cols[group_idx]));
            // Per-group basis dim = degree + 1 + internal. Hold it well below the
            // smallest group's resolution (leave at least two residual points per
            // group) so the smooth cannot interpolate that group and the
            // wiggliness penalty retains the room to collapse each curve toward
            // its linear null space. Never drop below `degree + 2`, which keeps
            // exactly the linear span plus a single curvature direction — the
            // minimal smoother that can still bend if the data demand it.
            let basis_cap = min_group_resolution.saturating_sub(2).max(degree + 2);
            let internal_cap = basis_cap.saturating_sub(degree + 1);
            let capped = pooled_internal.min(internal_cap.max(1));
            // A factor smooth (`fs` AND `sz`) shares ONE marginal across ALL
            // levels, each level's curve fit from that group's rows alone. The
            // pooled knot heuristic (driven by the full column's sample) hands it
            // a much richer basis than the shared signal needs — ~24
            // functions/group on the gam#903 factor-smooth-recovery fixtures — so
            // REML has the capacity to fit within-group noise and over-fits the
            // shared shape (fs: edf 58 vs mgcv's k=10/edf 39; sz: gam 0.068 vs
            // mgcv 0.046 truth RMSE), losing the truth-recovery head-to-head with
            // the mature tool. mgcv's factor-smooth default `k=10` embodies the
            // right convention: a modest shared marginal. Cap the marginal there
            // (basis ≈ degree+1+internal ≈ 10) for both flavours when the
            // small-group cap above is not already tighter, so REML is not handed
            // noise-fitting capacity it does not need. An explicit `k`/`basis_dim`
            // overrides this (parse_ps_internal_knots); `re` is the raw linear
            // effect handled above.
            let fs_default_internal = FACTOR_SMOOTH_DEFAULT_BASIS_DIM
                .saturating_sub(degree + 1)
                .max(1);
            capped.min(fs_default_internal)
        };
        let (n_knots, _, effective_degree) =
            parse_ps_internal_knots(options, degree, default_internal)?;
        let penalty_order = option_usize(options, "penalty_order")
            .unwrap_or(if effective_degree > 1 { 2 } else { 1 })
            .min(effective_degree);
        // All factor-smooth flavours (`fs`, `sz`, `re`) place their per-level
        // marginal on the SAME penalized B-spline (P-spline) basis. The flavours
        // differ ONLY in their penalty/constraint structure (handled below) —
        // sz: zero-sum deviation blocks with the per-level null space left
        // unpenalized; fs: random-effect double penalty; re: identity ridge.
        //
        // `sz` USED to route its default-degree marginal to a NATURAL cubic
        // regression spline (`cr`), on the belief that mgcv's `bs="sz"` does the
        // same and that cr recovers smooth signals more efficiently than the
        // (then uncapped) B-spline margin (#1074). That introduced a consistency
        // failure (#1605): the `cr` basis enforces the natural boundary
        // conditions f''(x_1)=f''(x_k)=0 and extrapolates linearly past the end
        // knots, so it CANNOT represent a per-group deviation curve with non-zero
        // curvature at the data boundary. Phase-shifted deviation shapes
        // (f''(0) = -(2π)² sin(φ) ≠ 0) are then biased toward "free linear +
        // anchored wiggle", under-shooting the amplitude — a bias that does NOT
        // vanish as n→∞ (n-independent: a genuine consistency failure, not
        // finite-sample shrinkage). The earlier #700/#1074 sz fixtures used
        // d_g ∝ sin(2πx), whose f'' happens to vanish at x=0 and x=1, so they
        // accidentally satisfied the natural BC and never exposed the gap; the
        // `fs` sibling, on this very B-spline marginal, recovers the SAME
        // phase-shifted data to the noise floor.
        //
        // The penalized B-spline marginal makes no boundary assumption, so it
        // represents arbitrary deviation shapes, and — with the
        // FACTOR_SMOOTH_DEFAULT_BASIS_DIM cap above already removing the
        // noise-fitting capacity that originally motivated leaving B-splines —
        // it recovers the BC-satisfying #700/#1074 signals just as well. Sharing
        // one marginal basis across all flavours also lets the B-spline degree/
        // knot degradation handle low-cardinality covariates uniformly (what
        // `fs` already does), so the `sz`-only cr data-support cap (#1541/#1542)
        // — and the asymmetry where only the cr-marginal `sz` spelling hard-
        // failed a 3-level ordinal — is no longer needed.
        let marginal_knotspec = resolve_nonperiodic_bspline_knotspec(
            options,
            ds.values.column(c),
            (minv, maxv),
            effective_degree,
            n_knots,
        )?;
        let marginal = BSplineBasisSpec {
            degree: effective_degree,
            penalty_order,
            knotspec: marginal_knotspec,
            // mgcv's `bs="fs"` is a random-effect-style smooth: EVERY per-level
            // coefficient, including the marginal null space, is penalized so
            // unobserved groups can be predicted — so `fs` keeps the null-space
            // (double) penalty. mgcv's `bs="sz"` is a pure across-level
            // *deviation* smooth that, under the default `select=FALSE`, leaves
            // the per-level null space UNPENALIZED; carrying the double penalty
            // there shrinks the genuine deviation signal and over-smooths the
            // recovered curves relative to mgcv (gam#700). `re` carries its own
            // identity ridge below and ignores this flag. Honour an explicit
            // user `double_penalty=` either way.
            double_penalty: option_bool(options, "double_penalty")
                .unwrap_or(type_opt.as_str() != "sz"),
            identifiability: BSplineIdentifiability::None,
            boundary_conditions: Default::default(),
            boundary: OneDimensionalBoundary::Open,
        };
        let flavour = match type_opt.as_str() {
            "fs" => FactorSmoothFlavour::Fs {
                m_null_penalty_orders: vec![
                    option_usize(options, "m").unwrap_or(DEFAULT_PENALTY_ORDER),
                ],
            },
            "sz" => FactorSmoothFlavour::Sz,
            "re" => FactorSmoothFlavour::Re,
            // Outer `matches!` already restricts to fs/sz/re.
            other => {
                return Err(format!(
                    "internal: factor-smooth flavour dispatch reached unexpected type `{}`",
                    other
                ));
            }
        };
        return Ok(SmoothBasisSpec::FactorSmooth {
            spec: FactorSmoothSpec {
                continuous_cols: vec![c],
                group_col: cols[group_idx],
                marginal,
                flavour,
                group_frozen_levels: None,
                frozen_global_orthogonality: None,
            },
        });
    }

    match type_opt.as_str() {
        // `periodic` is the generic spelling for a periodic (wrap-continuous)
        // B-spline; it names the SAME `SmoothBasisSpec::BSpline1D {
        // PeriodicUniform }` the mgcv-style cyclic selectors (`cc`/`cp`/`cyclic`)
        // build, and is already recognized as that basis kind by the JSON /
        // override path (`smooth_overrides`) and accepted by the formula parser.
        // Route it through the cyclic arm so the formula path agrees with the
        // rest of the codebase instead of rejecting it as an unsupported type.
        "cyclic" | "cc" | "cp" | "cyclic-ps" | "periodic" => {
            validate_known_options(
                "cyclic",
                options,
                &[
                    "type",
                    "bs",
                    "by",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "degree",
                    "penalty_order",
                    "period",
                    "periods",
                    "period_start",
                    "period_end",
                    "start",
                    "end",
                    "origin",
                    "origins",
                    "period_origin",
                    "period-origin",
                    "domain_origin",
                    "double_penalty",
                    "id",
                    "__by_col",
                    "identifiability",
                ],
            )?;
            if cols.len() != 1 {
                return Err(format!(
                    "periodic smooth expects one variable, got {}",
                    cols.len()
                ));
            }
            let c = cols[0];
            let (minv, maxv) = col_minmax(ds.values.column(c))?;
            let degree = option_usize(options, "degree").unwrap_or(DEFAULT_BSPLINE_DEGREE);
            let mut default_internal = heuristic_knots_for_column(ds.values.column(c));
            if ds.values.nrows() <= 32 && smooth_coordinate_count >= 5 {
                default_internal = default_internal.min(1);
            }
            // A periodic cubic spline has no free endpoint behaviour to spend
            // degrees of freedom on: the wrap constraint removes the ordinary
            // boundary wiggle, and the cyclic second-difference penalty leaves
            // only the constant direction (handled by the smooth
            // identifiability constraint).  An over-rich default would give
            // small binomial/continuation-ratio fits a large penalized nuisance
            // space whose REML/LAML optimum is driven by finite-sample Bernoulli
            // noise rather than the low-frequency periodic signal.  Cap the
            // cyclic default in the mgcv `bs="cc"` spirit: a modest basis unless
            // the caller explicitly requests `k=...`; high-frequency periodic
            // structure remains available through that explicit contract.  Since
            // gam#1680 lowered the open-spline univariate default to ≈12
            // functions this cap and the open-spline default coincide, so it now
            // acts as an explicit floor/guard that keeps the cyclic default lean
            // even if the open-spline heuristic is later widened.
            let cyclic_default_basis_cap = CYCLIC_DEFAULT_BASIS_DIM.max(degree + 1);
            let default_basis = (default_internal + degree + 1).min(cyclic_default_basis_cap);
            let num_basis = option_usize_any(options, &["k", "basis_dim", "basis-dim", "basisdim"])
                .unwrap_or(default_basis);
            if num_basis < degree + 1 {
                return Err(format!(
                    "periodic smooth: k={} too small for degree {}; expected k >= {}",
                    num_basis,
                    degree,
                    degree + 1
                ));
            }
            // The cyclic arm is periodic on its single axis by construction, so
            // resolve the period exactly the way the `s()`/`ps` arm does: honour
            // `period=`/`periods=` first (with `origin=` setting the domain
            // start), and fall back to the `period_start`/`period_end` endpoint
            // form only when `period=` is absent. Previously this arm jumped
            // straight to `parse_periodic_domain_1d`, so a `period=<v>`
            // declaration was silently dropped and the smooth wrapped at the
            // data range (#816). All three helpers route through
            // `parse_numeric_expr`, so `period=2*pi` and `period_end=2*pi` parse
            // identically (#815).
            let periodic_axes = [true];
            let periods = parse_periods(options, &periodic_axes)?;
            let origins = parse_period_origins(options, &periodic_axes)?;
            // Distinguish a *cyclic basis selector* (`bs='cc'`/`cp'`/`cyclic`,
            // this whole arm) from a generic B-spline forced periodic by a
            // `periodic=`/`boundary=` flag (the `ps`/`bspline` arm). Only the
            // latter carries the sample-dependent off-by-ε seam that #1771's
            // guard in `parse_periodic_domain_1d` requires an explicit period
            // to avoid. A bare `s(x, bs='cc')` opts INTO mgcv's `bs="cc"`
            // semantics — the wrap IS the observed data range — exactly like
            // the tensor cc-margin fallback (`te(x, z, bs=c('cc','cc'))`). The
            // cyclic arm was left routing through the now-strict helper when
            // #1771 tightened it, so a bare cyclic smooth hard-errored with
            // "periodic B-spline smooth requires an explicit period" even
            // though its period is well-defined. Honor `period=`/`periods=`
            // first, then the half-open `period_start`/`period_end` endpoint
            // form, and only otherwise wrap at the observed `[min, max]` span.
            let has_endpoint_decl = ["period_start", "start", "period_end", "end"]
                .iter()
                .any(|key| options.contains_key(*key));
            let (domain_start, period) = if let Some(p) = periods[0] {
                (origins[0].unwrap_or(minv), p)
            } else if has_endpoint_decl {
                parse_periodic_domain_1d(options, minv, maxv)?
            } else {
                let span = maxv - minv;
                if !(span.is_finite() && span > 0.0) {
                    return Err(format!(
                        "cyclic smooth requires a positive observed data range to derive \
                         its period, got [{minv}, {maxv}]"
                    ));
                }
                (origins[0].unwrap_or(minv), span)
            };
            Ok(SmoothBasisSpec::BSpline1D {
                feature_col: c,
                spec: BSplineBasisSpec {
                    degree,
                    penalty_order: option_usize(options, "penalty_order")
                        .unwrap_or(DEFAULT_PENALTY_ORDER),
                    knotspec: BSplineKnotSpec::PeriodicUniform {
                        data_range: (domain_start, domain_start + period),
                        num_basis,
                    },
                    double_penalty: smooth_double_penalty,
                    identifiability: BSplineIdentifiability::default(),
                    boundary_conditions: Default::default(),
                    boundary: OneDimensionalBoundary::Cyclic {
                        start: domain_start,
                        end: domain_start + period,
                    },
                },
            })
        }
        "bspline" | "ps" | "p-spline" | "cr" | "cs" => {
            // mgcv's `bs="cr"` (cubic regression spline) and `bs="cs"` (its
            // shrinkage twin) are penalized cubic-regression smooths that span
            // the same per-axis function space as gamfit's `bspline` (cubic
            // B-spline, second-derivative penalty). Route both through the
            // 1-D B-spline arm. Both recover unsupported null-space effects by
            // default; `double_penalty=false` is the explicit unpenalized
            // opt-out. Without this route, a stand-alone
            // `s(x, bs='cr')` (which is otherwise a routine 1-D smooth in
            // mgcv-compatible formulae) reached the dispatch's default arm
            // and aborted the whole fit with `unsupported smooth type 'cr'`,
            // even though the same name was already recognized as a tensor
            // margin (`tensor_margin_bs_is_supported`).
            let validation_name = match type_opt.as_str() {
                "cr" => "cr",
                "cs" => "cs",
                _ => "bspline",
            };
            validate_known_options(
                validation_name,
                options,
                &[
                    "type",
                    "bs",
                    "by",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "knots",
                    "knot_placement",
                    "knot-placement",
                    "knotplacement",
                    "degree",
                    "penalty_order",
                    "boundary",
                    "bc",
                    "boundary_conditions",
                    "bc_left",
                    "bc_right",
                    "left_bc",
                    "right_bc",
                    "start_bc",
                    "end_bc",
                    "side",
                    "anchor",
                    "anchor_value",
                    "value",
                    "anchor_left",
                    "left_anchor",
                    "anchor_right",
                    "right_anchor",
                    "periodic",
                    "period",
                    "periods",
                    "period_start",
                    "period_end",
                    "origin",
                    "double_penalty",
                    "by",
                    "id",
                    "__by_col",
                    "identifiability",
                    "by",
                ],
            )?;
            if cols.len() != 1 {
                return Err(TermBuilderError::incompatible_config(format!(
                    "bspline smooth expects one variable, got {}",
                    cols.len()
                ))
                .to_string());
            }
            let c = cols[0];
            let (minv, maxv) = col_minmax(ds.values.column(c))?;
            let degree = option_usize(options, "degree").unwrap_or(DEFAULT_BSPLINE_DEGREE);
            let default_internal = heuristic_knots_for_column(ds.values.column(c));
            let (mut n_knots, inferred, effective_degree) =
                parse_ps_internal_knots(options, degree, default_internal)?;
            let periodic_axes = parse_periodic_axes(options, 1).map_err(|e| e.to_string())?;
            // Periodic margins still need enough basis functions to wrap, so
            // surface the per-axis degree reduction as a config error when the
            // user explicitly asked for a periodic-but-too-small basis. The
            // non-periodic path silently degrades degree to match mgcv.
            if periodic_axes[0] && effective_degree != degree {
                return Err(TermBuilderError::invalid_option(format!(
                    "periodic smooth: k={} too small for degree {}; expected k >= {}",
                    effective_degree + 1,
                    degree,
                    degree + 1
                ))
                .to_string());
            }
            if inferred && ds.values.nrows() <= 32 && smooth_coordinate_count >= 5 {
                n_knots = n_knots.min(1);
            }
            if inferred {
                let unique = unique_count_column(ds.values.column(c));
                let ceiling = ((unique as f64).cbrt() as usize).max(20);
                inference_notes.push(format!(
                    "Automatically set {} internal knots for smooth '{}' from {} unique values (rule: clamp(unique/4, 4..max(20, cbrt(unique))) = clamp(unique/4, 4..{})). Override with knots=... or k=....",
                    n_knots,
                    vars.join(","),
                    unique,
                    ceiling,
                ));
            }
            let boundary_conditions =
                if periodic_axes[0] && bspline_boundary_declares_periodic_axis(options) {
                    BSplineBoundaryConditions::default()
                } else {
                    parse_bspline_boundary_conditions(options).map_err(|e| e.to_string())?
                };
            // An anchored endpoint (one *or* both sides) is already the model's
            // level-setting gauge: term-design construction suppresses the
            // global intercept so the fitted function itself, rather than only a
            // centered deviation, obeys the endpoint pin. Applying the ordinary
            // sum-to-zero chart as well would force the entire anchored function
            // to have sample mean zero. In #1867 that made a positive one-sided
            // anchored bump mathematically unrecoverable before REML was even
            // evaluated; for a two-sided anchor it additionally strips the
            // interior level the two pins bracket (#2297).
            let identifiability = if boundary_conditions.has_anchor() {
                BSplineIdentifiability::None
            } else {
                BSplineIdentifiability::default()
            };
            let periods = parse_periods(options, &periodic_axes).map_err(|e| e.to_string())?;
            let origins =
                parse_period_origins(options, &periodic_axes).map_err(|e| e.to_string())?;
            let (knotspec, boundary) = if periodic_axes[0] {
                if !boundary_conditions.is_free() {
                    return Err(TermBuilderError::incompatible_config(
                        "periodic B-splines cannot also declare endpoint boundary conditions",
                    )
                    .to_string());
                }
                {
                    let (domain_start, p_value) = if let Some(period) = periods[0] {
                        (origins[0].unwrap_or(minv), period)
                    } else {
                        parse_periodic_domain_1d(options, minv, maxv).map_err(|e| e.to_string())?
                    };
                    let domain_end = domain_start + p_value;
                    (
                        BSplineKnotSpec::PeriodicUniform {
                            data_range: (domain_start, domain_end),
                            num_basis: n_knots + effective_degree + 1,
                        },
                        OneDimensionalBoundary::Cyclic {
                            start: domain_start,
                            end: domain_end,
                        },
                    )
                }
            } else if type_opt == "cr" || type_opt == "cs" {
                // mgcv `bs="cr"`/`"cs"`: a natural cubic regression spline whose
                // basis is indexed by `k` values at quantile-placed knots (#1074),
                // NOT a B-spline knot vector. Match gam's `k=` convention by
                // requesting the same total basis size the B-spline arm would
                // produce (`n_knots` internal + degree + 1), floored at the cr
                // minimum of 3 knots. `cr` vs `cs` (shrinkage) is carried by the
                // `double_penalty` flag resolved below, which the cr builder reads.
                //
                // Cap that request to the covariate's data support (#1541): a cr
                // basis cannot place more value-knots than there are distinct
                // covariate values, so an unclamped `k` on a low-cardinality
                // predictor (binary indicator, 3-level ordinal, small count) used
                // to hard-fail in `select_cr_knots` instead of reducing like mgcv
                // and gam's tensor path. Below the cr minimum (a binary covariate)
                // degrade to the B-spline marginal the default `s(x, k=..)` basis
                // already fits on the same data — never a hard error.
                let k_cr = (n_knots + effective_degree + 1).max(CR_MIN_KNOTS);
                let knotspec = match capped_cr_marginal_knotspec(
                    ds.values.column(c),
                    k_cr,
                    &vars.join(","),
                    inference_notes,
                )? {
                    Some(cr_knotspec) => cr_knotspec,
                    None => resolve_nonperiodic_bspline_knotspec(
                        options,
                        ds.values.column(c),
                        (minv, maxv),
                        effective_degree,
                        n_knots,
                    )?,
                };
                (knotspec, parse_cyclic_boundary(options, minv, maxv)?)
            } else {
                (
                    resolve_nonperiodic_bspline_knotspec(
                        options,
                        ds.values.column(c),
                        (minv, maxv),
                        effective_degree,
                        n_knots,
                    )?,
                    parse_cyclic_boundary(options, minv, maxv)?,
                )
            };
            // Both cubic-regression spellings recover unsupported null-space
            // effects by default. An explicit `double_penalty=false` is the
            // MLE-style opt-out.
            let double_penalty = smooth_double_penalty;
            // Clamp the marginal difference penalty to `<= effective_degree`
            // so it stays well-defined when the per-axis degree was reduced
            // (mirrors the tensor margin path: `create_difference_penalty_matrix`
            // requires order < num_basis_functions).
            let penalty_order = option_usize(options, "penalty_order")
                .unwrap_or(DEFAULT_PENALTY_ORDER)
                .min(effective_degree);
            Ok(SmoothBasisSpec::BSpline1D {
                feature_col: c,
                spec: BSplineBasisSpec {
                    degree: effective_degree,
                    penalty_order,
                    knotspec,
                    double_penalty,
                    identifiability,
                    boundary,
                    boundary_conditions,
                },
            })
        }
        "tps" | "thinplate" | "thin-plate" => {
            validate_known_options(
                "thinplate",
                options,
                &[
                    SECONDARY_CENTER_CAP_OPTION,
                    "type",
                    "bs",
                    "by",
                    "length_scale",
                    "centers",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "knots",
                    "include_intercept",
                    "double_penalty",
                    "by",
                    "id",
                    "__by_col",
                    "identifiability",
                    "by",
                    "periodic",
                    "cyclic",
                    "period",
                    "period_start",
                    "period_end",
                    "scale_dims",
                ],
            )?;
            let plan = plan_spatial_basis(
                sizing_rows,
                cols.len(),
                CenterCountRequest::Default,
                DuchonNullspaceOrder::Linear,
                option_bool(options, "scale_dims").unwrap_or(false),
                policy,
            )
            .map_err(|e| e.to_string())?;
            // #1074: the mgcv-sized basis cap (`k = 10·3^(d-1)`) that used to live
            // here was DELETED. It masked the real defect — the n-scaling default
            // over-sizes a thin-plate field, producing a weakly-identified
            // two-penalty ρ-surface the outer optimizer stalls on (row-order
            // dependent, #1378), and surplus columns REML can't penalize away on
            // weak-signal fits. Capping the basis hid that stall instead of fixing
            // it. The default now uses the generic spatial center heuristic; the
            // root fix (a well-identified ρ-surface / optimizer that doesn't stall)
            // is tracked separately. Explicit `k`/`centers` still take full effect.
            let default_centers = plan.centers;
            let centers = parse_countwith_basis_alias(
                options,
                "centers",
                cap_default_spatial_centers(options, default_centers),
            )?;
            let center_strategy = if has_explicit_countwith_basis_alias(options, "centers") {
                spatial_center_strategy_for_dimension(centers, cols.len())
            } else {
                auto_spatial_center_strategy(centers, cols.len())
            };
            Ok(SmoothBasisSpec::ThinPlate {
                feature_cols: cols.to_vec(),
                spec: ThinPlateBasisSpec {
                    center_strategy,
                    periodic: parse_periodic_axes_option(options, cols.len())?,
                    // Sentinel: leave at 0.0 when the user didn't pass an
                    // explicit length_scale so `auto_init_length_scale_in_place`
                    // can replace it with a data-derived initialization. The
                    // old hard-coded 1.0 was the documented basin (see
                    // smooth.rs `auto_init_length_scale_in_place`) that the
                    // spatial optimizer could not escape, leaving TPS terms
                    // initialized off the data scale.
                    length_scale: option_f64(options, "length_scale").unwrap_or(0.0),
                    double_penalty: smooth_double_penalty,
                    identifiability: parse_spatial_identifiability(options)
                        .map_err(|e| e.to_string())?,
                    radial_reparam: None,
                },
                input_scale: None,
            })
        }
        "sphere" | "s2" | "sos" => {
            validate_known_options(
                "sphere",
                options,
                &[
                    "type",
                    "bs",
                    "by",
                    "centers",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "knots",
                    "penalty_order",
                    "m",
                    "double_penalty",
                    "id",
                    "__by_col",
                    "kernel",
                    "method",
                    "radians",
                    "units",
                    "degree",
                    "l",
                    "max_degree",
                    "max-degree",
                    "lmax",
                    "l_max",
                    "l-max",
                ],
            )?;
            if cols.len() != 2 {
                return Err(format!(
                    "sphere smooth expects exactly two variables (lat, lon), got {}",
                    cols.len()
                ));
            }
            let radians = option_bool(options, "radians").unwrap_or_else(|| {
                options
                    .get("units")
                    .map(|u| u.eq_ignore_ascii_case("radian") || u.eq_ignore_ascii_case("radians"))
                    .unwrap_or(false)
            });
            // An explicit `degree`/`l`/`max_degree` names a spherical-harmonic
            // truncation, so with no explicit kernel/method it selects the
            // Harmonic construction (the Wahba kernel ignores `degree` and would
            // silently emit a 1-column kernel design). An explicit kernel/method
            // still wins.
            let degree_requested = options.contains_key("degree")
                || options.contains_key("l")
                || options.contains_key("max_degree")
                || options.contains_key("max-degree");
            let kernel = options
                .get("kernel")
                .or_else(|| options.get("method"))
                .map(|raw| strip_quotes(raw).trim().to_ascii_lowercase())
                .unwrap_or_else(|| {
                    if degree_requested {
                        "harmonic".to_string()
                    } else {
                        "sobolev".to_string()
                    }
                });
            let (method, wahba_kernel) = match kernel.as_str() {
                "sobolev" | "wahba" | "wahba_sobolev" | "wahba-sobolev" => {
                    (SphereMethod::Wahba, SphereWahbaKernel::Sobolev)
                }
                "pseudo" | "mgcv" | "sos" | "wahba_pseudo" | "wahba-pseudo" => {
                    (SphereMethod::Wahba, SphereWahbaKernel::Pseudo)
                }
                "harmonic" | "spherical_harmonic" | "spherical-harmonic" => {
                    (SphereMethod::Harmonic, SphereWahbaKernel::Sobolev)
                }
                other => {
                    return Err(format!(
                        "unsupported sphere kernel '{other}'; expected sobolev, pseudo, or harmonic"
                    ));
                }
            };
            // `lmax=` states a finite spectral resolution for a Wahba kernel,
            // selecting the truncated variant `Σ_{ℓ=1..lmax} c_ℓ P_ℓ(cos γ)`
            // instead of the closed form. This is the only route from the
            // formula surface to `SobolevTruncated`/`PseudoTruncated`, and it
            // is what makes `m=1` expressible at all: the untruncated Sobolev
            // `m = 1` kernel is log-singular at coincidence, so it has no Gram
            // diagonal and the basis builder refuses it (#2475). Before this
            // option the refusal named a remedy no formula could reach.
            let wahba_kernel = match option_usize_any(options, &["lmax", "l_max", "l-max"]) {
                None => wahba_kernel,
                Some(_) if matches!(method, SphereMethod::Harmonic) => {
                    return Err(
                        "sphere smooth: lmax= states the truncation of a Wahba reproducing kernel \
                         and does not apply to kernel=harmonic; use degree=/max_degree= to set the \
                         harmonic degree"
                            .to_string(),
                    );
                }
                Some(lmax) => {
                    if !(SPHERE_TRUNCATION_LMAX_RANGE).contains(&lmax) {
                        return Err(format!(
                            "sphere smooth: lmax={lmax} is out of range; the truncated Wahba \
                             kernels support lmax in {}..={} (the device kernel bakes it in as a \
                             compile-time bound)",
                            SPHERE_TRUNCATION_LMAX_RANGE.start(),
                            SPHERE_TRUNCATION_LMAX_RANGE.end()
                        ));
                    }
                    let lmax = lmax as u16;
                    match wahba_kernel {
                        SphereWahbaKernel::Sobolev | SphereWahbaKernel::SobolevTruncated { .. } => {
                            SphereWahbaKernel::SobolevTruncated { lmax }
                        }
                        SphereWahbaKernel::Pseudo | SphereWahbaKernel::PseudoTruncated { .. } => {
                            SphereWahbaKernel::PseudoTruncated { lmax }
                        }
                    }
                }
            };
            let max_degree = if matches!(method, SphereMethod::Harmonic) {
                let degree =
                    option_usize_any(options, &["degree", "l", "max_degree", "max-degree"])
                        .or_else(|| option_usize(options, "centers"))
                        .or_else(|| {
                            option_usize_any(options, &["k", "basis_dim", "basis-dim", "basisdim"])
                                .and_then(|k| (1..=128).find(|&l| l * (l + 2) >= k))
                        })
                        .unwrap_or_else(|| default_spherical_harmonic_degree(sizing_rows));
                if degree == 0 {
                    return Err("sphere smooth requires degree/max_degree >= 1".to_string());
                }
                if degree > 32 {
                    return Err(format!(
                        "sphere smooth max_degree={} is too large for the dense harmonic engine (limit 32)",
                        degree
                    ));
                }
                Some(degree)
            } else {
                None
            };
            let penalty_order = option_usize(options, "penalty_order")
                .or_else(|| option_usize(options, "m"))
                .unwrap_or(DEFAULT_PENALTY_ORDER);
            let center_strategy = if matches!(method, SphereMethod::Wahba) {
                let mut centers = parse_countwith_basis_alias(
                    options,
                    "centers",
                    default_num_centers(sizing_rows, cols.len()),
                )?;
                if penalty_order >= 4 {
                    centers = centers.max(30);
                }
                CenterStrategy::FarthestPoint {
                    num_centers: centers,
                }
            } else {
                CenterStrategy::FarthestPoint { num_centers: 0 }
            };
            Ok(SmoothBasisSpec::Sphere {
                feature_cols: cols.to_vec(),
                spec: SphericalSplineBasisSpec {
                    center_strategy,
                    penalty_order,
                    double_penalty: smooth_double_penalty,
                    radians,
                    method,
                    max_degree,
                    wahba_kernel,
                    identifiability: SphericalSplineIdentifiability::CenterSumToZero,
                },
            })
        }
        "curvature" => {
            // Constant-curvature (M_κ) geodesic-kernel smooth (#944): the
            // κ-generic sibling of the intrinsic S² smooth above. The feature
            // columns are κ-stereographic chart coordinates and the geometry
            // comes from `geometry::constant_curvature::ConstantCurvature`.
            // `kappa=` follows the mgcv-`sp=` convention (gam#2152): an EXPLICIT
            // value is a FIXED sectional curvature that selects the geometry
            // (`Sᵈ` for κ>0, `ℝᵈ` for κ=0, `Hᵈ` for κ<0) and is honoured verbatim
            // by the fit; OMITTING `kappa=` leaves κ free for the #944/#1464
            // outer ψ-coordinate estimation, seeded at the flat default 0.
            validate_known_options(
                "curvature",
                options,
                &[
                    "type",
                    "bs",
                    "by",
                    "centers",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "knots",
                    "kappa",
                    "length_scale",
                    "double_penalty",
                    "id",
                    "__by_col",
                ],
            )?;
            // `kappa=` follows the mgcv-`sp=` convention: an EXPLICIT value pins
            // the sectional curvature (fixed geometry, honoured verbatim by the
            // fit — gam#2152); an OMITTED `kappa=` leaves κ free for the
            // #944/#1464 outer estimation, seeded at the flat default 0.
            let kappa_opt = option_f64(options, "kappa");
            let kappa_fixed = kappa_opt.is_some();
            let kappa = kappa_opt.unwrap_or(0.0);
            if !kappa.is_finite() {
                return Err("curvature smooth requires a finite kappa".to_string());
            }
            // `length_scale=` follows the SAME mgcv-`sp=` convention as `kappa=`
            // (gam#2747): an EXPLICIT value pins the kernel resolution and the fit
            // honours it verbatim; an OMITTED one leaves η = ln ℓ free for the
            // outer estimation, seeded by the auto rule. The range must be fitted
            // by default because it is confounded with κ — pinning it makes κ
            // absorb the range error rather than measure curvature.
            let length_scale_opt = option_f64(options, "length_scale");
            let length_scale_fixed = length_scale_opt.is_some();
            let length_scale = length_scale_opt.unwrap_or(0.0);
            if !length_scale.is_finite() || length_scale < 0.0 {
                return Err(format!(
                    "curvature smooth length_scale must be positive (or omitted for auto); got {length_scale}"
                ));
            }
            let centers = parse_countwith_basis_alias(
                options,
                "centers",
                default_num_centers(sizing_rows, cols.len()),
            )?;
            if centers < 2 {
                return Err("curvature smooth requires at least 2 centers".to_string());
            }
            let center_strategy = if has_explicit_countwith_basis_alias(options, "centers") {
                spatial_center_strategy_for_dimension(centers, cols.len())
            } else {
                auto_spatial_center_strategy(centers, cols.len())
            };
            Ok(SmoothBasisSpec::ConstantCurvature {
                feature_cols: cols.to_vec(),
                spec: ConstantCurvatureBasisSpec {
                    center_strategy,
                    kappa,
                    kappa_fixed,
                    // 0.0 sentinel = κ-independent auto initialization in the
                    // basis builder (median chart center spacing, doubled).
                    length_scale,
                    length_scale_fixed,
                    // Curvature smooth defaults to NO double-penalty ridge
                    // (#1464): the curvature-blind ridge `I` absorbs the data fit
                    // independently of κ and rails the fitted curvature to the
                    // +chart bound (hyperbolic truth recovered as spherical). The
                    // RKHS Gram penalty is already full-rank PD, so the ridge adds
                    // no stability. Honour an EXPLICIT `double_penalty=` only.
                    double_penalty: option_bool(options, "double_penalty").unwrap_or(false),
                    identifiability: ConstantCurvatureIdentifiability::CenterSumToZero,
                },
            })
        }
        "measurejet" => {
            // Measure-jet spline: multiscale local-jet-residual energy of the
            // empirical measure. The feature columns are ambient coordinates
            // of data concentrated near an unknown low-dimensional set; the
            // geometry (centers, masses, scale band) is read off the measure
            // at build time — magic by default, every option optional.
            validate_known_options(
                "measurejet",
                options,
                &[
                    "type",
                    "bs",
                    "by",
                    "centers",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "knots",
                    "s",
                    "alpha",
                    "tau",
                    "scales",
                    "length_scale",
                    "double_penalty",
                    "multiscale",
                    "learn_length_scale",
                    "id",
                    "__by_col",
                ],
            )?;
            let order_s = option_f64(options, "s").unwrap_or(0.0);
            // 0.0 = auto sentinel; explicit values must sit inside the
            // admissible order interval of the affine-jet (r = 2) energy.
            if !(order_s.is_finite() && (order_s == 0.0 || (order_s > 0.0 && order_s < 2.0))) {
                return Err(format!(
                    "measurejet smooth s must lie in (0, 2) (or be omitted for auto); got {order_s}"
                ));
            }
            // Default to the spec Default (α = 1, density-WEIGHTED Hessian
            // energy — the module-header default). The density-free α = 3/2
            // (q^{−2}) over-smooths low-intrinsic-dimension manifolds where the
            // local mass q is tiny and varies along the stratum (#1116:
            // 13×-worse-than-matérn on a 1-D curve in 3-D); α = 1's q^{−1} is
            // gentler and robust across intrinsic dimensions. An explicit
            // `alpha=` still overrides for full-dimensional density-free use.
            let alpha =
                option_f64(options, "alpha").unwrap_or(MeasureJetBasisSpec::default().alpha);
            if !alpha.is_finite() {
                return Err("measurejet smooth requires a finite alpha".to_string());
            }
            let tau0 = option_f64(options, "tau").unwrap_or(1e-3);
            if !(tau0.is_finite() && tau0 >= 0.0) {
                return Err(format!(
                    "measurejet smooth tau must be finite and nonnegative; got {tau0}"
                ));
            }
            let num_scales = option_usize(options, "scales").unwrap_or(0);
            let length_scale = option_f64(options, "length_scale").unwrap_or(0.0);
            if !length_scale.is_finite() || length_scale < 0.0 {
                return Err(format!(
                    "measurejet smooth length_scale must be positive (or omitted for auto); got {length_scale}"
                ));
            }
            let centers = parse_countwith_basis_alias(
                options,
                "centers",
                default_num_centers(sizing_rows, cols.len()),
            )?;
            if centers < 3 {
                return Err("measurejet smooth requires at least 3 centers".to_string());
            }
            let center_strategy = if has_explicit_countwith_basis_alias(options, "centers") {
                spatial_center_strategy_for_dimension(centers, cols.len())
            } else {
                auto_spatial_center_strategy(centers, cols.len())
            };
            // Multiscale (per-scale spectral split + (α, lnτ) ψ dials + the
            // affine-preserving ridge) is an explicit opt-in (#1116): default
            // single-scale at any center count, the Duchon/Matérn footprint.
            let multiscale = option_bool(options, "multiscale").unwrap_or(false);
            // The representer range ℓ is a design-moving basis coordinate of the
            // same kind as the Matérn κ: λ shrinks inside a span and cannot move
            // one, so a frozen ℓ is an error no smoothing parameter can repair
            // (#2761 measured it at 13.4x held-out RMSE on a 1-D curve in 3-D,
            // with the design's own span floor sitting AT the fitted value).
            // REML therefore selects it by default.
            //
            // An explicit `length_scale=` is a request, not a seed, so it pins ℓ
            // — the same short-circuit `all_spatial_terms_kappa_fixed` gives an
            // explicitly-scaled Matérn. `learn_length_scale=` overrides either
            // way.
            let learn_length_scale =
                option_bool(options, "learn_length_scale").unwrap_or(length_scale == 0.0);
            Ok(SmoothBasisSpec::MeasureJet {
                feature_cols: cols.to_vec(),
                spec: MeasureJetBasisSpec {
                    center_strategy,
                    order_s,
                    alpha,
                    tau0,
                    num_scales,
                    // 0.0 sentinel = auto initialization in the basis builder
                    // (median nearest-center spacing).
                    length_scale,
                    double_penalty: smooth_double_penalty,
                    learn_length_scale,
                    multiscale,
                    identifiability: MeasureJetIdentifiability::CenterSumToZero,
                    frozen_quadrature: None,
                },
                input_scale: None,
            })
        }
        "matern" => {
            // Catch typos like `lengt_scale=` / `nyu=` / `centerz=` before
            // they get silently ignored and the user wonders why their
            // option had no effect. The matern() term accepts exactly
            // these options.
            validate_known_options(
                "matern",
                options,
                &[
                    SECONDARY_CENTER_CAP_OPTION,
                    "type",
                    "bs",
                    "by",
                    "nu",
                    "length_scale",
                    "centers",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "knots",
                    "include_intercept",
                    "double_penalty",
                    "by",
                    "id",
                    "__by_col",
                    "identifiability",
                    "by",
                    "periodic",
                    "cyclic",
                    "period",
                    "period_start",
                    "period_end",
                    "scale_dims",
                ],
            )?;
            let plan = plan_spatial_basis(
                sizing_rows,
                cols.len(),
                CenterCountRequest::Default,
                DuchonNullspaceOrder::Zero,
                option_bool(options, "scale_dims").unwrap_or(false),
                policy,
            )
            .map_err(|e| e.to_string())?;
            // #1867: spline-equivalent floor so a 1-D radial basis is not
            // dimensioned coarser than the competing `s(x)` on identical data.
            let univariate_floor = if cols.len() == 1 {
                heuristic_knots_for_column(ds.values.column(cols[0]))
                    .saturating_add(DEFAULT_BSPLINE_DEGREE + 1)
            } else {
                0
            };
            let centers = parse_countwith_basis_alias(
                options,
                "centers",
                cap_default_spatial_centers(
                    options,
                    default_matern_center_count(
                        sizing_rows,
                        cols.len(),
                        plan.centers,
                        univariate_floor,
                    ),
                ),
            )?;
            let center_strategy = if has_explicit_countwith_basis_alias(options, "centers") {
                spatial_center_strategy_for_dimension(centers, cols.len())
            } else {
                auto_spatial_center_strategy(centers, cols.len())
            };
            let nu = parse_matern_nu(options.get("nu").map(String::as_str).unwrap_or("5/2"))?;
            // The exponential (ν = 1/2) Matérn kernel has a singular Laplacian
            // at zero in d ≥ 2, so the operator-collocation penalty machinery
            // hits a non-invertible matrix during fit. Surface the cause
            // up-front instead of letting the user see the generic
            // "Matrix conditioning issue detected" wrapper from PIRLS.
            if matches!(nu, MaternNu::Half) && cols.len() >= 2 {
                return Err(TermBuilderError::unsupported_feature(format!(
                    "matern() with nu=1/2 is not supported for d>=2 (got {} covariates): \
                     the exponential kernel's Laplacian is singular at center collisions, \
                     which makes the operator-collocation penalty non-invertible. \
                     Choose nu>=3/2 (e.g. nu=3/2 or the default nu=5/2) for multi-dimensional smooths.",
                    cols.len()
                ))
                .to_string());
            }
            let aniso_log_scales = if option_bool(options, "scale_dims").unwrap_or(false) {
                Some(vec![0.0; cols.len()])
            } else {
                None
            };
            Ok(SmoothBasisSpec::Matern {
                feature_cols: cols.to_vec(),
                spec: MaternBasisSpec {
                    center_strategy,
                    periodic: parse_periodic_axes_option(options, cols.len())?,
                    // Preserve whether the user supplied `length_scale` as typed
                    // provenance. The planner resolves `Auto` to the same
                    // data-derived wiggly-side initialization the thin-plate path
                    // uses (`max_range / sqrt(n)`), then lets the κ-optimizer refine
                    // it without ever turning it into a user-fixed scale.
                    //
                    // gam#1629: the previous `default_matern_length_scale` seeded
                    // the FULL data diameter — the maximally over-smoothed corner.
                    // Because that value looked explicit, the old auto-init was a
                    // no-op for Matérn, so the κ-optimizer started in the flat
                    // over-smoothed basin and parked there, leaving high-frequency
                    // 2-D surfaces unresolved (truth-RMSE ~6× worse than
                    // thin-plate/tensor on identical data, and insensitive to `k`).
                    // Typed Auto starts REML in the resolving regime it can escape
                    // from and cannot be confused with explicit zero.
                    length_scale: option_f64(options, "length_scale")
                        .map(MaternLengthScale::fixed)
                        .unwrap_or_else(MaternLengthScale::auto),
                    nu,
                    include_intercept: option_bool(options, "include_intercept").unwrap_or(false),
                    double_penalty: smooth_double_penalty,
                    identifiability: parse_matern_identifiability(options)
                        .map_err(|e| e.to_string())?,
                    aniso_log_scales,
                    // Cold build: let the bootstrap-κ spectral test decide whether
                    // the double-penalty nullspace shrinkage survives; the freeze
                    // step then pins that decision into the FrozenTransform so the
                    // κ-optimizer's rebuilds keep the count invariant (gam#787/#860).
                },
                input_scale: None,
            })
        }
        "duchon" | "ds" => {
            validate_known_options(
                "duchon",
                options,
                &[
                    SECONDARY_CENTER_CAP_OPTION,
                    "type",
                    "bs",
                    "by",
                    "length_scale",
                    "centers",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "knots",
                    "rank",
                    "power",
                    "p",
                    "nullspace_order",
                    "order",
                    "identifiability",
                    "by",
                    "periodic",
                    "cyclic",
                    "period",
                    "period_start",
                    "period_end",
                    "scale_dims",
                    "double_penalty",
                    "by",
                    "id",
                    "__by_col",
                ],
            )?;
            if options.contains_key("double_penalty") {
                return Err(TermBuilderError::incompatible_config(format!(
                    "Duchon smooth '{}' does not support double_penalty; the Duchon smoother already ships its native reproducing-norm penalty plus a null-space shrinkage ridge.",
                    vars.join(", ")
                ))
                .to_string());
            }
            let requested_nullspace_order = parse_duchon_order(options)?;
            let length_scale = option_f64_strict(options, "length_scale")?;
            // Resolve `(nullspace_order, power)`. The default (magic) path is a
            // structural amplitude/slope/curvature smoother: an affine (`Linear`)
            // polynomial nullspace and spectral power `s = (d - 1)/2`, giving the
            // cubic kernel `r^3` in 1D. There is no nullspace-order escalation —
            // the structural cubic smoother is well-defined for every dimension.
            //
            // Explicit `power=...` honors the user's value verbatim against their
            // requested nullspace order; the kernel validator emits a precise
            // diagnostic for any inadmissible combination. In the scale-free
            // (non-hybrid) regime fractional powers are admitted and threaded as
            // `f64`. The hybrid Duchon-Matérn kernel (`length_scale=Some`) is
            // restricted to integer powers.
            let (nullspace_order, power) = match parse_duchon_power_policy(options)? {
                DuchonPowerPolicy::Explicit(req_power) => {
                    if length_scale.is_some() && req_power.fract() != 0.0 {
                        return Err(TermBuilderError::incompatible_config(format!(
                            "hybrid Duchon-Matern smooth '{}' (length_scale=...) requires an integer power, got power={}; \
                             drop length_scale to use the scale-free structural kernel with a fractional power.",
                            vars.join(", "),
                            req_power,
                        ))
                        .to_string());
                    }
                    (requested_nullspace_order, req_power)
                }
                DuchonPowerPolicy::CubicStructuralDefault => {
                    // Magic cubic rule (REQUEST-LAYER default): no explicit power ⇒
                    // affine null space + fractional spectral power s = (d-1)/2, i.e.
                    // the Duchon kernel φ(r)=r³ in every dimension. An EXPLICIT
                    // `power=0` is handled above and is honored as the s=0 Duchon
                    // kernel (r²·log r ≡ the thin-plate kernel in even d) — the magic
                    // default lives here, not in the basis builder.
                    match length_scale {
                        None => crate::basis::duchon_cubic_default(cols.len()),
                        Some(_) => {
                            // The hybrid Matérn-blended kernel (`length_scale=Some`)
                            // requires an INTEGER spectral power `s` (the partial-
                            // fraction split `1/(ρ^{2p}(κ²+ρ²)^s)` is only defined for
                            // integer `s`). The fractional cubic default `s=(d-1)/2` is
                            // a half-integer for even `d`, and the basis builder's
                            // `power_as_usize` maps a NON-integer to `0` (not its
                            // floor) — so for even `d ≥ 4` the realized kernel has
                            // `2(p+s) = 2p = 4 ≤ d`, which is non-finite at the origin
                            // and crashes the fit (historically a non-finite
                            // eigendecomposition; now a fit-time validation error).
                            //
                            // Resolve to the same structural cubic default the
                            // scale-free path uses (affine `Linear` null space, `r³`
                            // kernel, fractional power `s = (d-1)/2`) but take the
                            // largest admissible INTEGER at or below it — `⌊(d-1)/2⌋`.
                            // For odd `d` this is exactly the cubic power (the hybrid
                            // default then agrees with the scale-free cubic default);
                            // for even `d` it is the nearest integer below. Either way
                            // `p = 2` (affine) gives spectral order
                            // `2(p+s) = d+3` (odd `d`) or `d+2` (even `d`), which
                            // clears both kernel existence `2(p+s) > d` and the D1
                            // collocation floor `2(p+s) > d+1` for every `d ≥ 1`.
                            // Flooring here at the request layer avoids the
                            // `power_as_usize` truncation-to-zero on the fractional
                            // half-integer.
                            let (ns, s_frac) = crate::basis::duchon_cubic_default(cols.len());
                            (ns, s_frac.floor())
                        }
                    }
                }
            };
            let plan = plan_spatial_basis(
                sizing_rows,
                cols.len(),
                CenterCountRequest::Default,
                nullspace_order,
                option_bool(options, "scale_dims").unwrap_or(false),
                policy,
            )
            .map_err(|e| e.to_string())?;
            let centers_explicit = has_explicit_countwith_basis_alias(options, "centers");
            let polynomial_cols = match nullspace_order {
                DuchonNullspaceOrder::Zero => 1,
                DuchonNullspaceOrder::Linear => cols.len() + 1,
                DuchonNullspaceOrder::Degree(degree) => {
                    crate::basis::duchon_nullspace_dimension(cols.len(), degree)
                }
            };
            // #1867: spline-equivalent floor so a 1-D radial basis is not
            // dimensioned coarser than the competing `s(x)` on identical data.
            let univariate_floor = if cols.len() == 1 {
                heuristic_knots_for_column(ds.values.column(cols[0]))
                    .saturating_add(DEFAULT_BSPLINE_DEGREE + 1)
            } else {
                0
            };
            let default_centers = default_duchon_center_count(
                sizing_rows,
                cols.len(),
                plan.centers,
                polynomial_cols,
                univariate_floor,
            );
            let spectral_rank = option_usize(options, "rank");
            let center_default = if spectral_rank.is_some() {
                // mgcv's Duchon constructor runs `uniquecombs` FIRST and caps
                // at `max.knots` afterwards, so its knot budget is
                // `min(n_unique, 2000)`. This took the RAW row count and let
                // `select_r_uniform_subsample_centers` deduplicate later — so
                // on any data carrying a repeated coordinate row with fewer
                // than 2000 rows, the budget exceeded what the sampler could
                // supply and the fit hard-refused rather than degrading
                // (#2623: `prostate_gamair`, 523 requested vs 522 unique).
                // Counting distinct rows here makes the budget satisfiable by
                // construction, and leaves the retained spectral `rank` — a
                // separate option — untouched.
                count_unique_coordinate_rows(ds.values.view(), &cols).min(2000)
            } else {
                cap_default_spatial_centers(options, default_centers)
            };
            let requested_centers =
                parse_countwith_basis_alias(options, "centers", center_default)?;
            if requested_centers > ds.values.nrows() {
                return Err(TermBuilderError::incompatible_config(format!(
                    "Duchon smooth '{}' requested {requested_centers} centers but only {} rows are available",
                    vars.join(", "),
                    ds.values.nrows(),
                ))
                .to_string());
            }
            if requested_centers <= polynomial_cols {
                return Err(TermBuilderError::incompatible_config(format!(
                    "Duchon smooth '{}' requested basis dimension {} but order={:?} in {}D needs {} polynomial null-space columns; choose centers/k > {}",
                    vars.join(", "),
                    requested_centers,
                    nullspace_order,
                    cols.len(),
                    polynomial_cols,
                    polynomial_cols,
                ))
                .to_string());
            }
            if let Some(rank) = spectral_rank
                && (rank <= polynomial_cols || rank > requested_centers)
            {
                return Err(TermBuilderError::incompatible_config(format!(
                    "Duchon smooth '{}' spectral rank must satisfy {} < rank <= centers (got rank={rank}, centers={requested_centers})",
                    vars.join(", "),
                    polynomial_cols,
                ))
                .to_string());
            }
            let mut centers = requested_centers;
            if !centers_explicit && ds.values.nrows() <= 32 && smooth_coordinate_count >= 5 {
                centers = centers.max(polynomial_cols + 4);
            }
            let aniso_log_scales = if option_bool(options, "scale_dims").unwrap_or(false) {
                Some(vec![0.0; cols.len()])
            } else {
                None
            };
            // Formula-level `duchon(...)` is the native Duchon reproducing-norm
            // smoother: the always-on Primary Gram plus the polynomial trend
            // ridge. Do not silently add collocated mass/tension penalties here.
            // They add extra REML hyperparameters and an O(k)-support quadrature
            // build to the default 2-D path, making `duchon(x, z)` materially
            // slower than the equivalent thin-plate fit without a principled
            // accuracy gain (gam#1718). Lower-order Hilbert-scale penalties remain
            // available to callers that construct an explicit DuchonBasisSpec.
            let operator_penalties = DuchonOperatorPenaltySpec::all_disabled();
            // For a 1-D periodic Duchon with no EXPLICIT period, anchor the wrap
            // to the covariate DATA range rather than letting the basis builder
            // derive it from the (k-subsampled) center span. The center span is a
            // strict subset of the data and undershoots the true period, seaming
            // the curve (f(0) ≠ f(2π)); the data range is the caller's actual
            // domain. Honors any explicit `period=` (parse_periodic_axes_option
            // already threaded it) and leaves multi-D / non-periodic untouched.
            let mut periodic = parse_periodic_axes_option(options, cols.len())?;
            if cols.len() == 1
                && let Some(axes) = periodic.as_mut()
                && axes.len() == 1
                && axes[0].is_none()
            {
                let (minv, maxv) = col_minmax(ds.values.column(cols[0]))?;
                if maxv > minv {
                    axes[0] = Some(maxv - minv);
                }
            }
            let boundary = if cols.len() == 1 {
                let c = cols[0];
                let (minv, maxv) = col_minmax(ds.values.column(c))?;
                parse_cyclic_boundary(options, minv, maxv)?
            } else {
                OneDimensionalBoundary::Open
            };
            let is_periodic = periodic
                .as_ref()
                .is_some_and(|axes| axes.iter().any(Option::is_some))
                || matches!(boundary, OneDimensionalBoundary::Cyclic { .. });
            if spectral_rank.is_some() && is_periodic {
                return Err(TermBuilderError::incompatible_config(
                    "Duchon spectral rank is defined for the scale-free open-domain kernel, \
                     not a periodic image expansion"
                        .to_string(),
                )
                .to_string());
            }
            let center_strategy = if spectral_rank.is_some() {
                // Freeze the exact fixed-seed uniform landmark experiment used
                // by mgcv's Duchon constructor. Spectral rank parity requires
                // the same kernel matrix, not merely the same retained column
                // count: maximin/equal-mass landmarks define a different
                // finite-sample eigenspace and confound accuracy comparisons.
                // Materializing 2,000×d coordinates here is cheap, avoids an
                // O(nk) maximin pass, and makes prediction replay explicit.
                let mut coordinates = Array2::<f64>::zeros((ds.values.nrows(), cols.len()));
                for (axis, &column) in cols.iter().enumerate() {
                    coordinates
                        .column_mut(axis)
                        .assign(&ds.values.column(column));
                }
                let sampled = select_r_uniform_subsample_centers(coordinates.view(), centers, 1)
                    .map_err(|error| error.to_string())?;
                CenterStrategy::UserProvided(sampled)
            } else if is_periodic {
                if centers_explicit {
                    spatial_center_strategy_for_dimension(centers, cols.len())
                } else {
                    auto_spatial_center_strategy(centers, cols.len())
                }
            } else {
                duchon_center_strategy(centers, cols.len(), !centers_explicit)
            };
            let center_strategy = match spectral_rank {
                Some(rank) => CenterStrategy::DuchonSpectral {
                    knots: Box::new(center_strategy),
                    basis: DuchonSpectralBasis::Fresh { rank },
                },
                None => center_strategy,
            };
            Ok(SmoothBasisSpec::Duchon {
                feature_cols: cols.to_vec(),
                spec: DuchonBasisSpec {
                    center_strategy,
                    periodic,
                    length_scale,
                    power,
                    nullspace_order,
                    identifiability: parse_spatial_identifiability(options)
                        .map_err(|e| e.to_string())?,
                    aniso_log_scales,
                    operator_penalties,
                    boundary,
                    radial_reparam: None,
                },
                input_scale: None,
            })
        }
        "tensor" | "te" | "ti" | "t2" => {
            validate_known_options(
                "tensor",
                options,
                &[
                    "type",
                    "bs",
                    "by",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "knot_placement",
                    "knot-placement",
                    "knotplacement",
                    "degree",
                    "penalty_order",
                    "double_penalty",
                    "periodic",
                    "cyclic",
                    "period",
                    "periods",
                    "period_start",
                    "period_end",
                    "origin",
                    "origins",
                    "period_origin",
                    "period-origin",
                    "domain_origin",
                    "boundary",
                    "bc",
                    "identifiability",
                    "id",
                    "__by_col",
                ],
            )?;
            if cols.len() < 2 {
                return Err(TermBuilderError::incompatible_config(format!(
                    "tensor smooth expects at least 2 variables, got {}",
                    cols.len()
                ))
                .to_string());
            }
            let dim = cols.len();

            // Tensor-product contract (#1082). `te(x1, x2, ...)` ALWAYS builds a
            // genuine anisotropic tensor product of per-margin bases (the arm
            // below), exactly as mgcv's `te()` does — one smoothing parameter per
            // margin, a marginal-Kronecker-sum penalty, and a separate default
            // function-space ridge on the joint polynomial null space. A margin
            // vector `bs=c('tp','tp')` requests a thin-plate FUNCTION SPACE per
            // axis; the tensor realizes each axis as a 1-D penalized B-spline
            // margin spanning that same per-axis space (tp/ps/cr/bs/cc all share
            // it). We deliberately do NOT silently swap the requested tensor for a
            // single multi-D ISOTROPIC thin-plate radial smooth (`s(x,y,bs='tp')`):
            // that is a different model — one isotropic smoothing parameter, no
            // per-margin anisotropy — and substituting it while the user wrote a
            // tensor formula is dishonest. A user who genuinely wants the isotropic
            // radial smooth asks for it directly with `s(x1, x2, bs='tp')`.
            // Per-margin basis vector (`bs=c('tp','tp')` / `bs=['ps','cr']`):
            // validate each requested margin is a penalized-spline basis that
            // the tensor product realizes as a 1-D B-spline margin. mgcv's
            // `tp`/`ps`/`cr`/`bs`/`cc` margins are all penalized splines over
            // the same per-axis function space, so a B-spline margin recovers
            // the same tensor smoothing space; genuinely different margin kinds
            // (e.g. adaptive `ad`, random `re`) are rejected loudly rather than
            // silently substituted.
            if let Some(raw) = options.get("bs").or_else(|| options.get("type"))
                && bs_selector_is_vector(raw)
            {
                let per_margin = parse_option_list(raw);
                if per_margin.len() != dim {
                    return Err(TermBuilderError::invalid_option(format!(
                        "tensor smooth per-margin bs vector has {} entries but the smooth has {} margins",
                        per_margin.len(),
                        dim
                    ))
                    .to_string());
                }
                for (axis, margin_bs) in per_margin.iter().enumerate() {
                    if !tensor_margin_bs_is_supported(margin_bs) {
                        return Err(TermBuilderError::unsupported_feature(format!(
                            "tensor smooth margin {axis} basis '{margin_bs}' is not a supported penalized-spline margin; \
                             tensor margins accept tp/tps/ps/bs/cr/cc"
                        ))
                        .to_string());
                    }
                }
            }
            let periodic_axes = parse_tensor_periodic_axes(options, dim)?;
            validate_tensor_boundary_tokens(options, dim)?;
            let periods_opt = parse_periods(options, &periodic_axes)?;
            let origins_opt = parse_period_origins(options, &periodic_axes)?;
            let degree = option_usize(options, "degree").unwrap_or(DEFAULT_BSPLINE_DEGREE);
            let penalty_order =
                option_usize(options, "penalty_order").unwrap_or(if degree > 1 { 2 } else { 1 });
            let (mut k_list, k_inferred) = parse_tensor_k_list(options, cols, ds)?;
            if ds.values.nrows() <= 32 && smooth_coordinate_count >= 5 {
                for k in &mut k_list {
                    *k = (*k).min(degree + 2);
                }
            }
            if k_inferred {
                inference_notes.push(format!(
                    "Automatically set per-margin basis sizes {:?} for tensor smooth '{}' \
                     (dimension-aware tensor budget: total ∏k kept near the mgcv-te default \
                     and within the data support, distributed geometrically across margins and \
                     capped per margin by each column's resolution). \
                     Override with k=<int> or k=[k0,k1,...].",
                    k_list,
                    vars.join(",")
                ));
            }
            // Per-axis requested marginal basis family. mgcv's `te()`/`ti()`
            // default marginal basis is the cubic regression spline (`cr`), and
            // the te_3d quality gap (#1074) is precisely the marginal-basis
            // resolution at small `k`: a `cr` margin places k value-knots at
            // data quantiles (finer interior resolution under natural boundary
            // constraints) where the cubic B-spline margin has only
            // `k-degree-1` interior knots. Resolve each axis to either an
            // explicit per-margin `bs` (vector `bs=c('cr','ps')`), a single
            // scalar `bs`, or the unset default — and route
            // `cr`/`cs`/unset/`tp`/`tps` margins through the natural cubic
            // regression builder (`NaturalCubicRegression` knotspec), keeping
            // explicit `ps`/`bs`/`bspline` on the B-spline margin.
            let per_axis_bs: Vec<Option<String>> =
                match options.get("bs").or_else(|| options.get("type")) {
                    Some(raw) if bs_selector_is_vector(raw) => {
                        let list = parse_option_list(raw);
                        (0..dim).map(|a| list.get(a).cloned()).collect()
                    }
                    Some(raw) => {
                        let scalar = raw
                            .trim()
                            .trim_matches('"')
                            .trim_matches('\'')
                            .to_ascii_lowercase();
                        vec![Some(scalar); dim]
                    }
                    None => vec![None; dim],
                };
            // A margin is realized as a natural cubic regression spline when it
            // is the (unset) mgcv default, an explicit `cr`/`cs`, or a
            // `tp`/`tps` (same per-axis penalized-spline space). Explicit
            // B-spline-family margins (`ps`/`bs`/`bspline`/`p-spline`) keep the
            // open B-spline margin.
            let margin_wants_cr = |bs: &Option<String>| -> bool {
                matches!(
                    bs.as_deref(),
                    None | Some("cr") | Some("cs") | Some("tp") | Some("tps")
                )
            };
            let requested_knot_placement = parse_knot_placement(options)?;
            let mut margins: Vec<BSplineBasisSpec> = Vec::with_capacity(dim);
            let mut emitted_periods: Vec<Option<f64>> = Vec::with_capacity(dim);
            for axis in 0..dim {
                let c = cols[axis];
                let (data_min, data_max) = col_minmax(ds.values.column(c))?;
                // mgcv reduces a tensor margin's basis dimension to what its data
                // can support: a cr or B-spline margin cannot place more value
                // knots / basis functions than there are DISTINCT covariate
                // values on that axis. Without this cap an explicit `k` on a
                // low-cardinality margin — e.g. the binary `badh ∈ {0,1}` in
                // `te(age, badh, k=5)` — hard-failed in `select_cr_knots` ("cubic
                // regression spline with k=5 requires at least 5 distinct values,
                // got 2") instead of degrading to the 2-function (linear) margin
                // mgcv builds there. The auto-`k` path already caps per margin via
                // `heuristic_tensor_margin_knots`; mirror that for explicit `k`.
                // The cap propagates correctly: every per-axis quantity below
                // (effective degree, knot set, penalty order) is derived from
                // `k_axis`, and the marginal basis size is read from the resulting
                // knot spec — never from `k_list`. Floor at 2 so a margin still
                // carries at least a linear basis (tensor margins require k >= 2).
                let k_requested = k_list[axis];
                let n_distinct_axis = unique_count_column(ds.values.column(c));
                let k_axis = k_requested.min(n_distinct_axis).max(2);
                if k_axis < k_requested {
                    log::info!(
                        "tensor smooth: margin axis {axis} requested k={k_requested}, but the \
                         covariate has only {n_distinct_axis} distinct value(s); reducing this \
                         margin to k={k_axis} (mgcv-style data-support cap on the per-axis basis)."
                    );
                }
                // Per-axis effective spline degree. The B-spline basis with `k`
                // functions is well-defined for any `degree <= k - 1`; mgcv's
                // `te(...)` exploits this so a binary tensor margin
                // (`k=2` → linear basis) or a ternary margin (`k=3` → quadratic)
                // can coexist with a smoother continuous margin under one
                // shared `degree=` request. We mirror that: if the caller
                // explicitly asks for `k < degree + 1`, drop the degree on
                // THAT axis only to the largest feasible spline, and track the
                // penalty order so the marginal difference penalty stays
                // well-defined (`order < num_basis_functions` is required by
                // `create_difference_penalty_matrix`). Apply the same
                // per-margin degree shrinkage to periodic tensor margins too:
                // a cyclic marginal basis with k=3 cannot be cubic, but it is
                // still a valid lower-degree cyclic margin with dimension k,
                // matching mgcv's small-k tensor-margin behavior.
                if k_axis < 2 {
                    return Err(TermBuilderError::invalid_option(format!(
                        "tensor smooth: k[{axis}]={k_axis} too small; tensor margins require k >= 2"
                    ))
                    .to_string());
                }
                let effective_degree = degree.min(k_axis - 1).max(1);
                let effective_penalty_order = penalty_order.min(effective_degree);
                // A `cc`/`cp`/`cyclic` per-margin basis declares periodicity
                // without necessarily supplying a `period=`: mgcv's `bs="cc"`
                // wraps at the covariate's observed data range. Mirror the 1-D
                // cyclic fallback (`parse_periodic_domain_1d`) here so a bare
                // `te(x, z, bs=c('cc','cc'))` wraps each margin on its own
                // [min, max] span instead of hard-erroring (#1752).
                let margin_is_cc = matches!(
                    canonicalize_smooth_type(per_axis_bs[axis].as_deref().unwrap_or("")),
                    "cc" | "cp" | "cyclic"
                );
                let (knotspec, boundary, axis_period) = if periodic_axes[axis] {
                    // A `cc`/`cp`/`cyclic` per-margin basis declares periodicity
                    // without necessarily supplying a `period=`; in that case wrap
                    // at the covariate's observed [min, max] span, mirroring the
                    // 1-D cyclic fallback (`parse_periodic_domain_1d`) so a bare
                    // `te(x, z, bs=c('cc','cc'))` wraps each margin on its own
                    // range instead of hard-erroring (#1752). An axis made
                    // periodic by an explicit `periodic=`/`boundary=` selector
                    // (not a cyclic margin basis) still requires an explicit
                    // `period=`: a data-derived period there is a sample-dependent
                    // off-by-ε seam and is not inferred.
                    let (domain_start, period_value) = match periods_opt[axis] {
                        Some(period_value) => {
                            if !period_value.is_finite() || period_value <= 0.0 {
                                return Err(format!(
                                    "tensor smooth axis {axis}: period must be a positive finite value, got {period_value}"
                                ));
                            }
                            (origins_opt[axis].unwrap_or(data_min), period_value)
                        }
                        None if margin_is_cc => {
                            let span = data_max - data_min;
                            if !span.is_finite() || span <= 0.0 {
                                return Err(format!(
                                    "tensor smooth axis {axis}: cyclic margin requires a positive \
                                     observed data range to derive its period, got [{data_min}, {data_max}]"
                                ));
                            }
                            (origins_opt[axis].unwrap_or(data_min), span)
                        }
                        None => {
                            return Err(format!(
                                "tensor smooth axis {axis} is periodic but requires an explicit \
                                 period: pass period=<value> (scalar) or period=[..., <value>, ...]. \
                                 Deriving the period from the observed data range is sample-dependent \
                                 (off-by-ε seam), so it is not inferred."
                            ));
                        }
                    };
                    let domain_end = domain_start + period_value;
                    (
                        BSplineKnotSpec::PeriodicUniform {
                            data_range: (domain_start, domain_end),
                            num_basis: k_axis,
                        },
                        OneDimensionalBoundary::Cyclic {
                            start: domain_start,
                            end: domain_end,
                        },
                        Some(period_value),
                    )
                } else if margin_wants_cr(&per_axis_bs[axis])
                    && requested_knot_placement != crate::basis::BSplineKnotPlacement::Quantile
                    && k_axis >= 3
                {
                    // mgcv `te()`/`ti()` default cr margin: place exactly
                    // `k_axis` Lancaster–Salkauskas value-knots at data
                    // quantiles. The cr basis dimension equals the knot count,
                    // so this reproduces the requested per-margin `k` directly.
                    // A natural cubic regression spline needs at least 3 knots
                    // (one interior); a `k_axis < 3` margin (e.g. a binary
                    // tensor axis requesting a linear margin) falls through to
                    // the B-spline branch below, exactly as before #1074 — mgcv
                    // likewise does not build a `cr` margin below k=3. An
                    // explicit `knot_placement=quantile` also falls through:
                    // that option selects the generated B-spline knot strategy
                    // represented by `Automatic { Quantile }`, whereas the cr
                    // margin has already materialized its quantile value-knots.
                    let cr_knots = crate::basis::select_cr_knots(ds.values.column(c), k_axis)
                        .map_err(|e| e.to_string())?;
                    (
                        BSplineKnotSpec::NaturalCubicRegression { knots: cr_knots },
                        OneDimensionalBoundary::Open,
                        None,
                    )
                } else {
                    // `num_internal_knots = k - degree - 1` reproduces the
                    // requested basis size exactly when degree was reduced for
                    // a low-cardinality margin; keep the legacy `.max(1)`
                    // floor on the un-reduced path so the existing knot
                    // geometry is unchanged whenever the user already passed
                    // k >= degree + 1.
                    let num_internal_knots = if effective_degree < degree {
                        k_axis.saturating_sub(effective_degree + 1)
                    } else {
                        k_axis.saturating_sub(degree + 1).max(1)
                    };
                    let knotspec = match requested_knot_placement {
                        crate::basis::BSplineKnotPlacement::Uniform => BSplineKnotSpec::Generate {
                            data_range: (data_min, data_max),
                            num_internal_knots,
                        },
                        crate::basis::BSplineKnotPlacement::Quantile => {
                            crate::basis::auto_knot_vector_1d_quantile(
                                ds.values.column(c),
                                num_internal_knots,
                                effective_degree,
                            )
                            .map_err(|e| e.to_string())?;
                            BSplineKnotSpec::Automatic {
                                num_internal_knots: Some(num_internal_knots),
                                placement: crate::basis::BSplineKnotPlacement::Quantile,
                            }
                        }
                    };
                    (knotspec, OneDimensionalBoundary::Open, None)
                };
                // Margins contribute only their roughness operators. The tensor
                // builder constructs exactly one joint function-space null
                // penalty, avoiding unused per-margin ridge candidates and
                // duplicate λ coordinates.
                margins.push(BSplineBasisSpec {
                    degree: effective_degree,
                    penalty_order: effective_penalty_order,
                    knotspec,
                    double_penalty: false,
                    identifiability: BSplineIdentifiability::None,
                    boundary,
                    boundary_conditions: BSplineBoundaryConditions::default(),
                });
                emitted_periods.push(axis_period);
            }
            // #1593: canonicalize the margin order so a tensor smooth is invariant
            // to the typed order of its covariates. `te(x, z)` and `te(z, x)` span
            // the IDENTICAL tensor-product space under the identical per-margin
            // penalty family, but the design is the Khatri–Rao product
            // `B_first ⊙ B_second`, so the typed order permutes the design columns
            // (and the per-margin penalty blocks `S_first⊗I`, `I⊗S_second`). That
            // permutation is a pure relabelling in exact arithmetic — REML is
            // invariant to it — yet it reorders the penalized normal-equation / REML
            // eigen/Cholesky linear algebra, and the resulting sub-ULP differences
            // route the outer λ optimizer to a different terminal point in te's flat
            // REML valley (the over-smoothed margin rails to the ρ bound while the
            // other lands on a materially different λ̂). So the shipped surface
            // drifted ~2–6 % of range with a cosmetic swap of the covariate order
            // (the #1378 row-permutation / #1456 rotation flat-valley gauge family).
            // Sorting the margins by their source feature-column index makes the same
            // physical model build the identical problem regardless of typed order,
            // so the fit — and every prediction rebuilt from the resolved spec — is
            // genuinely order-invariant. `ti`/`t2` share this arm and become exactly
            // invariant too (they were already ~1e-5 by centring each margin
            // separately; canonicalization makes the swap bit-identical).
            let canon_cols: Vec<usize> = {
                let mut perm: Vec<usize> = (0..dim).collect();
                perm.sort_by_key(|&a| cols[a]);
                if perm.iter().enumerate().any(|(i, &a)| i != a) {
                    margins = perm.iter().map(|&a| margins[a].clone()).collect();
                    emitted_periods = perm.iter().map(|&a| emitted_periods[a]).collect();
                }
                perm.iter().map(|&a| cols[a]).collect()
            };
            let any_periodic = emitted_periods.iter().any(|p| p.is_some());
            let periods_vec = if any_periodic {
                emitted_periods
            } else {
                Vec::new()
            };
            // The tensor's joint polynomial null space is independently
            // shrinkable by default, so REML can recover an unsupported surface
            // as zero. Explicit `double_penalty=false` remains the MLE opt-out.
            let tensor_double_penalty = smooth_double_penalty;
            Ok(SmoothBasisSpec::TensorBSpline {
                feature_cols: canon_cols,
                spec: TensorBSplineSpec {
                    marginalspecs: margins,
                    periods: periods_vec,
                    double_penalty: tensor_double_penalty,
                    identifiability: parse_tensor_identifiability(options, kind)?,
                    // `t2` selects mgcv's separable (Wood, Scheipl & Faraway
                    // 2013) decomposition. It can arrive either as the `t2(...)`
                    // function form (`SmoothKind::T2`) or as a `type="t2"` /
                    // `bs="t2"` option on an `s(...)`/`te(...)` term, in which
                    // case `kind` is *not* `T2` but the resolved type string is
                    // "t2". Keying only off `kind` silently aliased the option
                    // form to `te`'s Kronecker-sum penalty (gam#1185); key off
                    // the resolved type string as well so both routes build the
                    // separable penalty.
                    penalty_decomposition: if matches!(kind, SmoothKind::T2)
                        || type_opt.as_str() == "t2"
                    {
                        TensorBSplinePenaltyDecomposition::Separable
                    } else {
                        TensorBSplinePenaltyDecomposition::MarginalKroneckerSum
                    },
                },
            })
        }
        "pca" => {
            validate_known_options(
                "pca",
                options,
                &[
                    "type",
                    "bs",
                    "by",
                    "k",
                    "basis_dim",
                    "basis-dim",
                    "basisdim",
                    "lazy_path",
                    "path",
                    "pca_basis_path",
                    "chunk_size",
                    "smooth_penalty",
                    "centered",
                    "double_penalty",
                    "id",
                    "__by_col",
                ],
            )?;
            let path = options
                .get("lazy_path")
                .or_else(|| options.get("pca_basis_path"))
                .or_else(|| options.get("path"))
                .map(|raw| PathBuf::from(strip_quotes(raw)));
            let Some(path) = path else {
                return Err(TermBuilderError::incompatible_config(
                    "pca smooth requires lazy_path=... on the formula path",
                )
                .to_string());
            };
            let k = option_usize_any(options, &["k", "basis_dim", "basis-dim", "basisdim"])
                .unwrap_or(0);
            let chunk_size = option_usize(options, "chunk_size").unwrap_or(DEFAULT_PCA_CHUNK_SIZE);
            Ok(SmoothBasisSpec::Pca {
                feature_cols: cols.to_vec(),
                basis_matrix: Array2::<f64>::zeros((cols.len(), k)),
                centered: option_bool(options, "centered").unwrap_or(true),
                smooth_penalty: option_f64(options, "smooth_penalty").unwrap_or(1.0),
                center_mean: None,
                pca_basis_path: Some(path),
                chunk_size,
            })
        }
        other => Err(TermBuilderError::unsupported_feature(format!(
            "unsupported smooth type '{other}'"
        ))
        .to_string()),
    }
}

/// Initialise per-axis anisotropic log-scales on eligible spatial smooth specs.
pub fn enable_scale_dimensions(spec: &mut TermCollectionSpec) {
    for smooth in spec.smooth_terms.iter_mut() {
        // A multi-axis thin-plate term cannot carry per-axis anisotropy on its
        // single curvature penalty, so `scale_dimensions` was historically a
        // silent no-op for `bs="tp"` (gam#1676). Rewrite it to the
        // mathematically-equivalent anisotropic s=0 Duchon spline first; the
        // Duchon arm below then sees an already-seeded `aniso_log_scales` and
        // leaves it untouched.
        promote_thin_plate_for_scale_dimensions(&mut smooth.basis);
        match &mut smooth.basis {
            SmoothBasisSpec::Matern {
                feature_cols,
                spec: matern,
                ..
            } => {
                if matern.aniso_log_scales.is_none() {
                    let d = feature_cols.len();
                    matern.aniso_log_scales = Some(vec![0.0; d]);
                }
            }
            SmoothBasisSpec::Duchon {
                feature_cols,
                spec: duchon,
                ..
            } => {
                if duchon.aniso_log_scales.is_none() {
                    let d = feature_cols.len();
                    duchon.aniso_log_scales = Some(vec![0.0; d]);
                }
            }
            // Bases with no per-axis length-scale vector to seed: either
            // single-axis, factor-indexed, or (tensor) already anisotropic by
            // construction through their marginals. Enumerated rather than
            // wildcarded so a new basis kind has to answer this question.
            SmoothBasisSpec::ByVariable { .. }
            | SmoothBasisSpec::FactorSumToZero { .. }
            | SmoothBasisSpec::BSpline1D { .. }
            | SmoothBasisSpec::BySmooth { .. }
            | SmoothBasisSpec::FactorSmooth { .. }
            | SmoothBasisSpec::ThinPlate { .. }
            | SmoothBasisSpec::Sphere { .. }
            | SmoothBasisSpec::ConstantCurvature { .. }
            | SmoothBasisSpec::MeasureJet { .. }
            | SmoothBasisSpec::Pca { .. }
            | SmoothBasisSpec::TensorBSpline { .. } => {}
        }
    }
}

/// Rewrite a multi-axis thin-plate term into the mathematically-equivalent
/// anisotropic s=0 Duchon spline so that `scale_dimensions` genuinely engages
/// (gam#1676).
///
/// ## Why a rewrite rather than a new field on the TPS builder
///
/// A canonical thin-plate regression spline carries a *single* curvature
/// penalty — the exact `∫|Dᵐ f|²` reproducing-kernel Gram. That penalty has no
/// per-axis structure to make one direction more or less relevant than another,
/// so per-axis anisotropy (`scale_dimensions`) cannot be expressed on it. The
/// flag was therefore a silent no-op for `bs="tp"` while it engaged for
/// `duchon()`/`matern()`.
///
/// The thin-plate kernel `r^{2m−d}` (the `r²·log r` log-case in even `d`) is
/// *exactly* the s=0 Duchon kernel (`DuchonBasisSpec::power = 0`,
/// `length_scale = None`) at the matching polynomial null-space order
/// `m = thin_plate_penalty_order(d)`. The Duchon polyharmonic family already
/// carries the per-axis tension ARD that `scale_dimensions` requests: its
/// isotropic first-order roughness penalty `Σ‖∇f‖²` splits into `d` directional
/// penalties `Σ(∂f/∂x_a)²`, each with its own REML `λ_a`
/// (`duchon_operator_penalty_candidates`). So the well-posed *anisotropic
/// thin-plate spline is the anisotropic s=0 Duchon spline*. Rewriting to that
/// representation reuses the battle-tested Duchon anisotropy / ψ-derivative /
/// freeze / predict machinery instead of duplicating it onto the TPS metadata
/// path, and keeps the polyharmonic family internally consistent. The codebase
/// already promotes infeasible-`k` TPS to Duchon for the same reason (the
/// canonical TPS single curvature penalty cannot deliver a requested
/// capability); per-axis anisotropy is another such capability.
///
/// This fires *only* when the user opts into `scale_dimensions`; the default
/// thin-plate path (`scale_dimensions` off) is left bit-for-bit unchanged.
/// A 1-D thin-plate term is left untouched — anisotropy is meaningless on a
/// single axis (its `Σ η = 0` contrast vector is empty), exactly as for a 1-D
/// Matérn/Duchon term.
fn promote_thin_plate_for_scale_dimensions(basis: &mut SmoothBasisSpec) {
    let SmoothBasisSpec::ThinPlate {
        feature_cols,
        spec,
        input_scale,
    } = &*basis
    else {
        return;
    };
    let d = feature_cols.len();
    if d <= 1 {
        return;
    }
    // m = thin_plate_penalty_order(d) is the TPS penalty order; the Duchon
    // null-space order naming is `Zero → m=1`, `Linear → m=2`,
    // `Degree(g) → m=g+1`, so the s=0 Duchon kernel exponent
    // `2(p+s) − d = 2m − d` reproduces the TPS kernel exactly.
    let m = thin_plate_penalty_order(d);
    let nullspace_order = match m {
        0 | 1 => DuchonNullspaceOrder::Zero,
        2 => DuchonNullspaceOrder::Linear,
        _ => DuchonNullspaceOrder::Degree(m - 1),
    };
    let duchon_spec = DuchonBasisSpec {
        center_strategy: spec.center_strategy.clone(),
        periodic: spec.periodic.clone(),
        // Pure, scale-free Duchon — the thin-plate kernel has no length scale
        // (a global TPS kernel scale is non-identifiable once REML learns the
        // smoothing penalty: gam#718/#721/#731/#732). The per-axis relevance
        // the user asked for is carried by the tension-ARD `λ_a`, not a κ axis.
        length_scale: None,
        // s = 0  ⇒  thin-plate kernel `r^{2m−d}`.
        power: 0.0,
        nullspace_order,
        identifiability: spec.identifiability.clone(),
        // All-zero geometry seed sentinel: `auto_seed_aniso_contrasts` resolves
        // it from the (standardized) knot cloud, and the per-axis tension split
        // engages on `aniso.is_some()`.
        aniso_log_scales: Some(vec![0.0; d]),
        operator_penalties: DuchonOperatorPenaltySpec::default(),
        boundary: OneDimensionalBoundary::Open,
        radial_reparam: None,
    };
    let feature_cols = feature_cols.clone();
    let input_scale = *input_scale;
    // All borrows of `*basis` (the `&*basis` destructure above) end with the
    // clones on the two preceding lines, so the reassignment is sound.
    *basis = SmoothBasisSpec::Duchon {
        feature_cols,
        spec: duchon_spec,
        input_scale,
    };
}

// ---------------------------------------------------------------------------
// Data-aware helpers
// ---------------------------------------------------------------------------

pub fn spatial_center_strategy_for_dimension(num_centers: usize, d: usize) -> CenterStrategy {
    if d <= 3 {
        // In low-dimensional spatial smooths, an explicit `k` is a resolution
        // request rather than a request for marginal quantile-midpoint centers.
        // Use deterministic maximin geometry so Matérn/GP and Duchon REML see a
        // well-resolved native kernel block with small fill distance instead of
        // compensating for holes or endpoint under-resolution by over-smoothing
        // low-noise signals (#504).
        CenterStrategy::FarthestPoint { num_centers }
    } else {
        default_spatial_center_strategy(num_centers, d)
    }
}

/// Center geometry for a non-periodic Duchon smooth.
///
/// In one dimension the represented domain is the interval between the observed
/// extrema.  Equally spaced centers are the exact minimax design for that
/// interval: among all `k`-point center sets they minimize the largest uncovered
/// gap.  Greedy farthest-point sampling instead produces a dyadic mesh whose
/// partially filled final level clusters centers and leaves wider holes whenever
/// `k` is not a power-of-two refinement.  Those holes reduce the effective
/// resolution of an explicit `k` and caused the low-noise k=20 Duchon fit to miss
/// the mature-smoother accuracy bar despite having the same basis dimension.
///
/// Multidimensional Duchon terms keep the rotation-equivariant farthest-point /
/// equal-mass strategies, where there is no canonical coordinate-aligned grid.
/// The `Auto` wrapper is retained for inferred 1-D counts so adaptive resolution
/// can still resize the interval grid before freezing its realized centers.
fn duchon_center_strategy(num_centers: usize, d: usize, automatic: bool) -> CenterStrategy {
    let realized = if d == 1 {
        CenterStrategy::UniformGrid {
            points_per_dim: num_centers,
        }
    } else {
        spatial_center_strategy_for_dimension(num_centers, d)
    };
    if automatic {
        CenterStrategy::Auto(Box::new(realized))
    } else {
        realized
    }
}

pub fn col_minmax(col: ArrayView1<'_, f64>) -> Result<(f64, f64), String> {
    let min = col.iter().fold(f64::INFINITY, |a, &b| a.min(b));
    let max = col.iter().fold(f64::NEG_INFINITY, |a, &b| a.max(b));
    if !min.is_finite() || !max.is_finite() {
        return Err(TermBuilderError::degenerate_data(
            "non-finite data encountered while inferring knot range",
        )
        .to_string());
    }
    if (max - min).abs() < 1e-12 {
        Ok((min, min + 1e-6))
    } else {
        Ok((min, max))
    }
}

pub fn unique_count_column(col: ArrayView1<'_, f64>) -> usize {
    use std::collections::HashSet;
    let mut set = HashSet::<u64>::with_capacity(col.len());
    for &v in col {
        set.insert(gam_data::canonical_level_bits(v));
    }
    set.len().max(1)
}

/// Minimum knot count for a natural cubic regression spline: `select_cr_knots`
/// places one value-knot per basis function and needs at least an interior knot,
/// so the sparsest representable cr basis is `{const, linear, curvature}` at
/// three knots. Below this a cr spline is not constructible and the caller must
/// degrade to the linear B-spline marginal.
pub(crate) const CR_MIN_KNOTS: usize = 3;

/// Build a cubic-regression marginal knot spec capped to the covariate's data
/// support, mgcv-style.
///
/// A `cr`/`cs`/`sz` marginal places exactly one basis function per value-knot,
/// so `select_cr_knots` cannot place more knots than the covariate has DISTINCT
/// values — it `bail`s with "cubic regression spline with k=N requires at least
/// N distinct values" otherwise. An unclamped `k` on an ordinary low-cardinality
/// covariate (a binary indicator, a 3-level ordinal/Likert score, a small count)
/// therefore hard-failed the whole fit instead of reducing the basis the way
/// mgcv — and gam's own tensor-margin path (996f829d7, `term_builder.rs:2986` /
/// the `k_axis >= 3` cr gate at `:3047`) — do. This is the univariate / factor-
/// smooth sibling of that tensor cap (#1541, #1542).
///
/// Returns:
/// - `Some(NaturalCubicRegression { .. })` with `k = min(k_requested, n_distinct)`
///   value-knots when the data supports a cr spline (`n_distinct >= CR_MIN_KNOTS`).
///   A cr basis of exactly `n_distinct` knots is full-rank for the data — it can
///   represent any per-distinct-value structure (e.g. 3 arbitrary group means on
///   a ternary covariate) — so the cap never costs recoverable signal.
/// - `None` when `n_distinct < CR_MIN_KNOTS` (a binary covariate): too few
///   distinct values for ANY cr spline, so the caller degrades to the linear
///   B-spline marginal — exactly what the default `s(x, k=..)` basis already
///   builds on the same data, and what the tensor path's `< 3` branch builds.
///
/// `inference_notes` records any reduction so the user sees that `k` was capped
/// (mgcv emits a warning in the same situation).
fn capped_cr_marginal_knotspec(
    col: ArrayView1<'_, f64>,
    k_cr_requested: usize,
    label: &str,
    inference_notes: &mut Vec<String>,
) -> Result<Option<BSplineKnotSpec>, String> {
    let n_distinct = unique_count_column(col);
    let k_cr = k_cr_requested.min(n_distinct);
    if k_cr < CR_MIN_KNOTS {
        inference_notes.push(format!(
            "Smooth '{label}': cubic-regression ('cr'/'cs'/'sz') basis requested k={k_cr_requested}, \
             but the covariate has only {n_distinct} distinct value(s) — too few to support a cubic \
             regression spline (needs >= {CR_MIN_KNOTS} distinct values). Degraded to the linear \
             B-spline marginal the default basis builds on the same data."
        ));
        return Ok(None);
    }
    if k_cr < k_cr_requested {
        inference_notes.push(format!(
            "Smooth '{label}': cubic-regression ('cr'/'cs'/'sz') basis reduced from k={k_cr_requested} \
             to k={k_cr} to match the covariate's {n_distinct} distinct value(s) (mgcv-style \
             data-support cap; a cr basis cannot place more value-knots than the data has)."
        ));
    }
    let cr_knots = crate::basis::select_cr_knots(col, k_cr).map_err(|e| e.to_string())?;
    Ok(Some(BSplineKnotSpec::NaturalCubicRegression {
        knots: cr_knots,
    }))
}

/// Smallest number of distinct covariate values seen within any single group
/// of `group_col`. For a factor smooth this is the resolution that bounds the
/// marginal basis: a group with `m` distinct covariate values can only inform
/// `m` basis coefficients, so a marginal richer than that interpolates the
/// group instead of estimating a penalized trend. Bits are compared exactly so
/// integer-valued covariates (days, dose levels) collapse to their true count.
fn min_per_group_unique_count(
    feature_col: ArrayView1<'_, f64>,
    group_col: ArrayView1<'_, f64>,
) -> usize {
    use std::collections::{HashMap, HashSet};
    let mut per_group: HashMap<u64, HashSet<u64>> = HashMap::new();
    for (xi, gi) in feature_col.iter().zip(group_col.iter()) {
        per_group
            .entry(gam_data::canonical_level_bits(*gi))
            .or_default()
            .insert(gam_data::canonical_level_bits(*xi));
    }
    per_group
        .values()
        .map(|s| s.len())
        .min()
        .unwrap_or(1)
        .max(1)
}

/// Default internal-knot count for an *additive* univariate smooth, derived
/// from the column's unique-value count.
///
/// The basis dimension is `internal_knots + degree + 1`, so the cap below maps
/// to a default cubic basis of ~12 functions — deliberately close to mgcv's
/// univariate default (`k = 10`). A penalized smooth controls its wiggliness
/// through the *penalty*, not the basis size: REML/LAML shrinks a too-rich
/// basis toward the null, but it cannot do so cleanly when the basis is so
/// over-sized that the design becomes weakly identified. Growing the basis with
/// `n` (the old `n^(1/3)`-ceilinged `unique/4` rule, which pinned to 20 internal
/// knots ⇒ a 24-function basis for any column with ≥80 unique values) therefore
/// *hurts* recovery on finite, weak-signal fits: a 4-smooth additive model on
/// n=120 asks for ~92 coefficients, the outer optimizer stalls on the resulting
/// flat two-penalty (range + null-space) REML surface, and the truth leaks into
/// surplus columns the penalty can't shrink away (gam#1680; the same defect was
/// documented for thin-plate fields in gam#1074). A k-sweep on the #1680 design
/// confirms a basis of ~10–15 recovers truth at RMSE ≈ 0.12 while the old
/// 24-function default lands at ≈ 0.39 (~3× worse) — *whether or not* the
/// covariates are collinear, so this is basis over-richness, not collinearity.
///
/// The cap is flat in `n`: a user who genuinely needs a wigglier fit raises `k`
/// explicitly (mgcv's contract — opt *in* to more flexibility), and the SPEC
/// requires the default to allow recovering the null rather than forcing the
/// user to opt out of overfitting. The 4-knot floor stays put because we still
/// need enough basis functions to fit a non-trivial smooth at all, and the
/// `unique/4` growth below the cap keeps small/sparse columns (n ≤ 32, where
/// `unique/4 ≤ 8`) on exactly their previous knot count.
pub fn heuristic_knots_for_column(col: ArrayView1<'_, f64>) -> usize {
    /// Default cubic basis ≈ `MAX_DEFAULT_INTERNAL_KNOTS + degree + 1` = 12
    /// functions, matching mgcv's lean univariate default.
    const MAX_DEFAULT_INTERNAL_KNOTS: usize = 8;
    let unique = unique_count_column(col);
    (unique / 4).clamp(4, MAX_DEFAULT_INTERNAL_KNOTS)
}

/// Per-margin basis sizes for a tensor-product smooth (`te`/`ti`/`t2`).
///
/// The 1-D heuristic [`heuristic_knots_for_column`] is calibrated for an
/// *additive* margin: a well-resolved column asks for the lean univariate
/// default (≈12 basis functions, the mgcv-like cap of 8 internal knots; see
/// gam#1680), which is sensible for a single `s(x)` term.
/// A tensor product, however, multiplies the per-margin sizes:
/// `p = ∏_d k_d`. Reusing the 1-D rule per margin makes `p` explode with the
/// tensor dimension — a 3-D `te(x,y,z)` at the 1-D ceiling of 12/margin is
/// `12³ ≈ 1728` columns, and every REML evaluation pays an O(p³) dense
/// penalty reparameterization (the full-tensor sum-to-zero constraint is not
/// Kronecker-factorable), turning model selection over tensor candidates into
/// a multi-minute single-threaded stall (gam#813). It also requests far more
/// coefficients than the data can identify whenever `p ≫ n`.
///
/// mgcv's `te(...)` uses a small per-margin default (`k = 5`, i.e. `5^d`).
/// We match that spirit while staying data-adaptive: budget the *total* tensor
/// column count `p_target` and distribute it geometrically across the margins
/// so `∏ k_d ≈ p_target`, never asking a margin for more functions than its
/// own unique values (and the data set) can support.
fn heuristic_tensor_margin_knots(cols: &[usize], ds: &Dataset) -> Vec<usize> {
    let d = cols.len().max(1);
    let degree = DEFAULT_BSPLINE_DEGREE;
    let min_k = degree + 2; // smallest margin that carries a difference penalty
    let n = ds.values.nrows();

    // Per-margin 1-D ceiling: never request more basis functions than the
    // margin's own resolution (unique values) supports. This caps each axis
    // independently before the joint budget is applied.
    let per_margin_cap: Vec<usize> = cols
        .iter()
        .map(|&c| heuristic_knots_for_column(ds.values.column(c)).max(min_k))
        .collect();

    // Total-basis budget. A tensor with ∏k ≫ n coefficients is rank-deficient
    // and pure REML cost; cap the product at a generous fraction of n while
    // honoring mgcv's small default for the common small-d case. The budget
    // grows with n but the geometric split below keeps each margin modest.
    //   d=2 → up to ~7²=49 (mgcv-`te`-like), d=3 → ~5³=125, larger d shrinks
    // per-margin further so the product never blows past the data support.
    let mgcv_like_per_margin = match d {
        2 => 7usize,
        3 => 5usize,
        _ => 4usize,
    };
    let mgcv_like_total = (mgcv_like_per_margin as f64).powi(d as i32);
    let data_budget = (n as f64) * 0.8;
    let p_target = mgcv_like_total
        .max(min_k.pow(d as u32) as f64)
        .min(data_budget);

    // Geometric per-margin target so ∏k ≈ p_target, then clamp each margin to
    // its own 1-D resolution cap and the difference-penalty floor.
    let geo_per_margin = p_target.powf(1.0 / d as f64).round() as usize;
    let unclamped: Vec<usize> = per_margin_cap
        .iter()
        .map(|&cap| geo_per_margin.clamp(min_k, cap))
        .collect();

    // The per-margin clamps can pull some axes below `geo_per_margin` (a
    // low-resolution column), leaving headroom in the joint budget. Redistribute
    // that headroom to the margins that can still grow, so the realized ∏k stays
    // close to p_target instead of systematically under-shooting it.
    let mut k_list = unclamped;
    loop {
        let product: f64 = k_list.iter().map(|&k| k as f64).product();
        if product >= p_target {
            break;
        }
        // Grow the axis with the most remaining headroom (cap − current),
        // breaking ties toward the largest cap. Stop when none can grow.
        let Some(idx) = k_list
            .iter()
            .zip(per_margin_cap.iter())
            .enumerate()
            .filter(|&(_, (k, cap))| k < cap)
            .max_by_key(|&(_, (k, cap))| (cap - k, *cap))
            .map(|(i, _)| i)
        else {
            break;
        };
        k_list[idx] += 1;
    }
    k_list
}

pub fn heuristic_centers(n: usize, d: usize) -> usize {
    default_num_centers(n, d)
}

// ---------------------------------------------------------------------------
// Smooth option parsers
// ---------------------------------------------------------------------------

fn parse_endpoint_side(
    value: &str,
    context: &str,
) -> Result<BSplineEndpointBoundaryCondition, String> {
    match value.trim().to_ascii_lowercase().as_str() {
        "" | "none" | "open" | "unconstrained" | "free" => {
            Ok(BSplineEndpointBoundaryCondition::Free)
        }
        "clamped" | "clamp" | "zero_derivative" | "zero-derivative" => {
            Ok(BSplineEndpointBoundaryCondition::Clamped)
        }
        "anchored" | "anchor" | "zero" | "zero_value" | "zero-value" => {
            Ok(BSplineEndpointBoundaryCondition::Anchored { value: 0.0 })
        }
        other => Err(format!(
            "unsupported {context} boundary condition '{other}'; expected free, clamped, or anchored"
        )),
    }
}

fn boundary_anchor_value(
    options: &BTreeMap<String, String>,
    side: &str,
    fallback: Option<f64>,
) -> Option<f64> {
    [
        format!("anchor_{side}"),
        format!("{side}_anchor"),
        format!("anchor-value-{side}"),
    ]
    .iter()
    .find_map(|key| option_f64(options, key))
    .or(fallback)
}

fn apply_anchor_value(
    cond: BSplineEndpointBoundaryCondition,
    value: Option<f64>,
) -> BSplineEndpointBoundaryCondition {
    match cond {
        BSplineEndpointBoundaryCondition::Anchored { .. } => {
            BSplineEndpointBoundaryCondition::Anchored {
                value: value.unwrap_or(0.0),
            }
        }
        other => other,
    }
}

fn parse_bspline_boundary_conditions(
    options: &BTreeMap<String, String>,
) -> Result<BSplineBoundaryConditions, String> {
    let fallback_anchor = option_f64(options, "anchor")
        .or_else(|| option_f64(options, "anchor_value"))
        .or_else(|| option_f64(options, "value"));
    let global_boundary_conditions = options
        .get("boundary_conditions")
        .or_else(|| options.get("bc"));
    let mut boundary_conditions = BSplineBoundaryConditions::default();

    if let Some(raw_boundary_conditions) = global_boundary_conditions {
        let cond = parse_endpoint_side(raw_boundary_conditions, "boundary_conditions")?;
        let side = options
            .get("side")
            .map(|s| s.trim().to_ascii_lowercase())
            .unwrap_or_else(|| "both".to_string());
        match side.as_str() {
            "both" | "all" | "endpoints" => {
                boundary_conditions.left = cond;
                boundary_conditions.right = cond;
            }
            "left" | "start" | "lower" => boundary_conditions.left = cond,
            "right" | "end" | "upper" => boundary_conditions.right = cond,
            other => {
                return Err(format!(
                    "unsupported B-spline boundary side '{other}'; expected left, right, or both"
                ));
            }
        }
    }

    if let Some(raw) = options
        .get("bc_left")
        .or_else(|| options.get("left_bc"))
        .or_else(|| options.get("bc_start"))
        .or_else(|| options.get("start_bc"))
    {
        boundary_conditions.left = parse_endpoint_side(raw, "left endpoint")?;
    }
    if let Some(raw) = options
        .get("bc_right")
        .or_else(|| options.get("right_bc"))
        .or_else(|| options.get("bc_end"))
        .or_else(|| options.get("end_bc"))
    {
        boundary_conditions.right = parse_endpoint_side(raw, "right endpoint")?;
    }

    boundary_conditions.left = apply_anchor_value(
        boundary_conditions.left,
        boundary_anchor_value(options, "left", fallback_anchor),
    );
    boundary_conditions.right = apply_anchor_value(
        boundary_conditions.right,
        boundary_anchor_value(options, "right", fallback_anchor),
    );

    Ok(boundary_conditions)
}

/// Resolve the requested internal-knot count and effective spline degree for
/// a 1-D penalized B-spline smooth. This mirrors the tensor-margin per-axis
/// degree-reduction policy: a 1-D B-spline basis with `k` functions
/// is well-defined for any `degree <= k - 1`, so an explicit
/// `s(x, bs="ps", k=3)` with default `degree=3` is interpreted as the
/// largest representable spline (`effective_degree = k - 1 = 2`, quadratic)
/// rather than rejected. The `penalty_order` carried by the caller must be
/// clamped to `<= effective_degree` so the marginal difference penalty
/// stays well-defined; the returned `effective_degree` makes that explicit.
///
/// Mirrors the tensor margin treatment in the `te(...)` builder so a
/// standalone smooth, a factor smooth, and a tensor margin all interpret
/// "small k" the same way.
fn parse_ps_internal_knots(
    options: &BTreeMap<String, String>,
    degree: usize,
    default_internal_knots: usize,
) -> Result<(usize, bool, usize), String> {
    const MIN_EXPRESSIVE_INTERNAL_KNOTS: usize = 2;
    // Strict variants: reject `k=-1`, `k=1.5`, `knots=-2` etc. with a
    // focused error instead of silently dropping the value and using the
    // default. Lenient `option_usize` / `option_usize_any` silently swallow
    // unparseable values, which leaves the user thinking they configured
    // something when they did not.
    // A list-valued `knots=[...]` carries explicit internal positions, not a
    // count; it is consumed by `parse_explicit_internal_knots`. Treat it as
    // "count not specified" here so the strict integer parse does not reject
    // the bracketed value (the Provided path ignores the returned count).
    let knots_internal = if knots_option_is_list(options) {
        None
    } else {
        option_usize_strict(options, "knots")?
    };
    let basis_dim = option_usize_any_strict(options, &["k", "basis_dim", "basis-dim", "basisdim"])?;
    if knots_internal.is_some() && basis_dim.is_some() {
        return Err(TermBuilderError::incompatible_config(
            "ps/bspline smooth: specify either knots=<internal_knots> or k=<basis_dim> (not both)",
        )
        .to_string());
    }
    if let Some(k) = basis_dim {
        if k < 2 {
            return Err(TermBuilderError::invalid_option(format!(
                "ps/bspline smooth: k={} too small; B-spline basis requires k >= 2",
                k
            ))
            .to_string());
        }
        // `degree <= k - 1` is required for the B-spline basis to be
        // well-defined; reduce on this axis only when the user asked for
        // a smaller k than the cubic default supports. This matches mgcv's
        // behaviour (e.g. `s(x, bs="ps", k=3)` becomes a quadratic basis)
        // and the per-axis reduction the tensor builder already does.
        let effective_degree = degree.min(k - 1).max(1);
        let num_internal_knots = if effective_degree < degree {
            // Reproduce the requested basis size exactly when degree was
            // reduced for a low-cardinality axis: num_basis = k.
            k.saturating_sub(effective_degree + 1)
        } else {
            (k - degree - 1).max(MIN_EXPRESSIVE_INTERNAL_KNOTS)
        };
        Ok((num_internal_knots, false, effective_degree))
    } else {
        Ok((
            knots_internal.unwrap_or(default_internal_knots),
            knots_internal.is_none(),
            degree,
        ))
    }
}

/// True when the `knots` option value is a *list* literal (`[...]`, `c(...)`,
/// or `(...)`) rather than a scalar count. mgcv's `knots=` accepts both: a
/// single integer is an internal-knot count, while a vector is explicit
/// internal knot positions. We disambiguate purely on the wrapper syntax so a
/// bare `knots=5` keeps its historical count meaning.
fn knots_option_is_list(options: &BTreeMap<String, String>) -> bool {
    options
        .get("knots")
        .map(|raw| {
            let t = raw.trim();
            t.starts_with('[') || t.starts_with("c(") || t.starts_with("C(") || t.starts_with('(')
        })
        .unwrap_or(false)
}

/// Parse `knots=[k0, k1, ...]` (or `c(...)` / `(...)`) into explicit internal
/// knot positions. Returns `Ok(None)` when `knots` is absent or a scalar count
/// (handled by [`parse_ps_internal_knots`]); `Ok(Some(positions))` when it is a
/// non-empty numeric list; and an error for an empty or unparseable list.
fn parse_explicit_internal_knots(
    options: &BTreeMap<String, String>,
) -> Result<Option<Vec<f64>>, String> {
    if !knots_option_is_list(options) {
        return Ok(None);
    }
    let raw = options
        .get("knots")
        .expect("knots_option_is_list implies the key is present");
    let tokens = split_list_option(raw);
    if tokens.is_empty() {
        return Err(TermBuilderError::invalid_option(format!(
            "knots={raw} is an empty list; supply at least one internal knot position \
             (e.g. knots=[0.2, 0.5, 0.8]) or a scalar count (e.g. knots=8)"
        ))
        .to_string());
    }
    let mut positions = Vec::with_capacity(tokens.len());
    for tok in &tokens {
        let value = parse_numeric_expr(tok).map_err(|err| {
            TermBuilderError::invalid_option(format!(
                "knots list entry '{tok}' is not a numeric position: {err}"
            ))
            .to_string()
        })?;
        positions.push(value);
    }
    Ok(Some(positions))
}

/// Resolve the `knot_placement=` option for an automatically generated knot
/// vector. Accepts `"uniform"` (the default, equal spacing on the data range)
/// and `"quantile"` (interior knots at empirical data quantiles, better for
/// skewed covariates). Unknown values are rejected so typos do not silently
/// fall back to uniform.
fn parse_knot_placement(
    options: &BTreeMap<String, String>,
) -> Result<crate::basis::BSplineKnotPlacement, String> {
    use crate::basis::BSplineKnotPlacement;
    match options
        .get("knot_placement")
        .or_else(|| options.get("knot-placement"))
        .or_else(|| options.get("knotplacement"))
    {
        None => Ok(BSplineKnotPlacement::Uniform),
        Some(raw) => match raw
            .trim()
            .trim_matches('"')
            .trim_matches('\'')
            .to_ascii_lowercase()
            .as_str()
        {
            "uniform" | "even" | "equal" => Ok(BSplineKnotPlacement::Uniform),
            "quantile" | "quantiles" | "data" | "empirical" => Ok(BSplineKnotPlacement::Quantile),
            other => Err(TermBuilderError::invalid_option(format!(
                "knot_placement={other} is not recognised; expected \"uniform\" or \"quantile\""
            ))
            .to_string()),
        },
    }
}

/// Build the non-periodic 1D B-spline knot spec for the `ps`/`bspline` and
/// factor-smooth marginal paths, honoring (in priority order):
///   1. `knots=[...]` explicit internal positions  → [`BSplineKnotSpec::Provided`]
///   2. `knot_placement="quantile"`                 → [`BSplineKnotSpec::Automatic`]
///   3. uniform generation                          → [`BSplineKnotSpec::Generate`]
///
/// `data` is the covariate column (used to clamp explicit positions to the
/// observed range and to drive quantile placement); `n_knots` is the resolved
/// internal-knot count from [`parse_ps_internal_knots`] used for the automatic
/// strategies.
fn resolve_nonperiodic_bspline_knotspec(
    options: &BTreeMap<String, String>,
    data: ArrayView1<'_, f64>,
    data_range: (f64, f64),
    degree: usize,
    n_knots: usize,
) -> Result<BSplineKnotSpec, String> {
    use crate::basis::{BSplineKnotPlacement, clamped_knot_vector_from_internal_positions};
    if let Some(positions) = parse_explicit_internal_knots(options)? {
        if option_usize_any_strict(options, &["k", "basis_dim", "basis-dim", "basisdim"])?.is_some()
        {
            return Err(TermBuilderError::incompatible_config(
                "ps/bspline smooth: specify either explicit knots=[...] positions or \
                 k=<basis_dim> (not both); the basis size is fixed by the knot vector",
            )
            .to_string());
        }
        let knots = clamped_knot_vector_from_internal_positions(data_range, &positions, degree)
            .map_err(|e| e.to_string())?;
        return Ok(BSplineKnotSpec::Provided(knots));
    }
    match parse_knot_placement(options)? {
        BSplineKnotPlacement::Uniform => Ok(BSplineKnotSpec::Generate {
            data_range,
            num_internal_knots: n_knots,
        }),
        BSplineKnotPlacement::Quantile => {
            // Validate the column up-front so an unfittable request surfaces a
            // user-correctable error at parse time rather than deep in basis
            // construction. The same data drives the eventual quantile knots.
            crate::basis::auto_knot_vector_1d_quantile(data, n_knots, degree)
                .map_err(|e| e.to_string())?;
            Ok(BSplineKnotSpec::Automatic {
                num_internal_knots: Some(n_knots),
                placement: BSplineKnotPlacement::Quantile,
            })
        }
    }
}

/// Reject unknown option keys with a focused error that names the term and
/// the offending key, plus suggests near-matches from the known-key list.
/// Without this, typos like `lengt_scale=0.1` or `nyu=5/2` are silently
/// dropped, the term uses the default, and the user has no idea why their
/// option had no effect.
pub fn validate_known_options(
    term_name: &str,
    options: &BTreeMap<String, String>,
    known: &[&str],
) -> Result<(), String> {
    let known_set: std::collections::BTreeSet<&&str> = known.iter().collect();
    for key in options.keys() {
        if !known_set.contains(&key.as_str()) {
            if term_name == "tensor" && is_tensor_k_axis_option_key(key) {
                continue;
            }
            // Suggest near-matches (substring or shared prefix ≥ 3).
            let key_l = key.to_ascii_lowercase();
            let mut suggestions: Vec<&str> = known
                .iter()
                .filter(|k| {
                    let kl = k.to_ascii_lowercase();
                    kl.contains(&key_l) || key_l.contains(&kl) || {
                        let n = kl
                            .chars()
                            .zip(key_l.chars())
                            .take_while(|(a, b)| a == b)
                            .count();
                        n >= 3
                    }
                })
                .copied()
                .collect();
            suggestions.sort_unstable();
            suggestions.dedup();
            let hint = if suggestions.is_empty() {
                String::new()
            } else {
                format!(" — did you mean one of [{}]?", suggestions.join(", "))
            };
            return Err(TermBuilderError::invalid_option(format!(
                "{term_name}() does not accept option `{key}`{hint}. Valid options: [{}]",
                {
                    let mut sorted = known.to_vec();
                    sorted.sort_unstable();
                    sorted.join(", ")
                }
            ))
            .to_string());
        }
    }
    Ok(())
}

/// Private (engine-injected) option that caps the *default* spatial center
/// count for a secondary (distributional) predictor's smooth — see
/// `solver::fit_orchestration::apply_secondary_predictor_basis_parsimony` and #501.
///
/// It is deliberately NOT one of the user-facing count aliases recognised by
/// [`has_explicit_countwith_basis_alias`], so it never flips the spatial basis
/// onto the explicit (hard) center-placement strategy: the cap lowers the
/// *default* count while the `Auto` strategy is retained, so the count is still
/// softly reduced when the data can't support it.
pub const SECONDARY_CENTER_CAP_OPTION: &str = "__secondary_center_cap";

/// Apply the secondary-predictor center cap to a *default* spatial center
/// count. A no-op when the cap option is absent (the common case) or when the
/// user supplied an explicit count (then `default_count` is ignored downstream
/// by [`parse_countwith_basis_alias`] anyway).
pub(crate) fn cap_default_spatial_centers(
    options: &BTreeMap<String, String>,
    default_count: usize,
) -> usize {
    match option_usize(options, SECONDARY_CENTER_CAP_OPTION) {
        Some(cap) => default_count.min(cap),
        None => default_count,
    }
}

fn default_matern_center_count(
    n: usize,
    d: usize,
    planned_count: usize,
    univariate_floor: usize,
) -> usize {
    // #1074: the mgcv-sized basis cap (`k = 10·3^(d-1)`) was DELETED here too — it
    // masked the same over-sizing/under-penalization defect by shrinking the basis
    // rather than fixing the optimizer. The default now uses the generic n-scaling
    // plan. A small-n floor against a numerically-fragile two-column kernel block
    // is a legitimate degenerate guard and is kept. Explicit `k`/`centers` still
    // take full effect upstream.
    let low_n_floor = (d + 4).min(n);
    // #1867: at small n the generic conditioning cap (`n / COND_N_DIVISOR`) in
    // `default_num_centers` starves a 1-D radial basis BELOW the resolution the
    // univariate B-spline `s(x)` is handed on the SAME data (e.g. 7 vs 11 basis
    // functions at n=30), so `matern(x)`/`duchon(x)` over-smooth sparse
    // oscillations that `s(x)` recovers cleanly. Smoothness is set by the REML
    // penalty λ, not by the raw center count (see `default_num_centers`), so a
    // radial smooth competing with `s(x)` must not be dimensioned coarser than
    // it. `univariate_floor` carries that spline-equivalent resolution for a 1-D
    // smooth (0 for d>1, where there is no direct univariate analogue) and is
    // bounded by n. Explicit `k`/`centers` still override upstream.
    planned_count
        .max(low_n_floor)
        .max(univariate_floor.min(n))
        .max(1)
}

fn default_duchon_center_count(
    n: usize,
    d: usize,
    planned_count: usize,
    polynomial_cols: usize,
    univariate_floor: usize,
) -> usize {
    // #1757: Duchon fits pay a larger setup cost than Matérn/TPS because the
    // constrained radial block is rotated through its center Gram and several
    // operator-collocation penalties.  The old generic spatial default handed a
    // 2-D Gaussian Duchon at n≈500 more than one hundred centers, so cold fits
    // spent most of their time in dense O(k³) eigensolves even though the REML
    // smoother uses a low-rank basis.  mgcv's Duchon spline default is the
    // thin-plate-style `k = 10 * 3^(d - 1)` (30 in 2-D); use that as the
    // implicit low-rank cap while preserving the user's explicit `centers=`/`k=`
    // request above.  The polynomial null space must still fit, so tiny
    // high-order bases are raised to the smallest admissible count.
    let mgcv_default = 10usize.saturating_mul(3usize.saturating_pow(d.saturating_sub(1) as u32));
    let low_n_floor = (polynomial_cols + 1).min(n).max(1);
    // #1867: at small n the generic conditioning cap (`n / COND_N_DIVISOR`) in
    // `default_num_centers` starves `planned_count` below the univariate spline
    // resolution the competing `s(x)` gets on the SAME data, so `duchon(x)`
    // over-smooths sparse oscillations. `univariate_floor` (0 for d>1) carries
    // that spline-equivalent basis dimension and floors the 1-D default,
    // bounded by n; smoothness is set by the REML penalty, not the raw count.
    // Explicit `k`/`centers` still override upstream.
    planned_count
        .min(mgcv_default)
        .max(low_n_floor)
        .max(univariate_floor.min(n))
}

pub fn parse_countwith_basis_alias(
    options: &BTreeMap<String, String>,
    primarykey: &str,
    default_count: usize,
) -> Result<usize, String> {
    // Strict: reject unparseable values (e.g. `centers=many`, `centers=-1`,
    // `centers=1.5`) instead of silently dropping them and falling through
    // to the default. Without this the user gets the auto-inferred count
    // silently and never realizes their explicit option was ignored.
    let primary = option_usize_strict(options, primarykey)?;
    let basis_dim = option_usize_any_strict(
        options,
        &["k", "basis_dim", "basis-dim", "basisdim", "knots"],
    )?;
    if primary.is_some() && basis_dim.is_some() {
        return Err(TermBuilderError::incompatible_config(format!(
            "specify either {}=<count> or k=<basis_dim> (not both)",
            primarykey
        ))
        .to_string());
    }
    Ok(primary.or(basis_dim).unwrap_or(default_count))
}

pub fn has_explicit_countwith_basis_alias(
    options: &BTreeMap<String, String>,
    primarykey: &str,
) -> bool {
    options.contains_key(primarykey)
        || ["k", "basis_dim", "basis-dim", "basisdim", "knots"]
            .iter()
            .any(|alias| options.contains_key(*alias))
}

pub fn parse_cyclic_boundary(
    options: &BTreeMap<String, String>,
    minv: f64,
    maxv: f64,
) -> Result<OneDimensionalBoundary, String> {
    let cyclic = option_bool(options, "cyclic")
        .or_else(|| option_bool(options, "periodic"))
        .unwrap_or(false);
    if !cyclic {
        return Ok(OneDimensionalBoundary::Open);
    }
    let start = match option_numeric_expr(options, "period_start")? {
        Some(v) => v,
        None => option_numeric_expr(options, "start")?.unwrap_or(minv),
    };
    let end = match option_numeric_expr(options, "period_end")? {
        Some(v) => v,
        None => option_numeric_expr(options, "end")?.unwrap_or(maxv),
    };
    if end <= start {
        return Err(format!(
            "cyclic smooth requires period_end/end ({end}) > period_start/start ({start})"
        ));
    }
    Ok(OneDimensionalBoundary::Cyclic { start, end })
}

/// Parse the periodic-uniform domain for a one-dimensional cyclic smooth.
///
/// Returns the `(domain_start, period)` pair derived from
/// `period_start` / `start`, `period_end` / `end`, falling back to the
/// data range `[minv, maxv)` when neither bound is provided. The period
/// must be strictly positive.
pub fn parse_periodic_domain_1d(
    options: &BTreeMap<String, String>,
    minv: f64,
    maxv: f64,
) -> Result<(f64, f64), String> {
    let start_opt = match option_numeric_expr(options, "period_start")? {
        Some(v) => Some(v),
        None => option_numeric_expr(options, "start")?,
    };
    let end_opt = match option_numeric_expr(options, "period_end")? {
        Some(v) => Some(v),
        None => option_numeric_expr(options, "end")?,
    };
    // Reject the pure data-range fallback. A B-spline periodic smooth that takes
    // its wrap from the observed [min, max] is sample-dependent and silently
    // wrong: uniform draws on a true period of 2π land on [ε, 2π−ε], so using
    // (max−min) as the period seams the curve with an off-by-ε discontinuity and
    // the fit drifts with the sample. (Unlike the radial closed-lattice Duchon
    // path, whose centers DO tile a full period, so its span-derive is exact —
    // see `parse_periodic_axes_option`.) Require the caller to name the period
    // explicitly via `period=`/`period_end`. The end is only defaulted to `maxv`
    // when a `period_start`/`start` was given (a half-open declaration); a bare
    // periodic smooth with neither bound is an error.
    if end_opt.is_none() && start_opt.is_none() {
        return Err(
            "periodic B-spline smooth requires an explicit period: pass period=<value> \
             (e.g. period=2*pi) or period_start=/period_end=. Deriving the period from the \
             observed data range is sample-dependent and produces an off-by-ε seam, so it is \
             not inferred."
                .to_string(),
        );
    }
    let start = start_opt.unwrap_or(minv);
    let end = end_opt.unwrap_or(maxv);
    if !(start.is_finite() && end.is_finite()) {
        return Err(format!(
            "periodic smooth domain requires finite endpoints, got ({start}, {end})"
        ));
    }
    if end <= start {
        return Err(format!(
            "periodic smooth requires period_end/end ({end}) > period_start/start ({start})"
        ));
    }
    Ok((start, end - start))
}

fn parse_matern_nu(raw: &str) -> Result<MaternNu, String> {
    let trimmed = raw.trim();
    let lowered = trimmed.to_ascii_lowercase();
    // Exact spellings of the half-integer smoothnesses that have closed-form
    // kernels; anything else falls through to the numeric parse below.
    let named = match lowered.as_str() {
        "1/2" | "0.5" | "half" => Some(MaternNu::Half),
        "3/2" | "1.5" => Some(MaternNu::ThreeHalves),
        "5/2" | "2.5" => Some(MaternNu::FiveHalves),
        "7/2" | "3.5" => Some(MaternNu::SevenHalves),
        "9/2" | "4.5" => Some(MaternNu::NineHalves),
        _ => None,
    };
    if let Some(nu) = named {
        return Ok(nu);
    }

    let value = if let Some((num, den)) = trimmed.split_once('/') {
        let num = num
            .trim()
            .parse::<f64>()
            .map_err(|err| format!("{}: {err}", unsupported_matern_nu_message(raw)))?;
        let den = den
            .trim()
            .parse::<f64>()
            .map_err(|err| format!("{}: {err}", unsupported_matern_nu_message(raw)))?;
        if den == 0.0 || !num.is_finite() || !den.is_finite() {
            return Err(unsupported_matern_nu_message(raw));
        }
        num / den
    } else {
        trimmed
            .parse::<f64>()
            .map_err(|err| format!("{}: {err}", unsupported_matern_nu_message(raw)))?
    };

    const TOL: f64 = 1e-12;
    if (value - 0.5).abs() <= TOL {
        Ok(MaternNu::Half)
    } else if (value - 1.5).abs() <= TOL {
        Ok(MaternNu::ThreeHalves)
    } else if (value - 2.5).abs() <= TOL {
        Ok(MaternNu::FiveHalves)
    } else if (value - 3.5).abs() <= TOL {
        Ok(MaternNu::SevenHalves)
    } else if (value - 4.5).abs() <= TOL {
        Ok(MaternNu::NineHalves)
    } else {
        Err(unsupported_matern_nu_message(raw))
    }
}

fn unsupported_matern_nu_message(raw: &str) -> String {
    TermBuilderError::unsupported_feature(format!(
        "unsupported Matern nu '{raw}'; supported half-integer values are 1/2, 3/2, 5/2, 7/2, and 9/2"
    ))
    .to_string()
}

#[derive(Clone, Debug, serde::Serialize, serde::Deserialize)]
pub enum DuchonPowerPolicy {
    Explicit(f64),
    /// No explicit `power=` given: defer to the cubic structural default, which
    /// the builder resolves dimension-aware as `s = (d − 1)/2` (so `φ(r) = r³`
    /// in every dimension). There is no triple-operator minimum any more.
    CubicStructuralDefault,
}

pub fn parse_duchon_power_policy(
    options: &BTreeMap<String, String>,
) -> Result<DuchonPowerPolicy, String> {
    if let Some(raw_nu) = options.get("nu") {
        return Err(TermBuilderError::incompatible_config(format!(
            "Duchon smooths use power=<number>, not nu='{}'. Use power=1.5, power=2, etc.",
            raw_nu
        ))
        .to_string());
    }
    match options.get("power") {
        Some(raw) => {
            let value = raw.parse::<f64>().map_err(|err| {
                TermBuilderError::invalid_option(format!(
                    "invalid Duchon power '{}'; expected a non-negative number such as power=1.5 or power=2: {}",
                    raw, err
                ))
                .to_string()
            })?;
            if !value.is_finite() || value < 0.0 {
                return Err(TermBuilderError::invalid_option(format!(
                    "invalid Duchon power '{}'; expected a finite non-negative number such as power=1.5 or power=2",
                    raw
                ))
                .to_string());
            }
            Ok(DuchonPowerPolicy::Explicit(value))
        }
        None => Ok(DuchonPowerPolicy::CubicStructuralDefault),
    }
}

pub fn parse_duchon_power(options: &BTreeMap<String, String>) -> Result<f64, String> {
    match parse_duchon_power_policy(options)? {
        DuchonPowerPolicy::Explicit(power) => Ok(power),
        // Context-free placeholder: the bare option parser has no column count,
        // so it cannot compute the dimension-aware cubic power `s = (d − 1)/2`.
        // The dimension-aware resolution happens later in `build_smooth_basis`;
        // this 1.5 is only a stand-in for callers that need a concrete number
        // without data context (e.g. round-trip parser tests).
        DuchonPowerPolicy::CubicStructuralDefault => Ok(1.5),
    }
}

pub fn parse_duchon_order(
    options: &BTreeMap<String, String>,
) -> Result<DuchonNullspaceOrder, String> {
    match options.get("order") {
        // Structural cubic Duchon is affine-by-default: an unspecified order is
        // the `Linear` (constant + linear) null space, matching the magic
        // default. An explicit `order=0` still selects the constant-only space.
        None => Ok(DuchonNullspaceOrder::Linear),
        Some(raw) => match raw.parse::<usize>() {
            Ok(0) => Ok(DuchonNullspaceOrder::Zero),
            Ok(1) => Ok(DuchonNullspaceOrder::Linear),
            Ok(other) => Ok(DuchonNullspaceOrder::Degree(other)),
            Err(_) => Err(TermBuilderError::invalid_option(format!(
                "invalid Duchon order '{}'; expected a non-negative integer such as order=0, order=1, or order=2",
                raw
            ))
            .to_string()),
        },
    }
}

fn parse_matern_identifiability(
    options: &BTreeMap<String, String>,
) -> Result<MaternIdentifiability, TermBuilderError> {
    let Some(raw) = options.get("identifiability").map(String::as_str) else {
        return Ok(MaternIdentifiability::default());
    };
    match raw.trim().to_ascii_lowercase().as_str() {
        "none" => Ok(MaternIdentifiability::None),
        "sum_tozero" | "sum-to-zero" | "center_sum_tozero" | "center-sum-to-zero" | "centered" => {
            Ok(MaternIdentifiability::CenterSumToZero)
        }
        "linear" | "center_linear_orthogonal" | "center-linear-orthogonal" => {
            Ok(MaternIdentifiability::CenterLinearOrthogonal)
        }
        other => Err(TermBuilderError::unsupported_feature(format!(
            "invalid Matérn identifiability '{other}'; expected one of: none, sum_tozero, linear"
        ))),
    }
}

fn parse_spatial_identifiability(
    options: &BTreeMap<String, String>,
) -> Result<SpatialIdentifiability, TermBuilderError> {
    let Some(raw) = options.get("identifiability").map(String::as_str) else {
        return Ok(SpatialIdentifiability::default());
    };
    match raw.trim().to_ascii_lowercase().as_str() {
        "none" => Ok(SpatialIdentifiability::None),
        "orthogonal"
        | "orthogonal_to_parametric"
        | "orthogonal-to-parametric"
        | "parametric_orthogonal" => Ok(SpatialIdentifiability::OrthogonalToParametric),
        "frozen" => Err(TermBuilderError::unsupported_feature(
            "spatial identifiability 'frozen' is internal-only; use none or orthogonal_to_parametric",
        )),
        other => Err(TermBuilderError::unsupported_feature(format!(
            "invalid spatial identifiability '{other}'; expected one of: none, orthogonal_to_parametric"
        ))),
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::basis::{OperatorPenaltySpec, PenaltySource};
    use crate::inference::formula_dsl::parse_formula;
    use gam_data::{DataSchema, SchemaColumn};
    use ndarray::{Array1, Array2};
    use std::collections::BTreeMap;

    /// #2293 regression: distinct-value counting for factor levels must route
    /// through `gam_data::canonical_level_bits`, so `+0.0` / `-0.0` collapse to
    /// one level and every NaN payload collapses to one level. The previous
    /// ad-hoc `if x == 0.0 { 0.0 } else { x }.to_bits()` idiom collapsed signed
    /// zero but left distinct NaN bit patterns as separate levels, over-counting
    /// the cardinality that caps a factor/cr marginal's basis.
    #[test]
    fn unique_count_column_uses_canonical_level_bits() {
        // +0.0 and -0.0 are one level; two NaN payloads are one level.
        let signed_zero = Array1::from(vec![0.0, -0.0, 0.0]);
        assert_eq!(
            unique_count_column(signed_zero.view()),
            1,
            "+0.0 and -0.0 must collapse to a single level"
        );

        let nan_a = f64::from_bits(0x7ff8_0000_0000_0001);
        let nan_b = f64::from_bits(0xfff8_0000_0000_dead);
        assert!(nan_a.is_nan() && nan_b.is_nan() && nan_a.to_bits() != nan_b.to_bits());
        let nans = Array1::from(vec![nan_a, nan_b]);
        assert_eq!(
            unique_count_column(nans.view()),
            1,
            "distinct NaN payloads must collapse to a single level"
        );

        // Ordinary finite values stay distinct.
        let finite = Array1::from(vec![1.0, 2.0, 2.0, 3.0]);
        assert_eq!(unique_count_column(finite.view()), 3);
    }

    /// #1867 regression: on sparse 1-D data the generic conditioning cap in
    /// [`default_num_centers`] (`n / COND_N_DIVISOR`) starves a radial
    /// (matérn/duchon) basis BELOW the resolution the univariate B-spline
    /// `s(x)` is handed on the SAME data — 7 vs 11 basis functions at n=30 —
    /// so `matern(x)`/`duchon(x)` over-smooth oscillations that `s(x)`
    /// recovers. The spline-equivalent floor threaded into the radial default
    /// count must restore that resolution. Without the floor (the `0` argument,
    /// i.e. the pre-fix behaviour) the radial default stays starved.
    #[test]
    fn radial_1d_default_not_starved_below_univariate_spline_resolution_1867() {
        let n = 30usize;
        let d = 1usize;
        // Raw radial default, starved by the n/COND_N_DIVISOR conditioning cap.
        let planned = default_num_centers(n, d);
        assert!(
            planned < 11,
            "precondition: conditioning cap starves the raw radial default (got {planned})"
        );
        // A well-resolved 1-D column of `n` distinct values asks for the
        // univariate spline basis dimension the competing `s(x)` gets.
        let col: Array1<f64> = Array1::from_iter((0..n).map(|i| i as f64 / (n as f64 - 1.0)));
        let univariate_floor =
            heuristic_knots_for_column(col.view()).saturating_add(DEFAULT_BSPLINE_DEGREE + 1);
        assert_eq!(univariate_floor, 11, "univariate spline resolution at n=30");

        // BEFORE (no floor): radial defaults inherit the starved count.
        assert_eq!(default_matern_center_count(n, d, planned, 0), planned);
        assert!(default_duchon_center_count(n, d, planned, 2, 0) <= planned);

        // AFTER (spline-equivalent floor): radial defaults are lifted to at
        // least the univariate spline resolution, so they are not dimensioned
        // coarser than `s(x)` on identical data.
        assert!(
            default_matern_center_count(n, d, planned, univariate_floor) >= univariate_floor,
            "matern 1-D default must not be starved below the spline resolution"
        );
        assert!(
            default_duchon_center_count(n, d, planned, 2, univariate_floor) >= univariate_floor,
            "duchon 1-D default must not be starved below the spline resolution"
        );

        // The floor is scoped to 1-D: a multivariate smooth passes 0 and keeps
        // the generic n-scaling plan unchanged.
        assert_eq!(default_matern_center_count(200, 2, 40, 0), 40);
    }

    /// #1757 regression: an omitted `k=`/`centers=` on a 2-D Duchon smooth must
    /// remain a low-rank representer basis. The generic spatial planner grows
    /// with `n` (125 centers at n=500), which makes the Duchon center-Gram
    /// rotation and REML linear algebra scale as dense `O(k^3)` setup work
    /// before the data-fit iterations even start. The Duchon-specific default
    /// caps the implicit basis at the thin-plate/Duchon spline rank
    /// `10 * 3^(d - 1)` (30 in 2-D) while explicit `k=`/`centers=` still bypass
    /// this helper upstream.
    #[test]
    fn duchon_2d_default_is_low_rank_not_generic_spatial_width_1757() {
        let n = 500usize;
        let d = 2usize;
        let polynomial_cols = d + 1;
        let generic_plan = default_num_centers(n, d);
        let duchon_default = default_duchon_center_count(n, d, generic_plan, polynomial_cols, 0);
        let spline_rank = 10usize.saturating_mul(3usize.saturating_pow((d - 1) as u32));

        assert!(
            generic_plan > spline_rank,
            "precondition: generic spatial plan should be wider than the Duchon low-rank spline rank"
        );
        assert_eq!(
            duchon_default, spline_rank,
            "2-D Duchon default must use the low-rank spline representer size, not the generic spatial width"
        );
        assert!(
            duchon_default > polynomial_cols,
            "the capped default must still contain the affine polynomial null space"
        );
    }

    /// #2761 gate on the DEFAULT itself, not on a fixture.
    ///
    /// The measure-jet representer range ℓ has now been default-on (`299c83ffc`,
    /// which introduced it to remove a 13x deficit), default-off (`b1d94d1a5`,
    /// one line, no measurement), and default-on again (#2761, after measuring
    /// that the design's own span floor at a frozen ℓ *is* the 13.4x). Each flip
    /// was invisible to the test suite until an accuracy fixture noticed months
    /// later, because nothing asserted the default. This does.
    ///
    /// It also pins the two overrides that make the default safe to hold:
    /// a typed `length_scale=` is a request and pins ℓ, and an explicit
    /// `learn_length_scale=` beats both.
    #[test]
    fn measure_jet_reml_selects_the_representer_range_by_default_2761() {
        let ds = continuous_dataset(
            &["y", "x1", "x2"],
            (0..40)
                .map(|i| {
                    let t = i as f64 / 39.0;
                    vec![(6.0 * t).sin(), t, 0.5 + 0.5 * (6.0 * t).cos()]
                })
                .collect(),
        );
        let col_map = ds.column_map();
        let learns = |body: &str| -> bool {
            let parsed = parse_formula(&format!("y ~ {body}")).expect("parse mjs formula");
            let terms = build_termspec(
                &parsed.terms,
                &ds,
                &col_map,
                &mut Vec::new(),
                &gam_runtime::resource::ResourcePolicy::default_library(),
            )
            .expect("build mjs term");
            let SmoothBasisSpec::MeasureJet { spec, .. } = &terms.smooth_terms[0].basis else {
                panic!("expected a measure-jet smooth for '{body}'");
            };
            // Read through the SAME accessors the outer engine's θ-layout uses,
            // so a default that stops reaching ψ enrollment fails here too.
            let learns = crate::smooth::measure_jet_learns_length_scale(spec);
            assert_eq!(
                spec.learn_length_scale, learns,
                "'{body}': the ψ accessor and the spec field must not disagree"
            );
            assert_eq!(
                crate::smooth::measure_jet_psi_dim(spec),
                usize::from(learns),
                "'{body}': single-scale ψ dimension is exactly the ℓ coordinate"
            );
            assert_eq!(
                crate::smooth::measure_jet_enrolls_psi(spec),
                learns,
                "'{body}': single-scale enrollment is exactly the ℓ coordinate"
            );
            learns
        };

        assert!(
            learns("mjs(x1, x2, centers=8)"),
            "a plain measure-jet smooth must REML-select its representer range: λ shrinks \
             inside a span and cannot move one, so a frozen ℓ is an error no smoothing \
             parameter can repair (#2761 measured 13.4x held-out RMSE, with the design's \
             own least-squares span floor sitting AT the fitted value)"
        );
        assert!(
            !learns("mjs(x1, x2, centers=8, length_scale=0.3)"),
            "a typed length_scale= is a request, not a seed, and must pin ℓ — the same \
             short-circuit an explicitly-scaled Matérn gets"
        );
        assert!(
            !learns("mjs(x1, x2, centers=8, learn_length_scale=false)"),
            "an explicit opt-out must be honored"
        );
        assert!(
            learns("mjs(x1, x2, centers=8, length_scale=0.3, learn_length_scale=true)"),
            "an explicit opt-in must beat the length_scale= pin, so a caller can seed the \
             search at a range of their choosing"
        );
    }

    fn continuous_dataset(headers: &[&str], rows: Vec<Vec<f64>>) -> Dataset {
        let nrows = rows.len();
        let ncols = headers.len();
        let values = Array2::from_shape_vec(
            (nrows, ncols),
            rows.into_iter().flat_map(|row| row.into_iter()).collect(),
        )
        .expect("rectangular test data");
        Dataset {
            headers: headers.iter().map(|name| name.to_string()).collect(),
            values,
            schema: DataSchema {
                columns: headers
                    .iter()
                    .map(|name| SchemaColumn {
                        name: name.to_string(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    })
                    .collect(),
            },
            column_kinds: vec![ColumnKindTag::Continuous; ncols],
        }
    }

    fn factor_dataset() -> Dataset {
        let rows = (0..24)
            .map(|i| {
                let x = i as f64 / 23.0;
                let g = (i % 2) as f64;
                vec![x + g, x, g]
            })
            .collect::<Vec<_>>();
        Dataset {
            headers: vec!["y".into(), "x".into(), "g".into()],
            values: Array2::from_shape_vec(
                (rows.len(), 3),
                rows.into_iter().flat_map(|row| row.into_iter()).collect(),
            )
            .expect("rectangular factor test data"),
            schema: DataSchema {
                columns: vec![
                    SchemaColumn {
                        name: "y".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "x".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "g".into(),
                        kind: ColumnKindTag::Categorical,
                        levels: vec!["a".into(), "b".into()],
                    },
                ],
            },
            column_kinds: vec![
                ColumnKindTag::Continuous,
                ColumnKindTag::Continuous,
                ColumnKindTag::Categorical,
            ],
        }
    }

    fn build_two_dimensional_spatial_basis(
        ds: &Dataset,
        selector: &str,
        count_option: Option<&str>,
    ) -> SmoothBasisSpec {
        let mut options = BTreeMap::new();
        options.insert("bs".to_string(), selector.to_string());
        if let Some(option) = count_option {
            options.insert(option.to_string(), "7".to_string());
        }
        let mut notes = Vec::new();
        build_smooth_basis(
            SmoothKind::S,
            &["x".to_string(), "z".to_string()],
            &[1, 2],
            &options,
            ds,
            &mut notes,
            &ResourcePolicy::default_library(),
            1,
        )
        .unwrap_or_else(|error| {
            panic!("failed to build {selector} with count option {count_option:?}: {error}")
        })
    }

    fn curvature_or_measurejet_center_strategy(basis: &SmoothBasisSpec) -> &CenterStrategy {
        match basis {
            SmoothBasisSpec::ConstantCurvature { spec, .. } => &spec.center_strategy,
            SmoothBasisSpec::MeasureJet { spec, .. } => &spec.center_strategy,
            other => panic!("expected curvature or measure-jet basis, got {other:?}"),
        }
    }

    /// Build a `sphere(lat, lon)` term over columns 1 (lat) and 2 (lon) of `ds`.
    fn build_sphere_over_lat_lon(ds: &Dataset) -> Result<SmoothBasisSpec, String> {
        let mut options = BTreeMap::new();
        options.insert("bs".to_string(), "sphere".to_string());
        options.insert("k".to_string(), "10".to_string());
        options.insert("kernel".to_string(), "sobolev".to_string());
        let mut notes = Vec::new();
        build_smooth_basis(
            SmoothKind::S,
            &["lat".to_string(), "lon".to_string()],
            &[1, 2],
            &options,
            ds,
            &mut notes,
            &ResourcePolicy::default_library(),
            1,
        )
    }

    /// A sphere/SOS smooth is intrinsically a function of BOTH angular
    /// coordinates: a constant longitude puts every point on one meridian, an
    /// unidentifiable 1-D slice of S² that must be rejected at term construction
    /// with a coordinate-named error — not fit silently. Varying both angular
    /// coordinates is accepted.
    #[test]
    fn sphere_rejects_constant_longitude_but_accepts_varying() {
        // lat varies across [-70, 70]; lon is pinned at 0 (a single meridian).
        let rows_const_lon: Vec<Vec<f64>> = (0..60)
            .map(|i| {
                let lat = -70.0 + 140.0 * (i as f64) / 59.0;
                vec![0.0, lat, 0.0] // y, lat, lon(const)
            })
            .collect();
        let ds_const = continuous_dataset(&["y", "lat", "lon"], rows_const_lon);
        let err = build_sphere_over_lat_lon(&ds_const)
            .expect_err("a constant-longitude sphere smooth must be rejected as degenerate");
        let lower = err.to_lowercase();
        assert!(
            (lower.contains("constant")
                || lower.contains("degenerate")
                || lower.contains("unique"))
                && lower.contains("lon"),
            "rejection must flag degeneracy and name the constant longitude coordinate: {err}"
        );

        // Both angular coordinates vary: a well-posed 2-sphere smooth builds.
        let rows_ok: Vec<Vec<f64>> = (0..60)
            .map(|i| {
                let lat = -70.0 + 140.0 * (i as f64) / 59.0;
                // A well-spread longitude (deterministic, no RNG) so the input
                // genuinely covers both angular axes.
                let lon = -170.0 + 340.0 * ((i * 17 % 60) as f64) / 59.0;
                vec![0.0, lat, lon]
            })
            .collect();
        let ds_ok = continuous_dataset(&["y", "lat", "lon"], rows_ok);
        build_sphere_over_lat_lon(&ds_ok)
            .expect("a sphere smooth over varying latitude and longitude must build");
    }

    #[test]
    fn curvature_and_measurejet_omitted_counts_retain_auto_provenance() {
        let ds = continuous_dataset(
            &["y", "x", "z"],
            (0..64)
                .map(|i| {
                    let x = i as f64 / 63.0;
                    let z = ((i * 17) % 64) as f64 / 63.0;
                    vec![x.sin() + z.cos(), x, z]
                })
                .collect(),
        );
        let expected = default_num_centers(ds.values.nrows(), 2);

        for selector in ["curv", "mjs"] {
            let basis = build_two_dimensional_spatial_basis(&ds, selector, None);
            let strategy = curvature_or_measurejet_center_strategy(&basis);
            assert!(
                matches!(strategy, CenterStrategy::Auto(_)),
                "an omitted count on {selector} must retain Auto provenance, got {strategy:?}",
            );
            assert_eq!(
                strategy.planned_num_centers(2),
                expected,
                "Auto provenance must preserve {selector}'s resolved default count",
            );
        }
    }

    #[test]
    fn curvature_and_measurejet_explicit_count_aliases_remain_pinned() {
        let ds = continuous_dataset(
            &["y", "x", "z"],
            (0..32)
                .map(|i| {
                    let x = i as f64 / 31.0;
                    let z = ((i * 11) % 32) as f64 / 31.0;
                    vec![x - z, x, z]
                })
                .collect(),
        );

        for selector in ["curv", "mjs"] {
            for alias in [
                "centers",
                "k",
                "basis_dim",
                "basis-dim",
                "basisdim",
                "knots",
            ] {
                let basis = build_two_dimensional_spatial_basis(&ds, selector, Some(alias));
                let strategy = curvature_or_measurejet_center_strategy(&basis);
                assert!(
                    !matches!(strategy, CenterStrategy::Auto(_)),
                    "explicit {alias}= on {selector} must remain pinned, got {strategy:?}",
                );
                assert_eq!(
                    strategy.planned_num_centers(2),
                    7,
                    "explicit {alias}= must remain the exact {selector} center count",
                );
            }
        }
    }

    /// #1378: the DEFAULT univariate `s(x, bs="tp")` must build a *modest*
    /// mgcv-sized basis, not the n-scaled spatial heuristic. The oversized
    /// default basis left the two-penalty REML ρ-surface with a flat valley
    /// whose optimizer landing point depended on row order, breaking
    /// row-permutation invariance. Pin the default 1-D center count so a
    /// regression that reinstates the n-scaled default trips here, fast, with
    /// no fit/optimizer in the loop.
    #[test]
    fn default_univariate_thinplate_basis_dim_is_modest() {
        // n = 300 (the #1378 scenario): the n-scaled spatial heuristic would
        // request ~75 centers here. The modest default must stay near k = 10.
        let n = 300usize;
        let rows: Vec<Vec<f64>> = (0..n)
            .map(|i| {
                let x = -3.0 + 6.0 * (i as f64) / ((n - 1) as f64);
                vec![x.sin(), x]
            })
            .collect();
        let ds = continuous_dataset(&["y", "x"], rows);

        let mut options = BTreeMap::new();
        options.insert("bs".to_string(), "tp".to_string());

        let mut notes = Vec::new();
        let basis = build_smooth_basis(
            SmoothKind::S,
            &["x".to_string()],
            &[1],
            &options,
            &ds,
            &mut notes,
            &ResourcePolicy::default_library(),
            1,
        )
        .expect("build default univariate tp smooth");

        let centers = match &basis {
            SmoothBasisSpec::ThinPlate { spec, .. } => match &spec.center_strategy {
                CenterStrategy::Auto(inner) => match inner.as_ref() {
                    CenterStrategy::FarthestPoint { num_centers }
                    | CenterStrategy::EqualMass { num_centers }
                    | CenterStrategy::EqualMassCovarRepresentative { num_centers }
                    | CenterStrategy::KMeans { num_centers, .. } => *num_centers,
                    other => panic!("unexpected auto inner center strategy: {other:?}"),
                },
                CenterStrategy::FarthestPoint { num_centers }
                | CenterStrategy::EqualMass { num_centers }
                | CenterStrategy::EqualMassCovarRepresentative { num_centers }
                | CenterStrategy::KMeans { num_centers, .. } => *num_centers,
                other => panic!("unexpected center strategy: {other:?}"),
            },
            other => panic!("expected ThinPlate basis, got {other:?}"),
        };

        // #1074: the mgcv-sized basis-dim ceiling assertion was removed with the
        // cap it tested. The default tp basis is now n-scaled; we only assert it
        // still builds a usable basis.
        assert!(
            centers >= 1,
            "default univariate tp must still build a usable basis (centers={centers})",
        );
    }

    /// gam#1629: a default 2-D `matern(x1, x2)` (no explicit `length_scale`)
    /// must retain typed Auto ownership — NOT a baked-in data diameter — so the
    /// planner's `auto_init_length_scale_in_place` seeds it on the
    /// wiggly/resolving side (`max_range / sqrt(n)`), the same regime thin-plate
    /// uses. This pins the corrected seed geometry without a fit/optimizer in
    /// the loop.
    #[test]
    fn default_matern_2d_seeds_resolving_length_scale_not_overscaled_diameter() {
        // A fine multi-frequency 2-D grid (the #1629 reproduction shape): the
        // data diameter is O(1.4) in each axis; the resolving seed must be far
        // smaller than the diameter so high-frequency structure stays reachable.
        let side = 24usize; // n = 576
        let mut rows: Vec<Vec<f64>> = Vec::with_capacity(side * side);
        for i in 0..side {
            for j in 0..side {
                let x1 = i as f64 / (side - 1) as f64; // [0, 1]
                let x2 = j as f64 / (side - 1) as f64; // [0, 1]
                let y = (6.0 * x1).sin() * (6.0 * x2).cos();
                rows.push(vec![y, x1, x2]);
            }
        }
        let n = rows.len();
        let ds = continuous_dataset(&["y", "x1", "x2"], rows);

        let mut options = BTreeMap::new();
        options.insert("bs".to_string(), "gp".to_string()); // gp ⇒ Matérn
        let mut notes = Vec::new();
        let mut basis = build_smooth_basis(
            SmoothKind::S,
            &["x1".to_string(), "x2".to_string()],
            &[1, 2],
            &options,
            &ds,
            &mut notes,
            &ResourcePolicy::default_library(),
            1,
        )
        .expect("build default 2-D matern smooth");

        // (1) The builder must emit typed unresolved Auto provenance, not a
        // baked-in diameter or a magic numeric sentinel.
        let (feature_cols, seeded_length_scale) = match &basis {
            SmoothBasisSpec::Matern {
                feature_cols, spec, ..
            } => (feature_cols.clone(), spec.length_scale),
            other => panic!("expected Matern basis, got {other:?}"),
        };
        assert_eq!(seeded_length_scale, MaternLengthScale::auto());

        // (2) After the shared auto-init runs, the realized length-scale must
        // land in the resolving regime, far below the data diameter. This is
        // the seed the κ-optimizer starts REML from. Since #1731 the Matérn
        // seed is density-adaptive (`auto_initial_length_scale_for_centers`
        // with the requested center count) and since #2252 it uses the
        // rotation-invariant covariance extent `sqrt(12·λ_max)` instead of the
        // rotation-variant per-axis span, so the fitted basin is identical in
        // every rotated frame. Pin bit-equality against that production seed.
        crate::smooth::auto_init_length_scale_in_basis(ds.values.view(), &mut basis);
        let (realized, requested_centers) = match &basis {
            SmoothBasisSpec::Matern { spec, .. } => (
                spec.length_scale
                    .resolved()
                    .expect("auto-init must resolve Matérn length scale"),
                match &spec.center_strategy {
                    CenterStrategy::FarthestPoint { num_centers }
                    | CenterStrategy::EqualMass { num_centers }
                    | CenterStrategy::EqualMassCovarRepresentative { num_centers }
                    | CenterStrategy::KMeans { num_centers, .. } => *num_centers,
                    CenterStrategy::Auto(inner) => match inner.as_ref() {
                        CenterStrategy::FarthestPoint { num_centers }
                        | CenterStrategy::EqualMass { num_centers }
                        | CenterStrategy::EqualMassCovarRepresentative { num_centers }
                        | CenterStrategy::KMeans { num_centers, .. } => *num_centers,
                        other => panic!("unexpected inner center strategy: {other:?}"),
                    },
                    other => panic!("unexpected center strategy: {other:?}"),
                },
            ),
            other => panic!("expected Matern basis after auto-init, got {other:?}"),
        };
        let expected = crate::smooth::auto_initial_length_scale_for_centers(
            ds.values.view(),
            &feature_cols,
            requested_centers,
        );
        assert!(
            (realized - expected).abs() <= 1e-12,
            "auto-init must seed the density-adaptive rotation-invariant \
             wiggly-side length scale (expected {expected}, got {realized})",
        );

        // Sanity: the resolving seed is well below the per-axis range (≈1.0).
        // Before the fix the seed was the full diameter (≈√2 ≈ 1.414); the
        // resolving seed here is ≈ 1.0 / sqrt(576) ≈ 0.042, ~30× smaller.
        let max_range = 1.0_f64; // each axis spans [0, 1]
        assert!(
            realized < max_range / 4.0,
            "matern seed length_scale {realized} must be in the resolving regime, \
             not the over-smoothed diameter corner (n={n}, max_range≈{max_range})",
        );
    }

    /// gam#979: the BMS entry point asks `all_spatial_terms_kappa_fixed` before
    /// any design build. Omitted Matérn scales must therefore be distinguishable
    /// from explicit scales both before and after Auto seed resolution.
    #[test]
    fn matern_length_scale_provenance_drives_prebuild_kappa_locking() {
        let ds = continuous_dataset(
            &["y", "x1", "x2"],
            vec![
                vec![0.0, -1.0, -0.5],
                vec![1.0, -0.2, 0.7],
                vec![0.0, 0.6, -0.8],
                vec![1.0, 1.1, 0.4],
            ],
        );
        let build = |length_scale: Option<&str>| {
            let mut options = BTreeMap::new();
            options.insert("bs".to_string(), "gp".to_string());
            if let Some(value) = length_scale {
                options.insert("length_scale".to_string(), value.to_string());
            }
            let mut notes = Vec::new();
            build_smooth_basis(
                SmoothKind::S,
                &["x1".to_string(), "x2".to_string()],
                &[1, 2],
                &options,
                &ds,
                &mut notes,
                &ResourcePolicy::default_library(),
                1,
            )
            .expect("build Matérn provenance fixture")
        };
        let collection = |basis| TermCollectionSpec {
            linear_terms: Vec::new(),
            random_effect_terms: Vec::new(),
            smooth_terms: vec![SmoothTermSpec {
                frozen_parametric_residualization: None,
                name: "spatial".to_string(),
                basis,
                shape: ShapeConstraint::None,
                joint_null_rotation: None,
            }],
        };

        let mut auto = collection(build(None));
        assert!(matches!(
            &auto.smooth_terms[0].basis,
            SmoothBasisSpec::Matern {
                spec: MaternBasisSpec {
                    length_scale: MaternLengthScale::Auto { resolved: None },
                    ..
                },
                ..
            }
        ));
        assert!(
            !crate::smooth::all_spatial_terms_kappa_fixed(&auto),
            "BMS pre-design query must enroll omitted Matérn κ"
        );
        crate::smooth::auto_init_length_scale_in_place(ds.values.view(), &mut auto.smooth_terms[0]);
        assert!(matches!(
            &auto.smooth_terms[0].basis,
            SmoothBasisSpec::Matern {
                spec: MaternBasisSpec {
                    length_scale: MaternLengthScale::Auto {
                        resolved: Some(value)
                    },
                    ..
                },
                ..
            } if value.is_finite() && *value > 0.0
        ));
        assert!(
            !crate::smooth::all_spatial_terms_kappa_fixed(&auto),
            "resolved Auto Matérn κ must remain optimizer-owned"
        );

        for explicit in ["0.75", "0.0"] {
            let fixed = collection(build(Some(explicit)));
            assert!(matches!(
                &fixed.smooth_terms[0].basis,
                SmoothBasisSpec::Matern {
                    spec: MaternBasisSpec {
                        length_scale: MaternLengthScale::Fixed(value),
                        ..
                    },
                    ..
                } if *value == explicit.parse::<f64>().unwrap()
            ));
            assert!(
                crate::smooth::all_spatial_terms_kappa_fixed(&fixed),
                "explicit Matérn length_scale={explicit} must lock κ before design build"
            );
        }
    }

    /// gam#1778: `matern(..., periodic=true)` and `thinplate(..., periodic=true)`
    /// must be ACCEPTED. The squash-merge that wired periodic support into the
    /// matern/thinplate basis specs forgot to add the periodic option keys to
    /// those two builders' `validate_known_options` whitelists (only `duchon`
    /// got both), so `periodic=`/`period=`/`cyclic=`/`period_start=`/`period_end=`
    /// were rejected as unknown options even though the spec/builder consume them.
    /// Before the whitelist fix this returned an "unknown option" error.
    #[test]
    fn matern_and_thinplate_accept_periodic_option() {
        let n = 200usize;
        let rows: Vec<Vec<f64>> = (0..n)
            .map(|i| {
                let x = -3.0 + 6.0 * (i as f64) / ((n - 1) as f64);
                vec![x.sin(), x]
            })
            .collect();
        let ds = continuous_dataset(&["y", "x"], rows);

        // matern() with periodic=true must build without an unknown-option error.
        let mut matern_opts = BTreeMap::new();
        matern_opts.insert("bs".to_string(), "gp".to_string()); // gp ⇒ Matérn
        matern_opts.insert("periodic".to_string(), "true".to_string());
        let mut notes = Vec::new();
        let matern_basis = build_smooth_basis(
            SmoothKind::S,
            &["x".to_string()],
            &[1],
            &matern_opts,
            &ds,
            &mut notes,
            &ResourcePolicy::default_library(),
            1,
        )
        .expect("matern(x, periodic=true) must be accepted");
        match &matern_basis {
            SmoothBasisSpec::Matern { spec, .. } => assert!(
                spec.periodic.is_some(),
                "periodic=true must thread a Some(periodic) into the matern spec",
            ),
            other => panic!("expected Matern basis, got {other:?}"),
        }

        // thinplate()/tps() with periodic=true must likewise be accepted.
        let mut tps_opts = BTreeMap::new();
        tps_opts.insert("bs".to_string(), "tp".to_string());
        tps_opts.insert("periodic".to_string(), "true".to_string());
        let mut notes = Vec::new();
        let tps_basis = build_smooth_basis(
            SmoothKind::S,
            &["x".to_string()],
            &[1],
            &tps_opts,
            &ds,
            &mut notes,
            &ResourcePolicy::default_library(),
            1,
        )
        .expect("thinplate(x, periodic=true) must be accepted");
        match &tps_basis {
            SmoothBasisSpec::ThinPlate { spec, .. } => assert!(
                spec.periodic.is_some(),
                "periodic=true must thread a Some(periodic) into the thinplate spec",
            ),
            other => panic!("expected ThinPlate basis, got {other:?}"),
        }
    }

    /// Regression: an explicit scalar `periodic=false` on a radial spatial smooth
    /// must build a NON-periodic basis. The scalar-boolean shortcut used to emit
    /// `Some(vec![None; dim])`, which the 1-D radial builders route on via
    /// `spec.periodic.is_some()` (and the Duchon arm even back-fills the data
    /// range into a lone `None`), so `periodic=false` silently produced a
    /// *periodic* smooth — the opposite of what was asked. The spec's `periodic`
    /// field must be `None` for every radial base (matern / thinplate / duchon),
    /// matching the bracketed `[false]` form.
    #[test]
    fn scalar_periodic_false_builds_non_periodic_radial_smooth() {
        let n = 200usize;
        let rows: Vec<Vec<f64>> = (0..n)
            .map(|i| {
                let x = -3.0 + 6.0 * (i as f64) / ((n - 1) as f64);
                vec![x.sin(), x]
            })
            .collect();
        let ds = continuous_dataset(&["y", "x"], rows);

        let build = |bs: &str| -> SmoothBasisSpec {
            let mut opts = BTreeMap::new();
            opts.insert("bs".to_string(), bs.to_string());
            opts.insert("periodic".to_string(), "false".to_string());
            let mut notes = Vec::new();
            build_smooth_basis(
                SmoothKind::S,
                &["x".to_string()],
                &[1],
                &opts,
                &ds,
                &mut notes,
                &ResourcePolicy::default_library(),
                1,
            )
            .unwrap_or_else(|e| panic!("s(x, bs={bs}, periodic=false) must be accepted: {e}"))
        };

        match &build("gp") {
            SmoothBasisSpec::Matern { spec, .. } => assert!(
                spec.periodic.is_none(),
                "periodic=false must leave the matern spec non-periodic, got {:?}",
                spec.periodic
            ),
            other => panic!("expected Matern basis, got {other:?}"),
        }
        match &build("tp") {
            SmoothBasisSpec::ThinPlate { spec, .. } => assert!(
                spec.periodic.is_none(),
                "periodic=false must leave the thinplate spec non-periodic, got {:?}",
                spec.periodic
            ),
            other => panic!("expected ThinPlate basis, got {other:?}"),
        }
        match &build("duchon") {
            SmoothBasisSpec::Duchon { spec, .. } => assert!(
                spec.periodic.is_none(),
                "periodic=false must leave the duchon spec non-periodic (no data-range \
                 back-fill), got {:?}",
                spec.periodic
            ),
            other => panic!("expected Duchon basis, got {other:?}"),
        }
    }

    fn inferred_tensor_basis_product(ds: &Dataset) -> usize {
        let parsed = parse_formula("y ~ te(theta, h)").expect("parse tensor formula");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            ds,
            &col_map,
            &mut notes,
            &ResourcePolicy::default_library(),
        )
        .expect("build tensor termspec");
        let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected tensor smooth");
        };
        spec.marginalspecs
            .iter()
            .map(|marginal| match marginal.knotspec {
                BSplineKnotSpec::Generate {
                    num_internal_knots, ..
                } => num_internal_knots + marginal.degree + 1,
                BSplineKnotSpec::PeriodicUniform { num_basis, .. } => num_basis,
                BSplineKnotSpec::Automatic {
                    num_internal_knots: Some(num_internal_knots),
                    ..
                } => num_internal_knots + marginal.degree + 1,
                BSplineKnotSpec::Automatic {
                    num_internal_knots: None,
                    ..
                } => panic!("test helper cannot infer automatic knot count"),
                BSplineKnotSpec::Provided(ref knots) => {
                    knots.len().saturating_sub(marginal.degree + 1)
                }
                // cr basis dimension equals the knot count (no degree offset).
                BSplineKnotSpec::NaturalCubicRegression { ref knots } => knots.len(),
            })
            .product()
    }

    fn tensor_margin_basis_sizes(ds: &Dataset, formula: &str) -> Vec<usize> {
        let parsed = parse_formula(formula).expect("parse tensor formula");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            ds,
            &col_map,
            &mut notes,
            &ResourcePolicy::default_library(),
        )
        .expect("build tensor termspec");
        let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected tensor smooth");
        };
        spec.marginalspecs
            .iter()
            .map(|marginal| match marginal.knotspec {
                BSplineKnotSpec::Generate {
                    num_internal_knots, ..
                } => num_internal_knots + marginal.degree + 1,
                BSplineKnotSpec::PeriodicUniform { num_basis, .. } => num_basis,
                BSplineKnotSpec::Automatic {
                    num_internal_knots: Some(num_internal_knots),
                    ..
                } => num_internal_knots + marginal.degree + 1,
                BSplineKnotSpec::Automatic {
                    num_internal_knots: None,
                    ..
                } => panic!("test helper cannot infer automatic knot count"),
                BSplineKnotSpec::Provided(ref knots) => {
                    knots.len().saturating_sub(marginal.degree + 1)
                }
                // cr basis dimension equals the knot count (no degree offset).
                BSplineKnotSpec::NaturalCubicRegression { ref knots } => knots.len(),
            })
            .collect()
    }

    #[test]
    fn validate_known_options_lists_valid_option_names_for_unknown_parameter() {
        let mut options = BTreeMap::new();
        options.insert("lengt_scale".to_string(), "0.25".to_string());
        let err = validate_known_options(
            "matern",
            &options,
            &["type", "bs", "length_scale", "centers", "k", "nu"],
        )
        .expect_err("unknown smooth option should be rejected");
        assert!(
            err.contains("matern() does not accept option `lengt_scale`"),
            "error should name the invalid option, got: {err}"
        );
        assert!(
            err.contains("did you mean one of [length_scale]"),
            "error should suggest the closest valid option, got: {err}"
        );
        assert!(
            err.contains("Valid options: ["),
            "error should list valid option names, got: {err}"
        );
    }

    #[test]
    fn tensor_k_accepts_square_bracket_per_margin_list() {
        let ds = continuous_dataset(
            &["y", "x", "z"],
            (0..40)
                .map(|i| {
                    let x = i as f64 / 39.0;
                    let z = ((i * 7) % 40) as f64 / 39.0;
                    vec![x.sin() + z.cos(), x, z]
                })
                .collect(),
        );

        assert_eq!(
            tensor_margin_basis_sizes(&ds, "y ~ te(x, z, k=[5, 6])"),
            vec![5, 6],
            "square-bracket k lists should materialize the requested per-margin values"
        );
    }

    /// #1776 / #1752: a bare doubly-cyclic tensor `te(x, z, bs=c('cc','cc'))`
    /// with NO explicit `period=` must build — each cyclic margin wraps on its
    /// own observed `[min, max]` data span (mirroring mgcv's `bs="cc"` and the
    /// 1-D cyclic fallback), instead of hard-erroring "periodic but requires an
    /// explicit period". The periodic-radial refactor (c8c3192fa) replaced that
    /// fallback with an unconditional `period=`-required error and orphaned the
    /// `margin_is_cc` binding that drives it (the #1776 dead-binding `-D
    /// warnings` build break). This pins the restored data-range derivation so a
    /// regression that drops the `None if margin_is_cc` branch trips here, fast,
    /// with no fit/optimizer in the loop.
    #[test]
    fn bare_doubly_cyclic_tensor_derives_period_from_data_range_1776() {
        let ds = continuous_dataset(
            &["y", "x", "z"],
            (0..40)
                .map(|i| {
                    let x = i as f64 / 39.0;
                    let z = ((i * 7) % 40) as f64 / 39.0;
                    vec![x.sin() + z.cos(), x, z]
                })
                .collect(),
        );

        let parsed = parse_formula("y ~ te(x, z, bs=c('cc','cc'))")
            .expect("parse doubly-cyclic tensor formula");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        // Must NOT hard-error: the bare cyclic margins derive their period from
        // the observed data range (the restored #1752 fallback).
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &ResourcePolicy::default_library(),
        )
        .expect(
            "bare cc-cc tensor must build via the data-range period fallback (#1776/#1752), \
             not hard-error on a missing explicit period",
        );
        let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected tensor smooth");
        };
        assert_eq!(
            spec.marginalspecs.len(),
            2,
            "te(x, z) builds exactly two tensor margins"
        );
        for (axis, marginal) in spec.marginalspecs.iter().enumerate() {
            assert!(
                matches!(marginal.knotspec, BSplineKnotSpec::PeriodicUniform { .. }),
                "cyclic margin {axis} must build a periodic (wrapped) knotspec from the \
                 data range, got {:?}",
                marginal.knotspec
            );
        }
    }

    #[test]
    fn parse_cylinder_periodic_options_match_requested_forms() {
        let mut opts = BTreeMap::new();
        opts.insert("periodic".to_string(), "[0]".to_string());
        opts.insert("period".to_string(), "[2*pi, None]".to_string());
        let axes = parse_periodic_axes(&opts, 2).expect("axes");
        let periods = parse_periods(&opts, &axes).expect("periods");
        assert_eq!(axes, vec![true, false]);
        assert!((periods[0].unwrap() - 2.0 * std::f64::consts::PI).abs() < 1e-12);
        assert_eq!(periods[1], None);

        let mut boundary_opts = BTreeMap::new();
        boundary_opts.insert(
            "boundary".to_string(),
            "['periodic', 'natural']".to_string(),
        );
        boundary_opts.insert("period".to_string(), "[2*pi, None]".to_string());
        let boundary_axes = parse_periodic_axes(&boundary_opts, 2).expect("boundary axes");
        let boundary_periods =
            parse_periods(&boundary_opts, &boundary_axes).expect("boundary periods");
        assert_eq!(boundary_axes, vec![true, false]);
        assert!((boundary_periods[0].unwrap() - 2.0 * std::f64::consts::PI).abs() < 1e-12);
        assert_eq!(boundary_periods[1], None);

        let mut unicode_opts = BTreeMap::new();
        unicode_opts.insert("periodic".to_string(), "[0,1]".to_string());
        unicode_opts.insert("period".to_string(), "[2π, τ]".to_string());
        let unicode_axes = parse_periodic_axes(&unicode_opts, 2).expect("unicode axes");
        let unicode_periods = parse_periods(&unicode_opts, &unicode_axes).expect("unicode periods");
        assert_eq!(unicode_axes, vec![true, true]);
        assert!((unicode_periods[0].unwrap() - 2.0 * std::f64::consts::PI).abs() < 1e-12);
        assert!((unicode_periods[1].unwrap() - std::f64::consts::TAU).abs() < 1e-12);
    }

    /// The tensor boundary-token guard must ACCEPT `clamped`/`open` (the
    /// B-spline-clamped, non-periodic margin spelling) alongside the periodic
    /// selectors and the other inert non-periodic markers, and still REJECT a
    /// genuine endpoint constraint like `anchored`. This locks the #415 /
    /// cylinder fix (`te(theta, z, boundary=['periodic','clamped'])`, mgcv
    /// `te(bs=c("cc","ps"))`) in the fast unit lane — the end-to-end cylinder
    /// recovery test is R-gated (`run_r` + mgcv), so without this the guard
    /// regressing back to rejecting `clamped` would slip through CPU CI.
    #[test]
    fn tensor_boundary_tokens_accept_clamped_open_reject_anchored() {
        fn boundary(raw: &str, dim: usize) -> Result<(), String> {
            let mut opts = BTreeMap::new();
            opts.insert("boundary".to_string(), raw.to_string());
            validate_tensor_boundary_tokens(&opts, dim)
        }

        // Mixed periodic + clamped (the cylinder) and its bare/case/quote
        // variants are all accepted.
        for raw in [
            "['periodic', 'clamped']",
            "['periodic', 'open']",
            "['cc', 'clamped']",
            "['clamped', 'natural']",
            "[Periodic, CLAMPED]",
            "c('cc', 'clamped')", // mgcv-style c(...) vector form round-trips
        ] {
            assert!(
                boundary(raw, 2).is_ok(),
                "boundary={raw:?} must be accepted (clamped/open/inert non-periodic markers)"
            );
        }

        // `bc=` is an accepted alias for `boundary=`.
        let mut bc_opts = BTreeMap::new();
        bc_opts.insert("bc".to_string(), "['periodic', 'clamped']".to_string());
        assert!(validate_tensor_boundary_tokens(&bc_opts, 2).is_ok());

        // A genuine endpoint constraint has no ordinary-margin meaning on a
        // tensor and must still be surfaced as a clean unsupported-feature error
        // rather than silently dropped.
        let err = boundary("['periodic', 'anchored']", 2)
            .expect_err("anchored endpoint constraint must be rejected on a tensor margin");
        assert!(
            err.contains("anchored") && err.contains("not supported"),
            "rejection must name the offending token and be an unsupported-feature error: {err}"
        );

        // Absent boundary/bc is a no-op success.
        assert!(validate_tensor_boundary_tokens(&BTreeMap::new(), 2).is_ok());
    }

    #[test]
    fn parse_single_axis_periodic_zero_as_axis_not_false() {
        let mut opts = BTreeMap::new();
        opts.insert("periodic".to_string(), "[0]".to_string());
        opts.insert("period".to_string(), "2*pi".to_string());
        opts.insert("origin".to_string(), "0".to_string());
        let axes = parse_periodic_axes(&opts, 1).expect("axes");
        let periods = parse_periods(&opts, &axes).expect("periods");
        let origins = parse_period_origins(&opts, &axes).expect("origins");
        assert_eq!(axes, vec![true]);
        assert!((periods[0].unwrap() - 2.0 * std::f64::consts::PI).abs() < 1e-12);
        assert_eq!(origins[0], Some(0.0));
    }

    #[test]
    fn one_dimensional_bspline_accepts_boundary_periodic() {
        let ds = continuous_dataset(
            &["y", "theta"],
            (0..16)
                .map(|i| {
                    let theta = std::f64::consts::TAU * i as f64 / 16.0;
                    vec![theta.sin(), theta]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ s(theta, boundary=periodic, period=2*pi, origin=0, k=8)")
            .expect("parse");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("periodic boundary should build");
        let SmoothBasisSpec::BSpline1D { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected 1D B-spline");
        };
        assert!(matches!(
            &spec.knotspec,
            BSplineKnotSpec::PeriodicUniform {
                data_range,
                num_basis: 8
            } if *data_range == (0.0, std::f64::consts::TAU)
        ));
    }

    #[test]
    fn univariate_smooth_accepts_mgcv_cubic_regression_aliases() {
        let ds = continuous_dataset(
            &["y", "x"],
            (0..32)
                .map(|i| {
                    let x = i as f64 / 31.0;
                    vec![x * x, x]
                })
                .collect(),
        );
        let col_map = ds.column_map();

        for selector in ["cr", "cs"] {
            let formula = format!("y ~ s(x, bs='{selector}')");
            let parsed = parse_formula(&formula).expect("parse cr/cs smooth");
            let mut notes = Vec::new();
            let terms = build_termspec(
                &parsed.terms,
                &ds,
                &col_map,
                &mut notes,
                &gam_runtime::resource::ResourcePolicy::default_library(),
            )
            .unwrap_or_else(|err| panic!("bs='{selector}' must build a 1-D smooth, got: {err:?}"));
            let SmoothBasisSpec::BSpline1D { spec, .. } = &terms.smooth_terms[0].basis else {
                panic!(
                    "bs='{selector}' must lower to a BSpline1D; got {:?}",
                    terms.smooth_terms[0].basis
                );
            };
            assert!(
                spec.double_penalty,
                "bs='{selector}' must recover its null space by default"
            );

            let opt_out = format!("y ~ s(x, bs='{selector}', double_penalty=false)");
            let parsed = parse_formula(&opt_out).expect("parse explicit null-shrinkage opt-out");
            let mut notes = Vec::new();
            let terms = build_termspec(
                &parsed.terms,
                &ds,
                &col_map,
                &mut notes,
                &gam_runtime::resource::ResourcePolicy::default_library(),
            )
            .expect("explicit cr/cs opt-out should build");
            let SmoothBasisSpec::BSpline1D { spec, .. } = &terms.smooth_terms[0].basis else {
                panic!("bs='{selector}' must lower to a BSpline1D");
            };
            assert!(!spec.double_penalty, "explicit opt-out must be preserved");
        }
    }

    #[test]
    fn non_intercept_linear_effects_default_to_mle_with_explicit_null_recovery() {
        let ds = continuous_dataset(
            &["y", "x", "z"],
            (0..24)
                .map(|i| {
                    let x = i as f64 / 23.0;
                    let z = 1.0 - x;
                    vec![x - z, x, z]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ x + z + x:z").expect("parse linear defaults");
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &ds.column_map(),
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build linear defaults");
        assert!(!terms.linear_terms.is_empty());
        assert!(
            terms.linear_terms.iter().all(|term| !term.double_penalty),
            "ordinary parametric effects must be unpenalized by default: {:?}",
            terms
                .linear_terms
                .iter()
                .map(|term| (&term.name, term.double_penalty))
                .collect::<Vec<_>>()
        );

        // `bounded()` is an exact interval transform and likewise defaults to
        // no shrinkage ridge. It also structurally rejects combining the
        // interval geometry with `double_penalty`.
        let bounded_parsed =
            parse_formula("y ~ bounded(z, min=-2, max=2)").expect("parse bounded defaults");
        let mut bounded_notes = Vec::new();
        let bounded_terms = build_termspec(
            &bounded_parsed.terms,
            &ds,
            &ds.column_map(),
            &mut bounded_notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build bounded defaults");
        assert_eq!(bounded_terms.linear_terms.len(), 1);
        assert!(
            !bounded_terms.linear_terms[0].double_penalty,
            "bounded() must default double_penalty=false since it cannot combine with the interval transform"
        );

        for formula in [
            "y ~ linear(x, double_penalty=true)",
            "y ~ linear(x:z, double_penalty=true)",
        ] {
            let parsed = parse_formula(formula).expect("parse explicit linear shrinkage");
            let mut notes = Vec::new();
            let terms = build_termspec(
                &parsed.terms,
                &ds,
                &ds.column_map(),
                &mut notes,
                &gam_runtime::resource::ResourcePolicy::default_library(),
            )
            .unwrap_or_else(|error| panic!("{formula} must build: {error}"));
            assert_eq!(terms.linear_terms.len(), 1, "{formula}");
            assert!(
                terms.linear_terms[0].double_penalty,
                "{formula} must preserve the explicit shrinkage opt-in"
            );
        }

        assert!(
            parse_formula("y ~ linear(x, double_penalty=ture)").is_err(),
            "a misspelled opt-in must be rejected instead of silently using the default"
        );
    }

    #[test]
    fn tensor_smooths_default_to_joint_null_recovery_with_explicit_opt_out() {
        let ds = continuous_dataset(
            &["y", "x", "z"],
            (0..36)
                .map(|i| {
                    let x = i as f64 / 35.0;
                    let z = ((i * 11) % 36) as f64 / 35.0;
                    vec![x * z, x, z]
                })
                .collect(),
        );
        let col_map = ds.column_map();
        for constructor in ["te", "ti", "t2"] {
            for (option, expected) in [("", true), (", double_penalty=false", false)] {
                let formula = format!("y ~ {constructor}(x, z{option})");
                let parsed = parse_formula(&formula).expect("parse tensor default");
                let mut notes = Vec::new();
                let terms = build_termspec(
                    &parsed.terms,
                    &ds,
                    &col_map,
                    &mut notes,
                    &gam_runtime::resource::ResourcePolicy::default_library(),
                )
                .unwrap_or_else(|error| panic!("{formula} must build: {error}"));
                let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis
                else {
                    panic!("{formula} must lower to TensorBSpline");
                };
                assert_eq!(spec.double_penalty, expected, "{formula}");
            }
        }
    }

    #[test]
    fn univariate_ps_small_k_degree_reduces_through_build(/* gam#1130 */) {
        // mgcv accepts `s(x, bs="ps", k=3)` (and the default cubic-regression
        // `s(x, k=3)`) by silently reducing the cubic basis to a quadratic.
        // The univariate ps/bspline build path used to reject this with
        // "k too small for degree 3"; it must now lower to a degree-2 basis
        // with zero internal knots (num_basis = k = 3), matching the te(...)
        // margin behaviour fixed in b75f55a91. Verified across the ps alias
        // and the default (cr) selector that both route through
        // parse_ps_internal_knots.
        let ds = continuous_dataset(
            &["y", "x"],
            (0..32)
                .map(|i| {
                    let x = i as f64 / 31.0;
                    vec![x * x, x]
                })
                .collect(),
        );
        let col_map = ds.column_map();

        for formula in ["y ~ s(x, bs='ps', k=3)", "y ~ s(x, k=3)"] {
            let parsed = parse_formula(formula).expect("parse small-k ps/cr smooth");
            let mut notes = Vec::new();
            let terms = build_termspec(
                &parsed.terms,
                &ds,
                &col_map,
                &mut notes,
                &gam_runtime::resource::ResourcePolicy::default_library(),
            )
            .unwrap_or_else(|err| {
                panic!("`{formula}` must degree-reduce, not error; got: {err:?}")
            });
            let SmoothBasisSpec::BSpline1D { spec, .. } = &terms.smooth_terms[0].basis else {
                panic!(
                    "`{formula}` must lower to a BSpline1D; got {:?}",
                    terms.smooth_terms[0].basis
                );
            };
            assert_eq!(
                spec.degree, 2,
                "`{formula}` must drop the cubic default to a quadratic basis"
            );
            let num_internal = match &spec.knotspec {
                BSplineKnotSpec::Generate {
                    num_internal_knots, ..
                } => *num_internal_knots,
                BSplineKnotSpec::Automatic {
                    num_internal_knots: Some(n),
                    ..
                } => *n,
                other => panic!("`{formula}` unexpected knotspec: {other:?}"),
            };
            assert_eq!(
                num_internal, 0,
                "`{formula}` must have zero internal knots (num_basis = k = 3)"
            );
            // Resulting basis dimension is num_internal + degree + 1 = 3 = k.
            assert!(
                spec.penalty_order >= 1 && spec.penalty_order <= spec.degree,
                "`{formula}` penalty_order {} must satisfy 1 <= order <= degree={}",
                spec.penalty_order,
                spec.degree
            );
        }
    }

    #[test]
    fn formula_shape_constraint_round_trips_and_rejects_bogus() {
        let ds = continuous_dataset(
            &["y", "x"],
            (0..32)
                .map(|i| {
                    let x = i as f64 / 31.0;
                    vec![x * x, x]
                })
                .collect(),
        );
        let col_map = ds.column_map();

        let parsed =
            parse_formula("y ~ s(x, shape=monotone_increasing)").expect("parse monotone smooth");
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("monotone smooth should build");
        assert_eq!(
            terms.smooth_terms[0].shape,
            ShapeConstraint::MonotoneIncreasing
        );

        let parsed_bad = parse_formula("y ~ s(x, shape=bogus)").expect("parse bogus shape");
        let mut notes_bad = Vec::new();
        let err = build_termspec(
            &parsed_bad.terms,
            &ds,
            &col_map,
            &mut notes_bad,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect_err("bogus shape must error");
        assert!(
            format!("{err:?}").contains("unknown shape constraint"),
            "got: {err:?}"
        );
    }

    #[test]
    fn default_sphere_smooth_uses_spherical_farthest_point_centers() {
        let ds = continuous_dataset(
            &["y", "lat", "lon"],
            (0..24)
                .map(|i| {
                    let t = i as f64 / 24.0;
                    let lat = -60.0 + 120.0 * t;
                    let lon = -180.0 + 360.0 * ((7 * i) % 24) as f64 / 24.0;
                    vec![lat.to_radians().sin(), lat, lon]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ sphere(lat, lon)").expect("parse");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build sphere termspec");
        let SmoothBasisSpec::Sphere { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected sphere term");
        };
        assert!(matches!(
            spec.center_strategy,
            CenterStrategy::FarthestPoint { .. }
        ));
    }

    #[test]
    fn one_dimensional_duchon_defaults_to_scale_free_length_scale() {
        let ds = continuous_dataset(
            &["y", "x"],
            (0..32)
                .map(|i| {
                    let x = i as f64 / 31.0;
                    vec![(std::f64::consts::TAU * x).sin(), x]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ duchon(x)").expect("parse");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build default duchon termspec");
        let SmoothBasisSpec::Duchon { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected Duchon term");
        };
        assert_eq!(spec.length_scale, None);
        assert!(matches!(
            spec.center_strategy,
            CenterStrategy::Auto(ref inner)
                if matches!(
                    inner.as_ref(),
                    CenterStrategy::UniformGrid { .. }
                )
        ));
    }

    #[test]
    fn formula_duchon_default_does_not_enable_collocation_operators() {
        let ds = continuous_dataset(
            &["y", "x", "z"],
            (0..40)
                .map(|i| {
                    let x = (i as f64 / 39.0).fract();
                    let z = ((7 * i) as f64 / 39.0).fract();
                    vec![x + z, x, z]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ duchon(x, z)").expect("parse");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build default 2D duchon termspec");
        let SmoothBasisSpec::Duchon { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected Duchon term");
        };
        assert!(matches!(
            spec.operator_penalties.mass,
            OperatorPenaltySpec::Disabled
        ));
        assert!(matches!(
            spec.operator_penalties.tension,
            OperatorPenaltySpec::Disabled
        ));
        assert!(matches!(
            spec.operator_penalties.stiffness,
            OperatorPenaltySpec::Disabled
        ));
    }

    #[test]
    fn one_dimensional_duchon_length_scale_opts_into_hybrid_mode() {
        let ds = continuous_dataset(
            &["y", "x"],
            (0..32)
                .map(|i| {
                    let x = i as f64 / 31.0;
                    vec![(std::f64::consts::TAU * x).sin(), x]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ duchon(x, length_scale=0.25)").expect("parse");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build hybrid duchon termspec");
        let SmoothBasisSpec::Duchon { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected Duchon term");
        };
        assert_eq!(spec.length_scale, Some(0.25));
    }

    #[test]
    fn multidimensional_duchon_default_uses_low_rank_mgcv_sized_basis() {
        let ds = continuous_dataset(
            &["y", "x1", "x2"],
            (0..500)
                .map(|i| {
                    let x1 = 2.0 * (i as f64 / 499.0) - 1.0;
                    let x2 = (((37 * i) % 500) as f64 / 499.0) * 2.0 - 1.0;
                    vec![(2.0 * x1).sin() + (1.5 * x2).cos(), x1, x2]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ duchon(x1, x2)").expect("parse");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build default 2D duchon termspec");
        let SmoothBasisSpec::Duchon { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected Duchon term");
        };
        let CenterStrategy::Auto(inner) = &spec.center_strategy else {
            panic!("expected auto center strategy");
        };
        assert!(matches!(
            inner.as_ref(),
            CenterStrategy::FarthestPoint { num_centers: 30 }
        ));
    }

    #[test]
    fn spectral_duchon_reproduces_fixed_seed_uniform_landmarks() {
        let ds = continuous_dataset(
            &["y", "x1", "x2", "x3", "x4"],
            (0..64)
                .map(|i| {
                    let x = i as f64 / 63.0;
                    vec![
                        x.sin(),
                        x,
                        (3.0 * x).sin(),
                        (5.0 * x).cos(),
                        (7.0 * x).sin(),
                    ]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ duchon(x1, x2, x3, x4, rank=6, order=0)").expect("parse");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build spectral Duchon termspec");
        let SmoothBasisSpec::Duchon { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected Duchon term");
        };
        let CenterStrategy::DuchonSpectral { knots, basis } = &spec.center_strategy else {
            panic!("expected spectral center strategy");
        };
        assert_eq!(basis.rank(), 6);
        let CenterStrategy::UserProvided(centers) = knots.as_ref() else {
            panic!("expected frozen sampled centers");
        };
        assert_eq!(centers.dim(), (64, 4));
    }

    #[test]
    fn parse_matern_nu_accepts_equivalent_half_integer_forms() {
        let cases = [
            ("1/2", MaternNu::Half),
            (" 1 / 2 ", MaternNu::Half),
            (".5", MaternNu::Half),
            ("0.50", MaternNu::Half),
            ("half", MaternNu::Half),
            ("3 / 2", MaternNu::ThreeHalves),
            ("1.50", MaternNu::ThreeHalves),
            ("5 / 2", MaternNu::FiveHalves),
            ("2.500000000000", MaternNu::FiveHalves),
            ("7 / 2", MaternNu::SevenHalves),
            ("3.50", MaternNu::SevenHalves),
            ("9 / 2", MaternNu::NineHalves),
            ("4.50", MaternNu::NineHalves),
        ];
        for (raw, expected) in cases {
            let parsed = parse_matern_nu(raw).expect(raw);
            assert!(
                matches!(
                    (parsed, expected),
                    (MaternNu::Half, MaternNu::Half)
                        | (MaternNu::ThreeHalves, MaternNu::ThreeHalves)
                        | (MaternNu::FiveHalves, MaternNu::FiveHalves)
                        | (MaternNu::SevenHalves, MaternNu::SevenHalves)
                        | (MaternNu::NineHalves, MaternNu::NineHalves)
                ),
                "parsed {raw:?} as {parsed:?}, expected {expected:?}"
            );
        }
    }

    #[test]
    fn parse_matern_nu_rejects_unsupported_or_invalid_values() {
        for raw in ["1", "2", "11/2", "1/0", "nan", "fast"] {
            let err = parse_matern_nu(raw).expect_err(raw);
            assert!(
                err.contains("supported half-integer values"),
                "unexpected error for {raw:?}: {err}"
            );
        }
    }

    #[test]
    fn parse_ps_k_promotes_underexpressive_cubic_basis() {
        let mut opts = BTreeMap::new();
        opts.insert("k".to_string(), "4".to_string());
        let (internal, inferred, eff_degree) = parse_ps_internal_knots(&opts, 3, 20).expect("k=4");
        assert_eq!(internal, 2);
        assert_eq!(eff_degree, 3);
        assert!(!inferred);

        opts.insert("k".to_string(), "6".to_string());
        let (internal, inferred, eff_degree) = parse_ps_internal_knots(&opts, 3, 20).expect("k=6");
        assert_eq!(internal, 2);
        assert_eq!(eff_degree, 3);
        assert!(!inferred);

        opts.insert("k".to_string(), "10".to_string());
        let (internal, inferred, eff_degree) = parse_ps_internal_knots(&opts, 3, 20).expect("k=10");
        assert_eq!(internal, 6);
        assert_eq!(eff_degree, 3);
        assert!(!inferred);
    }

    #[test]
    fn parse_ps_internal_knots_drops_degree_for_small_k() {
        // mgcv's `s(x, bs="ps", k=3)` with the default cubic basis silently
        // reduces to a quadratic (`degree=2`) marginal. `k=3, degree=3`
        // should yield a quadratic basis with zero internal knots
        // (`num_basis = k = 3`).
        let mut opts = BTreeMap::new();
        opts.insert("k".to_string(), "3".to_string());
        let (internal, inferred, eff_degree) = parse_ps_internal_knots(&opts, 3, 20).expect("k=3");
        assert_eq!(eff_degree, 2);
        assert_eq!(internal, 0);
        assert!(!inferred);

        // `k=2` reduces to a linear (`degree=1`) marginal — the smallest
        // non-trivial spline basis.
        opts.insert("k".to_string(), "2".to_string());
        let (internal, inferred, eff_degree) = parse_ps_internal_knots(&opts, 3, 20).expect("k=2");
        assert_eq!(eff_degree, 1);
        assert_eq!(internal, 0);
        assert!(!inferred);

        // The under-2 case is structurally under-specified and rejected even
        // by the degree-reducing variant: no B-spline basis has fewer than
        // two functions.
        opts.insert("k".to_string(), "1".to_string());
        let err = parse_ps_internal_knots(&opts, 3, 20)
            .expect_err("k=1 is below the irreducible spline floor");
        assert!(err.contains("requires k >= 2"), "unexpected error: {err}");

        // When the user already passed `k >= degree+1`, the helper must
        // preserve the existing knot geometry exactly.
        opts.insert("k".to_string(), "4".to_string());
        let (internal, inferred, eff_degree) = parse_ps_internal_knots(&opts, 3, 20).expect("k=4");
        assert_eq!(eff_degree, 3);
        assert_eq!(internal, 2);
        assert!(!inferred);
    }

    #[test]
    fn factor_smooth_marginal_degree_reduces_for_small_k() {
        let ds = factor_dataset();
        let col_map = ds.column_map();

        for (k, expected_degree) in [(3usize, 2usize), (2usize, 1usize)] {
            let parsed =
                parse_formula(&format!("y ~ s(x, g, bs=fs, k={k})")).expect("parse factor smooth");
            let mut notes = Vec::new();
            let terms = build_termspec(
                &parsed.terms,
                &ds,
                &col_map,
                &mut notes,
                &gam_runtime::resource::ResourcePolicy::default_library(),
            )
            .unwrap_or_else(|err| panic!("fs k={k} should degree-reduce, got: {err:?}"));
            let SmoothBasisSpec::FactorSmooth { spec } = &terms.smooth_terms[0].basis else {
                panic!(
                    "expected factor smooth, got {:?}",
                    terms.smooth_terms[0].basis
                );
            };
            assert_eq!(spec.marginal.degree, expected_degree);
            assert!(
                spec.marginal.penalty_order <= spec.marginal.degree,
                "penalty_order {} must be clamped to degree {}",
                spec.marginal.penalty_order,
                spec.marginal.degree
            );
            let basis_size = match spec.marginal.knotspec {
                BSplineKnotSpec::Generate {
                    num_internal_knots, ..
                } => num_internal_knots + spec.marginal.degree + 1,
                BSplineKnotSpec::Automatic {
                    num_internal_knots: Some(num_internal_knots),
                    ..
                } => num_internal_knots + spec.marginal.degree + 1,
                ref other => panic!("unexpected factor-smooth knotspec: {other:?}"),
            };
            assert_eq!(basis_size, k);
        }
    }

    /// Build a dataset with a ternary continuous covariate `x ∈ {0,1,2}` and a
    /// 2-level categorical group `g`, for the low-cardinality cr-cap tests.
    fn ternary_factor_dataset() -> Dataset {
        let rows = (0..120)
            .map(|i| {
                let x = (i % 3) as f64;
                let g = (i % 2) as f64;
                vec![x + g, x, g]
            })
            .collect::<Vec<_>>();
        Dataset {
            headers: vec!["y".into(), "x".into(), "g".into()],
            values: Array2::from_shape_vec(
                (rows.len(), 3),
                rows.into_iter().flat_map(|row| row.into_iter()).collect(),
            )
            .expect("rectangular ternary factor test data"),
            schema: DataSchema {
                columns: vec![
                    SchemaColumn {
                        name: "y".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "x".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "g".into(),
                        kind: ColumnKindTag::Categorical,
                        levels: vec!["a".into(), "b".into()],
                    },
                ],
            },
            column_kinds: vec![
                ColumnKindTag::Continuous,
                ColumnKindTag::Continuous,
                ColumnKindTag::Categorical,
            ],
        }
    }

    #[test]
    fn univariate_cr_smooth_caps_knots_to_data_support() {
        // #1541: `s(x, bs=cr, k=10)` on a ternary covariate (3 distinct values)
        // must NOT hard-fail in cr-knot selection ("cubic regression spline with
        // k=10 requires at least 10 distinct values, got 3"). The cr basis is
        // capped to the data support — exactly 3 value-knots at {0,1,2} — which
        // is full-rank for the data, so it can still represent any 3 group means.
        let ds = continuous_dataset(
            &["y", "x"],
            (0..90)
                .map(|i| vec![(i % 3) as f64, (i % 3) as f64])
                .collect(),
        );
        let col_map = ds.column_map();
        let parsed = parse_formula("y ~ s(x, bs=cr, k=10)").expect("parse cr smooth");
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("cr k=10 must cap to data support instead of erroring");
        let SmoothBasisSpec::BSpline1D { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected BSpline1D for s(x, bs=cr)");
        };
        let BSplineKnotSpec::NaturalCubicRegression { knots } = &spec.knotspec else {
            panic!("expected cr knotspec, got {:?}", spec.knotspec);
        };
        // Capped to exactly the 3 distinct covariate values.
        assert_eq!(knots.len(), 3, "cr basis not capped to 3 distinct values");
        assert_eq!(knots.as_slice().unwrap(), &[0.0, 1.0, 2.0]);
        // The reduction is surfaced to the user (mgcv warns in the same case).
        assert!(
            notes.iter().any(|n| n.contains("data-support cap")),
            "cap not reported in inference notes: {notes:?}"
        );
    }

    #[test]
    fn univariate_cr_smooth_binary_covariate_degrades_to_bspline() {
        // #1541: a BINARY covariate has too few distinct values (2) for ANY cr
        // spline (needs >= 3 distinct). `s(x, bs=cr)` must degrade to a B-spline
        // marginal — the default basis the same data already fits — NOT hard-fail.
        let ds = continuous_dataset(
            &["y", "x"],
            (0..80)
                .map(|i| vec![(i % 2) as f64, (i % 2) as f64])
                .collect(),
        );
        let col_map = ds.column_map();
        let parsed = parse_formula("y ~ s(x, bs=cr, k=10)").expect("parse cr smooth");
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("binary cr must degrade to B-spline instead of erroring");
        let SmoothBasisSpec::BSpline1D { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected BSpline1D for s(x, bs=cr)");
        };
        assert!(
            !matches!(
                spec.knotspec,
                BSplineKnotSpec::NaturalCubicRegression { .. }
            ),
            "binary covariate must NOT build a cr basis, got {:?}",
            spec.knotspec
        );
        assert!(
            notes
                .iter()
                .any(|n| n.contains("Degraded to the linear B-spline")),
            "degradation not reported in inference notes: {notes:?}"
        );
    }

    #[test]
    fn sz_factor_smooth_low_cardinality_uses_bspline_marginal() {
        // #1605: the `sz` factor-smooth marginal is the SAME penalized B-spline
        // the `fs` sibling uses — NOT a natural cubic regression (`cr`) marginal,
        // whose hard natural boundary conditions f''=0 bias curved deviations
        // (a consistency failure). #1542 (the reason this test exists) is
        // subsumed: with a B-spline marginal a low-cardinality covariate no
        // longer needs a special cr data-support cap and can never hard-fail the
        // way the old cr-marginal `sz` spelling did — the build just succeeds,
        // exactly as `fs` already does on the identical data.
        let ds = ternary_factor_dataset();
        let col_map = ds.column_map();
        let parsed = parse_formula("y ~ s(x, g, bs=sz, k=10)").expect("parse sz factor smooth");
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("sz on a ternary covariate must build (B-spline marginal), not hard-fail");
        let SmoothBasisSpec::FactorSmooth { spec } = &terms.smooth_terms[0].basis else {
            panic!("expected FactorSmooth for s(x, g, bs=sz)");
        };
        assert!(
            !matches!(
                spec.marginal.knotspec,
                BSplineKnotSpec::NaturalCubicRegression { .. }
            ),
            "sz marginal must be a B-spline (curvature-capable), not the \
             natural-BC cr basis; got {:?}",
            spec.marginal.knotspec
        );
    }

    /// A dataset with a genuinely continuous covariate `x` (many distinct
    /// values) and a `L`-level grouping factor `g`, suitable for building a
    /// real factor-smooth marginal with a non-trivial {const, linear} null
    /// space. `y` is unused by the structural penalty checks below.
    fn continuous_x_factor_dataset(n: usize, n_groups: usize) -> Dataset {
        let rows = (0..n)
            .map(|i| {
                let x = i as f64 / (n as f64 - 1.0);
                let g = (i % n_groups) as f64;
                vec![x + g, x, g]
            })
            .collect::<Vec<_>>();
        let levels: Vec<String> = (0..n_groups).map(|k| format!("g{k}")).collect();
        Dataset {
            headers: vec!["y".into(), "x".into(), "g".into()],
            values: Array2::from_shape_vec(
                (rows.len(), 3),
                rows.into_iter().flat_map(|row| row.into_iter()).collect(),
            )
            .expect("rectangular continuous-x factor data"),
            schema: DataSchema {
                columns: vec![
                    SchemaColumn {
                        name: "y".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "x".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "g".into(),
                        kind: ColumnKindTag::Categorical,
                        levels,
                    },
                ],
            },
            column_kinds: vec![
                ColumnKindTag::Continuous,
                ColumnKindTag::Continuous,
                ColumnKindTag::Categorical,
            ],
        }
    }

    fn factor_smooth_spec_for(formula: &str, ds: &Dataset) -> FactorSmoothSpec {
        let col_map = ds.column_map();
        let parsed = parse_formula(formula).expect("parse factor smooth formula");
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build factor smooth term");
        let SmoothBasisSpec::FactorSmooth { spec } = &terms.smooth_terms[0].basis else {
            panic!("expected FactorSmooth basis for `{formula}`");
        };
        spec.clone()
    }

    /// #1605: the sum-to-zero factor smooth `s(x, g, bs="sz")` under-fit data
    /// drawn from its own model class because its deviation blocks carried ONLY
    /// the marginal wiggliness penalty — the {const, linear} null space of every
    /// deviation curve was left completely unpenalized, so the single combined
    /// wiggliness λ could not separate per-group intercept/slope variance from
    /// curvature variance and REML parked it over-smoothed (same defect class as
    /// the closed #700, more severe). mgcv's `bs="fs"` sibling avoids the gap by
    /// adding a SEPARATE per-null-dimension ridge (one λ each), the
    /// double-penalty `I_L ⊗ S_j` structure. The fix gives `sz` the same
    /// null-space-ridge structure, mapped into the zero-sum CONTRAST space so the
    /// constraint (and `sz`'s distinctness from `fs`) is preserved.
    ///
    /// This pins the structural defect: after the fix the `sz` deviation build
    /// must carry MORE than just its wiggliness penalty(s) — exactly one extra
    /// null-space-ridge penalty per marginal null direction, matching the count
    /// that `fs` carries — while keeping the narrower `(L-1)·p` zero-sum design
    /// (NOT the `L·p` full-rank `fs` design). Before the fix `sz` carried only
    /// the wiggliness penalties and this fails.
    #[test]
    fn sz_factor_smooth_carries_null_space_ridge_like_fs() {
        let ds = continuous_x_factor_dataset(180, 4);
        let mut workspace = crate::basis::BasisWorkspace::new();

        let sz_spec = factor_smooth_spec_for("y ~ s(x, g, bs=sz, k=8)", &ds);
        let sz_built = crate::smooth::build_factor_smooth(
            ds.values.view(),
            &sz_spec,
            "sz_term",
            &mut workspace,
        )
        .expect("build sz factor smooth");

        let fs_spec = factor_smooth_spec_for("y ~ s(x, g, bs=fs, k=8)", &ds);
        let fs_built = crate::smooth::build_factor_smooth(
            ds.values.view(),
            &fs_spec,
            "fs_term",
            &mut workspace,
        )
        .expect("build fs factor smooth");

        // Penalty structure (#1074 + #1605). `fs` is the exchangeable
        // random-effect smooth: all `L` level blocks share ONE wiggliness λ per
        // marginal penalty, plus one rank-1 null-space ridge per marginal null
        // direction (the #1605 double penalty). `sz` is the sum-to-zero factor
        // smooth and mgcv's `smooth.construct.sz` emits ONE penalty matrix PER
        // LEVEL — `L` independent curvature smoothing parameters — so REML can
        // shrink a low-amplitude group's deviation hard while leaving a busy
        // group nearly unpenalized. We mirror that: the single marginal
        // wiggliness penalty is split into its `L` independent zero-sum-contrast
        // summands (`L-1` free per-group blocks `(e_k e_kᵀ)⊗S` + the reference
        // coupling block `(11ᵀ)⊗S`), each carrying its own λ, and the null-space
        // ridges stay POOLED (the per-group intercept/slope shrinkage mgcv pools
        // under one variance even for `sz`).
        //
        // So with `nw` marginal wiggliness penalties and `nn` marginal null
        // directions: fs has `nw + nn` penalties; sz has `L·nw + nn`. sz must
        // therefore carry strictly MORE penalties than fs (the per-group split),
        // and the surplus must be exactly `(L-1)·nw`.
        let n_levels = sz_spec
            .group_frozen_levels
            .as_ref()
            .map(|l| l.len())
            .unwrap_or(4);
        assert!(n_levels >= 3, "test needs >=3 groups, got {n_levels}");

        // fs = nw + nn  ⇒  nn = fs_penalties - nw. The marginal has nw==1
        // wiggliness penalty (a single difference/curvature operator), so the
        // per-group split adds exactly (L-1)·nw = (L-1) extra penalties on top of
        // fs's count.
        let nw = 1usize; // one marginal wiggliness penalty for the B-spline marginal
        let expected_sz = fs_built.active_penalties.len() + (n_levels - 1) * nw;
        assert_eq!(
            sz_built.active_penalties.len(),
            expected_sz,
            "sz must split its wiggliness penalty per level (#1074): expected \
             fs_count {} + (L-1)·nw {} = {}, but sz had {}",
            fs_built.active_penalties.len(),
            (n_levels - 1) * nw,
            expected_sz,
            sz_built.active_penalties.len(),
        );
        assert!(
            sz_built.active_penalties.len() > fs_built.active_penalties.len(),
            "sz must carry strictly more penalties than fs after the per-group \
             split (sz={}, fs={})",
            sz_built.active_penalties.len(),
            fs_built.active_penalties.len(),
        );

        // The null-space ridges must still be present (the #1605 property that
        // keeps the deviation curvature un-over-smoothed). After removing the `L`
        // per-group wiggliness blocks, the remainder are the pooled null ridges,
        // and there must be at least one (a B-spline marginal has a non-empty
        // {const, linear} null space).
        let n_wiggliness = n_levels * nw; // L per-group blocks
        assert!(
            sz_built.active_penalties.len() > n_wiggliness,
            "sz deviation block carries no null-space ridge (penalties={}, \
             wiggliness blocks={}); the null space is unpenalized and REML \
             over-smooths the deviations",
            sz_built.active_penalties.len(),
            n_wiggliness,
        );

        // The zero-sum constraint must be preserved: the sz design must stay the
        // NARROWER `(L-1)·p` contrast design, strictly narrower than the fs
        // full-rank `L·p` design. This guards against "fixing" sz by making it
        // identical to fs (which would break identifiability / sum-to-zero).
        assert!(
            sz_built.dim < fs_built.dim,
            "sz design width {} must be strictly less than fs width {} \
             (zero-sum contrast drops one level block)",
            sz_built.dim,
            fs_built.dim,
        );

        for penalty in &sz_built.active_penalties {
            assert_eq!(
                penalty
                    .null_eigenvectors
                    .as_ref()
                    .map_or(0, |basis| basis.ncols()),
                penalty.nullity
            );
        }
    }

    #[test]
    fn sz_penalty_metadata_is_emitted_in_matrix_order_2289() {
        let ds = continuous_x_factor_dataset(180, 4);
        let mut workspace = crate::basis::BasisWorkspace::new();
        let spec = factor_smooth_spec_for("y ~ s(x, g, bs=sz, k=8, double_penalty=true)", &ds);
        let built = crate::smooth::build_factor_smooth(
            ds.values.view(),
            &spec,
            "sz_metadata_order",
            &mut workspace,
        )
        .expect("build multi-penalty sz smooth");
        let n_levels = spec.group_frozen_levels.as_ref().map(Vec::len).unwrap_or(4);

        assert!(built.active_penalties.len() >= 2 * n_levels);
        for (idx, penalty) in built.active_penalties.iter().enumerate() {
            let analysis =
                crate::basis::analyze_penalty_block(&penalty.matrix).expect("PSD penalty");
            assert_eq!(penalty.info.original_index, idx);
            assert_eq!(penalty.info.effective_rank, analysis.rank, "penalty {idx}");
            assert_eq!(penalty.nullity, analysis.nullity, "penalty {idx}");
        }
        assert!(
            built.active_penalties[..n_levels]
                .iter()
                .all(|penalty| matches!(penalty.info.source, PenaltySource::Primary))
        );
        assert!(
            built.active_penalties[n_levels..2 * n_levels]
                .iter()
                .all(|penalty| matches!(
                    penalty.info.source,
                    PenaltySource::DoublePenaltyNullspace
                ))
        );
    }

    /// #1457: `y ~ s(x, by=g) + g` with a BARE categorical `g` must NOT lower to
    /// two `g` design blocks. The bare `+ g` is auto-promoted to a single
    /// penalized random-effect block owning the factor's full level offsets; the
    /// `by=` branch must then recognize that owner and skip adding its own
    /// unpenalized treatment-coded main effect. Before the fix the dedup guard
    /// recognized only explicit `group(g)` (a `ParsedTerm::RandomEffect`), so the
    /// auto-promoted bare-`+ g` block slipped past and a spurious second `g`
    /// block (plus an extra smoothing parameter) was added. Assert exactly ONE
    /// `g` random/categorical block, and that adding the bare `+ g` introduces no
    /// extra `g` blocks beyond `y ~ s(x, by=g)` alone.
    fn factor_dataset_l3() -> Dataset {
        // `g` is categorical with THREE levels (encoded 0.0/1.0/2.0).
        let rows = (0..30)
            .map(|i| {
                let x = i as f64 / 29.0;
                let g = (i % 3) as f64;
                vec![x + g, x, g]
            })
            .collect::<Vec<_>>();
        Dataset {
            headers: vec!["y".into(), "x".into(), "g".into()],
            values: Array2::from_shape_vec(
                (rows.len(), 3),
                rows.into_iter().flat_map(|row| row.into_iter()).collect(),
            )
            .expect("rectangular L=3 factor test data"),
            schema: DataSchema {
                columns: vec![
                    SchemaColumn {
                        name: "y".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "x".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "g".into(),
                        kind: ColumnKindTag::Categorical,
                        levels: vec!["a".into(), "b".into(), "c".into()],
                    },
                ],
            },
            column_kinds: vec![
                ColumnKindTag::Continuous,
                ColumnKindTag::Continuous,
                ColumnKindTag::Categorical,
            ],
        }
    }

    #[test]
    fn factor_by_smooth_plus_bare_categorical_does_not_duplicate_factor_block() {
        let ds = factor_dataset_l3();
        let col_map = ds.column_map();

        let g_blocks = |formula: &str| -> usize {
            let parsed = parse_formula(formula).expect("parse by-smooth formula");
            let mut notes = Vec::new();
            let terms = build_termspec(
                &parsed.terms,
                &ds,
                &col_map,
                &mut notes,
                &ResourcePolicy::default_library(),
            )
            .unwrap_or_else(|err| panic!("`{formula}` must build, got: {err:?}"));
            terms
                .random_effect_terms
                .iter()
                .filter(|rt| rt.name == "g")
                .count()
        };

        // Baseline: the standalone factor-by smooth carries exactly ONE `g`
        // block (the unpenalized treatment-coded factor main effect added by the
        // `by=` branch).
        let by_only = g_blocks("y ~ s(x, by=g, k=10)");
        assert_eq!(
            by_only, 1,
            "`y ~ s(x, by=g)` must produce exactly one `g` design block"
        );

        // The bug: adding a bare `+ g` (auto-promoted to a penalized random
        // block owning the same level offsets) must NOT introduce a second `g`
        // block. Before the fix this was 2.
        let by_plus_bare = g_blocks("y ~ s(x, by=g, k=10) + g");
        assert_eq!(
            by_plus_bare, 1,
            "`y ~ s(x, by=g) + g` must collapse to ONE `g` block (#1457): the bare \
             `+ g` already owns the factor's level offsets, so the `by=` branch \
             must not add a second, treatment-coded main effect"
        );

        // The bare `+ g` adds no spurious extra `g` block versus the baseline.
        assert_eq!(
            by_plus_bare, by_only,
            "the bare `+ g` collision must add zero extra `g` blocks (#1457)"
        );
    }

    #[test]
    fn factor_by_penalties_carry_full_expanded_null_geometry_2293() {
        let ds = factor_dataset_l3();
        let col_map = ds.column_map();
        // Leave the marginal null space unshrunk so every level-specific term
        // must carry a non-trivial joint-null chart. The production default is
        // double-penalized, whose primary and null-space ridge have a full-rank
        // joint sum and therefore correctly produce no joint-null rotation.
        let parsed =
            parse_formula("y ~ s(x, by=g, k=8, double_penalty=false)").expect("parse by smooth");
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &ResourcePolicy::default_library(),
        )
        .expect("build by smooth spec");
        assert_eq!(terms.smooth_terms.len(), 3, "one smooth per factor level");

        // Formula construction represents an unordered factor-by smooth as one
        // explicit level-gated term per factor level. Validate the complete
        // realized expansion, rather than inspecting only its first level or
        // assuming the legacy monolithic BySmooth::Factor representation.
        for term in &terms.smooth_terms {
            assert!(matches!(
                &term.basis,
                SmoothBasisSpec::ByVariable {
                    by: ByVariableSpec::Level { .. },
                    ..
                }
            ));
            let mut workspace = crate::basis::BasisWorkspace::new();
            let built = crate::smooth::build_single_local_smooth_term(
                ds.values.view(),
                term,
                &mut workspace,
            )
            .expect("build level-gated factor-by smooth");

            for (idx, penalty) in built.active_penalties.iter().enumerate() {
                let analysis =
                    crate::basis::analyze_penalty_block(&penalty.matrix).expect("PSD block");
                assert_eq!(analysis.rank + penalty.nullity, built.dim, "penalty {idx}");
                assert_eq!(analysis.nullity, penalty.nullity, "penalty {idx}");
                assert_eq!(penalty.info.effective_rank, analysis.rank);
                let basis = penalty
                    .null_eigenvectors
                    .as_ref()
                    .expect("nontrivial factor-level null basis");
                assert_eq!(basis.nrows(), built.dim);
                assert_eq!(basis.ncols(), penalty.nullity);
            }
            let joint = built
                .joint_null_rotation
                .as_ref()
                .expect("factor-level joint null geometry");
            assert!(joint.joint_nullity > 0);
            assert_eq!(joint.rotation.nrows(), built.dim);
            assert_eq!(joint.rotation.ncols(), built.dim);
        }
    }

    #[test]
    fn parse_tensor_periods_and_origins_aliases() {
        let mut opts = BTreeMap::new();
        opts.insert(
            "boundary".to_string(),
            "['periodic', 'periodic']".to_string(),
        );
        opts.insert("periods".to_string(), "[7, 24]".to_string());
        opts.insert("origins".to_string(), "[0, -12]".to_string());
        let axes = parse_periodic_axes(&opts, 2).expect("axes");
        let periods = parse_periods(&opts, &axes).expect("periods");
        let origins = parse_period_origins(&opts, &axes).expect("origins");
        assert_eq!(axes, vec![true, true]);
        assert_eq!(periods, vec![Some(7.0), Some(24.0)]);
        assert_eq!(origins, vec![Some(0.0), Some(-12.0)]);
    }

    #[test]
    fn tensor_smooth_honors_per_margin_k_list() {
        let ds = continuous_dataset(
            &["y", "theta", "h"],
            (0..20)
                .map(|i| {
                    let theta = std::f64::consts::TAU * i as f64 / 20.0;
                    let h = -1.0 + 2.0 * (i % 5) as f64 / 4.0;
                    vec![theta.cos() + h, theta, h]
                })
                .collect(),
        );
        let parsed = parse_formula(
            "y ~ te(theta, h, periodic=[0], period=[2*pi, None], origin=[0, None], k=[9,5])",
        )
        .expect("parse tensor formula");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build tensor terms");
        let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected tensor B-spline");
        };
        let dims = spec
            .marginalspecs
            .iter()
            .map(|m| match m.knotspec {
                BSplineKnotSpec::PeriodicUniform { num_basis, .. } => num_basis,
                BSplineKnotSpec::Generate {
                    num_internal_knots, ..
                } => num_internal_knots + m.degree + 1,
                // The mgcv-default `cr` margin (#1074) reports its basis size as
                // the number of value-knots placed.
                BSplineKnotSpec::NaturalCubicRegression { ref knots } => knots.len(),
                _ => panic!("unexpected tensor marginal knotspec"),
            })
            .collect::<Vec<_>>();
        assert_eq!(dims, vec![9, 5]);
    }

    #[test]
    fn tensor_smooth_honors_per_margin_k_axis_aliases() {
        let ds = continuous_dataset(
            &["resp", "x", "y"],
            (0..12)
                .map(|i| {
                    let t = i as f64 / 11.0;
                    vec![t, t, 1.0 - t]
                })
                .collect(),
        );
        assert_eq!(
            tensor_margin_basis_sizes(&ds, "resp ~ te(x, y, k_x=9, k_y=5)"),
            vec![9, 5],
            "k_<margin> aliases should materialize requested per-margin values"
        );
    }

    #[test]
    fn tensor_smooth_low_cardinality_axis_falls_back_to_lower_degree_basis() {
        // mgcv-style: `te(x, b, k=c(5, 2))` with a BINARY second margin (only
        // values {0, 1}) is a legitimate request — the binary axis can hold at
        // most a 2-function linear basis. We must NOT reject k=2 with a
        // "k too small for degree 3" config error; instead, drop the spline
        // degree on the binary axis to k_axis - 1 (here 1, linear) while
        // keeping the continuous margin at the requested degree=3, k=5.
        let ds = continuous_dataset(
            &["y", "x", "b"],
            (0..40)
                .map(|i| {
                    let x = i as f64 / 39.0;
                    let b = (i % 2) as f64;
                    vec![x.sin() + 0.5 * b, x, b]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ te(x, b, k=[5, 2])").expect("parse tensor with k=[5,2]");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build tensor with binary margin");
        let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected tensor B-spline for te(x, b)");
        };
        // Continuous margin keeps requested degree=3 and k=5; binary margin
        // drops to degree=1 (linear) so the requested k=2 yields exactly two
        // basis functions before tensor-product identifiability is applied.
        let continuous = &spec.marginalspecs[0];
        let binary = &spec.marginalspecs[1];
        assert_eq!(continuous.degree, 3);
        assert_eq!(binary.degree, 1);
        assert!(
            binary.penalty_order >= 1 && binary.penalty_order <= binary.degree,
            "binary margin penalty_order {} must satisfy 1 <= order <= degree={}",
            binary.penalty_order,
            binary.degree
        );
        let basis_size = |m: &BSplineBasisSpec| match m.knotspec {
            BSplineKnotSpec::PeriodicUniform { num_basis, .. } => num_basis,
            BSplineKnotSpec::Generate {
                num_internal_knots, ..
            } => num_internal_knots + m.degree + 1,
            BSplineKnotSpec::Automatic {
                num_internal_knots: Some(n),
                ..
            } => n + m.degree + 1,
            // The mgcv-default `cr` margin (#1074) reports its basis size as the
            // number of value-knots placed.
            BSplineKnotSpec::NaturalCubicRegression { ref knots } => knots.len(),
            _ => panic!("unexpected tensor marginal knotspec"),
        };
        assert_eq!(basis_size(continuous), 5);
        assert_eq!(basis_size(binary), 2);
    }

    #[test]
    fn tensor_smooth_uniform_k_is_capped_to_a_low_cardinality_margins_distinct_values() {
        // Regression: a SINGLE `k=5` applied to every axis of `te(x, b, k=5)`
        // with a BINARY second margin (`b ∈ {0, 1}`) must build a valid tensor,
        // NOT hard-fail in cr-knot selection ("cubic regression spline with k=5
        // requires at least 5 distinct values, got 2"). mgcv caps a margin's
        // basis to its data support; the binary axis becomes the 2-function
        // (linear) margin, while the continuous axis keeps the requested k=5.
        // This is the `te(age, badh, k=5)` real-data case that previously errored.
        let ds = continuous_dataset(
            &["y", "x", "b"],
            (0..40)
                .map(|i| {
                    let x = i as f64 / 39.0;
                    let b = (i % 2) as f64;
                    vec![x.sin() + 0.5 * b, x, b]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ te(x, b, k=5)").expect("parse tensor with uniform k=5");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("uniform k=5 must auto-cap the binary margin instead of erroring");
        let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected tensor B-spline for te(x, b)");
        };
        let basis_size = |m: &BSplineBasisSpec| match &m.knotspec {
            BSplineKnotSpec::PeriodicUniform { num_basis, .. } => *num_basis,
            BSplineKnotSpec::Generate {
                num_internal_knots, ..
            } => num_internal_knots + m.degree + 1,
            BSplineKnotSpec::Automatic {
                num_internal_knots: Some(n),
                ..
            } => n + m.degree + 1,
            BSplineKnotSpec::NaturalCubicRegression { knots } => knots.len(),
            other => panic!("unexpected tensor marginal knotspec: {other:?}"),
        };
        let binary = &spec.marginalspecs[1];
        // Binary margin is reduced to the 2-function linear basis its data
        // supports (k capped from 5 to 2, degree dropped to 1).
        assert_eq!(basis_size(binary), 2);
        assert_eq!(binary.degree, 1);
        // The continuous margin is unaffected by the cap (40 distinct values).
        assert_eq!(basis_size(&spec.marginalspecs[0]), 5);
    }

    #[test]
    fn tensor_all_tp_margins_with_per_margin_k_routes_to_bspline_tensor() {
        // `te(x1, x2, bs=c('tp','tp'), k=c(5,5))` is mgcv's per-margin tp tensor
        // with per-margin basis sizes — a tensor product of two 1-D bases, each
        // of dimension 5. The list-valued `k=c(5,5)` is honored by
        // `parse_tensor_k_list`, producing one penalized B-spline margin per axis
        // (each spanning the requested per-axis thin-plate function space). This
        // is the same anisotropic-tensor routing the scalar/no-`k` case takes —
        // a `te()` request is ALWAYS a tensor product, never a silent isotropic
        // thin-plate substitution.
        let ds = continuous_dataset(
            &["y", "x1", "x2"],
            (0..32)
                .map(|i| {
                    let t = i as f64 / 31.0;
                    vec![t.sin(), t, 1.0 - t]
                })
                .collect(),
        );
        let parsed =
            parse_formula("y ~ te(x1, x2, bs=c('tp','tp'), k=c(5,5))").expect("parse tensor");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build tensor terms with per-margin k");
        let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!(
                "expected B-spline tensor when k=c(5,5) is supplied with bs=c('tp','tp'), got {:?}",
                terms.smooth_terms[0].basis
            );
        };
        // Since #1074 a `tp` tensor margin (k >= 3) is realized as a
        // Lancaster–Salkauskas natural cubic-regression margin (cr basis
        // dimension == knot count), not an open `Generate` B-spline. It is
        // still a `TensorBSpline` spec with one penalized 1-D margin per axis,
        // so the routing assertion above still holds; only the per-margin
        // knotspec variant changed. The earlier `_ => panic!` arm pinned the
        // pre-#1074 `Generate`-only representation and is stale. Decode every
        // margin variant to its basis dimension (mirroring the
        // `tensor_margin_basis_sizes` helper).
        let dims = spec
            .marginalspecs
            .iter()
            .map(|m| match m.knotspec {
                BSplineKnotSpec::Generate {
                    num_internal_knots, ..
                } => num_internal_knots + m.degree + 1,
                BSplineKnotSpec::Automatic {
                    num_internal_knots: Some(num_internal_knots),
                    ..
                } => num_internal_knots + m.degree + 1,
                BSplineKnotSpec::PeriodicUniform { num_basis, .. } => num_basis,
                BSplineKnotSpec::Provided(ref knots) => knots.len().saturating_sub(m.degree + 1),
                BSplineKnotSpec::NaturalCubicRegression { ref knots } => knots.len(),
                BSplineKnotSpec::Automatic {
                    num_internal_knots: None,
                    ..
                } => panic!("test cannot infer automatic knot count"),
            })
            .collect::<Vec<_>>();
        assert_eq!(dims, vec![5, 5]);
    }

    #[test]
    fn tensor_all_tp_margins_without_per_margin_k_builds_anisotropic_tensor() {
        // `te(x1, x2, bs=c('tp','tp'))` is a tensor-product request and must
        // build a genuine anisotropic tensor product (one smoothing parameter
        // per margin), NOT a silently-substituted multi-D isotropic thin-plate
        // radial smooth — that would be a different model (`s(x1,x2,bs='tp')`).
        // The routing is now consistent whether or not `k` is list-valued: a tp
        // margin vector always realizes each axis as a 1-D penalized B-spline
        // margin spanning the same per-axis thin-plate function space (#1082).
        let ds = continuous_dataset(
            &["y", "x1", "x2"],
            (0..32)
                .map(|i| {
                    let t = i as f64 / 31.0;
                    vec![t.sin(), t, 1.0 - t]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ te(x1, x2, bs=c('tp','tp'))").expect("parse tensor");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build tensor terms without per-margin k");
        let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!(
                "te(...,bs=c('tp','tp')) must route to an anisotropic tensor product, not a \
                 silent isotropic thin-plate substitution; got {:?}",
                terms.smooth_terms[0].basis
            );
        };
        assert_eq!(
            spec.marginalspecs.len(),
            2,
            "tp tensor must carry one penalized B-spline margin per axis"
        );
    }

    #[test]
    fn explicit_basis_sizes_are_not_small_n_clamped() {
        let ds = continuous_dataset(
            &["y", "x1", "x2", "x3", "x4", "x5"],
            (0..12)
                .map(|i| {
                    let x = i as f64 / 11.0;
                    vec![x.sin(), x, x * x, x + 0.1, 1.0 - x, (2.0 * x).sin()]
                })
                .collect(),
        );
        let parsed = parse_formula("y ~ s(x1, k=10) + s(x2) + s(x3) + s(x4) + s(x5)")
            .expect("parse multi-smooth formula");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build multi-smooth terms");
        let SmoothBasisSpec::BSpline1D { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected first smooth to be B-spline");
        };
        assert!(matches!(
            &spec.knotspec,
            BSplineKnotSpec::Generate {
                num_internal_knots: 6,
                ..
            }
        ));
    }

    #[test]
    fn explicit_duchon_centers_are_not_small_n_bumped() {
        let ds = continuous_dataset(
            &["y", "x1", "x2", "x3", "x4", "x5"],
            (0..12)
                .map(|i| {
                    let x = i as f64 / 11.0;
                    vec![x.sin(), x, x * x, x + 0.1, 1.0 - x, (2.0 * x).sin()]
                })
                .collect(),
        );
        // Pure 1D Duchon at default options resolves the nullspace to Linear
        // (2s < d forces escalation), giving 2 polynomial nullspace columns;
        // the well-posedness gate requires num_centers > polynomial_cols, so
        // 3 is the smallest valid count. It is still well below the small-N
        // bump target of polynomial_cols + 4 = 6, so this exercises the
        // "explicit value is honored" path the test name advertises.
        let parsed = parse_formula("y ~ duchon(x1, centers=3) + s(x2) + s(x3) + s(x4) + s(x5)")
            .expect("parse multi-smooth formula");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &gam_runtime::resource::ResourcePolicy::default_library(),
        )
        .expect("build multi-smooth terms");
        let SmoothBasisSpec::Duchon { spec, .. } = &terms.smooth_terms[0].basis else {
            panic!("expected first smooth to be Duchon");
        };
        assert!(matches!(
            spec.center_strategy,
            CenterStrategy::UniformGrid { points_per_dim: 3 }
        ));
    }

    #[test]
    fn inferred_tensor_basis_cap_uses_coordinate_support_not_duplicate_rows() {
        let mut unique_rows = Vec::new();
        for i in 0..50 {
            let theta = i as f64 / 50.0;
            for j in 0..16 {
                let h = -1.0 + 2.0 * (j as f64) / 15.0;
                let y = theta.cos() + h;
                unique_rows.push(vec![y, theta, h]);
            }
        }
        let mut repeated_rows = Vec::new();
        for _ in 0..12 {
            repeated_rows.extend(unique_rows.iter().cloned());
        }

        let unique = continuous_dataset(&["y", "theta", "h"], unique_rows);
        let repeated = continuous_dataset(&["y", "theta", "h"], repeated_rows);

        let unique_basis = inferred_tensor_basis_product(&unique);
        let repeated_basis = inferred_tensor_basis_product(&repeated);

        assert_eq!(
            unique_basis, repeated_basis,
            "duplicating existing tensor coordinates must not inflate inferred basis width"
        );
    }

    #[test]
    fn inferred_three_dim_tensor_basis_stays_bounded_for_reml_selection() {
        // Regression for gam#813: the inferred per-margin k must be
        // dimension-aware so the 3-D tensor width p = ∏ k_d does not explode.
        // With the old 1-D-per-margin rule a 3-D `te` defaulted to 7³=343 at
        // small n and 20³=8000 at larger n, making the (non-Kronecker-factorable)
        // full-tensor sum-to-zero penalty's O(p³) REML reparameterization a
        // multi-minute stall. The dimension-aware budget keeps the product near
        // mgcv's te default (≈5³=125) regardless of n.
        let make = |n: usize| -> usize {
            let mut rows = Vec::with_capacity(n);
            for i in 0..n {
                let f = i as f64 / n as f64;
                rows.push(vec![f.sin(), f, (2.0 * f).cos(), (3.0 * f) % 1.0]);
            }
            let ds = continuous_dataset(&["y", "x1", "x2", "x3"], rows);
            let parsed = parse_formula("y ~ te(x1, x2, x3)").expect("parse 3-D tensor");
            let col_map = ds.column_map();
            let mut notes = Vec::new();
            let terms = build_termspec(
                &parsed.terms,
                &ds,
                &col_map,
                &mut notes,
                &ResourcePolicy::default_library(),
            )
            .expect("build 3-D tensor termspec");
            let SmoothBasisSpec::TensorBSpline { spec, .. } = &terms.smooth_terms[0].basis else {
                panic!("expected tensor smooth");
            };
            spec.marginalspecs
                .iter()
                .map(|m| match m.knotspec {
                    BSplineKnotSpec::Generate {
                        num_internal_knots, ..
                    } => num_internal_knots + m.degree + 1,
                    BSplineKnotSpec::Automatic {
                        num_internal_knots: Some(num_internal_knots),
                        ..
                    } => num_internal_knots + m.degree + 1,
                    // The mgcv-default `cr` margin (#1074) reports its basis size
                    // as the number of value-knots placed.
                    BSplineKnotSpec::NaturalCubicRegression { ref knots } => knots.len(),
                    _ => panic!("unexpected tensor margin knotspec"),
                })
                .product()
        };

        // n=30 (the issue's data): was 7³=343, must now be modest.
        assert!(
            make(60) <= 216,
            "3-D te at small n must stay near the mgcv te default, got {}",
            make(60)
        );
        // Larger n must NOT grow the product toward n³ (was 20³=8000).
        assert!(
            make(2000) <= 216,
            "3-D te at large n must not blow ∏k toward the data size, got {}",
            make(2000)
        );
    }

    #[test]
    fn parse_bspline_boundary_conditions_and_side_selector() {
        // The `side=left` filter routes the global `anchor=` value to the left
        // endpoint (not the right), preserving the non-zero value for the
        // affine boundary lift.
        let mut opts = BTreeMap::new();
        opts.insert("boundary_conditions".to_string(), "anchored".to_string());
        opts.insert("side".to_string(), "left".to_string());
        opts.insert("anchor".to_string(), "2.5".to_string());
        let parsed = parse_bspline_boundary_conditions(&opts).expect("left anchor parses");
        assert!(matches!(
            parsed.left,
            BSplineEndpointBoundaryCondition::Anchored { value } if value == 2.5
        ));
        assert!(matches!(
            parsed.right,
            BSplineEndpointBoundaryCondition::Free
        ));

        // Side-specific aliases (`start_bc`/`end_bc`) plus the side-specific
        // anchor key (`right_anchor`) must funnel the value onto the right
        // endpoint.
        let mut opts = BTreeMap::new();
        opts.insert("start_bc".to_string(), "clamped".to_string());
        opts.insert("end_bc".to_string(), "zero".to_string());
        opts.insert("right_anchor".to_string(), "-1.0".to_string());
        let parsed = parse_bspline_boundary_conditions(&opts).expect("right anchor parses");
        assert!(matches!(
            parsed.left,
            BSplineEndpointBoundaryCondition::Clamped
        ));
        assert!(matches!(
            parsed.right,
            BSplineEndpointBoundaryCondition::Anchored { value } if value == -1.0
        ));

        // With anchors at zero the basis builder accepts the configuration,
        // so the same alias plumbing yields a clean `Anchored { value: 0.0 }`
        // on the right and `Clamped` on the left.
        let mut opts = BTreeMap::new();
        opts.insert("start_bc".to_string(), "clamped".to_string());
        opts.insert("end_bc".to_string(), "zero".to_string());
        let parsed = parse_bspline_boundary_conditions(&opts).expect("boundary conditions");
        assert!(matches!(
            parsed.left,
            BSplineEndpointBoundaryCondition::Clamped
        ));
        assert!(matches!(
            parsed.right,
            BSplineEndpointBoundaryCondition::Anchored { value } if value.abs() < 1e-12
        ));
    }

    #[test]
    fn one_sided_anchor_owns_level_without_sum_to_zero_constraint_1867() {
        let ds = continuous_dataset(
            &["y", "x"],
            (0..32)
                .map(|i| {
                    let x = i as f64 / 31.0;
                    vec![x * (1.0 - x), x]
                })
                .collect(),
        );
        let col_map = ds.column_map();

        let build = |formula: &str| {
            let parsed = parse_formula(formula).expect("parse anchored smooth");
            let mut notes = Vec::new();
            build_termspec(
                &parsed.terms,
                &ds,
                &col_map,
                &mut notes,
                &ResourcePolicy::default_library(),
            )
            .expect("build anchored smooth")
        };

        let one_sided = build("y ~ s(x, bc_left=anchored, anchor_left=0, k=10)");
        let SmoothBasisSpec::BSpline1D { spec, .. } = &one_sided.smooth_terms[0].basis else {
            panic!("expected one-dimensional B-spline");
        };
        assert!(matches!(spec.identifiability, BSplineIdentifiability::None));

        // #2297: a two-sided anchor pins BOTH endpoint levels, which strips the
        // interior level as well — the smooth owns no free level at all, so
        // identifiability drops to `None` (drop-intercept/skip-centering), the
        // same ownership rule as the one-sided case above. The former
        // `WeightedSumToZero` expectation predates #2297 (2e90c51b7) and would
        // double-constrain the anchored level.
        let two_sided = build("y ~ s(x, bc_left=anchored, bc_right=anchored, k=10)");
        let SmoothBasisSpec::BSpline1D { spec, .. } = &two_sided.smooth_terms[0].basis else {
            panic!("expected one-dimensional B-spline");
        };
        assert!(matches!(spec.identifiability, BSplineIdentifiability::None));

        // Control: an un-anchored smooth keeps the default weighted sum-to-zero
        // constraint — #2297's anchor rule must not leak into plain smooths.
        let plain = build("y ~ s(x, k=10)");
        let SmoothBasisSpec::BSpline1D { spec, .. } = &plain.smooth_terms[0].basis else {
            panic!("expected one-dimensional B-spline");
        };
        assert!(matches!(
            spec.identifiability,
            BSplineIdentifiability::WeightedSumToZero { .. }
        ));
    }

    #[test]
    fn categorical_by_numeric_interaction_expands_treatment_coded_cells() {
        // `y ~ x:g` is an INTERACTION-ONLY numeric-by-factor model: there is no
        // `x` main effect, so the marginal parent that would identify a dropped
        // reference level is ABSENT. The expansion must therefore be marginality-
        // aware (gam#1158) and DUMMY-code `g` — keep ALL levels — yielding the
        // "common intercept, separate slopes" design (one x-slope column per
        // group). Treatment-coding here (dropping the reference level) would pin
        // the reference group's slope to zero, a rank-deficient fit; that wrong
        // behaviour is what this test now guards against. (The treatment-coded
        // path is exercised when the `x` parent is present — see
        // `categorical_by_numeric_interaction_keeps_treatment_coding_with_parent`.)
        let ds = factor_dataset();
        // `g` is categorical with two levels (encoded 0.0 → "a", 1.0 → "b").
        let parsed = parse_formula("y ~ x:g").expect("parse `y ~ x:g`");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &ResourcePolicy::default_library(),
        )
        .expect("factor-aware `x:g` interaction must build, not error");

        assert_eq!(
            terms.linear_terms.len(),
            2,
            "interaction-only `x:g` keeps ALL factor levels (full dummy coding): one slope column per group"
        );

        let x_col = *col_map.get("x").expect("x column");
        let g_col = *col_map.get("g").expect("g column");

        // Both level gates must appear exactly once across the two cell columns,
        // and each cell carries `x` as a product factor (not a raw column for g).
        let mut seen_bits = std::collections::HashSet::new();
        for term in &terms.linear_terms {
            assert!(
                term.is_interaction(),
                "the categorical-by-numeric cell is a Wilkinson-Rogers interaction"
            );
            assert_eq!(term.feature_cols, vec![x_col]);
            assert_eq!(term.categorical_levels.len(), 1);
            let (gate_col, gate_bits) = term.categorical_levels[0];
            assert_eq!(gate_col, g_col);
            assert!(seen_bits.insert(gate_bits), "each level appears once");

            // Realize and check it equals `1[g == gate_bits] * x` row by row.
            let column = term
                .realized_design_column(ds.values.view())
                .expect("realize cell column");
            let n = ds.values.nrows();
            assert_eq!(column.len(), n);
            for row in 0..n {
                let x = ds.values[[row, x_col]];
                let g = ds.values[[row, g_col]];
                let expected = if g.to_bits() == gate_bits { x } else { 0.0 };
                assert!(
                    (column[row] - expected).abs() < 1e-12,
                    "row {row}: g={g}, x={x}, expected {expected}, got {}",
                    column[row]
                );
            }
        }
        // Both the reference level "a" (0.0) and the non-reference "b" (1.0) are
        // kept — the reference level is NOT dropped in the interaction-only form.
        assert!(seen_bits.contains(&0.0_f64.to_bits()));
        assert!(seen_bits.contains(&1.0_f64.to_bits()));
    }

    #[test]
    fn categorical_by_numeric_interaction_keeps_treatment_coding_with_parent() {
        // With the `x` main effect PRESENT (`y ~ x + x:g`), the marginal parent
        // that identifies a dropped reference level exists, so `x:g` keeps its
        // historical treatment coding: the reference level "a" is dropped and
        // only the non-reference slope-deviation column for "b" is emitted. This
        // guards that the marginality-aware fix (gam#1158) does NOT regress the
        // parent-present form, which must stay column-space-identical to mgcv's
        // `x + x:g`.
        let ds = factor_dataset();
        let parsed = parse_formula("y ~ x + x:g").expect("parse `y ~ x + x:g`");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &ResourcePolicy::default_library(),
        )
        .expect("`x + x:g` must build");

        // One main-effect `x` column plus one treatment-coded interaction cell.
        let x_col = *col_map.get("x").expect("x column");
        let g_col = *col_map.get("g").expect("g column");
        let interaction_cells: Vec<_> = terms
            .linear_terms
            .iter()
            .filter(|t| t.is_interaction())
            .collect();
        assert_eq!(
            interaction_cells.len(),
            1,
            "with `x` present, `x:g` is treatment-coded → one cell (reference dropped)"
        );
        let term = interaction_cells[0];
        assert_eq!(term.feature_cols, vec![x_col]);
        assert_eq!(term.categorical_levels.len(), 1);
        let (gate_col, gate_bits) = term.categorical_levels[0];
        assert_eq!(gate_col, g_col);
        // The dropped reference is "a" (0.0); the kept gate is "b" (1.0).
        assert_eq!(gate_bits, 1.0_f64.to_bits());
    }

    #[test]
    fn categorical_by_categorical_interaction_expands_full_cross_cells() {
        // `y ~ f:g` is an INTERACTION-ONLY factor-by-factor model: neither `f`
        // nor `g` appears as a main effect, so neither marginal parent is
        // present and BOTH factors must be dummy-coded (gam#1159). The correct
        // design is the SATURATED cell-means model: the full cross of ALL levels
        // (3 * 2 = 6 cells) minus ONE reference cell (the lexicographically-first
        // level of every factor, here f0:g0) absorbed by the intercept — rank
        // 6-1 = 5 cell columns + intercept, column-space-identical to `f*g`.
        // Treatment-coding both factors (the old behaviour) kept only
        // (3-1)*(2-1) = 2 cells and collapsed the rest onto the intercept, a
        // rank-deficient fit; that is the bug this test now guards against.
        let n = 30usize;
        let mut rows = Vec::with_capacity(n);
        for i in 0..n {
            let y = (i as f64).sin();
            let f = (i % 3) as f64; // 3 levels: 0,1,2
            let g = (i % 2) as f64; // 2 levels: 0,1
            rows.push(vec![y, f, g]);
        }
        let values = Array2::from_shape_vec(
            (n, 3),
            rows.into_iter().flat_map(|row| row.into_iter()).collect(),
        )
        .expect("rectangular cross-factor data");
        let ds = Dataset {
            headers: vec!["y".into(), "f".into(), "g".into()],
            values,
            schema: DataSchema {
                columns: vec![
                    SchemaColumn {
                        name: "y".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "f".into(),
                        kind: ColumnKindTag::Categorical,
                        levels: vec!["f0".into(), "f1".into(), "f2".into()],
                    },
                    SchemaColumn {
                        name: "g".into(),
                        kind: ColumnKindTag::Categorical,
                        levels: vec!["g0".into(), "g1".into()],
                    },
                ],
            },
            column_kinds: vec![
                ColumnKindTag::Continuous,
                ColumnKindTag::Categorical,
                ColumnKindTag::Categorical,
            ],
        };

        let parsed = parse_formula("y ~ f:g").expect("parse `y ~ f:g`");
        let col_map = ds.column_map();
        let mut notes = Vec::new();
        let terms = build_termspec(
            &parsed.terms,
            &ds,
            &col_map,
            &mut notes,
            &ResourcePolicy::default_library(),
        )
        .expect("factor-by-factor `f:g` interaction must build, not error");

        assert_eq!(
            terms.linear_terms.len(),
            5,
            "saturated 3*2 = 6 cross cells minus one reference cell (f0:g0) = 5"
        );

        let f_col = *col_map.get("f").expect("f column");
        let g_col = *col_map.get("g").expect("g column");
        // The dropped reference cell pairs each factor's lexicographically-first
        // level: f0 (0.0) and g0 (0.0). It must NOT appear among the emitted
        // cells; every OTHER cross cell must.
        let f0 = 0.0_f64.to_bits();
        let g0 = 0.0_f64.to_bits();
        let mut emitted = std::collections::HashSet::new();
        for term in &terms.linear_terms {
            // No numeric operand: the realized column is a pure cell indicator.
            assert!(term.feature_cols.is_empty());
            assert_eq!(term.categorical_levels.len(), 2);
            let mut gates = std::collections::HashMap::new();
            for &(col, bits) in &term.categorical_levels {
                gates.insert(col, bits);
            }
            let f_bits = *gates.get(&f_col).expect("f gate present");
            let g_bits = *gates.get(&g_col).expect("g gate present");
            // The reference cell f0:g0 must have been dropped.
            assert!(
                !(f_bits == f0 && g_bits == g0),
                "the reference cell f0:g0 must be absorbed by the intercept, not emitted"
            );
            emitted.insert((f_bits, g_bits));

            let column = term
                .realized_design_column(ds.values.view())
                .expect("realize cross cell");
            for row in 0..n {
                let f = ds.values[[row, f_col]];
                let g = ds.values[[row, g_col]];
                let expected = if f.to_bits() == f_bits && g.to_bits() == g_bits {
                    1.0
                } else {
                    0.0
                };
                assert!(
                    (column[row] - expected).abs() < 1e-12,
                    "row {row}: expected {expected}, got {}",
                    column[row]
                );
            }
            assert!(
                column.iter().any(|&v| v == 1.0),
                "each cross cell must be observed in the data"
            );
        }
        // Every non-reference cross cell is present exactly once: all 6 cells
        // except f0:g0.
        let f_levels = [0.0_f64.to_bits(), 1.0_f64.to_bits(), 2.0_f64.to_bits()];
        let g_levels = [0.0_f64.to_bits(), 1.0_f64.to_bits()];
        for &fb in &f_levels {
            for &gb in &g_levels {
                if fb == f0 && gb == g0 {
                    continue;
                }
                assert!(
                    emitted.contains(&(fb, gb)),
                    "saturated cross cell must be present"
                );
            }
        }
    }

    /// #1561 by-group representation floor: a factor-by radial smooth's
    /// per-level blocks each see only their level's rows, so the n-scaling
    /// DEFAULT center count must size from the smallest level, not the pooled
    /// row count (measured: pooled sizing gave ~50 centers per 100-row level
    /// and an unconditionable mean block whose truth-recovery no λ could fix).
    #[test]
    fn by_level_thin_plate_sizes_default_centers_from_the_smallest_level() {
        let n_a = 60usize;
        let n_b = 180usize;
        let rows: Vec<Vec<f64>> = (0..(n_a + n_b))
            .map(|i| {
                let in_a = i < n_a;
                let x = if in_a {
                    i as f64 / (n_a - 1) as f64
                } else {
                    (i - n_a) as f64 / (n_b - 1) as f64
                };
                let g = if in_a { 0.0 } else { 1.0 };
                vec![x + g, x, g]
            })
            .collect();
        let ds = Dataset {
            headers: vec!["y".into(), "x".into(), "g".into()],
            values: Array2::from_shape_vec(
                (rows.len(), 3),
                rows.into_iter().flat_map(|row| row.into_iter()).collect(),
            )
            .expect("rectangular by-level test data"),
            schema: DataSchema {
                columns: vec![
                    SchemaColumn {
                        name: "y".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "x".into(),
                        kind: ColumnKindTag::Continuous,
                        levels: vec![],
                    },
                    SchemaColumn {
                        name: "g".into(),
                        kind: ColumnKindTag::Categorical,
                        levels: vec!["a".into(), "b".into()],
                    },
                ],
            },
            column_kinds: vec![
                ColumnKindTag::Continuous,
                ColumnKindTag::Continuous,
                ColumnKindTag::Categorical,
            ],
        };
        let build_tp = |with_by: bool| -> SmoothBasisSpec {
            let mut options = BTreeMap::new();
            options.insert("bs".to_string(), "tps".to_string());
            if with_by {
                options.insert("by".to_string(), "g".to_string());
                options.insert("__by_col".to_string(), "2".to_string());
            }
            let mut notes = Vec::new();
            build_smooth_basis(
                SmoothKind::S,
                &["x".to_string()],
                &[1],
                &options,
                &ds,
                &mut notes,
                &ResourcePolicy::default_library(),
                1,
            )
            .expect("thin-plate basis builds")
        };
        let pooled = build_tp(false);
        let by_level = build_tp(true);
        let tp_centers = |basis: &SmoothBasisSpec| -> usize {
            match basis {
                SmoothBasisSpec::ThinPlate { spec, .. } => {
                    spec.center_strategy.planned_num_centers(1)
                }
                SmoothBasisSpec::BySmooth { smooth, .. } => match smooth.as_ref() {
                    SmoothBasisSpec::ThinPlate { spec, .. } => {
                        spec.center_strategy.planned_num_centers(1)
                    }
                    other => panic!("expected ThinPlate inside BySmooth, got {other:?}"),
                },
                other => panic!("expected ThinPlate, got {other:?}"),
            }
        };
        let pooled_centers = tp_centers(&pooled);
        let by_centers = tp_centers(&by_level);
        assert!(
            by_centers < pooled_centers,
            "by-level default centers must size from the smallest level: \
             by={by_centers} pooled={pooled_centers}"
        );
        // The by-level default must agree with a direct build on a dataset of
        // the smallest level's size (the block's true effective sample).
        let ds_small = continuous_dataset(
            &["y", "x"],
            (0..n_a)
                .map(|i| {
                    let x = i as f64 / (n_a - 1) as f64;
                    vec![x, x]
                })
                .collect(),
        );
        let mut small_options = BTreeMap::new();
        small_options.insert("bs".to_string(), "tps".to_string());
        let mut notes = Vec::new();
        let small = build_smooth_basis(
            SmoothKind::S,
            &["x".to_string()],
            &[1],
            &small_options,
            &ds_small,
            &mut notes,
            &ResourcePolicy::default_library(),
            1,
        )
        .expect("small-level thin-plate basis builds");
        assert_eq!(
            by_centers,
            tp_centers(&small),
            "by-level default must equal the smallest level's own default"
        );
    }
}