integral 0.4.0

//! Independent cross-check of the `c2s` coefficient matrix against a
//! high-precision (50-digit) **mpmath** reference.
//!
//! The Rust production path ([`c2s_matrix`]) builds the transform from an
//! explicit binomial closed form (Schlegel–Frisch / Racah real solid harmonics).
//! The reference below is derived by a *different* route — numerically projecting
//! `mpmath.spherharm` onto the Cartesian monomial basis at 50 digits, then
//! unit-normalizing against the raw (solid-harmonic) Gaussian self-overlap. Because the
//! two share only the *definition* of the quantity, agreement validates both and
//! catches a transcription error in the closed form that orthonormality alone
//! could not (orthonormality is invariant under relabeling the `(2l+1)` rows).
//!
//! Rows are compared **up to an overall sign** (each row sign-fixed so its first
//! nonzero entry is positive); the exact Condon–Shortley sign and the m-ordering
//! are pinned separately by the signed reference in
//! `c2s_independent_reference.rs`.

// The reference table below is machine-generated at 17 significant figures;
// digit separators would be noise and the trailing digits (beyond f64's ~16) are
// intentional so the literal rounds to the nearest representable value.
#![allow(clippy::unreadable_literal, clippy::excessive_precision)]

use integral::math::am::n_cart;
use integral::math::solid_harmonics::c2s_matrix;

/// Sign-fix a row in place: negate it so its first significant entry is positive.
fn sign_fix(row: &mut [f64]) {
    for &v in row.iter() {
        if v.abs() > 1e-12 {
            if v < 0.0 {
                for x in row.iter_mut() {
                    *x = -*x;
                }
            }
            break;
        }
    }
}

#[test]
fn c2s_matches_mpmath_reference() {
    let mut ref_rows = C2S_REF.iter();
    for l in 0..=6 {
        let mat = c2s_matrix(l);
        let nc = n_cart(l);
        for p in 0..(2 * l + 1) {
            let mut row: Vec<f64> = mat[p * nc..(p + 1) * nc].to_vec();
            sign_fix(&mut row);
            let refr = ref_rows.next().expect("missing reference row");
            assert_eq!(refr.len(), nc, "l={l} row={p}: width mismatch");
            for (i, (&a, &b)) in row.iter().zip(refr.iter()).enumerate() {
                assert!(
                    (a - b).abs() < 1e-12,
                    "l={l} row={p} col={i}: rust={a} mpmath={b} (Δ={:e})",
                    (a - b).abs()
                );
            }
        }
    }
    assert!(
        ref_rows.next().is_none(),
        "reference has more rows than c2s produced"
    );
}

// GENERATED by an mpmath script (50 dps). Do not edit by hand.
// Independent reference for c2s_matrix; rows sign-fixed (first nonzero > 0).
pub const C2S_REF: &[&[f64]] = &[
    // l = 0  (1 spherical x 1 cartesian)
    &[1.0000000000000000],
    // l = 1  (3 spherical x 3 cartesian)
    &[1.0000000000000000, 0.0, 0.0],
    &[0.0, 1.0000000000000000, 0.0],
    &[0.0, 0.0, 1.0000000000000000],
    // l = 2  (5 spherical x 6 cartesian)
    &[0.0, 1.0925484305920791, 0.0, 0.0, 0.0, 0.0],
    &[0.0, 0.0, 0.0, 0.0, 1.0925484305920791, 0.0],
    &[
        0.31539156525252001,
        0.0,
        0.0,
        0.31539156525252001,
        0.0,
        -0.63078313050504001,
    ],
    &[0.0, 0.0, 1.0925484305920791, 0.0, 0.0, 0.0],
    &[
        0.54627421529603954,
        0.0,
        0.0,
        -0.54627421529603954,
        0.0,
        0.0,
    ],
    // l = 3  (7 spherical x 10 cartesian)
    &[
        0.0,
        1.7701307697799305,
        0.0,
        0.0,
        0.0,
        0.0,
        -0.59004358992664351,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        0.0,
        0.0,
        2.8906114426405541,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.45704579946446574,
        0.0,
        0.0,
        0.0,
        0.0,
        0.45704579946446574,
        0.0,
        -1.8281831978578629,
        0.0,
    ],
    &[
        0.0,
        0.0,
        1.1195289977703462,
        0.0,
        0.0,
        0.0,
        0.0,
        1.1195289977703462,
        0.0,
        -0.74635266518023078,
    ],
    &[
        0.45704579946446574,
        0.0,
        0.0,
        0.45704579946446574,
        0.0,
        -1.8281831978578629,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        1.4453057213202770,
        0.0,
        0.0,
        0.0,
        0.0,
        -1.4453057213202770,
        0.0,
        0.0,
    ],
    &[
        0.59004358992664351,
        0.0,
        0.0,
        -1.7701307697799305,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    // l = 4  (9 spherical x 15 cartesian)
    &[
        0.0,
        2.5033429417967045,
        0.0,
        0.0,
        0.0,
        0.0,
        -2.5033429417967045,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        0.0,
        0.0,
        5.3103923093397916,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        -1.7701307697799305,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.94617469575756002,
        0.0,
        0.0,
        0.0,
        0.0,
        0.94617469575756002,
        0.0,
        -5.6770481745453601,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        0.0,
        0.0,
        2.0071396306718675,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        2.0071396306718675,
        0.0,
        -2.6761861742291567,
        0.0,
    ],
    &[
        0.31735664074561291,
        0.0,
        0.0,
        0.63471328149122582,
        0.0,
        -2.5388531259649033,
        0.0,
        0.0,
        0.0,
        0.0,
        0.31735664074561291,
        0.0,
        -2.5388531259649033,
        0.0,
        0.84628437532163443,
    ],
    &[
        0.0,
        0.0,
        2.0071396306718675,
        0.0,
        0.0,
        0.0,
        0.0,
        2.0071396306718675,
        0.0,
        -2.6761861742291567,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.47308734787878001,
        0.0,
        0.0,
        0.0,
        0.0,
        -2.8385240872726801,
        0.0,
        0.0,
        0.0,
        0.0,
        -0.47308734787878001,
        0.0,
        2.8385240872726801,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        1.7701307697799305,
        0.0,
        0.0,
        0.0,
        0.0,
        -5.3103923093397916,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.62583573544917613,
        0.0,
        0.0,
        -3.7550144126950568,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.62583573544917613,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    // l = 5  (11 spherical x 21 cartesian)
    &[
        0.0,
        3.2819102842008505,
        0.0,
        0.0,
        0.0,
        0.0,
        -6.5638205684017010,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.65638205684017010,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        0.0,
        0.0,
        8.3026492595241651,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        -8.3026492595241651,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        1.4677148983057512,
        0.0,
        0.0,
        0.0,
        0.0,
        0.97847659887050078,
        0.0,
        -11.741719186446009,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        -0.48923829943525039,
        0.0,
        3.9139063954820031,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        0.0,
        0.0,
        4.7935367849733238,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        4.7935367849733238,
        0.0,
        -9.5870735699466475,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.45294665119569692,
        0.0,
        0.0,
        0.0,
        0.0,
        0.90589330239139384,
        0.0,
        -5.4353598143483631,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.45294665119569692,
        0.0,
        -5.4353598143483631,
        0.0,
        3.6235732095655754,
        0.0,
    ],
    &[
        0.0,
        0.0,
        1.7542548368013539,
        0.0,
        0.0,
        0.0,
        0.0,
        3.5085096736027079,
        0.0,
        -4.6780128981369439,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        1.7542548368013539,
        0.0,
        -4.6780128981369439,
        0.0,
        0.93560257962738877,
    ],
    &[
        0.45294665119569692,
        0.0,
        0.0,
        0.90589330239139384,
        0.0,
        -5.4353598143483631,
        0.0,
        0.0,
        0.0,
        0.0,
        0.45294665119569692,
        0.0,
        -5.4353598143483631,
        0.0,
        3.6235732095655754,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        2.3967683924866619,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        -4.7935367849733238,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        -2.3967683924866619,
        0.0,
        4.7935367849733238,
        0.0,
        0.0,
    ],
    &[
        0.48923829943525039,
        0.0,
        0.0,
        -0.97847659887050078,
        0.0,
        -3.9139063954820031,
        0.0,
        0.0,
        0.0,
        0.0,
        -1.4677148983057512,
        0.0,
        11.741719186446009,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.0,
        2.0756623148810413,
        0.0,
        0.0,
        0.0,
        0.0,
        -12.453973889286248,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        2.0756623148810413,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.65638205684017010,
        0.0,
        0.0,
        -6.5638205684017010,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        3.2819102842008505,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    // l = 6  (13 spherical x 28 cartesian)
    &[
        0.0,
        4.0991046311514859,
        0.0,
        0.0,
        0.0,
        0.0,
        -13.663682103838286,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        4.0991046311514859,
        0.0,
        0.0,
        0.0,
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        0.0,
        0.0,
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    ],
    &[
        0.0,
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        11.833095811158760,
        0.0,
        0.0,
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        0.0,
        -23.666191622317520,
        0.0,
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        0.0,
        0.0,
        2.3666191622317520,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        2.0182596029148966,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        -20.182596029148966,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        0.0,
        -2.0182596029148966,
        0.0,
        20.182596029148966,
        0.0,
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        0.0,
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    ],
    &[
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        8.2908473356343115,
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        5.5272315570895410,
        0.0,
        -22.108926228358164,
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        0.0,
        0.0,
        0.0,
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        0.0,
        -2.7636157785447705,
        0.0,
        7.3696420761193880,
        0.0,
        0.0,
        0.0,
    ],
    &[
        0.0,
        0.92120525951492350,
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        1.8424105190298470,
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        -14.739284152238776,
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        0.92120525951492350,
        0.0,
        -14.739284152238776,
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        14.739284152238776,
        0.0,
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    ],
    &[
        0.0,
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        2.9131068125936569,
        0.0,
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        5.8262136251873139,
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        -11.652427250374628,
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        2.9131068125936569,
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        -11.652427250374628,
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        4.6609709001498511,
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    ],
    &[
        0.31784601133814213,
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        -5.7212282040865583,
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        7.6283042721154111,
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        -5.7212282040865583,
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        7.6283042721154111,
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        -1.0171072362820548,
    ],
    &[
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        2.9131068125936569,
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        4.6609709001498511,
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    ],
    &[
        0.46060262975746175,
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        -7.3696420761193880,
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        7.3696420761193880,
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        7.3696420761193880,
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        -7.3696420761193880,
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    ],
    &[
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        2.7636157785447705,
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        -7.3696420761193880,
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        -8.2908473356343115,
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    ],
    &[
        0.50456490072872416,
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        -2.5228245036436208,
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        -5.0456490072872416,
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        30.273894043723450,
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    &[
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    &[
        0.68318410519191432,
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