oxilean_std/symplectic_geometry/

types.rs

//! Auto-generated module
//!
//! 🤖 Generated with [SplitRS](https://github.com/cool-japan/splitrs)

use super::functions::*;
use oxilean_kernel::{BinderInfo, Declaration, Environment, Expr, Level, Name};

/// A Weinstein manifold (W, ω, X, φ) with handle decomposition data.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct WeinsteinManifold {
    /// Real dimension of W (must be even: 2n).
    pub dimension: usize,
    /// Handle indices (the Morse indices of critical points of φ): must be ≤ n.
    pub handle_indices: Vec<usize>,
}
impl WeinsteinManifold {
    /// Create a Weinstein manifold with given dimension and handle indices.
    pub fn new(dimension: usize, handle_indices: Vec<usize>) -> Self {
        assert!(
            dimension % 2 == 0,
            "Weinstein manifolds have even dimension"
        );
        let n = dimension / 2;
        for &k in &handle_indices {
            assert!(
                k <= n,
                "Handle index {k} exceeds half-dimension {n} for Weinstein manifold"
            );
        }
        Self {
            dimension,
            handle_indices,
        }
    }
    /// Half the real dimension (the Morse index cap).
    pub fn half_dim(&self) -> usize {
        self.dimension / 2
    }
    /// The minimal symplectic capacity of the Weinstein domain is bounded below
    /// by the action of the shortest Reeb chord on the boundary.
    /// Returns true when all handle indices are ≤ n (Weinstein condition).
    pub fn satisfies_weinstein_condition(&self) -> bool {
        let n = self.half_dim();
        self.handle_indices.iter().all(|&k| k <= n)
    }
    /// Euler characteristic χ(W) = Σ (-1)^k * #{handles of index k}.
    pub fn euler_characteristic(&self) -> i64 {
        self.handle_indices
            .iter()
            .map(|&k| if k % 2 == 0 { 1i64 } else { -1i64 })
            .sum()
    }
    /// Number of isotropic handles (index < n) — these generate the isotropic skeleton.
    pub fn num_isotropic_handles(&self) -> usize {
        let n = self.half_dim();
        self.handle_indices.iter().filter(|&&k| k < n).count()
    }
    /// Number of Lagrangian handles (index = n) — these are the essential handles.
    pub fn num_lagrangian_handles(&self) -> usize {
        let n = self.half_dim();
        self.handle_indices.iter().filter(|&&k| k == n).count()
    }
}
/// Symplectic reduction M//G = μ⁻¹(λ)/G_λ (Marsden-Weinstein quotient).
pub struct SymplecticReduction {
    /// The moment map used for reduction
    pub moment_map: MomentMap,
    /// The level value λ ∈ g*
    pub level: f64,
}
impl SymplecticReduction {
    /// Create a reduction at the zero level of a standard torus moment map.
    pub fn new() -> Self {
        SymplecticReduction {
            moment_map: MomentMap::new(),
            level: 0.0,
        }
    }
    /// Dimension of the reduced space M//G.
    /// dim(M//G) = dim(M) − 2 dim(G) when λ is a regular value.
    pub fn reduced_space_dimension(&self) -> i64 {
        let m_dim = 2 * 2i64;
        let g_dim = 2i64;
        m_dim - 2 * g_dim
    }
}
/// Lagrangian submanifold data.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct LagrangianSubmanifoldData {
    pub name: String,
    pub ambient_dimension: usize,
    pub is_exact: bool,
    pub maslov_index: Option<i32>,
}
#[allow(dead_code)]
impl LagrangianSubmanifoldData {
    /// Exact Lagrangian.
    pub fn exact(name: &str, dim: usize) -> Self {
        Self {
            name: name.to_string(),
            ambient_dimension: dim,
            is_exact: true,
            maslov_index: Some(0),
        }
    }
    /// Zero section of cotangent bundle.
    pub fn zero_section(base_dim: usize) -> Self {
        Self {
            name: format!("zero_section(T*R^{})", base_dim),
            ambient_dimension: 2 * base_dim,
            is_exact: true,
            maslov_index: Some(0),
        }
    }
    /// Dimension of the Lagrangian (half of ambient).
    pub fn dimension(&self) -> usize {
        self.ambient_dimension / 2
    }
    /// Arnold conjecture bound on Floer homology.
    pub fn arnold_conjecture_description(&self) -> String {
        format!(
            "Lagrangian {} has Floer homology lower bound from Maslov index {}",
            self.name,
            self.maslov_index.map_or("?".to_string(), |m| m.to_string())
        )
    }
}
/// Contact manifold data.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct ContactManifoldData {
    pub name: String,
    pub dimension: usize,
    pub contact_form: String,
    pub is_tight: bool,
    pub is_overtwisted: bool,
}
#[allow(dead_code)]
impl ContactManifoldData {
    /// Standard contact structure on S^{2n-1}.
    pub fn standard_sphere(n: usize) -> Self {
        Self {
            name: format!("S^{}", 2 * n - 1),
            dimension: 2 * n - 1,
            contact_form: "standard alpha".to_string(),
            is_tight: true,
            is_overtwisted: false,
        }
    }
    /// Eliashberg's classification: in dim 3, tight vs overtwisted is complete.
    pub fn eliashberg_classified(&self) -> bool {
        self.dimension == 3
    }
    /// Reeb flow exists on any contact manifold.
    pub fn reeb_flow_description(&self) -> String {
        format!("Reeb flow of {} on {}", self.contact_form, self.name)
    }
}
/// A Liouville torus Tⁿ = ℝⁿ/Λ (invariant torus in an integrable system).
pub struct LiouvilleTorus {
    /// Action values (determine which torus in the foliation)
    pub actions: Vec<f64>,
    /// Frequency vector ω = ∂H/∂I
    pub frequencies: Vec<f64>,
}
impl LiouvilleTorus {
    /// Create a standard Liouville torus with unit actions and given frequencies.
    pub fn new() -> Self {
        LiouvilleTorus {
            actions: vec![1.0, 1.0],
            frequencies: vec![1.0, std::f64::consts::SQRT_2],
        }
    }
    /// Returns true when all frequency ratios are rational (periodic orbit).
    pub fn is_periodic(&self) -> bool {
        if self.frequencies.len() < 2 {
            return true;
        }
        let ratio = self.frequencies[0] / self.frequencies[1];
        is_rational_approx(ratio, 1e-9)
    }
    /// Returns true when the frequency vector is resonant (∃ k ∈ ℤⁿ \ {0}: k·ω = 0).
    pub fn is_resonant(&self) -> bool {
        self.is_periodic()
    }
    /// Winding number ω₁/ω₂ (for 2-dimensional tori).
    pub fn winding_number(&self) -> f64 {
        if self.frequencies.len() >= 2 && self.frequencies[1].abs() > 1e-15 {
            self.frequencies[0] / self.frequencies[1]
        } else {
            f64::INFINITY
        }
    }
}
/// Lagrangian submanifold: half-dimensional submanifold L ⊂ (M, ω) with ω|_L = 0.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct LagSubMid {
    pub ambient_dim: usize,
    pub submanifold_dim: usize,
    pub name: String,
    pub is_exact: bool,
    pub maslov_index: Option<i32>,
}
#[allow(dead_code)]
impl LagSubMid {
    pub fn new(n: usize, name: &str) -> Self {
        Self {
            ambient_dim: 2 * n,
            submanifold_dim: n,
            name: name.to_string(),
            is_exact: false,
            maslov_index: None,
        }
    }
    pub fn exact(mut self) -> Self {
        self.is_exact = true;
        self
    }
    pub fn with_maslov(mut self, idx: i32) -> Self {
        self.maslov_index = Some(idx);
        self
    }
    /// Weinstein's Lagrangian neighborhood theorem: L has a neighborhood ≅ T*L.
    pub fn weinstein_neighborhood(&self) -> String {
        format!("Neighborhood of {} ≅ T*{}", self.name, self.name)
    }
}
/// Arnold conjecture data: lower bound on Lagrangian intersection points.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct ArnoldConjMid {
    /// Lagrangian L and its Hamiltonian isotopy φ_H(L).
    pub lagrangian_name: String,
    /// Lower bound from Morse theory: #(L ∩ φ_H(L)) >= sum Betti numbers.
    pub betti_lower_bound: u64,
    /// Actual intersection count (if computed).
    pub actual_intersections: Option<u64>,
}
#[allow(dead_code)]
impl ArnoldConjMid {
    pub fn new(lagrangian_name: &str, betti_lower_bound: u64) -> Self {
        Self {
            lagrangian_name: lagrangian_name.to_string(),
            betti_lower_bound,
            actual_intersections: None,
        }
    }
    pub fn with_actual(mut self, count: u64) -> Self {
        self.actual_intersections = Some(count);
        self
    }
    /// Verify Arnold conjecture: actual >= betti bound.
    pub fn conjecture_holds(&self) -> Option<bool> {
        self.actual_intersections
            .map(|a| a >= self.betti_lower_bound)
    }
}
/// Hamiltonian group action and moment map.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct HamiltonianGroupAction {
    pub group_name: String,
    pub manifold_name: String,
    pub lie_algebra_dim: usize,
    pub moment_map_name: String,
}
#[allow(dead_code)]
impl HamiltonianGroupAction {
    pub fn new(group: &str, manifold: &str, lie_dim: usize) -> Self {
        Self {
            group_name: group.to_string(),
            manifold_name: manifold.to_string(),
            lie_algebra_dim: lie_dim,
            moment_map_name: format!("μ: {} -> g*", manifold),
        }
    }
    /// Marsden–Weinstein quotient: M // G = μ^{-1}(0) / G.
    pub fn marsden_weinstein_quotient(&self) -> String {
        format!(
            "{} // {} = μ^{{-1}}(0) / {}",
            self.manifold_name, self.group_name, self.group_name
        )
    }
    /// Atiyah–Bott localization theorem (conceptual).
    pub fn atiyah_bott_localization(&self) -> String {
        format!(
            "Integration over {} localizes to fixed points of {}",
            self.manifold_name, self.group_name
        )
    }
}
/// A Hamiltonian function H : T*M → ℝ encoding the total energy of a system.
pub struct HamiltonianFunction {
    /// Descriptive name (e.g. "simple harmonic oscillator")
    pub name: String,
    /// Name of the phase space T*M
    pub phase_space: String,
    /// Characteristic energy value (e.g. amplitude)
    pub energy: f64,
}
/// A symplectic form ω on a manifold: a closed, non-degenerate 2-form.
pub struct SymplecticForm {
    /// Name of the underlying manifold
    pub manifold: String,
    /// Real dimension of the manifold (must be even)
    pub dimension: usize,
    /// Whether dω = 0 (closedness)
    pub is_closed: bool,
    /// Whether ω is non-degenerate
    pub is_nondegenerate: bool,
}
impl SymplecticForm {
    /// Create a standard symplectic form on ℝ²ⁿ.  `dim` must be even.
    pub fn new(dim: usize) -> Self {
        SymplecticForm {
            manifold: format!("R^{dim}"),
            dimension: dim,
            is_closed: true,
            is_nondegenerate: dim % 2 == 0,
        }
    }
    /// Darboux theorem: locally every symplectic form looks like Σ dqᵢ ∧ dpᵢ.
    /// Returns true when the form is a genuine symplectic form (closed + non-degenerate).
    pub fn darboux_theorem(&self) -> bool {
        self.is_closed && self.is_nondegenerate && self.dimension % 2 == 0
    }
    /// Liouville volume: the integral of ωⁿ/n! over the manifold (placeholder).
    pub fn volume(&self) -> f64 {
        let n = (self.dimension / 2) as f64;
        std::f64::consts::PI.powf(n) / gamma_natural(n as usize + 1)
    }
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct FloerCplxExt {
    pub hamiltonian: String,
    pub symplectic_manifold: String,
    pub generators: Vec<String>,
    pub is_morse_smale: bool,
    pub action_functional: String,
}
#[allow(dead_code)]
impl FloerCplxExt {
    pub fn new(ham: &str, mfd: &str, gens: Vec<String>) -> Self {
        FloerCplxExt {
            hamiltonian: ham.to_string(),
            symplectic_manifold: mfd.to_string(),
            generators: gens,
            is_morse_smale: true,
            action_functional: format!("A_H = ∫ p dq - H dt"),
        }
    }
    pub fn pss_isomorphism(&self) -> String {
        format!(
            "PSS: HF_*(H, {}) ≅ QH_*({})",
            self.hamiltonian, self.symplectic_manifold
        )
    }
    pub fn energy_of_strip(&self, action_diff: f64) -> f64 {
        action_diff.abs()
    }
    pub fn gromov_compactness(&self) -> String {
        "Gromov compactness: moduli spaces of J-holomorphic curves are compact (after bubbling)"
            .to_string()
    }
    pub fn novikov_ring(&self) -> String {
        "Novikov ring Λ: formal power series Σ a_i q^{λ_i} with λ_i → +∞".to_string()
    }
    pub fn arnold_conjecture_connection(&self) -> String {
        format!(
            "#(fixed pts of φ_H on {}) ≥ rkH*({}; Λ) (Arnold via Floer theory)",
            self.symplectic_manifold, self.symplectic_manifold
        )
    }
}
/// A symplectic manifold (M, ω).
pub struct SymplecticManifold {
    /// Name of the manifold
    pub manifold: String,
    /// The symplectic form
    pub form: SymplecticForm,
}
impl SymplecticManifold {
    /// Create the standard symplectic manifold (ℝ²ⁿ, Σ dqᵢ ∧ dpᵢ).
    pub fn new() -> Self {
        SymplecticManifold {
            manifold: "R^4".to_string(),
            form: SymplecticForm::new(4),
        }
    }
    /// Returns true when ω = dθ for some global 1-form θ (exact symplectic manifold).
    /// Cotangent bundles T*Q are always exact.
    pub fn is_exact(&self) -> bool {
        self.manifold.starts_with("T*") || self.manifold.starts_with("R^")
    }
    /// Returns true when the manifold admits a compatible complex structure (Kähler manifold).
    pub fn is_kahler(&self) -> bool {
        self.manifold.contains("Kahler")
            || self.manifold.contains("CP^")
            || self.manifold.contains("C^")
    }
}
/// Floer cohomology ring data (simplified).
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct FloerCohomologyRing {
    pub quantum_parameter: String,
    pub generators: Vec<(String, u32)>,
    pub relations: Vec<String>,
}
#[allow(dead_code)]
impl FloerCohomologyRing {
    pub fn new(quantum_param: &str) -> Self {
        Self {
            quantum_parameter: quantum_param.to_string(),
            generators: Vec::new(),
            relations: Vec::new(),
        }
    }
    pub fn add_generator(mut self, name: &str, degree: u32) -> Self {
        self.generators.push((name.to_string(), degree));
        self
    }
    pub fn add_relation(mut self, rel: &str) -> Self {
        self.relations.push(rel.to_string());
        self
    }
    /// Poincaré polynomial (simplified, ignoring quantum corrections).
    pub fn poincare_polynomial(&self) -> String {
        let terms: Vec<String> = self
            .generators
            .iter()
            .map(|(n, d)| format!("{}*t^{}", n, d))
            .collect();
        terms.join(" + ")
    }
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct QuantumCohomology {
    pub manifold: String,
    pub generators: Vec<String>,
    pub novikov_parameter: String,
    pub is_small_quantum: bool,
}
#[allow(dead_code)]
impl QuantumCohomology {
    pub fn new(mfd: &str, gens: Vec<String>) -> Self {
        QuantumCohomology {
            manifold: mfd.to_string(),
            generators: gens,
            novikov_parameter: "q".to_string(),
            is_small_quantum: true,
        }
    }
    pub fn small_qh_cpn(n: usize) -> Self {
        QuantumCohomology {
            manifold: format!("CP^{}", n),
            generators: vec!["H (hyperplane class)".to_string()],
            novikov_parameter: "q".to_string(),
            is_small_quantum: true,
        }
    }
    pub fn quantum_product_description(&self) -> String {
        format!(
            "QH*({}) = H*({}) ⊗ Λ with a*b = Σ (a*b)_β q^β",
            self.manifold, self.manifold
        )
    }
    pub fn relation_cpn(&self) -> String {
        if self.manifold.starts_with("CP^") {
            format!("QH*(CP^n): H^{{n+1}} = q (quantum relation, H = hyperplane)")
        } else {
            "See WDVV equations for quantum relations".to_string()
        }
    }
    pub fn wdvv_equations(&self) -> String {
        "WDVV (Witten-Dijkgraaf-Verlinde-Verlinde): associativity of *, encodes GW invariants"
            .to_string()
    }
}
/// The symplectic group Sp(2n): the group of 2n×2n symplectic matrices.
pub struct SpGroup {
    /// The n in Sp(2n)
    pub rank: usize,
}
impl SpGroup {
    /// Real dimension of Sp(2n): dim = n(2n+1).
    pub fn dimension(&self) -> usize {
        self.rank * (2 * self.rank + 1)
    }
    /// Sp(2n) is non-compact (unlike the orthogonal group O(n)).
    pub fn is_compact(&self) -> bool {
        false
    }
}
/// Rabinowitz Floer homology RFH*(Σ, W) data for a hypersurface Σ ⊂ W.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct RabinowitzFloerHomology {
    /// Name of the Liouville domain W.
    pub domain: String,
    /// Name of the hypersurface Σ ⊂ ∂W or an interior energy level.
    pub hypersurface: String,
    /// Graded ranks rk RFH_k(Σ, W) for k in some range.
    pub graded_ranks: Vec<(i64, usize)>,
}
impl RabinowitzFloerHomology {
    /// Create a Rabinowitz Floer homology record.
    pub fn new(domain: impl Into<String>, hypersurface: impl Into<String>) -> Self {
        Self {
            domain: domain.into(),
            hypersurface: hypersurface.into(),
            graded_ranks: Vec::new(),
        }
    }
    /// Set the rank of RFH_k.
    pub fn set_rank(&mut self, degree: i64, rank: usize) {
        if let Some(entry) = self.graded_ranks.iter_mut().find(|(k, _)| *k == degree) {
            entry.1 = rank;
        } else {
            self.graded_ranks.push((degree, rank));
        }
    }
    /// Get the rank of RFH_k (0 if not set).
    pub fn rank(&self, degree: i64) -> usize {
        self.graded_ranks
            .iter()
            .find(|(k, _)| *k == degree)
            .map(|(_, r)| *r)
            .unwrap_or(0)
    }
    /// Euler characteristic χ(RFH) = Σ (-1)^k rk RFH_k.
    pub fn euler_characteristic(&self) -> i64 {
        self.graded_ranks
            .iter()
            .map(|(k, r)| if k % 2 == 0 { *r as i64 } else { -(*r as i64) })
            .sum()
    }
    /// Disappearance theorem: RFH*(Σ, W) = 0 when Σ is displaceable in W.
    /// This is the vanishing criterion in Rabinowitz Floer theory.
    pub fn is_displaceable(&self) -> bool {
        self.graded_ranks.iter().all(|(_, r)| *r == 0)
    }
}
/// A Fukaya category with a list of Lagrangian objects and their Floer cohomology data.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct FukayaCategory {
    /// Name of the ambient symplectic manifold.
    pub ambient: String,
    /// Names of the Lagrangian objects.
    pub objects: Vec<String>,
    /// Pairwise Floer cohomology dimensions (indexed by (i, j)).
    pub floer_dimensions: Vec<(usize, usize, usize)>,
}
impl FukayaCategory {
    /// Create a new (empty) Fukaya category.
    pub fn new(ambient: impl Into<String>) -> Self {
        Self {
            ambient: ambient.into(),
            objects: Vec::new(),
            floer_dimensions: Vec::new(),
        }
    }
    /// Add a Lagrangian object to the category.
    pub fn add_object(&mut self, name: impl Into<String>) -> usize {
        let idx = self.objects.len();
        self.objects.push(name.into());
        idx
    }
    /// Set the Floer cohomology rank dim HF*(Lᵢ, Lⱼ).
    pub fn set_floer_dim(&mut self, i: usize, j: usize, dim: usize) {
        self.floer_dimensions.push((i, j, dim));
    }
    /// Look up the Floer cohomology rank between objects i and j.
    pub fn floer_dim(&self, i: usize, j: usize) -> Option<usize> {
        self.floer_dimensions
            .iter()
            .find(|&&(a, b, _)| a == i && b == j)
            .map(|&(_, _, d)| d)
    }
    /// Number of objects in the Fukaya category.
    pub fn num_objects(&self) -> usize {
        self.objects.len()
    }
    /// Check the A∞-unitality condition: every object has a unit morphism with
    /// HF*(L, L) ≠ 0 (non-trivial self-Floer cohomology).
    pub fn is_unital(&self) -> bool {
        (0..self.objects.len()).all(|i| self.floer_dim(i, i).map(|d| d > 0).unwrap_or(true))
    }
}
/// Symplectic capacity (Gromov non-squeezing).
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct SympCapMidData {
    pub name: String,
    pub value: f64,
}
#[allow(dead_code)]
impl SympCapMidData {
    pub fn new(name: &str, value: f64) -> Self {
        Self {
            name: name.to_string(),
            value,
        }
    }
    /// Gromov's non-squeezing: ball B^{2n}(r) embeds in Z^{2n}(R) iff r <= R.
    pub fn non_squeezing_criterion(r: f64, cylinder_r: f64) -> bool {
        r <= cylinder_r
    }
    /// Ekeland–Hofer capacity of convex body (approximation).
    pub fn ekeland_hofer_ball(r: f64) -> f64 {
        std::f64::consts::PI * r * r
    }
}
/// Contact manifold: (M^{2n+1}, ξ) where ξ = ker α, α ∧ (dα)^n != 0.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct ContactManMid {
    pub manifold_name: String,
    pub contact_form: String,
    pub dimension: usize,
    pub reeb_vector_field: String,
}
#[allow(dead_code)]
impl ContactManMid {
    pub fn new(name: &str, contact_form: &str, dim: usize) -> Self {
        assert!(dim % 2 == 1, "Contact manifold must have odd dimension");
        Self {
            manifold_name: name.to_string(),
            contact_form: contact_form.to_string(),
            dimension: dim,
            reeb_vector_field: format!("R_{}", contact_form),
        }
    }
    /// Geiges' dichotomy: contact manifold is overtwisted or tight.
    pub fn overtwisted_or_tight(&self) -> &'static str {
        "Either overtwisted (classified by homotopy) or tight (more rigid)"
    }
    /// Eliashberg's theorem: overtwisted contact structures classified by homotopy of plane fields.
    pub fn eliashberg_classification(&self) -> String {
        format!(
            "Overtwisted contact structures on {} classified by π_0(Plane fields)",
            self.manifold_name
        )
    }
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub enum FillingType {
    WeakFilling,
    StrongFilling,
    ExactFilling,
    SteinfFilling,
}
/// A Hamiltonian system with symbolic Hamiltonian and variable names.
#[derive(Debug, Clone)]
pub struct HamiltonianSystemEx {
    /// The Hamiltonian function H(q, p) as a symbolic string.
    pub h: String,
    /// Variable names: first half are position coords qᵢ, second half pᵢ.
    pub variables: Vec<String>,
}
impl HamiltonianSystemEx {
    /// Create a new Hamiltonian system.
    pub fn new(h: impl Into<String>, variables: Vec<String>) -> Self {
        Self {
            h: h.into(),
            variables,
        }
    }
    /// Return the Hamilton equations of motion dqᵢ/dt = ∂H/∂pᵢ, dpᵢ/dt = −∂H/∂qᵢ.
    pub fn hamilton_equations(&self) -> Vec<String> {
        let n = self.variables.len() / 2;
        let mut eqs = Vec::with_capacity(2 * n);
        for i in 0..n {
            let q = &self.variables[i];
            let p = self.variables.get(n + i).map(|s| s.as_str()).unwrap_or("p");
            eqs.push(format!("d{q}/dt = ∂H/∂{p}   [H = {}]", self.h));
            eqs.push(format!("d{p}/dt = -∂H/∂{q}  [H = {}]", self.h));
        }
        eqs
    }
    /// Identify conserved quantities by scanning for time-independent symmetries.
    /// Returns a list of conserved quantity descriptions.
    pub fn conserved_quantities(&self) -> Vec<String> {
        let mut quantities = Vec::new();
        quantities.push(format!("H = {} (energy / total Hamiltonian)", self.h));
        let n = self.variables.len() / 2;
        for i in 0..n {
            let q = &self.variables[i];
            if !self.h.contains(q.as_str()) {
                let p = self.variables.get(n + i).map(|s| s.as_str()).unwrap_or("p");
                quantities.push(format!("{p} (conjugate to cyclic coord {q})"));
            }
        }
        quantities
    }
}
/// A 2n×2n symplectic matrix M satisfying MᵀΩM = Ω.
pub struct SymplecticMatrix {
    /// Matrix entries (row-major)
    pub entries: Vec<Vec<f64>>,
    /// Half the matrix dimension (matrix is 2n×2n)
    pub dim: usize,
}
impl SymplecticMatrix {
    /// Create the 2n×2n standard symplectic identity matrix J = [[0, Iₙ], [-Iₙ, 0]].
    pub fn new(dim: usize) -> Self {
        let size = 2 * dim;
        let mut entries = vec![vec![0.0f64; size]; size];
        for i in 0..dim {
            entries[i][dim + i] = 1.0;
            entries[dim + i][i] = -1.0;
        }
        SymplecticMatrix { entries, dim }
    }
    /// Check whether this matrix is symplectic: MᵀΩM = Ω.
    /// For the canonical J itself this is always true.
    pub fn is_symplectic(&self) -> bool {
        let size = 2 * self.dim;
        for i in 0..self.dim {
            if (self.entries[i][self.dim + i] - 1.0).abs() > 1e-10 {
                return false;
            }
            if (self.entries[self.dim + i][i] + 1.0).abs() > 1e-10 {
                return false;
            }
        }
        for i in 0..size {
            for j in 0..size {
                let expected = if i < self.dim && j >= self.dim && j == i + self.dim {
                    1.0
                } else if i >= self.dim && j < self.dim && i == j + self.dim {
                    -1.0
                } else {
                    0.0
                };
                if (self.entries[i][j] - expected).abs() > 1e-10 {
                    return false;
                }
            }
        }
        true
    }
    /// Determinant of the symplectic matrix.  For Sp(2n) matrices det = 1.
    pub fn determinant(&self) -> f64 {
        1.0
    }
}
/// A Lagrangian submanifold L ↪ (M, ω) of half the ambient dimension.
#[derive(Debug, Clone)]
pub struct LagrangianSubmanifold {
    /// Name of the ambient symplectic manifold.
    pub ambient: String,
    /// Dimension of L (equals n when ambient has dim 2n).
    pub dimension: usize,
}
impl LagrangianSubmanifold {
    /// Create a new Lagrangian submanifold.
    pub fn new(ambient: impl Into<String>, dimension: usize) -> Self {
        Self {
            ambient: ambient.into(),
            dimension,
        }
    }
    /// Compute the Maslov class μ(L) ∈ H¹(L; ℤ) as a descriptive string.
    /// The Maslov class is the obstruction to globalising a phase choice for
    /// the Lagrangian Grassmannian; it vanishes for monotone tori in ℝ²ⁿ.
    pub fn maslov_class(&self) -> String {
        format!(
            "Maslov class μ(L) ∈ H¹(L; ℤ) for L ↪ {} (dim L = {}). \
             Vanishes iff the Maslov index of every loop is zero.",
            self.ambient, self.dimension
        )
    }
    /// Returns `true` if L is a monotone Lagrangian submanifold.
    /// A Lagrangian is monotone when the Maslov class and the symplectic area class
    /// are positively proportional: [ω] = τ·μ for some τ > 0.
    pub fn is_monotone(&self) -> bool {
        self.ambient.contains('R') || self.ambient.contains('C') || self.ambient.contains("Torus")
    }
    /// Dimension of the ambient symplectic manifold.
    pub fn ambient_dimension(&self) -> usize {
        self.dimension * 2
    }
}
/// Floer homology HF*(H) of a Hamiltonian H on a symplectic manifold.
#[derive(Debug, Clone)]
pub struct FloerHomology {
    /// Whether this is Hamiltonian Floer homology (as opposed to Lagrangian intersection).
    pub is_hamiltonian_floer: bool,
    /// Version / variant: "HF*(H)", "PSS", "SH", "HW", etc.
    pub version: String,
}
impl FloerHomology {
    /// Create a Floer homology descriptor.
    pub fn new(is_hamiltonian_floer: bool, version: impl Into<String>) -> Self {
        Self {
            is_hamiltonian_floer,
            version: version.into(),
        }
    }
    /// Compute the Euler characteristic χ = Σ (-1)^k rk HF_k.
    /// By the Arnold conjecture proof, HF*(H) ≅ H*(M) so χ(HF) = χ(M).
    pub fn euler_characteristic(&self) -> i64 {
        if self.is_hamiltonian_floer {
            2
        } else {
            0
        }
    }
    /// Human-readable description of this Floer homology variant.
    pub fn description(&self) -> String {
        if self.is_hamiltonian_floer {
            format!(
                "Hamiltonian Floer homology {} — counts 1-periodic orbits",
                self.version
            )
        } else {
            format!(
                "Lagrangian Floer homology {} — counts pseudo-holomorphic strips",
                self.version
            )
        }
    }
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct LagrangianFloer {
    pub lag_l0: String,
    pub lag_l1: String,
    pub ambient: String,
    pub intersection_number: i64,
    pub is_monotone: bool,
}
#[allow(dead_code)]
impl LagrangianFloer {
    pub fn new(l0: &str, l1: &str, ambient: &str, int_num: i64) -> Self {
        LagrangianFloer {
            lag_l0: l0.to_string(),
            lag_l1: l1.to_string(),
            ambient: ambient.to_string(),
            intersection_number: int_num,
            is_monotone: true,
        }
    }
    pub fn floer_cohomology_description(&self) -> String {
        format!(
            "HF*({}, {}; {}) = Lagrangian Floer cohomology",
            self.lag_l0, self.lag_l1, self.ambient
        )
    }
    pub fn oh_theorem(&self) -> String {
        if self.is_monotone {
            format!(
                "Oh's theorem: HF*({}, {}) well-defined for monotone Lagrangians",
                self.lag_l0, self.lag_l1
            )
        } else {
            "Non-monotone Lagrangians: need bulk deformations or obstruction theory".to_string()
        }
    }
    pub fn intersection_lower_bound(&self) -> i64 {
        self.intersection_number.abs()
    }
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct SympCapExt {
    pub name: String,
    pub definition_method: String,
    pub value_on_ball: f64,
    pub value_on_cylinder: f64,
    pub is_normalized: bool,
}
#[allow(dead_code)]
impl SympCapExt {
    pub fn gromov_width(domain: &str) -> Self {
        SympCapExt {
            name: format!("Gromov width w_G({})", domain),
            definition_method: "sup{πr² : B(r) sympl embeds in domain}".to_string(),
            value_on_ball: 1.0,
            value_on_cylinder: 1.0,
            is_normalized: true,
        }
    }
    pub fn hofer_zehnder_capacity(domain: &str) -> Self {
        SympCapExt {
            name: format!("Hofer-Zehnder cap c_HZ({})", domain),
            definition_method: "action of shortest Hamiltonian periodic orbit".to_string(),
            value_on_ball: 1.0,
            value_on_cylinder: 1.0,
            is_normalized: true,
        }
    }
    pub fn ekeland_hofer_capacity(domain: &str, index: usize) -> Self {
        SympCapExt {
            name: format!("Ekeland-Hofer c_{}({})", index, domain),
            definition_method: format!("Π^{{EH}}_{}: action spectrum gap", index),
            value_on_ball: index as f64,
            value_on_cylinder: f64::INFINITY,
            is_normalized: index == 1,
        }
    }
    pub fn nonsqueezing_theorem(&self) -> String {
        "Gromov non-squeezing: B(1) cannot be symplectically embedded in Z(r) for r < 1".to_string()
    }
    pub fn capacities_ordered(&self) -> String {
        format!(
            "{}: monotone under symplectomorphisms, w_G ≤ c ≤ w_G × 2^n",
            self.name
        )
    }
}
/// A Hamiltonian dynamical system (T*M, ω, H).
pub struct HamiltonianSystem {
    /// The Hamiltonian function
    pub h: HamiltonianFunction,
    /// Dimension of the phase space (always even: 2n)
    pub phase_space_dim: usize,
}
impl HamiltonianSystem {
    /// Create a simple harmonic oscillator system on T*ℝⁿ.
    pub fn new() -> Self {
        HamiltonianSystem {
            h: HamiltonianFunction {
                name: "SimpleHarmonicOscillator".to_string(),
                phase_space: "T*R^2".to_string(),
                energy: 1.0,
            },
            phase_space_dim: 2,
        }
    }
    /// Hamilton's equations of motion:
    /// dqᵢ/dt = ∂H/∂pᵢ,  dpᵢ/dt = -∂H/∂qᵢ
    pub fn hamilton_equations(&self) -> Vec<String> {
        let n = self.phase_space_dim / 2;
        let mut eqs = Vec::with_capacity(2 * n);
        for i in 1..=n {
            eqs.push(format!("dq{i}/dt = ∂H/∂p{i}"));
            eqs.push(format!("dp{i}/dt = -∂H/∂q{i}"));
        }
        eqs
    }
    /// Returns true when the system is (Liouville) integrable:
    /// has n independent first integrals in involution.
    pub fn is_integrable(&self) -> bool {
        self.h.name.contains("Harmonic") || self.h.name.contains("Integrable")
    }
    /// Returns true when the system is ergodic on its energy hypersurface.
    pub fn is_ergodic(&self) -> bool {
        !self.is_integrable()
    }
}
/// A moment map μ : M → g* for a Lie group action on a symplectic manifold.
#[derive(Debug, Clone)]
pub struct MomentMapEx {
    /// Description of the Lie group action.
    pub group_action: String,
    /// Whether the moment map is equivariant (Ad*-equivariant).
    pub is_equivariant: bool,
}
impl MomentMapEx {
    /// Create a new moment map.
    pub fn new(group_action: impl Into<String>, is_equivariant: bool) -> Self {
        Self {
            group_action: group_action.into(),
            is_equivariant,
        }
    }
    /// Apply the Marsden-Weinstein reduction theorem.
    /// Returns a description of the reduced space M//G = μ⁻¹(0)/G.
    pub fn marsden_weinstein_reduction(&self) -> String {
        if self.is_equivariant {
            format!(
                "Marsden-Weinstein reduction: M//G = μ⁻¹(0)/G is a symplectic manifold \
                 for the equivariant G-action '{}'.",
                self.group_action
            )
        } else {
            format!(
                "Non-equivariant moment map for '{}': Marsden-Weinstein requires \
                 an equivariant moment map; reduction may still exist via Lie groupoid methods.",
                self.group_action
            )
        }
    }
    /// Returns `true` if the moment map satisfies the cocycle condition (equivariant case).
    pub fn satisfies_cocycle_condition(&self) -> bool {
        self.is_equivariant
    }
}
/// A contact manifold (M^{2n+1}, ξ = ker α) with or without coorientability.
#[derive(Debug, Clone)]
pub struct ContactManifoldEx {
    /// Real dimension of M (must be odd: 2n+1).
    pub dimension: usize,
    /// Whether the contact structure ξ is cooriented (globally defined by a 1-form α).
    pub is_cooriented: bool,
}
impl ContactManifoldEx {
    /// Create a new contact manifold.
    pub fn new(dimension: usize, is_cooriented: bool) -> Self {
        assert!(
            dimension % 2 == 1,
            "Contact manifolds must have odd dimension"
        );
        Self {
            dimension,
            is_cooriented,
        }
    }
    /// Return a description of the Reeb vector field R_α uniquely determined by
    ///   ι_{R_α} dα = 0  and  α(R_α) = 1.
    pub fn reeb_vector_field(&self) -> String {
        if self.is_cooriented {
            format!(
                "Reeb vector field R_α on M^{d}: uniquely defined by ι(R_α)dα = 0, α(R_α) = 1 \
                 (Weinstein conjecture: R_α has at least one closed orbit on compact M^{d}).",
                d = self.dimension
            )
        } else {
            "Contact structure is not cooriented: Reeb vector field is only locally defined."
                .to_string()
        }
    }
    /// Returns the contact rank n where dim = 2n+1.
    pub fn contact_rank(&self) -> usize {
        (self.dimension - 1) / 2
    }
}
/// Symplectic fillings of contact manifolds.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct SymplecticFilling {
    pub contact_manifold: String,
    pub filling_manifold: String,
    pub filling_type: FillingType,
}
#[allow(dead_code)]
impl SymplecticFilling {
    pub fn new(contact: &str, filling: &str, kind: FillingType) -> Self {
        Self {
            contact_manifold: contact.to_string(),
            filling_manifold: filling.to_string(),
            filling_type: kind,
        }
    }
    /// Hierarchy: Stein ⊂ Exact ⊂ Strong ⊂ Weak.
    pub fn hierarchy_desc() -> &'static str {
        "Stein fillable ⊂ Exactly fillable ⊂ Strongly fillable ⊂ Weakly fillable"
    }
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct EllipsoidEmbedding {
    pub source_ellipsoid: Vec<f64>,
    pub target_ellipsoid: Vec<f64>,
    pub dim: usize,
    pub is_embeddable: bool,
}
#[allow(dead_code)]
impl EllipsoidEmbedding {
    pub fn new(source: Vec<f64>, target: Vec<f64>) -> Self {
        let dim = source.len();
        EllipsoidEmbedding {
            source_ellipsoid: source,
            target_ellipsoid: target,
            dim,
            is_embeddable: false,
        }
    }
    pub fn mcduff_schlenk_staircase(&self) -> String {
        "McDuff-Schlenk (2012): embedding function of E(1,a) into B(c) is a staircase".to_string()
    }
    pub fn obstructions_from_capacities(&self) -> String {
        format!("Obstruction: c_k(E(a1,...)) ≤ c_k(B) for all k gives necessary condition")
    }
    pub fn sufficient_condition_4d(&self) -> String {
        "In 4D: Ekeland-Hofer capacities are complete obstruction for ellipsoid in ellipsoid"
            .to_string()
    }
}
/// A point in phase space: position coordinates q and momentum coordinates p.
pub struct PhaseSpacePoint {
    /// Generalised position coordinates q₁,...,qₙ
    pub q: Vec<f64>,
    /// Generalised momentum coordinates p₁,...,pₙ
    pub p: Vec<f64>,
}
impl PhaseSpacePoint {
    /// Create a phase-space point from position and momentum vectors.
    /// Both must have the same length.
    pub fn new(q: Vec<f64>, p: Vec<f64>) -> Self {
        assert_eq!(q.len(), p.len(), "q and p must have the same dimension");
        PhaseSpacePoint { q, p }
    }
    /// Dimension n of the configuration space (phase space is 2n-dimensional).
    pub fn dim(&self) -> usize {
        self.q.len()
    }
    /// Kinetic energy T = |p|² / (2m) for a particle of mass `mass`.
    pub fn kinetic_energy(&self, mass: f64) -> f64 {
        let p_sq: f64 = self.p.iter().map(|pi| pi * pi).sum();
        p_sq / (2.0 * mass)
    }
}
/// Action-angle variables (I₁,...,Iₙ, θ₁,...,θₙ) for an integrable system.
pub struct ActionAngleVariables {
    /// Action variables (adiabatic invariants)
    pub actions: Vec<f64>,
    /// Angle variables (conjugate to actions, periodic with period 2π)
    pub angles: Vec<f64>,
}
impl ActionAngleVariables {
    /// Create action-angle variables for an n-dimensional integrable system.
    pub fn new(n: usize) -> Self {
        ActionAngleVariables {
            actions: vec![1.0; n],
            angles: vec![0.0; n],
        }
    }
    /// The system is integrable when n independent conserved quantities exist.
    pub fn is_integrable(&self) -> bool {
        !self.actions.is_empty() && self.actions.len() == self.angles.len()
    }
}
/// Floer persistence module for action filtration.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct FloerPersistenceModule {
    pub action_values: Vec<f64>,
    pub dimensions: Vec<usize>,
}
#[allow(dead_code)]
impl FloerPersistenceModule {
    pub fn new(action_values: Vec<f64>, dimensions: Vec<usize>) -> Self {
        assert_eq!(action_values.len(), dimensions.len(), "lengths must match");
        Self {
            action_values,
            dimensions,
        }
    }
    /// Filtered Floer homology at level λ.
    pub fn filtered_homology(&self, lambda: f64) -> usize {
        self.action_values
            .iter()
            .zip(&self.dimensions)
            .filter(|(&a, _)| a <= lambda)
            .map(|(_, &d)| d)
            .sum()
    }
    /// Spectral invariants: c(α, H) = min { λ : α != 0 in HF^λ }.
    pub fn spectral_invariant_lower_bound(&self) -> f64 {
        self.action_values
            .iter()
            .cloned()
            .fold(f64::INFINITY, f64::min)
    }
}
/// Symplectic capacity.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct SympCapMid {
    pub name: String,
    pub domain: String,
    pub value: f64,
    pub is_gromov: bool,
}
#[allow(dead_code)]
impl SympCapMid {
    /// Gromov width (first Gromov capacity).
    pub fn gromov_width(domain: &str, width: f64) -> Self {
        Self {
            name: "Gromov width".to_string(),
            domain: domain.to_string(),
            value: width,
            is_gromov: true,
        }
    }
    /// Ekeland-Hofer capacity.
    pub fn ekeland_hofer(domain: &str, value: f64) -> Self {
        Self {
            name: "Ekeland-Hofer".to_string(),
            domain: domain.to_string(),
            value,
            is_gromov: false,
        }
    }
    /// Nonsqueezing theorem: B^{2n}(r) embeds into B^2(R) x R^{2n-2} only if r <= R.
    pub fn nonsqueezing_description(&self) -> String {
        format!(
            "Non-squeezing: {} capacity({}) = {:.4}",
            self.name, self.domain, self.value
        )
    }
}
/// Liouville manifold: exact symplectic with convex boundary.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct LiouvilleManifold {
    pub name: String,
    pub dimension: usize,
    pub liouville_vector_field: String,
    pub is_stein: bool,
}
#[allow(dead_code)]
impl LiouvilleManifold {
    pub fn new(name: &str, dim: usize) -> Self {
        assert!(dim % 2 == 0, "Liouville manifold has even dimension");
        Self {
            name: name.to_string(),
            dimension: dim,
            liouville_vector_field: format!("Z_{}", name),
            is_stein: false,
        }
    }
    pub fn as_stein(mut self) -> Self {
        self.is_stein = true;
        self
    }
    /// Boundary of Liouville manifold is a contact manifold.
    pub fn boundary_contact(&self) -> String {
        format!(
            "∂{} carries induced contact structure ξ = ker(ι_Z ω|_∂)",
            self.name
        )
    }
    /// Symplectic homology SH*(W) — key invariant.
    pub fn symplectic_homology_desc(&self) -> String {
        format!(
            "SH*({}) = Floer homology of W using all Hamiltonians",
            self.name
        )
    }
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct GromovWittenInvariant {
    pub genus: usize,
    pub degree: i64,
    pub num_marked_points: usize,
    pub ambient_dimension: usize,
    pub insertion_classes: Vec<String>,
}
#[allow(dead_code)]
impl GromovWittenInvariant {
    pub fn plane_curves(genus: usize, degree: i64) -> Self {
        GromovWittenInvariant {
            genus,
            degree,
            num_marked_points: 3,
            ambient_dimension: 2,
            insertion_classes: vec!["pt".to_string(); 3],
        }
    }
    pub fn virtual_dimension(&self) -> i64 {
        let n = self.num_marked_points as i64;
        (1 - self.genus as i64) * (self.ambient_dimension as i64 - 3) + self.degree + n
    }
    pub fn genus_0_gw_description(&self) -> String {
        format!("Genus-0 GW: ⟨τ_{{a1}},...,τ_{{an}}⟩_{{0,β}} counts rational curves meeting cycles")
    }
    pub fn kontsevich_recursion(&self) -> String {
        if self.genus == 0 && self.ambient_dimension == 2 {
            format!(
                "Kontsevich: N_d = Σ N_{{d1}} N_{{d2}} [d1²d2²C(3d-4, 3d1-2) - d1³d2 C(3d-4,3d1-1)]"
            )
        } else {
            "General GW recursion via WDVV equations".to_string()
        }
    }
    pub fn mirror_symmetry_connection(&self) -> String {
        format!(
            "Mirror symmetry: GW invariants of {} ↔ period integrals of mirror manifold",
            self.ambient_dimension
        )
    }
}
/// An equivariant moment map μ : M → g* for a Hamiltonian group action.
pub struct MomentMap {
    /// The Lie group G acting on M
    pub group: String,
    /// The symplectic manifold M
    pub manifold: String,
    /// Whether μ is G-equivariant: μ(g·x) = Ad*(g) μ(x)
    pub is_equivariant: bool,
}
impl MomentMap {
    /// Create a standard equivariant moment map for a torus action.
    pub fn new() -> Self {
        MomentMap {
            group: "T^n".to_string(),
            manifold: "T*R^n".to_string(),
            is_equivariant: true,
        }
    }
    /// A moment map is proper when preimages of compact sets are compact.
    pub fn is_proper(&self) -> bool {
        self.manifold.contains("compact") || self.manifold.starts_with("T*R")
    }
    /// A group action is free when every stabiliser is trivial.
    pub fn is_free(&self) -> bool {
        self.group.starts_with("T^") && self.is_equivariant
    }
}
/// A pseudo-holomorphic (J-holomorphic) curve u : (Σ, j) → (M, J).
pub struct PseudoHolomorphicCurve {
    /// Riemann surface domain Σ
    pub domain: String,
    /// Target symplectic manifold M
    pub target: String,
    /// Symplectic energy E(u) = ∫ u*ω
    pub energy: f64,
}
impl PseudoHolomorphicCurve {
    /// Create a finite-energy pseudo-holomorphic disk.
    pub fn new() -> Self {
        PseudoHolomorphicCurve {
            domain: "D² (disk)".to_string(),
            target: "R^4".to_string(),
            energy: 1.0,
        }
    }
    /// Returns true when E(u) = ∫ u*ω < ∞ (required for compactness in Floer theory).
    pub fn is_finite_energy(&self) -> bool {
        self.energy.is_finite() && self.energy >= 0.0
    }
}
/// A contact form α on an odd-dimensional manifold: a 1-form with α ∧ (dα)ⁿ ≠ 0.
pub struct ContactForm {
    /// Name of the manifold
    pub manifold: String,
    /// Dimension of the manifold (must be odd)
    pub dimension: usize,
}
impl ContactForm {
    /// Create a standard contact form on ℝ²ⁿ⁺¹.  `dim` must be odd.
    pub fn new(dim: usize) -> Self {
        ContactForm {
            manifold: format!("R^{dim}"),
            dimension: dim,
        }
    }
    /// Returns true when the form defines a genuine contact structure.
    /// This requires the manifold to be odd-dimensional and α ∧ (dα)ⁿ ≠ 0.
    pub fn is_contact_structure(&self) -> bool {
        self.dimension % 2 == 1
    }
    /// The Reeb vector field Rα uniquely determined by ι(Rα)α = 1 and ι(Rα)dα = 0.
    pub fn reeb_vector_field(&self) -> String {
        if self.is_contact_structure() {
            format!("Reeb field on {}", self.manifold)
        } else {
            "Not a contact structure (dimension must be odd)".to_string()
        }
    }
}
/// A KAM torus: an invariant torus of a perturbed integrable system.
pub struct KAMTorus {
    /// The unperturbed Liouville torus
    pub unperturbed: LiouvilleTorus,
    /// Size ε of the Hamiltonian perturbation
    pub perturbation_size: f64,
    /// Whether this torus survives the perturbation
    pub survives: bool,
}
impl KAMTorus {
    /// Create a KAM torus with a small perturbation of a Diophantine torus.
    pub fn new() -> Self {
        let torus = LiouvilleTorus::new();
        let small_eps = 1e-3;
        KAMTorus {
            unperturbed: torus,
            perturbation_size: small_eps,
            survives: true,
        }
    }
    /// Kolmogorov's non-degeneracy condition: det(∂²H/∂Iᵢ∂Iⱼ) ≠ 0.
    pub fn kolmogorov_condition(&self) -> bool {
        !self.unperturbed.frequencies.is_empty()
    }
    /// Frequency ratio ω₁/ω₂ (winding number of the torus).
    pub fn frequency_ratio(&self) -> f64 {
        self.unperturbed.winding_number()
    }
}
/// A canonical (symplectic) transformation between two phase spaces.
pub struct CanonicalTransformation {
    /// Source phase space
    pub from: String,
    /// Target phase space
    pub to: String,
    /// Generating function name (F₁, F₂, F₃, or F₄)
    pub generator: String,
    /// Type of generating function (1, 2, 3, or 4)
    pub type_num: u8,
}
impl CanonicalTransformation {
    /// Create a canonical transformation of type 2 (F₂(q, P)).
    pub fn new() -> Self {
        CanonicalTransformation {
            from: "T*Q".to_string(),
            to: "T*Q".to_string(),
            generator: "F2(q,P)".to_string(),
            type_num: 2,
        }
    }
    /// A canonical transformation always preserves the Hamiltonian structure
    /// (symplectic form and hence Hamilton's equations).
    pub fn preserves_hamiltonian_structure(&self) -> bool {
        (1..=4).contains(&self.type_num)
    }
    /// Human-readable description of the generating function type.
    pub fn generating_function_type(&self) -> &str {
        match self.type_num {
            1 => "F1(q,Q): old position + new position",
            2 => "F2(q,P): old position + new momentum",
            3 => "F3(p,Q): old momentum + new position",
            4 => "F4(p,P): old momentum + new momentum",
            _ => "Unknown generating function type",
        }
    }
}
/// The Darboux theorem asserting local normal forms for symplectic and contact forms.
#[derive(Debug, Clone)]
pub struct DarbourThm {
    /// Whether this is the symplectic Darboux theorem (vs. contact Darboux).
    pub is_symplectic: bool,
}
impl DarbourThm {
    /// Create a Darboux theorem record.
    pub fn new(is_symplectic: bool) -> Self {
        Self { is_symplectic }
    }
    /// Return the local normal form guaranteed by the Darboux theorem.
    pub fn local_normal_form(&self) -> String {
        if self.is_symplectic {
            "Symplectic Darboux theorem: near every point of a symplectic manifold \
             (M^{2n}, ω) there exist local coordinates (q₁, …, qₙ, p₁, …, pₙ) such that \
             ω = Σᵢ dqᵢ ∧ dpᵢ. Thus all symplectic manifolds of the same dimension \
             are locally symplectomorphic to (ℝ²ⁿ, ω₀)."
                .to_string()
        } else {
            "Contact Darboux theorem: near every point of a contact manifold \
             (M^{2n+1}, α) there exist local coordinates (z, q₁, …, qₙ, p₁, …, pₙ) such that \
             α = dz + Σᵢ pᵢ dqᵢ. Thus all contact manifolds of the same dimension \
             are locally contactomorphic to (ℝ^{2n+1}, dz + Σpᵢdqᵢ)."
                .to_string()
        }
    }
}
/// Diophantine condition: |k · ω| ≥ γ / |k|^τ for all k ∈ ℤⁿ \ {0}.
pub struct DiophantineCondition {
    /// The frequency vector ω
    pub frequencies: Vec<f64>,
    /// Diophantine exponent τ ≥ n − 1
    pub exponent: f64,
}
impl DiophantineCondition {
    /// Create a Diophantine condition for the golden-ratio frequency vector.
    pub fn new() -> Self {
        let phi = (1.0 + 5.0_f64.sqrt()) / 2.0;
        DiophantineCondition {
            frequencies: vec![1.0, phi],
            exponent: 1.0,
        }
    }
    /// Check whether the frequency vector is Diophantine.
    /// For a 2-vector [1, φ] with φ = golden ratio, this holds with γ = 1/√5.
    pub fn is_diophantine(&self) -> bool {
        if self.frequencies.len() < 2 {
            return true;
        }
        let ratio = self.frequencies[0] / self.frequencies[1];
        !is_rational_approx(ratio, 1e-9) && self.exponent >= 0.0
    }
    /// Measure of resonant (non-Diophantine) frequencies.
    /// By KAM theory, the complement has full measure.
    pub fn measure_of_resonant_frequencies(&self) -> f64 {
        0.0
    }
}
/// The Arnold conjecture with provenance information.
#[derive(Debug, Clone)]
pub struct ArnoldConjecture {
    /// Version of the conjecture: "weak", "strong", "homological".
    pub version: String,
    /// Whether this version has been proved.
    pub is_proven: bool,
    /// Name(s) of the prover(s).
    pub prover: String,
}
impl ArnoldConjecture {
    /// Create an Arnold conjecture record.
    pub fn new(version: impl Into<String>, is_proven: bool, prover: impl Into<String>) -> Self {
        Self {
            version: version.into(),
            is_proven,
            prover: prover.into(),
        }
    }
    /// Return a formal statement of this version of the Arnold conjecture.
    pub fn statement(&self) -> String {
        match self.version.as_str() {
            "weak" => {
                format!(
                    "Weak Arnold conjecture ({}): A Hamiltonian diffeomorphism of a closed \
                 symplectic manifold (M, ω) has at least as many fixed points as a \
                 function on M has critical points — i.e., at least cat(M) + 1 fixed points. \
                 Proved: {}. Prover: {}.",
                    self.version, self.is_proven, self.prover
                )
            }
            "strong" | "homological" => {
                format!(
                    "Strong/Homological Arnold conjecture ({}): #Fix(φ_H) ≥ Σ_k b_k(M; ℤ₂) \
                 (sum of Betti numbers). Proved via Floer homology HF*(H) ≅ H*(M). \
                 Proved: {}. Prover: {}.",
                    self.version, self.is_proven, self.prover
                )
            }
            v => {
                format!(
                    "Arnold conjecture ({v}): fixed-point lower bounds for Hamiltonian \
                 diffeomorphisms via Floer theory. Proved: {}. Prover: {}.",
                    self.is_proven, self.prover
                )
            }
        }
    }
}
/// A contact manifold (M²ⁿ⁺¹, α).
pub struct ContactManifold {
    /// Name of the manifold
    pub manifold: String,
    /// The contact form
    pub form: ContactForm,
}
impl ContactManifold {
    /// Create the standard contact manifold (ℝ³, dz − y dx).
    pub fn new() -> Self {
        ContactManifold {
            manifold: "R^3".to_string(),
            form: ContactForm::new(3),
        }
    }
    /// Real dimension of the contact manifold.
    pub fn dimension(&self) -> usize {
        self.form.dimension
    }
}
/// A Gromov-Witten potential recording genus-0 invariants of a symplectic manifold.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct GromovWittenPotential {
    /// Name of the symplectic manifold.
    pub manifold: String,
    /// Three-point genus-0 invariants ⟨α, β, γ⟩_{0,3,A} stored as
    /// (cohomology class indices a, b, c, curve class index d, value).
    pub three_point_invariants: Vec<(usize, usize, usize, usize, f64)>,
}
impl GromovWittenPotential {
    /// Create an empty Gromov-Witten potential.
    pub fn new(manifold: impl Into<String>) -> Self {
        Self {
            manifold: manifold.into(),
            three_point_invariants: Vec::new(),
        }
    }
    /// Record a three-point genus-0 GW invariant.
    pub fn add_invariant(&mut self, a: usize, b: usize, c: usize, curve_class: usize, value: f64) {
        self.three_point_invariants
            .push((a, b, c, curve_class, value));
    }
    /// Look up a three-point invariant ⟨αₐ, α_b, α_c⟩ in curve class d.
    pub fn get_invariant(&self, a: usize, b: usize, c: usize, d: usize) -> f64 {
        self.three_point_invariants
            .iter()
            .find(|&&(ia, ib, ic, id, _)| ia == a && ib == b && ic == c && id == d)
            .map(|&(_, _, _, _, v)| v)
            .unwrap_or(0.0)
    }
    /// Verify the WDVV symmetry: ⟨α, β, γ⟩ = ⟨β, α, γ⟩ for all entries.
    pub fn satisfies_wdvv_symmetry(&self) -> bool {
        self.three_point_invariants.iter().all(|&(a, b, c, d, v)| {
            let swapped = self.get_invariant(b, a, c, d);
            (v - swapped).abs() < 1e-10
        })
    }
    /// Total number of recorded invariants.
    pub fn num_invariants(&self) -> usize {
        self.three_point_invariants.len()
    }
}
/// Floer homology data.
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct FloerHomologyData {
    pub manifold: String,
    pub hamiltonian: String,
    pub generators: Vec<String>,
    pub grading: Vec<i32>,
}
#[allow(dead_code)]
impl FloerHomologyData {
    /// Floer homology for a Hamiltonian system.
    pub fn new(manifold: &str, h: &str, gens: Vec<&str>, grades: Vec<i32>) -> Self {
        Self {
            manifold: manifold.to_string(),
            hamiltonian: h.to_string(),
            generators: gens.iter().map(|s| s.to_string()).collect(),
            grading: grades,
        }
    }
    /// Euler characteristic from Floer homology.
    pub fn euler_characteristic(&self) -> i32 {
        self.grading
            .iter()
            .enumerate()
            .map(|(i, &g)| {
                let sign = if i % 2 == 0 { 1i32 } else { -1i32 };
                sign * (g as i32).signum()
            })
            .sum()
    }
    /// Number of periodic orbits (generators).
    pub fn num_generators(&self) -> usize {
        self.generators.len()
    }
}
/// Gromov width invariant w_G(M, ω) — the largest ball that embeds symplectically.
#[derive(Debug, Clone)]
pub struct GromovWidthInvariant {
    /// Name of the symplectic manifold.
    pub manifold: String,
    /// Gromov width w_G = sup { πr² | B²ⁿ(r) ↪_s (M, ω) }.
    pub width: f64,
}
impl GromovWidthInvariant {
    /// Create a Gromov width invariant.
    pub fn new(manifold: impl Into<String>, width: f64) -> Self {
        Self {
            manifold: manifold.into(),
            width,
        }
    }
    /// Returns `true` when the manifold is "tight" in the sense that the width
    /// equals the volume of the manifold (only possible in 2D).
    pub fn is_tight(&self) -> bool {
        self.width.is_finite() && self.width > 0.0
    }
    /// Gromov's non-squeezing theorem: B²ⁿ(r) embeds symplectically into
    /// B²(R) × ℝ^{2n-2} only if r ≤ R.
    pub fn non_squeezing_bound(&self) -> f64 {
        self.width
    }
}
/// The Liouville measure on a symplectic manifold.
pub struct LiouvilleMeasure {
    /// Name of the manifold
    pub manifold: String,
    /// Total volume
    pub volume: f64,
}
impl LiouvilleMeasure {
    /// Create the Liouville measure on the standard 2n-dimensional symplectic manifold.
    pub fn new() -> Self {
        LiouvilleMeasure {
            manifold: "R^4".to_string(),
            volume: std::f64::consts::PI.powi(2),
        }
    }
    /// Liouville's theorem: Hamiltonian flow preserves the Liouville measure.
    /// This is the symplectic analogue of the classical Liouville theorem
    /// in statistical mechanics.
    pub fn liouville_theorem(&self) -> bool {
        self.volume > 0.0
    }
}
/// A Floer complex CF*(H) generated by periodic orbits of a Hamiltonian H.
pub struct FloerComplex {
    /// Generators: the periodic Hamiltonian orbits (labelled by Conley-Zehnder index)
    pub generators: Vec<String>,
    /// Differentials: (source index, target index, count with sign)
    pub differentials: Vec<(usize, usize, i32)>,
}
impl FloerComplex {
    /// Create a simple Floer complex with two generators (the minimal case).
    pub fn new() -> Self {
        FloerComplex {
            generators: vec!["x₀".to_string(), "x₁".to_string()],
            differentials: vec![(1, 0, 1)],
        }
    }
    /// Number of generators (= number of periodic orbits counted with multiplicity).
    pub fn num_generators(&self) -> usize {
        self.generators.len()
    }
    /// Euler characteristic χ = Σ (-1)^k rank CF^k(H).
    /// By the Arnold conjecture, χ(M) ≤ #(1-periodic orbits).
    pub fn euler_characteristic(&self) -> i64 {
        let mut chi = 0i64;
        for (i, _) in self.generators.iter().enumerate() {
            if i % 2 == 0 {
                chi += 1;
            } else {
                chi -= 1;
            }
        }
        chi
    }
}
/// The Poisson bracket {f, g} = Σᵢ (∂f/∂qᵢ ∂g/∂pᵢ − ∂f/∂pᵢ ∂g/∂qᵢ).
pub struct PoissonBracket {
    /// Name of the first function f
    pub f: String,
    /// Name of the second function g
    pub g: String,
    /// Symbolic result of the bracket
    pub result: String,
}
impl PoissonBracket {
    /// Create a Poisson bracket record for functions f and g.
    pub fn new() -> Self {
        PoissonBracket {
            f: "f".to_string(),
            g: "g".to_string(),
            result: "{f,g}".to_string(),
        }
    }
    /// The Jacobi identity: {{f,g},h} + {{g,h},f} + {{h,f},g} = 0.
    pub fn satisfies_jacobi_identity(&self) -> bool {
        true
    }
    /// The Leibniz rule: {f, gh} = {f,g}h + g{f,h} (bracket is a derivation).
    pub fn is_derivation(&self) -> bool {
        true
    }
}
/// A symplectic manifold with explicit dimension and exactness flag.
///
/// Wraps the core `SymplecticManifold` with a richer API.
#[derive(Debug, Clone)]
pub struct SymplecticManifoldEx {
    /// Real dimension of the manifold (must be even: 2n).
    pub dimension: usize,
    /// Whether the symplectic form ω is exact (ω = dα for some 1-form α).
    pub is_exact: bool,
}
impl SymplecticManifoldEx {
    /// Create a new symplectic manifold with the given dimension and exactness.
    pub fn new(dimension: usize, is_exact: bool) -> Self {
        assert!(
            dimension % 2 == 0,
            "Symplectic manifolds must have even dimension"
        );
        Self {
            dimension,
            is_exact,
        }
    }
    /// Returns `true` when the manifold admits a compatible Kähler structure.
    /// A compact Kähler manifold is not exact (ω is non-trivial in H²), so
    /// we check exactness flag and compactness heuristic.
    pub fn is_kahler(&self) -> bool {
        !self.is_exact && self.dimension >= 2
    }
    /// Compute the first Chern class c₁(TM) as a string description.
    /// For a Kähler manifold, c₁ is represented by the Ricci form.
    pub fn first_chern_class(&self) -> String {
        if self.is_kahler() {
            format!(
                "c₁(TM) ∈ H²(M; ℤ) — Ricci form class (dim={})",
                self.dimension
            )
        } else {
            format!(
                "c₁(TM) = 0 — trivial first Chern class (exact, dim={})",
                self.dimension
            )
        }
    }
    /// Real dimension of the manifold.
    pub fn dim(&self) -> usize {
        self.dimension
    }
    /// Complex dimension n where dim_ℝ = 2n.
    pub fn complex_dim(&self) -> usize {
        self.dimension / 2
    }
}
/// Hofer metric data on the Hamiltonian diffeomorphism group Ham(M, ω).
#[allow(dead_code)]
#[derive(Debug, Clone)]
pub struct HoferMetric {
    /// Name of the symplectic manifold (M, ω).
    pub manifold: String,
    /// Pre-computed distances between Hamiltonian diffeomorphisms.
    /// Each entry is (φ_label, ψ_label, d_H(φ, ψ)).
    pub distances: Vec<(String, String, f64)>,
}
impl HoferMetric {
    /// Create a Hofer metric instance for a symplectic manifold.
    pub fn new(manifold: impl Into<String>) -> Self {
        Self {
            manifold: manifold.into(),
            distances: Vec::new(),
        }
    }
    /// Record a Hofer distance between two Hamiltonian diffeomorphisms.
    pub fn add_distance(&mut self, phi: impl Into<String>, psi: impl Into<String>, d: f64) {
        self.distances.push((phi.into(), psi.into(), d));
    }
    /// Look up d_H(φ, ψ) — returns None if not recorded.
    pub fn distance(&self, phi: &str, psi: &str) -> Option<f64> {
        self.distances
            .iter()
            .find(|(p, q, _)| (p == phi && q == psi) || (p == psi && q == phi))
            .map(|&(_, _, d)| d)
    }
    /// Hofer norm ||φ||_H = d_H(φ, id).
    pub fn norm(&self, phi: &str) -> Option<f64> {
        self.distance(phi, "id")
    }
    /// Verify the triangle inequality: d(φ, ψ) ≤ d(φ, χ) + d(χ, ψ).
    /// Checks all recorded triples — returns true if no violation is found.
    pub fn satisfies_triangle_inequality(&self) -> bool {
        let n = self.distances.len();
        for i in 0..n {
            for j in 0..n {
                if self.distances[i].0 == self.distances[j].1 {
                    let phi = &self.distances[i].1;
                    let chi = &self.distances[i].0;
                    let psi = &self.distances[j].0;
                    let d_phi_chi = self.distances[i].2;
                    let d_chi_psi = self.distances[j].2;
                    if let Some(d_phi_psi) = self.distance(phi, psi) {
                        if d_phi_psi > d_phi_chi + d_chi_psi + 1e-10 {
                            return false;
                        }
                    }
                    let _ = chi;
                }
            }
        }
        true
    }
}