Struct CertifiedChart 

pub struct CertifiedChart {
    pub region: ChartRegion,
    pub lipschitz: f64,
    pub beta_center: f64,
    pub certified_radius: f64,
    pub amortized_jacobian: Option<Array2<f64>>,
    pub recon_center: Array1<f64>,
}

One offline-certified chart: a center, its Kantorovich constants, and the certified Newton-convergence radius R_c solved from h = β·η·L ≤ ½ at the worst-case in-chart start.

Fields

region: ChartRegion

lipschitz: f64

Closed-form Hessian-Lipschitz constant L over the chart.

beta_center: f64

β = ‖F'(t_c)⁻¹‖ at the chart center (worst-case in-chart start uses the center’s curvature; the radius is solved so the certificate holds for any start in the ball).

certified_radius: f64

Certified Newton radius: starts within radius of t_c satisfy h ≤ ½.

amortized_jacobian: Option<Array2<f64>>

Distilled amortized-encoder Jacobian for this chart (#1026 ladder item 3).

The exact encode map x ↦ t solves F(t; x) = J_m(t)ᵀ(m(t) − x) = 0. By the implicit function theorem its derivative at the converged root is dt/dx = −(∂_t F)⁻¹ (∂_x F) = H⁻¹ J_m (since ∂_x F = −J_m), so the first-order Taylor expansion of the encode map about this chart’s center t_c is the closed-form AFFINE predictor

t(x) ≈ t_c + (1/z) · A₁ · (x − z · m₁(t_c)),   A₁ = (J₁ᵀJ₁ + ridge·I)⁻¹ J₁,

with J₁ = Bᵀ J_Φ(t_c) and m₁(t_c) = BᵀΦ(t_c) the AMPLITUDE-1 reconstruction jets (the amplitude z factors out analytically, so the stored Jacobian is amplitude-free). This is the DISTILLED amortized encoder of the #1026 thread: the per-row Hessian factorization + Newton iteration is moved OFFLINE into this d × p matrix, leaving a single O(d·p) mat-vec online — no per-row eigendecomposition, no second-jet evaluation. The Kantorovich certificate is still evaluated AT the predicted start, so the amortized prediction is trusted iff h ≤ ½ and an uncertified row still routes to the exact multi-start solve (the encoder approximates inference, the certificate keeps it honest — the thread’s “encoder + certificate-gated exact fallback” deployment). None when the center’s Gauss–Newton block is singular (no certifiable amortization).

recon_center: Array1<f64>

Amplitude-1 chart-center reconstruction m₁(t_c) = BᵀΦ(t_c) (length p), the anchor the amortized predictor expands the encode map around.

Trait Implementations

impl Clone for CertifiedChart

fn clone(&self) -> CertifiedChart

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for CertifiedChart

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.