Function joint_encode_fallback_fraction 

pub fn joint_encode_fallback_fraction(
    atoms: &[SaeManifoldAtom],
    coords: &[Array2<f64>],
    amplitudes: ArrayView2<'_, f64>,
    amplitude_floor: f64,
) -> Result<f64, String>

The JOINT (multi-atom) encode-fallback fraction: the share of rows whose per-row joint reconstruction problem across CO-ACTIVE atoms is NOT covered by the composition of the per-atom certificates, so the row genuinely needs the exact multi-start solve (reviewer condition #3 — the honest encode-tax cost multiplier at scale).

The per-atom Kantorovich certificate certifies each atom’s coordinate encode IN ISOLATION — a block-diagonal view of the joint Hessian. The joint problem couples co-active atoms through the off-diagonal blocks H_kj = z_k z_j J_k(t_k)ᵀ J_j(t_j) (tangent-image inner products). When an atom’s own curvature block fails to dominate its coupling to the rest — Gershgorin: λ_min(H_kk) ≤ Σ_{j≠k} ‖H_kj‖ — the block-diagonal certificate no longer implies a joint root, a second basin can open, and the row must go to multi-start. This fraction GROWS with atom-image similarity and co-activation: no per-atom certificate covers the joint problem.

The curvature block uses the Gauss–Newton form z_k² J_kᵀ J_k (exact for flat/linear atoms, where the residual-curvature term vanishes identically); the off-diagonal is measured in Frobenius norm, which upper-bounds the operator norm, so a row DECLARED dominant is genuinely dominant and the fraction never under-reports the multi-start need. Rows with fewer than two co-active atoms have no cross blocks and are never counted as fallbacks.

amplitude_floor is the mass above which an atom counts as co-active; pass a small positive value (a domain threshold on the assignment mass, not a solver knob).