Scaled Chebyshev coefficients — proposed separate unit

Status: architecture approved; concrete implementation plan supersedes the open API choices below. No numerical production edit yet. Admission commit c5177c5 preserves the original recurrence/scaling. The parent-owned independent oracle contains reference acquisition, actual commands, frozen data and explicit cancellation limits. Its successful cross-method agreement is evidence, not an interval proof.

Contract to establish before code

For heat time tau finite and nonnegative, return K+1 coefficients c[0] = exp(-tau) I_0(tau) and c[k] = 2 (-1)^k exp(-tau) I_k(tau) for k>0. Zero time returns exactly [1,0,...]; zero order is valid. Reject negative/nonfinite heat time, K+1 overflow and impossible buffer capacity before any matvec output/scratch write. Retain the existing panic convention for these infallible entry points.

Proposed coefficient acceptance target: abs(computed - exact) <= 2^-21 * abs(exact) + 2^-149 (four f32 epsilons relative plus one smallest subnormal). This is a proposed measured/analytical target, not a present guarantee. Require finite signed coefficients; use the subnormal term only for true tiny results, never to conceal nonconvergence. K is caller-selected truncation order: accurate coefficients do not promise an accurate truncated heat action for every K. The matrix also needs the normalized symmetric-Laplacian assumptions for a spectral heat-error bound.

Do not claim support for the entire finite-f32 tau/order domain from moderate fixtures. In particular the oracle resolves low-order values at f32::MAX while Miller depth sufficient for normalization can be computationally infeasible. Before implementation, review an explicit work-admission/error policy for this case. A magic tau cutoff or silent zero vector is unacceptable. If practical work refusal is needed, its limit must be an explicit resource contract with failure before mutation; it must not be advertised as a mathematical singularity. Root selected a checked entry point with output/work limits plus a documented default-limits infallible wrapper. Prototype evidence records independent comparisons and explicit resource refusals.

Minimal numerical architecture

Keep one shared private recurrence family. Normalize the scaled sequence directly using q[0] + 2 sum(q[k],k>=1) = 1, where q=exp(-tau)I. Do not construct exp(+tau), or round an unscaled Bessel result to f32, on the coefficient path. Compute all requested orders together, rescale working values consistently, and round only final coefficients.

Replace fixed depth max(2k+60,80) with explicit depth/error admission. DLMF 3.6(iii) prescribes comparing normalized Miller solutions at increasing depth. Agreement alone is insufficient as a certified error bound: account separately for omitted normalization tail, boundary seed error and floating-point error. Positive-order monotonicity gives a terminal ratio in [0,1]; evaluate bounding terminal conditions or otherwise prove seed error. DLMF 10.37 supports this ratio bound.

One useful derived tail bound is 2 sum(q[k], k>N) <= 2 exp(tau*(cosh(s)-1) - s*(N+1)), s>0, obtained from the positive bilateral generating function at t=exp(s). Its minimizer is s=asinh((N+1)/tau) for positive tau. Evaluate this expression without cancellation for small s. This is an inference from DLMF 10.35.1/5, not a supplied library guarantee. It can expose why merely normalizing a shallow recurrence is insufficient.

Large-argument asymptotic seeding is a possible continuation if bounded work over the full finite-f32 domain is required. It needs its own validity/remainder proof; the asymptotic formula alone is not a uniform accuracy guarantee. DLMF 10.40, 10.41. Do not bolt on an unreviewed second kernel.

The separately public unscaled bessel_i must remain explicit: genuine f32 overflow is different from scaled coefficient overflow. Reconstructing it from already-rounded/underflowed q is unsafe numerically; review high-order and negative-argument behavior before sharing that path. No silent change is part of the present hygiene/admission work.

Frozen regression obligations

• Pin the independent tau=100/104 witnesses, small/subnormal/zero times, order zero, both coefficient parities, higher orders and wider practical tau. Preserve the unresolved cancellation-floor oracle entry as unresolved.
• Check the recurrence/normalization error criterion across selected depth transitions; specifically demonstrate where the old depth-80 rule is wrong.
• Test order/capacity arithmetic and negative/NaN/infinite tau rejection before matvec writes. Exercise explicit work/convergence refusal once its contract is reviewed; include f32::MAX as a reference or deliberate refusal witness.
• Compare heat actions against analytic zero/two-element operators and an independent dense reference with truncation error separated from coefficient error. Retain tolerances and describe their derivation.
• Run debug/release, strict acpu Clippy/docs and the owning workspace gates on the exact reviewed source. Version/package proposal follows this unit only after its actual supported domain and failures are settled.