Certify Gaussian expansion headroom for the physical chirp caller

[Dugongue]

PR3851 leaves Gaussian expansion and its norm join to PR3763 open. This package proves a sufficient numerical contract: for every positive integer source/target length and width squared u >= 1, downward dyadic Gaussian coefficients and inward complex p-grid rounding return norm at most 21/38 < 0.6 on every unit-disk input. The exact remaining margin is 9/190. Tensor products retain the product norm bound and an explicit contraction-weighted error recurrence.

The complete informal proof combines Gaussian Fourier duality, Poisson periodization, theta majorization, positive operator domination, directed rounding, tensor packing/telescoping and host window frames. Ten inspected linked-library sources are recorded with hashes and exact adaptation limits. Classical methods are credited; bounded source/DAG search does not establish first use, novelty or a fresh kernel check.

Registered Python 3.12 replay passes: 34 Gaussian rows, 340 complex line values and 80 tensor coordinates, compared with a separate residue-first oracle and direct Kronecker evaluation. They share Mason Hale's credited interval backend. Seven semantic control groups pass normally and with assertions disabled, including coefficient/center/deficit changes, invalid domain/cache aliases, missing periodic windings, coarse nearest-grid rounding, dirty exterior ownership, source-hash and actual exported CLI/receipt corruption.

This is an exact host reference and an informal universal sufficient proof. Physical expansion, protected encoding/handoff, catalogue and halo routing, the full tensor schedule, CRT and final integer recovery remain open. No new kappa, selected-frontier change, new Lean formalization, independent human review or practical speedup is claimed.

Credit Jose Manuel/jmbarrancoml's PR3851 and PR3763, Mason Hale's certified Gaussian recovery and retained PR2298 lineage, RaD/hipotures, Chafik Boukhalfa's PR481/486/489 and PR479, Ross Richardson's PR948, corrected PR3709/3787, Harvey–van der Hoeven, the retained OpenAI manuscript/Douglas Colkitt, Mathlib's Sebastien Gouezel and David Loeffler, Boris Alexeev/Codex and the shared MathProof sources. Exact predecessor documents and notices are preserved. Prepared by Dugongue with substantial OpenAI Codex assistance.

Reproduce: python3.12 -B scripts/research_package.py --package research/gaussian-expansion-headroom. Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. This round checked 578 open PR bodies and refreshed recent all-state records through PR3875, actual related sources and partial Discussion HTML. The selected baseline binding check passes; the unrelated selected cold replay was not rerun.