Complete physical Gaussian expansion tiles with final-grid error accounting

[jmbarrancoml]

Change and mathematical scope

I implement a complete physical normalized Gaussian expansion S/2 tile. The producer generates directed real coefficients and ordered pages; the consumer executes the actual signed convolution, normalization, final-grid extraction and overlapping source-window stream. This closes the coefficient-generation and complete normalized-output handoff left by PR1243 and the compression-core PR3851.

For integer p>100, integer 2<=alpha<sqrt(p), exact u=alpha², 1<t/s<2 and the retained halo domain, I give a conditional error bound <(sqrt(2)+2^-11)*2^-p<p²*2^-p, output norm <1, online time O(Lp log²(p+2)), and space/catalogue size O(Lp). I charge polynomial(t,p) preparation separately. The short-block factor bounds also cover u*(t/s-1)<1; I do not import the narrower combined S/E lemma. I retain PR371's separate final-grid truncation term.

I leave the selected frontier unchanged. The production-sized payload, full tensor/page-reuse/routing schedule, CRT/integer join and independent proof review remain open. The implementation retains the predecessor word-recovery, numerical and analytic premises.

Task and overlap

I started from canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision6. I refined uniform-transfer-audit to the complete expansion handoff and recorded its producer/consumer boundary, full bill, comparison and stop rules before implementation in Discussion3664.

I pin PR3851 at 7ab67b4a84d28cd562b9711aa315ca5b47822b37 through its complete authenticated export, PR371 at 988ef67986c313445936fc7103908b6163b9d47e, and PR1243's handoff at c8d2d7f685b9fded81212433970de29fe134f197. I retain the rad20 blocked-Gaussian, microbox and bulk-transfer sources.

The broad overlap search covered577 open PRs,2964 closed/merged PRs,2532 Discussion records and4126 local records, with complete pagination and zero warnings. At01:21 UTC on October19 I refreshed main with no guidance delta and checked the dependency heads, all of3664, the latest100 hub comments with their six replies, and100 recent all-state PRs covering all three updates since00:55:29. I found no competing complete expansion or dependency objection in that coverage. I did not inspect edits to older hub comments. Dugongue's PR3872 supplies a complementary rational-operation inverse update and leaves physical integration open.

Validation

I execute two nonzero blocks and two complete tiles against separate staged and periodic Gaussian references. I exercise equal source starts, a periodic seam, seven padding records and a clipped final block, and match the full word invoices of88 and130 calls. I also check78 exact factor/crop identities, production metadata at four tiles, ten domain rejections and nineteen ownership/failure controls. The short-input control found a scratch-lifetime bug, which I corrected before the successful reuse checks.

I passed source-history and committed-byte checks at 1a86a8fb73bb7a0982fff99852a031832c191902. All eight suites passed from its official export outside Git, and I verified that all37 files including the receipt stayed unchanged. The block, overlap, wrapped-tile and ownership commands took72.572,60.796,136.742 and37.514 seconds locally. CI39185264556 passed at the exact head. I matched all eight hosted result records to the export, verified the unchanged-input step and confirmed the tested merge ab2052a8be679654d967083a4f745d19a110226a has the publication tree. The hosted replay ran from01:28:17 to01:38:41UTC on October19.

• Reproduce: python3.12 -B scripts/research_package.py --package research/physical-gaussian-expansion.
• Environment: Python3.12.13, Darwin arm64; standard library. Eight commands,1180-second total timeout allowance.
• Export SHA256: 898dd6c8c1e53df8a1040918797b63128fc1d744f234d1f1e623b5acc85f198c.
• Receipt SHA256: 07ab9b5eec58f812793495ad76b6cede6a7e9912aeb3b53df3b82ec8bd08827d.
• Head tree: 99ba2a07ce55a2fa4c671eb43b71f255c4fa8e3e.

The small complete packets use the actual fixed-crossover word API's base/Karatsuba path. PR3787 supplies separate FFT-path evidence. The production checks inspect encoded metadata without allocating their P-grid payload. EVIDENCE.md distinguishes these finite checks from the quantified proof and records the earlier checker/runtime failures.

Credit

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance in mathematics, implementation, source-analysis agents, tests and prose. I credit eumemic and icekylinx for real-chirp work; Hanson Sun for PR1243; Artus Krohn-Grimberghe for PR371; Chafik Boukhalfa for the Gaussian/chirp predecessors; RaD/hipotures for the retained reports; Mason Hale for the numerical reference; Dugongue for word-ABI and inverse precedents; and Harvey–van der Hoeven, the OpenAI manuscript contributors and Douglas Colkitt for the retained framework. The complete source archives preserve their additional authors, licenses and notices. These are AI-assisted author checks, without independent human or formal-kernel review.

Open next questions

The package README and TASK.json give concrete follow-ups. The next caller must restore a complete natural target tensor before invoking PR3763 once, pay canonical global gathers and short mixed-radix moves, reuse coefficient pages and assemble compression cores. Count D actual axes in gamma=sum_i(2u_i+j_i+1) and retain both final S factors in integer recovery. These joins and their complete physical bill remain open.