Research: paid-copy E8 exact-RAM realization with an attributed DFT corollary

research/e8-paid-copy-ram-20261013/NOTICE.md

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+# Attribution, provenance and AI assistance
+
+This contribution is prepared for Artus's Tintagel mathematics verification
+campaign with substantial OpenAI Pro and Codex assistance. It is a research
+draft under this repository's Apache-2.0 license.
+
+OpenAI Pro supplied the tensor-E8 construction and its complete provider proof.
+PROVIDER-PROOF.md preserves that text verbatim (SHA-256 recorded in SOURCE.json).
+Codex recovered and independently audited its identities, prices and RAM proof,
+then bound the depth-one positive-C provider to the existing exact-Fourier
+compiler. A separate fresh Codex adversarial referee rederived the depth-one
+operator and transfer and wrote the exact checker adapted here. These roles and
+AI assistance are disclosed; the contribution is not credited solely to the
+packaging agent or described as a human-only or formal proof.
+
+The imported OpenAI exact-Fourier compiler, synchronized-sector traversal,
+canonical-root/CRT/chirp all-length transfer and source explanations are retained
+literally at the immutable pin in SOURCE.json. Their attribution and references
+to Newton interpolation, displacement methods, Bennett/dirty replay, Good's
+coprime factorization and Bluestein's chirp identity remain in the source files.
+OpenAI's Apache-2.0 LICENSE is included. No original h=100 supplier rate or
+synthesis theorem is imported as this provider's result.
+
+Prior E8 and community framework credit includes Jacob Sussman's E8 unit,
+phase-frame/cover and WHT/Fourier work; the Croc community contributors and their
+notices; Chafik Boukhalfa's explicit programs and exact-DFT transfer package;
+eumemic's batched recursion and transfer; Ryan Shea's community-transfer idea
+and independent audit; and the P390/P401/paired-cube antecedents described in
+the provider source. This package uses no contributor program as an executable
+dependency. It does not claim to invent E8 geometry, dirty response compensation,
+orthogonal covers, tensorization or cyclic transfer.
+
+The originally preserved P390 provenance notice follows unchanged. It attributes
+that historical word; its source claims are not automatically this draft's
+claimed theorems or added savings. The new provider rebuilds and pays its own
+larger helper core and private tableau. See NOVELTY.md for the exact overlap.
+
+---
+
+# Provenance and credits
+
+This package retains upstream code, data, licenses and notices. The new contribution is the target-prefix recovery construction and explicit dirty preconditioning for heterogeneous helper endpoints, together with the changed composition and checks.
+
+Direct sources in CrocSwap/integer-mult-bounds:
+
+- PR376: exact helper reuse and its reconstruction selections.
+- PR377: compressed nine-stage complex E8 supplier, bank padding, Clifford/cover framework.
+- PR378: the far-retimed complex word on which the new prefix construction operates.
+- PR381: nine-stage unsplit bit assembly, inherited exact moment and outer inequality machinery.
+
+Essential ancestry and directly used ideas include PR352's E8 local identity and original sources, PR363 and PR373's stage constructions, PR370's bit E8 supplier, PR375's unsplit degree-two halving, PR315's compiler/dirty response machinery, PR266's exact checkers, and PR234's geometry machinery. Their bundled authorship and licenses remain authoritative. PR365's coupled complex moves are part of the retiming lineage. PR371's rounding correction and PR379's signed-compensation discussion inform the explicit conditional scope.
+
+PR382 is the frozen stronger benchmark that invalidated our initial composition-only novelty claim; PR383 is the subsequent low-read composition reviewed in the same sweep. Neither is presented as our new mechanism. See research-notes/PR-SWEEP.md for every new PR surveyed in this round.
+
+All PR numbers refer to https://github.com/CrocSwap/integer-mult-bounds/pull/<number>. No publication, push or external message is performed by the supplied scripts.

research/e8-paid-copy-ram-20261013/NOVELTY.md

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+# Pre-submission overlap and attribution check
+
+This is a scoped current-source comparison, not a worldwide priority search.
+The public Croc main head checked for this draft is
+`17ce0c79b7b51c2634b439ea5a2b754f09909693`; current OpenAI Family130 scope
+still has Git blob `c4419a9add293ab5f0fd51c72a193238875affb3`.
+Complete selected source bodies and exact-head PR metadata were read. Search
+results and missing results are not used as proofs of universal absence.
+
+| Prior work | Inspected pin / source | Overlap and limit |
+| --- | --- | --- |
+| OpenAI Family130 / exact-DFT manuscript | `openai/math@adc7f1241b42e322a6451854ab7e4b4c146bf78a`; same imported blobs at `1581d4cbdd69a0faf34b2089614fe2a814316a6a` | Already supplies the exact-width compiler, synchronized sectors, CRT/chirp all-length transfer and a supplied-root exact-RAM power saving. Those mechanisms are imported with attribution. |
+| Community exact-DFT transfer, PR225 | `830971b6b157374ebf941d0ca0b73fd2b956e3e0`, `research/exact-dft-transfer/PROOF.md`, blob `a25684e4aa2d901b76a786579f635867fcd1d832` | Already transfers community suppliers and records larger numerical savings, including 7009/10^7. Credit eumemic's batched-transfer work, Ryan Shea's transfer idea, and Chafik Boukhalfa's package. This draft is not the first community transfer. |
+| PR256 and independent audit PR274 | PR274 `947726d431438fd73ee4c96231eb88aba2692360`; current PR descriptions and pinned notice | Report explicit program/Lean framework and a larger WHT/DFT saving near 7.54736e-4. These source-reported rates exceed this draft's 1/15000 tensor saving. No record-rate or newly completed Lean claim is made here. |
+| Sussman's E8 unit / Croc PR352 | `04b4c3c478b5bd8797d2663d89d8db4dfd0ad9f2`, `research/w3-p10b-e8-complex/README.md`, blob `b5f6f5acaa71745de31c84c6944cd414d336244e` | Earlier h=9, v=120, R=783 E8 unit and eight full-frame totals. Its documented complex rate is larger. This draft uses the same port/root geometry but an independently rebuilt, more expensive core, R=6968, m=27, and explicitly paid private response compensation. |
+| E8 prefix restoration, PR390 | `0e790f2dfb64e17bdee154aee151bb0c1dddb28e`, proof blob `4bc6d37b8bd9693e8b7bb5a6a6bd59f972493270` | Earlier target-prefix recovery and dirty preconditioning on the existing E8 word. This is not a claim to first restore dirty E8 helpers or remove its center bill. That word's improvements are not added to this invoice. |
+| Response-space gauges, PR401 | `7844f72c6051a4408284c3092490687061675c1a`, proof blob `c8ae460edfb1a68d656abb1591c359c45d496d29` | Existing helper-response entrance/exit mechanisms. They have different fixed words and cost profiles; no blanket acceptance or exclusion is inferred. |
+| PR631 E8 punctures | `fb893b5ee1317c2da3b297e6fe78eca0857d1a76`, open at the metadata check | Scoped bit-side star label-reader thresholds and ratio 63/111, explicitly excluding general pairing-rank decoders/helper tradeoffs. Its F2 reader space is not the real rank-eight response space in binary ambient dimension nine used here. This draft does not refute its stated theorem or copy its census. |
+| Croc selected multiplication frontier | Main's conditional PR626 lineage | A different target and dependency chain; its multiplication kappa is not this tensor exponent. This draft changes no selected witness or multiplier claim. |
+
+The distinct deliverable is a reproducible, independently scrutinized **paid-copy
+depth-one E8 positive-C realization** with its complete logical/phase/owner and
+private-memory proof, an explicit moment and all-matrix padding argument, and
+the attributed exact-DFT corollary in the stated RAM. It is useful as an
+alternative construction/reproduction and dependency audit, even though it has
+more helpers and a weaker rate than earlier community suppliers.
+
+The Pro-assisted provider text is preserved verbatim. Its broader tensor-family
+sections remain source material; the submission's main theorem and fresh
+adversarial review are restricted to architecture depth one. The cyclic compiler
+and all-length mechanisms are OpenAI's earlier work, not Pro/Codex inventions.
+The independently written referee checker is adapted only to this package's
+authentication and output interface, with explicit source hashes in SOURCE.json.
+
+No stronger supplier, global priority, best-known rate, first E8 construction,
+first dirty-helper method, new Lean proof, bit-complexity or integer-multiplication
+result is claimed. Earlier contributions and the Pro construction retain credit.
+
+Proposed publication scope is only
+`research/e8-paid-copy-ram-20261013/`, branch
+`codex/e8-paid-copy-ram-corollary-20261013`, in a draft PR to
+`CrocSwap/integer-mult-bounds`. The fourteen files are README.md,
+PROVIDER-PROOF.md, CYCLIC-COROLLARY.md, verify.py, RESULTS.json, SOURCE.json,
+NOTICE.md, REVIEW.md, NOVELTY.md, SHA256SUMS.json and the three imported
+OpenAI sections plus their LICENSE. No existing witness, source, workflow or
+upstream file is changed.