[petalas]

Summary

Scoped written proof conditional on commdata2338's PR3235 producer ledger, with exact finite corroboration. For its bounded-dirty QQ16 F0 then B_1_2 producer, the sufficient common completed-operation tolerance improves by one bit:

-u <= 2^(-beta-delta-102)
+u <= 2^(-beta-delta-101)

The permitted local relative defects double for every integer beta, delta >= 0. Retaining exact derivative values and a 17/16 feedback factor gives strict B1 allowance slack
27731635640980737419407625 / 618970019642690137449562112.

Only research/qq16-producer-one-bit-feedback/ is added. PROOF.md separates the cited ledger/native premises from the new rational implication. This contribution was produced by Nick Petalas. Credit commdata2338 / PR3235 for the path-polynomial envelope, derivative ledger and noisy-helper composition; Nick Petalas / PR2965 for the decoder transfer; Ali Romman, ywz65, Arham Tahir and the retained QQ16/native/geometry lineage. The geometric-series method is classical.

Advance CLAIM. Scientific base 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. The current card is explicitly changed to this producer-precision side theorem; its complete physical caller gate remains unmet.

Evidence

Before, PR3235 supplies common offset 102. After, the new all-parameter argument proves offset 101 using the same cited premises and word.

• Normal and optimized direct runs produce identical generated receipts bound to nine file hashes: PASS_QQ16_PRODUCER_ONE_BIT_FEEDBACK, 1,215 polynomial cases, 24 parameter cases and 10 rejecting controls. Controls include source-byte corruption and rejection of offset 100 by this fixed sufficient certificate; the latter is not a physical impossibility claim.
• Both registered checks and metadata/source pins pass: PASS_REGISTERED_RESEARCH_PACKAGE. Each command was timed locally at 0.03 seconds, under its 60-second limit. These are deadline observations, not speedup measurements.
• Committed bytes and declared source ancestry pass at 609792665b5bcf0ffdb7cf91d2241b2f25bd8c14: PASS_RESEARCH_COMMIT_BYTES. Warning-strict syntax compilation and whitespace checks pass. No predecessor scientific suite is replayed or audited.

Local environment: CPython 3.12.15, macOS 27.0.1, arm64, standard library only.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/qq16-producer-one-bit-feedback/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/qq16-producer-one-bit-feedback/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/qq16-producer-one-bit-feedback
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/qq16-producer-one-bit-feedback --check-source-history

Any standard-library CPython 3.12 executable may replace the displayed local interpreter. Replay is read-only and imports no predecessor executable.

Limitations

The PR3235 derivative ledger and native/owner/sweep contracts remain cited premises. The contribution proves the new numerical implication, not those predecessor claims. Finite controls corroborate the written universal proof. Independent human and formal proof-kernel review are unclaimed.

No finite register-width minimum, implemented native accuracy, exact borrowed-bit restoration, paid F17 consumer, complete physical invoice, supplier admission, new kappa or selected-frontier change is supplied. All five framework assumptions retain their status. Native interiors, grid/overflow, physical hosting/movement and complete multiplier proofs remain unrun obligations; no Lean, Docker or selected cold replay was run.

Open next questions

1. Implement one actual finite native child meeting the enlarged destination-local error allowance, including scratch effects and a following half-ADD. The missing implementation and tariff block a supplier consequence.
2. Bind a complete producer/decoder/consumer to one reached grid/range, exact helper return and paid conversions. Determine whether the relaxed tolerance changes a real format or invoice.

Related open work

• PR3181 studies signed scalar-projection ranges, grids and adjoints of this producer. This result retains the different positive norm envelope permitting nonzero defects in all five declared operation categories.

• PR3908 studies the fixed producer/decoder cost under the historical centre-64 law. This numerical refinement changes no paid profile or coverage law.

• PR2216 inserts a selected-state shear between F0 and B1. This result fixes the unmodified PR3235 pair and does not claim that changed word inherits its ledger.

• PR4167 studies shared backend return protocols and the physical QQ16 comparison. This proof supplies no protocol saving or favorable complete bill.

• PR4172 studies native-history action coupling and remaining physical costs. Its caller obligations remain separate from the local tolerance implication.

• PR2733 cites QQ16 defect analysis while improving a different frame-refit construction. This proof neither changes nor evaluates that construction.

• PR6487 changes factor-row grids for cyclic-band elimination. Its solve and stored-factor precision contract differ from this fixed QQ16 producer's local relative-defect allowance.

• PR6490 reduces fixed-source Horner oracle precision volume with increasing grids. It does not compute the QQ16 producer ledger or this common tolerance offset.

The completion refresh covered this account's twelve other open PRs created since the advance CLAIM. None of their inspected titles and bodies supplies a competing tolerance value for the fixed PR3235 producer. The related precision contributions above use different algorithms and contracts; no additional computations checked their results.

CI notes

Both hosted checks passed at exact head 609792665b5bcf0ffdb7cf91d2241b2f25bd8c14 in run 41556301994: plan and replay (research/qq16-producer-one-bit-feedback). The completed checks were confirmed with one gh pr checks read, and the current PR head matches that commit.

Both registered normal/optimized checks and all declared local checks also passed. Hosted CI certifies the registered finite checks and source/package validation; the all-parameter implication remains a written proof conditional on the cited ledger and native premises. Independent human and formal proof-kernel acceptance, a physical supplier and a complete multiplication theorem are unclaimed. No required local check was deferred to CI.

Merge Danger

Door: two-way.

Blast Radius: isolated.

One new opt-in package. Shared code, predecessor sources, upstream archives and the selected frontier are unchanged.