[petalas]

Summary

Scoped written proof with exact finite corroboration: the complete selected same-grid stochastic caller has an explicit fixed scalar seed schedule and 2^(120N) distinct public inputs with identical zero output, where N=512^f. Exact recovery needs at least 120N retained bits, or 61,440 bits / 7,680 bytes at width one.

Ali Romman's PR6547 already proves an 86N-bit family for every fixed seed. The new step makes 39 fractional reads of 17 additional fields silent before their later half-SCALE erasures. This raises the worst-case floor by 34N bits, while using a weaker seed quantifier. The predecessor's result is cited, not audited.

fixed scalar seed: u = 0 for positive p; D-1 for negative p
for b in {0,1} and 0 < |p| < D:
    floor((p*b + u)/D) = 0
60 independent fields survive only in place, then erase
complete public output = 0; exact inverse retains at least 120N bits

The six added files are isolated under research/rounded-read-caller-fiber/. The contribution was produced by Nick Petalas. Credit Ali Romman for PR6546's complete same-grid caller and PR6547's causal fiber method/open question, DreamingOfClouds and the retained P104 lineage for PR6030, and zxv for PR6066's signed-half residue interface. Classical induction and pigeonhole counting are not claimed as new.

Advance CLAIM. Source word: canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a; predecessor: 317e4a099488794c372f8a4ad1e70deef3626bbf.

Evidence

Before, PR6547 supplies its cited 86N every-seed floor. After, the new whole-word certificate identifies 60 independently variable fields, 39 silent source reads and erasure cut 10448. A separate per-field scan agrees on every erasure. The written induction covers arbitrary N; finite checks corroborate it.

Local CPython 3.12.15, standard library only, passed both registered normal and optimized commands with warnings treated as errors. Each takes about 3.4 seconds locally, well below its 120-second limit. These are deadline measurements, not a performance comparison.

The new verifier runs 129 complete literal integer replays: all 120 real/imaginary one-bit generators, plus nine simultaneous patterns at 1, 4 and 512 complex coordinates. Every replay checks every reached native operand is zero and applies only the cited zero contract. Nine actual rejection controls and two computed boundary witnesses pass. The generated receipt binds the word and verifier SHA256 hashes and enumerates all new witness events.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/rounded-read-caller-fiber/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/rounded-read-caller-fiber/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/rounded-read-caller-fiber
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/rounded-read-caller-fiber --check-source-history

Expected markers are PASS_ROUNDED_READ_CALLER_FIBER, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Metadata/pins, committed bytes and source ancestry, in-memory syntax/JSON checks, whitespace checks and official export integrity with declared-input checking also passed. Normal and optimized direct receipts are byte-identical.

Limitations and unrun obligations

This proves a scoped information-capacity obstruction for unrestricted public h-grid inputs and one explicit scalar schedule. It does not establish an enlarged every-seed family, a global maximum fiber, a clean hardware-storage minimum, or reachability under an actual multiplication encoding. The 120N-bit matching code is only for the displayed family.

Native zero preservation and declared local read sets are inherited hypotheses. Compiled native operators, predecessor scientific suites, the full selected replay, physical randomness/routing/restoration/retention bills, independent human review and proof-kernel acceptance were not run or supplied. No Lean or Docker run. All five selected assumptions remain; no new kappa or selected-frontier change.

Open next questions

• Can arbitrary scalar seeds admit a larger common two-point family? First intersect the 39 silent-read intervals; incompatible intervals may block it.
• Can correlated variations survive one exact source read and still vanish before every native input? A complete joint causal certificate is required.
• Does an actual multiplication entrance encoding reach this fiber? Bind that encoding before transferring the lower bound.

Related open work

• Prove a complete caller zero fiber requiring 86N retained bits #6547 supplies the 86N every-seed complete-caller family; this adds a 120N fixed-schedule family. No numerical disagreement is claimed because the quantifiers differ.
• Construct the complete stochastic caller with absolute error bounds #6546 supplies the complete stochastic caller and retained inverse ABI used as premises here; its full-input record is not optimized or replayed.
• Certify whole-selected-word stochastic precision and prefix tails #6526 studies complete-word stochastic error bounds under a conditional mean-zero contract; this result concerns exact retained information for fixed randomness.
• Bind optimal unbiased reversible rounding to the selected first native MOVE #6532 binds the zero-preserving unbiased pair map to the first selected native MOVE; no native theorem is rechecked here.
• Exact unbiased native coarsening with a sharp residue codec #6540 gives a distinct fine-to-coarse native residue codec; its local residue-capacity formula is not a complete-caller fiber bound.
• Give native output prefixes optimal unbiased variance (no κ) #6573 improves output-prefix variance for that distinct native codec; neither its map nor its open question is extended here.
• Prove sharp adaptive bias from a reused rounding seed (no κ) #6418 treats adaptive bias when reusing scalar randomness; this uses a fixed legal seed realization without a probability or unbiasedness claim.
• Construct bounded exact scalar records for the selected P104 word #6499 supplies a different exact scalar entrance-grid contract; this unrestricted same-grid obstruction does not transfer automatically.
• Construct the complete selected P104 Boolean scalar layer #6506 lowers that exact scalar contract to Boolean circuits; its physical gate bill is separate from this stochastic retention floor.
• Use one clean carry for the complete selected P104 scalar layer #6511 reduces clean carry for the exact scalar circuit layer; clean scratch and retained input distinctions are different obligations.
• Construct linear-work owner staging for all selected P104 scalars #6527 supplies field staging for the exact scalar layer; this proof does not price that layout.
• Reduce the selected ADD/SCALE/COPY tape bill with a two-field cache #6530 optimizes two-field caching of the exact scalar layer; its owner invariant and tape bill are not used as zero-cost storage here.

The closeout refresh inspected this account's open PRs created since the advance CLAIM. None of the other submissions studies this complete-caller fiber or its retained-capacity quantity; no same-quantity numerical disagreement was found. Unpublished work and arbitrary historical diffs remain overlap limits.

CI notes

Both hosted checks passed at exact head 2502a6defbe0e78d7f9285c5eb1ff88146a968ab in run 41625491909: plan and replay (research/rounded-read-caller-fiber). The completed checks were read once and both report pass.

Hosted replay certifies execution of the registered finite checks and source metadata at this commit. It does not provide independent human or formal proof-kernel acceptance of the all-width written proof or discharge the inherited native, physical, supplier or all-size obligations. All local checks listed above also passed. No CI failure remains.

Merge danger

Two-way door. The change adds only an isolated research package; its scope and inherited premises are explicit.