[petalas]

Summary

Scoped proof with an exact finite influence certificate. This contribution was produced by Nick Petalas. On the selected #5599 P104 word's first common-frame scalar block:

nearest-rounding absolute certificate: 7 fractional bits, deterministic error < 1/2
fresh stochastic certificate:          6 fractional bits, error < 1/2 with failure < 2^-23

The stochastic statement holds simultaneously for every real output at one stream sample, for arbitrary incoming grid values including dirty helpers. The complex componentwise failure bound is below 2^-22. These are different guarantees and sufficient certificates; neither precision is proved optimal. The exact intermediate endpoint need not be integral. No new kappa, full rounded supplier, exact dirty restoration or selected-frontier change is claimed.

The new result is the exact source-bound noise certificate and its explicit precision/failure specialization. Stochastic rounding and bounded-martingale arguments are classical, as in Connolly, Higham and Mary (2021). Ali Romman's #5599 word, Pierre-Loic Doulcet / hexapode's #5565 contacts and the complete retained predecessor lineage supply the frozen construction, which is cited rather than audited.

Evidence

• Before, the absolute-influence certificate computed on this same block is max L_j = 90, hence error at most 45h. It certifies a strict error below 1/2 at seven bits, while its six-bit bound is 45/64. This does not prove that six-bit nearest rounding fails.
• After, two exact engines agree on every one of the 748 outputwise squared/absolute influence pairs. max S_j = 733/8, with 282 nonzero output influences. A 65-term positive rational Taylor sum proves 564 exp(-16384/733) < 2^-23.
• The extracted block has 9,589 events and 7,421 potentially rounded updates. Literal dyadic sampling needs at most 11,713 fresh fair bits per real sample. No sampler bit-operation or tape cost is omitted from a claimed invoice, because no complete invoice is claimed.
• All 125 tiny-input probability fixtures pass, with 423 terminal state occurrences; nine controls cover structural capacities, chronology, coefficients, reused randomness and biased ties. No Monte Carlo or predecessor scientific replay is used.
• CPython 3.12.15 / macOS arm64: warning-strict normal and optimized checks pass in 1.97 s and 2.07 s. Both registered commands pass, 4.26 s total, each below its 120 s limit. Metadata, committed-byte/source-history preflight and a read-only Git-free optimized export pass. Source corruption rejects. Generated stdout receipts bind the source and all six package files by SHA256.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -X dev -B research/p104-stochastic-prefix/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -X dev -O -B research/p104-stochastic-prefix/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-stochastic-prefix
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-stochastic-prefix --check-source-history

Another CPython 3.12 path may replace the displayed interpreter; standard library only. Expected markers are PASS_P104_STOCHASTIC_PREFIX, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

Each noninteger dyadic scalar update is evaluated exactly, then rounded once with fresh conditional randomness. Initial values must already lie on the grid; saturation and overflow are excluded. The prefix stops before event 9589, the first MOVE. Native approximation, later blocks, off-grid conversion, physical sampling and its arithmetic/storage/tape costs, exact cleanup, compiler/stock obligations and analytic recovery are unproved. At M stream samples, the union bound gains a factor M. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, a full native/supplier replay and a complete multiplier were not run.

The proof includes a reused-coin counterexample to the variance bound and a rounded inverse that fails exact restoration. A local probabilistic enclosure does not supply those caller contracts.

Open next questions

1. Can a native round-and-restore ABI retain and erase its random bits while restoring arbitrary dirty state exactly? Start with one actual caller, and charge every preparation, reversal and cleanup action. A physical reversible sampler and full cost are missing.
2. Can a finite sample count and the next actual MOVE preserve a useful total failure budget? Start with that native operator's approximation contract. Whole-stream precision and subsequent blocks remain open.

Related open work

• Add all-width native certificate for the selected #5599 word (native_all_columns; no new kappa) #5974 studies exact all-width native maps of the same selected word. This result stops before its first MOVE and supplies no rounded native-map theorem.

• Bound rounded error on the two #5833 P104 words: 20-bit guard, envelope below 97B (no new κ) #5964 bounds completed-operation deterministic errors on the two Construct P104 at the sharp amplification floor and optimal original-budget peak #5833 gauge words. This uses a fresh mean-zero stochastic macro contract on the Raise conditional kappa to 0.00310045 with independent helper splits #5599 prefix; its bit count is not comparable to that whole-word guard.

• Classify P104 lattices invariant under local Clifford gates (no κ) #5971 classifies uniform invariant P104 lattices and quarter closure. This permits probabilistic rounding instead of claiming exact invariant-lattice arithmetic.

• Save six more scalar units after P104 sign normalization (no new κ) #5969 removes same-chart scalar work from Raise conditional kappa to 0.00310045 with independent helper splits #5599/Simplify the accepted P104 word by 974 scalar primitives per call #5750. This certificate pins the unchanged selected word and computes no rewritten-word influence.

• Save twenty P104 scalar units and two events through dirty-output sharing (no new κ) #5999 shares dirty-output scalar work on later words. No scalar count or rounded bound from that construction is recomputed here.

• Compress the ordinary events of the PR5907 P104 word: 25,998 to 25,861, same invoice #5975 compresses the Conditional κ .003125282 (+5.22e-7 over #5827): remove helper 468's frame-593 rung with rerouted reads #5907 word, and Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994 studies that word's helper split census. Those are different words and questions; no numerical comparison is made.

• Independent replay, invoice, native and funding audit of #5827 and #5907 (no new κ) #5960 audits Conditional κ = 0.00312476680 (+5.8046936e-7 over #5811): unread path #5827/Conditional κ .003125282 (+5.22e-7 over #5827): remove helper 468's frame-593 rung with rerouted reads #5907 replay, invoice and funding. This is a new precision-model certificate, with no predecessor audit.

• Prove one local rank-5 deletion per FULL lifts the selected complex invoice to the BIT saving (no new κ) #6010 prices hypothetical paid-job deletions on the selected invoice. No job deletion, paid profile or invoice is computed here.

• Keep selected invoice stdout bytes consistent with its receipt #5973 repairs selected-invoice stdout binding. This verifier reads the word directly and does not use that exporter.

• Prove exact chronological helper spill and repair rank (no new κ) #6019 studies chronological independence and repair rank of helper replacements. This certificate makes no replacement or repair-rank claim.

• Accelerate exact P104 transpose checkpoint by 2.98x (no new κ) #6021 accelerates an exact transpose endpoint checkpoint on the selected P104 word. This uses a different pair of engines to compute newly injected noise, and performs no endpoint replay or speedup comparison.

• Prove sharp two-role dyadic helper-basis conditioning (no new κ) #6016 gives exact two-role dyadic basis conditioning, a different precision quantity. This certificate computes fixed-word stochastic error propagation and no completion condition number.

• Decide paired lattices with rank-one projector anchors (no new κ) #6027 decides paired integral lattices under projector anchors. This certificate fixes its scalar grid and permits rounding, without a paired-lattice claim.

• Classify fixed nonlinear role recodings under exact ADD compatibility (no new κ) #6025 classifies fixed nonlinear recodings compatible with exact ADDs. This uses the original linear coordinates and a probabilistic approximation contract.

• Exclude finite-menu adaptivity for exact helper endpoints (no new κ) #6015 concerns finite-menu selection for exact helper endpoints. This rounds individual scalar updates and claims a probability bound rather than an exact endpoint from a menu.

• Prove zero-preserving polynomial helpers cannot reduce linear call minima (no new κ) #6017 extracts linear call profiles from zero-preserving polynomial helpers. This changes no nonlinear helper alphabet or operator-call count.

Completion refresh inspected all 15 currently open account PRs created since the CLAIM, #6014 through #6028 including this PR. The related entries are listed above; the remaining PRs concern other objects. No competing stochastic-prefix influence or failure value was found in those public bodies. No additional computations were run to compare results.

Related descriptions use PR bodies as context. No extra computation was run to validate another contribution or compare its numbers.

CI notes

Both hosted checks passed at head e5b18b6c00e7236700e6f46905d394dfdfef7506 in run 40592586749: plan in 4m35s and replay (research/p104-stochastic-prefix) in 1m26s. The check list was read once on completion; run metadata confirms this exact head and a successful completed run. The registered replay runs both normal and optimized commands. All registered checks and publication checks also passed locally; none was deferred solely to CI. Hosted success certifies the bounded replay, without claiming independent human or formal proof-kernel review.

Merge Danger

Door: two-way. Only six new files in an isolated research package are added.

Blast radius: package. Shared code, frozen evidence and selected-frontier files are unchanged.