[petalas]

Summary

Prove a scoped endpoint-conditioned simultaneous-rounding theorem. For a separate final norm bound C>=1 and exact prefix/inverse bound B, the sufficient fractional precision changes as follows:

-P = p + ceil(log2(8 K B^3))
+P = p + ceil(log2(8 K C B^2))

An exact closed word has C=1. The displayed guard then saves q bits when B=2^q. New determinant-one dyadic scale/shear words prove that quadratic dependence is necessary for closed words, and cubic dependence is necessary for unrestricted endpoints, including the gate-count factor. Every exact gate and inverse has norm at most two. Claim type: scoped mathematical proof with exact finite corroboration. No new kappa or selected-frontier change.

Credit ywz65's #2701 for the interaction-coordinate theorem and original guard, and Vlado Vojdanovski's #1673 for the precision interface and retained lineage. The new contribution is the endpoint-conditioned guard and its matching elementary-word examples, rather than a predecessor audit. Produced by Nick Petalas.

Evidence

• Before: Certify simultaneous gate rounding at the capped Toeplitz interface #2701's sufficient guard uses B^3 even when the final exact map is identity. After: PROOF.md retains its endpoint factor separately and proves the B^2 guard.
• Exact direct matrix products and a separate interaction recurrence pass 72 fixture runs, 2,298 literal prefixes, 24 simultaneous guard cases with noncommuting detours and arithmetic forcing, 18 lower-power rejection cases, 11 invalid-input rejections and one endpoint-only-premise rejection.
• Warning-strict normal and optimized registered commands pass locally in 0.46 and 0.45 seconds, respectively. The complete registered runner passes. CPython 3.12.15, macOS arm64, standard library only.
• Committed-byte/source-history preflight passes at 2768b0f99a915d7bb3cefd106c64cb266ed68c31, covering seven package/input files. Official export integrity and read-only Git-free optimized replay pass; all eight extracted files, including the export manifest, remain byte-identical. The optional receipt is generated by the verifier and binds all six package files by SHA256.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/closed-word-rounding-guard/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/closed-word-rounding-guard/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/closed-word-rounding-guard
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/closed-word-rounding-guard --check-source-history

Substitute another CPython 3.12 interpreter on a different machine.

Limitations and unrun obligations

The lower examples permit arbitrary bounded coefficient errors. They do not establish that a particular deterministic nearest-rounding rule produces those errors on dyadic coefficients. The result concerns final outputs; intermediate states retain their own norm factors. Optimal numerical constants are not claimed.

A production emitted caller, bounded-error reversible arithmetic, integer-bit and overflow budgets, physical tariffs, supplier admission and the complete multiplication theorem remain outside scope. Selected cold replay, native replay, Lean and Docker builds were not run. Independent human and formal proof-kernel review are unclaimed. The generic formal-interface card is explicitly changed to this original written precision theorem.

Open next questions

1. Bind one production caller whose certified endpoint norm C is much smaller than its prefix bound B, then substitute its actual arithmetic contract.
2. Determine whether a specified deterministic rounding rule excludes the correlated shear errors used by the lower examples and permits a better guard.

Related open work

• No open PR was located for the same endpoint-conditioned B exponent in the checked live list, cached complete PR/Discussion snapshot and distinctive main/merged searches. The closed Certify simultaneous gate rounding at the capped Toeplitz interface #2701 is the direct predecessor. Coverage excludes unpublished work and arbitrary code diffs.
• Prove sharp all-precision Neumann truncation at 226p² (no new κ) #5145 studies a scalar Neumann tail cutoff with its contraction and prefactor granted. It does not study this finite-word endpoint norm or coefficient-amplification exponent; no numerical comparison is claimed.
• Prove sharp theta order for broad Gaussian edge inverses (no κ) #5150 proves matching-order broad Gaussian inverse-norm growth at a phase-cell edge. It does not bound a finite word's accumulated coefficient error or certify the prefix/endpoint envelopes required here.

The completion refresh examined this account's nine other open PRs created since the advance CLAIM: #5145, #5146, #5148, #5149, #5150, #5151, #5152, #5154 and #5155. Their declared objects are scalar tails, guard-character sums, stable-lane reversal, prime supply/products, Gaussian inverse edges, a projection recurrence and E8/P112 factorization floors. None studies this endpoint-conditioned rounding exponent. No additional computation or same-number comparison was performed.

CI notes

Both hosted checks passed for 2768b0f99a915d7bb3cefd106c64cb266ed68c31 in CI run 39943696605: plan in 3m28s and the registered package replay in 1m19s. The completed run has conclusion success and its head SHA matches this contribution. The replay runs both registered normal and optimized commands. All registered commands, source-history preflight and export checks also passed locally; no local obligation was deferred to CI. Hosted execution is separate from independent acceptance of the written proof and inherited premises.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Adds one research package and its explicit replay registration. Frozen sources and the selected frontier are unchanged.