[petalas]

Summary

Scoped proof with exact finite corroboration: the unchanged PR1410 persistent z4 joint forward word has a 13-bit sufficient fractional guard under PR2502's explicit relative-ADD error model.

- previous sufficient forward guard: 14 bits
+ causal sufficient forward guard:   13 bits

For arbitrary exact dirty initial registers bounded by B, exact NEG, no overflow, and local ADD error at most u times the sum of the two approximate operand magnitudes, every register at every chronological cut has error at most B*2^-b whenever u <= 2^(-b-13), for every b >= 0.

The new step replaces a uniform feedback maximum with a strictly chronological triangular recurrence. Its integer upward-rounded certificate gives C_num=30186730759261 at scale 2^32, so C_num/2^32 < 7029 < 8192. The maximum is at record cut 1730, register 4. This improves a sufficient bound; it does not establish minimum actual precision or a new kappa.

This contribution is produced by Nick Petalas. Credit ywz65 / #2502 for the relative-error model, chronological impulses and prior sufficient guards; Cinna Davis / #1391 and #1410 for the unchanged word; Aaron Wang / #2464 for the precision handoff; Ross Richardson / #1287 and Sagar Thakrar / #1227 for the retained frame lineage. No predecessor verifier or search is replayed.

Evidence

Before: #2502's uniform feedback argument certifies 14 forward guard bits. After: the written causal proof and exact certificate certify 13 in the same local model.

The forward computation and a separate backward covector traversal agree on all 1,720 ADD recurrences and 1,904 updated rows, using 5,344 adjoint traversals. Tiny exact fixtures and controls reject omitted feedback, understated error, invalid gates and an insufficient guard. The generated receipt binds package bytes and pinned inputs.

From the repository root, exact local reproduction commands:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/z4-causal-error-guard
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/z4-causal-error-guard/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/z4-causal-error-guard/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/z4-causal-error-guard --check-source-history

Another CPython 3.12 interpreter may replace the machine-specific path. All commands pass locally on CPython 3.12.15. Each registered direct check took about 0.31 seconds; registered replay took about 0.86 seconds. Expected markers are PASS_Z4_CAUSAL_ERROR_GUARD q=13 upper_C<7029 allowance=8192, PASS_BACKWARD_CAUSAL_CHECK adds=1720 cuts=1904, and PASS_REGISTERED_RESEARCH_PACKAGE. The committed-byte/source-history check covers ten files and the reachable canonical base. Git-free isolated replay and source/receipt corruption and occupied-output controls also passed locally.

The environment record initially stated CPython 3.12.14. It was corrected to the actual 3.12.15 in the follow-up commit b75db5522 before PR publication, and its source-bound receipt was regenerated. The mathematical result and verifier are unchanged.

Limitations and unrun obligations

Exact initial inputs, exact NEG, the declared local relative-error assumption and no overflow are required. The upper bound need not be attainable by a specific rounding rule. No physical-host precision bucket, initial uncertainty, child-call rounding, integer range, workspace-lifetime or tape-cost contract is supplied. Separate-call and inverse guards are not improved here. Proof-kernel verification, full supplier replay, all-size implementation and analytic recovery remain unrun. No selected-frontier change is proposed.

Open next questions

• Bind this local guard to one physical host's input, child-error, range, lifetime and precision-bucket contract.
• For a specified rounding rule, determine whether 12 bits suffice or exhibit a reachable failure. This sufficient majorant settles neither alternative.

Related open work

• Prove sharp dirty Toeplitz shear gate-cut norm bounds (no new κ) #4829 sharpens dirty Toeplitz shear gate-cut operator norms; it treats a different word and does not compute this z4 causal relative-error bound.
• Prove exact all-depth binomial suffix norms (no new κ) #4831 gives exact binomial suffix norms for a parity compiler; it treats a different word and does not compute this guard.

The initial live snapshot of 413 open PR titles and bodies found no open PR targeting this z4 word's relative-ADD guard. Before DONE, a refreshed list of all 71 open PRs by this account included nine other PRs opened since the CLAIM; none studies this z4 word or computes its causal relative-ADD guard. The adjacent norm results #4829 and #4831 above remain distinct. Targeted main/merged searches and handoff claims were also checked. Sparse local notes, arbitrary diffs and unpublished work limit overlap coverage. The package builds on merged #2502 and #1410 rather than auditing them.

CI notes

Both hosted checks passed for commit b75db5522ac37375ba31f2085c364f959eccfebb: plan and the registered research/z4-causal-error-guard replay. CI run 39870102667 completed successfully. The package replay covers the normal and optimized registered commands. Local registered/direct/optimized checks and committed-byte/source-history preflight also passed. This finite CI evidence is separate from the written proof and does not certify physical-host admission or a complete multiplication theorem.

Advance CLAIM. The delivered mathematical scope matches the claim; the environment correction above resolves the subsequent publication handoff.

Merge danger

Two-way door. Blast radius: isolated research package. The change adds no shared runtime code or selected-result modification. Independent human and formal proof-kernel review are unclaimed.