Conditional accuracy: all-length cubic precision for the dirty stride inverse

[aliromman]

PR #985 uses a length admission and a conservative fifth-degree sufficient precision bound for its target-dependent rounded dirty word. This contribution proves an all-length admission and a cubic sufficient precision bound for that same constant-coefficient word. It retains the full recursion, exact source and scratch restoration, two physical data tapes, and all movement costs.

Review files: Proof and scope · Evidence · Source pins and credit.

For a certified prime-power radix r, initial norm at most 2^m, target error 2^-p, and incoming grid width w_input, the sufficient fractional width is at most max(p,w_input)+m+18r^3. The signed integer reserve is at most m+15r^3. The proof bounds inverse errors at the positive preimage and pays for that preimage in the integer guard. It proves a zero-subtree identity and a depth-independent internal magnitude bound for the actual dirty word; it does not assume the clean-scratch PR #977 identity transfers.

The common-operator subset and prefix estimates are credited to PR #862 and PR #905. PR #985 at 9df42a99dd8497b6e7d46cd2507bb89071372640 is retained byte-exact, with its full dependency closure. The derived solver changes admission and caches the two validation reads that the simultaneous phase lane (PR #1038) found were read twice but charged once. An independent array observer checks each literal indexed access. Every rounded data operation, recursive child, transport step, and head return is retained.

At r=64, p=128, and m=20, the explicit sufficient fractional budget is 1,420,902 bits; the pinned PR #985 full-word budget needs 517,236,859 bits with the same margin before any larger structural-depth admission. Both words have 4,000 leaf calls and six borrowed arrays. These are sufficient bounds, not minimum widths or timing gains.

Run:

python3 -I -O -B research/dirty-stride-precision/test_package.py

The package covers 5,220 budget certificates, 1,088 all-length thresholds, 180 coefficient extinction cases, 8,256 exponent geometries, four dirty zero subtrees, 2,322 retained power cases, 468 signed leaf cases, 80 certified kernel cases, and 18 actual uniform-preparation kernel cases. The 98 kernel cases execute 392 complete signed words and 24,848 leaf calls. A full L=5741, stride 2378, 12-bit example is newly admitted and checks its residual, exact inverse, and 3,952,237 paid tape moves. Twelve fault controls and a read-only relocated replay must reproduce the exact receipt. VALIDATION.json gives the measured local checks and omitted checks.

This is a conditional accuracy theorem with exact-rational supporting checks. The inherited supplied factor-preparation, analytic inverse, and fast scalar arithmetic hypotheses remain. It supplies no all-size factorizer, fast bit compiler, variable site phases, Lean proof, complete transform, new multiplication exponent, or measured end-to-end speedup. The selected certificate and upstream/ are unchanged.

Prepared for Ali Romman (@aliromman) with substantial OpenAI Codex assistance. NOTICE.md preserves predecessor credit and identifies the simultaneous phase-lane accounting finding. No independent human review is claimed.

Concurrent PR #1025 at 6ef2168ac90d42926bceb7ac9b2e029463800c76 improves the analytic circle exponent and applies it to PR #862, PR #977's clean-scratch estimates. Its pinned proof explicitly supplies no dirty-array circuit or new inverse execution. This contribution keeps its consumed analytic inputs and proves the PR #985 dirty-word transfer. A combined tighter dirty budget is a follow-up; OVERLAP.md records the distinction.

PR #985 later advanced to da1cdf8ed4ac9b1b8378dfec73f77ff4e81a4f70 with the same two read-once accounting corrections. This package intentionally retains the exact consumed 9df42a predecessor. The derived solver includes the corrections and the independent access observer; no moving source was imported.

The frozen standalone package replay returned zero in 740.81 seconds on macOS arm64 with Python 3.14.7 and one CPU. Peak child memory was 41,304,064 bytes. The selected-record check and all 25 selected tests passed. The full repository suite, selected cold replay, and hosted Linux matrix were not run. The local validation record also preserves an earlier successful package run whose external timing wrapper failed after completion because macOS blocked its sysctl query.

Later work: Philipp (@mightyOil)'s merged PR #1295, at 3a6773708f0d73f7dbee164f4c6e7658b0b41509, sharpens this conditional sufficient fractional envelope from max(p,w_input)+m+18r^3 to max(p,w_input)+m+15r^3. Its proof and certificate retain the common immutable commuting operator, leaf contract, inverse-preimage reserve and full operation bill. This is later evidence, not a change to this PR's pinned theorem or validation. The joint dirty-stride proof-interface audit remains open; neither result gives a new kappa.

[Mister-Meeseeks]

100-PR batch review at 6bc675be1554b3a3be771147f21f9dc2c6dae50a.

This remains open after scoped triage. Next review step: All-length dirty-stride accuracy theorem needs a dedicated audit of coarse-grid refinement, dirty magnitude and both orientations.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.

[aliromman]

A relevant later result is now incorporated: Philipp (@mightyOil)'s PR #1295 completes the commuting-interval substitution for this exact pinned word. Its proof and certificate reduce the sufficient fractional envelope to max(p,w_input)+m+15r^3. The radix-64 example changes from 1,420,902 to 1,162,598 fractional bits; preparation and the 4,000 leaves remain paid.

I added this link to the description and checked its source proof and the incorporation record. This is retained later evidence, not a fresh replay. The common immutable operator, leaf-error assumptions, inverse-preimage reserve and physical routing obligations remain explicit. It supplies useful evidence for the joint audit but does not resolve the requested full proof-interface review or establish new kappa.