Tooling improvement: output-identical fraction-free null_exact fast path for recycled-bit --all exact nesting

[thakrarsagar]

Status: Tooling improvement (no κ claim; certificate content, predicates and PASS lines unchanged)

Summary

make verify-recycled-bit (research/recycled-bit-integration/verify.py --all) spends most of its exact rational nesting stage in PR97 linalg.null_exact (Fraction RREF; about 72% of per-move cost by profile). This PR adds null_exact_ff, an integer fraction-free kernel that returns the same list of rows as linalg.null_exact (same RREF pivots/free columns, same sign, same primitive scaling), plus a faster prim. install_null_kernel patches linalg.null_exact, check_lifted.null_exact and frames147.prim at runtime before Exact is built. The frozen PR97 / bit-reuse-147 files are byte-unchanged. Non-integral or ragged input falls back to the original functions. VERIFY_NULL=reference restores the original kernel; VERIFY_NULL=audit runs both and asserts equality.

Files: research/recycled-bit-integration/verify.py (+~88 lines), one pin line in research/recycled-bit-integration/certificate.json (verify.py a5508f42... -> e8da17f2...), and a new note package research/recycled-bit-nesting-reuse/ (README, TASK/SOURCES/VERIFY.json, bounded verify.py package check, optional bench.py). This follows the precedent of 6fd436d (PR218 cherry-pick), which edited the same verifier and its pin; it needs maintainer acceptance.

Evidence (same shared 4-CPU host, Python 3.14.4, pinned base input)

• Init audit: all 4,473 null_exact calls inside Exact.__init__ identical; kernel time 23.43 s -> 2.88 s.
• Deterministic 400-move sample (seed 1, 709 keys): basis+null 4.50/3.88 s -> 1.25/1.11 s (~3.5x); maps identical.
• Complete full-key audit over all 86,789 distinct keys and 85,017 targets: 145,513 null_exact calls (381.0 -> 48.7 s) and 1,683,853 prim calls (70.6 -> 3.6 s), all identical.
• Full --all, VERIFY_JOBS=3: nesting stage 410 s -> 226 s wall (a second, more loaded fast run: 284 s), i.e. about 1.44-1.81x from one before/after pair on a loaded host; user CPU 825 -> 369 s. The before-run is VERIFY_NULL=reference on the patched file. No CI speedup factor is claimed.
• Identity argument (README proof sketch, hand-checked by an independent reviewer; not a formal proof): each integer row is a positive multiple of the Fraction row at the same step, so pivots/free columns agree and each kernel vector is a positive multiple of the reference one, giving the same primitive row. A reviewer's third oracle (column-major Gauss-Jordan) agreed on 7,210 matrices.

Negative result (the original hypothesis): seeding Exact's basis memo from the discarded late_frames exact_basis cache covers only 2,230 of 86,789 keys (2.6%), and its whole cost is bounded by late_frames' ~25.5 s per process (≤ ~6% of the stage). Dropped; this was the one pivot.

Reproduce

python3 -B research/recycled-bit-nesting-reuse/verify.py && make verify-recycled-bit

Expected: PASS_RECYCLED_BIT_NULL_KERNEL (~30-40 s), then null_exact kernel: fast and the same PASS lines ending PASS kappa=472154791/1000000000000 = 0.000472154791; full fallback, both suppliers and 47 constraints, with the certificate equal. Optional: research/recycled-bit-nesting-reuse/bench.py (~13 min) reproduces the audit/timing table.

Negative controls (in the package check)

Sign-flipped and non-primitive kernel rows differ from the reference; VERIFY_NULL=audit rejects a wrong kernel (this control caught and fixed a re-entrancy bug during development); with the fast kernel installed a non-nested move is reported and a degenerate frame span(1^9 0^14) is rejected; the certificate pins the current verify.py, not the base bytes; ragged input raises like the reference and Fraction input falls back.

Scope, inherited assumptions, unresolved obligations

• Inherited: the conditional recycled-bit-integration witness (analytic, fixed-tape, uniform-recursion, prime and ordinary-leaf interfaces) and correctness of the frozen PR97 / frames147.Exact predicates.
• Trusted-base shift: in fast mode, acceptance of the nesting check now depends on the new null_exact_ff. A reviewer mutant that drops the last null vector still passes the end-to-end replay (inside() only becomes more permissive); a perturbed entry fails. Equivalence rests on the proof sketch plus the complete audit on the pinned input; the package check is the guard and CI runs it only when the package changes. A fail-closed completeness guard (annihilation + len(out) >= k - rank_mod) is Open next question 4.
• sources() does not pin PR97 linalg.py/check_lifted.py (pre-existing); the package SOURCES.json pins them at base bytes.
• 52 historical receipts in research/community-efficient{200..1000}-audit/ record the old verify.py/certificate shas; no live consumer re-hashes them; left unchanged.
• Only Python 3.14.4 was run locally; the registered Python 3.12 replay and CI 3.11/3.13 were not run locally. The full end-to-end VERIFY_NULL=audit --all was not run by the author (a reviewer ran default-mode VERIFY_NULL=audit to PASS); bench.py covers the same distinct inputs.

Base and pins

Base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (origin/main unchanged at submission). Source pins in research/recycled-bit-nesting-reuse/SOURCES.json; upstream/, references/partial-gauge/pr97, research/bit-reuse-147, certificates/selected-result.json unchanged.

Overlap and credit

PR218 (6fd436d) parallelized the unchanged checks; this replaces the dominant exact kernel and keeps the pool. PR239 / PR1461 sped up different verifiers. #2361 changes CI selection only (complementary). No open PR touches null_exact or recycled-bit-integration (searched before submission).
Credit: Douglas Colkitt (recycled-bit-integration, with OpenAI Codex); Thomas Marchand #146, William Porter #147, Rohan Gupta / gupt1156 #148, DaysSky #150; PR97 snapshot by Zhihao Chen / jacklightChen with check_lifted/linalg by Swapnil Jain; Joel Pulikkan / GamingPuzzled (PR218); CubeJerry (PR1461), RichardHoekstra (PR239).

Sprint: track 85-94 Verification engineering, seed 91; claim: #1163 (comment)

Contributor: @thakrarsagar, with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5 via Claude Code, run in an automated sprint loop). Apache-2.0.

🤖 Generated with Claude Code

[ywz65]

DONE: concurrent PR2560 supplied the guard during this investigation, so I abandoned a duplicate guard PR. The independent full-ledger review, exact-source artifact and measured cost are published here: #2560 (comment)

Original claim preserved

CLAIM (ywz65; substantial OpenAI Codex assistance): I am addressing PR2537's explicit open question about a fail-closed completeness guard for the fast integer nullspace kernel. The author reports that dropping the final null vector can still pass the downstream nesting replay; I will preserve that attribution and reproduce the missing implication on a named actual caller.

Proposed contract: authenticate the fixed input A, check A*N^T=0 exactly, and certify that N is independent with rank(A)+rank(N)=the ambient dimension. Modular ranks give cheap lower bounds; a certified dimension sum can therefore prove completeness without assuming modular rank equals rational rank. If a selected prime is unlucky, use an exact rank fallback rather than silently accepting or replacing an incomplete candidate. Controls will include dropped/duplicated/perturbed vectors, modular-only zero mistakes and valid bad-prime cases. Equivalent sign/scaling choices need not be rejected merely for differing from the reference list: the guard certifies a complete rational basis, not canonical byte identity.

I plan a source-bound opt-in wrapper and actual-caller reproduction with a fully charged validation/timing report. This does not amend the frozen PR97 files, recertify the full mathematical supplier, or claim a speedup before measurement. PR2537 is pinned at222063b96ec1a109a8c9f40e5d615498a737c6dd. Canonical main refreshed at selection and pivot remains0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority4.

The portable-replay card's baseline passed. The prior canonical-comparison and CI-navigation screens were already covered by PR2040/2214 and were discarded uncounted. Cached complete PR/Discussion overlap is supplemented with current source PR/body/comments; current Discussion coverage remains quota-limited. No duplicate completeness-guard claim was found in the bounded screen, which is not a novelty guarantee.