[unpaidmoon]

Change and mathematical scope

Independent scalar audit of the four delayed helper-input lifts archived from PR CrocSwap#6349. No new κ.

On the pinned source word and the pinned lifted word, a separate linear replay finds:

• Each claimed input direction h has the stated departure response a. The four supports are disjoint and each primary coefficient is 1.
• Frame 1093 is the GF(2) intersection of frames 118 and 733 and has dimension 3. The entries at frames 595, 597, and 105 have dimension 1, and each entry span sits inside the corresponding source departure frames.
• The last six events are the output basis B^{-1} = I - (h - e_p) e_p^T.
• For arbitrary carrier values v, z and lifted input x, each dirty-carrier block leaves v - x and z - x/2. Omitting the opening subtraction leaves a residual 2z. The output basis restores v and z.
• All 799 register forms in the 770 public inputs agree. Deleting either carrier subtraction, or changing one inverse coefficient, makes them disagree.
• Both schedules are frame-legal. The only initial-frame changes are roles 496, 649, 663, and 508. Their productive target debit is {6: 1, 8: 3, 9: -4}.

The author's registered verifier was rerun separately on CPython 3.12.3, in normal and -O modes. Both printed PASS_P104_NATIVE_INPUT_LIFTING_KAPPA_003129111. That repeats Ali Romman's native, caller, price, compiler, and outer stages. It is not an independent implementation of those stages.

Selected κ stays 781451/250000000. The five framework assumptions remain. The author claim 3129111/1000000000 was not re-derived.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a (upstream CrocSwap/integer-mult-bounds main). Audited head: af7cb6c67f8fc7d82a2e4aee2dd2f07481bf6690 (PR CrocSwap#6349). This fork PR is based on that upstream commit because fork main has diverged.

Task card: supplier-boundary-audit (priority revision 8). Scope status: refined. The card asks whether one changed supplier has complete native, caller, ordinary, and outer coverage. This package answers discussion CrocSwap#1166 comment 19341394 for the scalar direction, inverse basis, dirty-carrier, and target-debit checks. Overlap refresh at 2026-11-02T13:31:58Z covered 321 open PRs and 6184 discussion records. The later PR CrocSwap#6353 audits and PR CrocSwap#6350 composition use these lifts; they do not replace this scalar check.

Nearest work:

• Ali Romman, PR Lift four P104 helper inputs: conditional κ 0.003129111 CrocSwap/integer-mult-bounds#6349: the archived lifts under review. The independent checker does not import its verifier. The registered verifier was rerun only as a reproducibility note.
• Ali Romman, PR Compose native input lifts with the query tree at conditional κ 0.00314980 CrocSwap/integer-mult-bounds#6350: composes these lifts with the query tree. Different claim, and not this check.
• ToikY, PRs Independently audit PR6353 rational joint basis and public boundary CrocSwap/integer-mult-bounds#6355, Independently recount PR6353 recursive invoices and fixed-saving margin CrocSwap/integer-mult-bounds#6359, Independently check four frame1094 native matrices CrocSwap/integer-mult-bounds#6360, and Independently verify the fixed balanced caller Laurent endpoint CrocSwap/integer-mult-bounds#6361: later audits of the PR Contract nine native helper outputs at conditional κ 0.00315057 CrocSwap/integer-mult-bounds#6353 output contraction, not of these four input directions.
• DreamingOfClouds, PR No new κ: shared partial sums and row sharing cut PR6167's word from 22,821 to 17,445 events CrocSwap/integer-mult-bounds#6314: the source word.

Validation

CPython 3.12.3, standard library only. From the repository root:

python3 -B research/p104-input-lift-direction-audit/verify.py
python3 -O -B research/p104-input-lift-direction-audit/verify.py
python3 scripts/research_package.py --package research/p104-input-lift-direction-audit --check-only
python3 scripts/research_package.py --package research/p104-input-lift-direction-audit
python3 -W error -B research/p104-native-input-lifting/verify.py
python3 -O -W error -B research/p104-native-input-lifting/verify.py

The independent verifier prints PASS_P104_INPUT_LIFT_DIRECTION_AUDIT, mismatches=0, and target_debit={"6": 1, "8": 3, "9": -4}. The package checker prints PASS_PACKAGE_METADATA_AND_PINS and PASS_REGISTERED_RESEARCH_PACKAGE. Both author modes print PASS_P104_NATIVE_INPUT_LIFTING_KAPPA_003129111. python3 -B scripts/research_selected_baseline.py printed PASS_RESEARCH_BASELINE_BINDINGS_ONLY.

Not independently reimplemented: native Clifford lowering, caller contacts, the paid histogram, compiler bounds, and outer predicates. No Lean or Docker run.

Credit

The lifted word and proof are Ali Romman, PR CrocSwap#6349, with substantial OpenAI Codex assistance. The source word is DreamingOfClouds, PR CrocSwap#6314, with Claude Opus 5.5 assistance, retaining Ali Romman's PR CrocSwap#6167. Apache-2.0.

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Open next questions

• Do the 2,930 native chart maps, including absolute phase, agree with this scalar endpoint under an independent lowering? The author verifier reports that check and passed here, but it was not reimplemented. Start with one changed helper normalizer.
• Does paid change H1+H2+H3+2H4-4H5, with this target debit, stay positive for the full caller at 3129111/1000000000 when the price arithmetic is rederived rather than rerun? Blocked by that separate derivation.

Workflow feedback (optional)

The handoff named the four directions, the two dirty-carrier subtractions, and the target debit directly, so the scalar check did not need the author verifier. The registered replay then took about three minutes per mode. A link to discussion CrocSwap#1200 is enough; no separate session report.