[petalas]

Scoped finite certificate; no new κ. Package research/p104-pr5718-shape-split-census, base 19fdb086e. The selected record is unchanged.

Result

Matarener's #5994 (research/p104-helper-split-exhaustion) excludes PR5599-type helper splits on the PR5907 word. Its first open question asks whether a split of the kind in flyingviper's #5718 remains: different input and output bases, with an output correction. This package decides the scalar part of that question exactly over Q.

The test covers the PR5718 shape. A group helper p retires at a strictly intermediate chart. A new independent helper takes over its continuation (unit injection b = e_p). The input functional may use the group, U-initial roles, or any role initial at the empty frame (PR5718's rule, main inputs included). Output normalisation is allowed on roles whose final chart is V. For every group, intermediate chart, cut after an event touching that chart, and group helper at the chart, the census checks three things: a pure prefix with a_p ≠ 0, no z-leak of d = S e_p to roles not final at V, and the pairing a·b = c·d' ≠ 0.

Word	Input rule	Evaluations	Leak	Pairing fails	Witnesses
PR5713 source (calibration)	group + U-initial	8,579	3,817	2	7
PR5713 source (calibration)	+ empty-frame roles	8,579	3,817	2	7
PR5907	group + U-initial	8,913	4,202	4	0
PR5907	+ empty-frame roles	8,913	4,206	4	0

• Calibration: on the PR5713 source word, the census recovers PR5718's own split: cut after 14330, chart 602, helper 708, ranks (2, 7). All 7 witnesses are on that chart and helper.
• PR5907: no class passes. Any further PR5718-type split there must cancel a z-leak with a mid-word correction. 425 classes leak to exactly one role. They are the nearest candidates and are listed per pivot in RESULT.json.
• PR5718's split is itself a boundary-relaxation witness of the Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994 census on its source word. So Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994's boundary exclusion already covers PR5718-shape splits on PR5907 whose defect stays on V-final roles. This package calibrates that link and locates the leaks.

Claim type: finite certificate of a scalar necessary condition on two fixed words. It is not a construction, an invoice or a complete multiplication theorem.

Evidence and reproduction

Interpreter: CPython 3.12 (~/git/integer-mult-bounds/.venv/bin/python3.12 locally), standard library only.

python3.12 -B research/p104-pr5718-shape-split-census/verify.py
python3.12 -B scripts/research_package.py --package research/p104-pr5718-shape-split-census
python3.12 -B scripts/check_research_commit.py --package research/p104-pr5718-shape-split-census --check-source-history

Local results: PASS_P104_PR5718_SHAPE_SPLIT_CENSUS, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed (34 s wall, 6 s CPU on a loaded machine; 3.6 s unloaded), and PASS_RESEARCH_COMMIT_BYTES with source history. The verifier authenticates the PR5907 word and the PR5713 word inside research/p104-two-boundary-helper-split/inputs/pr5713-source.tar. It also runs six synthetic controls of the pairing decision and compares RESULT.json byte for byte.

Limitations

• Only the z-part of the split is tested. The non-z part of the normalisation, native maps, owned lifetimes, caller, compiler and outer checks are not checked.
• Only unit injection on one group helper is covered, as in PR5718. Injection along a combination of several fields is not covered.
• Mid-word corrections (r += q·new at a later meeting with a leaked role) are not modelled; they are the open route. An exact Fraction implementation of that relaxation took more than ten minutes per word and was not run to completion. It is not in the package.

Open next questions

• Can a mid-word correction close the z-leak of one of the 425 PR5907 classes that leak to exactly one role? First step: list the later meetings of the pivot's chart with the leaked role, then test whether S_τ e_r reaches the leaked coordinate. Blocker: fast exact span maintenance.
• The four leak-free PR5907 classes (chart 549, helper 173, cuts 15974–15988) fail only a_p = d_p c_p. Does an input role outside the empty-frame rule make that solvable?

Related open work

• Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994 (Matarener): PR5599-type census on PR5907. This package answers its first open question for the PR5718 shape. The two censuses test different conditions (general injection versus unit injection on one helper), so their numbers are not compared.
• Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030 (DreamingOfClouds): a newer conditional word (helper 470 rung removal). This census was not run on it.
• This account's PRs opened since the CLAIM (Certify that balanced depth-three chain multiplicities break every unit symmetry for m ≤ 24 (no new κ) #6037, Prove the conjugate-pair target needs four calls in every group (no new κ) #6038, Classify mixed Q8 call-space shear identities; no pair of shear images separates (no new κ) #6039, Prove paired-stage loss at m = 4 under the chronology cut is exactly 10 (no new κ) #6041) study connector chains, conjugate-pair targets, Q8 shear identities and paired-stage loss; none studies P104 helper splits.
• Compress the ordinary events of the PR5907 P104 word: 25,998 to 25,861, same invoice #5975, Independent replay, invoice, native and funding audit of #5827 and #5907 (no new κ) #5960 (Matarener): other work on the PR5907 word (event compression, a replay audit). Neither studies PR5718-type splits.

CI notes

Both hosted checks passed at head 1889253fdbfd37e0f984ac412cf0ada4f36fa676: plan (4m32s) and replay (research/p104-pr5718-shape-split-census) (1m37s), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40611794147.

Credit

Matarener (#5994: criterion, pure-subspace construction, question), flyingviper (#5718: split shape), Ali Romman (#5713 calibration word, #5599 split criterion), DreamingOfClouds (#5907 target word) and the P104 lineage. Nothing of theirs is modified. Produced by Nick Petalas. Apache-2.0.