[unpaidmoon]

Change and mathematical scope

Prove that a contact-free helper split on a boundary from the zero subspace to a D-dimensional space has a convex rank bill. For m ≥ 1 paths and any exponent 0 < t < 1, the maximum of

G(x) = x^t + (D-x)^t - D^t - Σ_i ((x-a_i)^t + (b_i-x)^t - c_i)

on every closed feasible interval [A, B] is max(G(A), G(B)). The identity includes A = 0 and B = D. An interior dimension, including a join or meet of three or four paths, cannot raise the bill. When m ≥ 2, or when the single path is not the full span (0, D), every interior dimension is strictly below that maximum. The one-path full span is constant. Under the single-leg charge c = (b-a)^t, the endpoint values are k(a) and k(b), and k(u) > 0 for u ∈ (0, D).

This is a proof under those hypotheses. The program adds an exhaustive finite census at t = 1/2 on the integer domains named below. A positive G is a rank-bill signal. It is not a legal cut, not a paid-profile improvement, and not a change in the selected exponent. Selected κ stays 363005817/100000000000.

Remaining obligations: a proper helper boundary whose terminal lies strictly between 0 and D; the actual charges c_i on the published 68-split word; legality, ownership and the complete invoice.

Task and overlap

Base commit e54523517d0f03e58352e647c403a5af3e525a57. Priority revision 8. Card p104-side-correction, scope changed: the card asks whether a legal causal split improves the complete invoice, and this package only closes the endpoint case of the rank bill in research/p104-shared-native-helper-splits/EXPLORATION.md.

Session r271, item 4 of the ordered eligible list. Three open handoffs remain ahead of it (seven-call signed frames, comment 19377832; eight-source singleton, comment 19537798; six-source c = 7, comment 19537594). Conditional-κ review requests whose canonical artifact is an open pull request, including CrocSwap#7148, CrocSwap#7158, CrocSwap#7161 and CrocSwap#7244, are outside the list.

On 2026-11-11, scripts/find_overlap.py refreshed 566 open upstream PR records and 7234 discussion records with complete pagination. The phrases contact-free transport and endpoint rank bill had no matches. Closed work was searched through local notes and ledgers only. That report is a lead, not a novelty certificate. The predecessor exploration states convexity and leaves an extremum at 0 or D as a separate integer case. This proof shows that case is the same endpoint comparison.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B research/helper-split-endpoint-rank-bill/verify.py
python3 -O -B research/helper-split-endpoint-rank-bill/verify.py
python3 -I -B research/helper-split-endpoint-rank-bill/verify.py
python3 -B scripts/research_package.py --package research/helper-split-endpoint-rank-bill --check-only
python3 -B scripts/research_package.py --package research/helper-split-endpoint-rank-bill

Author-run marker:

PASS_HELPER_SPLIT_ENDPOINT_RANK_BILL powers=8200 half_rows=1729 positive_boundary_paths=132 full_span_zeros=90 two_path_rows=2057 two_path_strict_interior=406 kappa=363005817/100000000000 no_new_kappa=true

The census domains are complete for what they claim: every 1 ≤ D ≤ 12 and every one-path window; every ordered pair of windows for 1 ≤ D ≤ 6; every u^s with s = 2q-p, 2 ≤ q ≤ 5, 1 ≤ p < q and 1 ≤ u ≤ v ≤ 40. Signs of square-root sums use math.isqrt with a strict error bound. Failure controls reject a reversed power comparison, a forged interior win, and a dropped 0^t = 0 convention.

make frontier7000-verify, Lean, Docker and the 68-split discovery screens were not run. EXPLORATION.md is cited and is not a live verifier input.

Credit

The rank-bill formula and the open endpoint sentence are from @CubeJerry's shared-native helper-split exploration, on the shared two-chain word from DreamingOfClouds in CrocSwap#6902. The basis-splitting lineage is Ali Romman's CrocSwap#5599. Retained native and compiler authors of that construction stay credited there. Prepared by unpaidmoon with substantial Cursor Grok assistance. No formal-kernel review and no independent human review are claimed. New work is Apache-2.0.

Open next questions

1. Does the endpoint maximum survive when a paid path extends outside a helper boundary whose terminal dimension lies strictly between 0 and D? The curvature comparison uses both helper legs against the full dimension D. First step: one proper terminal and one path whose upper end passes it.
2. For the published 68-split word, is every remaining unsplit helper feasible interval nonpositive at both endpoints under its actual old charges? First step: evaluate G(A) and G(B) from the pinned witnesses, without a new geometry search. That payload is outside this package. A positive rank bill would still leave legality and the paid profile open.

Workflow feedback (optional)

CrocSwap#1200