[RossRichardson]

Change and mathematical scope

Proves the endpoint deficit for the common-two/disjoint-fresh-five relay with arbitrary admissible unary insertions at the current cut. Exact telescoping and row support replace an invalid absorbing-past argument. Dressed-swap and chirped-address counterexamples explain the boundary, and the paired-decoder proof corrects the general inverse-square normalization.

Extends the uniform path-count test to the necessary condition sum_j 2^(-d_j*f) >= 1, with its equality condition and an exact sharp Gaussian expansion. The expansion's full unshared rank histogram is stated and checked.

Only research/common-cut-endpoint-debt/ changes. The all-length operator theorem assumes the displayed relay expression; the retained scalar caller certificate covers B1, L=3. The failing all-odd-length completion bill is conditional on its stipulated profile. A separate full-native coefficient path shows why the theorem does not exclude changed coupled callers.

Finite exact checks and written proofs are distinguished throughout. There is no physical OPEN implementation of the sharp expansion, lower bound on shared circuits, favorable complete supplier, new kappa or selected-result change. Already published support code is credited rather than claimed as new.

Task and overlap

Base: 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7. Scope is explicitly changed from the current joint-query-controller card to finish the preserved C07/C08 investigation at the user's request. It isolates the endpoint obligation behind native-zero cut operations; it does not solve the installed QQ16 caller's matched bill.

PR #4203 supplies the preceding local carrier interface; PR #4055 supplies later changed callers outside this unary theorem. Their results are not republished. Eight later packages already retain the support utility; exact source reconciliation is included. The submission overlap screen paginated 375 open PRs and 3,148 Discussions, with no matching lead for the stated terms; this is bounded coverage, not a novelty certificate.

Validation

• Separate mathematical and source/verification review, including actual semantic defects and named-check deletion controls.
• Committed-byte and source-history preflight; registered Python 3.12 replay of all four commands.
• Official exact-commit export and all direct commands from a read-only Git-free path containing spaces; unchanged file inventory.
• Automatic package CI on the published exact head (run 39507667903).

Reproduce with python3.12 -B scripts/research_package.py --package research/common-cut-endpoint-debt. The three mathematical components include 58 actual semantic rejections and 58 named-check deletion controls; separate package controls check full receipts and captured source bytes. No unrelated frontier or later-caller replay is claimed.

Credit

Scientific compiler/source ancestry is pinned to 20ecf2891d80b5acf18d980b2164fb44ea7faa90; guidance and publication reconciliation use 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7. Immutable original sources, notices, complete provenance and the corrected proof lineage are retained. Ross Richardson with substantial OpenAI Codex assistance, including separate mathematical review contexts; no independent human or formal-kernel acceptance is implied. Apache-2.0.

Open next questions

The package's README.md and TASK.json specify two bounded follow-ups: price one changed physical cut in an already complete caller, or specify a shared-circuit interface for the sharp support witness. Missing full physical costs and source/temporary contracts block supplier conclusions from either local idea.

Reviewed and locally replayed head: 16801fc326a005af72644f4ea2c6650eccf905e5.