[aliromman]

Change and mathematical scope

Proves a necessary completion bound for a simple hypercube path and a functional that omits one coordinate. The bound counts unused matching edges and endpoint capacity, including the final edge to the start. It rejects prefixes that pass matching balance and return distance. This is an enabling proof with finite corroboration; there is no engine change, runtime claim or change to the complete kappa calculation (60403/20000000).

Proof, exact receipt, verifier, and source pins.

The written condition covers n >= 2, even 4 <= L <= 2^n, and simple prefixes with 2 through L vertices. Universality is not required. The returned splits are necessary conditions, not certificates that a completion exists.

Task and overlap

Answers Open next question 2 of PR #4907. The sharp cap in PR #4992 is related context, not a dependency. Base: bee95ce88f9f19802f0613f92838895cd0e14a1f, priority revision 7. Main was unchanged at submission. The refreshed search covered 308 open PRs, 4,096 Discussion records and 65 of 6,690 local notes, followed by relevant direct reads. Sparse omissions, arbitrary diffs and unpublished work remain outside that coverage.

Validation

CPython 3.12, standard library:

python3.12 -B scripts/research_package.py --package research/f7-prefix-matching-capacity

Expected: PASS_F7_PREFIX_MATCHING_CAPACITY in both normal and optimized commands, then PASS_REGISTERED_RESEARCH_PACKAGE.

Passed: 110,655 prefix/functional checks across four domains; exact descendant-split containment; 3,838 closed cycles; independent literal-word enumeration in two domains; 640 capacity cases; ten boundary controls. There are 704 strict rejections before the leaf relative to matching balance and return distance. Five actual code mutations fail with false completion prune, including omission of the closing edge or endpoint partners. Committed-byte checks, all four exact source pins and Git-free read-only optimized export replay pass.

The full-history wrapper was not run in the shallow checkout; the pinned base ancestry and all source blobs/SHA-256 values were checked directly. No seven-label census, maintained-engine comparison, selected cold replay, independent human review or formal proof-kernel check ran.

Credit

Nick Petalas supplied matching balance in PR #4894; Mihir Srivastava supplied the explicit handoff in PR #4907. Their retained proof notice credits Ross Richardson's prefix model and Nicholas Stone's parity correction. No predecessor code is imported. Prepared for Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Open next questions

Measure an opt-in engine implementation against unchanged leaf receipts and runtime. The cost of this condition can exceed the work saved; integration and a performance gain remain unproved.