[krzysztofwojt]

Change and mathematical scope

PR3244 checks propagation but explicitly leaves search/template completeness open. This package proves that every simple restored universal word in the pinned A/B multiplicity classes has a representation in the source's low templates, compressed high paths, gap vectors and initial domains. It adds a written representation argument and an independently implemented exact checker; no source defect was found in this scope.

The key new implication is that replacing a high path by its (visited set, endpoint, coordinate counts) representative preserves the whole word's prefix set, restoration, simplicity and multiplicities. Exact enumeration verifies the surrounding finite domains. This is a scoped proof/audit, not a completed search, full f(7)=24 validation, supplier admission or new kappa.

Task and overlap

• Task census-pruning-audit, priority revision 6, refined to connector-covering-code-simple-ab-domain-completeness. Task card and complete proof.
• Scientific and refreshed canonical main: 463113d675af7410ac91d53a4fed51ec6494eb9a. Contribution commit: aa47cff758c956304327f1fc67dac61e970cfb3d. Final fetch found no relevant upstream changes.
• Executable subject: conorholds' research/connector-covering-code/cycle_search.cpp, SHA256 597844fc32cde19c1d4104633fa6d3eb0d28c1ca140f8f3d2ba2cb632ecd30d1.
• Nearest work: Research: audit connector-covering-code's parity-capacity filter and its simple-word boundary #3224 (69567e40e8d92542248eda0d6a400d83adc1d3a3, capacity) and Research: audit singleton connector path-domain propagation #3244 (5964d7f22ad9cd9c27a1d2eb0b897f52b9425a13, propagation/emitted-template incidence). This closes their explicitly untested representation step; their executables are not imported or rerun.
• Canonical claim/handoff. Final overlap refresh covers 306 open PRs at 19:40:57 UTC and 2,093 Discussion records at 19:42:26 UTC, with complete pagination and no warnings, plus matching local packages and live Binary affine addresses: exact paid-call floor with arbitrary dirty registers #1317. No matching successor surfaced; unpublished work and arbitrary closed diffs remain outside coverage.

Validation

python3.12 -I -B scripts/research_package.py --package research/connector-domain-completeness
python3.12 -I -O -B research/connector-domain-completeness/verify.py --controls
python3.12 -B scripts/check_research_commit.py --package research/connector-domain-completeness

All pass on macOS arm64, CPython 3.12.12, Apple Clang 21.0.0. Exact evidence:

• All 2,520 A and 18,900 B labeled low words, yielding exactly 5/77 universal symmetry orbits; all 82 native constructors execute.
• All 112 simple Q3 paths, exactly 106 quotient classes and 848 translations; 48 translated full-word replacement controls.
• 256 initial-domain comparisons and every 102,552 A / 286,780 B gap vector, compared as complete ordered streams, with an independent count identity.
• Four compiled semantic mutants (missing orbit, discarded path counts, wrong carried parity, omitted gap seven) and three truncation/source-correspondence controls reject.
• Metadata/source pins, staged whitespace check, all nine committed package/input byte comparisons, archive integrity and optimized read-only Git-free export replay pass. A separate AI worker's independent low-word/path enumeration and targeted proof review agree with the primary results.

Catalogue/initial-domain checks compile unchanged authenticated source. Gap checks use two explicit, uniqueness-checked instrumentation replacements to capture the original recursive loop before capacity or word search; this is documented and is not a full native census. The dedicated Linux workflow declares the C++17 build; hosted CI is separate from the local evidence.

Unrun: full B exhaustion, complete branching/exhaustion validation, literature/classification and multiplicity premises, the non-simple branch, formal-kernel proof, selected cold replay and physical supplier obligations. Selected records, frozen predecessors and shared tools are unchanged.

Credit

Submitted for @krzysztofwojt with substantial OpenAI Codex assistance in derivation, implementation, checking and writing; a Luna worker supplied read-only overlap extraction and separate low-word/path enumeration. No independent human or formal-kernel review is claimed. New material is Apache-2.0.

Credit conorholds for the source search; commdata2338/#3224 and Nick Petalas/#3244 for the audits and handoff; Nicholas Stone, Aaron Wang, James Reed, Ross Richardson and Douglas Colkitt for the retained parity, engine, model and framework lineage. Original notices and evidence remain intact.

Open next questions

The README/TASK give the next bounded test: compare Engine::search against a small explicit Cartesian-domain oracle with satisfying, unsatisfiable and interrupted cases, then bind all template IDs/flags to complete source-pinned exhaustion receipts. That missing implication should be checked before a separately budgeted independent full B replay. Domain completeness alone does not establish an exclusion.