[petalas]

Summary

A scoped written proof and exact finite construction, produced by Nick Petalas. Optimize PR6071's closed cube-covering-walk construction in the released balanced-Gray direction of Josie Elliston's PR6107, Open next question 3.

C(k) = 2 ceil(2^(k-1)/k)
L(c) = max { k >= 1 : C(k) <= c }
N(c) >= L(c) floor(c/2) + 2(c mod 2)

Closure forces even flip counts, and the classical balanced cyclic Gray-code theorem attains this threshold. Applying apoursh / Armin Pourshafeie's existing upward/padding/reversed-downward split gives the new explicit unsigned connector guarantee. For c>=8 and b=floor(log2 c), L(c) lies between b+floor(log2 b) and that value plus two. This supplies the extra log2 log2 c source term. The written proof and exact scope are part of the isolated package.

Balanced Gray codes, the covering-walk split and block concatenation retain their predecessor credit. The new contribution is their explicit optimized connector threshold, capacity consequence and literal realization. The exact lower bound concerns closed covering walks, not every depth-two connector word. No new kappa or selected-frontier change is claimed.

Evidence

• Before: PR6071's displayed reflected-Gray guarantee gives 1,152 sources at c=256. After: the new exact bound gives N(256)>=1,408. These are two particular guarantees, not a claim of the best construction on main. Large composed batches are proved by decomposition, not executed.
• The literal nine-source witness at c=58 has 512 distinct cycle vertices, five flip counts 56 and four flip counts 58. Independent literal prefixes check all 511 nonzero masks. The complete word executes exactly 30,160 positive calls, each operator exactly 58 times, including all dirty-cancelling padding.
• All 27 complete dirty-input fixtures pass over Z/58Z. The replay also checks 120 small closed covering walks, 356 compiled small-mask cases, 1,023 exact capacity rows and ten rejecting controls. No predecessor scientific verifier is imported or replayed.
• Both registered warning-strict normal and optimized checks pass locally on CPython 3.12.15, macOS, standard library only. Their timed local durations are 0.148 s and 0.168 s. These are check durations, not a speedup claim. Generated receipts match and bind all seven package files.
• Committed bytes, modes and source ancestry pass for ten package/input files at 0c3ff82. Official export integrity and its three declared live inputs pass. Both read-only Git-free export modes pass with empty PATH and unchanged bytes/modes. An actual changed predecessor README is rejected.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/connector-balanced-walk-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/connector-balanced-walk-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/connector-balanced-walk-threshold
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/connector-balanced-walk-threshold --check-source-history --check-modes

Another CPython 3.12 executable can replace the local interpreter path. Expected markers are PASS_CONNECTOR_BALANCED_WALK_THRESHOLD, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and credit

This is an exact unsigned immutable-source/one-dirty-connector interface over any nonzero unital ring of finite characteristic. Arbitrary initial source, connector and target data are allowed. Physical admission/layout, coefficient preparation, instruction storage/dispatch, complete native prices, output volume, frames, restoration, precision/tape and an actual consumer remain separate obligations. The package writes the full conditional invoice and waives no unknown fee. The literal compiler is bounded to dimension ten and characteristic 512; the all-parameter proof uses the cited classical existence theorem.

No unrestricted depth-two optimum, new Gray-code existence theorem, physical supplier, complete multiplication theorem, selected cold replay, Lean or Docker result is supplied. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Credit Josie Elliston / PR6107 for the open balanced direction; apoursh / Armin Pourshafeie / PR6071 for the main covering-walk split; Nick Petalas / PR4231 and PR4248 for the unsigned interface and concatenation; Ross Richardson / PR934 and PR943, Riquelme Soares and retained lineage; Josie Elliston / PR6061 and zxv / PR6057 for finite blocks. Classical balanced-code existence and construction retain Bakos, Robinson-Cohn and Bhat-Savage's credit. Every predecessor file and credit line is unchanged.

Open next questions

• Can an open cube-covering path lower the characteristic threshold? Start by deleting each edge of a balanced cycle and comparing flip counts.
• Can unrestricted depth-two source words exceed this closed-walk capacity by an unbounded additive amount? Combine literal prefix geometry with PR6107's weight-count bound.
• Can one named consumer use the additional sources with a favorable complete I/P/D/call invoice? No physical consumer is supplied here.

Related open work

• PR6216 certifies PR6141's characteristic-five six-source exclusion. This contribution constructs a different closed-walk subclass and does not replay that exclusion or compute a competing six-source optimum.
• PR6566 proves read-set dominance for a fixed joint-root search. Its state-count objective and two-root interface differ from this one-connector characteristic threshold.
• PR6564 studies rounded signed dyadic conjugations. This contribution is exact, unsigned, and finite-characteristic, with no rounded root-carry quantity.
• PR6432 proves support-conditioned repetition bounds for singleton walks. This result optimizes a closed full-cube construction using classical balanced cycles and does not target its matching-prune question or compute a common numerical instance.
• PR4079 studies seven-source low-weight-mask coverage in characteristic two, a restricted mask family distinct from this all-mask construction.
• PR2134 studies one atomic TRIO event, changing the source alphabet that stays singleton here.
• PR2249 prices higher-arity toggles; this construction adds no higher-arity source call.
• PR3909 prices five-source arity-four calls. This contribution adds no batched source operation and computes no competing five-source tariff.
• PR3897 prices joint-output delivery. Separate output operators and all their padding calls remain paid here.

Canonical main was refreshed and remained 7c49fe3. Live open/merged searches, recent account work and claims, general-family main searches and reservations were checked before committing. Cached overlap covered 506 open PRs and 6,502 Discussion records with complete paging; local notes were sparse. Unpublished work and arbitrary historical diffs remain limits. Advance CLAIM.

Completion refresh inspected all 15 account open PRs created since the advance CLAIM, including this PR. PR6564 and PR6566 are the two adjacent connector contributions and are distinguished above. None of the other fourteen studies this exact closed-walk threshold or reports a competing value for this construction's source-capacity guarantee. No extra scientific computation audited another contribution or compared numbers from different contracts. The delivered scope matches the CLAIM.

CI notes

Both hosted checks passed at exact head 0c3ff8236c103c5960ef15c45b26432ad1603f8d in run 41610499679: plan in 3m34s and replay (research/connector-balanced-walk-threshold) in 1m25s.

The completed checks were read once. Run metadata confirms the matching head, completed status and successful conclusion. The registered replay executes both warning-strict normal and optimized commands. All registered commands, metadata/pins, committed bytes/modes/source history, export integrity and read-only Git-free replays also passed locally. Hosted execution certifies the registered finite checks and bindings; independent human and formal proof-kernel acceptance of the written universal argument remain unclaimed. No required package check remains unrun.

Merge Danger

Door: two-way. Blast radius: isolated research package. Seven new opt-in files; no shared implementation, frozen predecessor evidence or selected-result edit.