Research: one all-five toggle has no favorable price (no new kappa)

[1337isnot1337]

Result

This package gives an exact finite exclusion for adding one all-five-source call to the five-source universal dirty connector:

Q5: z += B0 x0 + B1 x1 + B2 x2 + B3 x3 + B4 x4

Q5 may be used at most once. It is charged optimistically as one complete dispatch and five elementary source additions. With all 31 nonzero output masks required, no schedule using Q5 has a strictly lower price than the existing singleton witness for any per-query overhead o >= 0 and elementary-add price p > 0, where an ordinary source or output call costs o + p and Q5 costs o + 5p.

The exact baseline witness has 74 complete queries and 74 elementary additions. The proof also allows output rows to use more than two readouts, which is necessary for a global pricing comparison.

Proof outline

At a source-call boundary, let q_t be the parity vector of source contributions already added to the dirty connector. A row read r times has dirty-connector coefficient r mod 2; generic correctness forces r even. Two readouts realize only a pairwise prefix difference q_s XOR q_t. If a required mask is absent from every such difference, that row needs at least four readouts. Thus c available nonzero pairwise differences imply at least 62 + 2(31-c) output readouts.

Restoring the connector after one Q5 forces every singleton count m_j to be positive odd. Hence m = sum(m_j) is odd and at least 5. The cases m=5,7,9 are the only ones below 11:

m	Maximum pairwise masks	Output readouts at least	Total queries at least	Total elementary additions at least
5	20 by the 7-prefix pair count	84	90	94
7	20, exact census of 33,600 labeled words	84	92	96
9	27, exact census of 1,159,200 labeled words	70	80	84
11 or more	parity lower bound	62	74	78

The m=7 and m=9 censuses enumerate every singleton ordering and every Q5 insertion slot, reduced only by source-label permutations that preserve the all-mask target and uniform prices. A separate recursive multiset enumeration checks the histograms and maxima. The computation is integer-only and exhaustive; it uses no sampling or floating point.

For m >= 11, Q5 plus the singleton calls costs at least 12 source dispatches and 16 source additions. With the 62 required output calls this gives at least 74 queries and 78 additions, so the baseline is still strictly cheaper when p > 0.

Reproduction and verification

Ran locally on CPython 3.12.15:

python3.12 scripts/research_package.py --package research/quint-toggle-connector --check-only
# PASS_PACKAGE_METADATA_AND_PINS
python3.12 scripts/research_package.py --package research/quint-toggle-connector
# PASS_QUINT_TOGGLE_CENSUS m7=33600:max20 m9=1159200:max27
# PASS_QUINT_TOGGLE_INDEPENDENT m7=33600:max20 m9=1159200:max27
# PASS_QUINT_TOGGLE_PRICING q5_lower_bounds=90/94,92/96,80/84,74/78 baseline=74/74
# PASS_REGISTERED_RESEARCH_PACKAGE research/quint-toggle-connector outcome=negative-result
python3.12 -I -B research/universal-connector-boundary/verify.py
# PASS_UNIVERSAL_CONNECTOR_BOUNDARY pairings=945 max_coverage=31 queries=74 leaves=182

SOURCES.json pins the baseline certificate. git diff --check passed. No repository-wide suite, Lean proof or selected-supplier integration is claimed.

Scope and exact remaining gap

The result assumes characteristic-two linear updates; five source labels; all 31 nonzero masks; arbitrary dirty source, connector and output rows; read-only sources; only the singleton source calls and at most one Q5; complete calls with scratch/head restoration; and uniform nonnegative per-call overhead with positive per-add price. Q5 receives no extra frame or implementation surcharge, making this a favorable price assumption for the proposed operation.

The remaining mathematical gap is repeated Q5 use or a mixed Q5/DUO/TRIO alphabet, plus other source counts. This result says nothing about physical supplier realization, compiler/tape costs, or the all-size multiplication theorem. The selected #1094 record remains κ = 836330098538459/500000000000000000; no kappa claim changes.

Current overlap review

The complete overlap snapshot at 2026-10-17 21:37 UTC covered 359 open PRs and 2,154 discussion records with no warnings. The immediately following live list showed 361 open PRs and one open issue, #913. No open PR matched the connector-alphabet overlap terms. The nearest completed work is the fixed-alphabet connector boundary (PR #943 lineage), the distinct two-source DUO (PR #1531), and the distinct three-source TRIO (PR #1518). Discussion #1884 closed its paid-frame investigation and released its other scopes. Exact Q5/all-five-toggle searches found no remote claim; broader quint and arity-five searches found unrelated P112 #2292 and R480 #597 records. The current active queue was focused on P104/P112, QQ16, T740 and selected BIT work. The overlap scan is a lead search, not a novelty certificate.

Assignment: connector-alphabet-change, priority revision 6, scope refined to one Q5 on the existing five-source batch; base 463113d675af7410ac91d53a4fed51ec6494eb9a; selector nonce f9384baf1db5ab998bf8643a7cf853d0.

Prepared by Cinna Davis with substantial OpenAI Codex assistance in scope selection, proof development, enumeration, verification and writing. No independent human or proof-assistant review is claimed. Apache-2.0.