Research: high-arity connector toggle cost floor

[1337isnot1337]

Result

This package adds a scoped exact bill for six-source hyperedge connector calls of arity four through six. Each such operation is charged one dispatch plus every constituent source addition; sources stay read-only, the connector is arbitrary dirty and restored, and the 63 output rows retain the existing separate-readout interface.

For any arity and any number of output readouts, the full batch needs at least 140 total queries and 142 total additions. If M nonzero masks are missing from pairwise source-prefix differences, dirty-connector restoration gives R >= 126 + 2M. The exact r((C2)^6)=14 basis value handles k=12,13; a binomial bound handles k<=11; k>=14 needs only the two-readout floor. Separately, expanding every batched source event into singleton toggles preserves its original boundaries, so the pinned f(6)=16 result gives at least 16 source additions. The attributed PR #2128 DUO word attains (140 queries,142 additions). No implementation surcharge is waived; any real routing/frame surcharge increases this favorable lower bound.

The package includes the complete proof, a 14-step witness replay, exhaustive dirty scalar checks, exact source pins, and three deliberate failure controls. It does not claim supplier admission or a new κ.

Reproduce

python3 -I -O -B research/six-source-higher-arity-toggle-floor-seedd45c17a9/verify.py
python3 scripts/research_package.py --package research/six-source-higher-arity-toggle-floor-seedd45c17a9 --check-only
python3.12 scripts/research_package.py --package research/six-source-higher-arity-toggle-floor-seedd45c17a9

Observed locally under CPython 3.14.7:

PASS_HIGH_ARITY_CONNECTOR_FLOOR arities=4..6 queries>=140 W>=16 readouts>=126 adds>=142 dirty_macro_cases=8064 query_cases=15 reverse=True controls=3
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)

The predecessor six-source singleton verifier also passed locally:

PASS_SIX_SOURCE_CONNECTOR f(6)=16 max14=59/64 max12=49/64 batch63=142 f(7) in [18,24]

Python 3.12 is unavailable in this environment; the package is registered for the Python 3.12 workflow. No hosted pass is claimed in advance.

Scope, overlap, and credit

Base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. The nearest work is PR #2128's DUO witness, PR #2156's DUO-domain full-batch analysis, and PR #2146's fixed one-TRIO profile. This package extends the charged source alphabet to the arity-four-through-six family and gives its explicit all-readout price argument; it does not claim the predecessor construction or census.

The basis-size input is Volodymyr Gavrylkiv, Bases in Dihedral and Boolean Groups, Table 3. The exact six-source singleton input and its provenance are in research/universal-connector-six-source. The #2128 witness head and certificate digest are recorded in EXTERNAL-PINS.json.

Prepared by Cinna Davis with substantial OpenAI Codex assistance. Aaron Wang/Navinor receives credit for the attaining DUO witness; the six-source singleton result and published group basis value retain their stated predecessor attribution. No independent human review is claimed.

[Mister-Meeseeks]

The enlarged readout model exposes a gap in the source-addition lower-bound argument. The claim W >= 16 does not follow from the predecessor’s two-readout difference-cover result once outputs may use more readouts.

An explicit counterexample to that intermediate claim is the closed singleton source word (0,1,2,3,4,5,0,1,2,3,4,5), with W=12. Let q_i be its prefix states. For each requested mask, read at both endpoints q_i,q_(i+1) of the first occurrence of each included source. Every output uses an even number of reads, cancels arbitrary dirty connector state, and receives precisely its requested source sum. The connector returns to its initial state. Repeated reads may be cancelled or retained; both are valid.

This does not disprove the combined W+R >= 142 bound (this example is expensive in output reads). It does invalidate the stated independent W>=16 proof and therefore leaves the combined-floor argument incomplete for the expanded model. Please prove the joint floor while accounting for missing pair differences and additional readouts, or restrict the claim to two-readout schedules. The query-floor argument can be retained separately if its cited basis theorem and model hypotheses hold. I am leaving this PR open for that focused repair.