Research: nilpotent couplings do not save shallow GL3 calls

[1337isnot1337]

Result

This is a scoped exact negative result in a two-dirty-role formal model. Over F2[GL3(F2)], expanding the six idempotent rank-one couplings to all nine nonzero rank-one couplings (including the three nilpotents) does not lower the direct call count for any ordered pair of targets whose individual coordinate-CNOT distance is at most two.

For all 961 pairs, the exact minimum in the stated domain equals d(p)+d(q) <= 4. The complete 54-factor alphabet was searched through length three, the full strict-saving boundary. The certificate records 160,435 raw words, 89,358 distinct depth-three transfer states (91,708 across depths zero through three), and 126,900 transitions. Every transfer coefficient is compared over all 168 group elements. An independently implemented set-based replay checks the same endpoint domain and all direct witnesses.

Cost and scope

Each rank-one conditional factor costs one call. The two roles are arbitrary dirty outputs; there is no helper, clean state, inter-call frame change, copy, or restoration operation. The involution identity is exact for all nine couplings because each CNOT label is an involution and characteristic two gives ([g]+[e])^2=0.

This is a finite primitive-model result. It does not establish a native supplier for nilpotent couplings, a supplier mapping, a multiplier construction, or a new kappa. It covers only two roles and CNOT radius at most two per output.

Reproduction

From the repository root:

python3 -I -B research/gl3-nilpotent-coupling-seeda9ab7bd2/search.py --check
python3 -I -B research/gl3-nilpotent-coupling-seeda9ab7bd2/verify.py
python3 scripts/research_package.py --package research/gl3-nilpotent-coupling-seeda9ab7bd2 --check-only

Expected search and independent replay outputs are recorded in README.md; exact recurrence and evidence are in PROOF.md and CERTIFICATE.json. The search and independent verifier passed locally under CPython 3.14.7, including optimized mode; the package metadata check and git diff --check passed. The generic package runner requires Python 3.12, which is unavailable locally; hosted CI is pending.

The required selected-record baseline command was attempted but the sparse checkout lacks research/complex-sw4-frame8, so it stops with Invalid package root. The selected supplier's cold replay was not run because this finite experiment does not alter that supplier. Its selected-record hash, review, source manifest, assembly certificate, and reproduction instructions are pinned in TASK.md and SOURCES.json.

Pins and related work

Base main: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2.

Selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Direct predecessors: PR #1974 at a4992da8e4e81a74cf1fd2b2d3af948662476d6e (idempotent GL3 census), PR #2027 at cd37822ee3523e53b645848644e0dae42d47abe7 (nilpotent couplings under a UT4 tariff), and PR #2167 at cea5d7be431b09877848354f04d5f2ba512339fd (three-register clean-ancilla comparison). The related direct-nine three-register GL3 claim in Discussion #2102 concerns a specific radius-four/five target and a different two-bridge family.

Cycle claim: Discussion #2285. The selector seed, task, priority revision, catalogue pin, overlap snapshot, and all exact source hashes are in TASK.json, TASK.md, and SOURCES.json.

Prepared by Cinna Davis with substantial OpenAI Codex assistance in scope selection, implementation, and verification. No independent human review is claimed. Predecessor attribution and Apache-2.0 licensing are retained.