Finite complex circuit: a shared ADD couples two temporal gauges

[thakrarsagar]

Joint temporal optimization of L225/L350 beats every composition of individually optimal schedules, including ties, by six complex scalar primitives per orientation in the declared [-4,4] zero-boundary family. A single shared ADD explains the gap: changing one helper's exponent costs one real primitive on its own path but avoids a four-primitive shared-operation penalty.

The fully priced finite candidate saves 474 complex scalar primitives per orientation against the original k8 word: 149,002 → 148,528 forward; 149,018 → 148,544 backward. Every transition/reset and original SCALE is charged. Both gauges are zero at every nonscalar event. Paid recursive profiles are unchanged.

Finite scalar construction and bounded optimization result; no new κ. Existing many-helper gauges have lower total scalar bills. The contribution is the exact shared-operation coupling result, with a small 9-by-9 conditional-cost table, not a scalar record claim. Native/hosted admission and the added macros' full adapter invoice remain open.

OS seed23 selects 16 interacting pairs from the sorted 2,414-pair graph. All 32 pair/orientation cases are complete. Separate distance-transform and dense/literal-price implementations agree on 49,693,014 transition candidates. A tight-edge product calculation and independent lexicographic DP both account for all tied single-helper optima. Both complete words pass 712-column replay, frame/boundary invariants, full invoices and local precision checks. Controls cover 729 tiny paths, zero schedules, twelve adverse cases and external saved-certificate corruption.

Reproduce with Python3.10+:

python3 -I -O -B research/k8-paired-temporal-seed23/verify.py

Expected: PASS_K8_PAIRED_TEMPORAL_NO_NEW_KAPPA pair=225,350 saved=474/474 strict_coupling_gain=6/6 pairs=16 orientations=2 columns=712.

Tested on macOS arm64 CPython3.14.8 and optimized3.11.15, including an outside-checkout working directory. Selected-record integrity and seven exact base Git-blob/SHA256 pins pass. No full cold replay or hosted CI success is claimed.

Exact base: bce54fb3f895cbe5cd4c0eb477c3afeed100190f. Eight new files are confined to research/k8-paired-temporal-seed23/; selected/frozen/upstream files are unchanged. Source pins, proof, scope, open next questions and detailed validation are in that package.

Credit @woklerx #1161, Hanson Sun/@bbMisk #1214, Sagar Thakrar #1259, Philipp/@mightyOil #1263/#1270/#1310/#1322, Cinna Davis/@1337isnot1337 #1350/#1353 and the retained Jeremy Kothe/Rohan Arun/icekylinx supplier lineage. The final refresh covered 850 open PRs. Concurrent #1369 covers zero-static single helpers; #1376 proves separation for disjoint L248/L253 schedules. This result supplies a strict shared-operation interaction for a different pair and background. Both newer heads are pinned and credited. Prepared by Sagar Thakrar with substantial OpenAI Codex assistance; Apache-2.0. No independent human review is claimed.

[Mister-Meeseeks]

100-PR batch review at c2bc6848cffb2e2203095d4c4bbc45b0848c22e7.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete bounded two-helper temporal optimization on 16 declared pairs, with dense transition/tie oracles and full word replay. L225/L350 saves 474 complex primitives and beats every tied independent composition by six; no all-integer/global or physical-supplier optimum is claimed.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.