Finite complex circuit: exact zero-boundary gauge cost and wrapped-helper admission

[mightyOil]

Varying helper gauges can remain nonzero while a recursive child or residual wrapper runs. This package imposes gauge zero at every non-scalar boundary, prices that restriction exactly, and admits representative complete words to the retained local scalar/frame/wrapper checker. Finite scalar construction and exact boundary-price comparison; no new kappa.

Sagar Thakrar's concurrent #1259 already supplies varying gauges and the clipping/shortest-path mechanism, independently rediscovered during this work. That overlap is credited prominently. The added result is the stricter boundary contract, its exact cost, wrapped-helper coverage and local admission evidence—not priority for varying gauges.

• All 32 pinned Finite complex circuit: varying L276 gauge saves 344 scalar primitives #1259 unrestricted minima reproduce exactly. Resetting at every non-scalar event costs 4 real primitives per complex call at L276 and L427, and zero at L359. The restriction therefore has a measured exact price and avoids new nonzero-gauge commutation through children/wrappers.
• Exhaustive exact optimization over all 477 no-COPY/clear helpers and all integer exponents gives L427: 522 primitives saved per complex call in each orientation. The proof clips to a finite interval without assuming an empirical cutoff.
• The domain includes 240 wrapped helpers, excluded by Finite complex circuit: varying L276 gauge saves 344 scalar primitives #1259. The best wrapped witness, L652, saves 72 and passes complete local scalar/frame/wrapper admission in both orientations.
• Adding a zero-boundary L359 schedule to the fixed Finite construction: exact two-helper k8 gauges save 576 scalar primitives #1214 two-helper word saves 518 extra, for 1,094 total: 149,002→147,908 forward and 149,018→147,924 backward. Its zero-boundary contract is relative to that fixed predecessor word; Finite construction: exact two-helper k8 gauges save 576 scalar primitives #1214's constant gauges and their inherited assumptions remain. This is not a joint three-helper optimum.

#1263 has the larger 13,000 saving with many constant gauges. No best bill across classes or composition with #1263/#1170 is claimed. Paid recursive profiles and 52/68-bit positive local envelopes remain unchanged under the inherited 192-bit allowance. No selected-frontier or upstream file changes.

Reproduce and evidence

python3 -I -B research/k8-phase-local-gauges/reproduce.py

Expected final marker: PASS_COMPLETE_K8_PHASE_LOCAL_GAUGES.

• Separate distance-transform DP uses literal PR696 lowerer prices: 1,910 minima, 759,448 padded vertices, all agree.
• Six complete literal words: 445,342 primitives, all 713 initial columns including dirty coefficient work. Macro replays separately check all 712 original columns.
• Three complete variants pass the unchanged source-pinned local frame/wrapper checker, 25 inherited mutation controls each.
• Ten package test groups, including 258 exhaustive tiny path problems; four additional predecessor-price/coverage controls; normal/optimized isolated checks; complete offline Python 3.12.13 replay from a path containing spaces; 25 selected-record tests and integrity pass.

Full native C++ compiler, hosted/global/controller admission, outer repricing, selected cold replay, remote CI and Lean were not run. All-size, realization, hosting, precision/tape, analytic and bit assumptions remain. Written clipping/telescoping arguments and finite exact checks do not formalize the multiplication theorem.

Base dadfea6e8a78ff6ba901705f87532a71c6f5bbb8, OS-random seed26, original complex-supplier track. Source pins include #1161 d210969d3158a1e6c505e41f67dd84e555c88564, #1214 26162c56ec9c50a80e49949f36d4dd9a137a76a6, and #1259 61367611154266a982fe25e1a6243a8aad77227d. Final broad gauge-overlap refresh covered860 open PR bodies through #1268. No pinned sprint issue existed; the local claim and final refinement are preserved.

Credit @woklerx, Hanson Sun/@bbMisk, Sagar Thakrar, icekylinx, Jeremy Kothe, Rohan Arun, Hao Yan, Douglas Colkitt and the retained supplier lineage. Prepared for Philipp/@mightyOil with substantial OpenAI Codex/GPT-6 Astra assistance in mathematics, code and writing. Both separate implementations are AI-assisted; no independent human review is claimed. Apache-2.0; original notices retained. Concrete next questions are in the README.

[Mister-Meeseeks]

100-PR batch review at 69bc8d601d4738dc511fc76f869e1f3c3d4bc3d4.

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

Reviewed scope: Exact per-helper signed-dyadic dynamic program with resets at every nonscalar event: 477 helpers, 240 wrapped choices, and a 1,094-operation composed local saving. Pinned reproduction and mutation controls pass; no new global native supplier admission or κ.

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.