Finite construction: exact two-helper k8 gauges save 576 scalar primitives

[bbMisk]

Change and mathematical scope

Scaling L251 and L427 saves 576 scalar primitives per complex invocation in each orientation. This is optimal among dyadic gauges on at most two of the 237 eligible helpers, with ADD topology and existing SCALEs fixed. Each pair's cost is affine between vertical, horizontal and diagonal break lines. Boundary charges make it coercive; unit discounts lower boundary values. A minimum therefore occurs at an integral intersection. Independent integer-grid enumeration confirms every pair minimum. Exact replay checks all 712 input columns. Native hosting remains unchecked; no new kappa is claimed. Hanson Sun, with substantial OpenAI Codex assistance, credits @woklerx's PR1161, Sagar Thakrar's lowerer and the retained supplier lineage.

Orientation	Baseline	Candidate	Scalar primitives saved
Forward	149002	148426	576
Backward	149018	148442	576

Status: finite scalar construction and scoped two-helper optimum. Paid profiles unchanged. Excluded: other helper interfaces, three or more scaled helpers, phase-varying gauges, changed original SCALEs, native/hosted admission, global invoice, new κ.

Task and overlap

• Base: 1b59b624d21924f8e3537cdc1ac52d3be6eda2fe.
• Seed: 25, OS randomness; complex-supplier track.
• Task card: research/k8-pair-dyadic-seed25/CLAIM.json.
• PR Exact single-helper dyadic rescaling saves332 complex primitives in selected k8 word #1161, d210969d3158a1e6c505e41f67dd84e555c88564: single-helper predecessor; its engine, proof and provenance retained unchanged. New scope: all 27,966 pairs among 237 cold/FULL, unwrapped, no-COPY/clear helpers, with both exponents in Z. Shared ADD interactions included.
• PR Consolidate selected complex stage scales with full primitive pricing #1170: existing stage-SCALE consolidation; no composition claim.
• PR Construction: complete complex factory lowering from 44 to 38 paid primitives #1064: local factory source-scale predecessor; different whole-provider domain.

Validation

python3 -I -B research/k8-pair-dyadic-seed25/verify.py
python3 -I -B research/k8-pair-dyadic-seed25/independent.py
python3 -I -O -B research/k8-pair-dyadic-seed25/test_package.py

• Exact all-column baseline/candidate endpoints; 735,975 vertices per orientation.
• Independent literal-lowerer pricing: every pair minimum; 19,162,332 interacting-pair grid evaluations; six exact dirty-state primitive replays.
• Sixteen rejection controls; actual source/program/certificate corruption; optimized offline relocation; gzip-header variation accepted by canonical payload equality.
• Local 52/68-bit positive envelopes; inherited 192-bit charge retained.
• Selected-record integrity and Ruff pass. Full native/global admission, cold replay, hosted CI and Lean unrun.
• Evidence: CERTIFICATE.json, PAIR-CENSUS.json.gz, PROGRAMS.json.gz, SOURCES.json, VALIDATION.json.

Credit

Hanson Sun (@bbMisk), substantial OpenAI Codex assistance. Credit @woklerx's PR1161; Sagar Thakrar's PR696 lowerer; icekylinx, Jeremy Kothe, Rohan Arun, Douglas Colkitt and the retained k8/T16 lineage. Original notices retained. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 26162c56ec9c50a80e49949f36d4dd9a137a76a6.

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

Reviewed scope: Two-helper dyadic gauges on 237 eligible helpers: 576 scalar primitives saved per orientation, all-integer pair optimum via integral break-line vertices and independent grid. Paid profile unchanged; no native/global admission.

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.