Refine PR #7 ternary star templates for conditional kappa = 3.8e-9

[rohanarun]

This refines Zhihao Chen (jacklightChen)'s ternary five-subset producer in #7, pinned at 6725c6a17b17871a35353fd29157f4ed851bc114. A different greedy tie break, comparison of three candidate circuits, and deletion of unused ancestors save 170,400 additional addition nodes in the four-point stars. Auxiliary roles per invocation fall from 11,840,940 to 11,670,540.

Under #7's retained general interfaces, the exact witness becomes kappa = 19/5,000,000,000 = 3.8e-9 > 2^-28, approximately 1.88% above #7's 3.73e-9 headline. This is a modest circuit refinement and an asymptotic exponent comparison, not a runtime benchmark or a global optimality claim.

Construction and proof boundary

The published star rule prioritizes maximum pair frequency and then smaller union support. The alternative prioritizes larger union support at equal frequency. For each canonical template, the driver chooses the smallest of the published rule and two alternative mask tie breaks, then prunes unused gates. Every expression remains a partition of its required source mask; every gate adds disjoint supports. The independent C++ pass checks every replacement boundary against the complete original producer.

Every new node still has its fixed common four-set, hence a common pair. The positive rational-source-span and complement-frame argument in #7 therefore applies. Retained-total outputs, center dimensions, stage joins, endpoint corrections, and decreasing dimensions remain the inherited ones; only the auxiliary width changes in their count formula. The proof explanation and exact parameter table are in research/prime-field-followup/README.md.

This is a conditional, unreviewed draft, for mathematical review. The prime-field transfer and complex network in #7, faster Gaussian arguments in #5, compact-control interfaces, and upstream multiplication theorem remain assumptions. There is no full formal verification or integrated manuscript patch. The current main headline is not changed.

Checks completed

• Complete h=28 producer enumeration, all 20,475 four-point stars and 365 canonical templates. Independent reconstruction checks every replacement gate and requested boundary sum; original demands reproduce byte-for-byte.
• Exact selected counts: 10,687,362 additions, 983,178 output uses, 11,670,540 roles.
• All selected templates' actual compiled forward/reverse role paths and 118,183 basis directions per orientation, including arbitrary target and dirty-scratch directions, checked simultaneously with exact F3 bit-plane arithmetic.
• Inherited h=8 controls rerun: all 1,156 scalar basis inputs, dirty-scratch restoration, corrected three-stage exchange, and both rational-frame schedules. These are controls of the inherited small producer, not a full simulation of the modified h=28 network.
• Exact saving a_b = 761/10^11; all 29 strict constraints and seven assembly margins. Minimum margin 4754049/1250000000000000, strict gap 4049/1250000000000000 above kappa.
• Source/dependency hashes checked; vendored reference files are byte-identical to the pinned Ternary five-subset networks for a conditional 373/10^11 > 2^-28 bound #7 sources. Baseline tests and patch checks previously passed; no existing generators, certificates, patches, or upstream files are modified by this PR.

Reproduce with Python 3.11+ and a C++17 compiler:

python3.11 research/prime-field-followup/verify.py
make verify

The full producer checker needs approximately 1.2 GB memory. Intermediates stay under build/prime-field-followup/. No third-party Python packages or network access are needed.

Attribution

The ternary motif, degree-three producer, initial star resynthesis, stronger complex producer, and accompanying general proof are Zhihao Chen (jacklightChen)'s contribution in #7. Preceding work by Douglas Colkitt, Bortlesboat, dleen, and eumemic remains credited. The pinned source files, their SHA-256 hashes, and provenance are included; their original code is unchanged and Apache-2.0 licensing is retained.

The template-selection refinement, additional exact controls, and updated parameter witness were prepared with substantial OpenAI Codex assistance at Rohan Arun's request. This is not independent mathematical review, a worldwide priority claim, or an OpenAI endorsement.

[Mister-Meeseeks]

Integrated and retained on main in 377e2bd. Thank you @rohanarun for this contribution.

The complete original source tree at cfd6a2baded9dba990b987934474a7721d6ca3fb is preserved in the historical source package, with original notices and predecessor credit. It can be extracted and replayed independently. Historical release wrappers do not overwrite current pinned suppliers.

Reviewed scope: Ternary star-template refinement: full h28 producer rebuilt twice; 20475 stars and 365 templates checked including arbitrary scratch; historical conditional 19/5000000000 assembly. Retains failed heuristic attempts and attribution.

The batch review and per-file dispositions cover every original changed path. Local focused checks passed; regenerated artifacts match the submitted sources. Attribution is also recorded in CONTRIBUTORS.md. This is integration of historical research and finite evidence, not a new selected κ or full formal verification of inherited tape/analytic interfaces. Linux replay jobs are tracked separately.