Finite construction: joint Horner saves 13,072 PR799 backward primitives (no new kappa)

[RossRichardson]

Two adjacent same-target additions with different sources can share one coefficient construction. This package applies exact joint binary Horner circuits to nonzero-frame blocks of the historical PR799 X4 backward program, saving 13,072 real-component primitives beyond accepted PR696 lowering.

Status: finite complex-supplier construction / research note; no new kappa.

Complete scalar projection	PR696	Candidate
Backward real primitives	143,674	130,602
Forward real primitives, retained recount	143,658	143,658
Maximum over orientations	143,674	143,658

The forward orientation then binds. With every original macro adapter charge retained, the conservative local arithmetic bound improves by only 16, from 9,875,380,343,098 to 9,875,380,343,082. This is not a global invoice run or runtime measurement.

The complete finite domain contains 6,112 adjacent pairs with a common target, two distinct sources and identical nonzero frames. Of these, 2,764 save more than merely removing one work clear. Exact path DP selects 1,586 disjoint pairs. The attained minimum is only for this declared binary-Horner pairing class; no unrestricted circuit optimum is claimed.

Every source read, copy, shift, signed addition, final clear and inherited recursive child is charged. No new helper or frame transition is introduced. Backward source reads remain 39,446; total arithmetic operand reads fall from 146,392 to 136,492. Full scalar prefix row-l1 remains 5,338,828, fractional depth 19, and the inherited positive envelope is unchanged.

Reproduce

python3 -I -B research/x4-joint-horner-seed29/verify.py

Expected: PASS_X4_JOINT_HORNER candidates=6112 pairs=1586 saved=13072 columns=737 no_new_kappa.

The command checks all candidate identities, recomputes the exact selection, evaluates both complete backward scalar primitive streams on 737 arbitrary input columns, verifies every group boundary and checks the frozen receipt. All 504 dirty helpers restore and private work clears. A separate integer implementation imports neither Fraction, the compiler nor predecessor code: it checks all 97,792 local matrix entries, independently emits the complete candidate stream/hash, and certifies the same optimum using a forward path recurrence.

Validation and scope

Normal and isolated optimized CPython 3.14.6/macOS replay pass, including 12 main and four integer adverse controls. Seven focused tests cover varied coefficient identities, exhaustive small matching comparisons, minimal relocation without Git in a path with spaces, source corruption, overlapping/suboptimal recipes and checkpoint integrity. Selected-record integrity, syntax/JSON/document links and staged whitespace pass. Detailed evidence and limitations are in VALIDATION.md.

Only research/x4-joint-horner-seed29/ is added. Scalar interpreters preserve but do not execute physical frame/phase/Clifford events. Work clears at fused pair boundaries rather than both old macro boundaries; typed production integration, fresh physical/global admission and receipt rebinding remain separate work. All inherited all-size/compiler, realization/hosting, precision/tape/recovery, bit and analytic obligations remain. Full repository/cold supplier suites, global assembly, Lean and hosted CI were not run.

Provenance and concurrent work

OS assignment seed 29, complex track. Starting base: 91a2094f2e022930069e8cfdc810a3ea1b81cd09; rebased integration base: 47481c89fd30a43f8f882eb4b391e89c29273fd0.

Main advanced during the sprint to PR890, conditional kappa 0.001661957774647029. This remains a completed PR799 checkpoint contribution; no transport to its newer U8 word is claimed. All six mathematical/source inputs are byte-identical, and the original selected bytes are accepted through the official #799 checkpoint with their unchanged hash. The selected frontier and upstream/ are untouched.

Source heads: PR799 80e6463157cfc80b85af09e7e2f14748fe2dac60, PR741 e516b1463606840a15f80aac1995d64ce0092e89, PR696 5a3f983715027c4fd83f50bc9001d4b18463bed1. Exact file/member pins, proof, completed census hashes, witness and local claim are included.

The final overlap refresh covered 710 open PRs through #939; no pinned sprint issue existed. #734/#837/#882 optimize individual coefficients; #793/#901/#932 change frame-zero compensation; #719 concerns repeated-source fixed-middle rank obstructions; #839 removes known-zero clears; #919 changes signs. This changes adjacent different-source preparation in nonzero frames. #922/#924/#931 and new #937 concern distinct historical factory/central maps. No pending code is imported or savings assumed additive.

Credit Jeremy Kothe's #799, Chafik Boukhalfa's #741, icekylinx's scalar/Horner architecture, Sagar Thakrar's #696, DreamingOfClouds, Rohan Arun, eumemic, Dugongue and retained predecessor notices. No priority claim for joint Horner arithmetic.

Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance in selection, implementation, mathematics, tests and writing. Both implementations were written by the same AI agent; no independent human review is claimed. Apache-2.0.