Finite construction: four-source Horner saves 8,976 beyond X4 pairs (no new kappa)

[ToikY]

Change and mathematical scope

Finite complex-supplier scalar construction; no new kappa. Sharing binary Horner work across three/four consecutive sources in historical PR799 X4 reduces the complete backward projection from PR696's 143,674 to 121,626 real-component primitives. This saves 8,976 beyond PR945's pair-only construction. Forward remains 143,658 and binds the maximum, so there is no additional orientation-maximum improvement beyond PR945.

The complete census covers 16,431 common-target, same-nonzero-frame windows of length 2–4. Exact interval DP selects 1,174 disjoint windows. Every source access, copy, shift, accumulation and clear is charged; the existing work register is used. One concrete quartet costs seven primitives per component versus at least nine for every pair/singleton partition, even after free intermediate-clear removal. This establishes a larger circuit class rather than reopening the pair search with more computation. Optimality is only within the declared filtered-window class.

Validation

python3 -I -B research/x4-four-source-horner/verify.py

Python 3.11+, standard library only. Expected:

PASS_X4_FOUR_SOURCE_HORNER candidates=16431 windows=1174 saved=22048 extra_vs_pairs=8976 columns=737 no_new_kappa

Validation: separate Fraction and integer-dyadic engines check all candidate identities, independent suffix/prefix DP, every chosen boundary, complete backward traces on all 737 arbitrary input columns, dirty-helper restoration and private erasure. Exact prefix row-L1 maximum 5,338,828, denominator depth 19 and positive envelope are unchanged. Both engines emit the identical full stream. Nine independent adverse controls, seven package test groups, optimized minimal offline relocation, all 38,394 baseline macro comparisons with PR696, selected-record verification and 17 selected-record regressions pass. Exact stream export and output preservation pass. Full repository/cold replay, hosted CI and Lean were not run.

Only eleven files in research/x4-four-source-horner/ are added: construction, proof, full recipe, two checkers, controls, pins, claim and validation. Non-scalar events are retained literally but not physically executed. Work clears at joint-window boundaries; typed production integration, native/hosted admission, physical source-access/global invoice, precision/tape and inherited compiler/realization/analytic obligations remain open. Finite scalar replay is not a proof of the full multiplication theorem.

Exact base: 0076643. OS assignment seed 22, complex-supplier track. No pinned coordination issue was found, so the claim is local. Package overlap refresh through #971 and immediate submission refresh through #974 found no duplicate. Newly arrived #972 concerns parity-frame completion, #973 bit frame509, and #974 U8/T21/T37 interface budgets. Main advanced to bbc600d, selecting PR942 U8/T37 at conditional kappa 0.001661960547713947; all nine historical source pins remain byte-identical and the patch applies cleanly there. The selected frontier and upstream/ are unchanged.

Credit Ross Richardson #945 at a3245bf for shared pair Horner work and the 13,072 comparator; Sagar Thakrar #696, Jeremy Kothe #799, Chafik Boukhalfa #741, icekylinx and retained predecessors. PR751/830 provide related fan-in precedent. The primary constructor adapts PR945's method; no pending executable source is imported at runtime. Savings are not composed with other pending circuits.

Prepared for ToikY with substantial OpenAI Codex assistance, including separate implementation and mathematical review; no independent human review claimed. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 4b40489c3e0f5819d00251de179a9ea7b1acfba8.

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

Reviewed scope: Four-source backward Horner accumulation in historical X4; forward cost still binds, so no additional orientation-maximum saving.

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.