Research: bidirectional Horner saves 312 SW4 primitives per orientation (no new kappa)

[bitplane]

A dyadic coefficient can sometimes be built more cheaply by processing its digits from low to high, halving between reads. This contribution chooses between binary/NAF Horner in both directions while preserving the original per-macro scratch boundary. Finite complex-supplier circuit construction; no new κ.

Beyond the credited #696 copy/unit and #723/#787 forward signed-digit methods, this saves 312 real primitives per orientation on the #741 SW4 supplier selected at the starting base, plus 928 for historical CK and 810 for historical JP8:

Program	Best forward lowering	Bidirectional	Additional saving
SW4 forward / backward	142,536 / 142,552	142,224 / 142,240	312 / 312
CK forward / backward	169,416 / 169,432	168,488 / 168,504	928 / 928
JP8 forward / backward	197,104 / 197,120	196,294 / 196,310	810 / 810

Every source read, copy, shift, accumulation and clear is charged. All 716+9 checkpoint or 728+9 SW4 arbitrary input columns pass exact primitive replay, including dirty helpers and private scratch. Fresh positive prefix bounds remain within the inherited 192-bit local allowance. Frames, paid children and macro clearing boundaries are preserved. These are circuit counts, not measured runtime improvements.

Only research/dyadic-horner-direction/ is added: executable lowering, induction/cost proof, exact receipt, pins, separate integer checker, tests, local claim and validation.

python3 -B -I research/dyadic-horner-direction/verify.py
python3 -B -I research/dyadic-horner-direction/crosscheck.py
python3 -B -I research/dyadic-horner-direction/test_verify.py

Expected: six programs, 962 distinct coefficients, additional savings 928,810,312 per orientation, 14 adverse controls; the separate integer checker agrees on every coefficient and 13,286 grid inputs; eight tests pass. Both implementations were written by one AI agent, not independent human reviewers.

Linux CPython 3.12.3 and 3.14.4 pass, including optimization, no site packages and minimal relocation. Source/receipt/typed-JSON/circuit corruption fails as intended; gzip/tar repackaging passes; existing output is preserved. Selected-record and staged whitespace checks pass. Full repository/cold supplier/global admission/phase-wrapper regeneration, Lean and hosted CI passes are not claimed.

Exact base: 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261. OS seed 39, complex supplier; no sprint coordination issue was available. Both reviewed records are bound through their manifests to the consumed SPEC/program payloads. SW4 uses the committed maintainer replay archive; the complete construction chain is not rebuilt here. The final overlap snapshot covered 646 open PRs through #833; upstream main was unchanged at that check. During publication main advanced to f567884aeb8851b6e794af1184119a3bfbe1e631, selecting #799 at κ = 0.001661634653669567. The complete receipt also passes against its actual inputs and new selected pointer; #741's identical pinned record is accepted at its new official checkpoint location. A corrupted checkpoint is rejected. This remains a #741/#626 contribution, with no transport claim for #799.

Credit Sagar Thakrar's #696/#723, Chris Pagolu's #734, our #787 replay, icekylinx, DreamingOfClouds, Jeremy Kothe, Chafik Boukhalfa, Rohan Arun, eumemic and the retained #622/#626/#741 lineage. #696's lowering is vendored verbatim with its source notice. #751 and #830 collect/transpose work across macro boundaries; this construction reverses digits within individual coefficients. #739/#769/#811 and the target/helper panel constructions are also distinct; their savings are not assumed additive.

Prepared for Gareth Davidson (@bitplane) with substantial OpenAI Codex assistance, Apache-2.0. Production integration and refreshed global invoices remain separate work. Uniform compiler, realization, hosting, recursive interface, precision/tape and analytic assumptions remain inherited. The selected frontier is unchanged; finite replay is not a full multiplication theorem proof.