Research: optimal clean root checkpoint saves 25–40% of binary product-shear calls

[FanPUYen]

Change and mathematical scope

Research note / scoped interface construction; no new kappa.

A genuinely clean root checkpoint removes one right-subtree cancellation from PR628's product-shear recursion. Keeping both child recursions balanced and all-dirty, an exact root-split formula saves 25–40% of primitive leaf calls with the same total scratch-array count. One array must now start clean. At the seed-selected generic factor count 54, split 22+32 uses 2,112 calls instead of 3,136 and one clean plus five dirty arrays instead of six dirty arrays.

The package includes the all-n proof, complete weighted/dispatch/preparation/storage/routing cost recurrence, executable adapter, source pins, deterministic certificate and meaningful negative controls. It establishes neither a new multiplication exponent nor a global scheduling optimum. Clean-array supply, all-size layout/compiler and the inherited analytic/precision interfaces remain explicit obligations.

Validation

Reproduce: python3 -B research/clean-checkpoint/verify.py

Expected: PASS_CLEAN_CHECKPOINT: 32640 splits; 498 noncommutative cases; n=54: 3136 -> 2112 leaves, 1 clean + 5 dirty arrays

Validation: 32,640 root splits, 498 free noncommutative cases, 216 rounded executions with independent dense rational comparison, 14 interface/algebraic controls, two rounded controls and four optimized package-corruption controls. Passes on macOS arm64 CPython 3.9.6 and 3.12.14, including isolated optimized and relocated replay. Selected-record integrity and clean-patch applicability pass. Root make verify on a detached base checkout passed seven historical groups then stopped at gen4-bank-verify on missing SymPy; no full-suite, Linux, remote CI or Lean pass is claimed.

Base: 66849e1. PR628: 0768f97. PR663: e0e546f. Initial overlap review through #675 distinguishes concurrent PR673's multiway dirty echo. Its phase-alias finding is credited and the corresponding root guard is included, with original patch pinned at 0e35f56.

Only research/clean-checkpoint/ is added; upstream/ and the selected frontier remain unchanged. OS-random seed 54 and local claim/result are retained; no pinned sprint issue was available. Prepared for FanPUYen with substantial OpenAI Codex assistance. Credit to chafreaky, @0xkilgore, @androolloyd and all retained predecessor notices under Apache-2.0. No independent human peer review is claimed.

Publication refresh reviewed recent PR metadata through #709 and repeated the clean-checkpoint search (no matches). Dillon Amburgey’s #689 optimizes balanced ternary placement in #673’s all-dirty echo; this contribution instead changes the root storage contract while retaining balanced binary dirty children. No #689 code is consumed.

[Mister-Meeseeks]

100-PR batch review at efad8a40094ab453c0525efef0b0eff2a5d8b850.

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

Reviewed scope: One supplied clean root checkpoint: restored nonlinear interface and sharp root split for balanced dirty children; clean storage and dispatch charged separately.

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.