Conditional κ = 7.51458925311258·10⁻⁴: addition reordering on #305's weighted gen5b composition (+0.0424% over #305)

[chafreaky]

Conditional κ = 7.51458925311258·10⁻⁴: addition reordering on #305's weighted gen5b composition (+0.0424% over #305)

One self-contained package, research/gen5b-weighted-repacked-reorder/ (#305's package with two added transcript stages), with its own workflow. No existing file is changed.

κ = 375729462655629/(5·10¹⁷) = 7.51458925311258·10⁻⁴, +3.184·10⁻⁷ over #305's 7.51140529238897·10⁻⁴. Bit coarse 7.52024·10⁻⁴, complex coarse 7.54736·10⁻⁴ (#233's word): the bit side binds.

What it adds

The reorder stage of #299 (where it gave +3.18·10⁻⁷ on the plain gen5b composition), re-screened on #305's final word (after its kernels, second descent, restorations and sinks) and appended after the sinks:

step	moves	local delta	φ	κ
#305	—	—	—	7.51140529238897·10⁻⁴
reorder	237: 212 root deliveries y += h moved past h's sibling deliveries from a rank-21 frame into y's next rank-22 frame; 25 forward gates into an earlier frame of a neighbouring incidence	{2: +237, 3: −217, 4: −25, 5: +5, 6: −20, 8: +20, 21: −212, 22: +212}	−495.07	7.51458164306940·10⁻⁴
reorder2 (screen re-run on the reordered word)	2 forward gates 18 → 16	{2: +2, 3: −2, 4: −2, 5: +2}	−1.19	7.51458925311258·10⁻⁴

Each move is a single addition moved, within its incidence window, into the frame of a neighbouring incidence; it is exact over Z because no crossed gate reads the moved addition's destination or writes its source, and rank mass is unchanged (the φ ledger, φ(r) = r·ln(120/r) at fixed deficit, predicted every κ exactly). The lever was found while testing Khattar–Gidney's conditionally clean ancillae (arXiv:2407.17966), whose literal class is empty on these words (every transcript operation is invertible, so no register's content is ever a combination of the others').

Fixed point. A re-census of every class on the final word (kernel families, early restorations, sinks, descent, reorder with a widened window) found nothing above 10⁻⁸ (best: 12 descent pair blocks, +2.4·10⁻⁹), so no further stage is added.

Checks

reorder_transform.py (byte-identical to #299's) re-derives every frozen move on the actual word — record content, the anchor frame's exact basis, the crossed-interval rule, disjoint operand sets — and runs the F₂ replay of all 18,944 columns forward and inverse (0 wrong rows), an omission control (999 / 66 wrong rows, rejected), an independent integer replay mod 2⁶¹ − 1 with random dirty inputs (identical on every register), the MOVE rebuild with both reflected ledgers, and #287/#291's exact source-span rule on every gate of its output. Each selection is bound to the hash of the word it was screened on (input 9c8dd18e… = #305's final word → 96fd1fa3… → 4b5edb89…). verify.py is #305's with the two stages required (14 stages, 14 omitted-stage controls); everything else of #305's package is unchanged, the new values are pinned in expected/kernel-pins.json.

Verify

python -m pip install -r research/gen5b-weighted-repacked-reorder/requirements.txt
python -B research/gen5b-weighted-repacked-reorder/verify.py --output /tmp/weighted-repacked-reorder   # about 6 minutes

Passes locally (PASS_IMMUTABLE_GEN5_KERNEL_RESTORE_SINK_FIVE_STAGE_BANKED_CONSTRUCTION, manifest b3c0a722…). The workflow (#305's, renamed) runs it and the weighted virtual word regeneration on Ubuntu.

Scope

Conditional finite construction under the retained public all-size hypotheses (no Lean certificate for the bit side). #305 and its stack (#275, #300, #296) are open; their word and stages are inherited unchanged. #304's re-annealed complex modules (7635/10⁷) would raise the complex cap further; not needed while the bit side binds.

Credits

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0; lineage in NOTICE.md. The base composition is eumemic's (#305; the weighted gen5b matching of #275, rohanarun's restart-packed kernels and second descent of #300); the gen5b word is ours (#296) on DreamingOfClouds' generator (#276/#285); the kernel, restoration and sink mechanisms are Dugongue's (#254/#259/#272/#283) and utcorvusvolat-dotcom's (#280); the descent and target stages are rohanarun's (#287/#291), the target-prefix mechanism eumemic's (#268). This package contributes the reorder stage (first in #299), its re-screen on #305's word and the fixed-point census.

[Mister-Meeseeks]

Thank you @chafreaky. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Commuting addition reorders after weighted gen5b kernels.

Review and limits: The exact crossed-interval rule and both scalar orientations are required. Retain the package's complete stage checks and producer regeneration.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 0314371983b8837f01723f5af2c70217f75b01cc. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.