Conditional κ = 7.696840393833340e-4: a stronger p = 10 bit word — re-annealed pair module, p = 10 local design, balanced cube orientation, tiered reuse matching (+0.71% over #315's word)

[DreamingOfClouds]

κ = 384842019691667/(5·10¹⁷) = 7.696840393833340e-4, conditional. The bit supplier binds.

compared with	κ	gain
#323, the highest open claim when I wrote this (18:20 UTC Oct 10)	7.69557449978832e-4	+0.0164%
#315, the base of this package	7.64230861320245e-4	+0.7136%
#285 (gen5 five-stage package, p = 12)	7.42785663120535e-4	+3.621%

The open frontier (#316, #317, #319–#323) stacks transcript stages, kernels and retimings on #315's p = 10 word.
Against #322 the gain is +0.0165%, against #321 +0.0617%, against #320 +0.0169%, against #319 +0.1857%, against #317 +0.1971%
and against #316 +0.5931%. This package does not stack those stages: it improves the word underneath them.

This package changes nothing after the bit word. Instead it replaces #315's p = 10 bit word with a
re-searched one, keeping #315's package, its checks and its centre-sharing complex supplier (7727/10⁷). Four things
changed:

1. The pair module is re-annealed with a debt penalty. The objective is the exact compile cost plus λ·debt,
   where the debt counts module additions that do not donate to a carrier arc. Debt 95 → 92, still 273 additions.
2. The local design is re-searched at p = 10. Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315 reused the p = 12 design. Now a cube takes 24 local
   additions instead of 27.
3. The cube orientation is balanced. Every coordinate is position 0 in exactly 12 cubes; sorted cubes gave 36
   cubes to coordinate 0 and none to coordinate 9. This lets all 960 rank-17 gauges become reuse recipients;
   Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315 matched 892.
4. The reuse matching is tiered. Donors are admitted by decreasing end-frame dimension, and the result is
   still a maximum matching. All 960 pairs are matched (Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315: 890), and none is dropped for bank divisibility.

The certified bit root rises from 7.64815357e-4 to 7.70276910e-4. That is still 0.32% below the complex leaf
cap, so the bit binds and κ follows it.

cd research/five-stage-p10b-banks
python3 -m pip install -r requirements.txt          # sympy==1.14.0
python3 -B verify.py --output /tmp/p10b-five-stage-verification     # about 3 minutes, no network

The four changes

1. Pair module (pm_p10.json). I re-annealed #315's module with simulated annealing on the exact one-instance
compile cost, with five-stage weights r·ln(m/r), plus a penalty λ·debt (λ = 150 to 300). Here the debt is
additions minus carrier arcs. The module keeps 36 inputs, the centre root and 273 additions. Its carrier arcs go
from 178 to 181, so its debt goes from 95 to 92. Probes with λ up to 1,000 bottomed out at debt 93 from #315's
module (about 115 from a random start); continued annealing reached 92.

2. Local design (local_design_p10.json). #315 reused, unchanged, a cube-local design found by a pair-aware
search on the p = 12 word. This design comes from 1-move steepest ascent at p = 10, starting from #315's design
and scored on the five-stage virtual price of the full word. It reached a local optimum after two moves, which
changed 2 of the 12 plane recipes. Because subtrees are shared, a cube takes 24 local additions and 8 local
carrier arcs instead of 27 and 11 (local debt 16 both times). On its own, under sorted cubes and Hopcroft–Karp,
this design lost physical reuse pairs in the research runs (884 → 644 with an intermediate module). It pays
only together with changes 3 and 4.

3. Cube orientation (cube_orient_p10.json, new). Position 0 of a cube carries face0, the generator's
4-target root, and the rank-17 reuse recipients are tied to it. Sorted cubes put coordinate 0 at position 0 of
C(9,2) = 36 cubes and coordinate 9 at none. The donors that fit the recipients followed that imbalance, leaving a
Hall deficit of 68: #315's maximum matching had 892 of 960. The new file lists each cube as a cyclic rotation of its
sorted triple whose position 0 follows the largest cyclic gap (mod 10). In the one orbit with gaps 4, 4, 2, position
0 is the element between the two 4-gaps. The rule is rotation-equivariant, so every coordinate is position 0, 1 and
2 in exactly 12 cubes each. Orientation alone left R and the virtual price unchanged in the research runs. The
generator reads the file when present and otherwise uses sorted cubes. It also stores the symmetric plane entries
that an oriented cube looks up; sorted cubes never read them.

4. Tiered reuse matching (make_physical.py). Hopcroft–Karp returns some maximum matching regardless of donor
value. With the balanced orientation it covered all 960 recipients but chose low-dimension donors: the summed
first-order pair value on the shipped word is about 40.2k against 45.1k for the tiered matching (in ledger units). The first-order
five-stage value of a pair grows with the donor's end-frame dimension e. The producer therefore admits donors in
tiers of decreasing e. After each tier it runs Kuhn augmenting paths from the current matching until none is left
among the donors admitted so far.

• Donors stay matched. An augmenting path never unmatches a matched donor, so this is the greedy algorithm on
  the transversal matroid of donors. For every threshold t, as many donors with e ≥ t are matched as any matching
  allows. Since the first-order value grows with e, this targets that value; it is not an exact optimum of the
  price.
• Still maximum. The last tier is the full candidate graph, run until no augmenting path remains, so the
  result is a maximum matching (Berge). Here that means all 960.
• Same pair rules. PR200's rules are unchanged; only the choice among maximum matchings changed.
• No drops. The residual width 4·1,200 + 20·6,900 = 142,800 is already ≡ 0 (mod 5). --bank-parity drops
  nothing, and the banks tile as pure (4²⁵) and (20⁵) banks.

Where the gain comes from. These are research-pipeline estimates (float, unbanked five-stage price, not
certified), measured against #315's modules and design regenerated through the same pipeline (892 pairs, before
#315's two bank drops):

step	reuse pairs	unbanked price	vs #315's word
#315's word	892	7.584707e-4	—
+ debt-penalised pair module (intermediate)	884	7.615053e-4	+0.400%
+ balanced orientation and tiered matching	960	7.625222e-4	+0.534%
+ p = 10 local design	960	7.640606e-4	+0.737%
+ final pair module (this package)	960	7.641048e-4	+0.743%

The new word

one invocation	#315, p = 10, m = 100	this package, p = 10, m = 100
virtual auxiliary roles	9,120	9,060
five-stage virtual price (research float, not certified)	7.4429e-4	7.4944e-4
compensated reuse pairs	890	960
independent dirty registers R	8,230	8,100
formal F₂ columns 2v + R	10,150	10,020
independent entrances (ranks)	1,270 (16, 17)	1,200 (16)
banks per stage	86,526	85,680
normalized stock / deficit	132,606 / 24,480	131,760 / 24,480
bit coarse saving (certified root)	7.64815357146385e-4	7.70276909563042e-4
complex coarse saving	7.72714351296671e-4	7.72714351296671e-4 (unchanged)
κ	7.64230861320245e-4	7.69684039383334e-4

What changed in the package

There is one new self-contained package, research/five-stage-p10b-banks/, plus its own path-filtered workflow.
No existing file changes; #315's package is unchanged.

The new package is #315's re-pinned for the new word, using #315's own discovery/repin.py flow:

• Changed data in bitword/producer/data/: pm_p10.json, local_design_p10.json, cube_orient_p10.json
  (new) and the frozen arcs arcs_p10.json (6,020 arcs).
• Two changed producers, which verify.py never runs: the generator (optional cube orientation) and
  make_physical.py (tiered matching).
• Re-pinned files: the five word files in bitword/selected/bit, word-pins.json and MANIFEST.json.
• Prose: updated to the new word.

Every verification file is byte-identical to #315's. That covers verify.py, every *_check.py, code/,
scalar/, bitword/bit and the vendored checker. The checks are exactly #315's; only the strict pins changed. Two
of those pins change what is exercised. The entrance census is now {16: 1,200}, so geometry527 checks two
distinct actual rank-16 bases; there is no other rank. The bank patterns no longer contain a mixed pattern, so the
tiler's mixed-width path is not used.

Verification

• Replays:

  • The package: Python 3.12 and 3.14, each from a fresh copy, give the same
    PASS_IMMUTABLE_PARITY_FUSED_P10_FIVE_STAGE_BANKED_CONSTRUCTION, the same κ and the same manifest. verify.py
    takes about 3 minutes.
  • regenerate.py reproduces the five pinned bit-word files byte for byte under both versions (about 45 s).

• Independent audit by a separate read-only Claude Opus 5.5 agent that I directed, with no blockers:

  • Fresh strict verify.py replays under Python 3.12 and 3.14 both PASS at κ = 384842019691667/(5·10¹⁷), with the
    same manifest. regenerate.py reproduces the five pinned word files byte for byte under both.
  • It re-checked the mod-2 decoder independently on all 960 targets, with no package code.
  • It recomputed the bit root (400-bit, with the 10⁻¹⁶ fallback, next grid point rejected) and κ from the
    normalized profile; both match the pins, and the bit binds.
  • On the dumped reuse candidate graph, it verified that for every donor-dimension threshold the shipped 960 pairs
    match as many donors as any matching allows.
  • It confirmed that the patched generator without the orientation file reproduces Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's generator output byte
    for byte.
  • It confirmed that 105 of the 124 files are byte-identical to Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's, including every check file.

• The complex program alone (from Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315; the program is unchanged): an earlier audit, also by a separate Claude Opus 5.5
  agent, re-derived its decoder over all 1,320 × 1,320 (target, source) pairs and re-applied the Graph.finish
  patch byte for byte. It replayed the scalar identity independently and confirmed 7727 is certified and 7728
  rejected, with and without the fallback.

• Lean (local, not pinned; unchanged from Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315):

  • Setup: Sussman's wht-power-saving-lean at f010392c (Lean v4.34.1, Mathlib d13f23b), with his unchanged
    tools/gx/regen_check.py.
  • What was checked: the exact gcert pinned here (sha256 bb0499ab…4d34, the same bytes as Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315) compiled
    and kernel-checked as OAI.PowerSaving.WHT.wht_main_block_B2Gcc27x, exponent 1 − 7727140/10¹⁰, 34/34
    modules. tools/Compare.lean passed against a challenge that differs from his ChallengeB2Gp193x only in
    exponent and names. The axioms are propext, Classical.choice and Quot.sound.
  • Not run: the official comparator and a clean build from nothing.
  • Scope: this covers the complex program's Walsh–Hadamard saving only. Because the bit binds, κ is unchanged
    if b is replaced by the Lean exponent.

Status, honestly

• Conditional: this is a conditional finite construction with Conditional κ = 7.42785663120535e-4: gen5 — compile-cost-annealed pair and all-but-one modules on #276's gen4 bit word, five-stage completed banks #285's retained interfaces: the all-size
  compiler, common weighted charts, restored rows, selectors, routing, prime supply, precision/recovery, complex
  symbolic correctness and analytic transfer. There is no Lean for the bit word, the bootstrap or the assembly, and
  no multiplier benchmark.
• Inherited texts: proof/, UPSTREAM-PR234-PROOF.md and the notices are inherited verbatim. They state
  p = 12 / source527 numbers and describe PR193's complex program. PROOF.md says so and gives this word's values.
• The complex program is Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's, unchanged, and unpublished outside Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315. It needs a 10-line patch to PR
  Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's Graph.finish and a 46-root loader. The patch and modules are shipped as provenance, but the loader is
  not. The gcert bytes are kept as emitted, because the Lean run checked exactly those bytes.
• Search scripts are not shipped: the debt-penalised annealer, the local-design scan and the orientation
  analysis are not included. The data files they produced are, and regenerate.py rebuilds the word from them.
• The margin over the open frontier is small (+0.0164% over κ = 7.69557449978832e-4: three rank-two kernels after frozen PR322 #323). It comes from the word alone, while Conditional κ = 7.65146115390441e-4: 480-gate concave-descent retiming on #315's p = 10 word (+0.120% over #315) #316–κ = 7.69557449978832e-4: three rank-two kernels after frozen PR322 #323
  stack stages on Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's word. The two are different levers.
• Follow-ups (not done here): the stage stacks on Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's word are word-specific choices (gates, kernels,
  pivots), so they would have to be re-searched on this word, not just re-pinned. These are Conditional κ = 7.65146115390441e-4: 480-gate concave-descent retiming on #315's p = 10 word (+0.120% over #315) #316 (concave-descent
  retiming), κ = 7.6817007781759e-4: composed p10 stages with full source-bound admission #317 and κ = 7.68257328708259e-4: shared-donor kernels after frozen PR317 #319 (composed stages, shared-donor kernels), Conditional κ = 7.69553898621543·10⁻⁴: the full transcript stage stack on #315's p = 10 word, with a shared-donor closure kernel (+0.697% over #315) #320 (the transcript stage stack with a
  shared-donor closure kernel) and Conditional kappa 7.692096978795988679e-4: five new p10 response kernels and source-bound finite assembly #321–κ = 7.69557449978832e-4: three rank-two kernels after frozen PR322 #323 (response kernels, retimings, rank-two kernels). On Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's word they added +0.12% to +0.70%. This
  word has more reuse pairs and no rank-17 entrances, so the yields will differ. I have not done this here. The
  p = 10 complex word mentioned in Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315 (7795/10⁷ centre-sharing) is still not needed while the bit binds.

CI note. This branch also carries Rohan Arun's #318 commit, cherry-picked unchanged (it re-pins the root
Makefile and README.md hashes in research/joint-dual/SOURCE.json). Without it the repo-wide verify (…, joint)
job fails on main and on every open PR. The commit is identical to #318's and drops out once #318 is merged.

Attribution

• Base package: Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315 (the p = 10 port of gen5, the centre-sharing complex supplier and the Lean run).
• Five-stage layout and gcert/gx: Henry Grant (hcg890)'s Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234, and Jacob Sussman's five-stage construction,
  gcert/1, gx.check1 and gxcore.
• Generator, checker and package: eumemic's Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 generator and checker, and the source527 package with its
  lineage. The notices of PR207, PR216, PR211, PR230 and PR210 are retained.
• Physical layer and complex re-pin: Chafik Boukhalfa (chafreaky)'s A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200 physical-word classes and pair
  rules. His A stronger complex supplier: PR200's reuse pairing on the source-assisted v4 word, complex saving 7.0991·10⁻⁴ (+1.28% over #194; no new κ, bit binds) #233 reuse pairing, The source-assisted complex word of #233 as one explicit numbered program (gcert/1): reference-checked, 7.547361·10⁻⁴ in the Lean-checked five-stage layout (no new κ) #256 gcert emitter and Conditional κ = 7.46709720962478·10⁻⁴: gen5b — exactly optimal local design and a debt-10 all-but-one module on #285's word, #287's stages re-derived, #233's complex supplier (+0.348% over #287) #296 re-pinning of the complex checks for a new gcert are
  also used.
• Banks: Evan McKinney's Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197 completed-bank method and Rohan Arun's Conditional κ = 7.08582410781108e-4: completed width-120 entrance banks on #234's five-stage bit supplier #237 width-120 banks, here at width 100.
• Complex word: Avi Eisenberg (ikeboy)'s Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193/Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194 complex helper, and icekylinx's Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144 framework and Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184
  frame flow.
• Earlier credits: everything credited in Conditional κ = 7.22869827347495e-4: a new paired-cube bit word (arc-aware local circuit, centre-sharing pair module) in the five-stage completed banks #276, Conditional κ = 7.42785663120535e-4: gen5 — compile-cost-annealed pair and all-but-one modules on #276's gen4 bit word, five-stage completed banks #285, A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304 and Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315 is retained. See NOTICE.md.

The debt-penalised module anneal, the p = 10 local-design search, the balanced cube orientation, the tiered reuse
matching and the re-pin were prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code (Claude Opus 5.5 agents).

🤖 Generated with Claude Code

[Mister-Meeseeks]

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

Contribution: Re-annealed p10b bit word and tiered reuse matching.

Review and limits: New scalar modules and balanced orientations are regenerated and the full finite construction is checked. Search scores nominate inputs; they are not admission evidence.

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: 0eca9340a3df6141b8e71a41638c3937b3522888. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.