A p = 10 complex supplier: five-stage complex saving 7795/10⁷ (centre-sharing) and 7737/10⁷ (#304 recipe), both kernel-checked in Lean (no new κ; raises the complex cap above 7727)

[DreamingOfClouds]

The complex supplier at cube size p = 10, with no new κ by itself. The source-assisted complex word of #184/#194
is compiled with #304's frozen recipe one cube size down. It has h = 20 and 960 ports, against h = 22 and 1,320 ports
before. The query modules were re-annealed at p = 10.

program (gcert/1, five-stage layout at h = 20, m = 100, W = 4v + R)	R	saving a	p = 11 counterpart	gain
centre-sharing modules, #233 pairs	6,455	7795/10⁷	#315/#325's centre-sharing program, 7727/10⁷	+0.88%
#304's recipe on annealed p = 10 modules, #233 pairs	6,515	7737/10⁷	#304, 7635/10⁷	+1.34%
control: PR #168 v4 modules cut to p = 10, #233 pairs	7,125	7375/10⁷

In each row the next 10⁻⁷ point is rejected, and both claims also hold with the 10⁻¹⁶ fallback. A five-stage
package would consume these fallback-inclusive coarse savings:

• centre-sharing: b = 779597612622781/10¹⁸;
• A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's recipe: b = 773718577669358/10¹⁸.

Each is certified by two independent rational moment engines.

Both headline programs were also kernel-checked in Lean with Jacob Sussman's unchanged generator, as
wht_main_block_B2Gcc95x (1 − 7795973/10¹⁰) and wht_main_block_B2Gcp37x (1 − 7737183/10¹⁰). Details are below.
As far as I know, these are the first h = 20 programs built with his chain.

Why this matters now. The open frontier is the community stacks on the p = 10 bit word (#315 and #325), and
they are approaching the 7727/10⁷ complex cap of #315/#325. A κ package that adopts this supplier raises that cap to
7.796e-4 (+0.89%). This PR does not make that change. #315's package pins its p = 11 complex program in five places,
which a κ package must generalize or re-pin first:

• code/complex_labels.py, complex_scalars.py and complex_splice.py (h = 22, v = 1,320, 22 centres);
• portable_complex.py;
• math_check.complex_supplier().

PROOF.md §5 lists what each pins. The programs are in the star form those checks read.

python -m pip install numpy==2.3.5 scipy==1.17.0
python3 -B research/complex-p10/verify.py      # about 12 minutes, no network

Why p = 10 on the complex side too

The five-stage price favours smaller cubes. With a ≈ D/L, the deficit per port D/v = 4 − 15/(p − 2) saturates,
while m = 5h keeps growing. On an ensemble of library-module draws, with one recipe for every size, the complex
word prices as follows. These are measurements on this pipeline, not certificates.

p	9	10	11	12
five-stage price (·10⁻⁴), mean ± se	6.48 ± 0.06	6.45 ± 0.04	6.30 ± 0.05	6.08 ± 0.05

Single draws mislead, because their spread (about 2%) is as large as the size effect. The edge comes from about 4%
fewer helper registers per port at p = 10. Cutting #304's p = 11 modules straight to p = 10 gives only 7235/10⁷, so
the gain comes from re-annealing at p = 10.

What changed

There is one new self-contained package, research/complex-p10/, plus its own path-filtered workflow (Python 3.12
and 3.14). No existing file changes. The package follows #304's structure.

verify.py checks all 41 source pins and rebuilds #202's tree from the vendored #233 package. It then runs four
anchors:

• (a) aligned_word_p.py at p = 11 reproduces source_aligned_local_v4.py's seven outputs byte for byte;
• (b) Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194's word reproduces 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's certificate sha and Sussman's Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193 histogram, and the generic idle ranks
  give (42, 21, 46, 4) at h = 22;
• (c) the control modules are re-derived byte for byte as fewest-addition cuts of PR Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 v4's files;
• (d) 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 Graph.finish centre patch builds the same graph as centre46.py, and changes nothing for a
  36-root pair module.

Then, for each of the six programs:

• Admission:
  • pair checks with five mutation controls;
  • PR Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's physical();
  • PR A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200's complex checker, which rejects early-read and missing-read mutations;
  • PR Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184's flow and exact lift;
  • the Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194 contract (13 checks).
• Program: the gcert emitter, then gx.check1 with the exact scalar identity and the gxcore mirror on the raw
  and normalized program, plus a flipped-sign control rejected by both on E5.
• Price: the exact five-stage price at h = 20, both without the fallback (A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304's convention) and with it, and
  the 10⁻¹⁸ coarse b by two engines.
• Reproduction: byte comparison of every committed certificate and of expected.json.

The layout at h = 20 is Sussman's bridged five-stage B₂ formula with the h-dependent idle climbs: H₅ = 5·H_inv +
2v(e_{2h−2} + e_{h−1} + e_{2h+2} + e₄), m = 5h, W = 4v + R. That gives ranks 38/19/42/4 and deficit 2,040 here.

• Earlier uses were all at h = 22: A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304, 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 complex supplier and his Lean build of Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193.
• His repository states the layout for every h: bankPrice h, bridge2_gcert, GX.wht_of_halves, and gxunit
  from BANK['B2'].
• 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 already applies the same generic layout to its p = 10 bit word.

Two latent bugs, fixed for p ≠ 11 and documented:

• PR 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's pairs_mw.py hard-codes the gauge rank f = 18, which equals h − 4 only at h = 22. At p = 9 it
  returns no pairs. pairs_mw_p.py uses f = h − 4, which is identical at h = 22, and on these layers it changes
  nothing.
• source_aligned_local_v4.py keeps the births of gauges it drops in its record. On the control's cut modules
  this makes PR A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200's checker reject the word. aligned_word_p.py reverses exactly those births. That is
  checked, not trusted: the checker recounts the record from the word. Nothing changes when no gauge is dropped,
  as anchor (a) shows byte for byte.

Verification

• Replays: verify.py passes in fresh copies under Python 3.12 and 3.14 (about 11.5 minutes each), and again
  after the final doc edits. The docs are not read by verify.py.
• Independent audit by a separate read-only Claude Opus 5.5 agent that I directed, with no blockers:
  • its own replay;
  • Sussman's refcheck.py and gxdry.py accepting both programs;
  • its own 150-digit pricing of all six programs: every claim holds, the next point fails and every coarse b
    floors correctly;
  • an independent decoder check of the control, anneal and centre graphs (921,600 target–source pairs each, 0
    mismatches, centre roots equal to their star sums);
  • the patch re-applied byte for byte, both bug fixes reproduced, and the module cuts reproduced.
• Lean (local; not pinned in the package):
  • Setup: wht-power-saving-lean at f010392 (Lean v4.34.1, Mathlib d13f23b) with his unchanged
    tools/gx/regen_check.py.
  • Build: 31 of 31 modules built per program with 0 errors, one kernel check at a time.
  • Comparison: tools/Compare.lean passes against challenges that differ from ChallengeB2Gp193x only in names
    and exponent.
  • Axioms: propext, Classical.choice and Quot.sound.
  • Hashes: the checked gcert hashes (370fae73…23ff and 433848305e…16e5, uncompressed) equal the
    certificate_sha256 values in expected.json.
  • Not run: the official comparator and a build from nothing.
  • Scope: this covers the Walsh–Hadamard saving only, not any κ assembly.

Status, honestly

• No new κ. The bit side binds everywhere today.
• The fallback envelope (10⁻¹⁶, 32·m² rank-one calls per paid child) is the inherited assumption, here at
  m = 100.
• Annealing effort is asymmetric: about 1.5 h at p = 10, against the p = 11 anneals of Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 and A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304. The p = 10
  prices are therefore not a bound.
• The centre-sharing set has no control of its own. Its gain over 7737 includes a different pair module.
• Not included: the annealing code. The module files are inputs, checked by the tree's contract loaders and
  priced by the chain.

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

• icekylinx: Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144, the Graph.finish framework and Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184's frame flow and exact lift.
• Avi Eisenberg (ikeboy): 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 and source_aligned_local_v4.py.
• Chafik Boukhalfa (chafreaky): A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200's complex checker, 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's 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's 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.
• Jacob Sussman: gcert/1, gx.check1, gxcore, the five-stage layout and the Lean chain.
• eumemic: Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's modules and checker, and the source527 moment engines (vendored unchanged).
• DaysSky: Extended carrier matching on #161's complex word: conditional κ = 5.901881e-4 #162's arcs.
• Our earlier PRs: A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ) #304 (re-annealed p = 11 modules), 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/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) #325 (the centre-sharing patch and the p = 10
  bit word).

The module annealing, packaging, audit and Lean runs were prepared by DreamingOfClouds with Claude Opus 5.5
(Anthropic), via Claude Code (Claude Opus 5.5 agents). Every upstream notice is retained; see NOTICE.

🤖 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: p10 centre-sharing and annealed complex suppliers.

Review and limits: Primitive improvement only. Six complete programs and their exact prices are checked; included historical Lean receipts are distinguished from a new maintainer kernel build.

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: 40ed0456ab1990beb3440e61bb883ccf95fdcea4. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.