A stronger complex supplier: compile-cost-annealed complex modules, five-stage complex saving 7635/10⁷ (+1.17% over #233; no new κ)

[DreamingOfClouds]

A stronger complex supplier (no new κ by itself): re-annealing the three query modules of the
source-assisted complex word gives these five-stage complex savings:

pair recipe	this word	PR #233's word (PR #256)	PR #193's word
PR #233's maximum-weight reuse pairs	7635/10⁷	7547/10⁷ (+1.17%)
PR #193's parity-avoiding pairs	7586/10⁷		7474547/10¹⁰ (+1.50%)

The current frontier is bit-bound, about 0.6% below #233's complex price (7.54736·10⁻⁴): #303/#300 at
κ = 7.49983·10⁻⁴ (12:00 UTC Oct 10), and #299 with bit coarse 7.50462·10⁻⁴. This
package raises the complex cap it is approaching from 7.547·10⁻⁴ to 7.635·10⁻⁴. It lifts nothing on
its own, because the bit supplier binds.

python -m pip install numpy==2.3.5 scipy==1.17.0
python3 -B research/complex-module-anneal/verify.py      # about 12 minutes, no network; -O is refused

What changed

There is one new self-contained package, research/complex-module-anneal/, plus its own path-filtered
workflow. No existing file changes. It is built in the mould of #256: the same vendored gcert_emit.py,
Jacob Sussman's gx/gx.py (unchanged) and the PR #233/#202 references.

The source-assisted complex word of #184/#194 calls three PR #168-v4 query modules:

• the pair module (n = 10, 22 instances);
• the all-but-one module (n = 9, 110 instances);
• the disjoint-triple module (one instance).

They were re-annealed against the exact compile ledger of the aligned word, as #285 did on the bit side.
The new modules are not smaller in total (2,462 additions against 2,353), but they are cheaper to compile:
the frame flow needs R = 9,160 slots instead of 9,412.

program (gcert/1, five-stage m = 110, W = 4v + R)	modules	pairs	R	N	gx.check1	certified a
control	PR #168 v4	PR #193 recipe	9,412	262,944	accepted	7469/10⁷
control	PR #168 v4	PR #233 recipe	9,412	262,944	accepted	7522/10⁷
result	re-annealed	PR #193 recipe	9,160	257,400	accepted	7586/10⁷
result	re-annealed	PR #233 recipe	9,160	257,400	accepted	7635/10⁷

Each certified value has exact rational bounds, and the next 10⁻⁷ grid point is rejected. The controls run
the same recipe on PR #168 v4's own modules: fresh carrier arcs, PR #200's descended frames, then the two
pair recipes. They measure the modules' effect alone: +1.50% and +1.57%.

The recipe is slightly weaker than PR #168's inherited physical layer on the old modules: −0.07% against
#193 and −0.33% against #233. The comparison above is therefore conservative.

What verify.py checks

verify.py checks every pin and rebuilds PR #202's 308 files from the vendored PR #233 package. It runs
two anchors:

• aligned_word.py reproduces source_aligned_local_v4.py's seven outputs byte for byte;
• PR 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.

Then, for each of the four words, it runs:

• the module contracts;
• the frozen arcs through compile_closure, and the frozen frames through the script's own forward
  closure;
• paircheck.py, an independent pair re-check with five mutation controls;
• PR Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's physical(), a randomized mod-2⁶¹−1 replay with three controls;
• PR A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200's complete complex checker as admission;
• PR Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184's flow and exact rational lift, and PR Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194's 13-check contract;
• PR 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 emitter, then a documented sign-convention normalization to the star-rule scatter;
• gx.check1 with the exact scalar identity, on the raw and the normalized program, plus Sussman's gxcore
  mirror on the committed program;
• the ledger and the exact five-stage price;
• byte comparison of the four committed certificates and expected.json.

Verification

• Replays: two fresh-copy replays in other working directories (Python 3.14, numpy 2.3.5, scipy
  1.17.0) reproduce every certificate byte for byte.
• Independent audit: read-only, no package code.
  • Exact two-sided moment bounds reproduce all four values: 7469, 7522, 7586 and 7635.
  • Its own GF(2) label and exact scalar replays agree, as does Sussman's gxcore mirror.
  • It re-checked the modules, pairs and layers, including make_layer.py reproducing all 12 frozen
    layer files.
  • It found no blocker.

Status, honestly

• κ: no new κ by itself; the bit side binds everywhere today.
• Lean: no Lean build. gx.check1, Sussman's Python reference checker, accepts all four programs. His
  Lean bridge, refcheck.py and gxdry.py were not run.
• Layout: the five-stage formula H5 = 5·H_inv + 2v(e42 + e21 + e46 + e4), W = 4v + R is applied at
  R = 9,160, as the bit packages apply it at several R. The deficit m·W − rank = 4v − 5·cst = 3,080 is
  unchanged. Confirmation is left to the layout's authors.
• Integration: to compose this into a five-stage package, the complex code there must accept a
  Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193-shaped program at R = 9,160. It is hard-wired to Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193 today, though 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 and Conditional κ = 7.50237961741364·10⁻⁴: kernels, early restorations, terminal sinks, a post-sink retiming and addition reordering on the gen5b word (+0.473% over #296) #299 already
  generalised it for 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 word. The programs here use the star rule, as that code expects.

Follow-up (not packaged)

A centre-sharing variant (pair-module root = centre, about 10 lines in Graph.finish) measured about
+2.7% in exploration. It needs a reviewed code change and is left for whoever owns Graph.finish.

Attribution

• icekylinx: Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144 and the Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184 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.
• 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 and 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.
• Jacob Sussman: gcert/1, gx.check1, the gxconv unit convention and the five-stage layout.
• eumemic: the Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 modules and paired_cube_physical.
• DaysSky: Extended carrier matching on #161's complex word: conditional κ = 5.901881e-4 #162's arcs.

The module annealing (exact compile-ledger evaluators and annealers, run by Claude Opus 5.5 agents) and the packaging
were prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. 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: Compile-cost annealing of complex query modules.

Review and limits: A stronger complex primitive, not a new multiplication bound. Four complete scalar programs and controls are replayed; the generic layout and inherited all-size interfaces retain the source qualifications. Saved Lean claims are separate from Python admission.

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: 7ac169c58c96cebdb5cd490842331dff3610af62. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.