Merged complex outputs and an annealed bit pair module on paired cubes: conditional κ = 5.878747e-4

[eumemic]

Conditional improvement

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{5878747}{10^{10}}=5.878747\times10^{-4}.$$

This is +27.54% over #144 (4.609169e-4, merged), +5.83% over #157 (5.555021e-4), which it supersedes, and +4.67% over #160 (5.616446e-4). The complex side binds: its saving is 5885669/10¹⁰, while the bit coarse saving is 5936323/10¹⁰. All of #144's interfaces remain assumptions.

Change

This is #157 with three changes: merged complex outputs on a new annealed pair module, an annealed bit pair module, and phase stop β = 10⁻⁹.

side	word	R	W per vertex	saving
complex	#157	17,529	16,539	5622769/10¹⁰
complex	this	16,011	15,681	5885669/10¹⁰
bit	#157 (witness-cut module)	25,624	29,144	5566382/10¹⁰ (coarse)
bit	this	23,368	26,888	5936323/10¹⁰ (coarse)

• Complex: Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's paired cubes at p = 11 (h = 22), with physical frames and late-read reuse.
  • The triple module is the same exact restriction of a pinned searched complex DAG as in Paired cubes at p = 11 with a searched pair module, physical frames and a paired-cube bit word: conditional κ = 5.555021e-4 #157. The pair module is a new pinned, annealed pair-disjoint module (365 additions), with its contract checked on load. Both are in references/paired-cube/sources.
  • Output merging. For every port T = (I, b), the face-2 output P and the edge-02 output Q are replaced by one signed node P + (2b₀ − 1)Q per (b₀, b₂), read with coefficient 1/2 by both ports b₁ = 0, 1 (merge_outputs in scripts/paired_cube/modules.py). The decoder contribution is unchanged and every addition keeps disjoint supports.
  • The carrier arcs are frozen: R = 16,011, with 2,970 rank-18 gauges.
  • scripts/paired_cube_physical.py moves 7,880 operation frames by physical descent (Physical frames and scratch reuse on Cayley covers: conditional κ = 3.32205398986e-4 #131's method under Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130's Clifford frames). It places every gauged role in a dead donor's slot (Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124), with the old-value read immediately before the role's first operation (Late compensation and joint frame refinement: conditional κ = 4.24830828577e-4 #143).
  • Per vertex: W = 15,681, deficit 1,320, largest child 20.
• Bit: the paired-cube bit word in research/paired-cube-bit (p = 12; partner-pair source mixing per LEMMA.md), now with an annealed pair-disjoint module (438 additions) in data/pair_module_p12.json.
  • The witness-cut module is kept as data/pair_module_p12_cut.json, and derive_pair_modules.py still regenerates it.
  • The carrier matching is re-solved and frozen in data/arcs_p12.json.
  • Searched pair module in the paired-cube bit word at p = 11: conditional κ = 5.616446e-4 #160 also uses a searched pair module in the bit word, at p = 11. This PR keeps p = 12 with a separately annealed module.
• Phase stop β = 10⁻⁹ (was 10⁻⁶), as re-instantiated by @geckods (Re-instantiated phase-stop on the p = 11 paired cubes: conditional κ = 5.108294e-4 #153, Re-instantiated atom exponent at the toll edge on the searched pair module: conditional κ = 5.557315e-4 #158).

#144's checks, finite bridge, row reserve and assembly are otherwise unchanged.

Verification

make paired-cube-verify
python3 research/paired-cube-bit/paired_cube_bit_word.py --p 12 --check
python3 research/paired-cube-bit/check_paired_cube_bit.py --dir research/paired-cube-bit/out --p 12

• Producer. It regenerates the complex input and passes all of Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's independent checks.
• Physical layer. It checks:
  • value spans and nested role chains, with every handoff spliced;
  • pair chronology;
  • target chains in actual read order;
  • the telescoping deficit;
  • an exact aliased dirty-scratch replay mod 2⁶¹ − 1, with three controls rejected.
• Bit checker. It recomputes the exact frames, the decoder identity, the chains, the partner-pair chronology, nondegeneracy, an F2 replay and the ledger, and five mutation controls are rejected. The generator regenerates the word and profile identically from the frozen arcs.
• Assembly. Both moments are strict, with the next grid points rejected. All 47 constraints and 7 margins hold.
• Every command above passes on Linux with a clean working tree, as do the maintainer's paired-cube entry-point tests and the full make verify.

Credits

• Paired-cube decoder, modules, shared cores, bit ledger and assembly: @icekylinx (Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144, building on Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130).
• Physical frames and reuse on covers: eumemic, with OpenAI Codex assistance (Physical frames and scratch reuse on Cayley covers: conditional κ = 3.32205398986e-4 #131).
• Birth-read reuse: @jamesyc (Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124).
• Late compensation reads: eumemic, with OpenAI Codex assistance (Late compensation and joint frame refinement: conditional κ = 4.24830828577e-4 #143).
• Completed-core sharing: @an664 (Completed-core sharing: conditional kappa > 2^-13 #128).
• Phase stop 10⁻⁹: @geckods (Re-instantiated phase-stop on the p = 11 paired cubes: conditional κ = 5.108294e-4 #153, Re-instantiated atom exponent at the toll edge on the searched pair module: conditional κ = 5.557315e-4 #158).
• Bit compile conventions: Zhihao Chen (Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97).
• Bit lineage of the pair modules (Interval strips and core-aware pair assembly: conditional κ = 4986133/10^11 (5101691/10^11 stacked with #57) #62): Swapnil Jain.
• Complex DAG, annealed pair modules, output merging, the p = 11 instance, the physical layer and the paired-cube bit word: eumemic, with Anthropic Claude assistance (Paired cubes at p = 11 with a searched pair module, physical frames and a paired-cube bit word: conditional κ = 5.555021e-4 #157).

🤖 Generated with Claude Code

[eumemic]

Superseded by #168 (κ = 6.096379e-4), which keeps this PR's construction and adds nested-prefix all-but-one modules on both sides, new annealed pair modules, #162's closure-compiled carrier arcs, a bundle-descent physical layer and the balanced prefix.