Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4

[eumemic]

Conditional improvement

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{6558894}{10^{10}}=6.558894\times10^{-4}.$$

This is +42.30% over #144 (4.609169e-4, merged), +7.00% over #173 (6.1297e-4, the best other open PR) and +10.42% over #163. It supersedes this PR's earlier versions (6.096379e-4, 6.457084e-4 and 6.489120e-4). The complex side binds: its saving is 6563200/10¹⁰, and the bit coarse saving is 6600253/10¹⁰ (6.5961531e-4 effective). All of #144's interfaces remain assumptions.

Change

This branch contains #161's construction (#161 is closed in favor of this PR) and adds the following.

side	word	R	W per vertex	saving
complex	#161	16,011	15,681	5885669/10¹⁰
complex	this	13,372	13,655	6563200/10¹⁰ (+11.51%)
bit	#161	23,368	26,888	5936323/10¹⁰ (coarse)
bit	this	19,788 (18,028 physical)	21,548	6600253/10¹⁰ (coarse, +11.18%)

Complex word (p = 11)

• Modules. The triple-disjoint, pair-disjoint and all-but-one modules are pinned annealed modules in references/paired-cube/sources. Each is loaded with its contract checked (triple_module_from, pair_module_from, all_but_one_from), in place of PR117's restrictions and Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's balanced tree. The all-but-one module was annealed from a nested prefix:
  • prefix sums p_{k+1} = p_k + x_k and suffix sums s_k = x_k + s_{k+1};
  • outputs y_0 = s_1, y_{n−1} = p_{n−1} and y_{i+1} = p_i + (x_i + s_{i+2});
  • each prefix is used twice with nested supports, which admits more carrier links than the balanced tree.
• Local channel circuit (local_L1.json). Every cube's 13 local outputs keep their values.
  • G01 (both modes) and G12 (mode 1) become sums of two shared long-diagonal differences, and F is built on axis 2: 27 additions per cube instead of 25. The (0,1) edges lose their G duty, so the carrier matching gains links.
  • Graph(p, local=None) runs Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's circuit unchanged.
• Fused outputs. Each port's single-target face-2, edge-02 and edge-12 outputs are fused into one signed node read at 1/2, in a fixed fold order per port class (merge_outputs, variant f8:00111100). This follows Fused paired cubes and paid balanced transfer: conditional κ = 5.93970203079492e-4 #163's same-singleton fusion.
• Carrier arcs. Extended carrier matching on #161's complex word: conditional κ = 5.901881e-4 #162's compile_closure compiles the frozen 10,704 arcs in the main producer. Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's verify.py independently checks acyclicity and span containment. There are 2,310 rank-18 gauges.
• Physical layer. 15,149 operation frames move in connected bundles that share a frame. The 2,310 gauged roles use late-read reuse (Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124, Late compensation and joint frame refinement: conditional κ = 4.24830828577e-4 #143), paired by maximum weight.
• Terminal sinks (research/terminal-sinks). 47 all-eight side roots are terminal sinks under Add exact dirty-target terminal compiler and 67-sink witness #166's lemma, and sinks_gate.py checks them.
  • A sink's register starts at frame 0, is only ever a destination outside phase one, and has a pivot target p with no deferred read of y_p during its writes.
  • The register is deleted. At the phase cut y_t −= y_p for the other targets, every write goes to y_p at its own frame, and after the last write y_t += y_p at the root frame.
  • The gate rechecks every sink condition, scans the literal stage words forward and reflected, runs a dirty-scratch replay mod 2⁶¹ − 1, and rejects four controls.
  • Per vertex: W = 13,655, deficit 1,320, largest child 20.

Bit word (research/paired-cube-bit, p = 12)

• Modules and merge. The word uses an all-but-one module (n = 10, 33 additions, pinned in data/qmod_p12.json) annealed from the nested prefix, and an annealed pair module with 442 additions. Its local channels use the L1 association: the G sums with keys (0,1,0), (0,1,1) and (1,2,1) add the two long face diagonals. Variant u merges each cube's face-1/edge-01 outputs into u(b0,b1) and its face-2/edge-02 outputs into w(b0,b2); each merged node is read by two single roots. There are 3,960 gauges.
• Physical layer (new). scripts/paired_cube_bit_physical.py applies the frozen word in references/paired-cube/bit-physical: late copies, 2,884 moved operation frames and 1,760 compensated late reads. It checks:
  • register contents against the package graph;
  • exact, G-nondegenerate frames;
  • spliced role chains and pair chronology;
  • target chains in read order and the partner chronology;
  • the ledger, by recount;
  • a 64-lane aliased F2 replay with three controls rejected.

Assembly

• Balanced prefix (scripts/paired_cube_assembly.py). It uses the RaD project's reviewed balanced positional transform, with prefix margin 1 − ε. The original prefix rejects this κ.
• Atom exponent at the toll edge. θ is the first 10⁻¹² grid point above a_0/(1 + a_0 − a_old), as in Re-instantiated atom exponent at the toll edge on the searched pair module: conditional κ = 5.557315e-4 #158: θ = 323062859/(5·10¹¹).

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

make paired-cube-verify runs the producer, the complex physical layer, the terminal-sink gate, the bit package checker, the bit physical layer and the network certificate.

• Complex. Every coefficient of H + K + B = I, frames, intersections, target chains, and the complex physical layer's spliced chains, read-order target chains, telescoping deficit and aliased mod 2⁶¹ − 1 replay with three controls.
• Bit. The package checker (five mutation controls), byte-identical regeneration from frozen arcs, and the bit physical layer above.
• 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. The full suite runs in CI.

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).
• Closure-compiled carrier arcs and pairing rounds: @DaysSky (Extended carrier matching on #161's complex word: conditional κ = 5.901881e-4 #162).
• Terminal-sink lemma: @jamesyc (Add exact dirty-target terminal compiler and 67-sink witness #166, closed by its author), audited here by its own gate.
• Same-singleton output fusion: @chafreaky (Fused paired cubes and paid balanced transfer: conditional κ = 5.93970203079492e-4 #163); the fold orders here come from our own search.
• Atom exponent and phase stop at their edges: @geckods (Re-instantiated atom exponent at the toll edge on the searched pair module: conditional κ = 5.557315e-4 #158, Re-instantiated phase-stop on the p = 11 paired cubes: conditional κ = 5.108294e-4 #153).
• Balanced positional transform: the RaD project (@hipotures, Source-framed h53 conditional witness above 2^-20: attributed research record #20); balanced assembly composition @jamesyc (Reversed two-stage basis and balanced assembly for conditional κ = 1.639226629e-5 #34), with the reviewed parameters of Chafik Boukhalfa's research/copied-fixed/balanced_assembly.py.
• Physical frames and reuse on covers, late compensation reads: eumemic, with OpenAI Codex assistance (Physical frames and scratch reuse on Cayley covers: conditional κ = 3.32205398986e-4 #131, Late compensation and joint frame refinement: conditional κ = 4.24830828577e-4 #143). Birth-read reuse: @jamesyc (Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124). Completed-core sharing: @an664 (Completed-core sharing: conditional kappa > 2^-13 #128).
• 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.
• Annealed triple, pair and all-but-one modules, the nested prefix, the local-circuit search, output merging (w02, u, the f8 fold orders), bundle descent, both physical layers, the terminal-sink gate, the paired-cube bit word and the assembly integration: eumemic, with Anthropic Claude assistance.
• Connected frame moves also appear, as parallel work, in Connected bit-frame descents on #163: conditional κ = 5.94035602295411e-4 #164's bit-frame descents.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you for this contribution. Closing as incorporated into the published historical construction, with attribution retained—not as a rejection.

We rechecked the current head 4a3c769e5c5430e7114c4d3e099ff34664677f17 and all 62 changed files. Every file is preserved byte-for-byte in the PR181 source archive used by the reviewed entrance-bank checkpoint on main. The historical implementation remains reproducible there; this closure does not select its standalone exponent as the current record.

• Incorporation evidence and exact file coverage
• Published archive and source pin
• Scoped review and contributor credit

The new central ledger separates incorporated work from pending review. Mathematical claims retain the limitations in the linked review.