Paired cubes at p = 11 with physical frames and a paired-cube bit word: conditional κ = 5.508074e-4

[eumemic]

Conditional improvement

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{5508074}{10^{10}}=5.508074\times10^{-4}.$$

This is +19.50% over #144 (4.609169e-4, merged), +7.83% over #152 (5.108289e-4) and +16.66% over #151 (4.72154791e-4). The complex side binds. The complex saving is 5514156/10¹⁰ and the bit coarse saving is 5566382/10¹⁰. All of #144's interfaces remain assumptions.

Change

• 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 and pair modules are exact restrictions of two pinned searched complex DAGs, in references/paired-cube/sources.
  • The carrier arcs are frozen: R = 18,035, with 3,630 rank-18 gauges.
  • scripts/paired_cube_physical.py moves 7,173 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 at a deadline immediately before the role's first operation (Late compensation and joint frame refinement: conditional κ = 4.24830828577e-4 #143).
  • Per vertex: W = 17,045, deficit 1,320, largest child 20.
• Bit: a paired-cube bit word, in research/paired-cube-bit.
  • It is a mod-2 analog of Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's decoder at p = 12: x_T = Σ_{c∈T} star(c) + Σ_{|S∩T|=1} x_S, with partner-pair sums mixed on the source registers.
  • LEMMA.md proves the mixing step. It is sequential, not lockstep.
  • Per vertex: m = 72, W = 29,144, deficit 1,936.

#144's checks, finite bridge, row reserve and assembly are 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, all chains, the partner-pair chronology, G-nondegeneracy, a literal F2 replay and the ledger. Five mutation controls are rejected, and the generator reproduces every output byte for byte.
• 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).
• 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).
• 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 DAGs, the p = 11 instance, the physical layer and the paired-cube bit word: eumemic, with Anthropic Claude assistance.

🤖 Generated with Claude Code

[ploniaaaaaaa]

Incredible

[eumemic]

Superseded by #157, which keeps the paired-cube bit word and replaces the complex modules.