Paired bit-output fusion and verified lifetime reuse: conditional κ = 5.96818007e-4

[huxint]

This adds a source-pinned, independently reconstructed contribution on PR161: paired singleton-output fusion on the bit word, collective physical-frame descent, and 1,760 compensated register reuses. The selected complex word adopts PR163's signed fusion and improves its frames.

The conditional assembled saving is κ = 596818007/10^12 = 0.000596818007, 1.52% above PR161. The new bit supplier certifies coarse saving 0.000628145878, with 19,848 physical auxiliaries instead of PR161's 23,368. Later concurrent PR168/169 report larger assembled bounds; this submission does not claim the current overall record. The bit construction can be composed with stronger complex suppliers.

The new package is research/paired-cube-lifetime/; prerequisite PR161 at d14e29157bc905be1ced0776dd893d0714013f3a is included and is not claimed as new work here.

• Bit: merge (face1, edge01) and (face2, edge02) per singleton target, retaining disjoint supports and the exact decoder. Move 1,872 operation frames and place all 1,760 compatible rank-21 recipients in dead donors' slots. Every handoff, retimed target transition and dirty-value cancellation is replayed. W=23,368, rank=1,680,560, deficit=1,936.
• Complex: collective equal-frame moves on PR163's fused signed word. All 2,640 inherited compensated pairs and execution order remain. W=15,681, deficit=1,320, certified supplier saving 0.000597531242.
• Verification binds each virtual opcode to its graph node, checks all rational/binary frame chains, replays the actual aliased bit word on every formal source/dirty input, checks the complex signed identity and exact integer compensation responses, and rejects adverse geometry/alias/scalar controls.
• Exact interval moments charge the entire rare-class fallback. The legal atom cutoff, full finite group/router/scalar bill, restored row reserve, 47 strict constraints and exact successor exclusions are retained.

Reproduce with standard-library Python 3.11+:

python3 -B research/paired-cube-lifetime/verify.py

Local validation: complete end-to-end package verification passed, including both reconstructed words, exact arithmetic and mutation controls. PR161's make paired-cube-verify passed. A broader inherited make verify run reached the historical positive-skip target and stopped because this machine lacks boost/multiprecision/cpp_int.hpp; the full gate is not claimed as passing. This draft includes a dedicated Python 3.11/3.13/3.14 workflow, and remains a draft while remote CI runs.

All retained analytic, general Clifford, weighted local-ring, uniform-recursion, recovery and fixed-tape interfaces remain assumptions. This is a finite conditional witness, not an unconditional multiplication theorem or a measured runtime result.

Credits: icekylinx (#144/#130, paired cubes and assembly); eumemic (#157/#161, modules, bit word, output merging and physical layer); Chafik Boukhalfa (#163, fused complex source and independent checker); James Chang (#124, compensated birth reuse); an664 (#128, completed-core sharing); Zhihao Chen and Swapnil Jain for the bit/compiler lineage; gupt1156/geckods for atom-toll re-instantiation. Detailed provenance and original notices are retained in NOTICE and selected/complex/.

Prepared by huxint with substantial OpenAI Codex assistance.