Contiguous-pair coordinate pricing and two-round circuit exchange: conditional κ = 5.2805219274162e-5

[Th0rgal]

Under the inherited analytic and fixed finite-alphabet multitape hypotheses,

$$T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad\kappa=\frac{26402609637081}{500000000000000000}=5.2805219274162\times10^{-5}.$$

This is a +0.026060% improvement over #98 ($+13,757,650,460$ grid steps over #98, $+14,123,369,682$ over #95, and $+15,602,338,941$ over #91) on the aligned-composition track, with bit saving 52808007812753/1000000000000000000.

Current comparison: #97 (6.3965813e-5) and #99 (6.3978725e-5) report higher conditional $\kappa$ values on the deferred-endpoint / signed-complex track. This PR is a cumulative refinement on the aligned-composition fixed-exterior track (#91 $\to$ #93 $\to$ #94 $\to$ #95 $\to$ #98), not a global record claim.

Two certified stages (research/contiguous-pair-exchange/)

1. Stage A — Contiguous-pair coordinate pricing with the pristine unchanged Aligned frame composition: κ=5.2789616935221e-5 with Lean arithmetic certificate #91 engine ($\kappa = \frac{26400279929441}{500000000000000000} = 5.2800559858882\times 10^{-5}$):

   • In Aligned frame composition: κ=5.2789616935221e-5 with Lean arithmetic certificate #91 (and inherited through Coordinate refinement of aligned composition: conditional κ = 5.2790906739644e-5 #93–Final-coordinate carrier pricing: conditional κ = 5.2791461623702e-5 #98), the odd singleton coordinate (22 at $h=23$, 24 at $h=25$) was placed at index 4, splitting the chain of consecutive coordinate pairs, and coordinates (17,18) at $h=25$ were transposed to (23,22), splitting both (16,17) and (18,19).
   • Moving the singleton out of index 4 ([6, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 17, 19, 20, 21, 22, 1, 2, 4, 3, 0] at $h=23$) and reuniting the split pairs ([4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 1, 0, 2, 3, 23] for oracle pricing and [5, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 1, 0, 2, 3, 23] for final relabeling at $h=25$) makes all coordinate pairs contiguous.
   • Passing this contiguous-pair order as ORACLE_COORDINATES to the unmodified scripts/experiments/aligned_composition_engine.py reclaims an additional role at $h=25$ ($R_{25} = 34,771$, down from $34,772$, reducing width to $W = 130,416,141$) and gains $+9,098,235,180$ grid steps over Final-coordinate carrier pricing: conditional κ = 5.2791461623702e-5 #98 with zero engine modifications.

2. Stage B — Two-round carry exchange and min-cost circuit reclamation ($\kappa = \frac{26402609637081}{500000000000000000} = 5.2805219274162\times 10^{-5}$):

   • Runs two outer rounds of the 1-for-1, 2-for-2, and 3-for-3 carry exchange pass (outer_exchange_rounds = 2) so 3-cycle updates in round 1 unlock further 1-for-1 and 2-for-2 exchanges in round 2.
   • Orders eligible pending live slots by (profile_cost(s, g), order[s]) and includes all zero-sum $\mathbb{F}_2$ circuits formed by anchor, eligible pending, and eligible retired slots when toggling the clearing support of a reclaimed retired slot. All frame inclusions, $\mathbb{F}_2$ independence checks, literal XOR realizations, and arbitrary-dirty restoration guards are unchanged.
   • Reclaims 4 roles at $h=25$ ($R_{25} = 34,768$, down from $34,772$) and sharpens the $h=23$ internal profile ($R_{23} = 26,399, W = 130,438,428$).

Every exterior, center-copy charge, side-growth contribution, and early/late restoration gate remains paid in full; the finite bridge formula and complex layer are unchanged.

Verification

• Fresh Stage A compilation (scripts/experiments/aligned_composition_engine.py) and Stage B compilation (research/contiguous-pair-exchange/engine.py), matching all committed words byte-for-byte, plus dense scalar support audit (scalar-23.json, scalar-25.json).
• Complete arbitrary-dirty physical replay in both orientations and 3-prime CRT profiles (crt_disagreements == 0) for all four Stage A and Stage B words.
• Exact rational and independent moment enclosures, 47 strict constraints, 7 margins, next-grid rejections, and complete profile exclusions of Final-coordinate carrier pricing: conditional κ = 5.2791461623702e-5 #98 at Stage A's saving and of both Final-coordinate carrier pricing: conditional κ = 5.2791461623702e-5 #98 and Stage A at Stage B's saving.
• Seven unit/adversarial controls (test_controls.py), including exact equivalence between profile_oracle_cached.cpp and the reference oracle, and 2,789 pinned source files (SOURCE.json).

python3 research/contiguous-pair-exchange/verify.py
python3 research/contiguous-pair-exchange/test_controls.py
python3 research/contiguous-pair-exchange/verify.py --regenerate
make verify

No search optimality, new Lean theorem, unconditional bound, or practical runtime speedup is claimed. All inherited all-size and analytic hypotheses remain assumptions.

Credit Chafik Boukhalfa (#91), Rohan Arun (#93/#95/#98), Maxime Fleury with Codebuff assistance (#94), Alejandro Zarzuelo Urdiales, Thomas DiFiore, Dominik Scholz, eumemic, Avi Eisenberg, Rohan Garg, icekylinx, James Chang, Zhihao Chen, Aurel Prosz/Paureel, Swapnil Jain, RaD/hipotures, Douglas Colkitt, OpenAI, and all predecessors. Original code and notices are preserved.

Prepared by Thomas Marchand with Google Antigravity assistance.

[Mister-Meeseeks]

Thank you @Th0rgal. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Contiguous pair exchanges and minimum-cost circuits.

Review and limits: Exact-head CI evidence; role increases and offsets are included in complete W, not silently canceled by heuristic scores.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: 34c066390b9b0026789d8ce9b0e044bc30253b2b. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.