Envelope-ordered carry exchanges: conditional κ = 5.1738676045141e-5

[DominikScholz]

This draft composes envelope-node ordering with profile-cost carried-signal exchanges on anchored split frames. Its conditional bound is

$$ T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{51738676045141}{10^{18}}=5.1738676045141\times10^{-5}. $$

This is 0.6287052980% above PR #75, whose reported conditional exponent is 257077122735047/(5*10^18) at da9b6cbebb33e8b8293a604dde0eace9de53406b. The exact comparison is κ − (201/200)κ₇₅ = 66174375396553/10^21 > 0. Bit saving is 25870676537201/(5*10^17). Complete finite compiler/profile/arithmetic checks, independent serialized replay, and deterministic regeneration of both words and the final certificate pass. Broad repository checks and CI review remain pending. All inherited all-size and transfer hypotheses remain conditional.

Change

Compose Thomas DiFiore's PR #74 scalar-node envelope ordering with Chafik Boukhalfa's frozen PR #71 anchored split construction and three passes of PR #70-style carried-signal exchanges. Each exchange preserves matching cardinality, unique uses, eligible dependencies and independence of output rows plus retained input rows. Integer profile-entropy scores guide discovery; acceptance uses the complete physical word and paid recursive profile.

Both axes use the same envelope ordering, preserving scalar forms and remapping all argument/output identifiers. The compiler checks every dependency, containment and dirty basis; reclamation pays literal XOR operations and frame raises for both targets and controls. PR #75's distinct nonlinear provider-circuit method is not incorporated.

Quantity	Frozen PR #71	This draft
h=23 auxiliary roles	27,455	27,256
h=25 auxiliary roles	36,015	35,720
Physical wires W	135,075,665	134,095,520
Conditional κ	51414646104039/10^18	51738676045141/10^18
Assembly backoff	10^-12	unchanged

The complete recursive rank is 77,103,077,100, with deficit 1,846,900. Copied centers, endpoints, data geometry and the complex layer remain charged. Because W crossed below 2^27, the bridge derives wire-bit width 27, row coefficient 843, and degree gap 7007/25; degree 2000 and suffix slope 8000 remain unchanged. Stale width fields are rejected rather than bypassed.

Verification

Passed: both complete compiler input/dirty-basis checks in both orientations; actual literal transition reconstruction; complete fixed-basis profiles with zero CRT disagreements; source/word bindings; independent serialized dirty-word replay and profile regeneration; complete paid child/rank identity; independent rational moment enclosures and adjacent bit-grid rejection; all 47 strict assembly inequalities and seven margins; and next-κ-grid rejection. The independent validation receipt records the completed checks; the frozen manifest retains its initial pending status. The same validation has also been reproduced in a clean checkout with identical results and three rejected stale-width controls.

Pending: broad repository checks and CI review. Both physical words regenerated byte for byte locally, with matching compiler results, literal transitions, complete profiles and the entire exact arithmetic certificate; see the reproduction receipt. Inherited repository CI remains unchanged. The local reproduction script recompiles both words, independently replays them, rebuilds complete profiles, checks exact arithmetic and rejects certificate drift. This draft is published before those follow-up checks. Finite evidence does not establish the inherited all-size compiler, analytic, routing, prime-selection, recovery, fixed-tape or transfer theorem, global optimality or practical runtime gain.

mkdir -p research/round6-pr71/baseline
git archive 1bef94fd40a746452548c84a4a8f8834670a3113 | tar -x -C research/round6-pr71/baseline
cd research/round7-scheduling
python3 verify.py
python3 verify.py --regenerate

The proof also provides sequential compiler/profile regeneration commands. The baseline archive is ignored; the immutable PR #71 commit is this branch's parent. Assert-enabled Python and a C++17 compiler are required; CXX may select a compiler/SDK.

Commit: 2333abcea79793fe301b479a7c971cc2be45596c. Proof and scope, Exact certificate, Source and witness manifest, Reproduction, Width-bridge controls.

Attribution

Thomas DiFiore's PR #74 supplies envelope-node ordering; Chafik Boukhalfa's PR #71 supplies the selected split/alignment and next-use pricing; Alejandro Zarzuelo Urdiales's PR #70 supplies carried-signal exchange methodology; Rohan Arun's PR #65/#67 supplies schedules, profile-cost selection and exact arithmetic; Dominik Scholz's PR #68 supplies compatible pending live controls. This composition and width-bridge derivation were prepared by Dominik Scholz with substantial OpenAI GPT-6 Astra assistance. Avi Eisenberg's PR #62 graph, eumemic's PR #57 reversible compiler, Chafik's PR #60 reclamation and all predecessor contributors retain credit. All Apache-2.0, source-specific, community, OpenAI and Harvey–van der Hoeven notices remain preserved.