Merged auxiliary exteriors: conditional κ = 5.5354656014473e-5

[eumemic]

Conditional improvement

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{55354656014473}{10^{18}}=5.5354656014473\times10^{-5}.$$

That is +6.97% over #79, which this builds on (head 79dd6e1), and +4.86% over #95, the best open PR when this was opened. The bit saving is 27678860161103/500000000000000000. All inherited all-size and analytic hypotheses remain assumptions.

Change

Every auxiliary role pays a separate exterior edge I − I_h⊗B (profile [h, m−2h]). Auxiliary source frames are free, so each role can merge that residual into one of its endpoint edges:

• creation edge I − (I_h − A_first)⊗B: source and sink shifted by −E;
• retirement edge I − A_last⊗B: non-output roles only.

The choice is made per role. #79's words, gates, XORs, W and deficit are unchanged.

Frame validity. No gate, XOR, frame raise or schedule changes. Only auxiliary source and sink frames move; these are free in the upstream contract, and PR29's stage two already uses source D_0. A retirement-merged role's final segment, from its last gate frame to the sink, is charged as the single residual it already was. The inherited accounting had split that segment into a cleanup with no gate event plus the exterior.

Every gate incidence, including the center scatter after the middle mixer, therefore keeps #79's frames. #79's replay covers them: dirty basis in both orientations, and frame raises rebuilt from literal XORs. The only new matrices are the merged endpoint residuals, which are idempotent and certified exactly.

	h23	h25
roles	27,297	35,684
merged at creation	15,410	20,111
merged at retirement	4,216	5,149
distinct merged edges certified	8,537	10,601

Corner-rank lemma. For M = I − UWᵀ with WᵀU = I, the NE corner rank is:

• (i−j+1) − r + rank U_{<j} + rank W_{>i} for j ≤ i+1;
• rank U_{≤i}W_{≥j}ᵀ above the band.

In the PR34 fixed basis this reduces to ranks of local submatrices X[S,T], so the result does not depend on the line. Each such submatrix is a 0/1 diagonal plus a rank ≤2 correction. Its rank is computed exactly over Q using 2×2 integer determinants, with no modular step.

Bridge. Merged edges create bit children up to 552, above 529. The bit halving degree goes from 9 to 17, the row coefficient becomes 1059, and the product row stock goes from p^2000 to p^2200 (gap 991/25). This affects only the row-product slack and the eventual cutoffs, not κ. The inherited validate_bridge recomputes all of these.

Proof and scope are in research/merged-exterior/PROOF.md.

Verification

make merged-exterior-verify takes about 4 minutes on 4 cores. It does the following:

• Checks the controlled permutations against the pinned PR34 module.
• Recomputes all 56,303 candidate edge profiles exactly and replays the per-role selection.
• Profiles the removed transitions with the unchanged inherited profiler.
• Rebuilds the full child list on top of Balanced coarse split frames: conditional κ = 5.1745625472866e-5 #79's verified profile.
• Checks the exact and independent moment enclosures, and that the next grid point is rejected.
• Checks the 47 constraints and 7 margins.

There is also a new CI group, merged.

The joint-dual, split-pair and balanced-split source manifests pin Makefile, README.md and the workflow file, so they are re-pinned. The only change to their certificates is the recorded manifest hash.

A full make verify passed on Linux (EC2) with a clean git diff.

As a discovery-time cross-check, all 19,138 selected edges were compared with direct NE elimination of the full 575×575 physical matrix (one line, mod 2^31−1): 0 mismatches.

Credits

Builds on @chafreaky #79 and the stack it composes:

• @eumemic Joint frame compiler: conditional κ = 4.669442391e-5 #57, Balanced coarse sums: conditional κ = 5.136837484e-5 #69
• @rohangar1 Split-pair recursion and paid clones: conditional κ = 4.598878089e-5 #59
• @ikeboy Interval strips and core-aware pair assembly: conditional κ = 4986133/10^11 (5101691/10^11 stacked with #57) #62
• @DominikScholz Rank-first pair assembly: conditional κ = 5.102757e-5 #63, Cost-aware live controls: conditional κ = 5.103512721785e-5 #68
• @rohanarun Reordered rank-pair regions: conditional κ = 5.1028985071975e-5 #65, Cost-aware retired-slot selection: conditional κ = 5.1033760584215e-5 #67
• @alejandrozu Profile-aware frame compilation: conditional kappa = 5.10403292611078e-5 above frozen #62 #70
• @chafreaky Ranked slot reclamation: conditional κ = 4.764513337e-5 #60, Anchored split frames with profile-cost live reclamation: κ = 5.1414646104039e-5 #71

It also uses the PR34 fixed basis (jamesyc) and the PR29 two-stage endpoint note. Merged exteriors by eumemic with Anthropic Claude assistance.

🤖 Generated with Claude Code

[eumemic]

Withdrawn: the frame argument in PROOF.md §2 is wrong, so the κ claimed here is invalid.

The inherited arbitrary-scratch schedule (notes/incidence-construction.tex) runs every auxiliary role's first gate at the background label D_0 (the early pass) and its last gate at D_1 (the late inverse mixer, gather and copy). The exterior is therefore separated from the role's local entrance and cleanup by gates shared with other roles.

• The merged entrance edge bridges the early-pass gates. Shifting the source by −E only moves E onto the source edge.
• The merged exit edge bridges the late-pass gates.

Neither can be charged as one edge. Closing; nothing here should be built on. Apologies for the noise.

🤖 Generated with Claude Code