Face-diagonal local circuits on both suppliers of PR168 v4 (κ = 6.65046903615388e-4)

[chafreaky]

Change and mathematical scope

The package applies to main at 3b6b66891c0ac888521cf591fe306c6286601d4f (which integrates #186 on top of my #181 and eumemic's #168) and is mergeable there; its construction inputs are PR168 v4 at 4a3c769e5c5430e7114c4d3e099ff34664677f17, and the twelve PR168 v4 files it executes or pins are vendored byte-identically with digests in references/pr168-v4, so no file outside the package changes. Two changes above PR168 v4. Complex: a new per-cube local channel circuit found by 240 random restarts over PR144's $3^{13}$ circuit options and two coordinate-descent rounds, scored by the numerical physical root: $A[0,0]$, $A[0,1]$ and $A[1,1]$ are assembled from the two face diagonals, $A[1,0]$ from direction-2 edges, $A[2,\cdot]$ from direction-1 edges; $G[0,1,\cdot]$ and $G[0,2,1]$ use edge differences and $G[0,2,0]$, $G[1,2,0]$, $G[1,2,1]$ use fixed-coordinate differences. The circuit has 33 additions per cube instead of 27, but the carrier closure admits 12,189 arcs instead of 10,704, so the virtual auxiliary stock falls from 13,372 to 12,877 roles. All modules, fused outputs and schedules of PR168 are retained; the carrier matching, gauge selection, endpoint frame descent, 2,310 compensated handoffs and 44 terminal-output deletions are regenerated for the new graph. Bit (unchanged from my #189): change PR168's bit local circuit too: the two sums $A[2,0]$ and $A[2,1]$ are assembled from the two face diagonals instead of direction-1 edges, so the bit virtual stock falls from 19,788 to 19,348 roles (modules, L1 association, partner mixing and output merge unchanged). For this graph a fresh carrier matching and gauge selection are regenerated by the retained generator rules, and a complete physical layer (6,138 descended operation frames, 1,760 compensated handoffs) is built by our own producer; then 34 face0 terminal sinks are deleted by James Chang's PR166 substitution. Both suppliers and the paid balanced composition are independently certified. Under the retained interfaces, the bound is

$$ T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{166261725903847}{25\cdot10^{16}} =0.000665046903615388. $$

This is +1.40% over PR168 v4 (6.558894e-4), +0.50% over my #189 (6.617552e-4), which it supersedes, and +0.48% over #186 (6.618855e-4). The refined complex profile has 10,523 physical auxiliary roles, 13,163 persistent roles, rank mass 867,438, deficit 1,320 and largest child 20; its saving is $665489485337/10^{15}=0.000665489485337$. The new bit profile has 17,554 physical roles, 21,074 persistent roles, rank mass 1,515,392, deficit 1,936 and largest child 60; its coarse saving is $668298937775631/10^{18}=0.000668298937776$ (PR168 v4: $6600253/10^{10}$), and the established atom wrapper gives ordinary saving approximately 0.000667878281961. The complex branch binds at saving 0.000665489485337.

The configurable circuit is icekylinx's (PR144); the complex part only selects a different configuration. Carrier arcs come from our own legal-edge Hopcroft–Karp matching on coordinate-padded frames, extended by DaysSky's PR162 rule, and are replayed by the retained compile_closure with its acyclicity and span checks. The new bit graph is admitted through the retained package checker (exact rational frames, mod-2 decoder, geometry, chains, partner chronology, F2 replay, ledger and five mutation controls). The physical bit layer follows PR131's descent rule (each frame between the join of its incoming frames and the meet of its outgoing frames) on exact rational subspaces, repaired to nondegenerate bases, with handoffs read at the recipient's first use; the 34 sinks are the face0 roots whose rank-21 caps contain every write, with the complete modified target chronology, all 17,554 dirty columns over F2 and over the integers, and exact prime witnesses for all 24,459 used frames. The same producer applied to PR168's unchanged bit graph reproduces a layer slightly above PR168's own (6.60082e-4 vs 6.60025e-4 coarse before sinks), which is the control for the bit-side pipeline. Scalar virtual stock, compensated handoffs, terminal deletions, the full orthogonal-group cardinality and all row reserves are charged separately. The exact four-file PR175 overlay at f1081fabbfadaaaaf7292a40d428dbbb865ce920 is retained in the verification baseline.

No mathematical hypothesis or ordinary-supplier interface is strengthened. General Clifford/tensor, uniform weighted compilation, restored-row, paid-layout, precision/recovery and analytic fixed-tape contracts remain the stated assumptions. The finite certificate is not an unconditional multiplication theorem.

Prepared by Chafik Boukhalfa with substantial Anthropic Claude and OpenAI Codex assistance. The prerequisite construction is eumemic's PR168 v4, retaining its Anthropic Claude/OpenAI Codex disclosures and earlier PR144/161 lineage. Terminal-output accumulation is James Chang's PR166; balanced-layout, atom-toll and earlier supplier attribution remain in research/paired-cube-diagonal-local-168/NOTICE. PR154 supplies the isolated verification technique; PR175 supplies the exact compiler overlay.

Validation

Validated exact commit 5b09c35ee837e0157342a7cd1fbd933467cfe90b (tree a4b41a5f4ce9fc3ae9afa905dee87fcc6abd23a6).

• The new signed graph passes the exact scalar identity, carrier closure and frame audits. Complete original and terminal-modified complex words pass in both directions, including every source/target/dirty formal column and the full target-frame chronology. The modified word checks 13,163 columns and restores all 10,523 dirty roles; three corruption controls are rejected.
• The new bit graph passes the retained package checker with its five mutation controls; our physical layer passes all 21,108 F2 and defining-integer-decoder columns, and the terminal-modified word passes all 17,554 dirty plus 3,520 data columns over F2 and over the integers with three sink controls rejected. All 24,459 used frame Gram witnesses preserve the retained prime range; word and factor controls are rejected.
• Exact supplier moments, full scalar/group/router/row formulas, all 47 strict inequalities and seven margins pass. Independent arithmetic reproduces the same bound with 30 rejected mutations; the package retains 21 assembly controls. Adjacent grid exclusions concern only the fixed certificates and parameters.
• The read-only aggregate reproduces the canonical certificate. SOURCE.json pins all 2,242 files of main plus the package, with its own digest reported separately; the vendored PR168 v4 inputs carry their own digest manifest.
• Full repository gate: PASS. All 16 native verification groups of main plus the selected package passed in isolated copies of the published tree using the PR154 technique (Linux, 632 seconds); every source file remained byte-identical.
• GitHub CI: all 56 checks passed on the exact commit above, including the Python 3.11, 3.13 and 3.14 matrices and all formal checks.

Canonical finite check:

python3 -B research/paired-cube-diagonal-local-168/verify.py

[Mister-Meeseeks]

Thank you @chafreaky. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Face-diagonal local circuits on both suppliers.

Review and limits: Prior exact-head complete replay of both suppliers, geometry and assembly; unchanged all-size assumptions.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 5b09c35ee837e0157342a7cd1fbd933467cfe90b. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.