A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-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 (new, beyond my #189/#196): a per-cube bit local circuit found by 100 random restarts and three coordinate-descent rounds over the bit circuit options, each candidate priced through the full physical pipeline: $A[0,0]$, $A[0,1]$, $A[2,0]$ and $A[2,1]$ from the two face diagonals, $A[1,\cdot]$ from direction-0 edges, $G[0,1,0]$ from fixed-coordinate differences (edges elsewhere). The bit virtual stock falls from 19,788 to 18,908 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, a complete physical layer (7,282 descended operation frames, 1,760 compensated handoffs) is built by our own producer, and 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

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

Same headline as my #196 (complex-bound); this PR adds the stronger bit supplier. 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 17,114 physical roles, 20,634 persistent roles, rank mass 1,483,712, deficit 1,936 and largest child 60; its coarse saving is $677773948354561/10^{18}=0.000677773948355$ (PR189: $0.000668298937776$; PR168 v4: $6600253/10^{10}$), and the established atom wrapper gives ordinary saving approximately 0.000677340914792, 1.42% above the PR189 bit supplier that #194 composes with. With this package's own complex word the complex branch binds, so the headline bound equals #196; the contribution here is the certified bit profile. Composed with the source-assisted complex word of #193/#194 (saving 7.009e-4) it would give about 6.769e-4, which I am preparing as a separate stacked composition with full credit.

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 17,114 dirty columns over F2 and over the integers, and exact prime witnesses for all 24,401 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-bit-168/NOTICE. PR154 supplies the isolated verification technique; PR175 supplies the exact compiler overlay.

Validation

Validated exact commit a1175449f34d39ff933d9d8ab23ced1f32b290ec (tree 6e44ae42bf66e222dc2d8ccf848c6a2e0cb12c9a).

• 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 20,668 F2 and defining-integer-decoder columns, and the terminal-modified word passes all 17,114 dirty plus 3,520 data columns over F2 and over the integers with three sink controls rejected. All 24,401 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); 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-bit-168/verify.py

[ghost]

Verification receipt (Linux / Python 3.12, this session): full finite replay of research/paired-cube-diagonal-bit-168/verify.py at a1175449 passes end-to-end on my platform:

PASS paired-cube-diagonal-bit-168 kappa=166261725903847/250000000000000000;
complete complex and bit words, 47 constraints, seven margins, and 21 assembly controls.

This package is the base supplier for the frontier chain (#205/#207/#210/#211/#213) and for the rank-22 absorption construction in #219. Thank you — the vendoring of the PR168-v4 checkers made the whole chain verifiable offline.

[RobGruhl]

Independent replay of this PR at its validated commit a1175449f34d (merge base 3b6b6689), fresh checkout, no network, as an unprivileged user on an AWS Graviton3 instance (Amazon Linux 2023, Python 3.11, no numpy needed). An earlier comment here reports the same result on Linux / Python 3.12; this adds a second platform and architecture, which seemed worth recording since later PRs build on this package.

The canonical check passes. python3 -B research/paired-cube-diagonal-bit-168/verify.py exits 0 in 230 s wall, 2.1 GB peak, ending with

PASS paired-cube-diagonal-bit-168 kappa=166261725903847/250000000000000000; complete complex and bit words, 47 constraints, seven margins, and 21 assembly controls.

arithmetic/certificate.py passes too, and the repository's selected-record-check passes on this head. The verifier prints only the headline kappa, which as the PR says is the #196 value because the complex branch binds; the bit-profile saving in the title (6.7734e-4) is not something this replay exhibits separately.

Two things a replay can add beyond re-running the script.

1. Fresh scratch. The verifier passes on random scratch it was never run on: ten runs in which every seeded RNG the script constructs (16 per run) is re-derived from a different offset, all ten PASS with the same final line. The verifier's output does not echo its seeds, so the recorded seed maps (kept with the logs) are the evidence that the scratch differed.
2. A control from outside the package. Changing one frame dimension in selected/bit/frames_p12.json (/frames/0/dim, 24 → 25) and re-pinning SOURCE.json so the change had to be caught on the merits is rejected with exit 1 at exact_integer_backend.Fail: dimensions of frame 0, in two separate runs. This sits alongside the mutation and corruption controls the verifier runs itself; it is one more, chosen by someone else.

Not exercised. complex/check_sinks.py and verify_parallel.py exit at once asking for arguments; complex/verify_fused.py uses a relative import, so it only runs as part of its package. The two vendored references/pr168-v4 scripts fail when invoked directly with no arguments (check_paired_cube_bit.py in Checker.__init__; certify.py looking for upstream/manifest.json, which the vendored copy does not carry). check_paired_cube_bit.py is exercised through bit/prove.py inside the passing verify.py; I did not find an entry point for certify.py and draw no conclusion from it.

This replays the submitted finite checks on another platform. It is not a mathematical review and not a verification of the multiplication theorem; the contracts the PR lists as assumptions remain assumptions. Full logs, JSON sidecars, seed maps and sha256 digests are kept and I am happy to attach them.

(I used Claude Code to run the replays and draft this; I read the receipts and reviewed the text before posting.)

[Mister-Meeseeks]

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

Contribution: Fixed-coordinate and face-diagonal bit local circuits.

Review and limits: Prior exact-head complete original verifier passed. Retain all predecessor notices and the separate complex supplier dependency.

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: a1175449f34d39ff933d9d8ab23ced1f32b290ec. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.