Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510)

[eumemic]

Conditional κ = 793733736881443/(5·10¹⁷) = 0.001587467473762886 with #535's complex supplier (complex-bound; +0.160% over #535's 0.001584931111785787, which is bound by #510's bit c).

This PR adds a bit supplier with R = 420 helpers. The best bit words so far have 480 (#503, priced at nine stages by #510) and 489 (#493). Priced by #510's nine-stage whole-residual chain (#463 template, 32 replicas, #442 hosting, #315 engines), unchanged:

c = 413456747036699736067/(2.5·10²³) = 0.001653826988146798944268 (10⁻²⁴ grid; the next point is rejected).

That is +4.18% over #510's R480 c (0.001587447106092548137108) and 4.01% above #535's complex b, so the bit side no longer binds with any cited complex supplier. #493's eleven-stage chain prices the same word at 0.00164031632456626881922 (+4.78% over #493's R489 in that chain).

Both compositions use the six-level, 47-inequality outer assembly of #493 (the one #510 uses; it reproduces #535's published κ values exactly):

complex supplier (cited from its receipt)	complex b	κ	binds
#535 (icekylinx; open)	1589991533606044174031/10²⁴	793733736881443/(5·10¹⁷) = 0.001587467473762886	complex
#531 (icekylinx; merged)	304013615566499291631/(2·10²³)	1517760977820331/10¹⁸ = 0.001517760977820331	complex

With #531's b nothing changes: main's headline was already complex-bound. Each κ is the same with the nine-stage and the eleven-stage c, because the complex side binds in both. The same pipeline files reproduce #510's R480 c, #493's R489 c, and the published κ values of #535 (0.001584931111785787 with #510's c, bit-bound; 0.001562995248531632 with #493's c, bit-bound) and #531 (0.001517760977820331, complex-bound).

Concurrent PRs. When this was prepared, #541 (Dugongue) reported κ = 0.001587212454640692, bit-bound with c = 0.001589735702947999. #542 (rohanarun; draft, physical verification pending) reported κ = 0.001587467473762886, the same complex cap as here, with c = 0.001590005077309866, just above #535's b. This word reaches that cap with c 4.0% above #535's b. The bit side therefore stays non-binding for any complex b below 0.001653.

The PR adds one package, research/e8-bit-r420/, with its own path-filtered workflow, and one .gitattributes line. No other existing file changes. The package runs on top of main: it uses research/e8-bit-carrier-skeleton (#493) and #531's receipt, both pinned by sha256.

Mechanism: keepers read by targets, then rewritten in phase B

• Partition. As in Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493: 5 complementary D4 pairs, 30 orthogonal 4-frames, 180 cells, 60 D4 quads, 180 D6 frames.
• Three registers per cell. Each source has a home cell, and two carriers copy it into its other two cells. Every cell holds exactly three registers (two line branches and one empty register): 120 sources + 240 carriers + 180 empty registers = 540, so R = 420. The 9 totals sit on dead registers (Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503's last-eligible-host rule).
• Phase A. Cells deliver two sums to their quads and keep one line, quads apply butterflies, and D6 nodes form o1 = o1 + o2. A D6 that does not host a Q-delivery takes a free keeper of the complementary block.
• Phase B.
  1. Each target t reads, at the cell C_t = Φ − {t, x(t)} of its 4-frame, the keeper of C_t. This is a source register holding t's single d-line (120 gates).
  2. The 12 keepers read in a block are exactly the block's 12 sources. They climb to the block's D4 frame, and one invertible F₂ transform per block turns them into the 12 Q-deliveries for the complementary block (477 gates).
  3. Each Q-delivery climbs to its target's host D6 and adds that D6's o1 register (120 gates).
  4. Each target climbs from its cell straight to t⊥ and reads its three needers (120 gates).
• Cleanup. The literal reverse, at the full frame, of all 2,661 source and helper gates (2,064 in A, 597 in B).
• Why it gets below 480. Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's README bound (R ≥ 480 with phase-A totals) used the fact that no register could both give a target its single d-line and feed the block Q-node. DreamingOfClouds' LP in Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's thread reaches 480 under three hypotheses: three branches per source, two transverse quads per cell, and matching-cell sources feeding the block frame. This word keeps the first two and drops the third. The sources that feed the block frame are the d-single keepers, and a target reads each of them at its cell before the Q-node runs. R = 420 equals Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's other bound for this partition: at least three registers per cell with three branches per source.

Review point: source registers rewritten in phase B, after targets read them

#493 writes sources only in phase A, and its interface.py checks that phase B writes only targets. This word has 597 phase-B source writes: 477 source ← source at block D4 frames, and 120 source ← helper at D6 frames. Every one writes a register a target has already read in phase B. No target feeds a source or helper.

Lemma. Let z = (x, s) be the sources and arbitrary helper contents, and y the targets. Assume:

• (H1) no z-destination gate reads a target;
• (H2) D is at frame 0, writes targets and reads helpers;
• (H3) A writes only z, and B interleaves z-only gates with target gates in any order;
• (H4) C is the literal reverse of all z-destination gates at the full frame;
• (H5) the target-to-target part V is the identity.

Then the raw word acts as (z, y) ↦ (U z, y + J x + K s), C acts as U⁻¹, and the whole word acts as (x, s, y) ↦ (x, s, y + J x + (K + N) s), where D adds N s. D is legal at frame 0 and C at the full frame. The proof is an induction over the gates: z evolves by itself, and each target gate adds a combination of the current z = U_k z. The order of z-gates and target reads enters only U and M. It is written out in the package README.

Reading of #425. #425's lemma (PROOF.txt §3, Zhihao Chen) has one structural hypothesis, "No target may feed a z register". It states the block form for the whole raw A/center/B word and executes U⁻¹ "by reversing and inverting the non-target gates", wherever they occur. So the interleaving is covered by #425's text as written. #493's check (4) was a property of the R489 word, not a hypothesis of #425. The bit-side realization is the role-independent common-label pullback of #472's PROOF.md, written out in #535's PR510-MUTABLE-SOURCE-CARRYOVER.md. It does not depend on the order either. Reviewers should confirm this reading.

The package checks the point in four ways:

• code/xcheck.py reads only the word. It checks the labels (even unimodular, rank 8), frame legality, and the exact F₂ map on all 660 columns with symbolic helper contents. It then evaluates (H1)–(H5) separately and checks the conclusion term by term: z′ = U z, J = I, K equal to D's helper sets, and C = U⁻¹.

• code/mutants.py runs nine negative controls, each rejected:

  • a target feeds a source, and a target feeds a helper (F₂-neutral pairs, caught by H1);
  • a dropped cleanup gate;
  • a corrupted phase-B source write with the cleanup changed alike, so every source is still restored;
  • a phase-B source write moved to the source's own line frame;
  • a single target-to-target read;
  • a corrupted frame-0 preconditioner;
  • a phase-B source write moved before the target read of that register;
  • a cleanup gate moved off the full frame.

  A cancelled pair of target-to-target reads (V = I) is accepted as a positive control.

• κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's stock checker refuses the word, and Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's relaxed copy passes it. Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's interface.py refuses it at check (4), as expected.

• The official cohort checkers accept the word. Three record-level mutants aimed at the phase-B source writes are rejected by the five-, nine- and eleven-stage column checkers.

Checks

bash research/e8-bit-r420/verify.sh DIR takes about 105 s here: it passed from fresh copies on Python 3.12 and 3.14. The two controls run in the background.

• Provenance.
  • Manifest check.
  • 19 files vendored from Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510 (its nine-stage chain, the Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463, Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443, Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 and Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315 files it carries, and Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503's R480 word) and 2 receipts from Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535, all checked by git blob id.
  • Main's files checked by sha256: Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's package through its own MANIFEST.sha256, and Direct delivery with 504 helpers for conditional κ = 0.001517760977820331 #531's receipt.
  • Every cited b checked against its receipt.
• Rebuild. The word is rebuilt from the frozen CP-SAT plan and must equal the shipped word.
• Independent replay and lemma. xcheck.py and the 9 + 1 controls. The R489 and R480 words also pass xcheck.py, as the lemma's special case.
• Official checkers.
  • Legality at NCOPY = 9: 29,874 records, rank mass 5,112.
  • Five-stage formal columns: 900 columns, prefix payload 100 bits < 104.
  • Nine-stage (columns_k, KPAIRS = 4): 1,380 columns, 18/18 omitted-bridge controls rejected.
  • Eleven-stage (KPAIRS = 5): 1,620 columns, 24/24 omitted-bridge controls rejected.
  • The KPAIRS = 2 regression, κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's 6 record mutants, the 3 phase-B record mutants, and Gram determinants with prime witnesses on all 672 frames.
• Nine-stage price.
  • Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's programme rediscovers the frozen selection, and all 120,960 residuals are hosted whole with 0 banks.
  • The template, parent audit (17 mutations) and hosted audit (8,603,712 instructions, 16 mutations) pass.
  • Invoice: prefix 189 bits (charged 192), two-level 416 (charged 512), maximum child 42 (2·42² < 64²), row 18,355.
• Eleven-stage price.
  • Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's audit files are adapted to R = 420 by the same six edits Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503 makes for R480.
  • 36,960 residuals are hosted whole; the audit replays 2,628,912 instructions and rejects 22 mutations.
  • Invoice: prefix 4,710 bits (charged 4,736), three-level 14,185 (charged 14,208), row 19,431 < 20,161. This is larger than Conditional kappa 0.001467289440740487: rank-sector factory on 112 ports #473's 3,456/10,304 and enters only the finite cutoffs; all 47 inequalities pass. The headline uses the nine-stage price.
• Controls. These run through the same files: Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510's R480 at nine stages and Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's R489 at eleven stages (Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's files unmodified). Every c and κ, and Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's emitted program hash, match their PRs' receipts.
• Local build. The C++ checkers were built against an overflow-checked 128-bit stand-in for boost::multiprecision::cpp_int. The workflow uses libboost-dev.
• Repository CI. Main already carries its own joint-dual pin update: its research/joint-dual/SOURCE.json matches main's scripts, and the certificates record that file's hash. So this branch does not carry Re-pin the root Makefile/README in research/joint-dual/SOURCE.json (fixes the failing 'Verify research artifacts' job on main and all PRs) #318's three files, and it changes no root file apart from .gitattributes.

Retained assumptions

• Every interface that Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's and Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510's compositions retain:
  • the all-size bit/Gaussian compiler;
  • the mutable-basis and target-label realization (the review point above);
  • whole-call correlated-helper hosting (Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442);
  • compressed windows and restored work;
  • the common grid and tape, precision, row and analytic contracts.
• Complex savings are cited from their PRs' receipts, not rebuilt. Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535 is an open PR, so the primary κ also depends on its review.
• The plan is a CP-SAT nomination and the hosting selections are nominations of Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's programme; none is claimed optimal.
• This is a conditional finite construction, not a Lean proof.

Credits

• Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493 lineage (eumemic): the D4 quad skeleton with sources as carriers, the generator modules and the checker chain this package runs.
• DreamingOfClouds:
  • Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503: totals on dead registers, hosted by the last eligible register;
  • Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510: the nine-stage pricing of these words, and the chain vendored here;
  • the LP in Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493's thread, whose third hypothesis this word drops;
  • Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463 and Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315: the nine-stage template and the moment engines.
• icekylinx:
  • Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535: the complex b of the headline, and the mutable-source carryover note;
  • Direct delivery with 504 helpers for conditional κ = 0.001517760977820331 #531 and Punctured Hamming ports for conditional κ = 0.001457945761723368 #523: the merged complex supplier and its baseline;
  • Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433: bit source rows.
• Zhihao Chen / jacklightChen: κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's scalar-interface lemma, Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386's nine-stage geometry.
• Dylan Crow / utcorvusvolat-dotcom: κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 (bitE8/1 format, independent checker, decoder totals, generalized official checkers).
• Chafik Boukhalfa / chafreaky: Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472, Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445 and Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443/Conditional κ = 0.001246073505487942: R740 replicated kernel with idle-data hosting #444 (eleven-stage emitter, audit, invoice and geometry, row bound, replica templates).
• DaysSky: Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's idle-axis hosting, κ = 9.95656966441037e-4: #370's E8 bit word in the nine-stage bridged word with #377's nine-stage complex E8 (above 2^-10; +2.94 % over #380) #381's columns_k.
• Jacob Sussman: the E8 unit.

Vendored files are byte-identical and listed in NOTICE.md and SOURCE-pr*.json.

Prepared by eumemic with substantial Anthropic Claude assistance. Apache-2.0.

🤖 Generated with Claude Code

[eumemic]

Note for review: main now selects #535 with #510 (1c1e42f), so the selected κ 0.001584931111785787 is bound by #510's bit c 0.001587447106092548. This PR's headline composition uses only that now-merged complex supplier: with #535's b = 1589991533606044174031/10^24 and this word's nine-stage c = 0.001653826988146798944268, κ = 793733736881443/(5·10^17) = 0.001587467473762886, complex-bound (+0.160%). verify.sh checks this row and reproduces #535's own κ as a control. #555 (punctured R439 word, c 0.001733052823) raises the bit cap further but depends on draft #545.

[Mister-Meeseeks]

Thank you @eumemic. This contribution at 2065fe4f72a914d907f1046603f5a36f17018535 is incorporated in the reviewed family through #588, published in 9594b61.

Interleaved scalar-interface lemma is valid by block triangular induction: no target feeds z; z update U depends only on z; V=I; D precedes raw map; reverse z gates restore U. Common-frame pullback commutes with XOR irrespective of role names. Whole-call correlated-helper hosting and all-size realization remain assumptions.

Preserved implementation and research:

• research/e8-bit-r420

The family review and individual disposition record the scope, evidence and attribution. Parallel and intermediate contributions remain credited even where a later composition has a stronger bound. The published multiplication bound remains conditional on the documented compiler, realization, hosting and analytic assumptions; these finite checks are not a full Lean multiplication proof.