Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543)

[eumemic]

Bit c = 1733052823330162099157/10²⁴ = 0.001733052823330162099157 at nine stages (R = 439 helpers, v = 111 live ports, decoder constant 63). That is +4.79% over #543's R420 word in the same pipeline and +4.43% over #545's punctured R489 unit. Every cited complex supplier binds: with #548's b the conditional κ = 49718254847979/31250000000000000 = 0.001590984155135328 (complex-bound; #548 is an open PR). The value of this word is bit headroom: κ stays complex-bound until a complex b exceeds 0.00173.

Concurrent PRs. While this was being verified, #551 (chafreaky) and #552 (Dugongue) reported κ = 0.001591968902833346 with stronger complex suppliers, #551 paired with #543's R420 word. Any of those complex suppliers is also complex-bound with this word, so they set κ. This PR does not claim a κ above them; it raises the bit ceiling from 0.00165383 (#543) to 0.00173305.

This PR stacks the punctured-star unit of #545 (DreamingOfClouds) on the R420 L3 skeleton of #543. Nine E8 lines are neither sources nor targets, and each of the nine retained totals is a pseudo-target that reads what the removed target would read, at frame rank 7.

quantity	value
bit c, nine stages (#510's chain, parameterised for v ≤ 120; 10⁻²⁴ grid)	0.001733052823330162099157
bit c, eleven stages (#493's chain as parameterised by #545's adapt.py, run unchanged)	427652201524503168379/(2.5·10²³) = 0.001710608806098012673516
#543's R420 word, nine stages	0.001653826988146798944268
#545's punctured R489 word, eleven stages (reproduced as a control)	0.00165947204563779652541

complex supplier (cited from its receipt)	complex b	κ	binds
#548 (Dugongue; open)	796759709641153381751/(5·10²³) = 0.001593519419282307	49718254847979/31250000000000000 = 0.001590984155135328	complex
#544 (chafreaky; open)	31864911873350496687/(2·10²²)	397677800014309/(2.5·10¹⁷) = 0.001590711200057236	complex
#535 (icekylinx; open)	1589991533606044174031/10²⁴	793733736881443/(5·10¹⁷) = 0.001587467473762886	complex
#531 (icekylinx; merged)	304013615566499291631/(2·10²³)	1517760977820331/10¹⁸ = 0.001517760977820331	complex

The same κ values hold with #543's word, since its c is also above every cited b (#547 reports the #548 value with it).

Why #545's construction needed a change on L3

In #493's skeleton the Q-node runs in phase A, so a pseudo-target can read everything a target reads before the scatter. In L3 the block node runs in phase B, after the targets have read its source registers at their cells. A retained total is read at the scatter, at the end of phase A, so the part of a target's sum that the block node delivers (the live lines of the complementary block B) is not available to a pseudo-target. In a 4-frame with one punctured line, the pseudo-target also needs an odd number of single d-line reads.

The pseudo-target total of a punctured line p (a fresh helper):

• reads the source of kline(p) at its line frame, before that line's cell gate;
• reads o1 at D6[p, qhost(p)] (it takes the place of the third register qdel[p]);
• reads p's needers at its two (z,z) D6s at p⊥. One of them is a free keeper of B that a phase-A mini-node at the block frame D4(B) rewrote to carry the pseudo-target's B-part. The mini-node reads the other free keepers of B, the sources of B that no target reads (the source in a punctured target's d-single cell), and extra carriers (helpers that read a source of B at its line frame) for the remaining lines.

The 9 totals are fresh helpers and 10 extra carriers are needed, so R = 420 + 9 + 10 = 439. The extra carriers also serve as junk wires of the block nodes. The decoder needs the classes of P in E8/2E8 to be affinely independent. A complementary D4 pair spans only 6 dimensions (affine rank 7), so P meets two block pairs: P = [10, 15, 51, 53, 56, 66, 72, 89, 111] in blocks (0,1) and (3,5). #545's own P is not eligible under #543's plan, because the frame (z,z) structure of its line 39 does not fit.

Rank-mass identity. At nine stages it reads T(8v − 9·cst) = 32·321 = 10,272. The hosting wall R ≤ 4v = 444 admits R = 439, and all 126,432 residuals are hosted whole with no banks.

Checks (bash research/e8-bit-p111/verify.sh DIR, about 4 minutes)

• Provenance. Manifest; 19 files from Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510 and 5 from Punctured star bit unit: bit c = 0.00165947204563779652541 (+4.54% over #510) #545, plus 2 receipts each from Conditional kappa 0.001590591090759927: observable pruning, Steinberg cancellation and response-owner composition #548, Conditional κ = 0.001588602246663212: static44 with selective bit frames #544 and Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535, all by git blob id; main's files by sha256. Every b is checked against its receipt. Conditional kappa 0.001590591090759927: observable pruning, Steinberg cancellation and response-owner composition #548's, Conditional κ = 0.001588602246663212: static44 with selective bit frames #544's and Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535's own κ are recomputed from their receipts, and all 47 constraint values equal their exact-assembly receipts.
• Rebuild. The word is rebuilt from Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543's plan and frozen parameters, byte for byte.
• Independent replay (code/xcheck_p.py, reads only the word).
  • The 111 labels are root lines of an even unimodular rank-8 lattice, and the 9 punctured lines are found by enumerating its roots.
  • Frames are closed in the live labels.
  • Exact F₂ replay on all 661 columns.
  • The 9 totals sit at the hyperplanes of the 9 punctured lines, with COPY ranks 7 × 9 = 63.
  • (H1)–(H5) of Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543's interleaved interface lemma hold, and its conclusion is checked.
  • Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543's 9 + 1 controls and 4 punctured-design controls are rejected.
• Hub checkers. Punctured star bit unit: bit c = 0.00165947204563779652541 (+4.54% over #510) #545's v ≤ 120 form of κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's checker refuses, its relaxed copy passes, 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 interface.py refuses at check (4), as for Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543.
• Official checkers at (111, 439).
  • Legality: 27,234 records, rank mass 5,129 = 8R + 63 + 14v.
  • Five-stage columns: 883, payload 93 bits.
  • Nine-stage columns: 1,327, 18/18 controls.
  • Eleven-stage columns: 1,549, 24/24 controls.
  • The KPAIRS = 2 regression, record mutants, and Gram witnesses on 636 frames.
• Nine-stage price (code/adapt9p.py: 27 listed edits to Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510's files).
  • The hosting selection is rediscovered, the parent and hosted audits pass, and all 126,432 residuals are hosted whole.
  • Invoice: prefix 177 bits (charged 192), two-level 393 (charged 512), row 18,355.
• Eleven-stage price (Punctured star bit unit: bit c = 0.00165947204563779652541 (+4.54% over #510) #545's adapt.py).
  • All 38,632 residuals are hosted whole and the audit passes.
  • Invoice: prefix 4,304 (charged 4,352), three-level 12,966 (charged 12,992).
• Controls.
  • On Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510's R480 word, the parameterised nine-stage files give the same template, audits and RESULT as Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510's vendored files, byte for byte. They reproduce Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510's c and κ values, and Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535's κ with Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535's b.
  • Punctured star bit unit: bit c = 0.00165947204563779652541 (+4.54% over #510) #545's word through Punctured star bit unit: bit c = 0.00165947204563779652541 (+4.54% over #510) #545's files reproduces Punctured star bit unit: bit c = 0.00165947204563779652541 (+4.54% over #510) #545's c, profile and program hash.

Retained assumptions

As for #543 (mutable-basis realization, interleaved interface lemma) and #545 (a 111-port bit supplier in the outer assembly; #442 whole-call hosting). Complex savings are cited from receipts. #548, #544 and #535 are open PRs, and #548 also retains #539's bad-class transport as a premise. This is a conditional finite construction, not a Lean proof.

Credits

• DreamingOfClouds:
  • Punctured star bit unit: bit c = 0.00165947204563779652541 (+4.54% over #510) #545: the punctured-star unit, the pseudo-target totals, the v/cst parameterisation and adapt.py, run here unchanged;
  • Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510, Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503, 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 chain and its pieces.
• eumemic: Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543 (the L3 skeleton) and Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493.
• Dugongue: Conditional kappa 0.001590591090759927: observable pruning, Steinberg cancellation and response-owner composition #548's complex b.
• chafreaky: Conditional κ = 0.001588602246663212: static44 with selective bit frames #544, and 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/Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443.
• icekylinx: Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535 and Direct delivery with 504 helpers for conditional κ = 0.001517760977820331 #531.
• Zhihao Chen / jacklightChen: κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425.
• Dylan Crow / utcorvusvolat-dotcom: κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370.
• DaysSky: Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442 and 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.
• Jacob Sussman: the E8 unit.

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

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @eumemic. This contribution at 65a6bc19f820c8b1873f65a517bcf70cd6887223 is incorporated in the reviewed family through #588, published in 9594b61.

Combines puncture with delayed block-node skeleton by adding phase-A mini-nodes and carriers, rather than assuming removed targets can read future phase-B values. H1-H5 includes phase-B helper writes as z updates; exact decoder and COPY rank63 checked. Pooled-cell/all-size contracts retained.

Preserved implementation and research:

• research/e8-bit-p111

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.