Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier)

[eumemic]

Conditional κ = 260706019542937/(2·10¹⁷) = 0.001303530097714685 with #471's complex supplier (complex-bound; +0.53% over #483), and 0.001325031497415498 with #484's draft complex supplier.

This PR adds a bit supplier with R = 489 helpers; the best bit word so far had 715 (#463). The bit side then no longer binds, so κ equals the complex cap of whichever complex supplier it is paired with.

Bit c, priced by the hub's eleven-stage whole-residual hosted pipeline (#445 geometry; #472 emitter, audit and finite invoice; #442 hosting; #315 engines):

c = 391360506745414227149/(2.5·10²³) = 0.001565442026981656908596 (10⁻²⁴ grid; the next point is rejected).

The same pipeline files reproduce three published c values exactly: #472's R715 (0.001293771240692396056124), #445's R740 (0.001264963576077378514188) and #451's R740 rank149 (0.001267536342017397153864). Our c is 20.6% above the best whole-residual bit supplier (#483, 0.0012982901) and 18.0% above the largest whole-residual complex b (#484).

Each composition below uses the six-level, 47-inequality outer assembly (#434/#315, as in #472). Every one is complex-bound.

complex supplier (cited from its receipt)	complex b	κ
#471 (Dugongue), the supplier of #473, #478 and #483	1305231506267591/10¹⁸	260706019542937/(2·10¹⁷) = 0.001303530097714685
#484 (DreamingOfClouds; draft, conditional on its Lemmas A and G)	663394767670133/(5·10¹⁷)	662515748707749/(5·10¹⁷) = 0.001325031497415498
#458 (chafreaky), the supplier of #472	1297203880559577927527/10²⁴	323880830669511/(2.5·10¹⁷) = 0.001295523322678044

When this was prepared, the best whole-residual κ was #483's 0.001296606691405309, bit-bound with #471's b. This word gives +2.19% over it with #484's supplier and +0.53% with #471's. The #484 figure carries that draft's two lemmas as review points. The #471 figure needs nothing beyond what #483 already uses, apart from the review point below.

Information only, not claimed: #448's cross-copy complex supplier (b = 718766148669433/(5·10¹⁷)) would give κ = 358867191156973/(2.5·10¹⁷) = 0.001435468764627892, complex-bound. That figure rests on #448's cross-family residual split, which #451 disputes (research/cross-family-residual-obstruction); the author's rebuttal is on #451's thread.

The PR contains one package, research/e8-bit-carrier-skeleton/, with its own path-filtered workflow. It also adds one .gitattributes line and the #318 joint-dual repair. No other existing file changes.

Mechanism: a D4 quad skeleton with sources as carriers

• Partition. The 120 root lines split into 10 D4 sub-root systems in 5 orthogonally complementary pairs. This gives 30 orthogonal 4-frames and 180 cells. A cell is a pair of lines in one 4-frame; it spans 2 dimensions and contains only those 2 lines.
• Sources. Each source keeps its own register in its matching cell. Two carriers copy it into its two cross cells, so each source has three branches. Three is the minimum: by Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463's carrier bound, two hyperplane sections cannot cover the 4-design s⊥.
• Quads. Cells deliver their sums to 60 quads. Each quad is a 4-dimensional D4 frame made of 6 cells from 6 blocks. A butterfly on the 6 cell sums gives one output per D6 the quad feeds.
• Tops. Each target arrives through its D6 triple {t, e}⊥. At the D6 it shares with its matching partner, it reads two quad outputs and a Q-delivery. At t⊥ it reads one register from each of its other two D6s.
• Q-nodes. In each block, the 12 sources (all in matching cells) undergo one invertible F₂ transform at the block's D4 frame. It yields 6 Q-deliveries for the complementary block's targets and 6 leftovers. These 577 gates are the only gates that write sources, and every one reads another source.
• Totals and cleanup. κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's 9 decoder totals are collected at the full frame on 9 fresh registers. With the totals on leftover registers instead, R = 480, but the prefix payload is 129 bits > 104. The cleanup is the literal reverse of all source and helper gates at the full frame.
• Register count. 120 sources + 240 carriers + 240 cell registers + 9 totals gives R = 489. The local histogram is {1:1200, 2:1440, 4:180, 6:120, 8:18}.

Review point: source registers as gate destinations

#370's independent checker asserts that sources are never gate destinations. #425's scalar-interface lemma (Zhihao Chen) shows that this assertion is sufficient, not necessary. If no target feeds a source or helper, the frame-zero preconditioner D and U⁻¹ at the common full frame (here C) give exactly (x, s, y) → (x, s, y + x).

• On the complex side, κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425 writes packets into sources below the full frame.
• On the bit side, Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433 already wrote source rows, but only at the full frame.
• This word writes sources at block D4 frames.

The package checks this point in five ways:

• relax.py builds the relaxed checker from κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's pinned file by removing that one assertion; the rest is unchanged. Exact F₂ replay on all 729 columns still requires every source to end holding exactly its own column. The stock file must refuse the word with "x register written".
• interface.py checks the lemma's hypotheses on the word:
  • no target feeds a source or helper;
  • D writes targets from helpers at frame zero;
  • C acts at the full frame and is the literal reverse of the source and helper gates;
  • B writes only targets.
• The relaxed checker rejects five mutants aimed at the interface:
  • drop a source cleanup gate;
  • corrupt a packet that targets read, while every source is still restored;
  • leave a source unrestored;
  • move a source write below the source's frame;
  • drop a source write.
• Both official column checkers reject the three matching record-level mutants.
• Two F₂-neutral edits, a target feeding a source and a target feeding a helper, pass the F₂ checker but are rejected by interface.py.

The official cohort checkers treat every register alike and accept the word unchanged. In #472's PROOF.md, the bit label realization does not depend on whether a register is called a source, helper or target. The mutable-basis realization remains a listed conditional interface, as in #472, #473 and #478.

Checks

bash research/e8-bit-carrier-skeleton/verify.sh DIR takes about 70 s. It passed from fresh copies on Python 3.12 and 3.14.

• Provenance. Manifest check, and 50 vendored files pinned by git blob id. adapt.py --check confirms the adapted hub files are Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's files plus 19 listed edits. Every cited b and frontier κ must equal its PR's own EXPECTED.json.
• Rebuild. The word is rebuilt from the frozen CP-SAT plan and must equal the shipped word.
• Independent checker. The relaxed κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 checker checks legality and exact F₂ on 729 columns. 671 frames have positive Gram determinants, with only the prime 2. Then interface.py runs, along with 11 word-level mutants and 2 hypothesis controls.
• Official checkers. Legality at NCOPY = 9: 33,990 ADDs, rank mass 5,664. Five-stage formal columns: 969 columns, prefix payload 100 bits < 104. Eleven-stage formal columns (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, KPAIRS = 5): 1,689 columns, 2,880 bridge additions, all 24 omitted-bridge controls rejected. The KPAIRS = 2 regression passes, and 6 + 3 record-level mutants are rejected.
• Gram witnesses. Exact Gram determinants and prime witnesses on all 672 frames.
• Eleven-stage hosting. All 43,032 helper residuals are hosted whole on idle data families of the eight pooled replicas, with no banks. 6,579 unbanked scalar columns are restored and 24 omitted bridge shears are rejected. The independent audit replays 3,252,216 renamed instructions and rejects 22 template mutations.
  • Physical prefix: 2,119 bits, charged at 2,176.
  • Three-level coefficient: 6,411 bits, charged at 6,464.
  • Maximum child 56; row coefficient 19,431 < 20,161.
  • Both moment engines, the 47 strict inequalities, and the next-grid, zero-η, zero-β and unit-β controls pass.
• Controls. Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's R715 word goes through Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's unmodified files. It reproduces Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's c and, with Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458's b, its κ = 0.001292099559442278. Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445's R740 word and Conditional κ = 0.001265931727546219: dual eleven-stage whole-residual hosting #451's R740 rank149 word go through their own emitters and reproduce their c values.
• Local build. The C++ checkers were built locally against an overflow-checked 128-bit stand-in for boost::multiprecision::cpp_int. The workflow uses libboost-dev.
• Repository CI. The branch carries rohanarun's 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 joint-dual pin repair (byte-identical to a45747d), as E8 bit supplier c ≈ 1.044064·10⁻³ in the nine-stage word (+2.2 % bit cap; no new κ, complex-bound) #417, Conditional κ = 1.128983115440183e-3: an E8 bit word with 740 helpers under #433's complex supplier #436 and Conditional κ = 0.001133928043872860: R740 shared-donor kernel with explicit residual bank routes #439 do.

Retained assumptions

• Every interface Conditional κ = 0.001292099559442278: R715 shared kernel, target mixing and whole hosting #472's eleven-stage composition retains:
  • the all-size bit/Gaussian compiler;
  • the mutable-basis and target-label realization;
  • 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. The primary κ also rests on Complex supplier b ≈ 1.327·10⁻³ (+4.7%) on whole-residual hosting: shared-donor kernel and entrance gauges with paid residual wrappers (conditional on two lemmas) #484's Lemmas A and G.
• The hosting selection is a frozen nomination of Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's integer programme, and the plan is a CP-SAT nomination; neither is claimed optimal.
• This is a conditional finite construction, not a Lean proof.

Bounds for this family (analysis, not run by verify.sh)

• R ≥ 120 for any design.
• R ≥ 300 for tops through consistent D6 triples: 14 registers per 4-frame at dimension 6.
• R ≥ 420 with three branches per source on this partition: 3 registers per cell.
• R ≥ 480 in this family with phase-A totals. A brute force over the d-parity patterns needs 4 single-line registers per 4-frame. The block Q-nodes need all 4 lines of each 4-frame, and no register can serve both roles.
• R = 480 exists but fails the 104-bit prefix bill.

Credits

[DreamingOfClouds]

Unverified note: an LP lower bound for R489's skeleton (exploratory, not reviewed or packaged)

Your README argues R ≥ 480 for this family (the "Bounds for this family" analysis). We ran a small linear program that reproduces that floor by computation. We haven't verified it independently, so treat it as a lead, not a result.

• Model. A register's frame path is a chain in the frame lattice, so it meets any antichain at most once (Csirmaz-style counting). Helper count is then bounded by a min-flow with node lower bounds on the Hasse diagram of the frames used by the skeleton; its dual is a max-weight antichain.
• Node bounds. From a junk-dimension argument: two dirty branches of a source span only {p, q, p+q}, and only p+q carries the cell sum, so a cross cell needs ≥ 3 registers and a matching cell ≥ 4. Without these bounds the LP returns only the carriers (8 per 4-frame).
• Result. With them, the LP optimum is 16 helpers per 4-frame, i.e. 480 over the 30 frames, plus the 9 totals if they are fresh (489, your shipped word). On the word with the totals on dead registers (our Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503, R = 480), levels 2 and 6 of the lattice are both tight, so there is no slack level within the family.
• Hypotheses the bound uses: 3 branches per source, 2 transverse quads per cell, and matching-cell sources feeding the block frame. A design that beats 480 has to break at least one of these.

We're happy to share the LP script if useful.

DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.