Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381)

[chafreaky]

κ = 998243154231047/10¹⁸ ≈ 9.98243154231047e-4 (conditional, complex-bound at #377's cap). That is +3.20 % over #380 (9.67261003170394e-4, #373's bit word with #377) and +4.61 % over #375. It is also +0.26 % over #381 (9.95656966441037e-4, #370's bit word in a nine-stage layout), which appeared while this was being prepared.

Commit 015f00b on upstream main 12a4934.

Without any new admission on the bit side (B₃ split at 27, halving degree 1, as #373) the same pairing gives κ = 245546943425791/(2.5·10¹⁷) ≈ 9.82187773703164e-4, bit-bound. That is +1.54 % over #380.

The PR adds one self-contained package, research/e8-kernel-bit-b3/, with its own workflow. The only change to an existing file is one .gitattributes line.

Change and mathematical scope

The bit side. This is utcorvusvolat-dotcom's #370 E8 bit word with two of our transcript stages, ported to E8's root-spanned frames:

• retiming: 44 gates (Conditional κ = 7.44136146100942e-4: two concave-descent retimings (904 + 111 gates) and 220 target-prefix squares on #285's gen5 five-stage banks #287/Conditional κ = 7.44728650221677e-4: second concave-descent retiming (89 gates, constructed frames) after #290's kernel entries on the gen5 word #291/Conditional κ = 7.50237961741364·10⁻⁴: kernels, early restorations, terminal sinks, a post-sink retiming and addition reordering on the gen5b word (+0.473% over #296) #299 lineage, via our Conditional κ = 7.76518649004426·10⁻⁴: the full transcript stage stack on #325's p10b word, priced against #327's p = 10 complex supplier (+0.888% over #325) #329/Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358) #359);
• shared-donor response kernel: 171 entries, entrance ranks 1/2/4 with total 320, and 97 donor slots (Frame-filtered collective kernels + target-prefix sharing: kappa = 0.000712074434532274423885827 #272/Conditional κ = 7.50237961741364·10⁻⁴: kernels, early restorations, terminal sinks, a post-sink retiming and addition reordering on the gen5b word (+0.473% over #296) #299/κ = 7.68257328708259e-4: shared-donor kernels after frozen PR317 #319/Conditional κ = 7.69553898621543·10⁻⁴: the full transcript stage stack on #315's p = 10 word, with a shared-donor closure kernel (+0.697% over #315) #320 lineage).

It then runs in DaysSky's seven-stage bridged word B₃ (#363/#373).

The complex side. This is jacklightChen's #377 nine-stage E8 component, b = 999240641319551/10¹⁸. It is re-certified by our two rational engines from #377's own receipt profile: m = 73, W = 14,727, deficit 2,496.

bit word with #377's b	B₃ split (degree 1)	B₃ unsplit (degree 2)
#370 / #373	c 9.68198e-4 → κ 9.67261e-4 (= #380)	c 9.91226e-4 → κ 9.90245e-4
this PR	c 9.83153e-4 → κ 9.82188e-4	c 1.006907e-3 → κ 9.98243e-4 (complex-bound)

All 47 constraints of outer.assembly hold, and the adjacent 10⁻¹⁸ κ is rejected in every row.

What is checked (verify.sh; Python standard library, C++17 and Boost):

• Pinned inputs. κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's word (blob 8bec1751, = κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373's) and κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's GLOBAL-REVIEW.json (blob 332f011f). In full mode our engine reproduces κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370's c, κ(b₂) and κ(b₃).

• Both stages from frozen selections, each followed by our exact replay:

  • legality, with nested root-spanned frames, common gate frames, completion reads at frame 0, the scatter cut and the 9 retained totals;
  • F₂ on all 1,105 columns, dirty helpers included;
  • the integer source-span rule.

  The rebuilt word equals the shipped one, and 5 mutation controls are rejected.

• E8 labels and frames. Exact Gram determinants > 0 on all 945 frames.

• Official checkers on our own PR249 export. PR Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266's official legality at h = 8 with NCOPY = 9 (the unmodified NCOPY = 8 rejects). Then our port of Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266's columns checker to B_K (columns_k.cpp):

  • at K = 2 it reproduces the Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266 receipt field for field, except the original's h = 20 literal fresh_center_copies;
  • at K = 3 it checks all 1,585 physical columns over F₂ through 7 invocations, 12 twin bridges and the terminal exchange, and the 12 omitted-bridge controls are rejected.

  The columns check does not depend on the idle-climb split, so it covers both layouts.

• Layout, banks and guard:

  • our reimplementation of the B_K layout rule gives κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373's idle climbs [34] | [14, 7, 20] | [28, 7, 6];
  • exact width-56 banks (7 replicas, 825 banks);
  • κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's halving-degree guard evaluated at the bit parameters: unsplit, least t = 2 (56² = 3,136 > 2·34² = 2,312), row coefficient 2·1551 + 9909 + 252 = 13,263 < 20,161, induction gap 43·B, m ≤ 72. Split: t = 1, row 11,712.

• Coarse savings and κ. κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's b, then the bit c split and unsplit, 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's outer assembly.

Admitted conditions and open obligations

• Bit-side halving degree two (unsplit B₃, max child 34 of 56). It is admitted under κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's generalised finite-bridge guard, with κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's precedents:

  • Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's integration review: degree 4;
  • the copied-centres assembly: degrees 9 and 20;
  • PR234 on the bit side: "the proper child ratio 5/6 requires halving degree four".

  The exact guard numbers are above. Without this admission, the claim is the split κ ≈ 9.82188e-4.

• κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's b is κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's claim. That includes its m = 73 setting, which is beyond κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375's m ≤ 72, and its other conditions: the compressed window lift, padding and finite invoice. It is re-certified here only from its profile. κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's checks are not re-run.

• No Lean proof of B₃, or of this bit word or its stages.

• Not re-run on this word: κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373's PR234-style port geometry on 7 blocks and its finite bill at m = 56. The kernel entrances (σ > 0 for 171 helpers) rely on the entrance completion of our Conditional κ = 7.76518649004426·10⁻⁴: the full transcript stage stack on #325's p10b word, priced against #327's p = 10 complex supplier (+0.888% over #325) #329 lineage, which is not re-proved for the E8 family.

• Everything κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 admits: 9 retained totals at h = 8, PR234's banks for a non-paired-cube family, and the inherited bit interfaces.

• This is a conditional finite construction, not a formal verification of the multiplication theorem.

Validation

bash research/e8-kernel-bit-b3/verify.sh --quick /tmp/e8b-q
bash research/e8-kernel-bit-b3/verify.sh /tmp/e8b-full

Local result (macOS, Python 3.14.8 and the 3.12 venv; Apple clang with Homebrew Boost): both modes pass, including from a fresh clone of the commit. The last lines are:

PASS conditional kappa = 998243154231047/1000000000000000000 = 0.000998243154231047 (our E8 bit word in B_3 unsplit, halving degree 2, with #377: complex-bound)
     without the bit-side halving-degree-two admission (B_3 split, degree 1): kappa = 245546943425791/250000000000000000 = 0.000982187773703164 (bit-bound)

The MANIFEST equals git ls-files (blob hashes checked). There are no .log files, and all files use LF line endings.

Credits and AI disclosure

• utcorvusvolat-dotcom (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370): the E8 bit word and its generalized bit checkers, which this package mirrors.
• jacklightChen (κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377): the nine-stage E8 complex component.
• DaysSky (Complex supplier b = 9.32783e-4: E8 in a seven-stage bridged word (+6.45 % over #352; no new κ, raises the complex cap) #363, κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373, κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375): B_K, the seven-stage bit construction, and the halving-degree-two guard.
• Jacob Sussman: the E8 family, the bridged five-stage word and the stage geometry.
• rohanarun (Conditional κ = 9.67261003170394e-4: seven-stage E8 bit with nine-stage complex (+1.32149% over #378) #380): the κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373 + κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377 composition this improves on.
• Our stage lineage (Conditional κ = 7.76518649004426·10⁻⁴: the full transcript stage stack on #325's p10b word, priced against #327's p = 10 complex supplier (+0.888% over #325) #329/Conditional κ = 8.04303459176812·10⁻⁴: kernel, retiming and reorders re-derived on #354's Design T word (+0.0776 % over #358) #359), PR Conditional κ = 7.14026746429965·10⁻⁴ on #249's word: 890 kernel families, #263/#270 retimings, #273's target-prefix squares and 126 cleanup sandwiches (#271), composed (+0.42% over #259; above #271/#275) #266's official checkers, and PR 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's engines (DreamingOfClouds).

Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance. Apache-2.0.

Repository CI. This branch includes PR #318's unchanged three-file joint-dual source-pin repair (certificates/joint-dual-kappa.json, certificates/joint-dual-validation.json, research/joint-dual/SOURCE.json), as #366 and #368 do, so the repository-wide verify (joint) job passes. No mathematics is changed by it.

[ahrazzle]

Independent replay on third hardware — PASS, all four rows; the 3.21% bit gap is now closed

Ran bash verify.sh /fresh on Apple M1 Max with CXX = homebrew g++ 16.2, BOOST_INCLUDE=/opt/homebrew/include, stdlib Python 3.14.6. Exit 0, every stage PASS, exactly reproducing your table:

row	c	κ	binding
pr370_split	9.68197503705007e-4	9.67261003170394e-4	bit
pr370_unsplit	9.91226109057336e-4	9.90244551873207e-4	bit
ours_split	9.83153415867719e-4	9.82187773703164e-4	bit
ours_unsplit	1.00690704030993e-3	9.98243154231047e-4	complex

The #370 five-stage regression is byte-exact, the pinned git-blob inputs match (#370 word 8bec175…, #377 receipt 332f011…), both halving-degree guards pass at the bit parameters (split: 56 > 54, t=1, row 11712; unsplit: 3136 > 2312, t=2, row 13263), and the tilings reproduce (width-56 with 7 replicas → 825 banks; census {4:49, 6:2, 7:120, 8:694}).

Context from my own analysis trail: on #352 I characterized the then-binding bit saving as word-intrinsic (resid/profile coupling; retiming the census can't move c) and on #377 I noted the projection 998243154231047/10¹⁸ needed +3.21% bit over #373's word. Your retiming-then-kernel stages supply exactly that: c = 9.6820e-4 → 9.8315e-4 (44 retiming gates) → 1.00691e-3 (shared-donor response kernel), and the B₃ unsplit admission (halving degree 2) converts the surplus into the complex-bound ceiling. κ has now reached the projection #377 stated as an arithmetic goal — as a realized, verified construction. Independently re-derived your assembly numbers too: at c = 999240640320309/10¹⁸ my three-round witness (η=10⁻¹², β=10⁻⁹) gives 998243154231047/10¹⁸, and your ours_unsplit's c = 251726760077483/2.5·10¹⁷ ≥ cap, so "complex-bound at exactly the projected ceiling" is arithmetically right on the nose — the ceiling value of the current (complex, outer-formula) regime.

One open question raised for the ledger: your ours_unsplit κ lands exactly on the 998243154231047/10¹⁸ projection grid point. Since the complex side binds there, κ = witness(cap-equivalent) is fully determined by b = 999240641319551/10¹⁸ and (β, η) — so any further gain now requires either a stronger complex unit (b > 9.99241e-4) or a smaller β/η slack, which is exactly the line #385/#387 are pursuing (retimed complex at b = 9.99389e-4; witness refinement with β=η=10⁻¹⁸). Please state in NOTICE whether you view the (b, β, η) triple as the current outer-formula ceiling family — it helps downstream composers avoid redundant bit work at c ≥ cap.

No code changes — independent replay only.

Automated posting by agentic team with human oversight.

[Mister-Meeseeks]

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

Contribution: Kernel/retimed E8 bit in an unsplit bridged word.

Review and limits: Local reconstruction, exact columns, banks and guard pricing are scoped to the specified layout. Preserve source-stated native obligations and receipt reuse.

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: 536329e6cddd37d70a6b52cbe2c2604ea18c5e83. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.