Conditional κ = 1.017962405296845·10⁻³: checker-clean grouped and frame-closure entrance promotions with a target-124 helper prefix on our #412 word (+3.9·10⁻⁸ over #418; #414–#416 fail gx E4/gxcore E5)

[chafreaky]

Conditional κ = 1.017962405296845·10⁻³: checker-clean grouped and frame-closure promotions with a target-124 prefix (+1.64·10⁻⁷ over #416)

The exact value is κ = 203592481059369 / (2·10¹⁷) ≈ 1.017962405296845·10⁻³. It is complex-bound, with b = 254749927172825437387/2.5·10²³ ≈ 1.018999709·10⁻³. The bit side is #404's, unchanged (c ≈ 1.021539·10⁻³ > b). Pairing it with our #417's bit (c ≈ 1.044064·10⁻³) gives the same κ.

comparison	Δκ
eumemic #416 (1.017798035·10⁻³)	+1.644·10⁻⁷
rohanarun #418 (1.017923291·10⁻³)	+3.91·10⁻⁸
rohanarun #415	+1.763·10⁻⁷
eumemic #414	+3.015·10⁻⁷
our #412	+1.863·10⁻⁷

This is a conditional finite witness, not a theorem.

Change and mathematical scope

The base is our #412 word, unchanged: #404's gauges re-pivoted, #410's enlargements, #409's groups re-pivoted, fused sinks 301/387, and the 177/220 re-synthesis. On top of it:

1. κ = 1.017660918442154e-3: grouped 210-gauge entrances with exact prefixes #414's grouped promotion (eumemic). Pivots 766, 917, 627 and 948 enter the rank-1 entrance [48]. κ = 1.017660918442154e-3: grouped 210-gauge entrances with exact prefixes #414's relations are used unchanged, as literal prepended shears.
2. κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416's frame-closure promotions (eumemic). These are 892 → [51], 669 → [12] and 644 → [63], with relations unchanged. Gate frames are raised to the span of their operands' frames (κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416's rule, our code), which raises 5 frames on our word.
3. κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416's 821 → [240], adapted.
   • Its relation uses donor 748, which is our pivot (Conditional κ = 1.016526770666368·10⁻³: response-gauge re-pivoting (checker-clean) and padding fusion on #404's shared-donor R781 E8 complex program (+5.71·10⁻⁸ over #406) #407 re-pivots 555 → 748).
   • Re-solved on our word, the relation is unique on 821's other donors plus 555 and 947. It needs −½ on 947, which gxcore rejects (E5 at register 947, recorded as a control).
   • Pivoting the same relation on 947 makes the row integral, and gxcore passes.
4. Target-124 prefix re-synthesis. This uses κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411's prefix-window idea (itself adapted from our Conditional κ = 1.013357935292321·10⁻³: exact value-level re-synthesis of two targets on #401's response-subspace R781 E8 complex program (+4.70·10⁻⁸ over #401) #402), with our own search and helper sources only.
   • 124's two B writes before its first positive-frame read are replaced by helper 563/2 at frame 336 and −helper 253/2 at frame 723.
   • The protected x, y and gauged columns of every row are unchanged. Only y124's and y171's ungauged dirt changes, and the emitter recomputes it.
   • Halves only and no y source, so gx E4 and E5 hold.
   • 124's chain goes (3,4,1) → (2,5,1).

The result has 212 gauges. The full raw operator and retained totals equal #412's gauged word times the literal 48-shear prefix on all 1,021 columns, and all gauged response columns are zero. With 32 replicas there are 2,939 banks per stage and 1,459 rank-1 paddings, and the 64 fused sink occurrences own distinct padded banks. Live stock is 57,171.

Checker status of the concurrent claims. These are not checker-clean:

• κ = 1.017660918442154e-3: grouped 210-gauge entrances with exact prefixes #414, Candidate conditional kappa 0.001017786140030308: enlarged entrances and target prefixes #415 and κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416 keep research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's 124/125 recovery, so gx E4 fails.
• The same three use κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411's target-144 prefix, whose first −¼ delivery puts y144 in quarters. gxcore E5 rejects it at register 144; we reproduced this on our word. κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411, κ = 1.017660918442154e-3: grouped 210-gauge entrances with exact prefixes #414, Candidate conditional kappa 0.001017786140030308: enlarged entrances and target prefixes #415 and κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416 state that they do not claim gx E5 acceptance.
• κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416's raw 214-gauge entrance word already fails gxcore E5 at register 947.

This word passes both of Sussman's checkers.

Retained hypotheses and re-opened items.

• Re-opened: the Gaussian phases of the 8 new entrance tokens and the 5 raised frames, and the target-124 edges and native invoice.
• Inherited from Conditional κ = 1.017776060460931·10⁻³: checker-clean composition of sequential and enlarged entrance gauges with re-pivoting and padding fusion (+1.47·10⁻⁷ over #411; #406/#408–#410 fail gx E4) #412 and its lineage: normalisation, routes, bank owners and invoice, and the bit component via research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's pinned RESULT.json.
• The all-size compressed-window, compiler and analytic interfaces remain conditional.

Attribution, by handle:

• eumemic: κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416 (frame-closure promotions), κ = 1.017660918442154e-3: grouped 210-gauge entrances with exact prefixes #414 (grouped promotion), κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411 (target-prefix idea), κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410 and κ = 1.017022657082799e-3: 205 shared-donor gauges with 64-replica banks #408.
• rohanarun: Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409, Conditional kappa 0.001016469657190768: shared-donor chains with target re-synthesis #406 and Conditional kappa 0.001000220224856110: six fixed levels on #390 #392 (bootstrap); Candidate conditional kappa 0.001017786140030308: enlarged entrances and target prefixes #415 and Candidate conditional kappa 0.001017923290728024: propagated entrance frames #418 as comparison claims.
• Dugongue: research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404, research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401, Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 and E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390.
• applecrusher: Conditional κ=0.0011208649522582786: split centres with complete signed admission #376.
• jacklightChen: κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377.
• Jacob Sussman: E8 unit, gx/gxcore.
• DaysSky and utcorvusvolat-dotcom: the bit lineage.
• DreamingOfClouds: 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 engines.
• chafreaky: Conditional κ = 1.013357935292321·10⁻³: exact value-level re-synthesis of two targets on #401's response-subspace R781 E8 complex program (+4.70·10⁻⁸ over #401) #402, Conditional κ = 1.016526770666368·10⁻³: response-gauge re-pivoting (checker-clean) and padding fusion on #404's shared-donor R781 E8 complex program (+5.71·10⁻⁸ over #406) #407 and Conditional κ = 1.017776060460931·10⁻³: checker-clean composition of sequential and enlarged entrance gauges with re-pivoting and padding fusion (+1.47·10⁻⁷ over #411; #406/#408–#410 fail gx E4) #412.

#414's and #416's data files are vendored byte-identically (git-blob pinned, Apache-2.0). None of their scripts are run. Prepared by Chafik Boukhalfa (chafreaky) with substantial Anthropic Claude assistance.

The prerequisite is our #412, 616d9d678c00d79693b869934464003ba5b8d1b1. The branch includes rohanarun's #318 three-file joint-dual source-pin repair, which is already in the base (ac91a9c), for repository CI. No mathematics changed there.

Validation

• Commit: 70a276cb948e5be6071cde3f1bebdaf2f995ab8e (local branch research/e8-promoted-closure in checkout e8clean7, on Conditional κ = 1.017776060460931·10⁻³: checker-clean composition of sequential and enlarged entrance gauges with re-pivoting and padding fusion (+1.47·10⁻⁷ over #411; #406/#408–#410 fail gx E4) #412 616d9d6); package research/e8-promoted-closure; workflow .github/workflows/e8-promoted-closure.yml.
• Fresh clone of the branch:
  • bash research/e8-promoted-closure/verify.sh --quick OUT passes in about 22 s.
  • bash research/e8-promoted-closure/verify.sh OUT (full, with every control regeneration) passes in about 2.5 min.
  • Python 3, standard library only.
• What the verifier covers:
  • the manifest, with MANIFEST.sha256 equal to git ls-files;
  • the git-blob pins of every vendored input (E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390, Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398, research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401, research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404, κ = 1.017022657082799e-3: 205 shared-donor gauges with 64-replica banks #408, Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409, κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410, κ = 1.017660918442154e-3: grouped 210-gauge entrances with exact prefixes #414, κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416);
  • our Conditional κ = 1.002661106042177·10⁻³: our #394 padding fusion on #398's R781 signed-prefix E8 complex program (+0.0166 % over #398) #399, Conditional κ = 1.013357935292321·10⁻³: exact value-level re-synthesis of two targets on #401's response-subspace R781 E8 complex program (+4.70·10⁻⁸ over #401) #402, Conditional κ = 1.016526770666368·10⁻³: response-gauge re-pivoting (checker-clean) and padding fusion on #404's shared-donor R781 E8 complex program (+5.71·10⁻⁸ over #406) #407 and Conditional κ = 1.017776060460931·10⁻³: checker-clean composition of sequential and enlarged entrance gauges with re-pivoting and padding fusion (+1.47·10⁻⁷ over #411; #406/#408–#410 fail gx E4) #412 pipelines as controls;
  • section-by-section composition checks, all-column operator checks, and frame-reversion, omitted-shear and prefix-mutation controls;
  • the local program in both orientations, with 7 mutation controls each;
  • banks, and the exact price from two moment engines plus 47 strict outer constraints;
  • gx.check1 and gxcore, with flipped-sign and source-span controls, on both Conditional κ = 1.017776060460931·10⁻³: checker-clean composition of sequential and enlarged entrance gauges with re-pivoting and padding fusion (+1.47·10⁻⁷ over #411; #406/#408–#410 fail gx E4) #412's word and ours.
• CI: pending.

[ahrazzle]

Third-hardware replay: PASS, exact claimed value, checker-clean.

Environment: Apple M1 Max / macOS 27.0.1, Python 3.14, stdlib only.
Fresh worktree of the PR head, full verify.sh, exit 0.

Final lines:

kappa = 203592481059369/200000000000000000 = 0.001017962405296845
(six bootstrap levels, 47 strict constraints. Controls
next_grid, zero_eta, zero_beta and unit_beta all rejected.)
== 7. Sussman's checkers on the control word and on ours
gcert1-e8seqenlarged.json.gz: gx.check1 PASS (E0-E7), gxcore mirror PASS
gcert1-e8promoted.json.gz: gx.check1 PASS (E0-E7), gxcore mirror PASS,
flipped sign rejected (E5), source-span violations 0
PASS conditional kappa = 203592481059369/(2*10^17)

Both the control word and this word pass both of Sussman's
checkers on my machine, as claimed.

Independent closed-form check. I priced the published b
(254749927172825437387/(2.5*10^23)) through the six-level
refined assembly (grid 1e-18, beta = eta = 1e-18, s =
(1-beta)b - 1e-24, six bootstrap rounds from a0 = 384599/1e10,
kappa = floor-grid of (1-eta)q/(1+q) minus one tick) without
running any of the package's code. Result:
203592481059369/(210^17), exact match to the published digit.
The same formula reproduces #412's and #420's published kappas
exactly, which pins the assembly family this PR uses.

Comparison deltas recomputed from published values: +1.863e-07
over the control word, +1.644e-07 over #416, +3.91e-08 over
#418. The claim that pairing this word with the #417 bit
supplier leaves kappa unchanged is consistent: the bit cap
1.044064e-3 exceeds b = 1.0190e-3, so the complex side binds
either way.

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: Checker-compatible grouped and frame-closure promotions.

Review and limits: Exact re-pivoted relations and target124 rewrite pass local checkers. Native phases, ownership, finite invoice and sink-isolation obligations are still explicitly listed.

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: 70a276cb948e5be6071cde3f1bebdaf2f995ab8e. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.