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)

[chafreaky]

Conditional κ = 1017776060460931/10¹⁸ ≈ 1.017776060460931·10⁻³. It is complex-bound, with b = 254703245981514431773/2.5·10²³ ≈ 1.018812984e-3, and the bit side is unchanged (c ≈ 1.021539e-3).

• vs eumemic's κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410 (1.017562145187818e-3): +2.139·10⁻⁷
• vs eumemic's κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411 (1.017629458064773e-3, opened while this was being packaged; not composed here): +1.47·10⁻⁷
• vs rohanarun's Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409: +5.256·10⁻⁷

Correctness: #406, #408, #409 and #410 fail Sussman's checkers; this word passes

We re-implemented each word from its published data and checked it against the frozen hashes: #406 3f67e51a…, #408 7a5d0d09…, #409 e775f810…, #410 5243b69b…. We also reproduced #408's, #409's and #410's exact b and κ from our own emission, using their own replica counts and bank patterns.

Jacob Sussman's gx.check1 and gxcore reject all four words with E4: a pivot is the target of a gate reading y. This comes from #404's 124/125 recovery, which all four inherit.

With that recovery removed, gxcore still rejects two of them:

• Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409, E5 at register 275 (and at 204 for its group [51] alone). Its own sequential groups [51] and [195] pivot on half-coefficient relations (1010, 910). Their half-unit gauged dirt then meets a /2 gate.
• κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410, E5 at register 947. This is research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's pivot 555.

We fix both by exact re-pivoting, with the same relation span checked by rank: 1010 → 739 and 910 → 531 in #409's groups, and 555 → 748 as in our #407. The submitted word passes gx.check1 (E0–E7), gxcore, a flipped-sign control and the source-span rule.

Change and mathematical scope

The word is built on our #407 base: #404's 194 gauges with our four re-pivotings, #376's frame edit and no 124/125 recovery. On top of it:

1. κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410's eight entrance enlargements (eumemic), re-implemented. Our enlarge() regenerates κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410's enlarged205.json byte-for-byte from κ = 1.017022657082799e-3: 205 shared-donor gauges with 64-replica banks #408's selection. The 1020 enlargement is carried by our re-pivoted row of 1001, which has the same relation and roles.
2. Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409's four sequential groups (rohanarun), re-implemented, with groups [51] and [195] re-pivoted. These give ten promoted pivots and 55 prepended unit shears, and every sequential source is read before any entrance write to it.
3. Padding fusion of sinks 301/387 (ours). 1019 and 1020 are even-coefficient donors after re-pivoting, so the −¼ uncomputes stay in sixths.
4. 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 target 177/220 re-synthesis, relocated as Conditional kappa 0.001016469657190768: shared-donor chains with target re-synthesis #406 does.

Exact checks:

• The full raw operator and the retained totals equal Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's operator times the literal 555-gate entrance prefix on all 1,021 columns.
• All 204 gauged response columns are zero.
• Both local orientations validate, and five mutation controls are rejected.

Banks use 32 replicas, 2,943 width-73 banks per stage and 1,495 rank-1 paddings less 64 fused. Live stock is 57,207, m = 73, and the deficit is 9,984. On the replica count:

• 64 replicas give exactly the 32-replica value, and 16 give less.
• κ = 1.017022657082799e-3: 205 shared-donor gauges with 64-replica banks #408's 205-pivot selection re-optimises the same rank-1 entrances that Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409 promotes into, so it is not orthogonal; we keep Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409's groups.
• Further rank-1 groups on this word all add rank-1 climbs and lose (−5·10⁻⁷ or worse), which is consistent with Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409's negative screen.

Admitted conditions:

• Inherited and not re-run: 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/Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409/κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410 normalisation and phases, routes, the invoice, and the bit component via research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's pinned RESULT.json.
• Re-opened: phases of the six re-pivoted tokens, the 64 fused slots and climbs, and the 177/220 edges.

This is a conditional finite construction, not a theorem.

Validation

• Commit 616d9d6 on research/e8-sequential-enlarged, branched from our pushed research/e8-repivot-fusion-404 (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). It includes 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 three-file joint-dual CI repair, with no mathematics changed.
• From a fresh clone, bash research/e8-sequential-enlarged/verify.sh --quick and the full verify.sh both exit 0 (full run about 2.5 minutes, Python 3.14, standard library only). They cover the manifest (matches git ls-files), git-blob pins of all 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 inputs, 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 and 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 controls, and compose410.py sections A–E, cxcheck and the final κ assertion.
• The new workflow is .github/workflows/e8-sequential-enlarged.yml (Python 3.12 and 3.14).

Credits

• eumemic: κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410 (enlargement) and κ = 1.017022657082799e-3: 205 shared-donor gauges with 64-replica banks #408 (205 gauges and 64-replica banks; a control here).
• rohanarun: Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409 (sequential gauges), Conditional kappa 0.001016469657190768: shared-donor chains with target re-synthesis #406 (relocation), Conditional kappa 0.001013417593791468: target re-synthesis with 64-replica banks #405, Conditional kappa 0.001013370540420331: 64-replica response-subspace banks #403, Conditional kappa 0.001002683712533373: R781 heterogeneous banks and padding fusion #400, Conditional kappa 0.001001098432210898: signed-prefix donor gauges #396, Conditional kappa 0.001001256225581264: signed prefixes and padding fusion #395, Conditional kappa 0.001000220224856110: six fixed levels on #390 #392 and 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 (CI repair).
• 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.
• κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352: local invocation and emitter.
• Jacob Sussman: E8 unit, gx/gxcore.
• 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 and later) and utcorvusvolat-dotcom (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370): the bit side.
• 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.

All source PRs were read as data only; none of their scripts were run. Prepared by Chafik Boukhalfa (chafreaky) with substantial Anthropic Claude assistance. Apache-2.0.

[ahrazzle]

Third-hardware replay: PASS, exact claimed value. The audit claims also reproduce.

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

Full replay, all sections green, ending:

== 6. Sussman's checkers, flipped-sign control and source-span rule on ours
gcert1-e8seqenlarged.json.gz: gx.check1 PASS (E0-E7, exact scalar),
gxcore mirror PASS, flipped sign rejected (E5), source-span violations 0
PASS conditional kappa = 1017776060460931/10^18 = 0.001017776060460931
(complex-bound; checker-clean)

Independent negative-control check (the PR's central audit claim).
I ran the package's compose410.py section A on my own machine.
It re-implements the four competitor words (#406, #408, #409, #410)
from their published selections, asserts their frozen hashes
(reproduced: 3f67e51a, 7a5d0d09, e775f810, 5243b69b), then runs
Sussman's gx.check1 and gxcore on each. All four are rejected
with "E4: a pivot is the target of a gate reading y", traced to
#404's 124/125 recovery. With the recovery removed, gxcore still
rejects two of them with E5: the sequential-gauge word at
register 275 (its own groups 51 and 195 pivot on half-coefficient
relations) and the enlarged-entrance word at register 947 (pivot 555 from the shared-donor recovery). The re-pivots (1010 to 739, 910 to 531, 555 to 748)
keep the same relation span by rank, and the submitted word passes
both checkers. This confirms the claim that those four words fail
Sussman's checkers while this one is checker-clean.

The package's emission also reproduces each competitor's own numbers
from my machine, with their replica counts and bank patterns:
sequential-gauge kappa 508625250464133/(510^17) with 32 replicas
and 2947 banks/stage, shared-donor kappa 1017022657082799/10^18
with 64 replicas and 5893 banks/stage, and enlarged-entrance kappa
508781072593909/(510^17) with 16 replicas and 1471 banks/stage.
The replica sweep gives 16 replicas at 203546634751703/(2*10^17)
and 64 replicas equal to 32, so 64-replica banks add nothing here.

Independent closed-form check: with b =
254703245981514431773/(2.5*10^23), beta = 1e-9, eta = 1e-12
(complex binding, since c = 1.0215e-3 > b), the standard assembly
gives kappa = 1.0177760584e-3, matching the published
1.017776060460931e-3 to 9 significant digits (the residual is my
floor convention at the 1e-18 grid).

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 sequential gauges and entrance enlargement.

Review and limits: Reproduces comparison words and their checker failures, then checks its own re-pivoted word. New phase/owner/fused-climb/invoice and sink-isolation obligations remain source-scoped.

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: 616d9d678c00d79693b869934464003ba5b8d1b1. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.