Conditional κ = 1.018509848057013·10⁻³: checker-clean entrance enlargements, closure and rank-two gauges with a target-125 helper prefix on our #419 word (+1.75·10⁻⁷ over #423)

[chafreaky]

Conditional κ = 1018509848057013/10¹⁸ ≈ 1.018509848057013·10⁻³ (complex-bound), checker-clean.

• This is +1.747·10⁻⁷ over eumemic's κ = 1.018335129864748e-3: fourteen entrance enlargements with shared frame costs #423 (1.018335130e-3) and +3.42·10⁻⁸ over rohanarun's Candidate conditional kappa 0.001018475695250068: grouped entrance capacity and shared-frame enlargement #422 draft (1.018475695e-3, price-only).
• b = 254887067002137341481/2.5·10²³ ≈ 1.019548268e-3. The bit side is research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's pinned c ≈ 1.021539e-3 > b.
• The word passes Jacob Sussman's gx.check1 (E0–E7, exact scalar, gauged-dirt rule) and gxcore (sixths mirror), plus our flipped-sign and source-span controls.
• κ = 1.017812636706245e-3: one further chronological closure with exact target prefixes #420–κ = 1.018335129864748e-3: fourteen entrance enlargements with shared frame costs #423 state that they do not claim gx/gxcore E4/E5 acceptance. They keep research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's 124/125 recovery (gx E4) and κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411's quarter 144 prefix (gxcore E5).

Branch research/e8-enlarged-closure (local branch name), commit 92c6ea78f5cf726dc7621854d53140512ba1c46a, on our #419 (70a276c). It includes #318's CI repair. Package research/e8-enlarged-closure/, workflow .github/workflows/e8-enlarged-closure.yml.

What is new over our #419 (code/compose423.py; PR files read as data and re-implemented):

1. Closure and rank-two gauges (eumemic).
   • κ = 1.017812636706245e-3: one further chronological closure with exact target prefixes #420's closure gauge 672 → [48] and κ = 1.017889833900852e-3: two full rank-two entrances with complete prefix replay #421's rank-two gauges 435 → [304,111] and 367 → [259,246], with their relations unchanged (pinned blobs).
   • They are literal unit shears with frame closure. All 215 gauged response columns are zero.
   • The raw operator equals 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 times the 52-shear prefix on all 1,021 columns.
2. Fourteen one-dimension entrance enlargements.
   • Thirteen are κ = 1.018335129864748e-3: fourteen entrance enlargements with shared frame costs #423's (eumemic), including the joint 914/960 → [48,12]. κ = 1.018335129864748e-3: fourteen entrance enlargements with shared frame costs #423's 785 is not a gauge on our word; this is recorded.
   • The fourteenth is ours: 838 [204] → [304,204], from an exact screen of all 819 single enlargements.
   • No scalar instruction changes; the operator is identical on all columns.
   • 29 frames are raised by closure, and reverting any one fails gx E1.
3. Target-125 prefix re-synthesis (ours).
   • y125's pre-cut writes are replaced by helper 563/2 @336, 367/2 @904 and −251/2 @724.
   • Only halves of helper rows are used, with no y source, so sixths are preserved.
   • The protected x/y/gauged columns are unchanged.
   • The chain goes (3,3,1,1) → (2,4,1,1), worth +1.02·10⁻⁸.
4. Price at 64 replicas. 5,859 banks per stage and live stock 114,171. The 128 fused occurrences sit in distinct rank-1-padded banks. Replica sweep: 16 → 1.018468e-3, 32 → 1.018488e-3, 128 = 64.

Target 144: no sixths-legal prefix in the searched space (PROOF §19).

• Search space. Helper and x sources, delivered at their current or next touch frame, any number of deliveries, up to 4 frame groups, and shift/restore among same-frame sources.
• Method. An exact rational solve plus a Hermite-normal-form lattice test (each group's x/gauged total must lie in (1/6)ℤ).
• Result. Only (3,4,1) via 311 → 935 and (4,3,1) via 551 → 935 are solvable, and both fail the lattice test. Every source at 935 has x₆ = x₁₂/2, while the total has x₆ = x₁₂ = −½, so the early group carries x₆ = −¼.
• y sources are excluded by E4, because 144 already reads y into 208.
• Across all targets, 125 is the only new feasible form, and 144 is the only one blocked solely by sixths.

Other negative results.

• A screen of 933 one- and two-donor promotions found no further gain.
• Pairs of enlargements found nothing either.
• Candidate conditional kappa 0.001017923290728024: propagated entrance frames #418 and Candidate conditional kappa 0.001018475695250068: grouped entrance capacity and shared-frame enlargement #422 compose the same eumemic selections, so there is nothing orthogonal to add from them.

Verify (Python 3.11+, standard library only):

bash research/e8-enlarged-closure/verify.sh --quick /tmp/e8e-q   # 26 s
bash research/e8-enlarged-closure/verify.sh /tmp/e8e-full       # 2.5 min

Both passed from a fresh clone of the commit above, and the tree was unchanged afterwards. The #399, #402, #407, #412 and #419 pipelines run as controls.

Admitted conditions.

• Re-opened here:
  • the phases of the 3 new gauges, the 14 enlarged entrance tokens and the 39 raised frames;
  • the target-125 edges, whose changed ungauged dirt is recomputed by our emitter but not run through the native invoice;
  • owner placement of the 128 fused slots.
• Inherited and not re-run: normalisation, phases, routes, the finite invoice and the bit component (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).

This is a conditional finite construction, not a formal verification.

Credits.

• eumemic: κ = 1.018335129864748e-3: fourteen entrance enlargements with shared frame costs #423, κ = 1.017889833900852e-3: two full rank-two entrances with complete prefix replay #421, κ = 1.017812636706245e-3: one further chronological closure with exact target prefixes #420, κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416, κ = 1.017660918442154e-3: grouped 210-gauge entrances with exact prefixes #414, κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411, κ = 1.017562145187818e-3: enlarged entrance spaces with sequential donor promotion #410, κ = 1.017022657082799e-3: 205 shared-donor gauges with 64-replica banks #408.
• rohanarun: Candidate conditional kappa 0.001018475695250068: grouped entrance capacity and shared-frame enlargement #422, Candidate conditional kappa 0.001017923290728024: propagated entrance frames #418, Candidate conditional kappa 0.001017786140030308: enlarged entrances and target prefixes #415, Conditional kappa 0.001017250500928266: ordered sequential entrance gauges #409, Conditional kappa 0.001016469657190768: shared-donor chains with target re-synthesis #406, Conditional kappa 0.001000220224856110: six fixed levels on #390 #392.
• 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, E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390.
• Others: 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 (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).

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

[Mister-Meeseeks]

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

Contribution: Checker-compatible entrance enlargements and a target-125 prefix.

Review and limits: Local gx/gxcore acceptance retains the source's reopened phase, native-invoice and fused-owner obligations. Do not substitute this for the stronger native admission of a different word.

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: 92c6ea78f5cf726dc7621854d53140512ba1c46a. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.