Best gx/gxcore-clean κ = 1.019752143318808·10⁻³: our #424 layers, 14 gate-frame retimings and a target-172 prefix on #426's R779 word (+9.85·10⁻⁷ over #426)

[chafreaky]

Conditional κ = 127469017914851/1.25·10¹⁷ ≈ 1.019752143318808·10⁻³ (complex-bound; gx.check1 + gxcore clean)

Package: research/e8-r779-retimed-closure
Commit: 642b702149b975e3a535f139243cd3afd286e868
Local branch: research/e8-r779-retimed-closure

This is the best result we know of that is clean under Sussman's gx.check1 (E0–E7, exact scalar replay, gauged-dirt rule) and gxcore (sixths mirror).

• +9.85·10⁻⁷ over ai-nanachi's Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426 (1.018766833e-3, checker-clean).
• +1.06·10⁻⁷ over rohanarun's Candidate conditional kappa 0.001019646575559504: R779 gauged helper inlining #429 (1.019646576e-3). Candidate conditional kappa 0.001019646575559504: R779 gauged helper inlining #429 is a price-only draft on Candidate conditional kappa 0.001018475695250068: grouped entrance capacity and shared-frame enlargement #422's word, which keeps research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's 124/125 recovery and κ = 1.017629458064773e-3: exact target-prefix resynthesis on sequential206 #411's quarter target-144 prefix, and neither claims gx/gxcore acceptance.
• +1.24·10⁻⁶ over our 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) #424.

For context: jacklightChen's #425 (≈1.048005508e-3) and its follow-up #431 use a Hamming-packet complex network under their own lemma, in place of gx's source-immutability and target-read rules. They are not comparable under these checkers.

The complex side binds: b = 255198274817418511443/2.5·10²³ ≈ 1.020793e-3. The bit side is #404's pinned c ≈ 1.021539e-3 > b.

Construction (code/compose426.py)

1. Input: Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426's R779 gauged word, pinned byte-for-byte (blob 0daa4323), with 202 gauges after the 556/577 helper inlining.
   • Control: our own emitter and price, at Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426's 8 replicas and bank patterns, reproduce Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426's complex coefficient 254951444354246755323/2.5·10²³ and its profile exactly.
   • Re-implementing the two substitutions directly on Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's signed R781 word (our 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) #424 base) is frame-infeasible, so we start from Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426's word.
2. Our 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) #424 layers, mapped to R779:
   • κ = 1.017660918442154e-3: grouped 210-gauge entrances with exact prefixes #414's group, κ = 1.017798035290733e-3: propagated frames admit 214 entrance gauges #416's promotions and our 947 re-pivot, κ = 1.017812636706245e-3: one further chronological closure with exact target prefixes #420's 672, and κ = 1.017889833900852e-3: two full rank-two entrances with complete prefix replay #421's 435/367: 52 literal shears, 213 gauges, operator identity on all 1,019 columns, every gauged response column zero;
   • the 14 enlargements (13 of κ = 1.018335129864748e-3: fourteen entrance enlargements with shared frame costs #423's and our 838);
   • fused sinks 301/387;
   • 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's 177/220 re-synthesis, relocated with restore source y221 (y180 fails gx E4 in Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426's signed layer);
   • our target-124 helper prefix.
3. Fourteen fixed-scalar gate-frame retimings: 12 single gates and 2 five-gate blocks.
   • The idea is eumemic's (κ = 1.018364819189613e-3: two joint gate-frame blocks with complete replay #427/κ = 1.018419742888184e-3: combined exact gate-frame retimings #428). The exact search on our word is ours.
   • Thirteen bases coincide with κ = 1.018419742888184e-3: combined exact gate-frame retimings #428's eleven and κ = 1.018364819189613e-3: two joint gate-frame blocks with complete replay #427's two. The fourteenth (B1157 → rank 7) is new.
   • Scalar words and the full raw operator are unchanged. --regen re-runs the search.
4. New target-172 helper prefix.
   • Deliveries: −h562/2 at frame 336, h358/2 at 904, −h367/2 at 904 and −h280/2 at 724.
   • Protected columns are unchanged, and the histogram moves by {2:+1, 3:−2, 4:+1}. Omitting a delivery fails E5.
5. Price at 64 replicas.
   • 5,845 width-73 banks per stage and live stock 114,045. The 128 fused occurrences own slot 0 of distinct padded banks.
   • The replica sweep (8–128) is at most the 64-replica value.

Negative screens on this word (exploration)

• Promotions: 1/2/3-donor promotions (5,659 candidates) found no gain.
• Enlargements: single, two-dimension, pair-joint and triple-joint enlargements (3,554 candidates) all break final frames under closure.
• Further inlining: extra equal-helper inlinings are frame-infeasible.
• Sinks: 324 still fails gxcore E5.
• Prefixes: an all-target prefix search finds only 124/172.
• Retimings: none are left after the fourteen.

Verify (fresh clone; Python 3.11+, stdlib only)

bash research/e8-r779-retimed-closure/verify.sh --quick /tmp/e8r-q   # ~30 s
bash research/e8-r779-retimed-closure/verify.sh /tmp/e8r-full        # ~90 s, adds the retiming-search regeneration

Both pass from a fresh clone, and the tree is unchanged afterwards. The script runs:

• the manifest and git-blob pin checks;
• our 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) #424 pipeline as a control;
• compose426.py;
• gx.check1, gxcore, our flipped-sign control and our source-span rule on both words.

Admitted conditions

• Re-opened here:
  • the phases of the 11 promoted gauges, the 14 enlarged entrance tokens, and the 39 raised and 22 retimed gate frames;
  • the 124/172 and 177/220 target edges with their changed ungauged dirt;
  • the fused-slot owner placement.
• Inherited and not re-run:
  • Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426's phases, routes and invoice;
  • 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 normalisation;
  • the finite invoice;
  • the bit component, via research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3 #404's pinned RESULT.json.

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

Credits

• @ai-nanachi: Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426 (R779 inlining, gauge layer, input word).
• @eumemic: κ = 1.018364819189613e-3: two joint gate-frame blocks with complete replay #427/κ = 1.018419742888184e-3: combined exact gate-frame retimings #428 (retiming idea), 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 0.001019646575559504: R779 gauged helper inlining #429 (comparison), 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 0.001013417593791468: 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 0.000998634658090835: finite bootstrap and exact coarse precision #387/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.
• @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 and 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.
• Ours (@chafreaky): Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378, Conditional κ = 1.000385913854264·10⁻³: bank padding fused into three new free sink helpers on #390's E8 complex program (+0.0166 % over #390) #394/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, 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, 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) #419, 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) #424, and here the R779 composition, the retiming search with the new B1157 retiming, the target-172 prefix and the negative screens.

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 R779 gauges, retiming and target-172 resynthesis.

Review and limits: New local checker acceptance does not discharge the source-listed phase, ownership and native invoice obligations. Bounded unsuccessful searches are not general optimality results.

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: 642b702149b975e3a535f139243cd3afd286e868. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.