Conditional κ = 0.00162128508944984: cold factory-scratch helpers end at 4-dimensional frames on the PR601 witness (+0.439%)

[DreamingOfClouds]

Conditional κ = 0.00162128508944984, +0.439% over #601 (0.001614202339036103). The conservative model gives 0.001621285086572193. The dispatch is ck ×44, atm3 ×22, jp8 ×6. The value comes from ASSEMBLY.json of the strict verify.py run on this package's literal owner map. Stacked on #601 (open); the diff includes #601's package until it merges.

The complex side binds. The bit supplier value (#543, 0.0016538) is above the complex bound, so κ is the same with or without the bit cap.

What changes relative to #601

In every call, 14 cold factory-scratch helpers (S b.10, one per factory block) end at a 4-dimensional frame instead of FULL, so their residual width drops from 9 to 4. One typed step, cold_end, is appended to each provider chain (ck, atm3, jp8) after hyperplane_end. It has three parts:

1. Premix: 142 rank-0 gates at frame [] at the head of section A, after the entry adapters and paid wrappers. They swap S b.0/S b.10 dirt and fan the candidate's initial value out over S b.1…S b.7. The compensation reads no cold-ended helper.
2. Low-frame cleanup: four multi-gate tail gates per candidate at its frame U_b (h += −½·P_a, h += P_b, P_c += h, P_d += h). After them, h's row and column of the state are unit vectors.
3. Exact FULL remainder: a dyadic Gauss–Jordan refactorisation of the rest of the inverse, touching no helper that ends below FULL.

The local profile changes by {1:+28, 5:+14, 6:−28} per call per provider. The literal owner map comes from an exact pattern-restricted CP-SAT model (one helper per port per call, expanded by Kőnig edge colouring, warm-started from an annealed map) under every inherited window, family and capacity constraint.

Necessary condition used throughout: h can end at U only if, at the end of B, the supports of its row in the inverse state map and of its dirt column both lie in the registers whose end-of-B frames lie in U. The premix collapses each candidate's dirt column to three low-frame registers.

Why the cold ends are legitimate

• Lean: [Lean] Hyperplane-end helpers: multi-gate cleanup tails and the cold specialisation (stacked on #593, no κ claim) #599 (hpe_labels_multi, hpe_word_multi, exact_cold) covers multi-gate tails and an unrestricted entrance (E = ⊥), building on [Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593. It is now merged.
• (N) for every (0, U_b) pair is checked exactly, global phase included (hpe_adapters_local.py), with wrapper mutants rejected.
• The refactored cleanup is not trusted. It is accepted only for words marked refactored_cleanup: non-target registers only, dyadic additions with signed binary scales, and the arbitrary-dirt endpoint replay of the full program exact in both orientations. The transport also checks the row and column cone conditions.
• Wrapper order is enforced twice: by the transport's zero-frame assertion, and by an event-level check in hpe_audit_local.py with a negative control (a premix gate hoisted before the pre-wrapper).
• The verifier changes are inert for words without cold ends.

Checks

research/complex-571-cold14/verify.sh /tmp/complex-571-cold14 ..   # kappa 0.00162128508944984

• Fresh-archive receipt: strict verify.py from a fresh git archive of this head, with no baseline reuse, gives PASS_COLD14_NATIVE_COMPLEX_AND_CITED_BIT and κ 0.00162128508944984 (exit 0).
• Full rebuild: verify.py rebuilds the three provider words through their typed chains, with pinned hashes and paired controls.
• Local replay: it runs exact scalar replay with the cone pre-check, both reflected ledgers, the independent local replay with mutants, exact wrappers and all phase edges.
• Global: it runs binding with the physical owner audit, a fresh finite invoice with both moment engines, and the assembly, phase-timing and generated-identity controls.
• Independent audit (a separate Claude agent's re-implementation): the three cold words replay exactly (forward, dual, inverse-reverse and random-dirt). There are 0 nesting violations, and the cone fits are tight at 60/60 per word. The compensation reads no wrapped helper. The chain-derived profile reproduces the conservative κ via b_exact. Every word-level mutant is rejected, with one exception: declaring a candidate FULL-ended is legal but unpriced, so only the profile/report check catches it.
• Extra scalar work (disclosed): the scalar primitive bound per call rises by about 31%; prefix bits are 41/46 of the 192 limit; and the FULL remainder has 4 addition numerators that are not powers of two.

Future optimization

quantity	value
LP ceiling, #601's dispatch, all three providers cold	≈ +0.49%
LP ceiling, free dispatch over {ck, atm3, jp8} cold (jp8-heavy; packager's estimate, not independently checked)	≈ +0.71%
this package's literal map (fixed dispatch ck 44 / atm3 22 / jp8 6)	+0.439%

• A jp8-heavy dispatch is the main open lever: an exact LP with one-helper-per-port-per-call constraints matches the aggregate LP, so time-refinability does not bind at the LP level, and the remaining gap is integrality and search. On this dispatch the owner-map headroom is ≈ 0.06 pt.
• End dimension: d = 4 is the best end dimension for these units. Raising it lifts one partner and adds 2ψ(d−4) + ψ(9−d) − ψ(5) to the local cost (ψ(r) = r·ln(73/r)), which exceeds the remaining hosting headroom.
• Unit supply: the 14 units per word are not certified optimal here.

Credits:

• jeremykothe2: Conditional κ = 0.001609915765452068: hyperplane-ended helpers on the PR588 witness #591, Conditional κ = 0.001612719427075359: hyperplane-ended helpers with ck/atm3 dispatch #592, [Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593, Conditional κ = 0.0016127213726553: improved owner map for the PR592 hyperplane-ended witness #596, Conditional κ = 0.001612722852614221: improved owner map for the PR596 hyperplane-ended witness #600 and Conditional κ = 0.001614202339036103: 46 hyperplane-ended helpers per call via commuting-prefix cleanup #601 (the hyperplane-end mechanism, verifier, Lean and the base packages).
• icekylinx: Direct delivery with 504 helpers for conditional κ = 0.001517760977820331 #531 and Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535.
• DreamingOfClouds: Lemma G of #484: clean proof and Lean 4 formalisation of its core #509, Lemma A of #484: the realization of paid wrappers and spectators, proved in Lean 4 #516 and [Lean] Hyperplane-end helpers: multi-gate cleanup tails and the cold specialisation (stacked on #593, no κ claim) #599 (Lean).
• The pinned Conditional κ = 0.001596133352130093: seven more cold-kernel helpers on PR564 #568, Conditional κ = 0.001600067494703216: width-seven providers and whole-owner packing #569, Conditional κ = 0.001600682403612293: joint-prefix kernels and structural bounds #571, Conditional κ = 0.001600770286423653: joint prefixes in all 72 complex calls #575 and Conditional κ = 0.001601661522127798: six paid-frame coalescences in the added-kernel provider #583 packages and their authors are listed in NOTICE.md.

The approach borrows from the uncomputation literature: Bennett 1989; Gidney 2015; Häner–Roetteler–Svore 2017.

DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

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