Conditional κ = 0.001736738651088946052684 (+0.00533%): pooled R437 slack across bit replicas on #3958

[rohanarun]

Author-checked conditional candidate, stacked on #3958. #3958 composes #3955's complex side with #3647's R437 bit supplier. This applies #3939's cross-replica owner map to R437. Draft.

	complex root lower	bit endpoint	binds	κ headline
#3958	0.0017411725885452521674224	R437 0.001736646042038446578551	bit	0.001736646042038446578551 − …
this	0.0017411725885452521674224	pooled R437 0.001736738651088946052684	bit	0.001736738651088946052684 − 10⁻⁴⁶ − 10⁻⁷⁰

+0.00533% over #3958. About 4.4e-6 of complex headroom remains.

Idea

Per replica, R437 keeps one two-residual window-6 family (idle 26) and 12 empty 42-families. Pooled over 32 replicas, the window-6 slack is 32·63 = 403·5 + 1 places. Residue t = 1 in mightyOil's bit-partial-width7-optimum table gives the pooled32 slot optimum: 403 empty 42-families plus one empty capacity-14 copy.

This package builds that optimum explicitly with 63 moves: the 62 two-family residuals of replicas 1–31, plus replica 0's window-8 residual (stage 8, inside window 6). They fill replica 0's two-residual family and its 12 empty families. The idle bill changes by {2:+13, 6:−1, 14:+1, 26:−32, 42:+19}. Rank mass and maxchild are unchanged.

Verification

• Inputs: Improve conditional BIT c to 0.001736646042 with composed R437 and credited slot coalescing #3647's GLOBAL-TEMPLATE.json.gz is vendored; its content SHA-256 equals template_content_sha256 in Improve conditional BIT c to 0.001736646042 with composed R437 and credited slot coalescing #3647's EXPECTED.json.
• Role audit: verify.py is independent of build.py. It rebuilds all 125,856 R437 hosted roles and checks window, inactive host pair, width-8 extent, disjoint cells, compact prefixes and hosted-exactly-once.
• Baseline: it reproduces Improve conditional BIT c to 0.001736646042 with composed R437 and credited slot coalescing #3647's EXPECTED profile and endpoint exactly.
• Pooled endpoint: accepted at c and rejected at c + 10⁻²⁴, with the fallback envelope.
• Composition: Conditional κ = 0.001734900431583947157341 (+0.00203%): hosted frame selection on #3939 #3955's complex root is rechecked; κ has the bit side binding.
• 6/6 mutation controls rejected.
• Registered replay: PASS_REGISTERED_RESEARCH_PACKAGE.

Scope

This carries the same cross-replica owner hypothesis as #3939; #3647's native hosted emission was not run with these owners. It inherits #3647's R437 interfaces (admission pending) and #3955's complex-side hypotheses.

Credit @ai-nanachi (#3647), Ali Romman (#1043), Sagar Thakrar (#3700/#2423/#1023), mightyOil (#1281) and the #3955 stack. Complex-side ideas in the stacked predecessors were done with help from OpenCode (Space bunny free). Prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy

[Clear1nsight]

CLAIM / pooled R437 compiler-boundary refinement / 2026-10-19 / expires08:49UTC. A separate agent is pairing reversible/native compiler correctness with replica-aware scheduling and uniform register-renaming proofs. We will bind the first actual cross-replica move, test the retained catalogue/emitter interface on its literal owner addresses, and construct or falsify the smallest replica-indexed refinement; expand to all63 moves only if the same bounded proof applies. The pooled owner map, slot optimum and conditional kappa remain credited to this PR. A streamed catalogue or renaming lemma will be distinguished from physical CALL/RETURN and supplier admission. PR3967 and new3980 concern the unpooled source; new3982/3983 cover different COPY/checkpoint obligations. No repeat pooling census, new kappa, merge or selected-frontier claim. Canonical main20ecf289/revision6 refreshed;679 open bodies plus newest-page deltas and rendered handoffs checked, broader coverage partial. Credit Rohan Arun/3959/3958, ai-nanachi/3647, Ali Romman/1043, Sagar Thakrar/3700/2423/1023, mightyOil/1281 and retained native/compiler lineage. Chris R/Clear1nsight with substantial OpenAI Codex assistance.

[Clear1nsight]

PARTIAL / completed emitted-IR refinement: #3995 (head ae29394). All63 pinned moves compile through the unchanged retained absolute-owner emitter; its complete7,814,016-event stream matches the separately retained audit encoder. Exactly2265 events change; newdigest68d691118d9ac43f79a35cd2d6662a8257e68eddfc6a0fd79524ca15145c775e. All125856 helper roles and63936 replica-indexed idle records check. Packing/profile/conditionalendpoint credit remains with this PR; we do not claim a new kappa.

The first literal move (stage6,callerreplica1,helper421) changes family1097/offset0 to family209/offset16. The compressed physical catalogue rejects this actual record, and its replica0-times32 request model cannot represent the pooled400877 rank2 and1 rank14 counts. Deleting its guard is insufficient. The new transport lemma proves arbitrary-value dirty-state renaming conditional on valid slot injections and charts; it does not make physical routing free. Pooled source access, charts, CALL/RETURN, consumers, parking, restoration, visited cells, precision and complete supplier joins remain open.

Normal registered, committed-byte/source-history, official export and124.8s read-only Git-free replay pass; hosted CI is tracked on #3995. No physical production execution is claimed. Six wrong-owner/replica/overlap/extent/pair/duplicate controls reject. Credit Rohan Arun, ai-nanachi, Ali Romman, Sagar Thakrar, mightyOil and the retained emitter/physical catalogue lineage. Substantial OpenAI Codex assistance.

Completed local-only source-compilation follow-up (not added to frozen #3995): the relocated helper is first read at localposition1763; first MOVE19236 is rank2/frame0->250/request18186. With the pinned #3967 adapter, both old/new permutation owner charts (axes22..29 versus38..45) compile both route endpoints at q=3, and a separately labelled relocated record has the identical complete4346-command common residual plan. Actual compile_node gathers/routes before MOVE(common=True); the changed chart belongs to those routes. C_native=C Q^-1 and high-key H->H Q give a uniform no-carry permutation identity, while physical source labels/descriptors still need explicit binding. The full8692*f two-sweep ADD bill plus every gather/route/scatter/native/preparation/return charge remains paid. No physical residual body or CALL/RETURN was executed. This is a source-bound handoff, not a second PR or production admission.

#3995 hosted CI passed at exact headae29394 in https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39221691337, including unchanged inputs. The reservation is released. Next test: the actual gathered source and physical return, retaining the preceding dirty read and complete consumers.

[ChaloupeMusic]

FINDING / no new κ. This chain's complex root comes from #3934 and #3955, which stack on #3251, retracted on 2026-10-18 after merged #3296/#3311. #3934's MOVES contain all 435,604 #3251 moves, and #3934 and #3955 both contain #3296's port-0 witness. With #3187/#3296's exact operator replay, each of #3934's 2,374 hosted-receiver moves changes the operator at its own owner and port (2,274 configurations, 140 owners, none preserved). At owner 0, all 55 identity-selection #3955 ports change it; the other 57 need a cell-to-axis map that #3955 does not give. Without these moves the complex side falls back to #3148 (0.0017298), which binds. Package and verifier: #4046.

[rohanarun]

Retracted, closing. This draft belongs to the #3934 → #3959 chain, which is built on #3251's idle-axis moves. Those were retracted on 2026-10-18 because #3190's donations change the exact eight-stream operator (see #3296/#3311/#3187). Native admission, the controller audit and the outer release never ran. #3995 and #4046 show the pooled move is rejected. The κ claimed here is not valid. Prepared by Rohan Arun with Anthropic Claude assistance.