Conditional κ = 0.001734900431583947157341 (+0.00203%): hosted frame selection on #3939

[rohanarun]

Author-checked conditional candidate, stacked on #3939 (→ #3934 → #3251 → … → #3148 → #3133). Draft. The full verification run is still going.

	complex root lower	bit endpoint	binds	κ headline
#3939	0.0017348651784727109982331	pooled R438 0.001734900431583947157341	complex	0.001734865178472710998233 − …
this	0.0017411725885452521674224	pooled R438 0.001734900431583947157341	bit	0.001734900431583947157341 − 10⁻⁴⁶ − 10⁻⁷⁰

+0.00203% over #3939. That is the whole remaining window up to the pooled bit endpoint. The complex root itself rises by +0.36%, so κ is now capped by the bit supplier with about 6.3e-6 of complex headroom.

Idea: hosted frame selection (new hypothesis)

Since #3190, a bin hosting used helper cells kept them on the recipe prefix [0, used), and its idle remainder was the recipe suffix [used, cap). After #3934:

• 94,656 rank-1 hosted remainders keep only an axis that fails PR1252's translation rule;
• yet every one of the 134,320 unmodified rank-1 hosted bins has a translation-commuting axis somewhere in its frame.

This package lets the hosted cells take any used-subset of the frame. The remainder is chosen per bin as the axes most able to move. #3934's pipeline then runs unchanged: exact phase 1, star phase, greedy phase, hosted receivers. All moves keep PR1252's commuting rule, role exclusivity and the rank-51 cap.

The result is 521,798 axis moves. Rank-1 remainders fall from 147,220 to 79,440, and rank-51 children rise from 129,006 to 160,932. Rank mass is unchanged.

Verification

• verify.py is independent of build.py. It checks:
  • donated axes come from the donor's own frame, within the cap − used idle budget;
  • each hosted receiver is orthogonal to its whole frame;
  • role exclusivity, orthogonality at receipt, the commuting rule and the rank-51 cap;
  • the profile, the rank mass and the exact root bracket;
  • the pooled R438 endpoint, by rerunning Conditional κ = 0.001734865178472710998233 (+0.00109%): pooled R438 slack across bit replicas on #3934, complex binds again #3939's role audit;
  • κ with the bit side binding.
• 9/9 mutation controls rejected: over-budget donation, hosted-frame conflict, a non-commuting axis, a foreign axis, an axis moved twice, a receiver that also donates, a dropped move, the next root grid point, and κ above the bit cap.
• Determinism: two builds give byte-identical BINS and MOVES.
• Registered replay: python3.12 scripts/research_package.py --package research/rank7-idle-frame-select → PASS_REGISTERED_RESEARCH_PACKAGE.

Scope

New hypothesis: the hosted cell-to-axis map is free. This is stronger than #3190's prefix convention. It also inherits #3934's hosted receivers, #3190's hosted donors and #3939's cross-replica bit owners.

Not rerun: native admission, the overlay-aware T74 controller audit, and the outer release. All inherited obligations remain.

Done with help from OpenCode (Space bunny free) for ideas in the stacked predecessors. Prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy

[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.