Conditional κ = 0.001734807021726127552033: greedy idle-axis phase on #3198 with the R438 bit supplier (+0.0021%)

[rohanarun]

Conditional κ = 0.001734807021726127552033 − 10⁻⁴⁶ − 10⁻⁷⁰, +3.646e-8 over #3198.

	complex root lower	bit	binds	κ
#3198	0.0017347705571115131604951	R438 0.001734846264120199151475	complex	0.001734770557111513160495 − …
this	0.0017348070217261275520343	R438 0.001734846264120199151475	complex	0.001734807021726127552033 − …

The complex side still binds, now 3.9e-8 below R438.

Change

#3198 left the complex side 7.6e-8 below the R438 bit endpoint. This PR keeps #3190's two phases (hosted remainders drained into the largest legal unhosted receivers, then #3184's exact stars on the untouched 48s). It adds a third greedy phase: repeat the single best one-axis move from a smaller unhosted child to an equal-or-larger unhosted receiver, as long as it lowers the moment cost.

Every move obeys #1252's commuting rule. Receivers are capped at 51, and no child both donates and receives. The result is 430,164 moves, with rank mass 65088·740 unchanged.

Variants tested:

variant	complex root
phases 1 + 2 (#3190)	0.0017347706
1 + 2 + 3	0.0017348070
1 + greedy only, no stars	0.0017347935

Fully idle cap-32/16/8 children don't exist in this fill. Rank-1 remainders, the last coordinate of a 48-bin, conflict with every other 48's axes and stay blocked.

Verification

verify.py (about 70 s, independent of build.py) re-derives:

• every move's donor-remainder membership, receiver hosting, roles and commuting rule;
• the rank cap;
• the profile and rank mass;
• the exact complex root bracket;
• the R438 endpoint, from its certificate's bit profile with the fallback envelope;
• κ = min(floor₂₄(complex), bit) − …, with the complex side binding;
• the gates over Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148, Conditional κ = 0.001730189061233850760095: idle rank-48 star exchanges on #3148's T740 fill (+0.0238%) #3184 and Conditional κ = 0.001734770557111513160495: R438 bit supplier with #3190's idle-remainder donors (complex binds again, +0.0775%) #3198.

Eight mutation controls are rejected. scripts/research_package.py passes locally under Python 3.12.

Scope

This stacks on #3198, so the diff includes #3190's package and its registration fix plus the R438 composition. It is conditional on that stack and on R438 (#1043 is open). Hosted donors and multi-donor receivers rely on #1252's per-axis rule; only unhosted pairs have an admitted controller audit. Native admission, the overlay-aware controller audit and the outer release were not rerun.

Credits

• Gonzalo-Mier (Idle pair exchanges improve the #1094 κ candidate #1252)
• Guéorgui Maissouradze (star class)
• Jeremy Diviney (Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148)
• dcharb78 (research(rank7-coupled-586-588-carrier-fold): lower helpers 586 & 588 via coupled B+restore carrier folds (κ = 0.0017297760894) #3133, Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042)
• Marius-Paul Dumitrean (Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979)
• Ali Romman (Finite BIT construction: R438 source boundary with full hosted invoice #1043) and Sagar Thakrar (Bit construction (no κ; conditional on #1043): R438 + one transport + optimal window-respecting re-hosting → c = 0.0017348463 (+0.082% vs #589); fixes two #1345 atlas crashes #2423)
• the r438-native-rehost authors
• DreamingOfClouds and eumemic

Prepared by Rohan Arun. Done with help from OpenCode (Space bunny free), with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy

[rohanarun]

Retracted. This PR reuses #3190's idle-remainder donations, and #3296 (merged, independently confirmed by #3311) shows those change the exact eight-stream physical operator. No operator-preserving replacement exists in this move class at any owner replica: 0 of 266,732 single-axis transfers survive, and the multi-copy search has found none. The full explanation is in #3251. Closing this together with #3184, #3190, #3198 and #3251.