Conditional κ = 0.001733427039589844234749 (bit-bound): idle-remainder donors on #3148's T740 fill (+0.187% over #3184)

[rohanarun]

Conditional κ = 0.001733427039589844234749 − 10⁻⁴⁶ − 10⁻⁷⁰. This is +3.238e-6 (+0.187%) over #3184 and +3.650e-6 over #3148, using #3148's own words, helpers and fill rows.

	complex root lower (10⁻²⁵)	binds	κ
#3148	0.0017297774946785970118869	complex	0.001729777494678597011886 − …
#3184	0.0017301890612338507600957	complex	0.001730189061233850760095 − …
this	0.0017347705571115131604951	bit (#589)	0.001733427039589844234749 − …

The complex supplier now exceeds the #589 bit saving (0.001733427039589844234749, as pinned in the T74 outer ledger), so κ is capped by the bit side.

Changed hypothesis

#3184 moved axes only among completely idle capacity-48 children. Every partially used bin also leaves an idle remainder child, whose axes are the recipe axes [used, cap). The hosted IDLE-REMAINDER-FRAMES on #1094 equal orthogonal_axes[hosted:].

#3148's idle bill is {1:173160, 3:48951, 5:13875, 48:282569}. Removing a small child is far more valuable than reshaping 48s. Moving one axis from a rank-1 remainder into a 48 saves 1·ln 73 of moment weight; a whole 48+48 → 45+51 pair saves 0.18.

Here any remainder, hosted or not, may donate axes to a fully idle, unhosted receiver in the same replica and port. Each moved axis follows #1252's rule:

• it commutes with the receiver's current axes;
• it commutes with all stage entry/exit X and Y translations and the terminal Y translation;
• it commutes with both terminal translations.

Receivers are capped at 51 (5329 > 5202). Moved axes are orthogonal to the donor's hosted prefix, and group axis multisets are unchanged.

The construction has two phases:

1. Drain remainders, smallest first, into the largest legal receivers.
2. Run Conditional κ = 0.001730189061233850760095: idle rank-48 star exchanges on #3148's T740 fill (+0.0238%) #3184's exact stars on the untouched 48s.

That gives 391,964 axis moves (138,369 donors into 167,112 receivers). The idle bill becomes:

{1:149514, 2:15880, 3:14951, 5:13855, 42:32303, 45:34720, 48:48434, 49:47026, 50:15320, 51:104766}

Rank mass is unchanged at 65088·740.

Verification

build.py (about 20 s) pins DATA-PHASES, the #3148 fill payload, ORDER, DISPATCH and the provider widths. It reproduces #3148's idle bill exactly.

verify.py (about 1 min) is independent of build.py. It re-derives:

• that donor axes come from the donor's own remainder;
• that receivers are unhosted, and no child both donates and receives;
• the unit and orthogonal-at-receipt checks, and the commuting rule;
• the rank cap;
• the profile and rank mass;
• the complex root bracket, using its own 10⁻⁷² interval arithmetic;
• κ = min(floor₂₄(complex), bit) − 10⁻⁴⁶ − 10⁻⁷⁰;
• gates against Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148 and Conditional κ = 0.001730189061233850760095: idle rank-48 star exchanges on #3148's T740 fill (+0.0238%) #3184.

It also rebinds against the fill. Eight mutation controls are rejected: a hosted receiver, an axis from the hosted prefix, 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.

python3 -B research/rank7-idle-remainder-donors-t740/verify.py --t740 /tmp/t740   # /tmp/t740 from #3148's materialize.py

Scope

This is an author-checked conditional candidate.

• New hypothesis beyond Idle pair exchanges improve the #1094 κ candidate #1252/Conditional κ = 0.001730189061233850760095: idle rank-48 star exchanges on #3148's T740 fill (+0.0238%) #3184: hosted remainders donate. No hosted-controller argument is given here; Idle pair exchanges improve the #1094 κ candidate #1252's admitted audit covered unhosted pairs only.
• Not rerun: Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148/research(rank7-coupled-586-588-carrier-fold): lower helpers 586 & 588 via coupled B+restore carrier folds (κ = 0.0017297760894) #3133's native admission, the overlay-aware controller audit and the outer release. The outer ledger asserts complex < bit, so it needs a bit-binding branch, and the halving pin must become [5329,5202].
• Since the bit side now binds, more complex packing does not raise κ until the bit supplier improves.
• All inherited obligations remain. Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148 is unreviewed.

Credits

• Gonzalo-Mier (Idle pair exchanges improve the #1094 κ candidate #1252): the move, the rule and the pricing.
• Guéorgui Maissouradze: the star class.
• Jeremy Diviney (Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148): the fill.
• 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).
• Jeremy Kothe (Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015).
• Nick Petalas (Attain the joint rank-seven packing cap exactly at T740 #3024).
• DreamingOfClouds and eumemic (Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589/Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555): the bit supplier.
• The data-phase and T37 lineage.

Prepared by Rohan Arun. Done with help from OpenCode (Space bunny free), whose partially-used-bin centre idea this generalizes, and with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy