Conditional κ = 0.001730189061233850760095: idle rank-48 star exchanges on #3148's T740 fill (+0.0238%)

[rohanarun]

Conditional κ = 0.001730189061233850760095 − 10⁻⁴⁶ − 10⁻⁷⁰. That is +4.11566555e-7 (+0.0238%) over the open #3148 (0.001729777494678597011886 − …), using #3148's own words, helpers and fill rows.

	complex root bracket (10⁻²⁵)
#3148	[0.0017297774946785970118869, …8870]
this	[0.0017301890612338507600957, …0958]

Changed hypothesis

#3133's RELAX-CAP and #3148's cap-attaining fill both charge one idle child per bin, of rank capacity − used. They never move idle capacity between bins. #3148's fill leaves 282,569 capacity-48 children completely untouched, in 82,880 (replica, port) groups.

#1252 (Gonzalo-Mier) showed on #1094 that two such children can exchange three axes, 48+48 → 45+51. The rule: each moved axis commutes with the receiver's axes, with every stage entry/exit X and Y translation, with the terminal Y translation and with both terminal translations. The group's axis set, and hence its Z4 phase polynomial, is unchanged. frontier1094-idle-star-ceiling extended this to stars.

The axis data is on main. The rank-seven T74/T740 lineage uses the same DATA-PHASES.json (c5916ff3…) as #1094. Its per-port idle children carry the orthogonal axis sets. For all 4,866 #1094 idle rank-48 children, these equal the hosted IDLE-REMAINDER-FRAMES that #1252 used, so legality is port-local.

Result

An exhaustive per-group star optimum finds 125,706 moves and changes 205,869 children:

Shape	Groups
[51,51,48,42]	37,080
[48,48] (no movable axes)	19,083
[51,51,45,45]	16,780
[51,51,42]	8,463
[51,48,45]	1,040
other	434

• The idle bill {48: 282569} becomes {42: 45543, 45: 34620, 48: 76700, 51: 125706}.
• Rank mass 65088·740 is unchanged.
• The halving guard holds: 73² = 5329 > 2·51² = 5202.
• The complex side still binds; Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's bit saving is 0.0017334….
• The within-group ceiling is 0.00173069…; stars capture 45% of it.

Verification

• build.py authenticates DATA-PHASES and Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148's decoded fill payload (8f4d6965…). It reproduces Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148's published root bracket exactly for the baseline profile.
• verify.py is independent of build.py. It re-derives:
  • every axis's commuting rule;
  • disjoint stars and axes;
  • orthonormal, independent final axis sets;
  • unchanged group axis multisets;
  • the rank cap and halving guard;
  • the profile and rank mass;
  • the root bracket, with its own directed-rounding interval arithmetic;
  • the κ gates over Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148 and Certify precision-40 κ above PR #3133 #3143.
• Nine mutation controls are rejected: non-commuting axis, axis moved twice, reused leaf, foreign axis, centre that is also a leaf, hosted child, two-axis move, wrong profile, next root grid point.

python3 -B research/coupled-carrier-cap740/materialize.py --output /tmp/t740   # on #3148
python3 -B research/rank7-idle-star-exchanges-t740/verify.py --fill /tmp/t740/COMPOSED-FILL.json.gz

Scope

This is an author-checked conditional candidate. 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 full native admission, controller/owner audit and outer release with the axis overlay. The T74 audit_mixed.py must read idle overlays, as Idle pair exchanges improve the #1094 κ candidate #1252 adapted the Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 auditor.
• Re-pinning the halving check from [5329,4608] to [5329,5202].

Star soundness is inherited from #1252's pair argument. All inherited all-size/compiler, realization, precision/tape, recovery and analytic obligations remain. #3148 is itself unreviewed; the same overlay applies to #3133's T74 fill.

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 T740 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).
• The retained data-phase and T37-envelope lineage.

Prepared by Rohan Arun. Done with help from OpenCode (Space bunny free), which identified the idle-exchange lever and the port to the rank-seven line, and with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy

[rohanarun]

Retracted. Merged #3187 shows the first committed star exchange changes the exact physical operator: at character 16, input Y0 leaks −1−i into output Y1. #3296 finds the operator changes in every star port at owner 0. The full explanation is in #3251. Closing this together with #3190, #3198, #3200 and #3251.