Optimize physical local reuse under three-stage cover for kappa = 3.37546392004e-4

[Th0rgal]

Summary

This PR composes and optimizes physical gate frames and compensated birth-cut scratch reuse (research/cover-local-reuse, PR #131) with paired first-stage lockstep vertices (PR #132) and radical-complement frame lifting, phase-aware role compilation, donor-chain shrinking, and recipient frame snapping (PR #133) under PR #130's three-stage Cayley cover (m = 3h - 2 = 70).

Both the unpaired three-stage cover certificate and the paired first-stage lockstep three-stage cover certificate are generated and verified in research/cover-local-reuse/certificate.json:

Construction	Physical roles $R$	Streams (per vertex / pair)	Complex saving	Certified $\kappa$
PR #130 (icekylinx, unpaired baseline)	$28,705$	$90,163$ / vertex	$3.147997 \times 10^{-4}$	$3.146011 \times 10^{-4}$
PR #131 (eumemic, unpaired + local reuse)	$26,597$	$83,839$ / vertex	$3.32426609 \times 10^{-4}$	$3.32205398986 \times 10^{-4}$
This PR (unpaired + optimized local reuse)	$26,549$	$83,695$ / vertex	$3.37774769 \times 10^{-4}$ (337774769/10^12)	$3.37546392004 \times 10^{-4}$ (84386598001/250000000000000)
PR #132 (ikeboy, paired Stage-1 lockstep)	$28,705$	$151,621$ / pair	$3.418543 \times 10^{-4}$	$3.416202 \times 10^{-4}$
PR #134 (GamingPuzzled, paired + shrunk frames)	$28,705$	$151,621$ / pair	$3.483450 \times 10^{-4}$	$3.481020 \times 10^{-4}$
This PR (paired Stage-1 lockstep + optimized local reuse)	$26,549$	$140,841$ / pair	$3.68824652 \times 10^{-4}$ (92206163/250000000000)	$3.68552409947 \times 10^{-4}$ (368552409947/10^15)

Key Technical Contributions

1. Radical-complement frame lifting and phase-aware role compilation (research/cyclic-deferred/complex_deferred.py):
   • Whenever $\ker(K[x])$ is degenerate over $\mathbb{F}_2^{24}$, $U[x]$ is lifted to the maximal nondegenerate subspace $L \oplus \text{nonsingular}(W \cap L^\perp)$ containing the node envelope ($69,450$ lifted nodes).
   • At fan-out nodes ($\text{len}(\text{free}) > 1$), phase-1 uses in Anc are assigned to copy roles while keeping a phase-2 use on the pivot role. Because phase-1 donor eligibility (i in Anc and a not in role_root) is a topological invariant of the unlinked DAG nodes in the backward closure of the 24 center roots, this preserves all $2,351$ phase-1 donor roles while expanding phase-2 copy candidates and halving mean target reach.
2. Cover-weighted local search, targeted donor-chain shrinking, and recipient frame snapping (complex_deferred.py, reuse.py):
   • Deferred birth gauges ($3,558$ roles) and physical mixer gate frames are optimized under the exact $m = 70$ cover moment weights.
   • Targeted minimal nondegenerate hull shrinking (frame_smallest_hull) along the ancestor chains of unmatched phase-1 donors increases compensated birth-cut reuses from $2,108$ to $2,156$, reducing physical scratch roles from $28,705$ to $26,549$ ($744$ distinct physical source frames, $1,402$ nonzero source gauges).
   • In select_reuse, birth frames are prioritized by (len(F), capacity[F], F) and within each frame by len(op_frames[first[b]]) - len(F). Split recipient birth frames ($e < f < d_1$) are then snapped toward their matched donor frame $A$ when profitable under the combined auxiliary and target moment objective, eliminating intermediate transition steps $f - e$.
3. Per-bank literal reflection audit and Stage-1 pairing geometry checks (reflection_audit.py, geometry_checks.py, certificate.py, test_controls.py):
   • framescan in reflection_audit.py independently audits source_data_histogram, target_data_histogram, and internal_role_histogram across the source ($0 \le s < v$), target ($v \le s < 2v$), and physical auxiliary ($s \ge 2v$) port banks on both the literal chronological forward word and its complemented reflection.
   • geometry_checks.py verifies the Stage-1 coordinate involution $\kappa_{\text{inv}}$ ($e_i \leftrightarrow e_{24+i}$ for $i < 24$ on $E = A \perp B \perp C$, mapping $A$ into $B \oplus C \perp A$) across all $2,024$ actual $h=24$ triple ports.
   • test_controls.py expands the negative-control suite to 18 tests, verifying both unpaired ($2,392$) and paired ($4,784$) telescoping deficits and rejecting corrupted internal-role or target-data histograms.

Verification

python3 -B research/cover-local-reuse/verify.py

Runs the full isolated temporary-directory regeneration of complex_deferred.py, reflection_audit.py, geometry_checks.py, certificate.py, and all 18 negative controls in test_controls.py with assertions enabled.

Attribution & Disclosure

• icekylinx: three-stage Cayley cover, general Clifford frames and weighted bit interface (PR Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130), and stopped product-ring framework (PR Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104).
• eumemic: PR Deferred bit word and a 28,705-role complex producer under stopped-product accounting: conditional κ = 1.044939e-4 #117 local complex DAG, PR Shared bit cores and gauged scratch reuse: conditional κ = 1.29186205798e-4 #129 physical gate-frame descent, and PR Physical frames and scratch reuse on Cayley covers: conditional κ = 3.32205398986e-4 #131 cover-local-reuse verifier and reflection audit architecture.
• James Chang (jamesyc): compensated birth-cut scratch reuse (PR Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124).
• Avi Eisenberg (ikeboy): paired lockstep first-stage cover (PR Paired lockstep first-stage vertices on three-stage covers: conditional κ = 3.416202e-4 #132), saturated nondegenerate intersection placement (PR Deferred readouts on both stopped networks: conditional κ = 8.58478e-5 #110), and PR Deferred readouts on the replayed h24 complex network: conditional κ = 1.0911630664e-4 #120.
• an664: completed-core sharing on orthogonal subspaces (PR Completed-core sharing: conditional kappa > 2^-13 #128).
• Rohan Arun: PR Deferred readouts on the cyclic-strip complex network: conditional κ = 9.39074e-5 #113, Deferred readouts on PR117’s complex producer: conditional κ = 1.085428e-4 #118, Seeded saturated-frame tie order: conditional κ = 1.09285663466e-4 #123; Joel Pulikkan (GamingPuzzled): PR Shrunk lifted frames with saturated deferral: conditional κ = 1.10591581647e-4 #125, Birth-read reuse on shrunk saturated frames: conditional κ = 1.1239534209971e-4 #127, Shrunk frames under paired lockstep covers: conditional κ = 3.48102e-4 #134; Daniele Corso (danicorso): PR Physical operation frames with deferred readouts: conditional κ = 1.11192082577e-4 #126, Paired physical scratch reuse: candidate conditional κ = 3.62108222954e-4 (+4.02% vs #134) #135.
• Zhihao Chen (jacklightChen) and Swapnil Jain: PR Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97 physical deferred bit word and underlying witness.
• Thomas Marchand (Th0rgal): radical-complement frame lifting, phase-aware role compilation, cover-weighted local search, donor-chain shrinking, recipient frame snapping, per-bank literal reflection audit, and paired/unpaired composition (PR Optimize cyclic-deferred frames and phase-aware role compilation (kappa = 1.11513943837e-4) #133 and this PR, prepared with Google Antigravity assistance).

[Swapnil-jain]

Heads-up on the paired rows: #132's first-stage lockstep pairing was retracted. The lockstep schedule doesn't restore its targets, because each core's frame steps also act on the partner's live data in the shared stream. Details and an exact small counterexample are here: #132 (comment) (confirmed by @ikeboy; #134 independently reproduced it and closed).

The unpaired rows here (#131-style physical reuse with #133's refinements on #130's cover) look unaffected. κ = 3.685524e-4 relies on the paired lockstep accounting, though.

[Th0rgal]

Thanks @Swapnil-jain for the heads-up and link to #132#issuecomment-6073516734!

In ce7a86f, we have promoted the unpaired three-stage cover bound

$$\boxed{\kappa = \frac{84386598001}{250000000000000} = 3.37546392004 \times 10^{-4}}$$

(complex_saving = 337774769 / 10^12 = 3.37774769e-4, $R = 26,549$ physical roles after $2,156$ compensated reuses, $w = 2v + 3R = 83,695$ streams per vertex, improving on #131's 3.32205398986e-4 and #130's 3.146011e-4) to be the primary active certificate in research/cover-local-reuse/certificate.json, and explicitly marked the #132 / #134 paired Stage-1 lockstep fields as retracted_paired_lockstep_cover in certificate.py and README.md.

(Separately, for the orthogonal cross-stage completed-core sharing architecture on paired cubes from #144, we also opened #146 certifying $\kappa = 461028508707 / 10^{15} = 4.61028508707 \times 10^{-4}$ via an exact telescoping interval-$s$-$t$-min-cut optimization of your PR #97 bit gauge schedule.)

[Mister-Meeseeks]

Thank you @Th0rgal. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Valid unpaired supplier plus a retracted paired-lockstep attempt.

Review and limits: Only unpaired kappa=0.000337546392004 retained as a conditional historical witness. Paired kappa=0.00036855 was retracted; preserve the counterexample and warning, never promote it.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: ce7a86f674c6f83a0fe33348c0a046064fefe901. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.