Deferred readouts on both stopped networks: conditional κ = 8.58478e-5

[ikeboy]

Change and mathematical scope

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{429239}{5\cdot 10^{9}}=8.58478\times10^{-5}.$$

This is +10.1% over #104 (7.79476e-5). It builds on #104 (head 854ba7d, included in this branch) and keeps all of its inherited hypotheses. The next κ grid point at denominator 10^10 is rejected.

Idea. #104 compiles every projector residual of rank r as one child of width r. Under that rule, Swapnil Jain's deferred readouts pay on both networks. A slot that reads its garbage out at a start frame σ_u starts its chain at σ_u instead of 0. Its entrance child shrinks by dim σ_u, and its exterior child grows to m − h + dim σ_u. The rank moves into a child near m, where it costs less.

	#104	here
bit network	positive-label producer, R = 38,776	Swapnil's round-seven deferred word, R = 28,866, unchanged
stopped bit saving	8.034e-5	1.2402e-4
complex network	aligned rational centres, R = 44,918	same DAG and roles, lifted binary frames, 8,893 deferred roles
complex saving (binding)	1949/25000000 = 7.796e-5	214657/2500000000 = 8.58628e-5
κ	7.79476e-5	8.58478e-5

Complex construction (new). On #104's unchanged h = 24 DAG:

• a maximum carrier matching under the inherited adjacency rules (8,966 links, R = 44,918);
• an explicit role compile;
• lifted binary frames: the F2 kernel of the functionals of every reachable root (t_S for outputs, e_i for centre A_i). The compile uses a lifted frame where it is nondegenerate and keeps the frames monotone along slot successors;
• phase one: the closure of the 24 centre completions;
• deferred frames σ_u ⊆ F0(u) ∩ ⋂ t_S^⊥ for roles that phase one does not touch and whose garbage does not reach a centre. An insertion rule nests them on every target.

The stage-one word follows Swapnil's design B. The readout coefficients come from the transpose sweep. Their only odd denominator is #104's 21.

Bit network. bit_round7.py derives the bit child list from Swapnil's two frozen round-seven data files, pinned by SHA-256. Their rank mass, 57,403,754,177, is the value that #97 reports.

Validation

Finite tests (research/deferred-stopped/complex_deferred.py):

• The compiled word computes all 8,120 root values.
• The deferred word, replayed over Z/(2^61−1) with arbitrary scratch and data (two seeds), restores every role and adds exactly x_S to every target.
• Exact F2 checks:
  • every role chain is nested;
  • every lifted frame and every deferral frame is nondegenerate;
  • every σ_u lies in its start frame and in t_S^⊥ for each reached target;
  • the deferral frames are nested on every target in readout order.
• The one-child histogram has total rank W m − N + L = 109,450,358,336 (as in Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104). Every child is below m (the largest is 574 of 576).
• certificate.py computes both exact moments with Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's routines, the stopped saving (1 − 1/1000)·coarse + (1/1000)·384599/10^10, the finite bridge and the 47-constraint assembly.

Proof arguments, not machine-checked here:

• Stage two is the complement time reversal of stage one with the same edge multiset, as in Swapnil's round seven and Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's two-stage counts. Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97 replays the reflected ledger for the bit word. For the complex deferred word, this PR states stage two but does not machine-check it.
• Deferred auxiliary endpoints use the translated gauges of notes/copied-centers-complex.tex and notes/endpoint-gauge-complex.tex (PR21/PR24).
• The larger children (528 of 529 and 574 of 576) raise the product-row stock to p^33000 (PR29 used p^47000). As in Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104 and Merged auxiliary exteriors: conditional κ = 5.5354656014473e-5 #96, the stock changes only the row slack and the cutoffs, not κ.
• All of Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's inherited hypotheses. See research/deferred-stopped/PROOF.md.

Reproduce:

python3 research/deferred-stopped/complex_deferred.py
python3 research/deferred-stopped/bit_round7.py DIR   # Swapnil's round-seven files or #97's copy
python3 research/deferred-stopped/certificate.py

DIR holds witness_23.json.gz and deferred_23.json.gz: certificates/round7/ of https://github.com/Swapnil-jain/integer-mult-kappa at commit 741e7aa078392553815df7926ee17ac5e25a8c38, or #97's copy in research/deferred-signed/swapnil-round7/certificates/round7/.

Credits

• icekylinx: the stopped product-ring interchange, opposite-bank factorization, rational centres and the h = 24 producer (Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104).
• Swapnil Jain: deferred readouts, V leaves, lifted frames and the round-seven bit witness.
• Zhihao Chen (jacklightChen): Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97's deferred bit integration and reflected ledgers, and the PR21/PR29 gauges.
• Aurel Prosz (Paureel): the two-stage topology.
• Everyone credited in Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's NOTICE and SOURCES.json.

Avi Eisenberg (ikeboy) prepared the complex deferral construction and this certificate with Anthropic Claude assistance.

🤖 Generated with Claude Code

[Mister-Meeseeks]

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

Contribution: Deferred complex stages plus stopped-depth bit accounting.

Review and limits: Fresh complex, supplied-input bit and assembly arithmetic retained. Original full complex stage-2 proof obligation is not discharged by randomized modular tests. Historical conditional research only; stale Makefile-pin CI failure and corrected local invocation recorded.

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: 09775e9f3fbf1900175010bcd0c3d6d1dbf65ffe. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.