Deferred bit word and a 28,705-role complex producer under stopped-product accounting: conditional κ = 1.044939e-4

[eumemic]

Conditional improvement

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{1044939}{10^{10}}=1.044939\times10^{-4}.$$

This is +5.87% over #116, the best open PR when this was opened (9.8699350338e-5), and +21.72% over #110 (8.58478e-5). It is +34.06% over #104, its base. The complex side binds. All of #104's retained analytic and tape interfaces remain assumptions.

#110 independently uses the same bit-side composition: Swapnil Jain's round-seven deferred words under #104's one-child rule, with stopped bit saving 1.2402e-4. #113, #114 and #116 reuse that bit side. This PR differs from them only on the complex side.

Change

• Bit side: Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97's deferred words under Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's one-child accounting.
  • Every charged residual is a nested projector residual, so each counts as one child.
  • Those residuals are: slot chains, gauged exits I−(I−σ)⊗Q, center copies, y_T and X_S chains, data connectors and copy corrections. Garbage readouts are scalar gates.
  • Coarse saving is 620523/(5·10^9); ordinary saving after Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's stopping conversion is 1.2401896e-4.
• Complex side: a new h=24 all-disjoint rational-center producer in Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's framework.
  • It uses 28,705 roles, against 44,918 for Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's (and Deferred readouts on both stopped networks: conditional κ = 8.58478e-5 #110's) producer, so a_c = 26129/(2.5·10^8).
  • It ships as a 91,770-addition DAG. Supports, labels and ranks are replayed from the operands, and Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's unchanged matcher rebuilds the row exactly.
• Bridge and assembly: Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's, at PHASE_STOP = 10^-6.
  • The largest bit child is 528, so the halving degree is 367 and the row stock is p^22000. This only affects eventual cutoffs.
  • All 47 constraints and 7 margins hold, and the next κ grid point is rejected.

Preconditions and scope are in research/deferred-product/PROOF.md.

Verification

make deferred-product-verify

This target does the following:

• Runs Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97's literal ledger and Swapnil Jain's check_word.py and check_frames.py.
• Replays the complex DAG.
• Replays the complete signed two-stage complex word at gate level: 0 violations, continuous reflection, and a histogram equal to the row's profile. The 3,308 alternating rank-2 residuals use Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104's one-child Gauss normal form.
• Recomputes the one-child list, both moments and the assembly.

A full make verify passed on Linux (EC2) with a clean git diff.

Credits

• Bit network: Swapnil Jain (round seven).
• Ledgers and integration: @jacklightChen (Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97).
• Stopped-product accounting, framework and assembly: @icekylinx (Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104), on its retained lineage.
• The same bit-side composition, found in parallel: @ikeboy (Deferred readouts on both stopped networks: conditional κ = 8.58478e-5 #110).
• Complex producer and this integration: eumemic, with Anthropic Claude assistance.

🤖 Generated with Claude Code

[Mister-Meeseeks]

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

Contribution: Replayed complex producer and stopped bit supplier.

Review and limits: Exact-head CI evidence retains 91,770 additions and 28,705 complex roles; full composition rather than isolated savings.

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: cbb05ce504d571546d9b7794c186a613c659c3bf. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.