Add staggered dual-paired rational-center and bit producers (κ = 8.032517e-5 complex-only / 8.800233e-5 joint)

[Th0rgal]

Summary

Supersedes #104 (research/stopped-product-rational-centers) and #107 (research/reversed-rational-centers) by replacing the binary-tree prefix/suffix schedules inside the $(h, d) = (24, 24)$ rational-center complex producer and the $h = 23$ positive-label bit producer with staggered dual-paired linear-chain schedules (research/staggered-rational-centers).

Both the complex-only bound (holding #104's frozen stopped-product-bit-axis.json and parent.AB = 803380799 / 10^13 fixed) and the joint bound (regenerating both the $h=23$ positive-label bit producer and the $h=24$ rational-center complex producer from source through the unmodified C++ maximum-cardinality bipartite matchers) are verified in research/staggered-rational-centers/verify.py:

Metric	PR #104 (stopped-product-rational-centers)	PR #107 (reversed-rational-centers)	This PR (Complex-Only)	This PR (Joint Bit + Complex)
Complex height / degree $(h, d)$	$(24, 24)$	$(24, 24)$	$(24, 24)$	$(24, 24)$
Complex addition gates $c$	$47,796$	$47,796$	$48,958$	$48,958$
Complex non-output non-input gates $q$	$6,958$	$6,958$	$8,120$	$8,120$
Complex matched pairs (match_complex_general)	$9,836$	$9,836$	$18,038$	$18,038$
Complex recursion roles $R = c + q - 2\cdot\text{matched}$	$44,918$	$44,918$	$39,040$ ($-5,878$)	$39,040$ ($-5,878$)
Complex rank sum / $\ell$	$976,842$ / $552$	$975,786$ / $552$	$938,064$ / $552$	$938,064$ / $552$
Certified complex floor $b$	$7.798287495 \times 10^{-5}$	$7.801725496 \times 10^{-5}$	$8.801801147 \times 10^{-5}$	$8.801801147 \times 10^{-5}$
Bit $h=23$ roles $R_{\text{bit}} = c + q - 2\cdot\text{matched}$	$38,776$	$38,776$	$38,776$ (inherited)	$34,014$ ($-4,762$)
Certified bit saving $AB$	$8.03380799 \times 10^{-5}$	$8.03380799 \times 10^{-5}$	$8.03380799 \times 10^{-5}$	$8.921551719638 \times 10^{-5}$
Headline certified saving $\kappa$	$7.7950234824805 \times 10^{-5}$	$7.7984487412405 \times 10^{-5}$	$8.0325171150015 \times 10^{-5}$	$8.8002329264725 \times 10^{-5}$

• Complex-only gain: $+2.374936 \times 10^{-6}$ ($+3.05%$) above PR Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104 and $+2.340684 \times 10^{-6}$ ($+3.00%$) above PR Reversed rational-center orders: conditional κ = 7.798412662809e-5 #107.
• Joint regenerated gain: $+1.005210 \times 10^{-5}$ ($+12.90%$) above PR Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104 and $+1.001784 \times 10^{-5}$ ($+12.85%$) above PR Reversed rational-center orders: conditional κ = 7.798412662809e-5 #107.

Key Mechanisms

1. Staggered Dual-Paired Prefix/Suffix Strip Schedules (StaggeredPairedTriple):

   • At even indices $i$, prefix advances via paired sums add(prefix[-2], add(values[i-1], values[i])); at odd indices $i$, suffix advances via paired sums add(suffix[-2], add(values[n-1-i], values[n-i])).
   • Every intermediate gate in a paired step has one fan-in at odd distance and one fan-in at even distance in the linear chain, making all paired partial sums eligible under match_complex_general.cpp (non_overlap, valid_S, valid_T) and match_positive_dag.cpp.
   • At the top level of two[u,v] (len(values) == 11, len(vals) == 10), keeping prefix unpaired while pairing suffix at odd indices $i \in {1, 3, 5, 7}$ preserves all 10 suffix[i] nodes in lookup[s] (so pre[2], pre[4], \dots, pre[20] in complex.py hit lookup with zero duplicate gates) while matching (suffix[2m], suffix[2m-1]) via suffix[2m+1] and q_m = add(vals[2m-1], vals[2m]) into prefix[2m+1] via vals[2m].

2. Linear-Chain Block, Coarse, and Coefficient Aggregations (rtotal):

   • Replaces balanced binary trees in block, coarse, and coeff with reverse linear fold rtotal while keeping binary-tree total inside result[excluded] so 1-, 2-, and 3-survivor inclusion-exclusion partial sums remain shared.

3. Leave-One-Pair-Out Readout Bridge & Same-Group Pair-Star Prefix Pairing:

   • In complex.py (StaggeredComplexProducer), when (a ^ 1) != b and 2 <= j < 18 and (i // 2 < a // 2 or (j // 2) % 2 == 0), missing[p][q] is assembled via the leave-one-pair-out bridge add(add(pre[jj], suf[jj + 2]), w[jj ^ (1 - (j & 1))]) (jj = j & ~1), aligning with the odd-step suffix pairs from two[u,v] and eliminating 680 high-rank gates.
   • For the 12 same-group pair stars (a ^ 1) == b (skipped by two[u,v]), pairing pre at odd idx_w < 20 reuses add(w[idx_w-1], w[idx_w]) from suf via lookup with zero extra gates and adds 54 matched pairs.

Verification

python3 -B research/staggered-rational-centers/verify.py
python3 -B research/staggered-rational-centers/test_controls.py
python3 -B research/reversed-rational-centers/verify.py
python3 -B research/reversed-rational-centers/test_controls.py

Prepared by Thomas Marchand with Google Antigravity assistance.

[Mister-Meeseeks]

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

Contribution: Staggered prefix/suffix pairs with compatible supports.

Review and limits: Exact-head CI evidence; joint and complex-only snapshots remain distinct.

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: 446dd07dbe14f6dcfcca629f3a9cb01cb148f003. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.