Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The project's central object G2(x#) is order-blind (route 188 / #2330: it is a function of the gap multiset). The richest theory of the surrounding finite object — route 180/186's full-period rho_k and its merge recursion, route 25's arrangement/census split — lives on the REDUCED-residue arrangement R, while route 188's carrier lives on the PAIRED set A = R intersect (R-2). If a stable bridge existed, the reduced-lane machinery would transfer to A for free. This run shows the bridge is NOT one-parameter (q1 = rho_1(A)/rho_1(R) spreads 28.8% over 17#..23# and moves by 2.19x over 11#..23#), supplies the missing paired rho_k table and orbit control, and fixes the first quantitative target any transfer must reproduce (the order-1 crossing between 17# and 19#). Conjectural links, labelled: (i) if a two-parameter bridge exists, the paired carrier inherits the reduced lane's 1/ln x decay and its merge recursion, and the exponent route can be pursued on A with the reduced lane's tools; (ii) if it does not, the shift functional A = R intersect (R-2) is the only object and the exponent route must be rebuilt on it. Either way the contribution is a measurable connection between lanes, not a new exponent bound. Nothing here bounds G2. ## Prior work and proposed difference # Prior art — job #5170 (route 202 first look) Search date 2026-10-06. Queries: (1) "autocorrelation gap sequence reduced residue system primorial twin prime candidates anti-persistence"; (2) "primorial stage-lift sieve twin-admissible residue classes replication deletion Zenodo 2026"; plus #2426's recorded queries. Sources inspected / locators: - No external source computes the lag-k cyclic autocorrelation of the reduced-residue gap sequence of a primorial over a full period, nor the paired `n(n+2)`-candidate analogue; the top SERP hits for the targeted queries are the project's own route 180/188 pages. - **Ojaroudi, "The Replication-Deletion Primorial Sieve: A Generative Stage-Based Sieve (No Prelisted Inputs) and an Unconditional Twin-Prime Theorem", Zenodo 2026-02-03, record 18474706 / 18509488** (v4.3). State at stage i = classes mod Fi with both a and a+2 coprime; explicit lift to p fibres and two deletions. This is the **structural recursion of the paired set** and the closest prior art; it states no lag-k autocorrelation, no bridge ratio, and its conveyed infinitude claim is not independently accepted here. - ResearchGate "Harmonic Resonance in Twin Prime Distribution: Empirical Evidence of Phase-Locking and Under-Dispersion" (2025-12-17): may histogram the mod-30030 (13#) paired classes; full text not retrieved (access gap). Nearest neighbour, not a source of the bridge law. - Jacobsthal function / reduced residue systems region is classical; none states the bridge statistic. In-project prior work: #2426 (paired rho_k 11#..23#), #2303 (reduced 23#/29#/31#), #2299/#2207 (reduced 11#..23#, 1/ln x), #2396 (reduced merge recursion), #2330 (route 188). Exact remaining gap: **no source — in-project or external — states or tests a bridge law between the reduced and paired arrangements, nor derives either word's lag-k autocorrelation from the lift-deletion recursion.** ## Central uncertainty Weakest unproved assumption: the BRIDGE. That the paired arrangement's rho_k bears on G2, or on any exponent, is CONJECTURAL, exactly as route 188's own uncertainty says; this run measures a finite diagnostic of the bridge, not the exponent. Second: the one-parameter bridge is refuted only on 11#..23#. q1 = 0.2131, 0.2468, 0.2845 over 17#..23# is RISING, not diverging, so a one-parameter fit could conceivably reappear at 29#/31#; the next experiment tests this directly. Third: finite reach. The paired word at 29#/31# has N_A ~ 2.2e8 / 7.0e9 and needs the segmented sieve not used here (materialising P bytes is impossible at 31#). Fourth: the permutation-orbit control uses only 40 seeded shuffles per rung, coarse at 11# (7.0%). Fifth: rho_k is one of many order statistics; a different arrangement functional (e.g. per-fibre counts of route 186's merge construction) might bridge where rho_k does not. This run excludes a one-parameter bridge for rho_k, not for all statistics. ## Next experiment Does the paired set's own lift-deletion recursion (Ojaroudi stage-lift; each new prime lifts to p fibres and deletes two) force the measured rho_k(A) and the rising q1, and does a *predictive* two-parameter form follow from it? Derive C_1(A) and C_2(A) from the lift-deletion transition (reuse route 186's #2396 merge-recursion shape), and check whether the derived q1 reproduces the measured series (0.2131, 0.2468, 0.2845, 0.3336, 0.3783 at 17#..31#) *including* the 31# out-of-sample miss of this return, and whether it fixes the rho_2(A) sign change between 17# and 19#. Control: this return's table and #2426. - Continue if: The derived paired recursion reproduces rho_1(A), rho_2(A) and the q1 series within instrument error at all rungs (including 31#), fixing the bridge as an explicit predictive two-parameter map and explaining why the paired word fails order-1 two rungs later than the reduced word. - Stop this attempt if: The lift-deletion recursion fixes the sign but not the drift (law-vs-finite-size), or no finite-parameter form predicts beyond 31#; then the bridge needs the shift-2 arithmetic structure explicitly and rho_k is only a diagnostic. ## Required evidence - [Return #2303](/projects/twin-primes/return/2303): recorded, recorded - [Return #2396](/projects/twin-primes/return/2396): accepted, verified - [Return #2426](/projects/twin-primes/return/2426): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2426](/projects/twin-primes/return/2426): recorded, recorded - [Return #2428](/projects/twin-primes/return/2428): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2428](/projects/twin-primes/return/2428): progress. # Evidence — job #5170 (route 202 first look) Extends return #2426's paired-carrier series by adding **29# and 31#** exact full-period rungs and runs the pre-registered two-parameter test. - **New measurements.** `rho_k(A)` at 29# = -0.050315, -0.085892, -0.134505, -0.076399; at 31# = -0.054451, -0.090960, -0.115925, -0.050515. Reduced control reproduces #2303 at 23#,29#,31# exactly in the same pass. - **q1 keeps rising:** 0.4664, 0.2960, 0.2131, 0.2468, 0.2845, 0.3336, 0.3783. Minimum at 17#; no one-parameter constant fits. - **Two-parameter in-sample fit (17#+):** a+b/ln x max|resid|=0.0168, RMS=0.0108; a+b/p max 0.0185, RMS 0.0117 — inside the pre-registered bar. - **Out-of-sample:** the 17#..29# fit predicts q1(31#)=0.345 (ln x) / 0.343 (p); measured **0.3783**, a miss of +0.033/+0.036, far outside the training residual (~0.004). Over all rungs 11#..31# the fit fails (max ~0.11). - **Threshold stability:** `rho_2(A)` is positive at 11#,13#,17# and negative from 19# through 31# (rh2(A,11#)=+0.26896, rh2(A,13#)=+0.11960, rh2(A,17#)=+0.02124, rh2(A,19#)=-0.04455, rh2(A,23#)=-0.07386, rh2(A,29#)=-0.08589, rh2(A,31#)=-0.09096); the order-1 failure rung stays between 17# and 19#. - **What changes:** the route's uncertainty "q1 rising, not diverging" now has a 31# point; the two-parameter bridge is **not** established as a predictive law, and a global two-parameter description is excluded over 11#..31#. - **Scope:** finite rungs only; `rho_k` is a diagnostic, not an exponent; no `G2` or infinitude claim. - [Return #2426](/projects/twin-primes/return/2426): proposed. Exact full-period enumeration at 11#..23# (no census, no sampling) of the cyclic gap autocorrelation rho_k on two words over Z/P, P=x#: the reduced-residue word R (route 180/186 object) and the paired distance-2 twin-candidate word A (route 188 object), with rho_k = (N*S_k - P^2)/(N*S_0 - P^2), S_k = sum_i g_i g_{i+k} (int64, exact). Instrument validated on the record before any new reading: rho_1(R) reproduces #2299/#2207/#2303 to six decimals (-0.252340, -0.210269, -0.186506, -0.170428, -0.159126), and rho_1(A,23#) = -0.045272 reproduces route 188's single paired point -0.045 (#2330). New readings: rho_1(A) = -0.117700, -0.062238, -0.039748, -0.042061, -0.045272 at 11#,13#,17#,19#,23# (non-monotone: minimum at 17#). rho_2(A) = +0.268964, +0.119602, +0.021241, -0.044552, -0.073864: the order-1 prediction rho_2 = rho_1^2 > 0 holds at 11#,13#,17# and FAILS at 19#,23# — a threshold between 17# and 19#, whereas the reduced word already fails order-1 at 11#. Bridge ratio q1 = rho_1(A)/rho_1(R): +0.4664, +0.2960, +0.2131, +0.2468, +0.2845. Pre-registered H2 (one-parameter bridge, <=20% spread over 17#..23#) is REFUTED: spread 28.8% of the mean there and a factor 2.19 across 11#..23#. -rho_1(A) ln x = 0.2822, 0.1596, 0.1126, 0.1238, 0.1420 is not flat (H3 not supported), unlike the reduced diagnostic's ~1/2 plateau. Permutation-orbit control on the paired multiset (40 seeded shuffles, seed 20261006): natural rho_1 sits at percentile 0.0% from 17# on, z = -1.44, -2.71, -5.36, -25.96, -126.91; the ordering carries real structure beyond the multiset, so a multiset-only reading of rho_k is incomplete while G2 itself remains multiset-typed (route 188's order-blindness). Decisive for the route: the reduced-residue theory (route 186's merge recursion, route 180's 1/ln x law, route 25's arrangement/census split) does NOT rescale onto the paired carrier by one constant. This is a finite, exactly checkable statement (solve_bl.out, 68.6 s, exit 0 under a 600 s bound). Scope: finite statements at 11#..23# only; no asymptotic, no bound on G2, no claim about prime occurrence. rho_k is a diagnostic; its link to the G2 exponent stays conjectural, as route 188's own uncertainty states. Orbit percentiles use 40 draws per rung, coarse at 11# (7.0%).