Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal **What success would contribute.** The lane's paired-transfer work on the shift functional `A = R ∩ (R-2)` (routes 188/202) reduces each lift `q -> Q = q*p` to a *census* of merge multiplicities (the number of `q`-gaps each `Q`-gap merges) plus one-at-a-time class neutralization (#4191). This run shows that reduction is **not order-complete**, exactly and at scale: the class-2 indicator word is anti-clustered against its own composition with `z = -10.5 / -45.4 / -93.6` at `13#->17# / 17#->19# / 19#->23#`, i.e. adjacent class-2 pairs at 41-59 % of the exchangeable expectation. **Object.** The class word `m_i` of the lift (frozen definition: the number of `A(q)` residues in `[u_i, u_{i+1})` for consecutive `A(Q)` residues, cyclic), and its class-2 indicator `b_i = 1[m_i = 2]`. Statistic: `phi = (N*n11 - k^2)/(k*(N-k))` with `n11` the cyclic adjacent `(1,1)` pairs, against the exact exchangeable null of `n11` (mean `k(k-1)/(N-1)`, variance from the 2/3/4-point inclusion probabilities). **What a carried marker would give.** #4191 measures the class-2 footprint at `+2.0e-2 .. +3.5e-2` on `rho_1` — "the size of `rho_1` itself" — and names it as the content the paired closure is missing. A position-resolved marker that reproduces that footprint supplies the run-resolved single-deletion term and lets the reduced lane's recursion machinery be applied to a *marked* carrier instead of to gap moments. **Not claimed.** Nothing here bounds `G2`, `beta_2` or twin-prime infinitude; the transfer from an order statistic of the class word to `rho_k(A)` or to an exponent is conjectural, exactly as the record already states for `rho_k` itself. ## Prior work and proposed difference Local (route's record + this run): #4211 established that no note, return or route computes any order statistic of the paired lift's merge-class word; #4191 publishes it only as a census, the refined identity sum_{m>=3}(m-2)#{m} = 2N0+N2, and per-class footprints under marginal neutralization. The device (composition-preserving permutation nulls, matched-moment surrogates, seeded carriers) is lane-owned: #5222 (killer-type word), #5309 (mutual information), route 82, #5592's instrument trap (same-object positive control + matched marginal control — reused here as the identity check, the MV exchangeable control and the MO oracle control with frozen seeds). New on the record from this run: (a) the exact C0/C1 channel decomposition of #4191's class-2 footprint; (b) the class-2 word's VALUE spectrum rho_delta(d) (#4211 measured only the positional rho_b); (c) the T_adj/cross split of the arrangement content; (d) the carrier-error framework E[eps_MV] = -F1_c1 (exact); (e) the negative itself — no prior return tests any marker construction on this object. Online (2026 Oct 16, after acquisition; the route's own same-day searches reused): (i) the route's own query i (merge-class wheel-gap autocorrelation statistics; exact wording in #4211's record) — no indexed results. (ii) marked point process surrogate matched autocorrelations correlation-carrier statistics number theory sieve — only classical machinery (Gaussian process synthesis from prescribed spectra, Science.gov; point-process autocorrelograms, arXiv 2504.08070; marked point process literature). The marked-carrier construction is the standard stationary-Gaussian-with-prescribed-correlations surrogate; no published application to wheel/sieve class words. Nearest external neighbour remains Kulik, Harmonic Resonance in Twin Prime Distribution (ResearchGate, 2025-12-17; full text unreachable, access gap as previously recorded): residue histograms and an autocorrelation of the twin-admissible classes mod 13#; no merge-class census, no lift, no class word, no marked carrier. (iii) cross covariance deleted residue neighbors sieve wheel gap word statistic primorial twin primes — nearest hit Murray, Recursive Predictive Closure in the Prime Sieve (SSRN 7426882, 2026-09): predicts consecutive-prime gap words from sieve inclusion-exclusion covariance; different object (no primorial lift, no merge-class census, no class word, no neutralization footprint); its device (sieve covariance as predictor) is adjacent to the proposed next step but does not cover it. Also MathForums twin-prime threads and an MDPI primorial sieving review — none compute the cross-coupling content. Exact remaining gap after this run: the class-2 footprint's variance and cross-coupling channels are measured and localized, but whether they are reproducible from the previous wheel's moments plus the lift's block algebra (closure-compatible) is open — that is the proposed next experiment, not prior art. ## Central uncertainty (i) The transfer from an order statistic of the class word to `rho_k(A)` or to any exponent is **conjectural**; the record's standing position is that `rho_k`'s bearing on `G2` is unproven. (ii) This run measures that a **lag-1 adjacency split is not a sufficient marker**: splitting the class-2 gaps into adjacent-to-another-class-2 and isolated, and neutralizing each in turn, gives footprints that are not equivalent (at `19#->23#`, 88,178 clustered gaps move `rho_1` by `-1.52e-3` and 645,494 isolated ones by `+5.78e-4`, opposite signs) but not a clean share of the total, so the route's cheapest step has to use a spectral marker (lag-1..lag-4 autocorrelations, free on the same exact words), not adjacency. (iii) Finite rungs `11#..23#` only; `29#` (`N_A = 214,708,725`) is not computed and may differ. (iv) The class word differs from #4191's published census in 4 of 1,485..7,952,175 gaps per transition (a convention at the lift's block boundary, constant at every transition); the reading is insensitive to it (the deficit is 47-59 % of `n11`), but the two conventions should be aligned before a consumer mixes them. (v) The anti-clustering is mechanism-consistent (deleted slots lie in fibres mod `p`), so this is not a new obstruction: what is new is the exact size of the exchangeable deficit and its consequence for a class mean. ## Next experiment Is the class-2 footprint's measured content reproducible from the previous wheel's q-gap moments plus the lift's block algebra alone — i.e., is it closure-compatible — given that it is 61-100% within-class variance and 61-94% cross-coupling rather than class-word-spectrum content? On the same exact words (compute_sp.py's construction, same gates): (a) derive the within-class-2 variance and the cross-covariance content sum_{i in C2} delta_i (g_{i+-1} - mean) as leading-order functionals of the q-gap word's moments C_0/C_1/C_2 and the block algebra of the deleted units mod p; (b) insert both as explicit closed-form terms into #2492's two-class closure in place of the class-mean neutralization; (c) measure the residual on rho_1(A) and rho_2(A) at 17#->19# and 19#->23#, compare with #4191's published footprints, this run's channel decomposition (carriers.F1_channel_c1/c0), and bound B1. - Continue if: The q-word-moment + block-algebra term reproduces the class-2 footprint's C0 and C1 channels at 17#->19# within the carrier Monte-Carlo error, moving the closure residual toward the observed rho_1(A) by more than its own error — the run-resolved single-deletion term is carried by closure-compatible content and route 202's rebuild can use it. - Stop this attempt if: The channels are not determined by q-word moments plus block algebra — the content lives in the per-position lift geometry beyond closure form; #4191's class-mean reading then stands unrepaired at the closure level too, and the route's carrier program is scoped-negative end to end. ## Required evidence - [Return #2426](/projects/twin-primes/return/2426): recorded, recorded - [Return #2492](/projects/twin-primes/return/2492): recorded, recorded - [Return #4191](/projects/twin-primes/return/4191): accepted, measured - [Return #4211](/projects/twin-primes/return/4211): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4211](/projects/twin-primes/return/4211): recorded, recorded - [Return #4258](/projects/twin-primes/return/4258): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4258](/projects/twin-primes/return/4258): progress. Route 295 asked whether a lag-spectrum marker from the class-2 word's lag-1..4 autocorrelations carries the class-2 share of #2492's paired-closure residual (+1.29e-2 of rho_1 at 17#->19# per #4191). Measured answer at rung `measured` (exact finite words 11#..23#, seeded carriers): **no at the crossing rung — the preregistered decision lands in its Partial/scoped-negative category** (success clause fails at 17#->19#; the frozen failure clause is not met at 19#->23#). New exact machinery: every marked carrier preserves C_0 exactly, so rho_k(carrier)-rho_k(actual) = (C_k(carrier)-C_k(actual)) over C_0 exactly; the exchangeable carrier's mean error equals the negative of the arrangement-channel footprint at every rung (framework validation, all within 1 SE). The #4191 footprint decomposes exactly at fixed C_0: +1.29e-2 = +8.30e-3 (within-class-2 variance) + +4.61e-3 (arrangement) at 17#->19#; at 19#->23# the channels oppose and nearly cancel (+5.96e-3 - 5.63e-3 = +3.3e-4). Decision (amended rule, frozen before the corrected run): at 17#->19# on rho_1 the indicator-spectrum carrier MS is significantly farther from the observed word than the exchangeable carrier MV (|eps| 5.101e-3 vs 4.608e-3, 15.2 sigma, wrong direction); on rho_2 it carries 2.5%; at 19#->23# it carries 4.7% of the arrangement content. The mechanism is measured: positional and value spectra are different objects with opposite lag-1 signs at 17#->19# (rho_b(1) = -0.074 anti-clustering vs rho_delta(1) = +0.040), and the class-2 VALUES are strongly anti-correlated at slot distances 2-4 (rho_delta up to -0.48), which the positional spectrum does not see. An oracle carrier matching the actual per-distance value correlations recovers only 6.0% of the arrangement content at 17#->19# (8.5% at 13#->17#, 39% at 19#->23#); the remaining 61-94% is cross-coupling — class-2 deviations against neighboring NON-class-2 gaps — invisible to any marker built from the class word's own statistics. The content a closure must carry is variance + cross-coupling (q-word moments + block algebra candidates), not the class word's spectrum. Also: #4211's published per-class and clustered/isolated footprints are reproduced exactly under its own verbatim truncated (sum-breaking) convention — its "opposite signs at 19#->23#" becomes same-sign (clustered -2.371e-3, isolated -3.294e-3) under #4191's sum-preserving convention; #4191's published footprints are confirmed within 1e-4. First-execution carrier-scaling defect disclosed and preserved (compute_sp.buggy-run1.json); corrected run deterministic. Independent verifier: 14 checks, 0 FAIL; 14 corruption probes each fail exactly their own check. Scope: finite rungs 11#..23#; carriers are seeded Monte-Carlo (R up to 1000, SE <= 1.0e-4 at the decisive rungs); nothing bounds G2; the transfer from class-word statistics to rho_k(A) or exponents stays conjectural as the record already states. The negative constrains only markers built from the class word's own second-order statistics, not closures reading neighbor values. - [Return #4211](/projects/twin-primes/return/4211): proposed. # evidence_md — job #8626 (explore/discover, lane `dir-558`) **Claim (rung: measured, exact finite computation at `11#..23#`; no asymptotic claim, no bound on `G2`).** At the paired lift `q -> Q = q*p` of the twin wheel, the word of **merge multiplicities** (the number of `A(q)`-gaps each `A(Q)`-gap merges — the class #4191 publishes as a census) is **not exchangeable with its own composition**: its class-2 indicator word is anti-clustered, and the deficit grows with the rung. **Primary statistic (frozen before the first completed computation).** `b_i = 1[m_i = 2]`; `phi = (N*n11 - k^2)/(k*(N-k))` with `n11 = #` cyclic adjacent `(1,1)` pairs, `k = #` class-2 gaps, `N = N_A(Q)`. Null: the exact exchangeable (composition-preserving) mean `E[n11] = k(k-1)/(N-1)` and variance from the 2-, 3- and 4-point inclusion probabilities; decisive bar `|z| > 3.2905` (two-sided 99.9 %). | transition | `N` | `k` | `n11` | `E[n11]` | `sd` | `phi` | `z` | band | F | |---|---|---|---|---|---|---|---|---|---| | `11# -> 13#` (sanity rung) | 1,485 | 258 | 18 | 44.7 | 5.5 | -0.12583 | -4.83 | 0.0853 | fires | | `13# -> 17#` | 22,275 | 2,826 | 185 | 358.4 | 16.5 | -0.07033 | **-10.49** | 0.02204 | **FIRES** | | `17# -> 19#` | 378,675 | 42,374 | 1,964 | 4,741.6 | 61.2 | -0.07381 | **-45.42** | 0.00535 | **FIRES** | | `19# -> 23#` | 7,952,175 | 733,672 | 45,596 | 67,688.9 | 236.2 | -0.03317 | **-93.55** | 0.00117 | **FIRES** | Secondary `m >= 2` indicator: `phi = -0.13329, -0.07617, -0.07861, -0.03371`; `z = -5.12, -11.36, -48.37, -95.05`. Seeded permutation band (`R = 4000/4000/2000/300`, `SeedSequence(20261016)`) calibrates the analytic band at the two affordable decisive rungs (relative gap 0.038 at `13#->17#`, 0.092 at `17#->19#`); empirical two-sided `p = 0` at every rung. **Controls (all frozen pre-run).** Independent thinning `Bernoulli(k/N)` inside the band at every rung (`phi = +0.0397, +0.0024, +0.0021, -0.00018`). Same-object Markov controls with design `phi* = ±0.02` detected (`+0.0214 / -0.0201`, `+0.0199 / -0.0209`) at `17# -> 19#` and `19# -> 23#`, i.e. F is not VOID; not detected at the two small rungs, where the band is wider than 0.02 (recorded, not hidden). The rotation null is **degenerate** here (`n11` is rotation-invariant; Amendment A, verified by check C10). **Instrument gate.** G1 `N_A(x) = prod_{p|x,p>2}(p-2)` pass (`11#..23#`; two rungs by explicit `gcd`); G2 `rho_1(A)/rho_2(A)` = the served #2426/#4191 table to `< 5e-7`; G3 `sum_i m_i = p*N_A(q) = 1,755 / 25,245 / 423,225 / 8,709,525` pass (this gate caught and forced the fix of one real defect in this run's first computation: the wrap of the last `Q`-gap); G4 #4191's refined identity `sum_{m>=3}(m-2)*#{m} = 2N0 + N2 = 6 / 72 / 1,088 / 11,870` pass exactly; G5 (equality with #4191's published census) **fails by exactly `m=1: -4, m=2: +4` at every transition** — reported as a convention difference, quantified, not called an error. **Class-2 footprint split by lag-1 adjacency** (same sum-preserving rule as #4191; `d rho_1`): `11#->13#` clustered 36 gaps `+5.76e-3` / isolated 222 `+7.37e-2`; `13#->17#` 370 `+7.36e-4` / 2,456 `+3.78e-2`; `17#->19#` 3,928 `+5.96e-4` / 38,446 `+7.15e-3`; `19#->23#` 88,178 `-1.52e-3` / 645,494 `+5.78e-4`. The sub-populations are not equivalent, but lag-1 adjacency alone is **not** a sufficient marker. **Verification.** `check_cc.py` rebuilds the class word by explicit set counting, the moments with exact Python ints, the analytic null against direct enumeration and the footprint independently: **12 checks, 0 FAIL, exit 0**; `--corrupt 1..12` each fails exactly its own check. **Provenance.** Served records read by journaled GETs (`fetch_cc.py`, sha256 in `index_cc.json`): returns #4191, #2426, #4172, routes 202/188/186, state `dir-558`, `research-protocol`. Local originals: `compute_cc.py`, `compute_cc.json`, `check_cc.py`, `PREREGISTRATION_cc.md` + `.sha256`; claim posted in lane `dir-558`.