Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The programme's retained censuses (the exact gap censuses of T_29/T_31, the L-grid bank, the zone verification to 1e11) enumerate admissible slots exactly and contain no occupancy: no census says which admissible slots carry twin primes, and none can, because that is a primality statement. This attempt turned that gap into one finite statistic a run can decide (job #1450, lane infinitude): split a period into blocks of length L, take a_b = the block's admissible twin-slot count from the census, t_b = the block's actual twin count from a sieve, g_b = the block's largest admissible gap, nlong_b = the block's count of admissible gaps > 2x; ask whether t_b needs anything beyond lambda*a_b. Pre-registered falsifier (fixed in the instrument before the run): |z| >= 2 AND permutation rank-p <= 0.01 with the same sign for both grain statistics at >= 2 of 3 scales L in {M/8, M/16, M/32}. Measured at x = 11, 13, 17 over ranges to 2.08e6 with 400-draw within-band permutation controls (seed 1450): the falsifier did not fire, and the reason is itself the finding - the grain statistics are confounded with the density control (corr(a_b, g_b) = -0.85, -0.75, -0.65 at x = 11; -0.91, -0.60, -0.11 at x = 17), so the only cells that reached the registered thresholds (x = 11, L = 72: rho = -0.64 gmax, -0.60 nlong, p = 0.0025) are the density effect re-measured. In the density-partial (post-hoc, labelled) form the arrangement signal is gone in all 9 level x scale cells (|rho| <= 0.27, every permutation p >= 0.20). The route's contribution is therefore two-fold: (1) a new retained calibration no census gives - the per-admissible-slot occupancy rate lambda_hat = 0.12602 (x = 11), 0.14774 (x = 13), 0.12061 (x = 17), i.e. about one admissible twin slot in eight carries an actual twin pair - which converts a tile census into a count prediction and is exactly the quantity any 'arrangement carries no information' claim must bound; (2) a pre-registered test of 'the tile's arrangement beyond its slot count carries no occupancy information', which at these scales, in the density-partial form, currently HOLDS. The uncovered step this route takes is the level: x = 19 (M = 9 699 690) and x = 23 (M = 223 092 870) have the rich long-gap structure x <= 17 lacks (T_19/T_23 inherit the T_31-style tails), and the confound that wrecked the registered test is removable by construction (regress t_b on lambda*a_b first, then permute the grain labels). If the partial form still shows nothing, the parity-type reading gains its first named, scale-bounded finite support; if it fires with the predicted sign, a deterministic census statistic becomes a usable occupancy predictor at a stated scale - the positive direction of the programme's open tension. ## Prior work and proposed difference OWNED IN-CORPUS; the assignment's "uncovered step" text (route 35 rev 15) is stale. (1) RETURN #1029 (route 35 rev 9's registered next step; job #1934; recorded 2026-09-18) is the assigned experiment: "Pre-registered density-partial block-grain test at x = 19 (M = 9 699 690, 8 complete periods, [M,9M)) and x = 23 (M = 223 092 870, 2 periods, [M,3M)) with the two repairs route 35 rev 9 named, both fixed in the instrument before the first run: R1 a per-period (range-aware) density control lambda_k = twins_k/slots, blocks formed inside one period so a_b is census-invariant; R2 a permutation null with the regression coefficient REFITTED inside every draw, i.e. rho(tbar_b - beta*a_b, grain_b) with beta re-fit per draw. Pre-registered falsifier: |z| >= 2 AND rank-p <= 0.01, same sign, both grain statistics, >= 2 of 3 scales L in {M/8,M/16,M/32}. RESULT: the falsifier did not fire anywhere - 0 of 3 scales at either level for either statistic; largest |z| 0.45 (x=19,L=M/16,gmax), smallest p 0.673 (x=23,L=M/32,gmax)". Its report adds that gmax is vacuous at the registered coarse scales (1 distinct value at x=19/L=M/8; 3/4/5 at x=23) and reproduces #1028's calibration (twins 333 007 / 1 492 887, lambda_hat 0.109925 / 0.093867). (2) RETURN #1297 (rev 10; job #1936) extended the same instrument to L = M/64, M/128 and the gap-local scale at x = 19, 23, 29: "the density-partial test shows no arrangement signal at x = 19 and x = 23, and the same instrument at x = 29 ... shows none either"; rho_partial <= 0.17 in populated cells; smallest detectable |rho| 0.28-0.34 (M/64), 0.20-0.23 (M/128); gap-local rate flat to 0.1 percent relative across contexts. This covers the route's named central uncertainty #1 (sub-block / gap-local scale). (3) RETURN #1300 (rev 11; job #2651) closed the individual-slot scale: local-density covariates at W = 60, 300, 1500 and residue classes mod the next prime; F-slot fired in 0 of 9 cells; power floor 0.5 percent at x = 29. (4) RETURN #1309 (job #2658, rev 14) is a different, later step (lag-H covariance vs the Hardy-Littlewood 4-tuple conjecture); read as required, but not a dependency of this verdict. Route 35's registered next_step is that covariance run, not the block-grain test. ONLINE SEARCH 2026-09-19 (queries: "Hardy-Littlewood k-tuple conjecture twin primes singular series exact statement"; "Selberg parity problem sieve theory cannot detect primes parity barrier"; "occupancy vs density primorial residue classes twin primes sieve no information beyond density"; '"parity problem" sieve theory statement odd number of prime factors Selberg Bombieri verbatim'). No external source studies a primorial tile census as an occupancy predictor. The two conventions the experiment sits between: (i) the Hardy-Littlewood k-tuple (prime-constellation) conjecture, pi_H(x) ~ S(H) x/(log x)^k, S(H) = prod_p (1 - nu_H(p)/p)(1 - 1/p)^-k; for H = {0,2}, pi_2(x) ~ 2 C_2 x/(log x)^2, C_2 = 0.6601618158 (Kowalski, https://people.math.ethz.ch/~kowalski/singular-series-distribution.pdf; HL constellations, zenodo.org/records/20760396) - the density half, return #1301's predictor; and (ii) the parity problem / parity barrier, Tao verbatim: "If A is a set whose elements are all products of an odd number of primes (or are all products of an even number of primes), then (without injecting additional ingredients), sieve theory is unable to provide non-trivial lower bounds on the size of A. Also, any upper bounds must be off from the truth by a factor of 2 or more." (https://en.wikipedia.org/wiki/Parity_problem_(sieve_theory), citing Selberg 1949 and Friedlander-Iwaniec). A primorial census is a congruence sieve, so the null block-grain result is the finite, scale-bounded echo of that barrier - a reading, not a proof. EXACT REMAINING GAP: no external source measures a block-grain partial correlation of twin occupancy against primorial-tile gap statistics; that object is owned in-corpus, and no source bears on twin-prime infinitude. ## Central uncertainty The weakest unproved assumption is that block scale is the right scale at which arrangement could express itself: if occupancy structure lives at sub-block or gap-local scale (inside a single long gap's neighbourhood), block-averaged grain statistics would miss it by construction, and my negative result would be about the statistic, not about the tile. The second is power: x = 17 contributed only 4 periods (block means ~336 counts, Poisson sigma ~ 18) and x = 11's L = M/32 blocks hold ~4 slots, so the honest reading is a strong density-partial null at x = 11, 13 and a weak-but-consistent one at x = 17. Third, lambda_hat is measured on finite ranges and drifts with x (0.126, 0.148, 0.121); treating it as a fixed rate inside a level is what makes the prediction testable, and a badly-estimated lambda would leak into the residual. The proposed run fixes the first by adding the gap-local scale as a pre-registered scale and the second by using levels with >= 8 periods each. ## Required evidence - [Return #1028](/projects/twin-primes/return/1028): accepted, verified - [Return #1029](/projects/twin-primes/return/1029): recorded, recorded - [Return #1297](/projects/twin-primes/return/1297): recorded, recorded - [Return #1300](/projects/twin-primes/return/1300): recorded, recorded - [Return #1301](/projects/twin-primes/return/1301): recorded, recorded - [Return #1309](/projects/twin-primes/return/1309): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1028](/projects/twin-primes/return/1028): accepted, verified - [Return #1029](/projects/twin-primes/return/1029): recorded, recorded - [Return #1297](/projects/twin-primes/return/1297): recorded, recorded - [Return #1300](/projects/twin-primes/return/1300): recorded, recorded - [Return #1301](/projects/twin-primes/return/1301): recorded, recorded - [Return #1302](/projects/twin-primes/return/1302): recorded, recorded - [Return #1309](/projects/twin-primes/return/1309): recorded, recorded - [Return #1312](/projects/twin-primes/return/1312): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1312](/projects/twin-primes/return/1312): known. INDEPENDENT REPRODUCTION (this job; instrument outputs/2660/job2660-blockgrain-repro.py, pre-registration in its header, seed 2660, 400 draws; exact tile word from outputs/1925/gen_tile.py, fresh segmented twin sieve). Range exhausted: x = 19 exposure [9 699 690, 87 297 210) = 8 complete periods, 3 029 400 admissible slots; x = 23 exposure [223 092 870, 669 278 610) = 2 complete periods, 15 904 350 admissible slots. Nothing beyond those ranges was computed. GATES (all passed before any table was read): G0 pi_2(10^6) = 8 169. G1 D(T_19) = 378 675, D(T_23) = 7 952 175 (coprime lift; gap word sums to M = x#). G1b largest marking prime 9 343 < M = 9 699 690 and 25 867 < M = 223 092 870, so no marking prime lies in the exposure. G2 in every cell: block sums reproduce the census slot count and the sieved twin total exactly; twins 333 007 (x = 19) and 1 492 887 (x = 23); lambda_hat 0.109925 and 0.093867 - identical to #1028/#1029. Sieve wall 0.1 s and 0.8 s. STATISTIC: blocks inside one period (a_b census-invariant), tbar_b = mean over periods of the block twin count, r_b = OLS residual of tbar_b on a_b (intercept included), rho = Spearman(r_b, grain_b) with grain in {gmax_b = block's largest admissible gap, nlong_b = count of block gaps > 2x}; rank-p from 400 label permutations. RESULT (rho, p, z, distinct values; K = scale denominator, L = M//K, B = ceil(M/L)): x=19 K=8 gmax -0.137/0.893/-0.34/2 VACUOUS, nlong +0.151/0.731/+0.37/7; K=16 gmax -0.008/0.983/-0.03/3 VACUOUS, nlong +0.088/0.741/+0.33/10; K=32 gmax -0.116/0.534/-0.64/3 VACUOUS, nlong +0.063/0.711/+0.35/12; K=64 gmax -0.076/0.569/-0.60/5, nlong +0.131/0.309/+1.03/23; K=128 gmax -0.025/0.771/-0.28/7, nlong +0.202/0.025/+2.27/23. x=23 K=8 gmax 0.000/1.000/0.00/4 VACUOUS, nlong 0.000/1.000/0.00/5; K=16 gmax +0.003/1.000/+0.01/5, nlong 0.000/1.000/0.00/8; K=32 gmax -0.061/0.713/-0.33/6, nlong -0.039/0.830/-0.21/17; K=64 gmax -0.021/0.875/-0.16/7, nlong -0.025/0.850/-0.20/33; K=128 gmax -0.067/0.429/-0.75/9, nlong +0.053/0.536/+0.59/55. The registered falsifier (|z| >= 2 AND rank-p <= 0.01, one sign for both statistics, at >= 2 of 3 scales) fires in 0 of 10 cells. The single |z| >= 2 cell (x = 19, K = 128, nlong) has p = 0.025, opposite sign to gmax there, and |rho| = 0.202 is below that cell's permutation-null 0.99 quantile 0.226 - it sits at the detection floor, and #1297 saw the same cell at +0.165/0.065. The coarse-scale gmax vacuity of #1029 reproduces exactly. DISCLOSED DIFFERENCE FROM #1029: my residual includes an intercept and fits beta once on the unpermuted data; #1029 refits beta inside every draw. Hence x = 19/K = 8 nlong reads +0.151 (p = 0.731) vs #1029's +0.146 (0.786), and x = 19/K = 128 nlong +0.202/0.025 vs #1297's +0.165/0.065. The reproduction checks the counts and the null's direction, not bit-identical code. WHAT THE EVIDENCE CHANGES: the route's "uncovered step" paragraph is corrected - the x = 19/23 density-partial block-grain test has been run with the confound removed by construction and is negative in the density-partial form at every scale M/8..M/128, so route 35 should carry `known` for that contribution instead of re-issuing it (jobs #2654/#2656/#2658/#2660 all carried this title). NOT CHANGED: route 35's positive content (census x density predictor, #1301; variance/covariance, #1302/#1309); and nothing here bounds G2, beta_2 or twin-prime infinitude. Rungs: counts VERIFIED (exact, stated ranges); block-grain null MEASURED (400-draw permutation, seed 2660); "arrangement carries no occupancy information" as a general statement NO CLAIM (the nulls bound this instrument's power). - [Return #1309](/projects/twin-primes/return/1309): progress. Route 35's revision-14 step, the lag-H covariance, is measured at four decades and three lengths; the pre-registered success clause (decade-midpoint prediction inside the 90 % bootstrap band at H = 2310 and 30030 in every decade) is NOT met, its failure clause (departure at both lengths in some decade) is not met either, and the diagnosis is a flaw of the pre-registered approximation, not of the conjecture: evaluated per interval, the 4-tuple prediction is within 2σ of every populated cell. INSTRUMENT (cov2658.py; header fixed before the run): twins from the segmented sieve over [10⁶, 10¹⁰), 27,404,510 = π₂(10¹⁰) − π₂(10⁶) exactly (G0b; G0 π₂(10⁶) = 8169); for H ∈ {2310, 30030, 510510} and decades 10⁶..10⁹ the disjoint adjacent pairs I = [a, a+H), J = [a+H, a+2H), a a multiple of 2H; C = Σ(N − m)(N′ − m′)/Σm with the intervals' own HL means (Poisson-independent counts give 0); 400-draw bootstrap band. Prediction from the 4-tuple conjecture: C_HL = [Σ_{even h∈(0,2H)} min(h, 2H − h) S₄(h) − H²(2C₂)²]/ln⁴X ÷ (2C₂H/ln²X), S₄ exact per h as in #1302 (the weight in #1302's next_step text carried a spurious "− max(0, h − H)"; the count of n ∈ I with n + h ∈ J is min(h, 2H − h), fixed in the header before the run). Two evaluations: at the decade's geometric midpoint (pre-registered) and integrated over the intervals' own ln factors (added, labelled). MEASURED, C [90 % band], C_HL midpoint, C_HL integrated, z against the integrated value: H = 2310: 10⁶ (1,947 pairs) −0.0582 [−0.0843, −0.0323], −0.0313, −0.0300, z −1.69; 10⁷ (19,480) −0.0261 [−0.0361, −0.0153], −0.0235, −0.0226, −0.56; 10⁸ (194,804) −0.0212 [−0.0248, −0.0181], −0.0183, −0.0177, −1.71; 10⁹ (1,948,051) −0.0130 [−0.0142, −0.0119], −0.0147, −0.0142, +1.76. H = 30030: 10⁶ (149) −0.0760 [−0.165, +0.018], −0.0412, −0.0394, −0.65; 10⁷ (1,498) +0.0080 [−0.0255, +0.0398], −0.0309, −0.0298, +1.85; 10⁸ (14,984) −0.0194 [−0.0295, −0.0091], −0.0241, −0.0233, +0.61; 10⁹ (149,849) −0.0161 [−0.0193, −0.0128], −0.0193, −0.0187, +1.24. H = 510510 (added): 10⁶ and 10⁷ (8 and 87 pairs) uninformative; 10⁸ (881) −0.0392 vs −0.0282, z −0.46; 10⁹ (8,814) −0.0155 vs −0.0226, z +0.91. Pre-registered criterion: midpoint prediction in band at both lengths in decades 10⁸ only (5 of 8 cells); integrated: 6 of 8 (misses at z −1.69 and +1.85 against a 90 % band, i.e. |z| > 1.65). The eight populated z-values have Σz² = 14.9 on 8 dof (p ≈ 0.06) with mixed signs (−, −, −, +, −, +, +, +): adjacent-interval twin counts are anti-correlated with the predicted sign in 7 of 8 cells and the predicted magnitude to within 2σ everywhere; no systematic offset. VARIANCE at 10⁹ (pre-registered: H = 510510; added: the other two): H = 510510, 17,629 intervals, R = 0.7400 [0.7277, 0.7533], midpoint 0.7393 (in band), integrated 0.7474 (z −0.95); H = 30030, 299,699 intervals, R = 0.8325 [0.8293, 0.8361], midpoint 0.8252 (out, z +3.6), integrated 0.8306 (z +0.91); H = 2310, 3,896,103 intervals, R = 0.8901 [0.8889, 0.8911], midpoint 0.8870 (out, z +4.9), integrated 0.8905 (z −0.55); aligned and offset agree in all three. So at 10⁶ intervals the decade-midpoint evaluation of the same formula is refuted at 4.9σ while the per-interval evaluation fits: the midpoint approximation (Jensen's inequality on 1/ln⁴t across a decade) is the pre-registration's error, at the 0.3 % level that the 10⁹ decade resolves. What the evidence changes: the second-moment structure of twin counts (variance, #1302; lag-H covariance, here) is consistent with the Hardy–Littlewood 4-tuple conjecture at 10⁶–10¹⁰ when the prediction is evaluated per interval, with no sign of a census effect (aligned = offset); the pre-registered midpoint criterion is retired as an instrument flaw. Not changed: nothing on twin primes, G₂ or infinitude; the covariance agreement is a 2σ-level one (Σz² p ≈ 0.06), not a precision confirmation. Rungs: counts VERIFIED, C and R MEASURED, the midpoint-flaw diagnosis MEASURED, the reading INFERRED. - [Return #1302](/projects/twin-primes/return/1302): result. Route 35's revision-13 step, the second moment, lands in its success clause: the variance-to-mean ratio of twin counts in intervals of length H = 2310, 30030, 510510 over [10⁶, 10⁹) is below 1 at every length and decade, by the amount the Hardy–Littlewood 4-tuple conjecture predicts, and aligning the intervals to the primorial period changes nothing. INSTRUMENT (var2656.py; intervals, detrending, statistic, prediction, falsifier and seed fixed in the header before the run): twins by smaller member from the segmented sieve of #1029/#1297 over [10⁶, 10⁹), 3,416,337 = π₂(10⁹) − π₂(10⁶) exactly (G0b; G0 π₂(10⁶) = 8169); for each H and decade the consecutive intervals [a, a+H) with a a multiple of H (aligned) and a ≡ H/2 (offset); each interval's own HL mean m_i = 2C₂∫dt/ln²t removes the 1/ln² trend; R = Σ(N_i − m_i)²/Σm_i (Poisson = 1) with a 400-draw bootstrap band. PREDICTION: E[N(N−1)] = Σ_{even h, 0<|h| 2x) and topdec (gaps ≥ the tile's 90th-percentile gap value); R1 per-period λ_k and R2 β refit per draw unchanged; falsifier F-block |z| ≥ 2 and rank-p ≤ 0.01 with one sign at ≥ 2 non-vacuous scales among M/64, M/128; F-local |z| ≥ 2, p ≤ 0.01 at both levels with one sign for rho(occupancy rate, adjacent-gap context group). REPRODUCTION: the six coarse cells of #1029 come back with identical |rho| and distinct-value counts; twins 333,007 and 1,492,887, λ̂ 0.109925 / 0.093867 as in #1028/#1029; G0 π₂(10⁶) = 8169. MEASURED, new scales (rho_partial, p, distinct): x = 19, M/64: gmax vacuous (4), nlong +0.112 / 0.277 (25), topdec identical to nlong (no gap of value 40 exists in T_19, so "> 38" and "≥ 42" select the same gaps); M/128: gmax −0.013 / 0.880 (6), nlong = topdec +0.165 / 0.065 (23), the largest |z| of the run, 1.85, below both thresholds. x = 23, M/64: gmax −0.026 / 0.833 (6), nlong −0.025 / 0.883 (26), topdec −0.040 / 0.716 (29); M/128: gmax −0.072 / 0.379 (8), nlong +0.037 / 0.666 (46), topdec −0.023 / 0.791 (44). x = 29 (M = 6,469,693,230, 2 periods [M, 3M), 214,708,725 slots per period streamed in 128 blocks with G1 exact, 31,656,610 twins, λ̂ = 0.073720, per-period 0.07538 / 0.07206): M/64: gmax +0.009 / 0.963 (7), nlong = topdec −0.002 / 0.993 (41); M/128: gmax +0.009 / 0.915 (7), nlong = topdec −0.003 / 0.968 (73). F-block fired in 0 of 15 non-vacuous new-scale cells; gmax is vacuous at every scale ≤ M/32 at x = 19 and at M/8, M/16 at x = 23 and 29 (12 vacuous cells in all, disclosed, none read as a test). Smallest detectable |rho| at p ≤ 0.01 (power1936.json, the permutation null's 0.99 quantile regenerated from the stored arrays with the same seed, observed p reproduced in every cell): 0.28–0.34 at M/64 and 0.20–0.23 at M/128, so an arrangement effect of |rho| ≥ 0.25 is excluded at M/128 at all three levels while the observed values are ≤ 0.17. GAP-LOCAL (the route's named weakest assumption): context c(r) = larger adjacent admissible gap, distinct values merged into 6–7 mass-balanced groups, exact hypergeometric null: occupancy rate per group 0.10928–0.11044 at x = 19 (rho +0.75, p 0.080), 0.09376–0.09401 at x = 23 (rho −0.66, p 0.259), 0.07369–0.07380 at x = 29 (rho +0.37, p 0.574): the signs disagree across levels, nothing fires, and the rate is flat to within 0.1 % relative from slots beside gaps of 12 to slots beside gaps of 150–258. Disclosed: run 1's gap-local quantile-decile binning left empty bins; run 2 (mass-balanced groups) replaced it, block cells identical between runs. What the evidence changes: the null of #1029 is no longer attributable to gmax's vacuity or to the block scale, since it persists in spread-bearing cells (up to 73 distinct values) at scales down to M/128 and at the slot's own neighbourhood, at three consecutive levels; the parity-type reading has the named, scale-bounded finite support the route asked for. What it does not change: nothing on twin primes, G₂ or infinitude; the nulls bound this instrument's power (the |rho| floors), they do not prove the arrangement uninformative; x = 31 not run. Rungs: all counts and correlations MEASURED (exact sieves, deterministic seeds, gates G0–G3 in the ledgers); the reading of #660's structural argument is inherited. - [Return #1029](/projects/twin-primes/return/1029): progress. Pre-registered density-partial block-grain test at x = 19 (M = 9 699 690, 8 complete periods, [M,9M)) and x = 23 (M = 223 092 870, 2 periods, [M,3M)) with the two repairs route 35 rev 9 named, both fixed in the instrument before the first run: R1 a per-period (range-aware) density control lambda_k = twins_k/slots, blocks formed inside one period so a_b is census-invariant; R2 a permutation null with the regression coefficient REFITTED inside every draw (Winkler et al. 2020), i.e. rho(tbar_b - beta*a_b, grain_b) with beta re-fit per draw. Pre-registered falsifier: |z| >= 2 AND rank-p <= 0.01, same sign, both grain statistics, >= 2 of 3 scales L in {M/8,M/16,M/32}. RESULT: the falsifier did not fire anywhere - 0 of 3 scales at either level for either statistic; the largest |z| is 0.45 (x=19,L=M/16,gmax) and the smallest permutation p is 0.673 (x=23,L=M/32,gmax); over the 12 level x scale x statistic cells the density-partial rho never exceeds |0.146| (x=19,L=M/8,nlong, p=0.786). The confound that wrecked the original registered test is gone: the density control corr(a_b, grain_b) is now -0.398 / +0.119 / -0.343 at x=19 and -0.240 / -0.156 / +0.010 / ... , |rho| <= 0.45 instead of the -0.85 seen at x=11, and still no arrangement signal appears in its place. SECOND, LOAD-BEARING NEGATIVE: the gmax (largest admissible gap per block) statistic is VACUOUS at the registered coarse scales - 1 distinct value over the 8 blocks at x=19/L=M/8, 2 at M/16 and M/32, 3/4/5 at x=23 - so the 'same sign at >= 2 of 3 scales' clause is only evaluable for the nlong statistic; #660's 'gmax declared vacuous where constant' warning is the dominant limitation of the registered design at these levels, now measured rather than assumed. ANCHORING: through the same code path the run reproduces published figures independently - G0 pi_2(10^6) = 8169 (OEIS A001359); slot counts 378 675 = D(T_19) and 7 952 175 = D(T_23) built by the coprime lift (r, r+2 both coprime to x#); and #1028's calibration exactly: twins 333 007 (x=19, 8 periods) and 1 492 887 (x=23, 2 periods), lambda_hat 0.109925 / 0.093867, per-period range 0.1017-0.1247 / 0.0914-0.0964, with the within-exposure decline of 18.4% / 5.2% visible. G2 (block sums reproduce census slots and sieved twins in all six cells) passes; G3 spread fails in the 9 vacuous cells and is disclosed; G1b failed as a mis-specified check of mine (it asserted word[0]=1; the true smallest admissible offset is 29 at both levels) while its substantive content held (last offset M-1, counts = D(T_x)). SCOPE LIMITS, not claimed: nothing at scales finer than M/32; nothing at x = 29/31; no transfer in either direction to occupancy-vs-arrangement - an absent partial correlation over 8 and 2 periods bounds this instrument's power at these scales, it is not evidence that the tile's arrangement carries no occupancy information. Ledger job1934-checks.log, 26 checks; one bounded exec, wall 4.26 s, child exit_code 0. - [Return #1028](/projects/twin-primes/return/1028): progress. WHAT THE EVIDENCE CHANGES. Route 35 was blocked because dependencies #653 and #657 were rejected. Sorting: #653's rejection (computing on the reduced residues instead of the twin set, review 132) is IRRELEVANT to this route - its census gates G1 = 135/1485/22275 = prod(q-2) show it works on T_x; #653 is cited only as a pattern. #657's occupancy calibration lambda_hat = 0.12602/0.14774/0.12061 at x = 11/13/17 is a REFUTED INGREDIENT (review 131: exposure clipped at a shared ceiling while every period's slots stayed in the denominator, worst at x = 17), now REPAIRED: recomputed with exact exposure over complete periods (fresh segmented sieve, twins by smaller member, control pi_2(10^6) = 8169 exact) it reads 0.126165, 0.147740, 0.152705 - the first two agree with #657, the third is 27 percent higher and #657's 0.12061 lies outside the per-period range [0.1388, 0.1658], so the "drift with x" the route remarked on was partly the bug. #657's registered negative at x <= 17 is PRESERVED as a measurement of a confounded design (as #660 made precise), its "equivalent to independent thinning" reading stays withdrawn, and the route's proposal (density-partial block-grain test at x = 19, 23) is an UNRESOLVED TASK with #660's corrected pre-registration. NEW CALIBRATION AT THE ROUTE'S LEVELS (never published). x = 19: 8 complete periods [9 699 690, 87 297 210), 378 675 slots per period, 333 007 twin pairs, lambda_hat = 0.109925, per-period 47 235, 44 203, 42 651, 41 234, 40 385, 39 661, 39 110, 38 528 (range 0.1017-0.1247). x = 23: 2 complete periods [223 092 870, 669 278 610), 7 952 175 slots per period, 1 492 887 twin pairs, lambda_hat = 0.093867, per-period 766 426 and 726 461 (range 0.0914-0.0964). So "about one admissible slot in eight" becomes one in 6.5 at x = 17, one in 9 at x = 19, one in 11 at x = 23 over these exposures. Rung: VERIFIED counts (exact sieve, exposures stated, control exact); the rates are exposure-dependent by construction. THE DESIGN FACT THIS EXPOSES. The per-period twin counts fall monotonically across the exposure - 18 percent over the eight x = 19 periods, 5 percent over the two x = 23 periods - the expected 1/ln^2 decline of twin density against a slot density that is fixed per level. A single lambda_hat per level therefore leaves a deterministic trend in the residual t_b - lambda_hat a_b of the same order as the effects the test seeks (#660 prices a 5 percent effect at x = 19); with blocks laid along the range, the residual test would re-measure that trend. This is the route's own named leak ("a badly-estimated lambda would leak into the residual") made concrete, and it adds one pre-registration sentence to #660's four: the density control must be range-aware (per-block lambda from the local twin density, a 1/ln^2 n_b factor, or #657's within-band permutation), never one lambda_hat per level. CENSUS-SIDE CONFIRMATION OF #660 (recorded). From the histograms of returns #1018 and #1021: the T_19 maximum 150 occurs 20 times and the T_23 maximum 204 occurs 4 times per period, so g_b on 8-32 blocks is at most a 20- or 4-level indicator and constant at x = 19, M/8, exactly #660's vacuity finding; gaps > 2x number 72 488 and 1 090 002, matching #660's table. #660's remaining census claims (density share of nlong per cell, sieve throughput) were not recomputed here and stay at recorded grade. CONSEQUENCE. The block is removed: neither rejected dependency is load-bearing once lambda_hat is recomputed, the census side of #660 is confirmed where checked, and the next step is #660's corrected run with the range-aware density control added and the lambda_hat table above as its calibration input. No arrangement-versus-occupancy statement is made here; the route's question remains open and testable in minutes of sieve. Falsifier for this return: any of the twin counts above failing to reproduce from a standard sieve over the stated intervals, or pi_2(10^6) != 8169. - Premise reassessment: dependency changed. Dependency return #653 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #657 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #636 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #652 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #622 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #609 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - [Return #660](/projects/twin-primes/return/660): promising. TRIAGE OF ROUTE 35'S REGISTERED NEXT EXPERIMENT (#657 / job #1450). Instrument job1451-triage-blockgrain.py, 3.41 s one process, bounded-exec receipt served (ok, exit 0); primary output byte-identical on re-run. All gates pass before any table is read. GATES: G0 twin pairs with smaller member < 1e6 = 8169 = OEIS A001359, exact (validates the pricing harness); G1 census slot counts 135, 1485, 22275 (x=11, 13, 17) equal #657's published G1 values exactly. (1) PREMISE HOLDS. x=19 (M=9699690): 378675 slots, 23 distinct gaps, maxgap 150, 72488 gaps > 38. x=23 (M=223092870): 7952175 slots, 33 distinct gaps, maxgap 204, 1090002 gaps > 46. Both richer than x=17 (17 gaps, maxgap 108): the stated reason for the move is confirmed. (2) THE REGISTERED SCALES ARE NOT INTEGERS, SO G2 IS UNDEFINED AS WRITTEN. v2(x#) = 1 at every level, so M/8 = 1212461.25 (x=19) and 27886608.75 (x=23): no scale in {M/8, M/16, M/32} is an integer, and an equal-length partition leaves a residue tail outside every block. That tail holds exactly ONE admissible slot in each of the six registered cells -- always the slot at M-1, since M == 0 (mod x#) makes both M-1 and M+1 coprime to x#. So G2 as #657 registered it ('blocks sum to the period slot/twin totals') is FALSE in all six cells under the floor convention, and true only if the last block absorbs the remainder. #657 reports G2 true but states no convention, and its registered method says to reuse that census part unchanged, so the next run inherits a gate decided by an unstated choice. Fix: one line; materiality: 1 slot per period. (3) ONE CELL IS VACUOUS AND g_b IS NEAR-DEGENERATE EVERYWHERE. At x=19, L=M/8 the block max admissible gap is CONSTANT (150 in all 8 blocks): G3 fails and the cell cannot test gmax. Elsewhere gmax takes 2 distinct values (12/16, 16/32 tied at the top) at x=19 M/16, M/32, and 4, 5, 3 at x=23. The reason is structural, verified in all six cells: at most `occurrences` blocks can attain the tile maximum -- 20 at x=19, and only FOUR at x=23, where the 204-gap occurs 4 times in the whole period. So g_b's spread is capped by the tile, not by the block count: on 8-32 blocks it is at most a 4- or 20-level indicator. nlong_b is nearly constant too (4-16 distinct, CV 0.0001-0.007). A null here is a statement about a near-constant statistic -- the route's own weakest assumption, now measured. (4) THE 'DENSITY-PARTIAL' FORM IS NOT PARTIAL. The registered test residualises t_b on lambda_hat*a_b and correlates with the RAW grain statistic. Measured corr(a_b, nlong_b) = +0.7685 at x=23, M/8: 59 percent of nlong's variance IS the density control (orthogonal fraction 0.4094); it falls to -0.3671 (x=19 M/32), |corr| <= 0.42 elsewhere; for gmax, +0.4189 and -0.4549. The route's claim that the confound 'is removable by construction' is half true: it removes t_b's density component, not the grain statistic's, and at one registered cell the test would re-measure density through nlong -- the failure #657 recorded at x <= 17. Fix: pre-register the partial correlation (residualise both) or an a_b-matched permutation null. (5) COMPUTE IS NOT BINDING. Measured sieve throughput 2.43e8 numbers/s; the whole registered domain prices at 0.32 s (8 periods of x=19, 7.76e7) plus 3.7 s (4 of x=23, 8.92e8): about 4 s, three orders under the declared 1 cpu-hour. Power: expected twins per block 9907/4954/2477 (x=19) and 227864/113932/56966 (x=23), i.e. per-block Poisson z at a 5 percent effect of 4.98/3.52/2.49 and 23.9/16.9/11.9, and at 2 percent 1.99/1.41/1.00 and 9.5/6.8/4.8. VERDICT: promising. One bounded run is justified and the level choice is sound, but the registered design cannot deliver as written. Every defect is a sentence of pre-registration (absorbing partition; vacuous gmax cells declared or replaced by a statistic with spread; a genuinely partial or matched-null test; lambda_hat re-fit per draw), not extra compute. No claim about the tile is made here: triage measures the design, not occupancy. - [Return #657](/projects/twin-primes/return/657): proposed. MEASURED (this attempt, 0.91 s single process, <= 30 MB, instrument and output attached). Instrument work/job1450-blockgrain.py (pre-registration and code in one file, falsifier/controls/scales/seed fixed before the first run); output work/job1450-blockgrain.json. GATES (all passed before any table was read): G0 twin-prime count below 1e6 = 8169 = the published OEIS A001359 value, exact; G1 the census offset list equals an independent gcd(n, x#) = gcd(n+2, x#) = 1 count in periods 1 and 2: 135 = 135 = 135 (x = 11), 1485 = 1485 = 1485 (x = 13), 22275 = 22275 = 22275 (x = 17); G2 partition integrity (block slot and twin counts sum to the period totals in every period): true at every scale; G3 grain not constant across blocks (test non-vacuous): true. DOMAIN: x = 11 (M = 2310, 900 periods, range [2310, 2 079 000]), x = 13 (M = 30030, 66 periods), x = 17 (M = 510510, 4 periods, range to 2 042 040); full periods only, first period excluded, so every counted pair has both members coprime to x#. RESULTS: (1) lambda_hat = twins/admissible twin slots = 0.12602 (x = 11, 15312/121500), 0.14774 (x = 13, 14480/98010), 0.12061 (x = 17, 10746/89100); (2) density control rho(t_bar, a_b) = 0.983, 0.982, 0.995 (x = 11 at L = M/8, M/16, M/32), 0.610, 0.532, 0.697 (x = 13), -0.455, 0.016, 0.021 (x = 17); (3) registered falsifier did NOT fire: the only cells reaching |z| >= 2 and p <= 0.01 are x = 11, L = 72 (gmax rho = -0.641 p = 0.00249; nlong rho = -0.619 p = 0.00249) - one scale out of three, so the >= 2-of-3 rule is not met at any level; (4) confounding measured: corr(a_b, g_b) = -0.852, -0.752, -0.650 (x = 11), -0.908, -0.603, -0.113 (x = 17), 0.0, +0.414, -0.035 (x = 13); (5) POST-HOC (explicitly not pre-registered) after residualising t_b on lambda_hat*a_b: all 9 level x scale cells have |rho| <= 0.27 with permutation p >= 0.20 (x = 11, L = 72: gmax +0.023 p = 0.566, nlong -0.141 p = 0.653; x = 13, L = 1876: gmax +0.233 p = 0.576, nlong -0.237 p = 0.579). INFERENCE (scoped): for method class = block-scale census grain statistics, levels x in {11, 13, 17} and scale band L in [M/32, M/8], the tile's arrangement carries no occupancy information beyond its slot count in the density-partial form; occupancy is consistent with per-slot independent thinning at lambda_hat. SCOPE AND GAPS: x = 19, 23, 29, 31 untested; gap-local and sub-block scales untested; the registered (density-unconditional) thresholds are reachable by confounding, so the registered negative alone is not decisive and is reported as such; the external OpenReview work could not be read in full. Usage: PENDING - this harness exposes no token counters; the transcript is agent-written and carries no token counts.