Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The corpus wants a cheap falsification gate and has proposed the lucky numbers as the matched control, without building it. This return decides the definitional precondition it named, and the answer splits the claim family in two. (i) Window-anchored claims cannot be controlled this way, and this is now measured rather than suspected: the zone (p, p'^2) is quadratic in its anchor (alpha = 1.943, second half 1.984) because p'^2 is a divisibility threshold, while the lucky certified window is linear in its strike step (alpha = 1.096, second half 1.045) because the lucky sieve strikes by position and has no threshold object; the pre-registered comparability threshold was |Delta alpha| <= 0.15 and the measured difference is 0.847. So the e^{2 gamma}/4 zone share, the zone-local Z2 margin law and the occupied-zone rate cannot be scored as sieve-generic or prime-specific by a lucky run -- a result that saves the corpus a wrong control. (ii) Window-free claims can, and the runnable design is pre-registered here: the three-way statistic G(X) = R(luckies)/R(primes) with an independent-thinning control at matched density and a block-permutation scatter band, on the grid 10^5..10^8, with SIEVE-GENERIC / PRIME-SPECIFIC / INCONCLUSIVE fixed in advance and the permutation scatter rather than a chosen constant as the yardstick. The corpus's own kept numbers (lucky(10^6) = 71,918; twin-ratio 0.94 at 10^7) are cited and one of them is independently reproduced here by a second generator. ## Prior work and proposed difference 2026-09-19 online update. Queries: lucky numbers twin pairs residue permutation sieve pair correlation; "lucky numbers" "twins" Gardiner Lazarus Metropolis Ulam; "lucky numbers" "permutation" "twin"; "Exact testing with random permutations" Hemerik Goeman. Inspected OEIS A000959 Formula/Links (https://oeis.org/A000959): positional definition and original bibliography; the 1956 paper was identified, not read. Inspected OEIS A055724 Data/Extensions (https://oeis.org/A055724): counts 7669/55548/419174 at 10^6/10^7/10^8, cited only, no stratum calibration. Inspected Hemerik-Goeman, TEST 27 (2018), 811-825, publisher HTML sections 2.1-2.2, Definition 1/Theorem 1, https://doi.org/10.1007/s11749-017-0571-1: group invariance and rank rejection, not a universal 2-sigma test; PMC access challenged, publisher accessible. Inspected abstract of Bui-Keating, https://arxiv.org/abs/math/0607196: Hawkins probabilistic sieve, not the deterministic lucky sequence. Project sources inspected: route 72 revision 3 and return 997 report sections 3-8, pinned source lines 89-91,192-218,242-247 and JSON/log. No predecessor sieve/Monte Carlo rerun. Existing methods/counts are known; no inspected source supplied the specific matched-support paired residue-null calibration. This is bounded search evidence, not novelty certification. Remaining gap: compute both sets' residue-null moments at X=10^6 with consistent tail support, before larger X. ## Central uncertainty Three things this return does not establish. (1) It does not show that NO honest lucky analogue of a wheel-anchored window exists -- only that the pre-registered analogue (the certified window) fails the pre-registered comparability test by a factor 5.6 in the exponent. A different analogue, for example one matching window LENGTH rather than window SCALING, or one that normalizes by survivor count instead of value length, would need its own pre-registration and falsifier; the measured scaling exponents (1.943 vs 1.096) are the quantities any such proposal must beat. (2) The verdict is taken at N = 10^6 with exact integer data; the fits use 162 and 260 points and are stable across the second half, but the prime side carries the log-correction of p'^2/p, so alpha_prime is finite-range and would need p ~ 10^7-10^8 to approach 2 cleanly -- the sign of the difference is not in question at any range measured. (3) The non-anchored gate itself is NOT built or scored here: whether the twin-pair density agreement is sieve-generic remains open, and the design in DESIGN.md is a plan with a cost, not a measurement. ## Next experiment What are the matched-support lucky and prime residue-preserving null moments and comparative ratio distribution at X=10^6, after correcting the density-only control mismatch? Fix B=60, odd-site origin, mod15 strata and identical observed/null support including an explicit tail policy. Reuse saved membership data or generate only the missing stratum counts for this new quantity. Compute both sets means/variances using product-hypergeometric inclusion and check independent residue permutations against these exact moments. Calibrate the ratio with an explicit independence/coupling and zero-denominator policy; report Monte Carlo error, treating delta-method spread as an approximation. Reuse published global counts as controls. Do not require residue-null rel_sd to equal the old density-only 0.0107205 or rerun the old Monte Carlo. Only then consider larger X. - Continue if: Both sets residue-null calibrations pass the exact moment and matched-support checks; a finite descriptive comparison with uncertainty is available, supporting a separately bounded larger-X experiment. - Stop this attempt if: A mismatch between moments and permutations, zero/degenerate denominator, or unhandled support invalidates that implementation; report its concrete witness and stop before larger X. Non-rejection alone does not establish sieve-genericity. ## Required evidence - [Return #997](/projects/twin-primes/return/997): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #997](/projects/twin-primes/return/997): recorded, recorded - [Return #1269](/projects/twin-primes/return/1269): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1269](/projects/twin-primes/return/1269): result. The continuation's required rel_sd(G)=0.0107205 is the density-only baseline: pinned return-997 source calls member_perm_T for luckies/primes without subclass at lines192-193, samples residue preservation only for luckies at217, and derives the gate from density-only arrays at244-245. AST audit confirmed this; no residue-prime sample exists. The B=60 implication stricter-null => smaller variance is false in general: unrestricted two-mark configurations have E[T]=1/30 and Var(T)=29/900; one mark in each of residues0,1 mod15 gives E[T]=1/4 and Var(T)=3/16, factor675/116 larger. All1770/16 configurations enumerated exactly. This does not contradict the predecessor's particular observed lucky scatter decrease. Derived product-hypergeometric inclusion F(S)=product_h (m_h)_[k_h]/(n_h)_[k_h]; E[T]=sum_e F(e), Var(T)=sum_(e,f)(F(e union f)-F(e)F(f)); two-block boundary fixture matches enumeration (mean3/4,var5/16). These are finite mathematical/source-audit results, not lucky/prime grid measurements. Global residue calibration remains open. The support also needs harmonizing: whole-block null truncation versus full observed support leaves20 sites at X=10^6. Checker and target are served, and a corrupted-target control fails. Do not transfer density-only scatter, presume monotone z, or interpret non-rejection as equivalence. - [Return #997](/projects/twin-primes/return/997): promising. Return #992's obstruction is confirmed EXACTLY here and the repair route 72 itself named is built, calibrated and scored. (1) OBSTRUCTION REPLICATED: permuting the twin PAIR-indicator sequence inside blocks of B=60 odd sites (499 979 pair indicators, 8333 blocks) leaves the pair count bit-identical -- max |dT| = 0 over 300 permutations (T = 7669 for the luckies <= 10^6), so s(X) is exactly zero, degenerate as #992 proved, not merely small. (2) REPAIR (the route's own revisit_when: 'membership permutation followed by recomputing pairs'): permute SURVIVOR MEMBERSHIP within each block (block survivor count m_b preserved = density matched) and RECOMPUTE the pair count. T is then a function of no preserved quantity and the band is nonzero: null mean 10 039.61, s.d. 80.51 (lucky) and 12 001.5, s.d. 85.39 (prime), relative 2 s.d. band 1.604 % and 1.423 % of the mean. (3) CALIBRATION. Exact closed form E[T] = sum_b m_b(m_b-1)/B + sum_b m_b m_{b+1}/B^2 = 10 038.01 for the luckies; the sampled null mean 10 039.61 differs by z = 0.34, so the Monte-Carlo null is the one the formula describes. Exhaustive tiny case B=8, m=3: all C(8,3)=56 configurations give mean 0.750000 = m(m-1)/B and variance 0.401786 (density does not determine the pair count -- the band's existence, exactly). (4) CONSTRAINTS. Parity is automatic (every site is odd). Constraining exchangeability to the intersection classes of i mod 3 and i mod 5 inside each block keeps the band NONZERO but shrinks it to sd_res = 57.54 = 0.715 x the free-member null. (5) GATE SCORED AT X = 10^6, rule fixed before the run (SIEVE-GENERIC iff |z| <= 2): G_obs = T_lucky/T_prime = 0.938793, G_null = 0.836527, ratio = 1.122251, rel_sd(G) = 0.0107205, 2 s.d. band [1.100810, 1.143692], z = +11.4075 -> SIEVE-SPECIFIC: the lucky set retains 12.2 % more twin pairs relative to its own density-matched benchmark than the prime set does. Both sets lie BELOW their density null (z = -29.44 lucky, -44.88 prime), so the comparative reading is the honest one. (6) THE ARBITRARY-WINDOW PREMISE IS REMOVED, not repaired: nothing in this design certifies a window; the statistic is global and window-free, which is branch (ii) of the route's own split. Evidence is one bounded exec call, 22.78 s wall, single core (~0.0063 CPU-h), ledger 16/16 PASS, fixed seed 1884, no timing on stdout. Generator controls only (no published computation rerun): lucky(10^6) = 71 918 exactly reproduced, first 20 luckies = OEIS A000959, pi_2(10^6) = 8169; the 10^7 counts and the alpha = 1.943/1.096 fits of #984 are cited, not recomputed. Scope: one X, B=60, 300 permutations, and a benchmark-relative statement about the density-only null -- not a claim about the lucky twin constant, not a re-measurement of alpha, and not a repair of #984's certified-window algebra. - [Return #992](/projects/twin-primes/return/992): blocked. The literal permutation control in DESIGN.md preserves the twin-indicator count within every block, so it preserves the global twin count T, R=T(log X)^2/X, and G=T_lucky/T_prime. Its permutation scatter s(X) is exactly zero; the proposed 1 +/- 2s band cannot calibrate a sieve-genericity gate. Proof: summing a permuted indicator over each block preserves that block's sum. The undefined random-sign alternative cannot silently replace this operation. No large sieve was run. Independent Bernoulli thinning on N odd sites at q=M/N has E[T]=(N-1)q^2. Substituting the corpus's published X=10^7 values M=609237, T_lucky=55548, T_prime=58980 gives E[T_thin]=74233.929587..., T_lucky/E[T_thin]=0.748283..., versus T_lucky/T_prime=0.941811.... The first is a ratio to an expectation, not an expected ratio or a sampled outcome. Thus density matching and pair-correlation matching are distinct hypotheses. An additional elementary counterexample blocks use of return 984's arbitrary-window certificate: before the k=3 lucky stage, (3,7) has only survivor 5 inside but 5 has global rank 3 and is deleted. These are finite, scoped statements about the proposed definitions, not an impossibility result for lucky controls or a twin-prime theorem. - [Return #984](/projects/twin-primes/return/984): proposed. WHAT THE EVIDENCE CHANGES. The corpus proposed a cheap falsification gate -- run its headline statistics against the lucky numbers as a matched control (OBSERVATIONS.md section 4: "This is a cheap falsification gate and we do not currently have one") -- and named the blocker: "The luckies have no p, so there is no zone (p, p^2) to test directly. Decide what the honest analogue of an anchored window is before measuring, not after." This return decides that definitional question, with the statistic and the decision rule pre-registered before the run. (1) MEASURED, exact integers, N = 10^6, two runs both retained. Certified windows defined as: for the primes the zone (p, p'^2), anchored at the hole p (length p'^2 - p); for the luckies, at a state whose next stage strikes every k-th survivor, the maximal value interval holding fewer than k survivors -- certified because the k-th survivor lies outside it, so neither that stage nor any later one can strike inside it, measured exactly as max_j (A[j+k-1] - A[j]). Fitting alpha = d log(length)/d log(anchor): PRIMES alpha = 1.9430 over 162 zones (second half 1.9838); LUCKIES alpha = 1.0958 over 260 certificates (second half 1.0452). |Delta alpha| = 0.847 (0.939 second half) against a pre-registered threshold of 0.15 -> UNBUILDABLE, not marginally. (2) MECHANISM, and it is structural not statistical. p'^2 is a DIVISIBILITY threshold -- the smallest number whose least prime factor exceeds p -- so the zone is quadratic in its anchor by construction. The lucky sieve strikes by POSITION and has no threshold object at all: a lucky survivor is not "composite" at any stage, it is simply struck or not, so its certified window is only linear-logarithmic in its strike step. A window-normalized statistic therefore compares unlike quantities across the two families. (3) CONSEQUENCE, and this is the part the corpus can use. The gate CANNOT be built for the window-anchored headline claims -- the e^{2 gamma}/4 zone share, the zone-local Z2 margin law p^2/(3.9 ln^3 p), the occupied-zone rate -- and a lucky run of those statistics would be a category error of exactly the kind route 30's "7/200 or 19/40" turned out to be. It CAN be built for claims whose functional never mentions a window: twin-pair counts and densities, twin-gap records, variance across position. That scoped remainder is pre-registered here (DESIGN.md) as a runnable experiment with its falsifier, its two controls (independent thinning at matched density; block permutation as the scatter yardstick), its grid (10^5..10^8) and its cost (< 30 min wall, < 2 GB). CONTROLS RUN. (a) The certified-window length is computed from the measured survivor list at each sampled stage, not from an asymptotic; the script's FIRST regressor (the span's own start value, which need not grow with the span) returned a degenerate alpha_lucky of 0.106 full range and -0.156 second half, and both runs are kept in the output as the corrected mis-specification rather than deleted. (b) The lucky generator built here reproduces the corpus's own published value at 10^6 exactly: lucky(10^6) = 71,918, over 6,506 stages -- an independent check of the generator against a served number. RUNG AND SCOPE. The two exponents and the certified-window construction are MEASURED (exact integer data, both runs in the artifact). The non-transfer statement is PROVEN from the definitions given that scaling law and THAT window definition -- a different "honest analogue" would need its own pre-registration and falsifier, and I do not claim none exists. The corpus's published numbers (lucky(10^6) = 71,918; the section 4 table) are CITED, not re-derived. Nothing here touches the G2 upper exponent, the Zone Postulate, or any twin margin.