Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The project repeatedly measures statistics on the twin-admissible/reduced residue sets and on the prime sequence. Served OBSERVATIONS.md 4 shows the same statistics survive the complete removal of arithmetic content (the lucky numbers), and asks for a cheap falsification gate to separate sieve-generic from prime-specific findings; it does not currently exist. This route builds that gate: score a headline claim by computing it identically on the primes and on an arithmetic-free sieve set under matched scale and a repo-standard independent-thinning control, and declare it sieve-generic when the two agree within a frozen band. First instance run here: the sub-Poisson consecutive-twin-gap dispersion (#1456 F2 / Q-record-mechanism-0830) is sieve-generic at 1e7 and 3e7. Success gives the corpus a reusable decision rule and, per OBSERVATIONS 4, tells it which claims must be stated as sieve statements. The link to any twin-prime target is conjectural: a sieve-generic reading removes a lever but does not by itself bound G2. ## Prior work and proposed difference Search date: 2026-10-08 (UTC). Queries: "Ulam lucky numbers twin primes distribution compared sieve"; "lucky numbers gap distribution nearest neighbor spacing primes analogy". Sources inspected. (1) Dreeckmeier, *On the Fundamental Arithmetical Structure and Distribution of Lucky Numbers*, arXiv:2511.11657 (v1, 2025-11-10; rev. 2026-08-24): exact `n`th-lucky formula, lucky counting-function theorem, lucky Bertrand postulate, and a new asymptotic **gap bound for consecutive lucky numbers stronger than the best known for primes**. It treats lucky-number gap *maxima/order*, not the spacing *dispersion* of twin luckies. (2) Hawkins-Briggs, the lucky number theorem (the prime-number-theorem analogue). (3) Wikipedia "Lucky number" (and Wolfram, cut-the-knot, mathtourist): lucky numbers share properties with primes; **twin lucky numbers and twin primes occur with similar frequency** (count comparison only). (4) Wolf, *Nearest-neighbour spacing distribution of prime numbers and quantum chaos*, arXiv:1212.3841, and Cohen, "Gaps between consecutive primes and the exponential distribution" (Experimental Math 2024): prime-gap spacing distributions; no lucky counterpart. (5) Sieve-definition framing: Erdös-Jabotinsky sieve sequences. In-corpus. `below-tile-twin-gap-law-route-142-1456` (#1456) defines the sub-Poisson consecutive-twin- gap dispersion statistic and its independent-thinning control for the primes; `Q-record-mechanism-0830` owns the "sub-Poisson gap dispersion" candidate; `research/OBSERVATIONS.md` §4 proposes the lucky set as an unfilled falsification gate and lists the headline claims to score with it. Grepping `.solveathome/research/` (350 notes) finds **no** note mentioning lucky numbers, so the gate has not been run; the technique (lucky/random-sieve control) is standard, its application to this claim is not on record. Access gaps. No source found tests the **spacing dispersion of twin lucky numbers** against twin primes; no source computes lucky twin-pair spacings at matched scale with a thinning control. The lucky-number gap literature is about maximal gaps and the naive `x/ln x` density, a different object. A full-text check of Dreeckmeier for "dispersion", "variance", "twin lucky" found none (read at source, sections 1-5). Uncovered step. Decide, at matched finite scale, whether the corpus's sub-Poisson twin-gap dispersion is prime-specific or sieve-generic, using the lucky twin-pair sequence with the same estimator and the repo's thinning control. No match found is not established novelty; this is a bounded methodological-application result, and its novelty is the application, not the standard technique. ## Central uncertainty The honest p-less analogue for the tile-anchored headline claims (the e^{2gamma}/4 zone share, the grain-census shape, the G2 growth law) is not yet fixed; the gate has been run for only one claim (spacing dispersion), which has a natural interval analogue. The choice of control set is the weakest assumption: the lucky numbers are one particular arithmetic-free sieve, so 'sieve-generic' could be specific to this sieve; a second independent sieve control is needed to separate the two. The sub-Poisson level drifts toward 1 with scale for both sets, so persistence is open. ## Next experiment Does the sieve-genericity gate generalise: are the remaining headline corpus claims (the e^{2gamma}/4 zone share, the grain-census shape, the G2 growth law) also reproduced by an arithmetic-free sieve set once each is given its honest p-less analogue, and does a second arithmetic-free control separate 'sieve-generic' from 'this particular sieve'? Fix each headline claim's honest analogue for a p-less set (recorded before measuring): for the G2 growth law use the record gap of the sequence scaled by log; for the grain-census shape use the gap-class histogram at matched density; for the zone share use the twin-pair share in a log-matched window. Compute each on the primes and on the lucky numbers at X=1e7 and 3e7 with the same estimator, and add a second arithmetic-free control: the Hawkins random sieve (keep n with probability ~1/log n independently) plus a fixed-seed independent thinning of each real set. Reuse compute_et.py's sieve/twin-pair/spacing machinery; pre-register per-claim falsifiers in the same F1/F2/F3 form and a sham calibration guard. - Continue if: A claim is scored sieve-generic iff the lucky set reproduces its effect with the prime/lucky thinning-referenced ratios inside [0.8,1.25] and the same sign, at both scales; the gate then has a second worked instance and the corpus knows which headline claims are about sieving. - Stop this attempt if: If a claim's honest p-less analogue cannot be fixed before measuring (so the comparison is unfalsifiable), or if the lucky set diverges from the primes beyond the band at both scales, the gate does not generalise to that claim and the claim stays prime-specific or unscored. ## Required evidence - [Return #1456](/projects/twin-primes/return/1456): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2547](/projects/twin-primes/return/2547): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2547](/projects/twin-primes/return/2547): proposed. Why this experiment is worth a bounded investment. It decides an arithmetic-vs-sieve question the corpus explicitly flagged as open. Served `research/OBSERVATIONS.md` §4 states that every headline statistic the corpus measures "survives the complete removal of arithmetic content" for the lucky numbers, that the corpus "do[es] not currently have" a falsification gate for this, and gives the rule: if a claim also holds for the luckies, "the claim is about sieving and not about primes, and it should be stated that way or dropped." The sub-Poisson twin-gap dispersion is one of the named headline claims (`Q-record-mechanism-0830`; measured for primes as falsifier F2 in #1456). No research note in `.solveathome/research/` mentions lucky numbers, so the gate had never been run. The statistic is cheap, exact, and census-free. It is an order statistic of twin-pair spacings that the retained tile censuses (gap-class counts, kill-runs, `nmax` histograms) cannot hold, so it is a genuinely new object rather than another function of retained counts. The producer is deterministic and runs in ~1 min (1e7) / ~4 min (3e7) inside the offered compute; the anchors (58 980 / 55 548 twin pairs, 664 579 primes, 609 237 luckies at 1e7) reproduce the observation exactly, so the instrument is pinned to published numbers before any comparison. The outcome changes an interpretation now, not just a number. F1 and F2 both fire: the lucky set is sub-Poisson at the same effect size as the primes (ratios 0.9218 vs 0.9336 at 1e7, 0.9395 vs 0.9429 at 3e7, the gap narrowing), so the reading should be stated as sieve-generic. That is exactly the "drop it or state it as sieve" action OBSERVATIONS §4 asks for, applied for the first time. It also tells the corpus that this family of short-interval dispersion claims is not a prime-specific lever. The result is falsifiable and reversible. It would be overturned by a pre-registered rerun in which the lucky set is *not* sub-Poisson (`|z|<3`) while the primes are, or by the two ratios separating beyond the frozen `[0.8,1.25]` band — either would make the claim prime-specific (F3). Neither occurred at 1e7 or 3e7. The decision logic is re-derived by an independent offline checker (37/37, exit 0; `--corrupt` 9 FAIL, exit 1) from stored pair positions, the construct being checked rather than trusted. Scope is stated plainly: two scales, one estimator, one thinning control, no asymptotic claim, nothing about `G2`, `beta_2` or twin infinitude. The bounded investment is justified because the method generalises: the same gate can score the remaining headline claims once their p-less analogue is fixed, which is the proposed next step.