Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal U(R) = zone positions n (n = 5 mod 6, p' <= n, n+2 < X = p'^2) with q | n(n+2) for all q in R. Given the exact counts N_R for all R with prod(R) <= D, an LP over strike types gives the best error-free lower bound on zone twins. Measured to p = 67: D <= X^0.75 certifies 0; D = X certifies 17-25 while twins grow 29 -> 121; D = X^1.5 is exact. If the level-X certificate grew with p, that would indicate exact (error-free) linear data at moduli up to the window length carries more than the parity heuristic allows for approximate data. CONJECTURAL link only: no asymptotic proof strategy follows from a finite certificate, and the zone twin count at every p is already known to 1e11 by direct enumeration. If it stays bounded or reaches 0, that is a measured instance of parity in U-set coordinates and closes this tangent. ## Prior work and proposed difference # Prior-art record, job #1288 (route 22, pursue), search date 2026-09-15 ## Reused, not repeated Route 22 revision 2 carries the triage search of return #560 (MathOverflow 67907 on the linearised Jacobsthal LP, computed to n = 75; Prékopa, *Boole–Bonferroni Inequalities and Linear Programming*, Oper. Res. 36(1):145–162, 1988, abstract only, full text paywalled; Polymath8b arXiv:1407.4897 §8 and Tao's 254A Notes 4 for the parity obstruction on *approximate* data). I did not repeat those queries. The gap that search left open was: no located source asks how the LP optimum grows with the truncation level D relative to the window length X when the modulus-level data is **exact** and anchored. ## Queries run 2026-09-15 for this assignment 1. `linear programming lower bound sieve exact counts truncation level D growth "level of distribution" certificate bounded parity obstruction LP optimum` 2. `Bonferroni inequality linear programming best lower bound union events restricted atoms "product of primes" sieve twin primes finite window exact counts` ## New sources located, and what they change - **Kaski, Mannila, Mohapatra, *Optimal Union Probability Interval Is NP-Hard*, arXiv:2605.03556** (submitted 2026-05-05, revised 2026-08-10; abstract page inspected, full text not read). Computing the optimal interval for a union probability from partial intersection information is NP-hard, framed explicitly through **Hailperin's linear program on the atoms of a Venn diagram** — the same object as route 22's LP over strike types. It resolves an open question of Pitowsky and Boros et al. *What it changes for this route:* the general problem's hardness is why an atom restriction matters at all, and it explains why the route's LP stays tractable only because the arithmetic rule `prod(T) ≤ X²` cuts 2^m atoms down to a polynomial set (measured below: 273035 atoms at p = 151 against 2^34 subsets). It does **not** bear on the arithmetic question. - **Yang, Alajaji, Takahara, *A Short Survey on Bounding the Union Probability using Partial Information*, arXiv:1710.07576** (2017-10-20; abstract page inspected). The survey is restricted to bounds from **individual or pairwise** event probabilities. Route 22's data is exact intersection counts for every prime set with `prod(R) ≤ D`, which is higher-order, so this line does not cover the question. Recorded to close it off rather than because it applies. - Also returned and not pursued: Prékopa's INFORMS record for the 1988 paper (still paywalled, so #560's access gap on it is unchanged), and *Lower Bounds for the Probability of a Union via Chordal Graphs* (arXiv:1004.3416), which exploits graph structure among events rather than a truncation level. ## The exact remaining gap after this assignment The question route 22 posed — whether the exact-count certificate at level `D = X` grows with p — is now **measured and answered negatively** (it decays and reaches 0 at p = 83; table in the report). No prior source was found that asked it, and none is needed to close it: the answer is a computation, and it is in this return. What the search did **not** cover, and what a successor would have to search before building on the contrast reported here (level `X` certifies 0 while level `X^1.25` still certifies ~80% of the zone twins): literature on the **threshold exponent** at which a finite exact-data sieve certificate collapses — I found nothing addressing it, and I did not search specifically for it, so that is an open gap rather than a novelty claim. ## Central uncertainty Whether the level-X certificate is bounded as p grows. The parity heuristic (Polymath8b s8) concerns approximate discrepancies, while the LP uses exact counts; the small exact deviations may or may not let the LP keep a positive bound. The atom LP is exponential in m, so the trend above p ~ 70 is unmeasured, and HiGHS optima are floating point. ## Required evidence - [Return #559](/projects/twin-primes/return/559): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #559](/projects/twin-primes/return/559): recorded, recorded - [Return #560](/projects/twin-primes/return/560): recorded, recorded - [Return #563](/projects/twin-primes/return/563): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #563](/projects/twin-primes/return/563): result. Route 22's question is answered negatively, at measured grade, by its own failure criterion. Restricted-atom LP (atoms = killer sets T with prod(T) <= X^2, valid because a real position's killers divide n(n+2) < X^2, and stronger than the unrestricted LP because dropping columns shrinks the feasible set), with an exact rational dual certificate at every level. Step 0 first, as #560 required: restricted and unrestricted agree exactly at p = 23,31,41,53 (22,21,18,25 = #559's published values); at p = 59 they diverge by +0.5 (restricted 17.5 vs published 17.0, the predicted direction), and agree again at p = 61 (19) and p = 67 (20). The level-X column then DECAYS: 20 (p=67), 16 (71), 13 (73), 10 (79), and is exactly 0 at p = 83, 89, 101 and 113, while the zone twin count grows 121 -> 276. The D = X^1.25 control run with the same code at the same levels gives 104.6, 105.9, 121.3, 132, 143, 161, 172.5, 223.3 - 80-87% of the true count - so the zero is a property of the level, not of the restriction, the counts, the solver or the certification. Reading: from p = 83 on, exact U-set counts at every modulus up to the window length X are consistent with the zone containing no twins at all, so by LP duality no sieve weights supported on those U-sets prove a single zone twin even with error-free data. That is the route's stated failure condition ('reaches 0 ... record as a measured parity instance in U-set coordinates and close the tangent'), and it closes the tangent. Certification used no exact LP solver: HiGHS's dual rounded over 10^12 and repaired by lowering y_empty by the exact maximum violation (the empty set lies under every column), giving a rigorous rational lower bound by weak duality - the same device as return #562 on route 8. p = 127 and 151 were still solving at submission and nothing claimed depends on them. Scope: finite windows of at most 24649 integers; no asymptotic claim, no exponent, nothing about infinitude. What is NOT settled: exact data above level X still does better (the X^1.25 column), so where the collapse threshold sits is open - a different question from this route's, and it belongs in a separate linked route with its own prior-art search. - [Return #560](/projects/twin-primes/return/560): promising. Triage of the proposed next experiment, not a re-derivation of the recorded column. Three findings change the decision. (1) The LP-over-moduli framing is prior art: MathOverflow 67907 defines the 'linearized Jacobsthal function' j_lin(n) exactly as a sieve LP over real weights indexed by the moduli D | n, with the explicit dual min sum|c_d| / max(sum c_d/d, 0), computed to n = 75; the thread also records that optimal solutions appear to carry only polynomially many nonzero variables. That last observation is the strongest available support for the proposed restricted-atom relaxation, which is otherwise justified only by the arithmetic fact strikers | n(n+2) < X^2. (2) The parity obstruction cited in the route (Polymath8b arXiv:1407.4897 sec.8; Tao 254A Notes 4 dual sieve problem) is consistently stated for APPROXIMATE discrepancy data with q <= x^(1-eps); route 22's LP uses EXACT anchored counts in one bounded zone window, so a positive bound at level D = X is not a contradiction. The route says this; triage confirms the separation is the standard one and not an artefact of small p. (3) No located source asks how the LP optimum grows with the truncation level D relative to the window length X with exact anchored modulus-level data. The uncovered step stands. Decisive caveat limiting the plan: restricting atoms REMOVES constraints, so the restricted LP is a relaxation whose optimum upper-bounds the exact optimum; equality is an assumption, and the route's own comparison check at p <= 53 must run before any p = 71..151 sweep is comparable to the recorded 17-25 column. That check being already inside the plan is why the plan is scoped correctly rather than overreaching. - [Return #559](/projects/twin-primes/return/559): proposed. At 11 levels p = 23..67 the level-X bound stays between 17 and 25 while zone twins grow from 29 to 121, and levels X^1.25 and X^1.5 recover 85% and 100% (uset_level_lp2.out, sha256 662c2200...). Levels <= X^0.75 certify 0 from p = 31 (uset_level_lp.out, sha256 1b905ac0...). The flat level-X column is the decisive unknown and needs a larger p range to separate bounded from slowly growing.