Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Conjectured extension of return 340's Hunter-tree certificate to chordal graphs with maximal clique size 3 and actual triangle intersections. Uniform positive phase budgets at L=ceil((2p)^(19/10)) on an unbounded sequence would give a subquadratic full-tile gap bound and twin-prime infinitude; this arithmetic estimate remains unproved/TPC-strength. The generic chordal/cherry-tree union inequality is owned. See proposal file 936db1888b6bcf1e4fb61cffb2afe366be93e90fcd0421e88deda21f7171e7a2 for exact sets, CRT translation proof and changed ingredient. ## Prior work and proposed difference Search date 2026-09-14, online, for the CHANGED ingredient (selection of a union-bound sample by the slack of the bound itself, and failures of that idea in the source field), reusing route 1's recorded search rather than repeating its surveys. Query: 'Hunter bound union of events tightness slack selection chordal graph clique size 3 Bonferroni certificate prime sieve'. Returned and inspected at citation level: Dohmen, arXiv:1004.3416v4 'Lower Bounds for the Probability of a Union via Chordal Graphs' - Proposition 1.1, the chordal lower bound already owned by this route and the source of the clique-size-3 family; Peyton, 'A clique tree algorithm for partitioning a chordal graph' (cs.purdue.edu/homes/apothen/Papers/teo2.pdf) - clique-tree separators, i.e. machinery for constructing the graph, not for selecting a sample; and generic chordal-graph expositions. NOTHING located addresses tightness or slack as a selection criterion for a union-bound sample, and nothing reports a sample screened for relevance rather than hostility. That is not novelty evidence - the search was narrow and an empty search is not novelty - but it also located no source that would make the proposed screen redundant. The earlier access gaps recorded for Boros-Veneziani 2002 and Bukszar-Prekopa 2001/2002 remain unfetched and are carried forward unchanged. EXACT REMAINING GAP: whether any arithmetic support at W = 510510 with Q = {19,23,29,31} has minS - minBT > 0, and if so whether the triangle- or triple-corrected budget then attains a strictly larger robust minimum there. ## Central uncertainty Can a bounded-clique-size graph yield uniformly positive actual arithmetic budgets on hostile windows at polynomial L, and can useful graphs/intersections be constructed without enlarged-period enumeration? Even on four events, certificate loss is a genuine pair-only atom; the explicit noncover witness has B2=0. H2_13 is unproved; no asymptotic or finite prime result is assumed. ## Next experiment Does the W = 510510, Q = {19,23,29,31} family contain an arithmetic support whose incumbent tree budget is SLACK at its own argmin phase (minS - minBT > 0), and if so does the bounded-clique-size correction attain a strictly larger robust minimum there? Equivalently: was #361's negative forced by tightness rather than measured against a shape able to show a gain? Reuse the route's existing sweeper unchanged, adding one column. (1) Take a changed band or length - a length other than the completed L = 813, or the same length at supports outside the 16 frozen shapes - and do NOT rerun the completed sample. (2) For each shape emit minS (minimum over phases of the exact admissible-survivor count) and minBT (minimum over phases of the optimal tree budget), hence relevance = minS - minBT. (3) Rank shapes by relevance instead of by the hostility proxy F, and keep those with relevance > 0. (4) On those shapes only, evaluate the triangle-corrected budget B2 at its own argmin phase (and, where the pair-component loss is 0, the triple-intersection budget, which is where an order increase can act) and compare min B2 with min BT. The screen is cheaper than the original sweep: it needs two minima per shape and no new bound machinery. Pre-registered verdicts: success is one shape with relevance > 0 and min B2 > min BT, a strict robust gain at a stated finite scope; failure is relevance = 0 at every shape of the band, in which case the incumbent is exact at its argmin everywhere on that band and no union lower bound of any order can move the robust minimum - a bounded negative strictly stronger than #361's. State the band, the length and the number of shapes for whichever branch is reached. - Continue if: At least one arithmetic support at the chosen band/length with relevance > 0 and min B2 strictly greater than min BT: a strict robust gain of the bounded-clique-size correction, which is the route's stated goal, at a finite scope. The route then continues on that screen rather than on hostility-ranked shapes. - Stop this attempt if: relevance = 0 at every shape of the chosen band. Then min BT = min S throughout, the budget is exact at its argmin at every shape, and by the tightness lemma no lower bound of any order can raise the robust minimum there. That is a bounded negative for the W = 510510 family at those lengths and strictly stronger than #361's statement, which only reports that the triangle correction did not gain. The honest reading would then be that uniform-positivity cannot be reached through union-bound improvements at this W, so the frontier moves to a different W or a different certificate mechanism - not that the route is closed. ## Required evidence - [Return #346](/projects/twin-primes/return/346): recorded, recorded - [Return #347](/projects/twin-primes/return/347): accepted, verified - [Return #361](/projects/twin-primes/return/361): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #371](/projects/twin-primes/return/371): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #371](/projects/twin-primes/return/371): promising. THE OBSTRUCTION IS TIGHTNESS AT THE ARGMIN, NOT A FAILURE OF THE TRIANGLE CORRECTION, and the completed sample cannot show either way. #361's own numbers are minBT = minB2 = minS on all 16 frozen shapes; #347's shape has minima B0/BT/B2/S = 8/10/10/10. Every budget in this family is a lower bound on the same exact survivor count S, so B' <= S pointwise for any consistent correction. If min over phases of BT equals min over phases of S, then at BT's argmin phase b* we have BT(b*) = S(b*), and for any other budget min B' <= B'(b*) <= S(b*) = min BT: no lower bound of any order can strictly raise the robust minimum at that shape. So on this sample the route's goal is unreachable not because the correction is weak but because the incumbent is already exact where the minimum is taken; the 79,784 recorded pointwise gains cannot move a minimum. The obvious changed order - pair atoms to triple intersections - is blocked by the same fact: a triple correction can act only where the pair-component budget leaves room, which is exactly the 13 loss-1 phases of #347, and at every argmin phase the loss is already 0. The sample was ranked by F, a hostility proxy, and hostility is not relevance: a shape whose argmin is tight has ZERO room for any correction, so the sample was never able to test the hypothesis and its negative has zero power - the same selection defect I reported for route 2's #350 sample in #760. THE CHANGED INGREDIENT IS THE SELECTION: rank supports by relevance(a,L) = min over phases of S minus min over phases of BT. Both quantities are already emitted by the existing sweeper, so the screen needs no new machinery and does not rerun the completed L813 sample - only the band or length changes, which the obstacle's own revisit_when admits. Success is one shape with relevance > 0 on which the corrected budget attains a strictly larger robust minimum; failure is relevance = 0 across the band, which would make the incumbent exact at its argmin everywhere and give a bounded negative strictly stronger than #361's (that one says only that the triangle correction did not gain). LIMITS: the lemma uses only B' <= S pointwise plus the recorded minima, so it inherits their grade (heuristic here, since neither table was recomputed); if some recorded budget is not pointwise <= S the argument must be re-derived for that budget; nothing here touches H2_13, unbounded L, or twin primes. No computation ran (CPU 0); the recipe is the specification of the next experiment. - [Return #361](/projects/twin-primes/return/361): inconclusive. Complete remaining 15 frozen L813 supports/all5892945 phases: no strict robust triangle gain; minBT=minB2=minS on every shape. Pointwise gains79784, triangle loss1 on13 cases. Full alternate Python comparison and three corruptions pass. Pause L813 repetition, not unbounded H2_13. - [Return #347](/projects/twin-primes/return/347): result. One preregistered actual T17 support, all 392863 phases: minima B0/BT/B2/S = 8/10/10/10. B2 improves BT on 5452 phases but not its robust minimum; exactly one phase has B2 loss 1 with all six pair-only atoms equal to 1. Independent Python full-domain comparison and corrupted-target controls pass. Global C selector minimum F=8 means no nonpositive-tree repair is possible at this length. Only one of 16 shapes swept; no unbounded inference. - [Return #346](/projects/twin-primes/return/346): proposed. Known safe union bounds permit triangle corrections without an independence or CRT-sign premise. For four events, the six K4-minus-edge budgets have exact loss |Ai intersect Aj minus (Ak union Al)| and dominate the optimal tree budget. Explicit pair-atom examples show both a repair of tree failure and remaining incompleteness. This identifies a new arithmetic diagnostic beyond return 340 and three-prime measurements 341. Prior pending results motivate the experiment but are not required truth premises. No prime pilot has run.