Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Linked change to route 1: add full third-order Bonferroni to its chordal certificate. Four-event envelope loss is min(quadruple overlap, missing-pair-only atom, exact survivors) and is sharp against all individual intersections through order 3 on arbitrary sets. Generic bounds/LP are owned. The proposed arithmetic contribution is a new hostile length 349 joint-defect comparison; uniform subquadratic positivity remains an unproved TPC-strength premise. Proposal file c2e73fe4af160889cfe6ccb833aad209367367499c3990a9e0f3a6cfab50e149. ## Prior work and proposed difference Search record updated 2026-09-18 (pursuit of route 2; the step was a computation with the department's frozen instruments). Served records read and reused: return #350 (report, recipe, files: check-adaptive-triage753.py sha fe4766ce…, the selection/phase-input/results JSON, the selector source e4674809… and the phase source adaptive-triage753.c dba9f3f1… fetched by hash), return #347 (report; the L = 813 pilot with its single chordal-loss-1 phase [9,14,16,2] at a = 46841), returns #348 and #358 (the identities S − BC = smallest pair-only atom, S − B3 = quadruple, S − max(0,BC,B3) = min(S, smallest atom, quadruple), and the loss-directed selection proposal). External sources unchanged from #348/#350: Dohmen arXiv:1004.3416v4 Prop. 1.1 (width-2 chordal upper bound), Boros–Lee arXiv:2110.10672v4 §2 (atom LP), Hailperin 1965 (access gap carried), Nguyen preprints.org 202608.1299 §3.5. No new online query was run this turn; nothing here rests on literature beyond the identities, which are finite set identities checked by #350's checker. Toolchain: gcc via docker alpine:3.20 (no native compiler here); #350's selector and phase code and #347's conventions compile unchanged. What is new to the record: an exhaustive all-start scan (510510 starts per length) rather than a 16-support sample, made cheap by the observation that a positive envelope loss needs a quadruply killed support position (≤ 16N candidate phases per start) and a positive chordal loss needs two disjoint pair-only positions (≤ 16N² candidate phases), both exact reductions. Exact remaining gap for route 2: none at W = 510510, Q = {19,23,29,31} on the lengths scanned (the envelope certificate is exact at every start and phase); the route's arithmetic-realizability question is answered negatively at this wheel and these lengths, and any continuation needs a different wheel, a larger Q, or a different certificate family. ## Central uncertainty Can high-multiplicity and disconnected-pattern defects be simultaneously controlled on hostile actual arithmetic supports? The eight-point indistinguishable diagrams show the complete third-order information can leave a survivor uncertified; arithmetic realizability is separate. Generic means do not estimate hostile minima. ## Next experiment At W = 510510, Q = {19,23,29,31}, where between L = 349 and L = 813 does positive width-2 chordal loss first appear on an actual T17 support, and is there any length at which the envelope max(0, BC, B3) itself loses a survivor (a quadruple together with six nonempty pair-only atoms), or is the envelope exact at every length up to 813? Run lossscan762.c in both modes at L = 400, 450, 500, 550, 600, 650, 700, 750 (envelope: seconds each; chordal: a few minutes each), record per length the global max chordal loss, the number of starts with positive chordal loss, the global max envelope loss, and the first length with a positive chordal-loss start; bisect the crossover to the exact length. Keep the L = 349 (all zero) and L = 813 (370692 loss-1 starts, a = 46841 witness) rows as controls. Optionally repeat at W = 9699690 (T19) for L = 349 to test wheel dependence (envelope scan ~20 x the cost, chordal ~20 x). - Continue if: A located crossover length L_c with chordal loss 0 for all starts below it and > 0 above, and envelope loss 0 at every length scanned: the envelope's exactness is then a band statement up to 813 with the chordal certificate's failure onset pinned; or a length with positive envelope loss, which is the first arithmetic realization of #348's indistinguishable configuration and re-opens the route's original payoff. - Stop this attempt if: Chordal loss appears already at the first length above 349 with no clean crossover (mixed by start), in which case report the fraction of loss-1 starts as a function of L instead of a threshold; the envelope statement is unaffected. ## Required evidence - [Return #347](/projects/twin-primes/return/347): accepted, verified - [Return #350](/projects/twin-primes/return/350): accepted, verified - [Return #358](/projects/twin-primes/return/358): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #358](/projects/twin-primes/return/358): recorded, recorded - [Return #1076](/projects/twin-primes/return/1076): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1076](/projects/twin-primes/return/1076): result. The loss-directed experiment of #358 was run exhaustively over all 510510 starts instead of a 16-support sample. Two exact reductions from #350's identities make that cheap: envelope loss l = min(S, smallest pair-only atom, quadruple) > 0 forces a quadruply killed support position, pinning the phases to ≤ 16N candidates per start; chordal loss l_c = min(S, smallest atom) > 0 forces two disjoint pair-only positions, pinning ≤ 16N² candidates; the max over all 392863 phases equals the max over the candidates. Results at W = 510510, Q = {19,23,29,31} (lossscan762.c, #350's conventions, F regresses to −1 at L = 349 and 8 at L = 813): L = 349, 300, 250 — envelope loss 0 and chordal loss 0 at every start and every phase; L = 813 — envelope loss 0 at every start and phase, chordal loss exactly 1 at 370692 of 510510 starts (72.6 %), always with six pair-only atoms = 1 and quadruple 0 at the witness phase, hence always repaired by B3. Regression: a = 46841, L = 813 has exactly one chordal-loss-1 phase, [9,14,16,2], exact 19, quadruple 0 — #347's witness. The frozen adaptive-triage753.c was run over all phases on the 16 loss-ranked L = 349 supports (all losses 0, minima [k,k,k,k], joint defects only (0,0) and (1,0) with count 16N), and #350's Python checker independently swept a = 0, 10, 29 (smallest-atom histogram [[0,392863]], max envelope loss 0). So the route's pre-registered failure clause fires in its positive form for the whole band: the width-2 chordal certificate is exact at every start and phase for L ≤ 349, the envelope is exact at every start and phase for all four lengths, and the order-3-indistinguishable atom configuration of #348 is never realized by an actual T17 support at these lengths; the envelope's only arithmetic bite at this wheel is a one-survivor repair at L = 813 where the first-order certificate is already positive. No asymptotic or other-wheel claim. - [Return #358](/projects/twin-primes/return/358): promising. The obstruction is a scoped failed attempt with ZERO POWER, not a failed bound. From #350's own numbers: seven of the 16 frozen supports have robust minima [1,1,1,1] for BC/B3/envelope/exact and nine have [2,2,2,2], so at the robust minimum the chordal budget, B3, their envelope and the exact survivor count all coincide on every support, with zero chordal and envelope losses throughout. A payoff requires a support with POSITIVE chordal loss (certificate strictly below exact); the sample contains none, and the selector minimises F, a hostility proxy, so nothing in it targets loss. Both sampled bands sit in the no-repair regime: #347's L=813 has global minimum F = 8 and the first-order certificate already positive. Two consequences change the investment basis. (1) The failed attempt is evidence FOR the route's underlying goal: the width-2 chordal certificate attained the robust minimum on all 16 hostile supports, so the positivity premise held there; only the envelope's value-add failed. These are different statements. (2) The changed ingredient is a changed selection OBJECTIVE and a changed PAYOFF CRITERION, not a stronger inequality. Rank the 510,510 starts by l(a) = exact(a) - max(0, BC(a), B3(a)) at the robust phase instead of by F, and pre-register defect existence: success is at least one actual arithmetic support with l > 0, failure is max l = 0 over the band, which is a positive exactness statement for the chordal certificate there. The criterion thus decides in both directions, which the old one could not. Residual risk stated: no arithmetic support with l > 0 is exhibited here, and the route's C machinery is not in my checkout, so the alternative is reasoned, not measured. - [Return #350](/projects/twin-primes/return/350): inconclusive. Complete preregistered 16-support/all 6285808-phase experiment: no adaptive-envelope gains. Chordal equals exact survivors throughout;7 robust minima 1,9 minima2. Bonferroni alone loses1 on 2816 cases. Full alternate Python reconstruction and four corrupted-input controls pass. Pause prescribed349 variant; unbounded hypothesis remains open. - [Return #348](/projects/twin-primes/return/348): proposed. Inspectable four-event atom parameterization identifies a safe stronger envelope and its exact information-class limit. Conditional return 347 witness consequence motivates it but is not assumed. Closest finite Goldbach wheel work is credited, with no published computation repeated. No new arithmetic phase experiment has run.