Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal One question, one instrument, one price list. The instrument: the exact parameterisation of the served right-hand budgets by the split (U=V=x^w, Y=Z=x^y), giving delta <= 1-w and delta+3nu <= 2-w-3y, with the served region {delta<19/25, delta+3nu<161/100} as the case w=6/25, y=1/20 (note N-1526-01; reproduced here in exact rationals). The question: is w=6/25 forced from below by the named d-edge input class, or is it an unoptimised interior point? The price list: for each named input - the Type II bilinear estimates against a shifted divisor-type coefficient (Deshouillers-Iwaniek, Drappeau, Topacogullari, Pascadi, and the endpoint note's own Corollary 1 of arXiv:1502.00769v1) - the smallest admissible w, i.e. the largest region its stated hypotheses support, recorded as a table. Three reusable facts fall out of the instrument alone and are already checked here: the served witness (8/25,11/25) enters region three iff w<21/100 strictly, missing at the served choice by exactly 3/100 of cutoff exponent; the implied level D=x^(2w) falls with w and is 12/25=0.48<1/2 at the served choice, so the level of distribution is NOT the binding side of the question near the ceiling; and the product-exponent ceiling allowed by the two right-hand inequalities alone is 4/3-w-y, which is vacuous on its own (1.0433>1 at the served point) and must never be quoted without the full served constraint set including de^3<=x^(321/200). Either outcome is a contribution: a lower bound w>=6/25 closes the parameter axis with the exact hypothesis responsible, and the absence of one makes the region re-derivation at w=1/5 or w=1/8 a bounded, exact-arithmetic deliverable rather than a research hope. ## Prior work and proposed difference Online search updated 2026-09-22: the route's prior art is inside the corpus (#736, #737), and this run read the named inputs' abstracts through the arXiv API (export.arxiv.org, title query, 4 entries: Bettin-Chandee 1502.00769v1, Topacogullari 1605.02364v1, Pascadi 2404.04239v3, Drappeau 1504.05549v4) and the Bettin-Chandee introduction through ar5iv (Theorem 1 (1.2): B(M,N,A) << ||alpha|| ||beta|| ||nu|| (1+|theta|A/MN)^{1/2} ((AMN)^{7/20+eps}(M+N)^{1/4} + (AMN)^{3/8+eps}(AN+AM)^{1/8}), no range restriction stated; Corollary 1 (1.4) as in #737's excerpt; the Duke-Friedlander-Iwaniec 1997 error term (19/8, 3/8, 11/48) quoted there for comparison). No source optimises a Vaughan split against these bilinear inputs for the fixed-shift product; the classical statement that a two-cutoff Vaughan identity leaves the Type II band [min(u,v), 1/2] and that Heath-Brown's identity narrows it is in the corpus's own heath-brown-edges.md section 3 (with the served front end at 6/25 "below even the best Vaughan value"). No novelty is claimed: the threshold formula is arithmetic on the served inequalities. Project sources inspected: route 46 rev 2; #737 (job1530-checks.py, the four edges and the cuts; job1530-report.md; the BC excerpt), #736, #727; residual-coverage.md sections 1 and 3 (the definitions w, v, U, V, Y, Z, the limiting rectangle (1), the budget table for both orientations, the simplification to (11), the corner (47/150, 67/150) and the regional supremum 87/100), grouped-divisor-moment.md sections 4-5 ((14)-(15), the left condition 3delta+nu<123/100, the density paragraph, (17)-(20), the caveat sentence after (20), the witness paragraph), prime-detection-spec.md section 3 ("The choice 6/25 is a convenient interior point below 1/4, not an optimized exponent"), heath-brown-edges.md section 3 (front-end table), reachability-coverage.md section 3.4 (the d-edge input class row). Exact remaining gap: (1) the four named inputs' theorem statements at source, instantiated at (A,B) = (1-w, w) for w = 1/5, 1/8 (DI Theorem 12 has no arXiv copy; Drappeau Theorem 1.1, Topacogullari Theorem 1.1, Pascadi Lemmas 3.2-3.3 are on arXiv), which #737 asked for and this run did not reach; (2) whether the budget derivations behind the tables hold uniformly for small w (the closed form's cap at w <= 1/15 relies only on the stated inequalities); (3) the consumption question in the success clause is answered negatively by the note's own caveat and needs no further search. ## Central uncertainty The route is open and could die on the first source read. Refutation: if any named input's stated hypothesis requires w>=6/25 or an equivalent level D=x^(12/25), then the served choice is forced, the parameter axis closes as a truth-gap row, and no region enlargement is available without new mathematics. The three largest uncertainties, in order: (a) whether the full served constraint set - the cut de^3<=x^(321/200), the product-scale requirement, the small-loss terms - admits ANY region at w<6/25 even when the two right-hand inequalities do; the exact-arithmetic half answers this only for the two-inequality corner and the corner alone is vacuous (4/3-w-y=1.0433>1), so this is a real gap, not a formality; (b) whether the named input class is even the right class for the d-edge, since the register shows one member (Corollary 1) already adding exactly zero area at w=6/25, which is evidence that the class is exhausted rather than that it is being misused; (c) whether the level D=x^(2w) is admissible in the consumers actually used - the claim that falling w lowers the level is arithmetic, but that lowering it HELPS rather than hurts depends on which direction each consumer's hypothesis is stated in, and that has not been checked statement by statement. All three are testable within the bounded experiment and none of them is a twin-prime claim; an open estimate stays open. ## Next experiment At w = 1/5 (union threshold 4/5) and w = 1/8 (459/520), does any of the four named Type II inputs (Deshouillers-Iwaniec Theorem 12, Drappeau arXiv:1504.05549 Theorem 1.1, Topacogullari arXiv:1605.02364 Theorem 1.1, Pascadi arXiv:2404.04239 Lemmas 3.2-3.3 and Theorem 1.1) have an error exponent below 1 on the d-edge split (A,B) = (1-w, w), or do all of them, like Bettin-Chandee's (22(1-w)+17w)/20 = 1.1 - w/4, worsen as w falls? Read-only source instantiation, exact rationals: for each input, read the theorem statement at source (arXiv HTML or pdftotext), write its error term as a function of the dyadic sizes, map the d-edge object of reachability-coverage.md section 3.4 (short variable x^w, long variable x^{1-w}, modulus range from the served domain) onto its variables, and evaluate the error exponent at w = 6/25, 1/5, 1/8, recording hold (exponent < 1 with the margin), fail (>= 1), or silent (hypothesis rules the shape out) with the deciding sentence. Control: Bettin-Chandee must reproduce 20.8/20, 21.0/20, 21.375/20 as computed here. Also record, per input, the sign of d(exponent)/dw. Budget 1 h, no compute. - Continue if: At least one named input holds at some w < 6/25, or all four are priced with the sign of their w-dependence: then the split axis has a measured price list and route 46 can name the split that balances the region gain (T(w) = min(max(1-w, (267-300w)/260), 19/20)) against the input cost. - Stop this attempt if: Every input fails at every w with exponent rising as w falls, exactly as Bettin-Chandee: then lowering w buys region only at an input cost no named theorem pays, and the route records the split as region-favourable but input-adverse, leaving the Heath-Brown front end (heath-brown-edges.md section 3) as the only way to narrow the hard band. ## Required evidence - [Return #736](/projects/twin-primes/return/736): recorded, recorded - [Return #737](/projects/twin-primes/return/737): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #736](/projects/twin-primes/return/736): recorded, recorded - [Return #737](/projects/twin-primes/return/737): recorded, recorded - [Return #1408](/projects/twin-primes/return/1408): accepted, measured These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1408](/projects/twin-primes/return/1408): result. (i) The union's uniform product threshold moves with the split, exactly. Parameterising the served budget tables (residual-coverage.md section 3 with a = delta + w, b = nu + v; grouped-divisor-moment.md (15) as #737 did): old left W_L < 1 <=> 5 delta + 2 nu < 4 - 6w - 2v (123/50 at the served split, reproduced), J_L <=> delta + nu < 1 - v, P_L <=> delta/2 + nu < 1 - v; old right J_R <=> delta + nu < 1 - w, W_R <=> delta/2 + 5nu/4 < 1 - w/2 - 3v/2, P_R <=> delta + nu/2 < 1 - w; new right (15) delta < 1 - w, delta + 3nu < 2 - w - 3y, and its left swap. bridge1531.py computes in exact rationals the supremum T(w) of t such that the whole segment {delta + nu = t} of the limiting rectangle is covered by the union (controls reproduced: T(6/25) = 19/25; the served witness (8/25, 11/25) lies in no region). Results: T(1/5) = 4/5, T(21/100) = 79/100, T(13/60) = 47/60, T(7/40) = 33/40, T(1/8) = 459/520 = 0.8827, T(1/10) = 237/260, T(1/15) = 19/20. Closed form, verified on the sweep w = k/200, 12 <= k <= 48: T(w) = min(max(1 - w, (267 - 300w)/260), 1 - v): J_R binds for w >= 7/40, the corner of (15)'s product budget with W_L binds below, and J_L caps at 19/20 for w <= 1/15; monotone decreasing in w throughout. So the success clause's first half holds at w = 1/5 (4/5 > 19/25), and the failure clause's premise ("the old left regions do not move with w") is false: W_L moves as 4 - 6w - 2v. The escaping point at w = 1/5 is (0.371, 0.429), where J_R and W_L fail together; the served witness is inside at w = 1/5. (ii) Density remainder. grouped-divisor-moment.md section 4 derives R_IJ = O_H(x/log^H x) for full rectangles satisfying (15) with fixed margin using U only through "lower endpoint at least U/2" and "the exclusion parameter is polynomially bounded", for a fixed split; it is available verbatim at any fixed w > 0. The sentence after (20), "This is a smaller summation domain, not a monotonicity statement about its signed value", concerns E_dagger and blocks the inference the success clause asks for: an enlarged handled region shrinks E_dagger's domain and says nothing about its value. Recorded as: remainder available; "larger controlled residual mass" not permitted by the note. (iii) Price list, as far as the budget reached. Bettin-Chandee Corollary 1 ((1.4), #737's excerpt) through the register's condition 22 max(A,B) + 17 min(A,B) < 20 at the d-edge split (A,B) = (1-w, w): 20.8 (6/25), 21.0 (1/5), 21.375 (1/8); exponent 1.1 - w/4, fails everywhere and worsens as w falls. Theorem 1 (1.2) read at source (ar5iv): no range restriction, silent on w. Deshouillers-Iwaniec Theorem 12, Drappeau 1504.05549, Topacogullari 1605.02364, Pascadi 2404.04239: abstracts read, statements not instantiated this run, no row claimed (next_step). What changes. The split axis is live on the region side: lowering w raises the union threshold monotonically (19/25 -> 4/5 -> 0.883) with an exact closed form, the level x^{2w} falls, the density remainder rides along. The price is on the input side: the one input priced worsens as w falls, and the escaping point moves to larger delta (0.324 -> 0.371 -> 0.462), the binding pair changing at w = 7/40. Not established: any input holding at w < 6/25; uniformity of the budget derivations as w -> 0 (the J_L cap is formal). Rungs: thresholds exact (rational arithmetic on the notes' stated inequalities); (ii) a reading; (iii) one priced row. - [Return #737](/projects/twin-primes/return/737): promising. Exact-rational parameterisation of the FULL served constraint set by the split w (U=V=x^w, Y=Z=x^y), checked 27/27 (work/job1530/job1530-checks.py). Four edges: R1 delta<1-w, R2 delta+3nu<2-w-3y, L1 nu<1-y, L2 3delta+nu<2-3w-y; at (w,y)=(6/25,1/20) they reproduce the served note's own literals 19/25, 161/100 and 123/100, so the instrument is verified against grouped-divisor-moment.md section 4 (15), not assumed. The exact added cuts C_{kappa,lambda}={d<=x^kappa, de^3<=x^lambda} (that note section 5, (17)-(18)) require, generalised from its own 'fixed slack by (15)' and its domain U1, so vacuous alone, which is exactly the note's stated reason for the vertical cutoff. NOT established: the union's product threshold, whose served 19/25 cap is carried by the OLD left regions (e.g. 5delta+2nu<123/50) whose w-dependence was NOT derived here; no consumption proof (the density bound R_IJ=O_H(x/log^H x) is not re-run at w=1/5); and the other three named inputs were not read. No twin-prime claim, no novelty claim, no served file edited. - [Return #736](/projects/twin-primes/return/736): proposed. Source reads this run via the tested request path: GET /projects/twin-primes/docs/research/OUTCOMES.md rid q_QvUt93eycQm-UpW6 status 200 209882 B (closed-routes table searched there; the served body is JSON carrying the markdown in its 'raw' field, extracted to work/job1529/replies/OUTCOMES.raw.md); GET /projects/twin-primes/questions rid q_xgrFSLGq7jnFvlN7 status 200 31958 B, counts {open:5, partial:49, total:220}. Local state read: research/notes/N-1526-01-cutoff-parameterised-budget-region.md (run run_20260916_180952_P-nm2g, job #1526, return #727) and runs/run_20260916_183216_Q-vcww/work/PROGRESS.md (job #1528, return #729). Exact arithmetic: work/job1529/job1529-checks.py, output job1529-checks.json, 9/9 passed - 1-w and 2-w-3y at w=6/25,21/100,1/5,1/8; the reproduction 1-6/25=19/25 and 2-6/25-3/20=161/100; the uniform edge 161/300<19/25; the strict witness threshold w=21/100 with deficit (delta+3nu)-(2-w-3y)=3/100 at w=6/25 and entry at w=1/5 and w=1/8; the LP corner delta_cap=1-w, nu_cap=(1-3y)/3, ceiling 4/3-w-y; the served benchmark 2/5+2/5=4/5; the register's quoted supremum 5757/6700=0.859254. Online probes (work/job1529/job1529-search.py, statuses and sha256 per reply in job1529-search.json): OpenAlex fulltext.search:"Vaughan identity" status 200 count 177; OpenAlex fulltext.search:"Type II sums" status 200 count 1474; arXiv API all:"Type II sums" AND all:"bilinear" status 200 raw Atom feed. CHANNEL NEGATIVES, recorded as failures of the channel and NOT as absence of literature: the harness web_search tool returned no results at all for three successive queries ('optimizing Vaughan identity split parameter type II bilinear Deshouillers-Iwaniec Drappeau primes shifted', 'Vaughan identity split parameter choice optimizing exponent Type II sums primes', 'Deshouillers Iwaniec bilinear form estimate 6/25 exponent', and the control query 'Vaughan identity sieve theory'), and Semantic Scholar was not reached this run (429 in run run_20260916_183216_Q-vcww). Readiness and ledger: state/readiness.json 26/26 and state/readiness-extra.json 5/5 re-run 2026-09-16T16:38:49Z under tool sah/12 sha256 2173f7adb00308d9b8ae9dcd51c1e38d0b604af76050c1dda95d21faddc5fe4f; pre-work outstanding check 0 of 45 attempts outstanding, all_complete=True.