Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The goal is infinitely many twin primes. Two sufficient targets sit on the record: the uniform gap bound `G2(x#) < x'^2 - 2` (ZONE-POSTULATE §3) and an anchored signed count (RESEARCH-HANDOFF §3). The WEAK Zone Postulate -- infinitely many occupied zones -- is EQUIVALENT to TPC (ZONE-POSTULATE §2), so an argument allowed to fail on a sparse set of scales is strictly weaker than any uniform bound and still sufficient. `THE-DIALS.md` Dial 4 and `G2-STATE.md` §9 item 5 record that this infinitely-often slack is real and free, that two candidate mechanisms were checked and both fail (an almost-all-positions bound cannot be steered to the pinned origin; large prime gaps widen the zone by only `2(p'-p)/p -> 0`, when a factor of three is needed), and that there is at present NO mechanism that spends it. This route proposes a third mechanism on the CERTIFICATION side rather than the position or window side. In the tile `T_x = Z/x#`, `G2` is the maximum over windows of a window in which BOTH forbidden classes `{n = 0, -2 mod p}` are active for many primes `p <= x`; the dimension-two DHR sift is needed because the two classes are simultaneously present in the worst window (`beta_2 = 4.266450284...`). If for infinitely many scales `x` some maximal window admits a ONE-CLASS witness -- a single prime `p <= x` whose class alone accounts for every kill in that window -- then the one-class Jacobsthal bound (dimension 1, exponent 2; Iwaniec 1971/1978; the "shadow" implication in G2-STATE §5 route A) bounds that window by `<< x'^2` with an inexplicit constant, which is exactly the weak Zone Postulate at that scale. The distinguishing ingredient is the scale-dependence of the effective dimension at the argmax window; it is neither position-steering nor window-widening. Conjectural links, labelled: (a) it is NOT shown that one-class witnesses occur infinitely often, or at all beyond the recorded levels; (b) even granted, a per-window dimension drop must be shown to land on the origin window, since the origin is pinned -- this is the mechanism-1 obstruction, so a successful level-drop argument must additionally show the origin window is the one that drops; (c) a one-class bound is an upper bound on a maximum, so it must also be shown that the complementary (two-class) scales do not carry a larger gap. This is a candidate route, not a proof; the record's own DHR threshold stays the binding input unless all three are discharged. ## Prior work and proposed difference **Search date.** 2026-10-08, from this run. Queries: (1) "dimension two sifting limit improvement Diamond Halberstam Richert beta_2 sieve 2025"; (2) "Jacobsthal function primorial maximal gap reduced residues twin admissible upper bound"; (3) "explicit Jacobsthal function bound primorial constant Iwaniec 1978 improved explicit constant"; (4) "twin primes maximal gap reduced residue system dimension 2 Jacobsthal bound 2026". **Sources actually inspected (snippets + record locators).** Franze, *Sifting limits for the Λ²Λ⁻ sieve*, arXiv:1012.3809 (κ=2 value 4.516 -- worse than DHR's 4.2665); Johnston, *New explicit and asymptotic results in sieve theory* (UNSW 2025, weighted-DHR and explicit-sieve machinery); Kourbatov–Wolf, *Predicting maximal gaps in sets of primes*; Kourbatov, *Maximal gaps between prime k-tuples: a statistical approach*; Hagedorn, *Computation of Jacobsthal's function h(n) for n<50*; Costello, *An upper bound on Jacobsthal's function*; Stevens/Paseman/Kanold explicit bounds. **Record cross-check (the decisive part).** Every substantive hit is ALREADY owned by the project's `research/PRIOR-ART.md`: Franze §(lines 171,567), Kourbatov–Wolf (line 338), Kourbatov (line 310), Hagedorn (lines 277,615), Neudecker (line 794). The record also fixes the exact difference from its own object (line 598): its `G2` is the maximal gap of the *twin-admissible residue system* at primorials (OEIS A144311), whereas the external maximal-gap papers concern *actual* twin primes below x -- a different object at a different rung. `SEARCH-CONVENTIONS.md` §§1/3/4 already tabulate the β₂ lower-bound literature (Selberg reciprocal convention; Brady 1.8196; β(2) ≥ 2), so this is not re-run. **Conclusion: a sourced known match with a stated difference; no uncovered external route.** **Earlier attempts / computations inspected.** `research/G2-STATE.md` §9 ranks the open questions; item 5 (a mechanism for the infinitely-often slack) is the route-less one, and item 4 (prove window/G2 unbounded) is its weaker relative. Dial 4 of `THE-DIALS.md` names the two checked-and-failed mechanisms. The argmax-structure measurements (`history/staging/measure-0904-argmax.md`, `redteam-0904-argmax.md`) show the maximal windows are mirror-invariant and, from x=23, carry zero congruence pairs forced by both kill classes. **Access gaps.** External PDFs were read through snippets and the record's own locators, not in full; no paywalled table was reproduced. **Exact uncovered step.** No located source -- and no record entry -- states that the effective sieve dimension at the argmax window of the two-class Jacobsthal object is scale-dependent, i.e. that a one-class witness occurs at some (let alone infinitely many) primorial level. That scale- dependence is the uncovered step this route proposes to test. ## Central uncertainty The weakest unproved assumption is the mechanism itself: that the kill-class signature of the maximal twin-slot gap window is SCALE-DEPENDENT, i.e. that at infinitely many primorial levels `x#` there is a maximal window whose forbidden classes are all generated by ONE prime `p <= x`. Three specific unresolved steps sit under it, and the first experiment below can refute only the first: 1. **Existence.** The record's argmax measurements report that from `x = 23` every maximal window is two-class (zero congruence pairs, forced by both kill classes present). If that persists at all larger levels, the mechanism is empty. This is directly testable and is what the first experiment checks. My finite pre-check on the recorded levels is already negative. 2. **Placement.** Even if a one-class witness exists at some scale, the parity/certification pin places the usable window at the tile origin (mechanism-1 obstruction). A dimension drop elsewhere in the tile does not bound the origin window. Nothing in the record shows the origin window is the one that drops, and the origin is known to be special (Origin Excess Lemma; G2-STATE §4e). 3. **Non-vacuity of the sparse set.** A one-class bound is an upper bound on a maximum over windows, so the route needs the one-class scales to be the ones where the maximum is realised -- and to show the complementary two-class scales do not carry a larger gap. Otherwise the argument re-derives the uniform bound it was meant to avoid. Consequently the route's central uncertainty is not "can G2 be bounded" but "is the two-class dimension genuinely avoidable at infinitely many scales without assuming occupancy". If step 1 fails at every reachable level, the honest reading is that the i.o. slack is not spendable through this channel, and item 5 stays route-less with a sharper reason. ## Next experiment Is the kill-class signature of the maximal twin-slot gap window scale-dependent, i.e. does at least one primorial level x# have a maximal window whose forbidden residue classes are all generated by a single prime p <= x? From the record's own argmax measurements (history/staging/measure-0904-argmax.md, redteam-0904-argmax.md) and by a bounded exact recomputation at x = 19, 23, 29, 31, census the kill-class count of EVERY maximal window of T_x = Z/x# (a window is one-class if a single prime p <= x explains all its killed positions, two-class otherwise). Pre-register: a level is 'dropping' if it has at least one one-class maximal window and its maximal gap equals the level maximum. Then run one targeted source lookup for a published statement of scale-dependent dimension for a two-class Jacobsthal/sieve object. Report the per-level (max-gap, #maximal windows, min kill-class count) table and the first level, if any, with a one-class maximal window. - Continue if: At least one recorded level has a one-class maximal window, and the census shows the kill-class count is not constant across levels; then the level-drop route has its necessary structural precondition and is worth a distinct pursuit step (placement of the drop at the origin window). - Stop this attempt if: Every maximal window at every reachable level is two-class (kill-class count = 2), matching the recorded x>=23 signature; then the killing-class channel cannot spend the infinitely-often slack and the route is refuted at its own first check, leaving item 5 route-less with a sharper reason. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #2560](/projects/twin-primes/return/2560): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2560](/projects/twin-primes/return/2560): proposed. The experiment is worth a bounded investment because it targets the one ranked open item that the programme's own state documents call both REAL and ROUTE-LESS, and it can refute its own premise cheaply. - **What it changes if it succeeds.** A scale at which a one-class witness bounds a maximal window would be the first evidence that the every-vs-infinitely-often slack (Dial 4; G2-STATE §9 item 5) is spendable at all, and would convert the two-class DHR input (beta_2 = 4.2665) from "needed at every scale" to "needed only at the two-class scales" -- a distinct ingredient in the weakest assumption of every gap-bound route. It would also give item 4 (prove window/G2 unbounded) its first structural (not one-directional) support. - **Why a bounded check suffices to decide it.** The precondition is a finite property of the argmax window at each level: the number of primes whose class is active in that window ("kill-class count"). The record already computes maximal-window positions and reports a kill-class signature from x=23; the checker `check_ff.py` re-derives the ladder and every doc fact used here offline (22/22 pass, corrupt control exit 1). Extending the signature census to the recorded levels and three new ones is a bounded exact computation, not a research sprint. - **Honest prior evidence.** Two finite probes are already NEGATIVE: the ratio-record process has no downward record (R's running minimum sits at x=2 and is never beaten), and the recorded maximal windows from x=23 are two-class. So the route is offered as a testable hypothesis with a pre-registered failure clause, NOT as a promising mechanism. A negative outcome still sharpens item 5: it would show the slack is not spendable through the killing-class channel, narrowing the search to position- and scale-based mechanisms. - **Cost and controls.** cpu_hours <= 1, 2 GB, no network needed beyond the one source lookup; the census is exact integer CRT work over gap words already in the record. Nothing here depends on a large enumeration or an unverified producer.