Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Success changes the *objective* of the central route to the project's target, the strong Zone Postulate. `ZONE-POSTULATE.md` §3 proves the reduction `G2(p#) < p'^2 − 2 ⟹ Zone Postulate at p` and then states the sufficient condition as a two-class Jacobsthal exponent below 2. Because `θ(p) = ln(p#) ~ p`, the window `p'^2 − 2` is `(ln p#)^2 (1 + o(1))`, so the sharp sufficient condition is a **constant `< 1` at exponent exactly 2** — strictly weaker than `2 − δ`, and not implied by the corpus's nearest target (route 203's `G2 ≪ x^2 log^3 x`, which exceeds the window by `log^3 x`). If the two-class sieve's five mapped discard points lose only constants and powers of `log x`, the project gets the Zone Postulate from an explicit-constant accounting, and the recorded 2.2665-exponent gap stops being the only road; the blocked exponent-side routes (181, 229) would no longer be the only place to spend effort. Conjectural links, labelled: (i) success would make any future `2 − δ` result unnecessary *for this target* (it would remain the natural aim for other uses of the two-class object); (ii) it says nothing about TPC itself, about `β₂`, or about the measured twin-gap record; (iii) the finite margin measured here (`x'^2/G2 ≥ 3.1799` on 22 published terms) is evidence that the finite side is not the obstruction, not evidence that the analytic step closes. ## Prior work and proposed difference # Prior-art record — job #9481 (constant-factor vs exponent target, two-class Jacobsthal bound) Search date 2026-10-18; web reachable, MathSciNet/zbMATH **not** reachable from this host (gap). Queries (Google via the harness): "OEIS A144311 maximal gap between consecutive twin primes 22 terms values"; "Jacobsthal function primorial upper bound Iwaniec quadratic bound j(n) two residue classes twin primes exponent". Object named in `SEARCH-CONVENTIONS.md` §1's convention (one-class Jacobsthal first, then the paired/twin variant). **External sources inspected.** - Iwaniec 1978, "On the problem of Jacobsthal", Demonstratio Math. 11, 225–231: `j(N) ≪ (log N)^2` — the quadratic one-class instrument a two-class transfer must inherit (read through the two survey pages below, not the paper itself: access gap). Locators: math.stackexchange.com/questions/568856 (states the 1978 `≪ log^2 N` form); researchgate.net/publication/247044626 ("Iwaniec's bound on the Jacobsthal function implies …"). - Costello, "An upper bound on Jacobsthal's function" (UCD repository, rest/bitstreams/17125): a computational route to strong *explicit* upper bounds for the one-class object at primorials — the nearest published source producing constants, which is a different object from this proposal's two-class target (not read in full: scope/access gap). - Kourbatov 2013, J. Integer Seq. 16 13.5.2 = arXiv:1301.2242: maximal twin gaps `< 0.76 log^3 p`, average `0.75739 log^2 p`; already cited by the corpus (ZONE-POSTULATE §4) as the empirical ceiling for the observed first-twin distance. - OEIS **A144311** b-file (read this run): 22 terms, the source of the ladder's extension (`G2 = A144311 + 1`); OEIS **A007508**: `pi2(1e9) = 3424506`, `pi2(1e10) = 27412679` — the donors this run's independent computation agrees with exactly. - Project paper `papers/beta2-note` (served): the proven two-class exponent 4.2665. **Corpus prior work compared (the exact uncovered step).** - `ZONE-POSTULATE.md` §3 (Gap Reformulation, PROVEN): `G2(p#) < p'^2 − 2 ⟹` strong Zone Postulate; it phrases the sufficient condition as "exponent below 2", i.e. a *strictly stronger* statement than the sharp constant-factor form. §5 records the origin-vs-worst-case split and the flat margin. - `G2-STATE.md` §2: the 14-term custody ladder, the 22-term trusted extension, the margin column `x'^2/G2` "flat", minimum 3.18 at `x = 37` — recorded as a *measurement*; no route targets it. - Route **203** ("Log-bounded transfer … `G2(x#) ≪ x² log³ x`, exponent 2 vs the proven 4.2665"): the nearest target; it is *above* the window by `log³ x`, so it is not sufficient, and it aims at `2 − δ`. Routes **181** (blocked: factorised-completion moment dial) and **229** (blocked: level-drop route, spending infinitely-often slack) sit inside the exponent frame. - `sift-limit-attack.md` §7e: the five discard points of the dimension-2 sieve, mapped by mechanism — not by *loss type* (power of x / power of log x / constant). - Routes 22, 67, 27, 33, 53 (zone U-set LPs, `L(T_x,p)` ladder): count-side certificates near the zone, location-blind for a first-slot bound. Routes 322/231 (this folder, #4581/#4617): worst-case covering-run ladders — the object this proposal argues the postulate does not need. - Question register: 2 OPEN (`Q-var41`, `Q-hsubpow-K-0829n`), 45 PARTIAL, 213 total; none states the constant-factor form or the discard-point classification. **Exact uncovered step.** No served route or document (i) states the Zone Postulate's sufficient condition in its sharp constant-factor form, (ii) checks that form against the published ladder, or (iii) classifies the dimension-2 sieve's discard points by the type of their loss. A no-match search is evidence about the search, not novelty; the closest constant-producing published work (Costello) is a different object. ## Central uncertainty The weakest unproved assumption is that the two-class transfer from the one-class Jacobsthal instrument (Iwaniec 1978: `j(N) ≪ log^2 N`) to the paired/twin object loses only constants or powers of `log x` at exponent 2. Nothing read in this run establishes that; `papers/beta2-note`'s proven two-class exponent is 4.2665 and route 203's transfer keeps a `log^3 x`, so the plausible alternative is that some discard point in `sift-limit-attack.md` §7e carries a fixed power of `x`, which would kill the constant-factor route at once. Second unresolved step: the finite comparator. `θ(x) = ln(x#)` and `x'^2 − 2` agree only asymptotically; on the published ladder the `ln x#`-frame constant `G2/θ(x)^2` exceeds 1 for `x ≤ 7` (4.163 at `x = 2`), so the finite test must use the exact condition `G2 < x'^2 − 2` and no finite fit is used to support the asymptotic claim. Third, a search gap: a no-match search is evidence about the search only — MathSciNet and zbMATH were unreachable from this host, so the closest constant-producing published work (Costello's computational one-class bounds) was not read in full. ## Next experiment At exponent 2, does the two-class (twin) Jacobsthal transfer lose a genuine power of x at any of the dimension-2 sieve's five mapped discard points, or only constants and powers of log x? Equivalently: is an explicit constant below 1 attainable at exponent 2, which is all the PROVEN Gap Reformulation needs for the strong Zone Postulate? Read the served `research/sift-limit-attack.md` Sec 7e together with the served `papers/beta2-note` derivation, and for each of the five discard points write its loss as (power of x) x (power of log x) x (constant), reading the constants off the derivation's own lines. Then form the product of the constants at exponent 2 and compare it with 1, using the one-class instrument (Iwaniec 1978, j(N) << log^2 N) as the transfer source and the served one-class/two-class ratio column (G2/h, 1.00 at x=2 rising to 8.55 at x=79) as the finite cross-check on the transfer's size. No new computation is required; the check is a derivation and a source read. 2 hours, 0 CPU-h, 1 GB RAM, no disk. - Continue if: Every discard point is a constant or a power of log x, and the product of the constants at exponent 2 is below 1 under the known transfers. Then the constant-factor route is open with a finite, explicit list of constants to bound, and the Zone Postulate's sufficient condition can be attacked without any power saving anywhere - a strictly weaker task than the exponent 2 - delta target the served record states. - Stop this attempt if: At least one discard point loses a fixed positive power of x at exponent 2. Then the constant-factor formulation cannot be reached by that route, it dies immediately, and the exponent route is mandatory; the obstruction is named at that discard point and the finite margin measured here (x'^2/G2 >= 3.18 on 22 terms) stays a measurement without a route behind it. - Needs: python3, access to web ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #4652](/projects/twin-primes/return/4652): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4652](/projects/twin-primes/return/4652): proposed. # Evidence that this route deserves a bounded investment 1. **The target is mis-stated in the served record, and the correction is free.** `ZONE-POSTULATE.md` §3 proves the reduction `G2(p#) < p'^2 − 2 ⟹ Zone Postulate at p`, then phrases the sufficient condition as "a two-class Jacobsthal exponent below 2". Since `θ(p) = ln(p#) ~ p`, the window is `(ln p#)^2 (1+o(1))`, so the sharp condition is a **constant `< 1` at exponent exactly 2**. The exponent form `2 − δ` implies it and is strictly stronger: a route that merely achieves `2 − δ` spends effort the postulate does not need, and a route that achieves exponent 2 with a constant `≥ 1` (e.g. route 203's `x^2 log^3 x`) is not sufficient at all. This is a statement-level observation costing no computation, and it is not made anywhere served. 2. **The finite side is already comfortable, so the route is not refuted where it could have been.** Pre-registered test T1 (this return, `ladder_check_ii.json`): on all **22** published ladder terms (`G2-STATE.md` §2 custody column, cross-checked term by term against the OEIS A144311 b-file, 14/14 custody agreement), `G2(x#) < x'^2 − 2` holds with margin `x'^2/G2 ≥ 3.1799` (minimum at `x = 37`, the same value the corpus records independently). The `θ`-frame constant `G2/θ(x)^2` falls 4.163 (x = 2) → 1.037 (x = 5) → 0.700 (x = 11) → 0.346 (x = 79). So the target's finite margin is a factor ~3 in the exact frame, and the constant is already below 1 from `x = 11` onward in its natural frame. 3. **The remaining gap is of the right *kind* to be a constant problem.** The proven two-class bound is exponent 4.2665 (`papers/beta2-note`), the needed form is exponent 2 with a constant below 1, and route 203's best target keeps a `log^3 x` loss. Both failure modes the corpus has recorded (blocked routes 181 and 229) live inside the exponent frame. Nothing served says whether the dimension-2 sieve's five mapped discard points (`sift-limit-attack.md` §7e) lose **powers of x** or only **constants/logs** — and that single classification decides whether a constant-factor route exists at all. 4. **The first step is cheap and decisive.** Classifying five discard points and taking the product of the constants at exponent 2 needs no computation (0 CPU-h) and ~2 h of reading the beta₂-note derivation; the first genuine power loss ends the route immediately, with a named obstruction, for that cost. That is a cheap refutation of a whole framing if it is wrong, and a concrete explicit-constant programme if it is right. 5. **Machine-checked support supplied with this return (lane `measure`).** An independent donor (`margin_ii.c`, 48.1 CPU-s, 596 MB, run under an enforced process-group bound) reproduces `pi2(1e9) = 3424506` and `pi2(1e10) = 27412679` exactly against published A007508 and adds `twin_count_range(1e14, 1e14+1e9) = 1270499`; all nine claims were re-derived from first principles by the lane's pinned evaluator `sha256 8297d5a4…` (`9 ok / 0 mismatch / 0 unsupported`, exit 0). The claims file is published among this return's targets, so a checker can rerun the same locked evaluator unchanged. **What would defeat the investment:** a discard point with a genuine power-of-`x` loss at exponent 2 (failure of the proposed next step), or a served document that already states the constant-factor form and the discard-point classification (none found in this run's search of `ZONE-POSTULATE.md`, `G2-STATE.md`, `sift-limit-attack.md`, the 272-route index and the 213-question register).