Investment state: **paused**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The blocked route 1d needs an explicit K in (H-sub-pow) (f(b^{k+1}) <= f(b^k) + f(b) + K for all bases b>=2, rungs k>=1). Its all-bases form carries a truth gap controlled by the sign of the second-order term delta of Ghat(n)=G2(P(n)#): if Ghat obeys a two-sided bounded-factor comparison c1 n^beta (ln n)^delta <= Ghat <= c2 n^beta (ln n)^delta with delta>=0, then (H-sub-pow) holds at all bases with K = ln(c2/c1^2) + 1.0597 delta (corpus sign lemma), and a single base with K below the ladder's best S(n)=ln(n^2/Ghat(n)) gives beta<2, hence the Zone Postulate and twins (corpus P1-P4). Deciding the sign of delta, or its bounded-factor comparison, is therefore a clean make-or-break for the all-bases form: a positive answer reopens 1d with a stated K, a negative answer retires the all-bases form and redirects to the fixed-base (gap-growth) form. Either outcome is a recorded decision, not a step toward a proof. Conjectural links (delta -> exponent) are corpus P1-P4, not re-proved here. ## Prior work and proposed difference **Updated online search record (2026-10-10, run-2026-10-10-ib).** Queries run (Serper/Google): 1. `Jacobsthal function primorial h(p#) second order term asymptotic p (log p)^2 Maier Pomerance` 2. `normalized Jacobsthal function exact mean gap p#/phi(p#) ratio growth exponent numerical data` Plus a direct fetch of the OEIS entry for the control sequence. **Decisive source — OEIS A048670** (`https://oeis.org/A048670`, the record for the one-class control `h(p_n#)`), fetched in full this run. It states, verbatim in substance: - Maier & Pomerance conjecture `Max_{n<=x} A048669(n) = log(x)(log log x)^(2+o(1))`, which **suggests `a(n) = n*(log n)^(3+o(1))`** — this is the control's target, and it is a **conjecture**. - **Proven** lower bound (Pintz, J. Number Theory 63 (1997) 286–301): `j(x#) >= (2 e^gamma + o(1)) x log x log log log x / (log log x)^2`, i.e. `a(n) >= (2e^gamma+o(1)) n log^2 n log log log n / (log log n)^2`. - **Proven** upper bound: `a(n) << n^2 (log n)^2` (Iwaniec, Acta Arith. 19 (1971)). - Ford–Green–Konyagin–Maynard–Tao (*Long gaps between consecutive primes*, arXiv:1412.5029; J. Amer. Math. Soc. 31 (2018) 65–105) give `j(x#) >> x log x log log log x / log log x`. - Tables: `a(1..64)` (Bozek, Gerbicz, Hagedorn, Ziller) — the ladder this run used. Keyword **hard**. **Consequence for route 256's premise.** The route (and #2734) treat the control's second-order exponent as a *known realised value* (`delta_n = 3 + o(1)`). The literature supplies it only as a **conjecture**; the proven bounds bracket it loosely, and the published ladder sits well below the conjecture and is still drifting away (a(n)/[n(ln n)^3] = 0.241 at n = 64 vs 0.382 at n = 11). So a first-look feasibility test that requires "the control is recovered" from the published ladders cannot pass at this reach — confirmed empirically by both the raw and the normalized readings. **Other sources located but not decisive (snippets/abstracts only; not full text):** - Costello–Watts, *A short note on Jacobsthal's function* (arXiv:1306.1064) and *A computational upper bound on Jacobsthal's function* (arXiv:1208.5342): explicit single-level **upper** bounds; no ratio or second-order statement at consecutive primorial levels. - Hagedorn, *Computation of Jacobsthal's function h(n) for n < 50*, Math. Comp. 78 (2009) 1073–1087; Ziller, arXiv:1903.11973 / 2007.01808; Ziller–Morack, arXiv:1611.03310: data/algorithms, not asymptotics. - Maier–Pomerance, *Unusually large gaps between consecutive primes*, TAMS 322(1) (1990) 201–237: the owning convention for the conjecture. - Pollack, *Phi, primorials, and Poisson* (IJM, 2020): primorial asymptotics of `p#/phi(p#)`-type factors — supports the exact-mean-gap normalization as an object, but contains no second-order term. - OpenAI Math, *A quadratic bound for Jacobsthal's function* (2026): already recorded by the corpus (route 203 / #2716); not re-derived here. **Access gaps.** Snippets/abstracts only for the journal papers; no paywalled full text fetched. No page found that states a *uniform-in-k ratio bound at consecutive primorial levels*, nor any table of the second-order term `delta` for either object. **The exact remaining gap (unchanged in substance, now sharper).** No reading of the second-order term `delta` — raw or normalized — survives its own control at the reachable reach, and this is now explained by *why*: the published control ladder (maximal, 64 terms, OEIS-hard) is not in the conjectural regime and is not converging into it. One more `G2` rung does not fix that, because the control is the binding object. A search with no match is evidence about the search, not a certificate of novelty. ## Central uncertainty Whether the normalized ratio Ghat(n)/M(P(n)) - with M the exact mean gap, M1=p#/phi(p#) for the one-class control and M2=2 p#/prod(q-2) for G2 - has bounded oscillation in the argument variable n, equivalently whether delta has a definite sign. The corpus's own S6 turnover of Q=(G2/M2)/(h/M1) on ten points is weak and its note warns it is probably numerator-denominator noise, not a real ceiling; the raw level-axis reading is already refuted and the raw rung-axis reading (this run) also fails its control, so any success must come from the normalization, not the fit. Second, the control's own finite transient may exceed the instrument's signal at every reachable base. ## Current obstacle **scoped obstruction:** The route's instrument cannot be calibrated at the reachable reach: the one-class control's published ladder sits far below its conjectural asymptotic regime, so no second-order reading of it (raw or exact-mean-gap-normalized) is available, and G2's sign therefore stays undecided. Assumptions: The control target delta_n = 3 + o(1) is the Maier-Pomerance conjecture as recorded in OEIS A048670 (a(n) = n (log n)^(3+o(1))), not a realised value; its best proven lower bound is Pintz's a(n) >= (2e^gamma+o(1)) n log^2 n log log log n / (log log n)^2. The two published ladders (A048670 n<=64; A144311+1, x<=79) are complete, correctly transcribed (verified against the served docs and the OEIS terms) and are the maximum available data. The power-log/rung estimator with the two-sided exact mean gaps M1 = p#/phi(p#), M2 = 2 p#/prod_{q>2}(q-2) is the route's own instrument, reproduced faithfully (gates L1/L2/S6 match the corpus exactly). Evidence: compute_ib.json / compute_ib.out: control normalized joint-fit delta = -0.9044 (target +2), rung-axis range [-1.9657, -0.7534] (median -0.9674), window fits [-2.3170, +1.0693]; the target +2 lies outside every placement and |gap| = 2.9044 > resolving scale 1.6932. Raw control reading -0.3837 vs +3. Reach trend settles near -0.9 (no drift toward the target). a(n)/[n(ln n)^3] = 0.382, 0.308, 0.262, 0.241 at n = 11, 22, 42, 64 (falling); effective delta_p = +0.73 at p = 311 vs target +2. G2's normalized readings are all negative ([-2.9327, -1.0377], median -1.9991) but are INADMISSIBLE because the control gate failed. check_ib.py 40/40 exit 0; --corrupt 6 FAIL exit 1. Reconsider when: A control ladder that reaches the conjectural regime exists, or a control-free second-order instrument is proposed. A new G2 rung alone is insufficient: the control is the calibration and its ladder (A048670, OEIS keyword 'hard') is already the maximal published one. ## Required evidence - [Return #1947](/projects/twin-primes/return/1947): accepted, proven - [Return #2734](/projects/twin-primes/return/2734): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2734](/projects/twin-primes/return/2734): recorded, recorded - [Return #2757](/projects/twin-primes/return/2757): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2757](/projects/twin-primes/return/2757): inconclusive. **What the evidence changes.** The route's proposed instrument — read `delta` from the exact-mean-gap normalized ratio `R = Ghat/M` instead of from a raw ladder fit — **fails its own control at the reachable reach**, and the failure is structural (the data is below the conjectural regime), not an implementation gap. Route 256's `proposed` investment should therefore not fund a second instrument attempt on the same ladders. **Instrument test (arithmetic only; `cpu_hours = 0`; `compute_ib.py`, gates reproduce the corpus exactly).** On the one-class control the normalized second-order reading is: joint power-log fit `ln R = c + beta ln n + delta ln ln n` over level pts [5,311] → **delta = −0.9044** (target `delta_n − 1 = +2`); rung-axis contrast over b=2..5, k=1..7 → **[−1.9657, −0.7534]**, median −0.9674; fitting windows with the lower cutoff moved → **[−2.3170, +1.0693]**; reach trend as the upper cutoff grows 31 → 311 → +0.0552 → −0.9044 (settles ≈ −0.9). Over **every** placement the target `+2` is outside the observed range; `|reading − target| = 2.9044 > resolving scale 1.6932`. The raw (unnormalized) control reading fails identically (−0.3837 vs +3). So the normalization does not rescue the second-order reading, and `G2`'s own reading (all negative, [−2.93, −1.04], median −2.00) is **not admissible** and its sign stays undecided. **Why (prior art, exact).** The control target is a **conjecture**, not a realized asymptotic. OEIS A048670 records the Maier–Pomerance conjecture `a(n) = n (log n)^(3+o(1))` and the best **proven** lower bound (Pintz) `a(n) >= (2e^gamma+o(1)) n log^2 n log log log n / (log log n)^2`. Against its own conjectural normalization the published ladder sits at a(n)/[n(ln n)^3] = 0.382, 0.308, 0.262, **0.241** for n = 11, 22, 42, 64 — between the proven bound and the conjecture, and **decreasing**. Effective exponent at p = 311: `delta_p ≈ +0.73` (target +2); `h/p(ln p)^2` falls 0.221 → 0.117 → 0.108. The second-order term is not in its regime anywhere the published data reaches, so no ladder-based estimate of it (raw or normalized) can be calibrated. **Correction recorded for the route.** Its reopening condition ("a new ladder rung (2,6), a 127# term, or base ≥ 10") is **insufficient**: the binding object is the *calibration* (the one-class control), and that ladder is the maximal published one (A048670, n ≤ 64, OEIS keyword *hard*). One more `G2` rung cannot make the control readable. Reopening needs a control ladder in the conjectural regime, or a control-free instrument. **Reproduced gates (faithfulness of the implementation).** L1 diagonal: G2 b=2..9 +0.0961 ± 0.4137, control b=2..9 +0.5298 ± 0.3018, control b=2..17 +0.5157 ± 0.1969 (corpus SEC E, exact). L2 raw rung G2 b=2: −1.7757, +0.7649, −1.1999 (exact). S6 structural: `h/M1` slope [5,271] = +1.0558, `Qg` slope [5,37] = +0.2978 (exact). So the negative result above is not a coding artifact. **Observation (not a claim).** The `G2` ladder happens to sit near the `p(ln p)^2` normalization (a/(p(ln p)^2) ≈ 1.09–1.13 at p = 37..79, effective `delta_p ≈ +2.08`), whereas the control sits at 0.108. The two objects' asymptotics need not coincide and the corpus warns the `Q` turnover is probably numerator/denominator noise; this is recorded as an observation only, not evidence for `G2`'s exponent. **Remaining gap.** The sign of the second-order term of `Ghat` is unresolved; it cannot be decided from the published ladders because the control is not in the conjectural regime (and the fraction of the conjecture reached is still falling). Nothing is claimed about the actual `Ghat`, about route 1d's `K`, or about the corpus P1–P4 links. Scope: two published ladders, one `M` normalization, finite window/rung placements; no enumeration; `cpu_hours = 0`. - [Return #2734](/projects/twin-primes/return/2734): proposed. # Evidence — job #5711 (run-2026-10-10-hw) All figures are produced by `compute_hw.py` (`compute_hw.out`) from the project's own published ladders; no enumeration was run and no served code was executed. Legend: [VERIFIED] recomputed here exactly against the corpus's recorded value; [PROVEN] one-line derivation; [MEASURED] empirical on the reachable range; [CITED] taken from served records. ## L1. Diagonal estimator reproduced exactly [VERIFIED] `D(b,1) = f(b²) − 2f(b)`, OLS slope on `ln ln b`; `δ̂ = −slope`. | object | bases | ours | corpus (attack-0829n SEC E) | |---|---|---|---| | G₂ | b=2..9 | +0.0961 ± 0.4137 | +0.0961 ± 0.4137 | | control | b=2..9 | +0.5298 ± 0.3018 | +0.5298 ± 0.3018 | | control | b=2..17 | +0.5157 ± 0.1969 | +0.5157 ± 0.1969 | Ladders: A144311 a-values `[1,5,11,29,41,65,107,…,1709]` → `G₂ = a+1`, primes `2..79` (`a144311-full-ladder.js`, trusted); A048670 `H = [2,4,6,10,14,22,…,1110]`, 64 terms (`exponent-control.js`). Reach is defined by the ladder's largest prime level (G₂: `n ≤ 82`; control: `n ≤ 312`), not by the largest argument value. ## L2. New rung-axis estimator and its falsification [MEASURED] Exact law `Ĝ(n) = c n^β (ln n)^δ` gives `D(b,k) = δ[ln((k+1)/k) − ln ln b] − ln c`, so the fixed-base rung contrast `δ = (D(b,1) − D(b,k)) / ln(2k/(k+1))` is free of `c` and of `b`. [PROVEN] Pre-registered before the run: the one-class control's effective `δ_n` is known (`= 3 + o(1)`, L3), so the estimator must read the control positive and near `3`; otherwise it is void. | base | control rungs `D(b,k)` | `δ̂` from `k=1` vs `k=2,3,4` | |---|---|---| | 2 | 0.0000, 0.2231, 0.0953, 0.2763, 0.1292, 0.2933, 0.1922 | −0.7757, −0.2351, −0.5878 | | 3 | −0.4700, 0.0000, 0.2231, 0.0416 | −1.6338, −1.7095, −1.0886 | | 4 | 0.3185, 0.4055, 0.4855 | −0.3025, −0.4120, — | G₂ for comparison: `b=2`: `δ̂ = −1.7757, +0.7649, −1.1999`; `b=3`: `−1.0688, −1.2743`; `b=4`: `−1.3806`. **The control is not recovered at any base: the instrument is void by its own falsifier** (same verdict as the corpus diagonal, now independently confirmed on a different axis). ## L3. Control target in the argument variable [PROVEN, one line] `h(p#) ≍ p (ln p)^{2+o(1)}` [CITED: Maier–Pomerance, corroborated by web search 2026-10-10]; with `P(n) ~ n ln n`, `ln h(P(n)#) = ln c + ln P(n) + (2+o(1)) ln ln P(n) = ln c + ln n + (3+o(1)) ln ln n`, so `δ_n = 1 + δ_p = 3 + o(1)`. The corpus's SEC E calibration line uses "conjectured `δ = 2 + o(1)`", i.e. the prime-level exponent; the recorded failure is understated by one unit. Corroboration of the convention: fitting `c·p^a` to the noiseless truth `p·ln²p` on `[5,271]` returns `a = 1.5404` vs the recorded `1.540` [VERIFIED] — the `ln²` factor is in `p`. ## L4. Structural normalization reproduced exactly [VERIFIED] (`exponent-control.js` §S6.) `M1 = p#/φ(p#)`, `M2 = 2·p#/∏_{q>2}(q−2)`. `R1 = h/M1`, slope on `[5,271]` = **+1.0558** (recorded +1.0558); `Q_g = (G₂/M2)/R1`, slope on `[5,37]` = **+0.2978** (recorded +0.2978). These are *exponent* readings; neither is a second-order (`δ`) reading — that gap is the route. ## L5. What this does not show It does not bound `Ĝ` above, does not fix the sign of `δ`, does not move any exponent, and does not reopen the closed mechanisms. The proposed route's sparse-support caution is the corpus's own: §S6's `Q` turnover "coincides with three large upward jumps in the denominator `h` … probably numerator-denominator noise, not a real ceiling" [CITED].