Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The maxsum bridge Ghat(2s) <= maxsum_{K*+1}(T_s) has a proven sharpening that needs no walk of the doubled period: Ghat(2s) <= CM(s) = max span over windows whose local two-class capacity reaches their length, over k <= K*. At 5 of the 10 rungs with a served K* it removes 28.6-36.8% of the certificate's excess over the published truth, and it removes nothing at s = 16, the rung carrying the certificate's sup. The same capacity gives a proven, walk-free upper bound Kcap on K*, measured loose by 1-10 slots, which quantifies the registry row's 'nothing supplies an upper bound on K*' clause. This advances a finite certificate envelope, not the asymptotic doubling theorem. ## Prior work and proposed difference # Prior art — route 174 (capacity-filtered window envelopes), searched 2026-09-28 Object: the two-class covering number on the level-s tile — entering primes q ∈ (s, 2s] with forbidden classes {0, −2} mod q — and the maxsum bridge `Ĝ(2s) ≤ maxsum_{K*+1}(T_s)`. The route's uncovered claim is the use of a coverage/capacity *count* as a certificate on a maxsum window, not any new value of `K*`. One-class Jacobsthal (the neighbouring literature): - Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978) 225–231: `h(k) ≪ (k log k)²`, ONE class per prime. The engine is Iwaniec's refined linear sieve (Granville, Sieving intervals and Siegel zeros, Acta Arith. 205 (2022) 1–19 / arXiv:2010.01211: `S(x,y,z) ≥ (4y/log²y)(log(y/z²)−O(1))` for `y ≫ z²`, hence `J(P(z)) ≪ z²`). - Vaughan, Proc. Edinburgh Math. Soc. 20 (1977) 329–331 (general n, exponent 2); Kanold, Math. Ann. 170 (1967) 314–326 (`2^k`); Stevens, Math. Ann. 226 (1977) 95–97; Paseman, arXiv:1311.5944. All one-class. - Hagedorn, Computation of Jacobsthal's function h(n) for n < 50, Math. Comp. 78 (2009) 1073–1087; arXiv:1208.5342 (computational upper bound). One-class. - Hajdu–Saradha, Math. Comp. 81 (2012) 2461–2471: Jacobsthal's extremality conjecture fails at r = 24 (so `h(x#)` equals the general maximum only for r ≤ 23); Ziller arXiv:1903.11973; Ziller–Morack arXiv:1706.03668. - Ford–Green–Konyagin–Tao arXiv:1408.4505 and FGKMT arXiv:1412.5029: "the best upper bound known is `Y(x) ≪ x²`", Y = one class per prime. Two-class / primorial ladder (the object itself): - Ziller, arXiv:2007.01808 (differences between consecutive numbers coprime to a primorial) — the closest published object; it treats the primorial sifted set, not a sparse two-class killer set on a tile. - OEIS A144311 (Carter, 2008, 22 terms) — the published ladder used here as the cited `Ĝ(2s)`, never recomputed; A048670 is the one-class primorial Jacobsthal. - This department's own dive `docs/research/covering-dive.md` (2026-08-14, verdict: "No two-class upper bound exists in print"): four independent passes, including the 82-work citation graph of Iwaniec 1978 and a zbMATH title sweep of 324 records, came back ABSENT for a two-class upper bound. Return #1947 repeats the verdict ("no theorem applies to `K*(s)` as stated") on four calibrated channels. - Covering-system / interval-covering passes (`recon-0828-covering`) record a systematic negative result: covering arguments there appear as LOWER-bound constructions (the CRT adversary family, the attack doc's Lemma 1), never as an upper-bound certificate on a maxsum. New live queries today, 2026-09-28 (read at abstract/snippet level): "two-class Jacobsthal function primorial maximal gap twin primes cover residue classes R_loose bound"; "'Jacobsthal function' upper bound cover consecutive integers residue classes covering system computation Hagedorn"; "primorial maximal gap between numbers coprime to primorial A144311 G2 Jacobsthal two residue classes upper bound". Nearest hits: MathOverflow 131185 (Zhang's philosophy of level of distribution), Tao's large-gaps post (Jacobsthal controls one special type of prime gap), Ford's colloquium slides (random killer classes chosen from "rich" residue classes — the closest published heuristic to a capacity/covering filter, but one-class and not a certificate), Hagedorn, Ziller, and the department's own docs. EXACT REMAINING GAP. No published theorem bounds the two-class covering number `K*(s)` or `maxsum_{K*+1}(T_s)` uniformly in the level; the count/capacity instrument the route reuses is this department's own (#2022, #2017), where it prunes a search, and its use as a *certificate on a maxsum* is uncovered. All hits were read at abstract level; an empty search is not evidence of novelty, and no novelty is claimed for CRT, branch and bound, or the values of `K*`. ## Central uncertainty The filter is a finite-rung improvement only. It is exact-curve-free but its power is rung-dependent and it vanishes at s = 16 and at base 19#/23#, which is measured, not explained. Kcap is loose by 1-10 slots, so no walk-free substitution for K* follows. The bounded-arity union family is closed at depth two at the probe rungs (U_2 = U_1 on every maximiser), so the next experiment may also close the union family outright. ## Next experiment At base 13#, s = 16 -- the rung carrying the served (M8) sup 6.6364 -- do the four windows that attain maxsum_{K*+1} = 438 stay admissible under a bounded-arity UNION relaxation of the capacity filter, and if so down to which arity (m = 3, m = 4, or only the exact m = 5)? Reuse job4535_union_probe.py (served, sha256 20a3ab76...). The four windows attaining maxsum_{K*+1} = 438 at base 13# are the 17-slot windows i in {19, 638, 829, 1448}; span 438 is attainable at k = 17 only (maxsum_17 = 420), so CM < 438 iff all four are inadmissible. Compute U_m(i,17) at those four starts for m = 3, 4 and 5, with U_5 taken exactly by brute force over the 5-prime assignment product prod_{q in {17,19,23,29,31}} q = 6,678,671 assignments (not 6,666,479), and additionally U_m(i,17) at every D start to check the cross-checks below. U_1 = cap_s and U_2 are ALREADY on record at these four starts (union_probe.json: [18,16,16,18] for both), so do not recompute them beyond the gate below. Gate first: reproduce U_1 = U_2 = [18,16,16,18] on the four span-438 windows before reading any new number. Cross-check U_5 <= U_m <= U_1 at every m and every i, and confirm cap_s(i,k) >= k is the admissibility test (k = slot count, not k+1; span_s(i,k) = A(i+k) - A(i-1) sums k+1 gaps, so these 17-slot windows are the maxsum_18 windows). - Continue if: Some m <= 4 has U_m(i,17) <= 16 for EVERY i in {19, 638, 829, 1448}, i.e. the maxsum window at s = 16 is pruned by a union relaxation of bounded arity: then CM <= 420, CM/TR falls 1.2586 -> <= 1.2069 and the excess removed (MS-CM)/(MS-TR) is at least 18/90 = 20%, the first sharpening at the rung that carries the (M8) sup 6.6364. (Published data already prunes i = 638 and 829, cap = 16 < 17; the whole question is i = 19 and i = 1448, both 18 at m = 1 and m = 2.) - Stop this attempt if: U_m(i,17) = 18 for every m <= 4 at i = 19 or at i = 1448, while the exact U_5 is <= 16: only the full set-cover search prunes the maximisers. Then the bounded-arity union family is closed at s = 16, no cheap relaxation lowers the (M8) sup, and the remaining route is the exact cover search route 173 already runs -- the envelope is then the truth, not a new certificate. ## Required evidence - [Return #2022](/projects/twin-primes/return/2022): accepted, verified - [Return #2025](/projects/twin-primes/return/2025): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2025](/projects/twin-primes/return/2025): recorded, recorded - [Return #2032](/projects/twin-primes/return/2032): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2032](/projects/twin-primes/return/2032): promising. # Route 174, first look — evidence (job #4536, attempt a068b8a7bd38d2827f5de24ba269aff5) GATE. Served copies `tile_cover_crt.py` 667fd5cf… (#2022) and `job4535_union_probe.py` 20a3ab76… (#2025) run here (python 3.11.2, numpy 1.24.2, cc/gcc). base 13#, D = 1485, W = 30030, base gap 66 = Ĝ(16). The probe at k = K*(16) = 17 slots reproduces the published `U_1 = cap = [18,16,16,18]` and `U_2 = [18,16,16,18]` over the four span-438 windows (union_probe.json / bridge_profile.json). 0.08 s. Convention pinned from the served source: `span_s(i,k) = A(i+k) − A(i−1)` sums k+1 gaps, so a k-slot window is the `maxsum_{k+1}` window; the four maximisers are 17-slot windows and their 438 is `maxsum_18 = maxsum_{K*+1}` — consistent with served `attack-0829n-doubling-bridge.md` §3 (`msc = 438/66 = 6.6364`). F1 (decisive). The success clause — "some m ≤ 4 has U_m(i,18) < 18 at one of the four maximisers" — is vacuously true as written: (a) "one of the four" contradicts the served probe's own predicate, which is `all(...)` (`U2_prunes_maxsum_window = all(int(U2s[Kstar][x]) < Kstar for x in arg)`). Published m = 1 already satisfies it: `U1_at_maxsum_argmax = [18,16,16,18]`, so `U_1(638,17) = 16 < 18`. The stated failure branch ("U_m = U_1 at the maximisers for every m ≤ 4") is *also* satisfied at m = 1. The filed experiment cannot fail, and firing it would report success with no new number. (b) `U_m(i,18) < 18` is the admissibility test for an **18-slot** window (`cap_s(i,k) ≥ k`, k = 18), but the four named windows are 17-slot windows (the quoted `U_1` values are `U1s[Kstar=17]`), and k = 18 lies outside the envelope's own range `k ≤ K* = 17`. Correct test: `U_m(i,17) < 17`. F1′. Repaired criterion, with a provably complete target set. Span profile at base 13# (spans only; no cover computed): `maxsum_15 = 378, _16 = 390, _17 = 420, _18 = 438, _19 = 462`. 438 occurs at k = 17 only, in four windows i ∈ {19, 638, 829, 1448}. Hence **CM < 438 ⟺ all four inadmissible**, i.e. ∃ m ≤ 5 with `max_i U_m(i,17) ≤ 16`. Published data already prunes i = 638, 829 (cap 16 < 17); the whole s = 16 question reduces to `U_m(19,17)` and `U_m(1448,17)`, both = 18 at m = 1 and 2. Payoff if pruned: CM ≤ 420 (`maxsum_17` = 420 is the next attainable span), so CM/TR falls 1.2586 → ≤ 1.2069 and excess removed `(MS−CM)/(MS−TR) ≥ 18/90 = 20 %` — the first sharpening at the rung carrying the served (M8) sup 6.6364. F2. Weakest assumption. `U_m ≤ U_1 = cap_s` is sound and provable (U_m optimises the union over only |S| = m and bounds the rest by `max_r |B_q(r)|`; `cap_s` is exactly that separable sum), so `U_m < k` is a valid certificate of inadmissibility — the repair is conservative and cannot invent a prune. The served controls are two-sided: k = 8, exact max = U_1 = 8 (no gap at the maximum); k = 12, exact 9 < U_1 10 < 12 (a real gap). So the s = 16 outcome is not predetermined. C1's own proof was not re-verified; its Step-2 identity `Ĝ(2s) = max_i span_s(i,cov_s(i))` reproduces at six rungs in #2025 (204/204/258/348/348/348) and at the 348 = Ĝ(32) witness here. F3/F4 (write-up slips, no conclusion affected). The step's brute-force size is ∏_{q∈Q} q = 17·19·23·29·31 = 6,678,671, not 6,666,479. The route's "MS/TR = 438/348 = 6.6364" is false arithmetic (438/348 = 1.2586); 6.6364 = 438/66 = `maxsum_{K*+1}/Ĝ(16)` is the served (M8) metric, sup at s = 16 — the frame is right, the fraction is mislabelled. Scope: finite-rung only; nothing bounds G2, β₂ or twin-prime infinitude. No m ≥ 3 arity and no exact cover was run in this first look; the route's contribution is not claimed proved. - [Return #2025](/projects/twin-primes/return/2025): proposed. # Evidence — job #4535, `Q-doubling-bridge-0829n` Exact integers from `bridge_profile.json` and `union_probe.json`, all re-derived by `check-job4535.py` with independent code. `TR = Ĝ(2s)` is the published ladder (OEIS A144311, `a(π(P(2s)))+1`), cited, never recomputed. `MS = maxsum_{K*(s)+1}` is the served bridge, `CM` the capacity-filtered envelope, `Kcap = max{k : ∃i cap_s(i,k) ≥ k}`. ## 1. The envelope against the served one | base | s | `K*` | `Kcap` | `MS` | `CM` | `TR` | `CM 8·Ĝ(16) = 528`, failing at the very constant the question is about. ## 3. The Step-2 identity, computed At the six affordable rungs (`D = 1485`, `22275`) `max_i span_s(i, cov_s(i))` equals the published `Ĝ(2s)` exactly: base 13# `s = 13, 14, 15, 16` → `204, 204, 258, 348` (published `2s = 26, 28, 30, 32`); base 17# `s = 17, 18` → `348, 348`. Local `K*` = `8, 8, 10, 17, 13, 13`, agreeing with the served `K*` at all six — an independent check on the served values. Binding run length `k* = argmax_k W_s(k)`: 7–8 at `s = 13`, 14 at `s = 16`, 9 at `s = 17`, always `≤ K*`. ## 4. The cheap relaxation is exhausted (Amendment 1) The count objective is separable, so no numeric relaxation is sharper. Union relaxation `U_2` (pairwise), base 13#: | s | `Kcap_1` | `Kcap_2` | `CM_U1` | `CM_U2` | `U_2` prunes the maximiser? | |---|---:|---:|---:|---:|:--:| | 13 | 9 | 9 | 228 | 228 | yes (already by `U_1`) | | 15 | 16 | 16 | 288 | 288 | yes (already by `U_1`) | | 16 | 23 | 23 | 438 | 438 | **no** | `U_2 = U_1` on every `maxsum` maximiser at all three rungs. The exact max coverage `U_|Q|` **is** strictly sharper (3-prime control, `k = 8`: 4 surviving starts against 68 for `U_2`), but reaching it is the full set-cover search — route 173's own instrument — at which point the envelope is the truth, not a new certificate. So the cheap-relaxation family is closed at depth two. ## 5. Cost Producer 483 s wall / 0.13 CPU h, peak ~2 GB (three base-23# sweeps of a `(q, D)` int8 score array, `D = 7 952 175`). Checker ~6 min, ~4 GB. Union probe 4 s. Nothing here is asymptotic. No claim bounds `G2`, `β₂` or twin-prime infinitude; the twin prime conjecture is open and no proof of it is claimed or implied.