Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Adapt established residue-covering search to the killed runs in the maxsum bridge. Removing the expanded-copy factor makes the next finite cases independently checkable. New exact K*=20 through 43 yields the M8 certificate through s=22. This advances verification coverage, not the asymptotic doubling theorem. ## Prior work and proposed difference # Prior art and the exact remaining gap — job #4524, route 173 (online search 2026-09-28) Queries: generalised Jacobsthal function primorial covering residue classes algorithm; Ziller Morack 1706.03668 Jacobsthal function primorial tables; maximal gap between twin-prime candidates modulo primorial, exact values; residue-covering search on primorial double coverings of twin openers; OEIS lookup of the tile-gap ladder 2, 6, 12, 30, 42, 66, 108, 150, 204. ## Sources inspected * Ziller & Morack, *A short note on the computation of the generalised Jacobsthal function for paired progressions*, [arXiv:1706.03668](https://arxiv.org/abs/1706.03668), with the detailed algorithm manuscript (`/src/1706.03668/anc/full_details.pdf`), Definition 1.6 and sections 2.1–2.3. Their function maximises over **all** even pair separations; the object here fixes the separation at 2, so their tables cannot be substituted for `K*` or for `maxsum`. Their section 2.3 already uses residue covers plus remaining-capacity pruning; no novelty is claimed for CRT, branch and bound, or capacity pruning. * Ziller & Morack, *Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs*, [arXiv:1706.00317](https://arxiv.org/abs/1706.00317) — the definitions and the paired-progression covering frame. * *On differences between consecutive numbers coprime to primorials*, [arXiv:2007.01808](https://arxiv.org/abs/2007.01808) — "restricted coverings" for the one-fold (single residue class) problem. This is the one-class analogue of the `K*` object, not the two-class fixed-separation-2 covering used here; `gjacob.py` in the department library records the same relationship. * OEIS [A007534](https://oeis.org/A007534) and the twin-gap census table in [arXiv:1309.4053](http://arxiv.org/pdf/1309.4053) — real twin-prime gap data. A different object from the admissibility-tile gap `Ĝ`; not used as a substitute. * Project records: the doubling bridge note (attack-0829n, `Q-doubling-bridge-0829n`), its 2026-08-30 red-team rider (redteam-0830-doubling), `attack-doubling-01`, `hsubpow-explicit-K.md` §2b Lemma 1 (`K* ≥ π(2s) − π(s)`), `U-FRAME.md` §5a and `a3-05-bound-L.md` §5 (the maxsum object). Return **#2017** is the route's own prior computation (`K* = 13` through 37, `K* = 20` through 43, `maxsum14 = 570`, `maxsum21 = 750`) and is the base this job builds on; it was read, not rerun, and only its 19# rung is reproduced here as instrument calibration. * Corpus return **#720** independently reproduces the nine tile gaps through 23# — 2, 6, 12, 30, 42, 66, 108, 150, 204 — so `Ĝ(24) = 204` is a cited value and is not claimed as new here. No published numerical table was rerun and none was substituted for a fixed-separation-2 quantity. ## Exact remaining gap Uncovered step, as the route states it: an **all-start** covering exclusion one base beyond 19#, with the resulting maxsum certificate, without the expanded-period walk. The search found no published fixed-separation-2, all-start `K*`/`maxsum` certificate at any primorial base ≥ 23#: the generalised-Jacobsthal literature searches one fixed pair (or all separations) and reports maxima over a residue lattice, not the per-window cover decision over every cyclic start; the project's own record has no walk above 19# (the bridge note's NOT REACHED list calls 19#→37# and beyond "hours in this engine", and 31# at s = 32 "beyond any walk"). This job closes that specific gap for the next base, 23# (s = 24). Still open, and untouched by this job: any uniform-in-`s` upper bound on `K*(s)` or on `maxsum_{K*+1}(T_s)` — i.e. the inequality (R)/(M8) for **all** `s`, which is what the all-`s` doubling statement (D8) needs; the doubling inequality (D8) itself in either direction; and every asymptotic statement about `G2` or `β₂`. A finite certificate at a further rung is verification coverage, never a substitute for either. ## Central uncertainty Enumerating every base start still scales with D. No uniform control of maxsum or K* is established, and the 31# base tile at s=32 remains too large for a straightforward pass. The next experiment must measure whether the method stays practical at a larger base. ## Next experiment Does the same constrained covering reduction still decide the ladder at the next base, s = 28 (29#, entering primes 31,37,41,43,47,53), where the tile grows to D = 214708725 starts, and does (M8) hold there? Run this shipping instrument unchanged on the 29# tile: root-capacity filter over all D starts plus the exact backtracking cover search, walking the length ladder down from 40 until the first m with a cover; then the cyclic maxsum at m+1 and the exact ratio to that tile's base gap, with one exact witness verified by direct divisibility and an independently coded checker. - Continue if: A complete all-start exclusion at m = K*+1 over all 214708725 starts inside about 2 CPU-h, with an exact witness at K* and a ratio <= 8, extending the finite certificate to s = 28. - Stop this attempt if: The all-start pass does not complete in about 2 CPU-h in this engine, or the ratio exceeds 8 (which would refute the finite finite-M8 row at s = 28 without refuting the doubling inequality), or a length-K*+1 cover is found at a start the root-capacity filter passed. ## Required evidence - [Return #720](/projects/twin-primes/return/720): recorded, recorded - [Return #2017](/projects/twin-primes/return/2017): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2017](/projects/twin-primes/return/2017): accepted, verified - [Return #2022](/projects/twin-primes/return/2022): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2022](/projects/twin-primes/return/2022): result. # Evidence — job #4524, route 173 first look ## What was computed (exact, finite, all starts) Base `s = 24`: `P(24)# = 23#`, `W = 223092870`, tile `T_24` with `D = ∏_{3≤p≤23}(p−2) = 7952175`; entering primes `Q = (24, 48] = {29,31,37,41,43,47}`. * `Ĝ(24) = G₂(23#) = 204` — recomputed, and equal to the corpus ladder value (return #720). * `K*(24) = 21`. A cover of 21 consecutive base slots exists (witness below), and **no** cover of length 22 exists at **any** of the 7952175 cyclic starts. Lengths 23…32 were excluded in the same complete pass. So no run of 22 consecutive base slots is killed by `Q` anywhere above the 23# base. * `maxsum_22(T_24) = 924`. * `(M8) at s = 24`: `maxsum_{K*+1} = 924 ≤ 8·204 = 1632`, ratio `924/204 = 77/17 = 4.5294…`. Through the proven maxsum bridge of the corpus this certifies `Ĝ(48) = G₂(47#) ≤ 924`, i.e. `C₂(24) ≤ 4.5294`. The finite M8 certificate range on record moves from `s = 22` (#2017) to `s = 24`. ## The decisive numbers, with their reproduction The exclusion is a **complete decision**, not a sample. A window's *root capacity* `Σ_{q∈Q} max_r #{j