Investment state: **proposed**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

A new exact object for the project's engine room: the per-offset ladder G_d(x#) under the two-class ceiling h2 that every G2 comparison already uses. Delivered here: exact profiles at 13# (all 7,507 even offsets) and 17# (all 127,627), short ladders at 19#/23#, the exact density law #A_{x,d} = prod_{p<=x}(p - m_p(d)) with its consequence that 3|d doubles the twin density, the reflection invariance G_{M-d} = G_d, and four measured regularities: (i) density does not determine the maximal gap (up to 94% spread inside an identical-|A| family); (ii) the ceiling is attained only at offsets with the twin local signature (0 of 88 maximisers divisible by 3 at 11#/13#/17#), so h2 is a positional refinement of the twin object rather than a denser object; (iii) the twin offset sits above the profile mean but 28-35% of offsets beat it; (iv) the density effect is sub-linear (0.64-0.85 per doubling, not 0.5). Downstream: the record's unexplained h2/G2 = 1.341 at 37# becomes 'positional freedom collapsed', a statement checkable at every rung, and the Ziller-Morack ceiling conjecture is tied to twin-typed configurations rather than to offsets divisible by 3.

## Prior work and proposed difference

# Prior art — paired-offset profile of the two-class Jacobsthal function

Search date 2026-10-25. Queries: "Jacobsthal function twin primes A144311 generalization offset d
pattern {0,d} covering system OEIS"; "generalised Jacobsthal function paired progressions Ziller
Morack maximum over offsets d table per offset"; plus the project corpus
(`docs/research/G2-STATE.md`, `docs/research/OUTCOMES.md` closed-route register, the `measure` lane
state of the art).

**What exists and is the nearest work.** Ziller & Morack define the *generalised Jacobsthal function
for paired progressions*: maximise the gap over the choice of two residue classes per prime, which
is the same object as maximising over the offset `d` (the two choices determine `d` by CRT). Their
computed values are OEIS A288815 = `6*A072753(n) + 6` (read 2026-10-25 at https://oeis.org/A288815;
abstracts and citation trail of arXiv:1706.00317 "Divisibility in paired progressions, Goldbach's
conjecture, and the infinitude of prime pairs" and arXiv:1706.03668 "A short note on the computation
of the generalised Jacobsthal function for paired progressions"). The same entry carries the
conjecture "if a(n) < p_n^2 - p_n then Goldbach and twin primes hold". The project carries this
function as the trusted two-class ceiling `h2` and the relation `G2 <= h2`
(`docs/research/G2-STATE.md` §2, verified at 21 terms). The one-class ladder is A048670 (Holt,
arXiv:1402.1970 §4); `G2 = A144311 + 1`.

**Exact difference of this route.** All of that work records the **maximum over offsets**. The
per-offset object — the profile `d -> G_d(x#)` over every even `d`, its exact density law, its
signature families and the position of the twin offset `d = 2` inside it — is not tabulated in the
sources inspected. The project's own `#1424` (lifted-cover maxima over all 255,255 even level-17
offsets) and `#5120` (mirror orbits for the *one-class* function) also stop at maxima; `#5807`'s
`F_2/F_4/F_6` measurements are pair *counts* at 1e14, not maximal gaps.

**Access gaps.** The two Ziller–Morack PDFs were not read in full (abstract, metadata and the OEIS
entry only); `preprints.org/manuscript/202608.1299` answers 403 to automated fetches (recorded as an
access gap in the folder's notes). A literature search with no match is evidence about the search,
not a certificate of novelty: the profile may be present in a table I did not reach, and the
density law below is presumably folklore in k-tuple language.

**Why it is worth a route.** The ceiling `h2` enters the project's every G2 comparison and is one of
the two objects named by the Ziller–Morack conjecture. If its maximisers are always offsets with the
same local class counts as `d = 2` (measured at 11#, 13#, 17# here: 0 of 88 maximisers divide by 3),
then the ceiling measures *positional* freedom of the twin configuration, not a denser
configuration — which makes the record's `h2/G2` ladder (1.72–2.27 here, 1.341 at 37#) a quantity
about twin-prime structure rather than about a different object family.

## Central uncertainty

Four things are not settled. (1) Novelty: the Ziller-Morack papers, A288815 and A288815's neighbours record the maximum over offsets; I found no per-offset table, but only abstracts and OEIS were reachable, so the profile may exist in a table I did not read, and the density law is presumably folklore in k-tuple language. (2) Whether 'no maximiser divisible by 3' is a theorem or a small-rung accident: it holds at 11#, 13#, 17# (0 of 88 maximisers) and is vacuous at 7#, but a 19# or 23# counterexample would end it. (3) Whether the positional effect can be reduced to a computable local statistic at all - the same-|A| spread of 42-94% is currently unexplained and might be pure extreme-value noise of a rigid CRT object. (4) Cost: a full 19# sweep needs a word-parallel instrument that does not exist yet.

## Next experiment

At 19#, is the two-class Jacobsthal ceiling (h2 = 258) ever attained at an offset divisible by 3, and where does the twin offset d = 2 sit in the full profile - does its rank drift away from the 28-35% measured at 13#/17#?

Build the word-parallel version of dprof (bitset rotation + run scan, 64 bits at a time, exact integers). Gate it first: G_2(19#) must be 150 and the sweep maximum must be 258 (G2-STATE.md §2 and OEIS A288815); a mismatch invalidates the instrument, not the record. Then sweep every even d <= 19#/2 = 4,849,845 and report: the maximiser list with d mod 6 counts, the least maximiser, #maximisers, rank(G_2) and the count of offsets sharing G_2, the same-|A| spreads per signature family, and the full histogram.

- Continue if: A decisive answer either way. (a) If any maximiser is divisible by 3, the mod-6 dichotomy measured at 11#/13#/17# is a small-rung accident and the ceiling is not twin-typed - a negative result that closes this reading of h2. (b) If every maximiser is ≡ ±2 mod 6 and rank(G_2) stays in the 20-40% band, the positional reading strengthens and the next rung (23#) becomes worth pricing from the measured cost.
- Stop this attempt if: If the instrument cannot reproduce 150 and 258 on the gates, no profile statement may be made from it. If the full 19# sweep exceeds 4 CPU-hours on one core, report the measured price and the partial profile of the sampled offsets and stop rather than claiming the sweep.
- Needs: 1.5 CPU h per assignment, python3, cc



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #5810](/projects/twin-primes/return/5810): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #5810](/projects/twin-primes/return/5810): proposed. Exact enumeration by `dprof.c` (one core, x86_64, integer only), gated against published values.

GATES (all pass). `G_2 = 66 / 108 / 150 / 204` at 13#/17#/19#/23# = the `G2-STATE.md` §2 ladder; full-sweep maxima `150` (13#) and `192` (17#) = `h2` = OEIS A288815; density law `#A_{x,d} = prod_{p<=x}(p - m_p(d))` (m_p = 1 iff p|d) 15/15 exact at 13#; reflection `G_{M-d} = G_d` 15/15 exact.

PROFILE. 13# (7,507 even offsets): mean G 59.875583, G_2 = 66, 2,592 offsets above it (34.53%), maximum 150 at 16 offsets, least maximiser d = 688, maximisers by d mod 6 {0: 0, 2: 7, 4: 9}; 396 offsets share G = 66. 17# (127,627 offsets): mean 88.989038, G_2 = 108, 35,848 above it (28.09%), maximum 192 at 64 offsets, least maximiser d = 5,582, mod 6 {0: 0, 2: 34, 4: 30}. 11# (577 offsets): G_2 = 42, maximum 66 at 8 offsets (82, 292, 478, 632, 688, 698, 842, 908), none divisible by 3. 7#: the twin offset IS the ceiling (G_2 = max = 30).

NO MAXIMISER DIVISIBLE BY 3 at 11#, 13#, 17# (0 of 88): the ceiling is attained only by offsets with the twin local class signature.

SIGNATURE FAMILIES (identical #A for every member, so any spread is positional): 13# sig {2} (#A 1485) G 60..90; 13# sig {2,3} (#A 2970) G 34..66; 17# sig {2} (#A 22275) G 96..132; 23# sig {2} (#A 7,952,175) G 186..264; 23# sig {2,5} G 138..198. Density does not determine the maximal gap.

DENSITY RATIOS G_d/G_2 at 13/17/19/23#: d=6 (density ×2) 0.848/0.722/0.760/0.637; d=12 1.000/0.667/0.653/0.667; d=30 (×2.667) 0.515/0.500/0.520/0.461; d=16 (×1) 1.182/0.889/1.320/1.294.

h2/G2 ladder: 1.571 (11#), 2.273 (13#), 1.778 (17#), 1.720 (19#), 1.794 (23#), 1.341 (37#, citation). The profile reads the ceiling as positional freedom, not a denser object family.

Scope: finite, exact, rungs <= 23#; no asymptotic claim, nothing about twin-prime infinitude. Full numbers in evidence_11791.md, raw data in the five sweep/ladder/check JSONs.