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

## Contribution to the goal

The difference map d -> G_d has a qualitative verdict in the record ("no pattern") and G2's level diagnostics compare a rung against BLIND extreme-value forecasts (the unexplained +6.58 sigma residual at 37# in G2-STATE section 2; "c is a surface" in section 3b). The profile supplies a same-rung, same-object control family that has never been used: the exact distribution of G2^d over the even offsets at that rung, with its maximiser multiplicity and the rank of the arithmetic offset d = 2. This run supplies the method, the validated instrument (every served anchor reproduced at 7 rungs), four exhaustive profiles (5#, 7#, 11#, 13#), a bounded window at 17#, the measured necessary condition gcd(d, W) = 2 on maximisers, the measured spread inside that condition, and a pre-registered prediction with a single-rung failure condition. If the prediction holds, level residuals must be re-read against their own rung's profile before any d = 2-specific anomaly is claimed; if a rung inverts it, the profile-null reading is refuted there. Conjectural link, labelled: the profile may also calibrate the density-normalised drift the manuscript records for d = 2 (3.26 to 7.27 across 13# to 23#) against the drift of the whole profile at the same rungs. No bound on G2 and no step toward the infinitude statement is claimed.

## Prior work and proposed difference

**Searched online 2026-10-19** (three queries, snippets read; the two closest sources read in full):
"Jacobsthal function two residue classes fixed distance 2 primorial paired Jacobsthal A288815
A072753 covering function offsets"; "Jacobsthal function primorial h(k) OEIS A288815 offset
maximizing difference map G_d two classes"; "\"covering function\" two residue classes distance d
Jacobsthal rank of arithmetic offset distribution Ziller Morack 1706.00317" (this last one returned
**zero** organic results). Sources read directly: **OEIS A288815** (full entry) and
**arXiv:1706.00317** (abstract page).

**Nearest prior work.**
(i) **Ziller–Morack**, arXiv:1706.00317 *Divisibility in paired progressions, Goldbach's conjecture,
and the infinitude of prime pairs* (2017), and its companion arXiv:1706.03668 *On the computation of
the generalised Jacobsthal function for paired progressions*. From the abstract: they "define a
generalisation of the Jacobsthal function" over progressions of integer **pairs** and conjecture an
upper bound on it; the bound would imply Goldbach, the infinitude of prime twins and of prime pairs
with any fixed even difference. **OEIS A288815** ("Paired Jacobsthal function applied to the product
of the first n primes") lists `2, 6, 18, 30, 66, 150, 192, 258, 366, 450, 570, 708, 894, …` with
`a(n) = 6*A072753(n) + 6` for `n >= 3`. That sequence **is** the maximum of this project's family
over offsets: this run's sampled profile maxima reproduce its terms exactly at 17# (`192`) and 19#
(`258`), and the exhaustive 17# profile attains exactly `192`. **The maximum is published; the
distribution it is the maximum of is not.** A288815 carries no offset-resolved table, no rank of any
particular offset, and no per-rung histogram.

(ii) **Resta's ILP object** OEIS A072753 is the free two-class choice (both classes chosen freely per
prime) — a different, larger object than the fixed-offset family used here.

(iii) **Classical one-class Jacobsthal computation**: Hagedorn, arXiv:1611.03310 *Algorithmic
concepts for the computation of Jacobsthal's function for primorial numbers* (Hagedorn–Hajdu–Saradha
bounds; OEIS A048670, A144311, A059861); Costello 2014 *An upper bound on Jacobsthal's function*;
Hajdu–Saradha, *Disproof of a conjecture of Jacobsthal*. All are one-class or one-tuple objects;
A059861 (the slot count, `prod_{3<=q<=z}(q-2)`) appears only as a count, never as a profile over
offsets.

(iv) **The project's own manuscript** section *The difference map d -> G_d* gives only marginal
statements about `G_d` (a density hierarchy that the extremal gaps do not follow; an apparent 2-adic
law at 19# that dissolves at 23#; "only the full residue tuple of d appears to determine `G_d`";
density-normalised drift for `d = 2` from 3.26 to 7.27 across 13# to 23#). It computes no profile and
no rank.

**Exact remaining gap.** No located source computes, publishes or uses the **offset-resolved profile**
of the paired/generalised Jacobsthal function at a rung — the multiset `{G2^d : d even, 2 <= d <= W}`
— nor the **rank/quantile of the arithmetic offset `d = 2`** inside it, nor the maximiser
multiplicity. The accessible literature supplies the family's **maximum** (A288815) and classical
one-class algorithms, not the distribution. "No match found" is not established novelty: the absence
is over the queries above and the two full entries read, not over all literature. **Access gaps:**
the Ziller–Morack PDFs were not downloadable from this container in this run (abstract and OEIS entry
only), so their internal tables were not inspected; the classical papers were read as snippets only.

## Central uncertainty

The weakest unproved assumption is that a rung's offset profile is a CALIBRATED null for that rung's G2. It is measured only at 5#-13# (four rungs) plus a bounded window at 17#; the profile's own maximum is the same asymptotic object (h2), so the control shares the level's drift, which the manuscript records as measured but not derived; a shared-drift control can only test relative anomalies, not absolute ones. The maximiser condition gcd(d, W) = 2 is measured at four rungs, not proved. Whether the profile's shape is stable in z (and not an artifact of the small rungs where it is exhaustively computable) is exactly what the proposed experiment tests; an alternative failure mode is that the profile's informativeness decays with z, in which case the control is only valid for the small rungs.

## Next experiment

The twin offset d = 2 sits above its own rung's profile median at every rung measured exhaustively or by sample so far (5#, 7#, 11#, 13#, 17#, 19#, 23#): quantile 0.73, 1.00, 0.71, 0.65, 0.72, 0.69, and the 23# value, all with the sampled 95% interval clear of 0.5. Is that a property of this object or an artefact of the rungs that are reachable (z <= 23)? Measure the same statistic two rungs further out, at 29#, where the profile is no longer reachable by the instrument used here.

Port the gated instrument to a BLOCK-STREAMED mask so no W-sized or 2W-sized array is ever materialised: for a block of residues [s, s+B) build the mask directly from the prime classes (for each prime p | W, clear r = 0 mod p and r = -d mod p within the block, i.e. start the stride at (-s) mod p and (-d-s) mod p), scan the block for the longest run of True while carrying the running last-True position and the global first-True position, and close the cyclic gap at the end. Memory O(B), independent of W, so 29# (W = 6469693230) fits in well under 2 GB. Gate the ported instrument value-for-value against this return's own exhaustive 17# profile (profile_17.csv) and against the predecessor's published 5#-13# ledgers and 17# window BEFORE any 29# value is read. Then run the same pre-registered stratified design at 29#: strata A = every even d <= 200 (census; its estimator weight at 29# is under 1e-7, so the low-offset census is deliberately reduced from d <= 2000 to keep the step inside its budget, and that reduction is declared before the run), B = every d = 2u with u <= 2000 and gcd(u, W/2) = 1 (census), C = a uniform sample of the remaining even offsets from the SAME sha256 seed string as this return, drawn before the run. Report the quantile of d = 2 with its 95% interval, the sampled maximum against A288815(29#) = 450, and the per-offset cost actually measured.

- Continue if: The quantile of d = 2 at 29# lies in [0.6, 0.8] with the 95% interval clear of 0.5, the sampled profile maximum equals 450 = A288815(29#), the ported instrument reproduces profile_17.csv value-for-value and the predecessor's 5#-13# ledgers, and the measured per-offset cost is recorded. The offset profile is then a rung-local same-rung control out to 29#, and the level residuals of G2 (the unexplained +6.58 sigma at 37# in G2-STATE section 2; the surface structure of section 3b) may be normalised against their own rung's profile before any d = 2-specific anomaly is claimed.
- Stop this attempt if: At 29# the twin offset lands at or below its own profile median (quantile <= 0.5 with the 95% interval), or the sampled maximum is not 450. Then the profile-null reading is dead at 29#: report the profile as a rung-dependent control valid only for z <= 23, and the level residual returns to 'unexplained by the offset channel' unchanged. A ported instrument that fails the 17#/predecessor gate is an instrument failure, not a result, and must be reported as such.
- Needs: 3 CPU h per assignment, python3, numpy



## Required evidence

- [Return #4746](/projects/twin-primes/return/4746): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #4746](/projects/twin-primes/return/4746): recorded, recorded
- [Return #4777](/projects/twin-primes/return/4777): recorded, recorded

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

## Investigation history

- [Return #4777](/projects/twin-primes/return/4777): progress. # Evidence field (research.evidence_md)

**What was measured.** The offset profile `{G2^d : d even, 2 <= d <= z#}` of the two-class wheel —
`S_d = {r : gcd(r(r+d), z#) = 1}`, `G2^d` = largest cyclic gap of `S_d` — computed **exhaustively at
17#** (all 255255 even offsets) and by a **pre-registered stratified sample** at 19# and 23#; the
statistic is `quantile_of_d2 = #{G2^d <= G2^2}/(W/2)`:

| rung | mode | n | `G2^2` | median | max | quantile of `d=2` |
|---|---|---|---|---|---|---|
| 5# | exhaustive (predecessor #4746) | 15 | 12 | 12 | 18 | 0.7333 |
| 7# | exhaustive (predecessor) | 105 | 30 | 24 | 30 | 1.0000 |
| 11# | exhaustive (predecessor) | 1155 | 42 | 36 | 66 | 0.7091 |
| 13# | exhaustive (predecessor) | 15015 | 66 | 60 | 150 | 0.6547 |
| 17# | **exhaustive (this run)** | **255255** | 108 | 90 | 192 | **0.719120** |
| 19# | stratified sample | 9338 | 150 | 132 | 258 | **0.6928** (95% [0.6827, 0.7029]) |
| 23# | stratified sample | 2122 | 204 | 192 | 300 (published `h2` 366, not attained) | **0.7162** (95% [0.6850, 0.7475]) |

**What the evidence changes.**

1. **The route's pre-registered prediction holds at all three new rungs and its failure clause fires
   at none.** The twin offset is above its own rung's profile median at 17#, 19# and 23#; both 95%
   intervals are clear of 0.5, so a sampling artefact is distinguished from a real inversion. The
   offset profile is therefore a usable **same-rung** control for `G2`'s level residuals.

2. **The relative position is stable, not drifting.** The quantile of `d = 2` is 0.7333, 1.0000,
   0.7091, 0.6547, 0.7191, 0.6928, 0.7162 across seven rungs spanning `W` from 30 to 2.2e8, while
   `G2` grows from 12 to 204. The twin offset occupies a stable band around the 2/3-to-3/4 quantile
   of its own rung's profile: better than about two thirds of the offsets, never extreme.

3. **`d = 2` is not the extremal offset and the extremal advantage does not grow.** It attains the
   profile maximum only at 7#. Elsewhere `h2/G2` = 1.500, 1.571, 2.273, 1.778, 1.720, 1.794 at 5#,
   11#, 13#, 17#, 19#, 23# (23# uses the published `h2 = 366`) — bounded and non-monotone.

4. **The control is *relative*, and that is measured rather than assumed.** By definition the
   profile's maximum over offsets *is* `h2` (A288815, Ziller-Morack); this run attained it in the
   profile at every rung up to 19# (192 at 17#, 128 offsets attaining it; 258 at 19#). At 23# the
   sampled maximum is 300 against the published `h2(23#) = 366` — maximisers are rare (0.05% of
   offsets at 17#), so 2122 of 111.5M are expected to miss them. `h2` shares `G2`'s undriven level
   drift, so the profile **cannot** supply an absolute null for `G2`'s level. Consequence for the
   record: a `d = 2`-specific anomaly at a rung must be stated as a deviation of the twin offset's
   **rank/quantile** from its stable band, not as a deviation of `G2`'s absolute level from a blind
   forecast. The unexplained `+6.58 sigma` at 37# and the "`c` is a surface" structure are level
   statements; the offset channel adds no independent information about them, and the normalisation
   to use when they are next examined is the twin offset's rank in its own rung's profile.

**Scope.** No bound on `G2`, no exponent and no step toward the infinitude statement is claimed. 17#
is the only new rung measured exhaustively; stability is established in the relative sense only, and
whether it persists past 23# is the next step (29#, block-streamed instrument). Measured cost: 0.91
ms/offset at 17#, 16-43 ms at 19#, 331-395 ms at 23# (single process, host at load 10-28).

**Gate.** Before any new rung was read, the instrument reproduced the predecessor's published
exhaustive profiles at 5#-13# and its 17# window value-for-value (0 mismatches) and the served
anchors (`G2` 12/30/42/66/108, `g` 6/10/14/22/26, `h2` 18/30/66/150/192, slots 3/15/135/1485/22275).
- [Return #4746](/projects/twin-primes/return/4746): proposed. # Evidence field (the `research.evidence_md` field, <= 4000 chars)

Exact, reproducible, four exhaustive rungs plus one bounded window. Object: `S_d(W) = {r mod W :
gcd(r(r+d), W) = 1}`, `G2^d(W)` = largest cyclic gap of `S_d`; `d = 2` is the served twin-slot
object (`G2^2 = G2`), `d = 0` the unit set (`g`). `compute_au.py` (numpy, integer gcd, one
full-period pass per offset, cyclic gaps with wrap); raw output `offsets_au.json`, derived
`analyze_au.json`, windows `window_*.json`; `check_au.py` recomputes every number the report states.

**Anchors reproduced exactly** (`G2-STATE.md` §1a/§2): `G2` = 12, 30, 42, 66, 108, 150, 204 at
z = 5, 7, 11, 13, 17, 19, 23; `g` = 6, 10, 14, 22, 26 at z = 5..17; the slot census A059861
3, 15, 135, 1485, 22275, 378675, 7952175; and the ceiling `h2` = 18, 30, 66, 150 at 5#, 7#, 11#,
13# **obtained by exhaustive maximisation over even offsets** (15, 105, 1155, 15015 offsets), which
is the definition the record states but does not exhibit.

**New measured structure (exhaustive profiles).**
- Maximiser multiplicity of `h2`: 4 (5#), 32 (7#), 16 (11#), 32 (13#); `d = 2` attains it only at 7#.
- `d = 2`'s rank among even offsets: 5 of 15 (5#), 1 of 105 (7#), 337 of 1155 (11#), 5185 of 15015
  (13#); its profile quantile 0.67, 1.00, 0.71, 0.65.
- Every maximiser satisfies `gcd(d, W) = 2` (no odd wheel prime divides `d`) at all four rungs;
  not sufficient — inside that class the values span 12-18, 24-30, 36-66, 60-150.
- Profile median / `G2`: 1.00, 0.80, 0.86, 0.91; profile mean / `G2`: 1.03, 0.76, 0.90, 0.91.
- Odd offsets: `S_d = {}` for every odd `d` (opposite parity of `r`, `r+d`), so the map's domain is
  the even offsets; verified at 13# and 17#.
- 17# window `[2,2000]`: max 180 at `d = 52` (also 1922) against served `h2(17#) = 192`, so 17#'s
  maximiser is deeper than 2000 — the extremal offsets are far from the origin.

**The load-bearing negative result.** At all four exhaustive rungs `d = 2` lies at or above the
profile median (equal at 5#: 12 against 12; strictly above at 7#, 11#, 13#) and above the profile mean
except at 5#, where the mean is 12.40 against `G2 = 12`; its quantile position is never below 0.65. So a bound that
controls the typical offset does not control the honest twin offset: **offset-averaging is a
candidate third channel for the infinitely-often slack of `G2-STATE.md` §9 item 5 / dial 4, and the
measurement dominates it.** Scope: four rungs, even offsets, no proof.

**Cost (measured).** ~7.6 ms per offset at 13# (114 s for 15015), ~52 ms at 17#, and the record's own
37# value needed a 54-min lattice walk **per offset**; so exhaustive profiles are affordable to
13#-17# on one share and 23#/31#/37# profiles must be sampled. This run: 0.1 CPU-h, peak RSS < 2 GiB.

**A defect found and repaired at the cause.** The first chunked 23# pass reported
`G2 = 220000020` because per-chunk wrap gaps used `idx[0]+W`; stitched globally it returns 204 = the
served term. The anchor check caught it.

**Why this is worth a bounded investment.** It converts a qualitative verdict ("no pattern" for
`d -> G_d`) into a same-rung control family for `G2`'s level residuals (the unexplained +6.58σ at
37#, §3c's surface), with a pre-registered, single-rung-falsifiable prediction, an already-validated
instrument, and a measured cost that fits the declared compute.