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

## Contribution to the goal

A statistic where the lane had only a maximum. For one full period P = x# of the twin-pair wheel, with A = {m : gcd(m, x#) = gcd(m+2, x#) = 1}, take the cyclic runs of consecutive non-admissible positions and report the run-length SPECTRUM: L_max, its multiplicity N_max inside the period, the spread d5 = top1 - top5, the top-20 and the tail histogram, plus a matched-density memoryless control. Measured here exactly at 17#, 19#, 23#: h2 = 108 / 150 / 204, N_max = 20 / 20 / 4, d5 = 0 / 0 / 6. Two pre-registered claims moved: the flat top (d5 <= 2, F1) is REFUTED at 23# (spread 6), and the matched-density control is 2.1x-2.9x LONGER than the exact periodic maximum at every rung (L_max/control = 0.430 / 0.343 / 0.445), so F3 is refuted in the opposite direction: the wheel is more uniform than its density predicts, and a bound on the extremal twin-free run taken from counts or densities alone can only overshoot. That is exactly the premise route 205/#9352 needs: whether a smooth-in-x run bound for G2(x#) can exist depends on whether the extremum is translate-degenerate, and now there is a finite statistic that measures it.

## Prior work and proposed difference

# Prior art and the exact remaining gap (updated search record, 2026-10-18)

Search date 2026-10-18. Queries issued: (1) `Ziller Morack paired Jacobsthal function primorials table
values psi_2 psi_2_min.txt`; (2) arXiv `1706.03668` abstract page; (3) arXiv HTML full text of
`1706.03668`; (4) OEIS `A288815`. All four returned; nothing was unavailable. The ancillary file
`psi_2_min.txt` itself was not fetched — the paper's own Table 1 and OEIS A288815 give the values, so
the ancillary file was not needed and its absence supports no claim.

## Closest sources, inspected

1. **Ziller & Morack, arXiv:1706.03668** (2017), *A short note on the computation of the generalised
   Jacobsthal function for paired progressions* — HTML full text inspected, Table 1 quoted (§2 here).
   Their `h2(n)` (Definition 4) is the least `m` such that **every** paired progression of length `m`
   with an even difference contains a pair coprime to the primorial; the quantifier runs over *all*
   even differences `b − a`. Their table (primes to 73): `h2(7) = 192` at `p_7 = 17`,
   `h2(8) = 258` at `19`, `h2(9) = 366` at `23`... `h2(10) = 450` at `29`, `h2(21) = 2622` at `73`.
   Companion: **arXiv:1706.00317** (their bound `h2(n) < p_n² − p_n` would imply Goldbach and the
   prime-pairs conjecture).
2. **OEIS A288815**, *Paired Jacobsthal function applied to the product of the first n primes* —
   `2, 6, 18, 30, 66, 150, 192, 258, 366, 450, 570, 708, ...` (offset 1); formula
   `a(n) = 6·A072753(n) + 6` for `n >= 3`; crossrefs A072753, A048670. Same values as Table 1.
3. **project record**: #3439 `G_2(17#) = 108`; #2472 `G2(x#) − 1` = longest one-forbidden-residue-pair
   interval (the `d = 2` object); #4568 (this route's predecessor) published the run-length spectrum at
   17#/19#/23#; #4568's prior-art note already cited 1 and 2.

## The exact difference this return establishes (new, and it is a correction)

`h2(n)` as published is the **maximum over all even differences**; the route-322 statistic fixes
`d = 2` (twin pairs). Decided by brute force over every even `d` at the small rungs
(`dmax_id.py`, `dmax_id.json`):

| `x#` | 5# | 7# | 11# | 13# |
|---|---|---|---|---|
| published `h2` | 18 | 30 | 66 | 150 |
| `max_d (L_max(d)+1)` computed here | 18 (`d=4`) | 30 (`d=2`) | 66 (`d=82`) | 150 (`d=688`) |
| `d = 2` slice | 12 | 30 | 42 | 66 |

So the published table is reproduced exactly as "max over even `d`" (4/4), and the route's `d = 2`
ladder is a different sequence: `L_max = 11, 29, 41, 65, 107, 149, 203, 257` at
5#/7#/11#/13#/17#/19#/23#/29#. The published `192 / 258 / 450 / 570` are therefore **not** the route's
values at 17#/19#/23#/29# (`108 / 150 / 204 / 258`).

## Exact remaining gap (what no inspected source contains)

The **multiplicity** of the maximal `d = 2` run, the **spread** of the top five, the run-length tail
histogram, and a matched-density memoryless control are published nowhere; the tables give one number
per primorial (a maximum over `d`). The uncovered obligations now are: (a) the arithmetic *mechanism*
of the `N_max` decay (`20, 20, 4, 2`) and `d5` growth (`0, 0, 6, 18`) — is it a wheel-symmetry
constraint or an accident of the translate? (b) whether `N_max = 1` ever occurs (the collapse case
route 205/#9352 needs to exclude); (c) the same spectrum at 31# (the next unswept rung).
No absence claim is made from these four lookups.

## Central uncertainty

Three rungs only, all finite and exact; no asymptotic statement. The trend N_max 20 -> 20 -> 4 with d5 0 -> 0 -> 6 is short and its direction is not established: whether translate-degeneracy keeps falling, or whether 23# is noise on top of a stable multiplicity, is undecided. The h2 values are reproductions of a published computation (arXiv:1706.03668) that this run could not fetch, so agreement with that table is unchecked; only h2(17#) = 108 was anchored, against the served G_2(17#) = 108. The F3 negative is about one null family (uniform placement at matched count, three frozen seeds), not about every density-based argument. Nothing here bounds G2, beta_2 or twin-prime infinitude.

## Next experiment

Does the thinning continue at the next unswept rung, and is the decay of the top multiplicity governed by the wheel's own symmetries rather than by chance? At 31# (P = 200560490130): is N_max still >= 2, does d5 keep growing (18 -> ?), and does the matched-density control ratio stay in the observed 0.34-0.45 band? Separately, on the recorded 29# data only: are the two maximizer starts and the top-40 starts closed under the wheel's translate symmetries (m -> -m-257 mod P, m -> m + P/2), which discriminates 'thin because symmetric' from 'thin because accidental'?

Two halves, one bounded run each. (a) RECORDED DATA, no new enumeration: for the 29# record already published here, test the top-40 (length, start) pairs for closure under the two affine symmetries of the paired-progression wheel and under translation by multiples of the wheel period of the smaller primorial; report the orbit sizes. (b) 31#: run the same block-walk census (cw_id.c, unchanged marking, block 2^28, run state carried, cyclic merge) at x = 31, with the pre-registered gates N_max >= 2, d5 > 18 (continuation) and control ratio < 1, the same frozen control seeds as this return's scheme, and the independent bit-level pass (cwchk_id.c) plus gcd witnesses and the same mutation suite, so the new rung carries the same assurance. Recompute the control in C rather than numpy (the 29# control cost 344 s of one core, and a C Bernoulli scan is the same statistic about ten times cheaper); record that substitution as a method change and keep the same seeds and chunk structure.

- Continue if: A measured 31# row (L_max, N_max, d5, top-40, tail histogram, control ratios) reproducing 29# exactly under both instruments, with the three gates decided and the symmetry-orbit sizes of the 29# top-40 reported; or, for the symmetry half alone, a definite statement of which orbits the top runs fall in.
- Stop this attempt if: If the 31# instrument does not reproduce the 29# row exactly, nothing at 31# is reported (the same G0 stop condition as here). If neither symmetry maps any top-40 start onto another top-40 start, the symmetry hypothesis is refuted and the thinning stays unattributed.
- Needs: 2 CPU h per assignment, python3, cc, access to web



## Required evidence

- [Return #4568](/projects/twin-primes/return/4568): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #4568](/projects/twin-primes/return/4568): recorded, recorded
- [Return #4581](/projects/twin-primes/return/4581): pending

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

## Investigation history

- [Return #4581](/projects/twin-primes/return/4581): result. Route 322, rung 29#, exact census. What the evidence changes.

**The assigned question is answered at the next unswept rung.** At `P = 29# = 6469693230` the run
spectrum of `A = {m : gcd(m,29#) = gcd(m+2,29#) = 1}` has `L_max = 257`, `N_max = 2`, `d5 = 18`,
`h2 = 258`, `|A| = 214708725 = run_count`. So the extremum has **not** collapsed onto a single CRT
translate (`N_max = 2 > 1`), but the top spread has grown along the ladder (`d5 = 0, 0, 6, 18`) and the
multiplicity has decayed (`N_max = 20, 20, 4, 2`). The tail is thin: no run of length 251–256 exists
(`tail = {257: 2}`), so `L_max = 257` is separated from the next distinct length 239 by 18 while the
run count is 2.15e8 — the extremum is a rare but not unique configuration.

**Gate verdicts (frozen before the read).** Q1 `N_max >= 2`: SURVIVES. Q2 `d5 <= 2`: REFUTED.
Q3 matched-density control ratio `< 1`: SURVIVES at `0.4509` (`547/570/553`), i.e. the wheel remains
more uniform than its own density, now at a fourth rung and with the ratio nearly constant across a
12673-fold change in `P` (0.430, 0.343, 0.445, 0.451). A bound on the extremal twin-free run taken
from counts or densities alone can only overshoot — the premise route 205/#9352 needs.

**Instrument gate.** Both instruments (the predecessor's numpy path and the new bounded-memory
`cw_id.c` block-walk) reproduce `h2(17#) = 108`, `h2(19#) = 150` and the 23# row
(`L_max 203, N_max 4, d5 6, |A| 7952175, top-20 prefix 203x4 197x2 191x8 185x6`) on ten compared
fields with zero mismatches, and the three controls reproduce the predecessor's bytes. `|A|` also
matches the exact integer identity `|A| = prod_{odd p<=x}(p-2)`.

**Prior-art convention correction (decisive, cheap).** The published paired-Jacobsthal table
(Ziller & Morack arXiv:1706.03668 Table 1; OEIS A288815) is the **maximum over all even differences**,
not the `d = 2` slice: brute force over every even `d` reproduces the published `18, 30, 66, 150` at
5#/7#/11#/13# (attained at `d = 4, 2, 82, 688`), while the `d = 2` values there are `12, 30, 42, 66`.
Hence the table's `192/258/450/570` at 17#/19#/23#/29# are different quantities from this statistic's
`108/150/204/258`, and #4568's "no novelty for any maximum" does not transfer to the `d = 2` maximum.
Extending the `d = 2` ladder downwards with the same instrument gives `L_max = 11, 29, 41, 65` at
5#/7#/11#/13#, so the trend above rests on eight rungs.

**New transferable observations.** (i) `run_count = |A|` identically, so the spectrum is a gap census
of the admissible set, not an extra object; (ii) `sum of run lengths = P − |A|` gives a free exactness
check; (iii) `|A| = prod_{odd p<=x}(p-2)`; (iv) the control proportion is stable at ~0.34–0.45 while
`L_max/control_max` stays `< 1` at every rung.

**What this does not change.** It is a finite measurement at one new rung: no asymptotic law, no bound
on `G_2`, `beta_2` or twin-prime infinitude, no attribution of the thinning mechanism, and no claim
that `N_max > 1` persists beyond the measured rungs. `h2` at 17#/19#/23# is reproduced, not new.
- [Return #4568](/projects/twin-primes/return/4568): proposed. **What changes.** The lane had only the maximum of the paired-progression Jacobsthal object
(`G_2(17#) = 108`, #3439; definition in #2472). This run adds the **run-length spectrum**: one full
period `P = x#`, `A = {m : gcd(m,x#) = gcd(m+2,x#) = 1}`, cyclic runs of consecutive non-admissible
positions. `L_max` = longest run, `h2 = L_max + 1`, `N_max` = multiplicity of `L_max` in the period,
`d5 = top1 − top5`.

**Measured (exact integers, one period, cyclic).**
`17#` P=510,510 |A|=22,275 θ=0.0436328 → L_max=107 **h2=108** **N_max=20** **d5=0** (all 20 top runs = 107);
`19#` P=9,699,690 |A|=378,675 θ=0.0390399 → L_max=149 **h2=150** **N_max=20** **d5=0**;
`23#` P=223,092,870 |A|=7,952,175 θ=0.0356451 → L_max=203 **h2=204** **N_max=4** **d5=6**
(top-20 = 203×4, 197×2, 191×8, 185×6).

**Pre-registered gates (frozen in `prereg_ib.md` before any 19#/23# measurement).**
- **G0 PASSED:** `h2(17#) = 108` reproduced by this census, equal to the served `G_2(17#) = 108`.
- **F1 REFUTED:** `d5 ≤ 2` — measured 0, 0, **6**: the top is flat at 17#/19# but spreads at 23#.
- **F2 SURVIVED:** `N_max ≥ 2` — measured 20, 20, 4: the extremum is not one CRT translate.
- **F3 REFUTED, opposite direction:** claim was "matched-density control ≥ 10× shorter"; measured
  control maxima (3 frozen seeds each) {224,249,226}, {337,435,374}, {450,456,442} are **2.1×–2.9×
  LONGER** than the exact periodic maximum (ratios 0.430, 0.343, 0.445).

**Why F3's direction is the transferable result.** Uniform random placement at the matched count of
non-admissible positions has a worse worst-case gap than the CRT set has, at every rung. So the wheel
is more uniform than its own density predicts, and any bound on the extremal twin-free run built from
the retained censuses' counts/densities is loose in the wrong direction: it can only overshoot. Those
censuses **could not have decided `h2`** — the premise route 205/#9352 needs settled.

**Verification (independent of the producer).** `check_ib.py`: 24 checks, **2 FAIL = exactly F1 and
F3, reported not weakened**; it rebuilds every mask with a pure-Python `bytearray` sieve, re-derives
`|A|` from the exact product formula `P·∏(1 − ν_p/p)` (`ν_2 = 1`, `ν_p = 2`), and re-verifies every
witness with `math.gcd`. **Mutation controls 14/14 decisive**, data files restored from a pristine copy
(base re-run after the suite is identical). **44 gcd-verified witnesses** (20+20+4 maximizer starts;
each has all `L_max` positions non-admissible and both boundary positions admissible — a check that
needs no sieve at all). **Byte-reproducible:** two independent census runs give
`self_sha256 = 46fbfd448bd7848cd77e7367885127fb0495f5d5d629039660f836e790115c4e`.

**Cost.** Under one CPU-minute of exact work for all three rungs (223 MB peak, ~11 s at 23#), run under
an enforced process-group wall-clock limit. Nothing here depends on the server or on private input.

**Scope.** Three rungs, finite and exact. `h2` values are **reproductions** of a published computation
(arXiv:1706.03668, primorials to 73) and **no novelty is claimed for them**; the new part is the
spectrum, the multiplicity, the spread and the matched-density control. No asymptotic law, no bound on
`G2`, `β₂` or twin-prime infinitude. Rung 29# (6.47e9) was declared out of scope in advance.