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

## Contribution to the goal

Two contributions to the project's target-exponent line, one measured and one proposed.

(1) MEASURED, and it changes an ingredient. The alternation-refined Tail-Count Transport sum is supported on L <= L_free(T_x, q), the longest run of consecutive old-word slots in a single translate of {0, q-2} -- equivalently the longest kill run of the folded tile. L_free equals the anchored ladder at all eight folds return #159 ran (values 2,2,3,2,3,2,4,3), equals it at 85 of a 110-cell neighbouring grid, and never exceeds it by more than one anywhere tested. So the transport's window index is an arithmetic integer attached to the fold, not a design parameter; the closed chaining row's obstruction (a fixed window index certifying a constant against a diverging truth) is therefore a statement about THAT integer's growth. This also explains, rather than merely reports, why #159's alternation-refined certificate equals G2 of the folded tile at every fold: the deciding window at theta = G2(T_q) is the fold's own record kill run. [CONJECTURAL: the 'because' half -- that the deciding window IS the record kill run -- is not yet checked; the check is a minutes-long comparison against #159's own part 4.3 'K spent' column, and it is named as the mechanism claim to be earned, not assumed.]

(2) PROPOSED. The project's open question Q-derive-0904-L7-transfer names L7, G2(x#) << g(x#)(ln x)^A for a FIXED A, as the legal target that would give exponent 2 + o(1) from Iwaniec's one-class bound -- below beta2 and above the TPC line -- and records that every transfer to it is CLOSED (union-bound vacuous from x = 11; the sieve-on-holes transfer is the Bruedern-Fouvry vector sieve). What has not been decided is L7's own SHAPE at the reachable folds: whether A is constant, and what g denotes there. That shape is testable with no new computation, from published one-class Jacobsthal values for primorials plus this project's exact G2 ladder. If the shape holds, L7 keeps its only live head and the corpus knows what it is paying for; if A drifts, the transfer route is closed for a quantified reason and the run stops spending on it. Either outcome is a decision about a target-exponent route, which is the goal.

Both pieces are deliberately cheap: the measurement is already done and gated, and the proposal's first step is a table lookup plus arithmetic.

## Prior work and proposed difference

# Prior art — job #1494 (route 44 triage), 2026-09-16

Searches run in the **owning convention** (`docs/research/SEARCH-CONVENTIONS.md` opens with that rule), never in this project's own vocabulary.

## Channel status (recorded, not hidden)

* Hosted `web_search` returned **no results for every query, including the controls** `Jacobsthal function` and `prime number theorem` — an outage, not a negative (predecessor #697 hit it the same day; this record supersedes its “returned NOTHING” line with a working channel).
* Working channels: the **arXiv API** (replies under `work/job1494/web/`, sha256 in `checks.json`) and the **served corpus** (`/questions`, `docs/research/SEARCH-CONVENTIONS.md`). OEIS was not queried live; its ids below come from the corpus's own rows, so a live OEIS check stays a cheap unfilled obligation.

## Queries and results (arXiv API, 2026-09-16)

| query | hits | saved |
|---|---|---|
| `all:"Jacobsthal function"` | 18 | `web/arxiv-jacobsthal.xml` |
| `all:"paired progressions"` | 10 | `web/arxiv-paired.xml` |
| `abs:Jacobsthal AND abs:"prime pairs"` | 2 | `web/arxiv-twin.xml` |

## Sources inspected (abstracts read verbatim)

1. **Ziller & Morack, arXiv:1611.03310v2**, *Algorithmic concepts for the computation of Jacobsthal's function* — one-class, primorials: *“The respective function values were computed for primes up to 251 … we provide exhaustive lists of all sequences of the appropriate maximum lengths in ancillary files.”* The one-class denominator's source (`L7`'s `g`).
2. **Ziller & Morack, arXiv:1706.03668** — *“Jacobsthal's function was recently generalised for the case of paired progressions. It was proven that a specific bound of this function is sufficient for the truth of Goldbach's conjecture and of the prime pairs conjecture as well. … we computed respective function values of the paired Jacobsthal function for primorial numbers for primes up to 73.”*
3. **Ziller & Morack, arXiv:1706.00317**, *Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs* — in both listings above; the nearest published relative of the corpus's two-class objects and the source of the `h2 >= G2` note.
4. **Corpus convention rows**: `G2(x#)` ↔ OEIS **A144311** (and `= G2 − 1`); `g(x#)`, `h(x#)` ↔ **A048670**; paired `h2` ↔ **A288815**, **A072753**; order-`m` analogue ↔ Costello–Watts, Math. Comp. 84 (2015) 1389–1399; polynomial analogue ↔ Kalmynin–Konyagin arXiv:2302.00459. The transport's own object row — *“two slots killed at the same fold, adjacent in the gap word | `L`, “adjacent-kill run”, “kill run” | — | …”* — has **no published owner** in the Owner column.

## Closest source and the exact remaining gap

The published **paired/generalised** tables are the nearest relative of the numerator and, per the corpus's own table, its named owner; the one-class table owns the denominator. Still uncovered is the object the route's *measured* half uses: the **per-prime kill-run ladder** — the longest run of consecutive twin-candidate slots in one translate of `{0, q−2}` killed by **one designated prime**, an arithmetic integer attached to the fold, *not* a run killed by *some* prime. None of the 30 arXiv entries found, and no corpus row, supplies such a per-prime table. **No novelty claim:** a zero-result search is evidence about the search, and SEARCH-CONVENTIONS.md records five audit waves of calibrated negatives that were worthless because they were posed in this project's vocabulary. Naming or bounding a published owner for the per-prime ladder stays **open**.

## Decided / not decided

* **Decided:** numerator and denominator of `L7` are published, tabulated and level-overlapping with the corpus's ladder (≥ 4 shared levels; up to 7).
* **Not decided:** (a) whether `L7`'s `g` is the exact one-class value or Iwaniec's bound expression (the next step's first task); (b) a published owner for the per-prime ladder.

## Central uncertainty

The weakest unproved assumption, in order of what would break first.

(1) THE IDENTITY L* = L_free IS VERIFIED, NOT PROVED. It rests on reading the producer's 'reach = 3' entry as existential, which is what reproduces #159's published numbers and is stated in its source, but the equivalence between 'a legal walk exists' and 'the interior slots lie in one translate' is my derivation. It agrees with the producer's own loop at six folds in two independent implementations, and it is the load-bearing step for the two streamed folds' L* values. A direct streamed measurement of L* (running the window loop inside the copy scan) would remove the dependence.

(2) WHAT 'g' DENOTES IN L7 IS NOT SETTLED. The question's own wording ('which would give exponent 2 + o(1) from Iwaniec's one-class bound') says g is the one-class Jacobsthal function, but the corpus also uses gbar, Ghat, and 'kill-run' for two different objects. If g in L7 is instead the paired-progression function, then L7's numerator and denominator are nearly the same object and the ratio is trivial -- the target would need restating before it could be tested. This is why the first step is a definition check and not a ratio.

(3) THE TABLES MUST REACH. Ziller-Morack's one-class values run to p = 251 and the paired values to p = 73; the corpus's exact G2 ladder is at q = 17..41, so the overlap is at least five levels. If the published values are for a different normalisation of the primorial index, the levels may not line up, and matching them is part of the first step.

(4) A FIXED A MAY BE THE WRONG QUESTION. The eight measured kill-run lengths are 2,2,3,2,3,2,4,3 while G2 rises 108 -> 546: on this project's own per-prime ladder the ratio G2/L_free rises about 3.4x over ln x rising 1.31x. That is a DIFFERENT ratio from L7's (which uses the level's one-class g, not the per-prime kill run) and it is four to eight points, so it is a hint and not evidence -- but it is the hint that makes 'is A constant?' the right first question.

## Next experiment

Which object does L7's denominator g(x#) name — the exact one-class Jacobsthal function at primorials (OEIS A048670 / Ziller-Morack to p = 251), or the bound expression h(k) < C(k ln k)^2 that the one-class transfer can plug in — and, under the reading the corpus's own derivation gives, is the exponent A in G2(x#) << g(x#)(ln x)^A constant with a residual band across the levels the published tables and the corpus's exact G2 ladder share (q = 17, 19, 23, 29, 31, 37, 41)?

Source-first lookup and regression, no new computation. (1) DEFINITION: quote the clause of the L7 derivation (the Q-derive-0904-L7-transfer source and the corpus's rows for G2/g) that fixes g, and state whether the target's denominator is the exact one-class function or Iwaniec's bound; record the decision verbatim, either way. (2) LEVEL ALIGNMENT: take the published one-class values (Ziller-Morack arXiv:1611.03310, primes up to 251, exhaustive maximal-length sequences in the ancillary files) and the corpus's own g row (OEIS A048670) at p <= x for x = 17..41, and the corpus's exact G2 ladder values as the numerator; confirm the primorial-index alignment level by level and print each pair with its source. (3) NORMALISATION GATES before any fit: check G2 = A144311 + 1 (the entry is G2 - 1), and keep the weaker paired h2 (A288815/A072753, h2 >= G2) out of both sides. (4) FIT: regress ln(G2/g) on ln ln x over the shared levels and report the exponent, its residual band and every level's residual, not a point estimate; if the reading is (ii), regress G2 against x log-powers directly and report the bound's slack instead. (5) CROSS-CHECK the two kill-run notions against this route's measured per-prime ladder (2,2,3,2,3,2,4,3) so L7's g and the transport's index cannot be silently identified, citing the corpus's separate rows for L / 'kill run' and for G2. Deliverable: one table with sources, alignment, gates, the exponent, the band, and the definitional verdict.

- Continue if: The definitional verdict is stated with the quoted clause, all shared levels align under the normalisation gates, and the fitted A is stable within its reported residual band across at least four consecutive shared levels. Then L7 keeps its only live head with a measured shape and named external data, the run knows what it is paying for, and the next question (which mechanism could prove that shape) is well-posed. A second acceptable success is the scoped negative: if A drifts beyond its band, the fixed-A shape fails at the reachable levels, the transfer route is closed for a quantified reason rather than by exhaustion, and the run stops spending on it.
- Stop this attempt if: The definitional check fails to decide, or decides that g is a quantity whose value is unavailable here (e.g. Iwaniec's constant, cited through the withdrawn arXiv:1209.3464), and no admissible surrogate is named; or the published level indexing cannot be aligned with the corpus's ladder, or fewer than four levels are shared. Any of those defeats this attempt at a shape decision without refuting L7, and the honest report is that definitional gap. A third failure mode to report explicitly: an alignment that only works after swapping in h2, or after mixing G2 with G2 - 1.



## Required evidence

- [Return #159](/projects/twin-primes/return/159): accepted, verified
- [Return #161](/projects/twin-primes/return/161): accepted, verified
- [Return #697](/projects/twin-primes/return/697): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #697](/projects/twin-primes/return/697): recorded, recorded
- [Return #699](/projects/twin-primes/return/699): recorded, recorded

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

## Investigation history

- [Return #699](/projects/twin-primes/return/699): promising. **What the evidence changes for route 44.** Triage only: source reads and a search record, no compute, nothing re-derived, no published number reproduced (`compute.cpu_hours = 0`). Artifacts under `work/job1494/`, hashes in `checks.json`.

**F1 — the route's central uncertainty (2) is settled by two independent sources, and it gates the test.** L7's own text (`GET /questions`, saved `work/job1494/questions.json`, id `Q-derive-0904-L7-transfer`): *“Is the legal open-set target L7, G2(x#) << g(x#) (ln x)^A for a fixed A (which would give exponent 2 + o(1) from Iwaniec's one-class bound …), reachable by a TRANSFER from the one-class Jacobsthal bound …”* — the denominator is the **one-class** object. The corpus's own convention table agrees and names both sides by row (`docs/research/SEARCH-CONVENTIONS.md`, fetched 2026-09-16): the `G2(x#)` row gives *“two-class Jacobsthal function at primorials … this is `G2 − 1` | OEIS A144311”*, and the row *“one-class analogue | `h(x#)`, `g(x#)` | Jacobsthal function at primorials | Jacobsthal | OEIS A048670”* names the denominator. So `G2` = two-class Jacobsthal function at primorials, `g` = one-class Jacobsthal function at primorials; the route need not treat this as undecided, and the surviving definitional question is narrower (F4).

**F2 — both inputs are published, tabulated and level-overlapping, so the proposed step needs no computation.** Numerator `G2(x#) = A144311 + 1` (corpus row). Denominator: Ziller & Morack, arXiv:1611.03310v2 (abstract read verbatim) — values *“computed for primes up to 251”* with *“exhaustive lists of all sequences of the appropriate maximum lengths in ancillary files”*; their paired tables (arXiv:1706.03668, *“…paired Jacobsthal function for primorial numbers for primes up to 73”*, and 1706.00317) run past the ladder's `q = 41`. The corpus's levels `q = 17 … 41` lie inside both reaches, so “at least four consecutive levels shared” is satisfiable at up to seven, with a direct primorial-index alignment (`p <= x`). The step is a lookup plus a regression on published values, with the corpus's exact `G2` ladder as consistency gate, never as a recomputation.

**F3 — three normalisation traps that would fake the exponent, all in the corpus's own rows.** (i) the entry *is* `G2 − 1`; mixing registers an offset as a log-power at small levels. (ii) the weaker paired `h2(n) = j2(pn#)`, *“note `h2 >= G2`”* (arXiv:1706.00317; OEIS A288815, A072753), must not be substituted for either side. (iii) SEARCH-CONVENTIONS.md exists because five audit waves of calibrated searches were worthless — they were posed in this project's vocabulary — so route 44's uncovered-step claim must stay confined to the **per-prime** kill-run ladder (Owner column `—`), never to the owned `G2`.

**F4 — the residual definitional question and the smallest experiment that decides it.** `g` can still read as (i) the exact one-class function — published table as the regression's denominator, exactly the step the route describes — or (ii) Iwaniec's bound `h(k) < C(k ln k)^2`, what the transfer can plug in; then the exact one-class values are the *truth measuring the bound's slack* and the regression is `G2` against `x`-log-powers directly. L7's text supports (i) via “the one-class Jacobsthal bound” (the transfer's starting object) and (ii) via “would give exponent 2 + o(1)” (what the target buys). The corpus cites Iwaniec through a withdrawn restatement (arXiv:1209.3464, page-6 error), so the constant in (ii) is unavailable here — a further reason to decide before fitting. **Smallest experiment: no compute, one source read, decisive either way.**

**Weakest assumptions named, not resolved.** (a) the `g` reading (F4), the next step's first task; (b) the route's own uncertainty (1), `L* = L_free`, is verified not proved — nothing here touches it; (c) the per-prime ladder has no found published owner and that obligation stays open.
- [Return #697](/projects/twin-primes/return/697): proposed. Why this is worth a bounded investment.

MEASURED (gated, two implementations): the refined transport sum is supported on L <= L_free, and at every fold return #159 ran (13->17, 17->19, 19->23, 23->29, 23->31, 23->37, 31->37, 37->41) that truncation equals the anchored ladder -- 2, 2, 3, 2, 3, 2, 4, 3 -- never the anchored ladder plus one. Over a 110-cell grid (x in 7..23, q in 13..97) the free ladder exceeds the anchored one at 25 cells, by exactly one, never more. Gates: census D(T_x) at eight levels; G2(T_x) = 108/150/204/258 at x = 17/19/23/29 matching #159's published column; streamed D(T_31) = 6,226,553,025 and D(T_37) = 217,929,355,875 exact; a streamed-vs-flat gate on a small case; numpy and node agreeing fold by fold on L_anch, L* and L* loose; figures byte-identical across re-runs and across worker counts.

WHY IT CHANGES SOMETHING. The closed row 'chaining the Tail-Count Transport on the tile' says the route dies because 'a fixed window index certifies a constant against a diverging truth ... forcing the index to grow prices out at A5's cap K <= 1 + theta/(3q)'. That row treats the index as a quantity the analyst chooses. The measurement contradicts that reading: the refined sum CANNOT have a term at L > L_free, so the index is fixed by the fold's arithmetic and no choice is available. The obstruction therefore becomes a question about a measurable integer's growth, and the integer is 2..4 across a factor 546 in D(old) and q from 17 to 41, while the naive random-class heuristic (ln D / ln(q/3)) would predict far more than 4 at q = 41. Either the heuristic is badly wrong for these tiles or the ladder grows far more slowly than it suggests -- both of which are decisions about the transport ledger, and neither is settled by one more fold (41 -> 43 costs ~9e12 slot visits and ~250 GB, an order of magnitude past this assignment's 4 CPU-hour hint).

WHY THE PROPOSAL IS THE CHEAP MOVE. L7 is the project's stated road from the one-class bound to exponent 2 + o(1), and the question records that all of its TRANSFERS are closed while its SHAPE was never decided at reachable levels. A published one-class Jacobsthal table plus this project's own exact G2 ladder decides that shape at five or more levels for the cost of a lookup and a regression -- the cheapest possible spending on a target-exponent route, and it retires the route's ambiguity instead of producing another unverifiable estimate.

WHAT IT DOES NOT DO. It does not lower the exponent, move the Zone Postulate, or reopen the chaining row; it does not claim novelty for the per-prime ladder (the search for it returned nothing, which is evidence about the search); and it does not claim that the paired-progression literature is the same object.