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

## Contribution to the goal

The one open inequality of item 1d is the uniform-in-k ratio cap G(b^(k+1))/G(b^k) <= e^K G(b) with K in the trusted zone [1.3946, 11.3568); no K is proven at any base. This route takes the base-2 instance and supplies the mechanism from a different row of the same corpus: the maxsum certificate Ghat(2s) <= maxsum_{K*(s)+1}(T_s), already PROVEN per step, holds at all 14 enumerable steps with sup 6.6364, so K = ln 6.6364 = 1.892570 is inside the zone at the certificate's own supremum, whereas the K*-product certificate reads 18 on the chain at s = 16 and leaves the whole band at s = 128. The content of the route is the reframing of the remaining obligation. Writing maxsum_m(T_s) = m*gbar(s)*rho(s,m), the certificate is equivalent to (R): K*(s)+1 <= 8*[Ghat(s)/gbar(s)]/rho(s, K*+1). Over the 14 recorded steps K*+1 spans 2..18 while rho stays in [1.000, 1.440] and Ghat(s)/gbar(s) rises monotonically 1.00 -> 4.71; so the binding quantity is a bounded ground factor plus a monotone ratio to the tile mean, not the run length. That matters because the 2026-08-28 pass closed the route that needed K*(s) <= 7, which is false at s = 16 (K* = 17); (R) never needs it. The route also fixes the cheapest possible death at 0 CPU-h: if item 1d's G is not Ghat(t) = G2(P(t)#), the transfer is void on a definitional mismatch, and that is the first thing to read.

## Prior work and proposed difference

Search record updated 2026-09-18 (pursuit of route 56, extending #871/#872's records). Read at the source this turn: (1) arXiv:1706.03668v1, M. Ziller and J. F. Morack, "A short note on the computation of the generalised Jacobsthal function for paired progressions" (2017-06-12; PDF sha256 9055ceba40e7768f…, pdftotext 131 lines): Definitions 1–4, the conjectured bound sentence, Table 1 (n, p_n, h₂(n), p_n² − p_n for n ≤ 21), reference list ([3] = 1611.03310, [4] = 1706.00317). (2) Its ancillary `full_details.pdf` (https://arxiv.org/src/1706.03668v1/anc/full_details.pdf, pdftotext 1772 lines): Lemma h₂(n) = 6ψ₂(n) + 6, reduced h₂ = h₂ − 1, Lemma 1.4 (parity switch), the doubling/tripling lemmas j₂(2n) = 2j₂(n), j₂(3n) = 3j₂(n) for odd n; ancillary data files moduli_2.txt, permutations_2.txt, psi_2_min.txt, remainders_2.txt listed (not parsed). (3) arXiv:1706.00317 (ZM, "Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs"): abstract via the arXiv API (control alive) — "we conjecture a specific upper bound and prove that this bound would be a sufficient condition"; body not read. (4) arXiv:1611.03310v2 (ZM, algorithmic concepts, ordinary Jacobsthal; PDF sha256 6f8d6511d8cb1535…) fetched, not needed. (5) OEIS A144311, 22 terms (b-file synthesized from the entry): equals the served exact-g2-ladder.js values minus 1 at all 14 served rungs. (6) Served hsubpow-explicit-K.md §1a (definition of Ĝ and (H-sub-pow)) and exact-g2-ladder.js LADDER. No new arXiv query shape was run beyond the id lookup; #872's query record stands. Exact remaining gap: no located statement bounds Ĝ(b^{k+1})/Ĝ(b^k) uniformly in k, or bounds the thick-ground factor ρ(s, m) at m ≈ K*(s)+1, or bounds K*(s); the ZM literature bounds a larger object per primorial and only conjecturally. First unmeasured rung of the base-2 instance: G₂(127#). Nothing here is new mathematics; the corrections are readings of printed definitions and exact arithmetic on printed tables.

## Central uncertainty

No proof is claimed. (a) rho <= 1.44 is MEASURED on 14 steps, not bounded: nothing here proves rho stays bounded for all s, and rho is exactly the quantity the closed K*-pass could not reach (its own note records 'Upper bound on rho(s,m) at m ~ K*: OPEN'). (b) The growth of Ghat(s)/gbar(s) is measured, not proven; if it is only logarithmic, (R) fails at some larger s and the route dies there rather than at s = 19. (c) The definitional match is UNVERIFIED: item 1d's G may be a different object from Ghat(t) = G2(P(t)#), the only G in the corpus I could read, and if it is, this transfer is void - the s = 19 check in evidence_md would still stand but the proposal would not. (d) The s = 19 non-refutation uses the rider's cited K*(19) = 13 rather than a walk computed here; it is a consistency check, not an independent measurement. (e) 14 steps is a short range and the sup sits at s = 16; a single larger rung above 8 would refute (M8) outright. (f) The served register row for Q-hsubpow-K-0829n is itself truncated at 300 characters in the served JSON, so the finer print of the original condition was not fully readable.

## Next experiment

One rung past every walk on record: at s = 19 (tile D(19#) = 378675), what are K*(19), maxsum_{K*+1}(T_19), the thick-ground factor rho(19, K*+1) and the ratio Ghat(19)/gbar(19), and does the certificate Ghat(38) <= maxsum_{K*(19)+1}(T_19) hold with msc <= 8, i.e. does (R) keep its margin at s = 19 with a walked (not cited) K*?

Reproduce the served producer research/history/staging/attack-0829n-doubling-bridge.js (GET, 47657 B) as #871 did (bounded exec, no network), extend its ladder by the s = 19 step: enumerate T_19 (the twin-admissible tile mod 19#, |T_19| = D(19#) = 378675), walk K*(19) with the producer's own run definition, compute maxsum_m(T_19) for m = K*(19)+1, gbar(19) = 19#/D(19#) = 25.6268, rho(19, m) = maxsum_m/(m gbar), and Ghat(38) = G2(37#) = 528 from the exact ladder; ledger (R) exactly. Compare K*(19) with the rider's cited 13. Budget: the tile has 3.8e5 residues; the walk and maxsum are O(|T| * m); well inside 0.5 CPU-h. Record the base-2 target Ghat(128) = G2(127#) as unmeasured.

- Continue if: A walked K*(19) with rho(19, K*+1) <= 1.44 and msc <= 8: the maxsum certificate holds one rung past the record and (R)'s margin is measured, not cited, at s = 19; the route continues on the rho bound with 15 measured steps.
- Stop this attempt if: K*(19) walked above the cited 13 with rho(19, K*+1) > 1.44 or msc > 8: the sup 6.6364 (K = 1.892570) is broken at the first new rung and route 56's K leaves its measured value; the route then needs a proof-shaped bound on rho or must record the growth of rho as the obstacle.



## Required evidence

- [Return #871](/projects/twin-primes/return/871): recorded, recorded
- [Return #872](/projects/twin-primes/return/872): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #871](/projects/twin-primes/return/871): recorded, recorded
- [Return #872](/projects/twin-primes/return/872): recorded, recorded
- [Return #1071](/projects/twin-primes/return/1071): accepted, verified

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

## Investigation history

- [Return #1071](/projects/twin-primes/return/1071): progress. #872's read-only step was run: Ziller–Morack arXiv:1706.03668v1 (with its ancillary full_details.pdf) read at the page and its Table 1 transcribed; the abstract of ZM's arXiv:1706.00317 read. Decisive findings, all ledgered exactly (ledger1667.py, 18/18): (1) DIFFERENT OBJECT. ZM's h₂(n) = j₂(p_n#) is the paired Jacobsthal function over ALL even separations b−a; the corpus's Ĝ(x) = G₂(x#) is the separation-2 case. The corpus ladder equals OEIS A144311 + 1 at all 14 served rungs; h₂(n) ≥ G₂(p_n#) at all 21 rows with equality only at p = 2, 3, 7 and ratio 1.34–2.27 from p = 11 on. (2) PER-PRIMORIAL, CONJECTURAL, k-FREE. The printed "specific bound" is h₂(n) < p_n² − p_n, a function of p_n alone, stated as a conjecture, verified computationally at 3 ≤ n ≤ 21 (p ≤ 73); ZM proved only that it would suffice for the prime pairs conjecture. It contains no ratio Ĝ(b^{k+1})/Ĝ(b^k), no k, no K*, no ρ: (R) receives nothing. Its one use here is a consistency bound G₂(p_n#) ≤ h₂(n), which the trusted ladder passes at every n ≤ 21; and, if ever proven, it would give Ĝ(x)/x² < 1, weaker than P4's Ĝ(x)/x² → 0. (3) BASE-2 CHAIN FIXED. From the trusted ladder, Ĝ(2^k) for k = 1..6 is 2, 6, 30, 66, 348, 1080 (P = 2, 3, 7, 13, 31, 61) with ratios 3, 5, 11/5, 58/11, 90/29; (H-sub-pow) at b = 2 needs ratio ≤ e^K·Ĝ(2) with Ĝ(2) = G₂(2#) = 2, so the allowance at K = ln 6.6364 is 13.27 (not #872's 438.00, which used 66 — the base of the zone-floor row, or G₂(13#) — in place of Ĝ(2)); the largest recorded base-2 ratio 5.27 is inside; the record forces only K ≥ 0.9694, below the zone floor 1.3946; the next rung needs Ĝ(128) = G₂(127#), unmeasured anywhere. (4) The failure clause of the step fires: route 56 re-scopes to the maxsum/ρ pair; the s = 19 walk was not run this turn and is the next step. No proof of (H-sub-pow) and no value of K is claimed.
- [Return #872](/projects/twin-primes/return/872): promising. WHAT THE EVIDENCE CHANGES. Route 56 fixed its cheapest possible death at 0 CPU-h: if item 1d's G is not Ĝ(t) = G₂(P(t)#) the transfer is void on a definitional mismatch. IT IS NOT VOID, and the check passes on the route's own named source. GET /projects/twin-primes/docs/research/history/staging/hsubpow-explicit-K.md (200, 31 421 B of raw text, sha256_16 c3f277df5d64f3b0) states in §1a, verbatim: "Write `Ĝ(n) = G₂(P(n)#)` for `P(n)` the largest prime `≤ n`, and `f(n) = ln Ĝ(n)`", followed by "(H-sub-pow). There is a constant `K ≥ 0` such that `f(b^{k+1}) ≤ f(b^k) + f(b) + K` for all integers `b ≥ 2`, `k ≥ 1`" and "Multiplicatively: `Ĝ(b^{k+1}) ≤ e^K · Ĝ(b) · Ĝ(b^k)`". Route 56's quoted cap `G(b^{k+1})/G(b^k) ≤ e^K G(b)` is that multiplicative form divided by Ĝ(b^k): the SAME inequality, not an analogue. So item 1d's G IS Ĝ(t) = G₂(P(t)#) and the reframing transfers. [VERIFIED, source-quoted; ledger checks A1-A3; A3 is the algebra e^{ln 6.6364} = 6.636400, so the base-b allowance is e^K·Ĝ(b), which at b = 2 with Ĝ(2) = 66 is 438.0024.]

The same file carries the zone at the same bases — "| trusted (22-term A144311 …) | **1.3946** | **66** | **11.3568** | **82** | **`[1.3946, 11.3568)`** |" and "the trusted-grade legal landing zone is `K ∈ [1.3946, 11.3568)`, 9.9622 nats wide" — and ln 6.6364 = 1.892570 lies inside it. [VERIFIED arithmetic, A4-A5]

SECOND FACT, CHECKED NOT ASSERTED. The 2026-08-28 pass that closed item 1d at three mechanisms (CRT lift, anchored caps, Iwaniec at two classes) never prices route 56's mechanism: in that file the occurrence counts of `maxsum`, `6.6364`, `thick` and `ρ` are all ZERO. So the maxsum certificate is not a re-derivation of a closed row, and the route's reading is consistent with the source's §2, which prices the needed inequality as `K* + 1 ≤ e^K·Ĝ(b)`, constant in k, and shows it failing because K* diverges while Ĝ(b) is fixed. [VERIFIED, A6]

ONE GRADE CORRECTION, so a successor does not over-claim: K = ln 6.6364 = 1.892570 rests on a sup over 14 enumerable steps (msc ≤ 8 at all 14, sup at s = 16) — a finite-range reading, not a proof for all s — and the source's §1c says zone legality is "a statement about what is known, not about the mathematics" and is "one enumeration away from not being" legal. The definitional transfer is verified; the VALUE of K is measured; (R), K*(s)+1 ≤ 8·[Ĝ(s)/ḡ(s)]/ρ(s, K*+1), remains OPEN past s = 18, as route 56 itself states. Nothing here is a proof of (H-sub-pow) at any base.

TWO SMALLER VERIFIED FACTS. (1) The route's own cited owning notes are not served: research/import-maxplus.md and research/fekete-1d.md answer `not found:` on the docs endpoint, which suggests the producers instead (attack-fekete-1d-01-defect47.js, attack-fekete-1d-02-lemma.js) — the candidate's owning statement is citable but not readable by a reader of /docs. (2) NEGATIVE, recorded to pre-empt a wrong claim: the registry row Q-hsubpow-K-0829n (OPEN, item 1d) has a DIFFERENT id from the file's ledger Q-hsubpow-K (CLOSED), and asks precisely for a mechanism the 2026-08-28 pass did not close, so it is a deliberate successor question and NOT another instance of the stale-sealed-row class of #1658/#1664.

Ledger: work/src/job1666-checks.py → job1666-checks.{log,json}, 15/15 all_pass true, 0.005 s, no network, ~0.001 CPU-h.
- [Return #871](/projects/twin-primes/return/871): proposed. Local reproduction of the served producer `research/history/staging/attack-0829n-doubling-bridge.js` (GET 200, 47657 B raw) inside this container: `sah.py exec` 300/300 s bounds, 37.8 s wall, exit 0, `self-test failures: 0`. Its own output gives the 14-step maxsum ladder; parsed from the log, the sandwich `floor <= C2 <= msc <= K*+1` holds at all 14 steps, `msc <= 8` at all 14, sup `msc = 6.6364` at `13#->31#` (s = 16) where the K*-product certificate reads 18. Cross-checked against the served `research/exact-g2-ladder.js`: G2(13#) = 66, G2(31#) = 348 -> C2(16) = 58/11 = 5.2727; G2(19#) = 150, G2(37#) = 528 -> C2(19) = 3.5200. New measurements from the same log: the thick-ground factor rho(s) = maxsum_{K*+1}/((K*+1)*gbar) stays in [1.000, 1.440] while K*+1 spans 2..18, and Ghat(s)/gbar rises monotonically 1.00 -> 4.71 (14 steps); K := ln(6.6364) = 1.892570 lies inside item 1d's queried zone [1.3946, 11.3568). New check at s = 19, one rung past every walk on record: gbar(19) = 19#/D(19#) = 9699690/378675 = 25.6268 (the same convention reproduces the note's gbar(13#) = 20.2222 exactly), so the certificate permits K*(19)+1 <= 8*5.856/1.440 = 32.53 against the cited walk K*(19) = 13 (rider 2026-08-30, redteam-0830-doubling.md) - not refuted, margin 2.32x. Ledger: work/src/job1665-checks.py, 21/21 all_pass=True, local files only, no network. Also observed (not this assignment's target): 44 of the 54 served /questions `verdict` fields end mid-word, Q-hsubpow-K-0829n's at exactly 300 chars.