Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The all-bases (H-sub-pow) f(b^(k+1)) <= f(b^k) + f(b) + K (f = ln G2(P(n)#)) is the single remaining gap of the bounded-defect route, and the owner note's PROVEN sign lemma says it holds with a finite K if and only if the second-order exponent delta of Ghat ~ c n^beta (ln n)^delta is >= 0; the legal zone [1.3946, 11.3568) then consumes K >= 1.0597 delta - ln c (its T7 is a necessary condition), so an upper bound on delta is exactly the quantity the zone needs. Today the sign of delta for Ghat is unknown two-sided, and the one instrument the corpus built for it - the k = 1 diagonal - failed its control. This return measures that the second slice of the same identity fails the same control by more (control readings -0.3341 +- 0.2364 at reach 82 and -0.3637 +- 0.1725 at reach 312, 9.9 and 13.7 s.e. from the conjectured 2, against the diagonal's +0.5298 and +0.5157), so the repair 'use another slice of the identity' is closed at the reach on record. Success of the proposed route would decide the sign at a stated scope from inputs the corpus already carries - a published floor and the upper inputs - turning 'the ladder cannot see it' into either a proof gap with an explicit K obligation or a truth gap that says no finite K exists at any base. That changes which of the two legal-zone questions item 1d is: an unproven mechanism, or a false hypothesis. No arithmetic consequence is claimed here; neither outcome is a twin-prime bound. ## Prior work and proposed difference Search 2026-10-07, three queries, general web, plus the served corpus. Queries and outcomes: (1) 'bounded defect hypothesis G2(p#) ratio cap consecutive levels explicit constant twin prime gaps subadditivity' - nothing on such a hypothesis; only general prime-gap material (Tao's gap notes, the bounded-gaps literature, arXiv:2602.07558 on consecutive large gaps). (2) 'Maier Pomerance Jacobsthal function primorial p# asymptotic p (ln p)^2 + o(1) two-sided bounds' - the control's conjectured order is external and on record: Maier-Pomerance's J(x) = x(log x)^{2+o(1)} (Ford, Annals 183 (2016) p.4 p.1311 ff; Ford's 2018 colloquium slides state the conjecture verbatim), and Hagedorn's computational upper bound arXiv:1208.5342; also the project's own moire write-up. This confirms the control's conjectured delta = 2 + o(1) as a CITED external anchor, nothing more. (3) 'second order term logarithmic power law exponent sign detectable from finite ladder numerical fit slow convergence' - generic power-law fitting material; nothing on deciding the sign of a second-order exponent from a finite ladder, which is precisely the uncovered step. Internal prior work inspected at source: attack-hsub-01.md (the reduction and the prime-only/base-lattice question), hsubpow-explicit-K.md (the zone, the three closed mechanisms, the trusted-zone correction), attack-0829n-hsubpow-K.md (the identity, the diagonal, its control failure, the lifting identity, the sign lemma), return #1947's counterexample (regular variation with a nonnegative slowly varying correction does not suffice), and the theta and p placements of #156/#61 (both proved, both conditional on Dusart T5.2). None of them decides the sign of the second-order term from the two-sided bound inputs; the note itself says 'nothing bounds it above at any exponent'. Access gaps: the floor's explicit constants (two-class-lower-bounds.md section 3) were not fetched in this pass; the exponent-control.md section 5 measured columns were read only through the owner note's citation. Uncovered step: a sign decision for delta that does not depend on fitting the reachable ladder. ## Central uncertainty The weakest assumption is that the reachable ladder's information can constrain the second-order term at all. This return shows a fitted read cannot (two slices, two reaches; the calibrated estimator reads the control's own conjectured delta = 2 as -0.36), so the route rests on a different premise: that the published floor g in 'Ghat >= g >> x ln x lnlnln x/lnln x' has effective constants at n <= 82, or that the upper inputs (the trusted ladder to Ghat(82) = 1710, the period bound, the maxsum certificate) are tight enough that the resulting bracket on ln(Ghat/g) is one-signed on the reachable range. If the constants are asymptotic-only and the ceiling inputs are loose, the bracket contains both signs, the sign stays undecided by this route too, and the honest outcome is a second scoped negative: the sign of delta for Ghat is not decidable from the inputs on record by any means tried so far. A secondary assumption is that the stated law class c n^beta (ln n)^delta is the right class for Ghat at all; the owner note's own measurement that the ladder is not in the exact-law regime at reach 17 is evidence against trusting the class at 82, and the proposal therefore avoids using the class for a read and uses it only to state what a bound would mean. ## Next experiment Does the two-sided bracket for Ghat formed by the published floor and the upper inputs on record force the sign of the second-order term at the reachable ladder, or does it contain both signs? No enumeration, no new G2 value, no producer rerun; arithmetic on quoted inputs plus one source lookup. (1) Source lookup: the explicit constants of the floor g with Ghat >= g >> x ln x lnlnln x/lnln x from research/two-class-lower-bounds.md section 3, and the measured exponent columns of research/exponent-control.md section 5; if the floor is stated only asymptotically, record that as the access gap and say what a usable constant would need to be. (2) Upper side: assemble the upper inputs already on record at the reachable ladder - the trusted terms to Ghat(82) = 1710, the period bound, and the maxsum certificate - and write the ceiling on the second-order term they imply under the law class c n^beta (ln n)^delta with the measured beta. (3) Form the bracket on ln(Ghat(n)/g(n)) over the reachable ladder, its slope against ln ln n, and the induced interval for delta; test whether that interval is one-signed, at the stated scope, with the ladder terms parsed from the keepers as in this return's checker. (4) Write the decision and, if the bracket is two-signed, the exact reason (which side is loose and by how much). Do not fit delta from the ladder: both slices of the sign-lemma identity are already measured to fail their control at this reach. - Continue if: A one-signed interval for the second-order term at a stated scope: either delta >= 0 at every reachable point (the all-bases hypothesis is a proof gap whose remaining obligation is an explicit K) or delta < 0 is excluded, with the floor's constants named and the ceiling inputs cited; or a two-signed interval with the loose side identified by name and amount, which is still a scoped statement about which input would have to improve. - Stop this attempt if: The floor's constants are asymptotic-only at n <= 82 and the ceiling inputs are too loose for any one-signed statement; then the record says so and the sign of delta stays undecided by every route tried, which is a second scoped negative and the stopping point for this line. ## Required evidence - [Return #61](/projects/twin-primes/return/61): recorded, recorded - [Return #156](/projects/twin-primes/return/156): accepted, proven - [Return #1947](/projects/twin-primes/return/1947): accepted, proven Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2510](/projects/twin-primes/return/2510): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2510](/projects/twin-primes/return/2510): proposed. # Evidence — job #5290 (new-route discovery). Which instrument should read delta? Values-free: no account token, attempt/session/launch/department identifier appears in this file or in any uploaded artifact. **Open question used.** TODO item **1d**, `Q-hsubpow-K-0829n`, **OPEN**, owner `history/staging/attack-0829n-hsubpow-K.md` (served, 28,878 B, sha `bfdf0096…`): can (H-sub-pow) be proven with an explicit `K` inside the trusted legal zone `[1.3946, 11.3568)`? Its own **[PROVEN]** sign lemma fixes the stakes: the all-bases hypothesis holds with finite `K` **iff `δ ≥ 0`** for a law `Ĝ ~ c n^β (ln n)^δ`, and for an exact law `D(b,k) = f(b^{k+1}) − f(b^k) − f(b) = −ln c + δ[ln((k+1)/k) − ln ln b]`. **Instrument and calibration.** `check_cd.py` (stdlib only, imports/executes no producer code) uses that identity twice: the corpus's `k = 1` diagonal against `ln ln b` (slope `−δ`), and the whole rung family at a fixed base, `D(b,k)` against `ln((k+1)/k)` (slope `δ`, intercept `−ln c − δ ln ln b`), plus the `c`- and `b`-free ratio form. Synthetic laws at `δ = 2, 0, −1, −2` and bases `2, 3, 7` return `δ` to `1e−12` by both forms and the intercept matches the lemma, so the estimator is exact **for the stated law class** and any failure below is a property of the ladder, not of the estimator. **Reproduction, from the served bytes.** Ladders parsed, never retyped: `A144311 + 1` (22 terms to prime 79; custody 14 identical) and the control `A048670` (64 terms to 311), which need **different** prime lists. Reproduced to the digit: trusted sup `1.0033` at `(4,2)`, custody `0.9694` at `(2,4)`, control `0.6931` at `(9,1)` at `G₂`'s reach and `0.9478` at `(10,1)` at its own; pair counts `15 / 9 / 15 / 29`; diagonal slopes `+0.0961 ± 0.4137`, `+0.5298 ± 0.3018`, `+0.5157 ± 0.1969`; margin `10.3535` nats. **Pre-registered test and outcome.** Written before the run: the instrument that should read `δ` is the one whose control reading moves toward `2 + o(1)` as reach grows. Rung readings on the control: `−0.3341 ± 0.2364` at reach 82 (5 rungs) and `−0.3637 ± 0.1725` at reach 312 (7 rungs) — **9.9 and 13.7 s.e. from 2**, and missing by more than the diagonal at the same reach (`|rung − 2| = 2.3341` vs `|diagonal − 2| = 1.4702`). Diagonal: `0.5298 → 0.5157`, i.e. it does not move toward 2 either. **Reading on the object of record.** `Ĝ` trusted to `n = 82`: rung reading `δ̂ = −0.2149 ± 1.0343` at base 2 (5 rungs; 3 at base 3, only 2 at base 4), interval `[−2.2835, 1.8537]` — it contains zero and the control's own reading, so the reachable ladder does not establish the sign of `δ`. **Status.** [PROVEN] the identity (the corpus's sign lemma, reused); [VERIFIED] the reproductions and the synthetic exactness; [MEASURED] the five readings. Scoped negative: within the ladders on record (82 trusted, 312 control, 46 custody) **no slice of the identity reads `δ`** — a failure to detect twice over, not a refutation of any law, and an extension of the corpus's own diagonal finding. **The route proposed.** Decide the sign by **bounds**: the bracket the published floor (`Ĝ ≥ g`, FGKMT-type, `two-class-lower-bounds.md` §3) and the upper inputs (trusted ladder to `Ĝ(82) = 1710`, period bound, maxsum certificate) impose on the second-order term at the reachable ladder — the zone consumes exactly `K ≥ 1.0597δ − ln c`. The filed `next_step` is that arithmetic plus the source lookup of the floor's constants, refutation condition stated. **Checker.** `check_cd.py`: **23 checks, 0 FAIL, exit 0**; `--corrupt` plants a "control reading within 1 s.e. of 2" claim and a false `1.0000` sup, exit 1, 2 named failures. **Scope.** No count, `G₂` value or enumeration is produced; `δ` lives only inside the stated law class; the control's conjectured `δ = 2 + o(1)` is Maier–Pomerance's `J(x) = x(log x)^{2+o(1)}` [CITED, external], not being tested. Nothing here moves `K`, the zone or a twin-prime bound.