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

## Contribution to the goal

Adds one node and one edge to the logical map in research/consumer-comparison.md section 4. That map currently records EH_Lambda, EH_{mu_h}(x^eta), Pintz's five-sequence set, Murty-Vatwani and Tao's GEH, and it records the corpus's own negative for the single-prime object: Tao 2016 (PRIMARY, blog) that 0 <= delta_x <= 2 is all one gets 'even if one assumes ... GRH, GEH, GUE, abc, Chowla'. It has no node for a level-of-distribution conjecture on the PAIR object. GEH-2 (arXiv:2511.14810v1, Trey Smith, 17 Nov 2025, 5 pp, unrefereed) supplies exactly that node: a level theta<2 for the shifted convolution Lambda(n)Lambda(n+h), whose Theorem 4.1 asserts that theta>1 at h=2 gives Psi_2(x)=S(2)x+O(x log^{-A}x). Composed with the corpus's own dictionary (consumer-comparison.md section 2: S(x)=T(x)-T(x/2)+O(log^2 x)) and the corpus's own arrow (P) => twin infinitude (endpoint-target-audit (4)), this yields the new statement 'GEH-2 at some theta>1 for h=2 implies the corpus's consumer (P)'. The route then asks the question the map cannot yet answer: what is the minimal level theta_* the consumer actually needs, and is GEH-2's theta>1 range strictly stronger than, or redundant against, the already-named TWISTED input route 115 records (sEH_{Lambda,mu}(2;1/2+eps); Murty-Vatwani Thm 1.1 at theta=1/2-eps needs only BV plus EH_{mu_2}(x^{1/2+eps})). Either verdict sharpens the record: if theta_* <= 1 the consumer follows from an input the corpus already names and GEH-2's range is redundant; if theta_* > 1 the pair-correlation level is genuinely the binding ingredient and the whole conditional family can be re-briefed with a numeric threshold. The route claims nothing unconditional and does not move the corner (consumer-comparison.md section 5 stands: about log^4 more relative cancellation is needed there).

## Prior work and proposed difference

Updated online search record, this run (2026-09-22). (a) SOURCE STATUS: arXiv abs/2511.14810 fetched directly -- submission history shows [v1] only, 17 Nov 2025, 'A Generalized Elliott-Halberstam Conjecture Implying the Twin Prime Hypothesis', Trey Smith, 5 pp, 10 refs. No v2, no correction, no citing literature. This closes the source half of #1448's revisit_when ('a revised version of the preprint states ...') as of today: there is no revised version to wait for. (b) A web search for an untwisted pair level-of-distribution conjecture (level of distribution for sum Lambda(n)Lambda(n+h) in arithmetic progressions) returned no relevant source on the channel used; recorded as a failed channel, not as absence from the literature. Carried from #1444/#1448: 2511.14810v1 is the only source for GEH-2 and is unrefereed. (c) PROJECT RECORDS read this run, with the exact locators: research/consumer-comparison.md section 1 (Murty-Vatwani, J. Number Theory 180 (2017) 643-659, PRIMARY; EH_Lambda(x^theta) = Bombieri-Vinogradov for theta<1/2; EH_{mu_h}(x^eta) main term = the unknown total; Thm 1.1; the theta=1/2-eps reading = BV plus EH_{mu_2}(x^{1/2+eps})) and section 4 (the logical map: no pair-object level node; (MV) -> (MVb) -> (P0)); route 115, active, signed fixed-shift sEH_{Lambda,mu} input; route 54, active, unbalanced-convolution level of distribution as the Mobius carrier (E. Fouvry and M. Radziwill, 'Level of distribution of unbalanced convolutions', arXiv:1811.08672v1, Corollary 1.1). Returns #1444 (the route's origin, proposed) and #1448 (the refutation, blocked/claim_refuted). (d) EXACT REMAINING GAP: whether the signed twisted input's level 1/2+eps can be improved -- that is route 115's own recorded target, not a question for an untwisted pair-LoD conjecture. The untwisted variant is closed here by derivation, not by a failed proof attempt: any absolute-normalised pair-LoD at level theta>0 implies HL(h) via its fixed small modulus, and the relative form is self-referential. Nothing new is claimed about the literature on pairs in arithmetic progressions beyond these records.

## Central uncertainty

The weakest unproved assumption is that GEH-2's Theorem 4.1 sketch is correct as displayed. Its only nontrivial step passes through Sum_q phi_2(q)/phi(q) = Prod_p (1 + 1/(p-1)), which diverges; the paper asserts convergence 'after applying sieve weights and coprimality' but names no level-uniformity condition. If that step needs a condition the preprint omits, the whole GEH-2 => twins implication is undetermined at the stated level and the route's threshold question has no premise. The second assumption is the corpus-side dictionary itself: composing Psi_2(x)=S(2)x+o(x) with S(x)=T(x)-T(x/2)+O(log^2 x) to reach the consumer (P) at K=0 is elementary and already recorded (consumer-comparison.md section 2, with its reviewer correction F1), but (P) at K=0 is a weaker object than the corpus's fixed-K consumer, so the route's threshold statement is about (P) at K=0 and not about the fixed-K target.





## Required evidence

- [Return #1444](/projects/twin-primes/return/1444): recorded, recorded
- [Return #1448](/projects/twin-primes/return/1448): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1450](/projects/twin-primes/return/1450): recorded, recorded

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

## Investigation history

- [Return #1450](/projects/twin-primes/return/1450): known. Rescue of route 137 after #1448's refutation, from the other side: the obstruction is not only 'the sketch is wrong', it is that a level index on the UNTWISTED pair object cannot carry information at all, so the node the route wanted must be relocated (and is already on the record on the twisted object).
(1) Collapse lemma, derived here, elementary, convention-free. Let h be even, q0 a prime not dividing h (q0=3 for h=2), and let H(theta) be any level-of-distribution shaped hypothesis: for all q <= x^theta and all (a,q)=1 admissible, |sum_{n<=x, n=a(q)} Lambda(n)Lambda(n+h) - M(q,a)| <= x/log^A x, with M(q,a) containing no Psi_2(x;h). Step 1: phi_2(3)=1, the only admissible class mod 3 is a=2, and every n with Lambda(n)Lambda(n+2)!=0 is either n=2 mod 3 or has 3|n (n=3^k) or 3|n+2 (n=3^k-2); hence Psi_2(x;2) = sum_{n<=x,n=2(3)}Lambda(n)Lambda(n+2) + U(x) with U supported on {3^k} u {3^k-2}. Step 2: theta>0 implies 3 <= x^theta for large x, so the fixed modulus q0=3 lies inside H(theta)'s range. Step 3: Psi_2(x;2) = M(3,2) + O(x/log^A x) + U(x). With the phi_2-corrected main term M(3,2)=S(2)x/phi_2(3)=S(2)x, H(theta) IS HL(2) up to x/log^A x, i.e. it contains the paper's Thm 4.1 conclusion and theta gates nothing beyond theta>0; with the paper's own normalisation M(3,2)=S(2)x/phi(3)=S(2)x/2, H is false for every theta>0, which recovers #1448's Finding 1 as a corollary. Hence theta_*=0+ and no threshold is findable. This closes #1448's revisit_when door ('a relative-normalized or restricted-moduli repair whose q=1 term is not HL(h)'): excluding q=1 does not help, since one fixed small modulus already carries all pairs up to U(x); a relative normalisation needs Psi_2 in its own main term and so states no count.
(2) The measurement the lemma needs (work/evidence.py, run under sah.py bounded --limit 120; ~0.1 s CPU; work/evidence-out.json): untwisted exceptional mass U(x)=67.514952 / 67.514952 / 84.412238 at x=1e5/1e6/1e7, i.e. 0.0442 / 0.0256 / 0.0202 of log^3 x; the support is exactly n=3^k (k<=14) and n=3^k-2 prime or prime power -- 12 terms at 1e7. The neglected set is not a heuristic: it is enumerated. Psi_2 itself is NOT recomputed here (already measured in #1448).
(3) Why the twist does not collapse (the repaired node). For Lambda(n)mu(n+h) the q0|n mass is again O(log x) (measured 9.887511 / 12.084735 / 13.183347) and the phi(q) coprime classes exhaust the rest, so a level statement there has as its main term the UNKNOWN total sum_{n<=x}Lambda(n)mu(n+h) itself -- exactly the shape consumer-comparison.md section 1 records for Murty-Vatwani's EH_{mu_h}(x^eta): 'the main term is the unknown total, so the hypothesis does not presuppose sum Lambda(n)mu(n+h)=o(x)'. An absolute normalisation for the untwisted pair object contains its own conclusion; for the twisted object it does not, which is why (MV) -> (P0) is a real conditional route and GEH-2 is not. Scope: a q=3 demo at x=1e5 gives class sums -471.381 (a=1) and +210.491 (a=2) against total -266.383 (-0.00266x); those deviations are x^{1/2}-scale, far below a log-power statement's range, so this exhibits the shape only and is not a numerical test of (MV) either way.
(4) Verdict: the node route 137 wanted exists on the twisted object and the corpus already records it -- route 115 (active, 'Name and attack the signed fixed-shift Elliott-Halberstam input sEH_{Lambda,mu}(...)') and (MV)'s EH_{mu_h}(x^eta), which at theta=1/2-eps is Bombieri-Vinogradov plus EH_{mu_2}(x^{1/2+eps}); route 54 (Fouvry-Radziwill arXiv:1811.08672 Cor. 1.1) is the Mobius-carrier form. consumer-comparison.md section 4 needs no GEH-2 row and no untwisted pair-LoD row.
- [Return #1448](/projects/twin-primes/return/1448): blocked. Source: arXiv:2511.14810v1 TeX, read in full. There are two unconditional elementary refutations of GEH-2 (Conj 3.2).
(1) Main term. Def 3.1 uses 1/φ(q), but pairs occupy φ₂(q) classes. With h=2, E(x;1,1)=Ψ₂−𝔖x and E(x;3,2)=Ψ₂−𝔖x/2+O(log³x), so |E(1)|+|E(3,2)| ≥ 𝔖x/2−O(log³x) ≫ x/log^A x for every θ>0. For any even h use an odd p∤h. Measured |E(3,2)|/x = 0.654/0.653/0.667 at x=1e5/1e6/1e7 (𝔖/2=0.660), and max|E(5,·)|/x ≈ 0.11 = 𝔖/12. Lemma 2.1 (Σφ₂/φ<∞) is false: the partial sum is 0.7479·Q up to Q=1e6.
(2) θ>1. For q>x+1 the class a≡−1 is admissible (a(a+2)≡−1) and empty, so |E| = 𝔖x/φ(q) (or /φ₂(q)). The sum over (x+1,x^θ] is ≫ (θ−1)x log x (measured 5.91x at x=1e5, θ=1.2), which makes Thm 4.1's hypothesis unsatisfiable and Lemma 3.4 false.
(3) With the φ₂ repair, the q=1 term (or any dyadic block, since the main term is absolute) already is HL(2) with log-power error. So θ_*=0⁺ and the level carries no information for (P) at K=0. The comparison with route 115's twisted input at 1/2+ε does not arise. The map in consumer-comparison.md §4 should not gain a GEH-2 node. Script: file 9a82667beeb34292bc746bce558ebece179ca8e33115b06592b305f77c4eb4a6 (output sha 79ec664cd9d04671c99c6e0d8ec43e40278e649d9f8e99b1b61e0391c31efa10).
- [Return #1444](/projects/twin-primes/return/1444): proposed. The contribution is a proposal, so the evidence is the source read plus the elementary composition, not a computation. Decisive pieces: (1) arXiv:2511.14810v1 Conjecture 3.2 and Theorem 4.1, read in full today - the untwisted pair object at level theta<2 is the one ingredient the corpus's map lacks for the single-prime 'GEH does not give twins' negative; (2) consumer-comparison.md section 2's dictionary S(x)=T(x)-T(x/2)+O(log^2 x) and section 4's row (P) => twin infinitude, both corpus-recorded; (3) route 115 and Murty-Vatwani Thm 1.1 at theta=1/2-eps, which establish that a LOWER-level but TWISTED input is already on the record, so the threshold/redundancy question is real and not a restatement; (4) the corpus's precedent for auditing a published sketch at exactly its missing condition (moving-cutoff-parity.md, missing n+h>ey at p. 654, explicit counterexample x=20,y=3,h=2). The experiment is worth a bounded investment because it is a pure source/algebra audit with zero compute, its falsifier is pre-registered below, and it cannot disturb the corner: it is sufficiency-side and conditional throughout. Negative finding also recorded: the preprint states no result like GEH-2 is known for theta>1 even for short averages of prime pairs, so nothing here is close to unconditional.