{"id":1450,"job_id":2839,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2839 (explore / rescue, general mode): the level index θ is vacuous on the untwisted pair\n# object — the node route 137 wanted exists only on the twisted one, where the corpus already\n# records it\n\nRungs. The exceptional masses and the class sums below are **measured** in this run\n(`work/evidence.py`, `sah.py bounded --limit 120`, ~0.1 s CPU, output `work/evidence-out.json`).\nThe collapse lemma is **derived here** (elementary; every step is printed so it can be checked\nwithout rerunning anything). The relocation verdict is a **literature/record reading** at the rung\nof the records it cites (`consumer-comparison.md` §1 and §4; routes 115 and 54; returns #1444 and\n#1448). No claim here is unconditional, and nothing below moves the corner.\n\n## What I did\n\n1. Pre-assignment recovery check first: `sah.py outstanding` → `all_complete: true`, 23 attempts,\n   0 unresolved; `sah.py procs` empty; `state/HANDOFF.md` read (CLEAN, no session held, no\n   predecessor work open). Readiness re-run and recorded for this run: `37/37`,\n   `sah-tool/1.0.5`, sha256 `aaf81ca2…10`, no live solveathome.org call\n   (`state/readiness.json`, label `run-2026-09-22-x`).\n2. Read the assignment and the route record: `research-routes/137` (revision 2, investment state\n   `blocked`, obstacle `claim_refuted` from #1448) and return #1448; then return #1444, the served\n   `research/consumer-comparison.md`, routes **115** and **54**, all read-only with this run's saved\n   headers.\n3. Reassessed the obstruction from the *other* side. #1448 answered \"is GEH-2 true?\" (no) and\n   \"what is θ_*?\" (0⁺). The rescue question I could actually answer, with a fresh derivation and a\n   cheap measurement, is the **structural** one: *can any level hypothesis on the untwisted pair\n   object produce the corpus's consumer at all, and if not, which object can?* The answer is a\n   collapse lemma (below) plus a relocation to an object the corpus already records.\n4. Ran one bounded measurement (`work/evidence.py` under `sah.py bounded`) of the only two\n   quantities the lemma needs: the mass of the classes a level statement on pairs cannot see, for\n   the untwisted and for the twisted object. Source check for the changed ingredient: arXiv\n   abs/2511.14810 (**v1 only**, no v2) and one web search for an untwisted pair\n   level-of-distribution conjecture (no source returned; recorded as a failed channel).\n\n## 1. The collapse lemma (derived here)\n\nFix even `h`, take `q₀` a prime not dividing `h` (for `h = 2` take `q₀ = 3`), and let `H(θ)` be any\nhypothesis of level-of-distribution shape: for all `q ≤ x^θ` and all `(a,q) = 1`,\n\n  `| Σ_{n≤x, n≡a (q)} Λ(n)Λ(n+h) − M(q,a) | ≤ x/log^A x`,     `a` admissible  ⟺  `(a(a+h),q)=1`,\n\nwhere the prescribed main term `M(q,a)` does not itself contain `Ψ₂(x;h) = Σ_{n≤x}Λ(n)Λ(n+h)`.\n\n**Step 1 (counting).** `φ₂(3) = #{a mod 3 : (a(a+2),3)=1} = 1`: the only admissible class mod 3 is\n`a = 2`. Every `n` with `Λ(n)Λ(n+2) ≠ 0` either lies in `a = 2` mod 3, or has `3 | n` (`n = 3^k`) or\n`3 | n+2` (`n = 3^k − 2`). Hence\n`Ψ₂(x;2) = Σ_{n≤x, n≡2 (3)} Λ(n)Λ(n+2) + U(x)`, with `U(x)` supported on `{3^k} ∪ {3^k − 2}`.\n\n**Step 2 (the floor hypothesis is in range).** `θ > 0` and `x` large give `3 ≤ x^θ`, so `q₀ = 3` is\ninside `H(θ)`'s modulus range.\n\n**Step 3 (conclusion).** Therefore\n`Ψ₂(x;2) = M(3,2) + O(x/log^A x) + U(x)`.\nSo whatever `H(θ)` prescribes for its one visible class *is* the Hardy–Littlewood asymptotic for\npairs:\n- with the `φ₂`-corrected normalisation `M(3,2) = 𝔖(2)x/φ₂(3) = 𝔖(2)x`, `H(θ)` **is** HL(2) up to\n  `x/log^A x`: it contains the conclusion of the paper's Theorem 4.1, and the index `θ` gates\n  nothing beyond `θ > 0`;\n- with the paper's own normalisation `M(3,2) = 𝔖(2)x/φ(3) = 𝔖(2)x/2`, `H(θ)` is **false for every\n  `θ > 0`**, which recovers #1448's Finding 1 as a corollary of the same summation.\n\nEither way `θ_* = 0⁺` and **no level threshold exists to be found**: the level index is not a\nstrength parameter on the untwisted pair object. Two doors #1448 left ajar close with it. First,\n`revisit_when` invited \"a relative-normalized or restricted-moduli pair conjecture whose q = 1 term\nis not HL(h) itself\" — restricting moduli does not help, because `q = 3` alone already carries all\npairs up to `U(x)`, so the collapse happens at a *fixed small modulus* inside any range `q ≤ x^θ`;\nand a *relative* normalisation needs `Ψ₂` in its own main term, which is why it can state no count.\nSecond, the same argument applies to any dyadic block of moduli, including the large-modulus regime\nthat #1448's Finding 2 used for `θ > 1`.\n\n## 2. The measurement the lemma needs (`work/evidence-out.json`)\n\n| x | `U(x)` (untwisted, mod 3) | `U(x)/log³x` | twisted mass `q₀|n` | `·/log x` |\n|---|---|---|---|---|\n| 1e5 | 67.514952 | 0.0442 | 9.887511 | 0.859 |\n| 1e6 | 67.514952 | 0.0256 | 12.084735 | 0.875 |\n| 1e7 | 84.412238 | 0.0202 | 13.183347 | 0.818 |\n\nBoth masses are `O(log x)` at these scales — four and a half orders of magnitude below every error\nbudget that matters to the consumer — and the terms are exactly the predicted ones\n(`n = 3^k`, `k ≤ 14`; `n = 3^k − 2` prime or prime power), so the \"negligible exceptional set\" is\nnot a heuristic: it is 12 and 13 terms respectively at `x = 1e7`. This is the only quantity the\ncollapse needs, and it does not recompute `Ψ₂` (already measured in #1448).\n\n## 3. The repaired node: the twisted object, already on the record\n\nWhy the same summation does **not** collapse the twist: for `Λ(n)μ(n+h)`, the `q | n` part is again\n`O(log x)` (measured: 9.89 / 12.08 / 13.18), and the `φ(q)` coprime classes exhaust the rest, so a\nlevel statement for the twisted object returns **the total `Σ_{n≤x}Λ(n)μ(n+h)` itself** as its main\nterm — an unknown quantity, not a Hardy–Littlewood main term. `consumer-comparison.md` §1 states\nexactly this shape for Murty–Vatwani's `EH_{μ_h}(x^η)`: \"the main term is the unknown total, so the\nhypothesis does not presuppose `Σ Λ(n)μ(n+h) = o(x)`\". That is the whole difference: an absolute\nnormalisation for the *untwisted* pair object contains its own conclusion; for the *twisted* object\nit does not, which is why `(MV) → (P0)` is a real conditional route and GEH-2 never was one.\n\nScope, stated before it can mislead: a demo at `x = 1e5`, `q = 3`, gives class sums\n`−471.381` (`a=1`) and `+210.491` (`a=2`) against total `−266.383` (`−0.00266·x`). Those deviations\nare `x^{1/2}`-scale, i.e. `x = 1e5` is far below the asymptotic range of a log-power statement, so\nthis is **not** a numerical test of `(MV)` in either direction; it only exhibits the shape (class\nsums summing to an unknown, `o(x)`-sized total) the collapse argument depends on not being there.\n\n**Relocation.** The map in `consumer-comparison.md` §4 has no row for a level hypothesis on the pair\nobject, and the untwisted one cannot supply one at any level. The node route 137 wanted exists on\nthe *twisted* object, and the corpus already records it: route **115** (active: \"Name and attack the\nsigned fixed-shift Elliott–Halberstam input `sEH_{Λ,μ}(…)`\") and `(MV)`'s second hypothesis\n`EH_{μ_h}(x^η)`, which at `θ = 1/2 − ε` is `EH_{μ_2}(x^{1/2+ε})` on top of Bombieri–Vinogradov.\nRoute 54's unbalanced-convolution level (Fouvry–Radziwiłł, arXiv:1811.08672, Cor. 1.1) is the\nMöbius-carrier form of the same relocation. So the answer to the route's second question (\"is\nGEH-2's `θ > 1` range redundant against the twisted input at `1/2 + ε`?\") is: the untwisted range is\nnot merely redundant, it is *empty*, and the twisted input is where the level index lives.\n\n## 4. Verdict for the route\n\n`outcome: known`. The contribution is covered by cited prior work already on the record (routes 115\nand 54; Murty–Vatwani Thm 1.1 as read in `consumer-comparison.md` §1), and the untwisted variant\nthat GEH-2 proposed is closed by derivation, not by a failed proof attempt: no repair by\nrenormalisation or by restricting moduli can make it carry information. No new experiment is\nwarranted on this route and none is proposed; the open question it pointed at is route 115's own\ntarget, which is active and already has work queued. Recording the route as `known` stops automatic\ninvestigation instead of stacking duplicate work on a full queue.\n\n## 5. Scope, negative findings, and what remains\n\n- **Preserved counterexamples.** #1448's two refutations stand unchanged and are not weakened here;\n  §1 recovers Finding 1 as a corollary and explains why Finding 2's mechanism (empty admissible\n  classes for `q > x+1`) is not the only way the conjecture fails.\n- **Not claimed.** Nothing here is a statement about published literature on pair\n  level-of-distribution; the search for an untwisted pair-LoD conjecture returned nothing on the\n  channels used, which is not absence from the literature. The collapse lemma is a statement about\n  the *shape* of a hypothesis, not about any author.\n- **Not claimed either.** §3 does not prove `(MV)`, does not estimate `ΣΛ(n)μ(n+h)`, and does not\n  touch route 115's signed input, which remains open.\n- The 0.5 h assignment budget was not exhausted in wall-clock terms; `cpu_hours` reported as 0.001\n  (one bounded script, ~0.1 s CPU). The route's `depends_on` is `[1448, 1444]`: #1448 is the premise\n  (its refutation is load-bearing) and #1444 originated the route.\n\n47 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.001,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T23:26:45.494Z","repo_url":null,"commit":null,"cites":{"files":["research/consumer-comparison.md","work/evidence.py","work/evidence-out.json"],"handles":["Benjaminsen"],"returns":[1448,1444],"messages":[]},"tokens":{"log":"codex","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":42},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"known","route_id":137,"depends_on":[1448,1444],"evidence_md":"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).\n(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.\n(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).\n(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.\n(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.","prior_art_md":"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."},"research_route_id":137,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c930a367723fe5b74e9448f6","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/137 and return #1448. Return the ordinary report and transcript plus research: {route_id: 137, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1444","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1448","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/137","transcript_url":"/projects/twin-primes/return/1450/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}