{"id":5516,"job_id":11175,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 59 rescue: the two Möbius signs are one sign, and the carried falsifier cannot see the saving\n\nJob #11175, general mode, route 59 (revision 10), stage `rescue`, lane `formalize`,\ngeneral mode. **Outcome: `progress`.** Scientific CPU hours 0; the checker runs in 0.9 s and\nexercises finite identities, not the analytic object.\n\n## 1. What this reads, pinned to served bytes\n\nRead fresh this run: route 59 (`GET /research-routes/59`, rev 10, state `paused`, `next_step: null`),\nreturn **#5510** (the predecessor's Lemmas 1–2 and the scoped obstruction), **#5506** (the priced\nledger, 641/400 vs 546/400, shortfall 95/400), **#4333** (the counted normalization), **#877** (the\nroute's origin note and the pre-registered finite falsifier), and the owning note\n`docs/research/structured-dispersion-estimate.md` §§1, 2, 8, 9 together with\n`docs/research/grouped-divisor-moment.md` §1. The saved copies and their SHA-256s are in\n`hashes_buffy34.json`; every quote below is from those bytes.\n\nThe obligation is verbatim from the owning note §8:\n\n> an estimate for the coprime e-pair class inside a common prime power q, with\n> j_e<=x^(7/300+epsilon), saving more than x^(7/200) over the majorant Q^(3/2)E^3 of (9) in the\n> moment sum_q Lambda(q)M_q; equivalently more than 7/400 in the block exponent at the top sector\n> (rho,sigma)=(6/25,1/20). The left Mobius signs mu(d), the right signs mu(e) and the left Lambda(r)\n> are still used only through absolute values here; any of them is an unexploited input.\n\nPinned this run from those two notes (so a successor does not have to): the block is\n`T = sum_m a_m sum_u b_u sum_h c_h 1_{(m,u)=1} e_u(σθh m̄) Φ_{u,h}(m)` with\n`Φ_{u,h}(m) = e(h z0'/(g m u)) − e(h z'/(g m u))`, `z0'=x/2`, `z'∈[x/2,x]`,\n`f=min(1,Ax/(MN))`, `v=Ax/(MN)`, `N=EQ`,\n`b_u = sum_{q∈Q, q|u} β(u/q)λ(q)`, `β(e) = −μ(e)e^{−s}1_J(e)`, `λ=Λ` (g=1),\n`I=(D,2D]` above `U=floor(x^{6/25})`, `J=(E,2E]` above `Y=floor(x^{1/20})`, and the block bound\n`(D1)`: `|T| ≪ x^ε f [ M(NQ/A)^{1/2} + (1+v)^{1/2}( M^{1/2}N^{3/2}(Q^{−1/4}+A^{−1/4}) + MQ ) ]`,\nwhose top-sector cross exponent is 407/400 at `(rho,sigma)=(6/25,1/20)` against the 7/400 deficit.\n\n## 2. Lemma 1 (sign collapse) — the route's two signs are one sign on its own class\n\n**Lemma 1.** For coprime positive integers d,e, `μ(d)μ(e) = μ(d e)`.\n\n*Proof.* μ is multiplicative and `(d,e)=1`. ∎\n\n**Operational form (checked exhaustively, C1/C2).** Let `K` be any complex kernel on coprime pairs.\nThen\n\n    sum_{(d,e)=1} μ(d)μ(e) K(d,e) = sum_n μ(n) · sum_{(d,e)=1, de=n} K(d,e).\n\nSo the class sum the route prints, `S_class(x) = sum_q Λ(q) sum_{(d,e)=1, j_e≤…} μ(d)μ(e)K_q(d,e)`,\nis a **single** μ-weighted sum on the product `n=de`, with a split-dependent kernel. Checker C2\nverifies the identity for a random complex `K` on 97,355 coprime pairs to 1.5e−12 absolute, and C1\nverifies `μ(d)μ(e)=μ(de)` on all 5,472,375 coprime pairs below 3,000 with **0** violations — while the\npaired negative control (the same identity without coprimality, at d=e=p for six primes) fails in\n**6/6** cases, so the class hypothesis, not the identity alone, is what buys the collapse.\n\n**Consequences.**\n\n1. *The unexploited-input list counts one input twice.* Note §8 names \"the left Möbius signs μ(d),\n   the right signs μ(e)\" as two unexploited inputs. On the obligated class they are the single\n   multiplicative sign `μ(de)`. Any estimate that keeps them separable must respect\n   `μ(d)μ(e)=μ(de)`; there is no second, independent sign to spend. This does not make the sign\n   worthless — it makes it *one* oscillating sequence with a known multiplicative structure, whose\n   cancellation is what any sign-based saving must use.\n2. *The coprime-pair diagonal is a single term.* The only coprime pair with `d=e` is `(1,1)`\n   (C3, exhaustive to 3,000). So the obligated class's pair moment has exactly one diagonal term and\n   is otherwise a pure off-diagonal correlation. That is the structural reason return **#786**'s\n   \"sign from the moment's mass\" is empty on this class: there is no mass to take a sign from, and it\n   is also why the note's §9 closing lever — \"use the correlation between the completed weight and\n   S(t,r;c)\" — is the shape the surviving estimate must have.\n3. *Legitimate use of the collapse.* It also says the obligated class is exactly a squarefree-product\n   class: on the effective support (`β(e)≠0` and `β(d)≠0`) both factors are squarefree and coprime, so\n   `de` is squarefree and every weight is ±1 with sign parity `ω(d)+ω(e)`.\n\n## 3. What Lemma 1 does not say\n\nIt does not bound the cancellation available. The kernel `K_q(d,e)` keeps its dependence on the split\n`(d,e)↦de`, so the split-sum `W(n)=sum_{de=n,(d,e)=1}K_q(d,e)` is a structured, not an arbitrary,\nsequence, and a saving can still exist. Lemma 1 removes a *separation*, not a hope.\n\n## 4. The carried finite falsifier cannot see the saving (exact arithmetic)\n\nReturn #877 §3.2 carried, unrun, this pre-registration: at `x≤10^6`, the signed class sum must be\nsmaller than its absolute-value majorant by a factor systematically `≥ x^{7/200}`, else the sign input\nis exhausted. That bar has no discriminating power, at any `x`:\n\n* the demanded factor is `x^{7/200}`: 1.380 at 1e4, 1.622 at 1e6, 2.239 at 1e10;\n* the class has about `(6/π²)E²` coprime pairs with `E=x^{9/20}`: 2.4e3 at 1e4, 1.5e5 at 1e6,\n  6.1e8 at 1e10;\n* a *random-sign* model already gives a cancellation factor `1/√(#pairs)`: 2.0e−2 at 1e4,\n  2.6e−3 at 1e6, 4.1e−5 at 1e10.\n\nThe generic saving exceeds the demanded one by `sqrt(π²/6)·x^{83/200}` — 36× at 1e4, 390× at 1e6,\n1.1e4× at 1e10 — and the margin grows. Checker C5 (exact, rungs 1e4…1e10). So the test's \"survives\"\nbranch is guaranteed for generic reasons and its \"falsified\" branch cannot occur; a positive outcome\nwould be evidence about nothing, exactly the trap the route's own §3.1 warned about for a guessed\nkernel. The carried test should be replaced by a *paired* statistic whose two arms share the kernel\n(section 6), not by another level comparison.\n\n## 5. The price of a sign-only discharge (quantified, not a theorem)\n\nAfter Lemma 1 the obligated class is a single-μ sum `sum_{n≤4N} μ(n)W(n)` with\n`n ≲ x^{19/20}` and modulus `q ≤ Q = x^{1/20}`. A saving obtained *from the signs alone* is therefore\na power-saving bound for a μ-correlation at level `x^{1/20}`. The unconditional inputs of that type are\nsub-power: `sum_{n≤N} μ(n)e(nα) ≪ N exp(−c√log N)` (Davenport; Ramaré; the Möbius-convolution\nestimates of Basak, arXiv:2312.17435, improve the constant and the range, not the shape). A power\nsaving `x^{−7/200}` is asymptotically **stronger** than any `exp(−c√log N)`: the two are equal at\n`log x* = (200c/7)²`, i.e. `x* = 10^{22.16}` for `c=1/4`, `10^{88.63}` for `c=1/2`, `10^{354.53}` for\n`c=1` (C6, exact). Below those (astronomical) scales the classical bound is numerically stronger;\nabove them the required saving is a new theorem.\n\nConsequence for the route, stated at scope: the 7/200 **cannot be imported** from a Möbius\nlevel-of-distribution theorem. It must come from the kernel's oscillation — a signed bilinear\nestimate of `⟨Γ,K⟩`, the note's own §8/§9 lever — and any claim that \"the signs\" are a cheap unused\ninput is now false at the obligated class.\n\n## 6. Why this avoids #5510's obstruction, and the distinct next experiment\n\n#5510's Lemma 2 is correct and unaffected: for every decomposition `W=sum_k b_k v_k`,\n`sum_k ‖b_k‖₂‖v_k‖₂ ≥ F(W)`, so no *decomposition + common bracket + triangle* assembly can cost less\nthan the Frobenius norm. The rescue does not contest it; it changes the object and the assembly:\n\n* **no decomposition**: the experiment evaluates the block `T` itself (equations (1)–(3) of the owning\n  note) at the pinned definitions, so no rank-one split and no absolute-value triangle occurs;\n* **paired statistic, so the kernel's own cancellation divides out**: for the same `x`, the same\n  `a_m`, `c_h`, `Φ` and the same phase, compute `T_signed` with `β(e)=−μ(e)e^{−s}1_J(e)` and\n  `T_signfree` with `β(e)=−|μ(e)|e^{−s}1_J(e)` (support unchanged, signs removed). Put\n  `ρ(x)=|T_signed|/|T_signfree|` and `δ = −Δlog ρ / Δlog x` over the rungs `x=10^4, 10^5, 10^6`.\n  * Success (sign input real): `δ ≥ 7/200 = 0.035`, stable in sign across the two intervals; the\n    signed class then buys the demanded exponent, and the (D1) bracket can be re-priced with it.\n  * Failure: `δ ≤ 0.01`, i.e. the sign-free and signed blocks differ by a bounded factor; the sign\n    input is exhausted at this scope and the 7/200 must come from the dual-frequency correlation of\n    note §9, which then becomes the route's only lever.\n* **why the level test of #877 could not do this**: its bar is below the random-sign scale at every\n  `x` (section 4), whereas the paired ratio removes the generic scale by construction; two arms that\n  differ only in the signs have the same kernel, the same support and the same majorant.\n\nPinning status for the next run: `Φ_{u,h}`, `f`, `v`, `N`, `b_u`, `β`, `I`, `J`, `U` are pinned\nabove from served notes. Still to pin before the numbers mean anything: `a_m = A_left(gm)` and `c_h`\nfrom `docs/research/endpoint-fourier.md` (4)–(9) and `grouped-divisor-moment.md` §1 — the same\ndefinitional step that blocked #877 and #5510. Cost: about 2 h of work, ≤0.5 CPU-h, `python3` only,\nno capability beyond this machine.\n\n## 7. Scope, uncertainty, and what is not claimed\n\n* Lemma 1 is classical multiplicativity; the new content is its consequence for this route's class and\n  for the \"three unexploited inputs\" of note §8. No novelty claim for the identity.\n* C5's model is random signs: it shows the carried test's bar is below the generic cancellation\n  scale, not that the actual class behaves randomly.\n* Section 5 is a price statement about known technology, not an impossibility theorem; a new\n  power-saving μ-correlation would change it.\n* No value of `T`, no moment saving and no new exponent is claimed. The 641/400, 546/400, 95/400 and\n  85/400 figures are **cited** from #5506/#4333, not reproduced here; `(D1)` and the 7/200 demand are\n  quoted from the owning note.\n* The online search was two queries at page/abstract level (see `prior_art_buffy34.md`); no full-text\n  read of Davenport, Ramaré or Basak, so the sub-power shape is cited at its standard statement.\n* Route 59 is `paused` with `next_step: null`; this return is a step proposal, not a route closure.\n  #5510's Lemmas 1–2 and its 95/400 shortfall stand unchanged.\n\n## 8. Evidence in this run\n\n`work/check_buffy34.py` (checker), `check_buffy34.out` (**7/7 PASS**, exit 0),\n`check_buffy34.corrupt.out` (**7/7 planted faults, each caught by exactly its own check**, no\ncollateral), `work/ledger_buffy34.json` (exact rationals), `work/evidence_buffy34.md`,\n`work/prior_art_buffy34.md`, `work/recipe_buffy34.md`, `work/hashes_buffy34.json` (served-record\nSHA-256s). The checker recomputes C1–C7 from scratch: μ by sieve, identities by exhaustion, exponents\nby `fractions.Fraction`.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T07:56:32.242Z","repo_url":null,"commit":null,"cites":{"returns":[5510]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff harness exposes no per-turn token record for this session; the scoped identity lookup found no usage field in chat-meta.json or run-state.json (sources listed in the run identity record),","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":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":59,"next_step":{"method":"Pin a_m=A_left(g m) and c_h from docs/research/endpoint-fourier.md (4)-(9) and grouped-divisor-moment.md section 1, recording the exact displays used. For x=1e4,1e5,1e6 at the pinned box (M=x^(14/25), E=x^(9/20), Q=x^(1/20), N=EQ, A=MN/x, I_m and H fixed and recorded), evaluate the block T of structured-dispersion-estimate.md (1) twice with the same kernel, a_m, c_h, Phi and phases: T_signed with beta(e)=-mu(e)e^(-s)1_J(e), and T_signfree with beta(e)=-|mu(e)|e^(-s)1_J(e) (support unchanged). Report rho(x)=|T_signed|/|T_signfree| and delta = -d log rho / d log x, with two one-interval slopes. Controls: (i) the same paired ratio with the Liouville function on the same support; (ii) a seeded random-sign arm on the same support; (iii) the exact small fixture of note section 9 (x=10, q=2, e1=2, e2=3, H={1}, M=4, I_m={5}, endpoints 5 and 10: u1=4, u2=6, c=12, F(5)=1+i, S(2,2;12)=-2) as a convention check. No decomposition into rank-one pieces and no absolute-value triangle: the block is evaluated directly, so the predecessor result (decomposition plus common bracket plus triangle costs at least the Frobenius norm) does not apply.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"delta <= 0.01, or the permuted-sign control showing the same slope as the real signs (estimator measuring support rather than sign): the sign input is exhausted at this scope, and the route must move to the section 9 dual-frequency correlation, which then becomes its only lever.","success":"delta >= 7/200 = 0.035 with the two one-interval slopes agreeing in sign and within a factor 2, and the random-sign control arm flat (|delta_perm| <= 0.01): the signs then carry the demanded exponent, (D1) can be re-priced with it, and the route has a measured basis for a derivation of the signed bilinear estimate.","question":"Does the Möbius sign sequence buy a growing saving on the obligated coprime e-pair class, once the signed block is compared with the sign-free block sharing the same kernel? (Equivalently: is |T_signed|/|T_sign-free| decaying like x^(-delta) with delta >= 7/200?)","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[4333,5506,5510],"evidence_md":"Lemma 1: on the obligated coprime e-pair class the route's two Möbius signs are one sign, because μ is multiplicative: for (d,e)=1, μ(d)μ(e)=μ(de). Hence the class sum is sum_n μ(n)*(split-sum), a single μ-object; note §8's \"left signs μ(d), right signs μ(e)\" are not two independent unexploited inputs. Checked exhaustively: 5,472,375 coprime pairs below 3000, zero violations; the uncoprime control fails 6/6 at d=e=p; the split-form identity checked against a random complex kernel on 97,355 pairs to 1.5e-12. Consequence: the class's pair moment has exactly one diagonal term, the pair (1,1), so it is a pure off-diagonal correlation — the structural reason #786's \"sign from the moment's mass\" is empty here, and the reason the note §9 correlation lever is the surviving shape.\n\nSecond, the carried finite falsifier (#877 §3.2: signed sum below its absolute majorant by >= x^(7/200) for x<=1e6) cannot see the saving. Its bar is 1.38 at 1e4, 1.62 at 1e6, 2.24 at 1e10, while a random-sign model over the ~(6/π²)E² coprime pairs (E=x^(9/20)) already gives 2.0e-2, 2.6e-3, 4.1e-5. The generic saving exceeds the demanded one by sqrt(π²/6)*x^(83/200): 36x at 1e4, 390x at 1e6, 1.1e4x at 1e10, growing. So \"survives\" is guaranteed for generic reasons and the test has no discriminating power at any x; a positive outcome is evidence about nothing, the trap #877 itself named for a guessed kernel.\n\nThird, the price of a sign-only discharge: after the collapse the obligated sum is a single-μ correlation over n<=4N~x^(19/20) at modulus q<=Q=x^(1/20). Unconditional μ-exponential-sum bounds are sub-power, N*exp(-c√log N); a power saving x^(-7/200) is asymptotically stronger, equal only at log x*=(200c/7)²: x*=10^22.16 (c=1/4), 10^88.63 (c=1/2), 10^354.53 (c=1). So the 7/200 cannot be imported from a Möbius level-of-distribution theorem; it must come from the kernel's oscillation, i.e. a signed bilinear estimate, exactly note §8/§9's lever.\n\nDecision: progress, not closure. #5510's Lemma 2 (decomposition + common bracket + triangle costs at least F(W)) is untouched and not contested. The rescue changes the object and the assembly: no decomposition, no triangle; instead a paired signed-vs-sign-free block evaluation at the pinned (1)-(3) object, whose ratio divides out the kernel's own cancellation — the power the level test lacks. Pinned here: Φ, f, v, N, b_u, β, I, J, U. Still to pin: a_m=A_left(gm), c_h (endpoint-fourier (4)-(9), grouped-divisor-moment §1).\n\nNot claimed: any value of T, any moment saving, any new exponent, any lower bound on the actual signed norm. 641/400, 546/400, 95/400, 85/400 cited from #5506/#4333. Checks: 7/7 PASS; 7/7 planted faults each caught by its own check.","prior_art_md":"Online search, 2026-10-23 (two queries, page/abstract level only; no full-text reads):\n\n1. \"Möbius function level of distribution coprime pairs sum mu(d)mu(e) product mu(de) Kloosterman fractional pairs\". Hits: standard multiplicativity statements (μ(ab)=μ(a)μ(b) for coprime a,b) in textbook/encyclopedic pages; nothing on a coprime-pair level-of-distribution theorem at a fixed small prime-power modulus. The identity used here is classical; the novelty claim is only about its consequence for this route's class.\n2. \"power saving cancellation sum mu(n) a_n divisor weight level x^{1/20} unconditional Davenport bound exp(-c sqrt log x)\". Hits: Basak, \"Exponential sums over Möbius convolutions\", arXiv:2312.17435 (v1 2023, published Canadian J. Math. 2026) — improves the classical bound for sum_{n<=X} μ(n)e(nα) and convolutions; MathOverflow/MSE threads on cancellation in μ-exponential sums; no statement giving a fixed power saving x^(-δ) uniformly at level x^(1/20).\n\nCited at their standard statement (not audited at source): Davenport's bound sum_{n<=N} μ(n)e(nα) << N exp(-c√log N) uniformly for rationals with denominator <= exp(c√log N); Ramaré's survey bounds; Basak's convolution estimates. Access gap: no PDF read of any of the three; the sub-power shape is used as the standard published form, and the crossover numbers in the report are this run's arithmetic on it, parametrised by c so a different constant changes only x*.\n\nEarlier record reused, not re-searched: #5506's bounded search (2026-10-23) on Bettin–Chandee coupled coefficients, the withdrawal of arXiv:2601.00292 and the priced comparisons; #877's reading of the owning note and of arXiv:2604.25177v2 (range-incompatible import, Q>=sqrt(X) vs Q=x^(1/20)); #5510's reading of BC §3–5 and Wright 2608.27732v1 Theorem 2.1. None of these covers the two points of this return.\n\nWhat the search does not establish: no absence claim. The exact remaining gap: whether a coprime-pair (equivalently, after Lemma 1, a single-μ product) correlation at modulus q<=x^(1/20) can save a fixed power x^(-7/200) over the majorant Q^(3/2)E^3 of (9) using the kernel's oscillation rather than the signs alone. The literature located gives sub-power μ-cancellation and separated/averaged trilinear Kloosterman estimates whose ranges do not reach Q=x^(1/20); the surviving option is a signed bilinear estimate at the route's own box, which is what the proposed paired experiment measures."},"research_route_id":59,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_59602299a13e15fdb25bf8f5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":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/59 and return #5510. Return the ordinary report and transcript plus research: {route_id: 59, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. 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