{"id":"7a587822-fa1a-4ec3-92a2-d87d8336a1a7","arxiv_id":"2603.25564","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The murmuration density function is computed for Hecke forms of weight k and level N=ℓ^a with ℓ a fixed odd prime and a→∞.","lead":"The authors compute the murmuration density for Hecke modular forms of fixed weight and prime-power level as the power tends to infinity. This extends the recent murmuration program from the usual level aspect into the depth aspect of a fixed prime.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Body is encoding-corrupted; depth-aspect error control for the claimed murmuration density cannot be inspected.","rationale":"The readable abstract states a natural extension of the murmurations program to the depth aspect, but the body is unusable for verification. The reader's weakest assumption (uniform validity of the analytic machinery as a\to∞) is exactly the condition that would have to hold for the density to exist and match the claimed closed form; nothing in the corrupted extract allows that condition to be confirmed or refuted. No independent support (code, formal proof, numerical tables) is present. Therefore the verdict remains UNVERDICTED with low confidence until a clean manuscript is available; no stronger mathematical objection can be raised from the given text, and none needs to be manufactured.","tokens_in":8467,"tokens_out":472,"duration_ms":17830,"concrete_test":"Obtain a clean PDF/TeX of arXiv:2603.25564; extract the main theorem (including any error terms and ranges of a,k,ℓ) and verify whether the depth-aspect bounds used for the a\to∞ limit are stated explicitly and tend to zero in the claimed regime. If they do not, or if the density formula is not derived under those bounds, the claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an explicit murmuration density for Hecke forms of fixed weight k and level N=ℓ^a (ℓ fixed odd prime, a\to∞). That claim rests on passing from averages of Hecke eigenvalues (via Petersson/trace formulae or related spectral sums) to a limiting oscillatory density, which requires error terms that remain controllable uniformly as a\to∞. The supplied full-text extract is almost entirely garbled (replacement characters, mixed non-NT content including a cs.CV arXiv stamp, unreadable formulae), so the main theorem statement, the precise range of a, and the depth-aspect estimates cannot be checked. Without those, existence and closed form of the density remain unverified; the analytic uniformity flagged by the reader is still the load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper claims to compute the murmuration density function for the family of Hecke forms of fixed weight k and prime-power level N=ℓ^a, where ℓ is a fixed odd prime and a\to∞. From the abstract and the few readable fragments, the intended result is an explicit limiting averaged oscillatory density for Hecke data (eigenvalues or related spectral sums) in the depth aspect, obtained via standard analytic tools such as Petersson/trace formulae. The supplied full-text extract is almost entirely corrupted by encoding garbage, mixed non-NT material (including a cs.CV arXiv stamp), and unreadable formulae, so the precise main theorem, error estimates, and derivation cannot be inspected.","tokens_in":8580,"tokens_out":671,"duration_ms":6554,"significance":"If the claimed density is correctly derived with uniform error control as a\to∞, the result would be a genuine contribution to the emerging literature on murmurations of modular forms, extending known conductor-aspect phenomena to the depth aspect for prime-power levels. Explicit closed-form densities of this type are of interest for understanding average oscillatory behaviour of Hecke eigenvalues and for comparisons with other families. The abstract states a clean computational claim of the sort that is typically valued in analytic number theory; however, because the body is unreadable, no machine-checked proofs, reproducible code, or fully checkable error estimates can be credited on the present extract.","major_comments":[{"comment":"The full-text extract is severely corrupted (replacement characters throughout, unreadable formulae, and intrusion of unrelated cs.CV material with arXiv stamp 2603.25565). Consequently the main theorem statement, the precise range of a, and the depth-aspect error estimates cannot be verified. The central claim that a limiting murmuration density exists and is given by a closed form therefore remains uncheckable on the supplied manuscript.","section":null},{"comment":"From the abstract and readable fragments, the argument must pass from averages of Hecke eigenvalues (via Petersson or trace formulae) to a limiting density as a\to∞ for fixed odd prime ℓ. Uniform control of the error terms in this depth aspect is load-bearing; without a readable derivation or explicit error hypotheses, it is impossible to confirm that the claimed density is justified.","section":null}],"minor_comments":[{"comment":"Even the readable fragments contain mixed or garbled section headings and formula scraps that prevent assessment of notation consistency or reference completeness.","section":null},{"comment":"The abstract is clear, but the body must be re-supplied in a clean, properly encoded form before any further technical review is possible.","section":null}],"recommendation":"uncertain","confidential_remarks":"The extract appears to be a corrupted or concatenated file (NT abstract mixed with cs.CV stamp and encoding garbage). I cannot produce a substantive mathematical report until a clean PDF or source is provided. Recommendation is therefore uncertain pending a usable manuscript; if the clean version confirms the abstract claim with standard depth-aspect estimates, the paper would likely be suitable for minor or major revision rather than outright rejection."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this paper claims an explicit murmuration density for Hecke forms of fixed weight k and level N = ℓ^a (ℓ fixed odd prime, a → ∞). That is a natural next case in the murmurations program, not a new framework, and the abstract states it cleanly.\n\nWhat is new is the depth-aspect specialization itself: previous murmuration densities have been worked out in other aspects, and this fills in the prime-power level family with a → ∞. From the readable fragments one can see the usual ingredients—averages of Hecke data, spectral sums, and a limiting oscillatory density—so the setup looks standard for the subfield. The claim is external-benchmark style rather than circular, and there are no free parameters or invented objects in the abstract.\n\nThe soft spot is severe and practical: the full-text extract we have is almost entirely garbled (encoding garbage, mixed non-NT material, unreadable formulae). I cannot inspect the main theorem statement, the precise range of a, or the error-term control that would justify uniformity as a → ∞. That analytic uniformity is load-bearing; without it the density is only a claim. This is not a detected mathematical error—it is an inability to check the paper at all from the supplied source. Confidence therefore stays low.\n\nWho it is for: specialists already working on murmurations or averages of modular forms in the depth aspect. They will want the closed form if the derivation holds. It is not for a general number-theory audience or for anyone expecting a conceptual breakthrough.\n\nI would send it to peer review once a clean manuscript is available. The abstract is sharp enough and the problem is real enough that a referee who can read the proofs should decide. Until then I would not cite it or bring the garbled extract to reading group. If the clean version checks out, it is solid subfield progress; right now we simply cannot tell.","headline":"Clean abstract claim of an explicit depth-aspect murmuration density for prime-power levels, but the supplied body is too corrupted to verify the argument.","tokens_in":9250,"tokens_out":493,"would_cite":false,"duration_ms":7323,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","11F30","11F66","11M41"],"pacs":[],"model":"grok-4.5","headline":"The murmuration density for Hecke modular forms of fixed weight and prime-power level is computed explicitly as the power of the prime tends to infinity.","keywords":["murmurations","Hecke forms","modular forms","depth aspect","prime-power level","density function","L-functions","spectral averages"],"falsifier":"Compute the empirical average of the relevant Hecke eigenvalues (or root-number-weighted coefficients) for forms of level ℓ^a with successively larger a and check whether the resulting histogram converges to the closed-form density claimed in the paper.","tokens_in":9327,"feed_emoji":"〰️","tokens_out":580,"duration_ms":15399,"temperature":0.7,"pith_summary":"Murmurations are the oscillatory patterns that appear when you average arithmetic data (such as Hecke eigenvalues) over families of modular forms or L-functions and then look at the averages as a continuous function of a scaling parameter. This paper establishes that such a limiting density exists, and writes it down in closed form, for the family of Hecke forms of fixed weight k whose levels are pure powers of a fixed odd prime. The computation is carried out in the depth aspect: the level N = ℓ^a grows by letting the exponent a tend to infinity while the prime ℓ stays fixed. A sympathetic reader cares because an explicit density turns a mysterious numerical oscillation into a concrete, checkable analytic object that can be compared with other families and used to predict averages at large depth.","feed_headline":"Murmuration density computed for Hecke forms at prime powers","feed_subtitle":"An explicit limiting oscillation appears as the level power tends to infinity with the prime fixed.","key_machinery":"The passage from discrete averages of Hecke eigenvalues to a continuous limiting density, carried by a Petersson/trace-formula identity (or equivalent spectral average) that remains controllable uniformly in the depth aspect a → ∞.","core_discovery":"For the family of Hecke forms of weight k and level N = ℓ^a (ℓ a fixed odd prime), the murmuration density function exists in the limit a → ∞ and is given by an explicit closed-form expression obtained by averaging the relevant Hecke eigenvalues (or spectral sums) and passing to a continuous density.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Murmuration density for Hecke forms as prime-power level grows","Explicit murmuration density of Hecke forms at levels ℓ^a","Closed-form murmuration density as a→∞ for fixed odd prime","Depth-aspect murmuration density for weight-k Hecke forms","Limiting murmuration density exists for Hecke forms at ℓ^a"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The analytic estimates that convert averages of Hecke data into a limiting density stay valid uniformly as the exponent a goes to infinity for a fixed odd prime.","fun_headline_variants_meta":{"raw":{"variants":["Murmuration density for Hecke forms as prime-power level grows","Explicit murmuration density of Hecke forms at levels ℓ^a","Closed-form murmuration density as a→∞ for fixed odd prime","Depth-aspect murmuration density for weight-k Hecke forms","Limiting murmuration density exists for Hecke forms at ℓ^a"]},"model":"grok-4.5","effort":"low","cost_usd":0.005834,"raw_usage":{"total_tokens":1371,"prompt_tokens":550,"num_sources_used":0,"completion_tokens":97,"cost_in_usd_ticks":58340000,"prompt_tokens_details":{"text_tokens":550,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":724,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":550,"tokens_out":97,"duration_ms":6934,"temperature":1.0,"reasoning_tokens":724,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T18:10:04.431542+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the empirical average of the relevant Hecke eigenvalues (or root-number-weighted coefficients) for forms of level ℓ^a with successively larger a and check whether the resulting histogram converges to the closed-form density claimed in the paper.","supporting_citations":[],"review_version":1}