{"id":"96d958c5-00df-4a3f-8ea2-1d75393e6a2e","arxiv_id":"2604.00764","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For connected simply connected unimodular solvable Lie groups of exponential growth, there exist coherent frames whose index sets have zero lower Beurling density.","lead":"The authors prove that coherent frames can exist over index sets with zero lower Beurling density for square-integrable projective representations of certain solvable Lie groups. This shows that a standard localization assumption in density theorems for frames cannot be dropped in general.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The supplied manuscript is the math.FA note whose Theorem 1.2 is the strongest claim identified by the reader; the hep-ph abstract is a metadata mismatch and is ignored. The reader already isolates the precise external dependencies (Lemma 3.4 + continuous-frame/discretization results). Those dependencies are load-bearing in the sense that the paper does not re-prove them, yet they are applied inside their published scope and the projective-to-central-extension reduction is handled carefully. No stronger internal gap appears. Consequently the CONDITIONAL verdict (accept-shaped pending specialist confirmation of the cited inputs) remains appropriate; no adjustment is required.","tokens_in":11266,"tokens_out":499,"duration_ms":4558,"concrete_test":"Verify that the continuous Parseval frame of [11, Thm. 0.2] for the regular representation of the affine/Grélaud group, after the unitary intertwiner of Theorem 3.1 and the orthogonal projection onto the coefficient space of π_σ, still yields a frame vector η whose orbit under the projected discrete set remains a frame for H_π; if the frame bounds collapse under this composition the construction fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.2) is an existence result for zero lower Beurling density coherent frames. The proof reduces the projective case to a genuine representation of a unimodular central extension G_σ of exponential growth, embeds a closed simply-connected nonunimodular subgroup H ≅ affine or Grélaud (Lemma 3.4), obtains a continuous Parseval frame of translates for λ_H ([11]), discretizes it ([10]), transfers via unitary equivalence of regular representations (Theorem 3.1), and projects back via the coefficient map of π_σ. The zero-density conclusion then follows because the image of H is a proper closed nonunimodular subgroup, so G cannot be covered by finitely many right translates of a compact set. The steps are standard and carefully adapted; the external inputs are used within their stated hypotheses. No internal contradiction or missing hypothesis that would invalidate the existence claim is visible.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript proves that every irreducible square-integrable projective unitary representation of a connected, simply connected unimodular solvable Lie group of exponential growth admits a coherent frame whose index set has vanishing lower Beurling density (Theorem 1.2). The argument reduces the projective case to a genuine representation of a unimodular central extension G_σ of exponential growth, embeds a closed simply-connected nonunimodular subgroup H isomorphic to the affine or Grélaud group (Lemma 3.4), obtains a continuous Parseval frame of translates for the regular representation of H, discretizes it, transfers the frame via unitary equivalence of restricted regular representations (Theorem 3.1), and pulls it back through the coefficient map of the associated representation of G_σ. Zero density follows because the image of H is a proper closed nonunimodular subgroup, so G cannot be covered by finitely many right translates of a compact set. A short final section constructs a concrete class of unimodular exponential groups admitting square-integrable representations modulo the centre to which the result applies.","tokens_in":11403,"tokens_out":1109,"duration_ms":24807,"significance":"The result is a clean and useful negative theorem: it shows that the localisation hypothesis η ∈ B_π cannot be dropped from the density lower bound for coherent frames on unimodular amenable groups (Theorem 1.1), and that several related structural results (stability under weak limits of translates, frame measure formulae) likewise require localisation. The reduction through central extensions is carefully adapted to the projective setting, where genuine square-integrable representations are known not to exist for the groups under consideration. The existence proof is non-constructive but rests on standard structural theorems used within their stated hypotheses; if correct, it settles a natural open question in the density theory of coherent systems.","major_comments":[{"comment":"In the proof of Theorem 1.2 (Section 3.3), the claim that the closed simply-connected subgroup H of G_σ isomorphic to the affine or Grélaud group necessarily satisfies H ⊆ G × {1} is asserted in a single sentence (“as H is simply connected”). Because the circle factor is central and compact, the projection of H onto T must indeed be trivial, but this deserves an explicit one-line argument (e.g., continuous homomorphisms from a simply-connected exponential group into T are trivial). Without it the subsequent identification Γ = j(Λ) and the transfer of the frame from λ_{G_σ}(Λ) to π(Γ)η are not fully justified.","section":null},{"comment":"The same proof invokes [11, Thm. 0.2] for a continuous Parseval frame of translates for λ_H and [10, Thm. 1.3] (or [5, Thm. 3.4]) for discretisation. Both results are stated for genuine regular representations of nonunimodular groups. While the reduction to G_σ makes the application formally legitimate, a brief remark confirming that the hypotheses of those theorems (type-I, exponential, existence of admissible vectors) remain satisfied for the concrete affine/Grélaud subgroups of G_σ would close a potential gap for readers who do not consult the cited works.","section":null}],"minor_comments":[{"comment":"Notation for the universal cover in the proof of Lemma 3.4 is typographically garbled (“eS”, “eH”, “e_S”). Standard notation (tilde or widehat) would improve readability.","section":null},{"comment":"In Section 2.1 the formal-degree identity is written with d_π^{-1} on the right-hand side; some authors place d_π on the left. A parenthetical consistency note with the convention of [1] would avoid confusion.","section":null},{"comment":"The phrase “projective regular representations” appears in the introduction; the body works exclusively with ordinary regular representations of the central extension. Aligning the terminology would prevent a minor ambiguity.","section":null},{"comment":"Reference [5] is listed as a 2026 preprint (arXiv:2603.10423). If it remains unpublished, a short statement that the discretisation can equally be taken from the published [10] alone would future-proof the citation.","section":null},{"comment":"Section 4 ends abruptly after recovering the Heisenberg example of [28]. A single sentence indicating that the same construction yields infinitely many non-isomorphic groups would strengthen the “class of examples” claim.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The supplied package pairs an hep-ph abstract (arXiv:2604.00764, electron-proton scattering) with a pure math.FA manuscript (arXiv:2604.00765, coherent frames). I have refereed the mathematical text that constitutes the full manuscript body, which is the only coherent document present. The abstract/title mismatch should be corrected by the authors or the editorial office before any further processing; it is not a scientific flaw of the mathematics itself. The mathematical argument appears sound and suitable for a short note in a harmonic-analysis or Lie-group journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The paper that actually arrived is the math.FA note (arXiv:2604.00765), not the hep-ph abstract in the metadata. Ignore the mismatch; the math is what we have.\n\nWhat is new is Theorem 1.2: for an irreducible square-integrable projective representation of a connected simply-connected unimodular solvable Lie group of exponential growth, there is a coherent frame whose index set has lower Beurling density zero. That immediately shows the localization assumption η ∈ B_π cannot be removed from the density lower bound (Theorem 1.1) and that several earlier structural results (stability under weak limits, frame measure formula) need it. The construction is careful: analytic cocycle → unimodular central extension of exponential growth → closed simply-connected non-unimodular subgroup isomorphic to the affine or Grélaud group (Lemma 3.4) → continuous Parseval frame of translates, discretized, transferred by unitary equivalence of restricted regular representations, then pulled back via the coefficient map. Zero density follows because the image of that subgroup is proper and non-unimodular, so the ambient group cannot be covered by finitely many right translates of a compact set. They also note the set can sit inside a subgroup of dimension ≤ 3 and treat the non-analytic cocycle case by similarity.\n\nThe argument is pure existence from cited structural theorems; no free parameters, no circularity. The external inputs (Führ continuous frames, discretization, absolute closedness, Jenkins growth) are used inside their stated hypotheses. Soft spots are ordinary for a short note: several steps are black-box citations rather than re-proofs, and a specialist should confirm the continuous-frame/discretization results transfer cleanly through the central-extension/projective setting. That is a check, not a hole visible on the page. Section 4 supplies a concrete class of examples with square-integrable representations modulo the centre, which is useful.\n\nThis is for people working on density theorems for coherent systems and frames of translates on Lie groups. The math is solid enough that a serious editor should send it to referees. I would bring it to reading group and would cite the sharpness statement.","headline":"Clean sharpness result: zero lower Beurling density coherent frames exist for square-integrable projective representations of unimodular exponential solvable groups, so localization cannot be dropped from the density lower bound.","tokens_in":12046,"tokens_out":537,"would_cite":true,"duration_ms":4346,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E25","22E27","43A80","46B15"],"pacs":[],"model":"grok-4.5","headline":"Coherent frames can exist with zero lower Beurling density on certain solvable Lie groups, so the usual density lower bound needs a localization assumption.","keywords":["Beurling density","coherent states","frame","solvable Lie group","projective representation","square-integrable representation","exponential growth"],"falsifier":"Exhibit a connected simply-connected unimodular solvable Lie group of exponential growth whose every square-integrable projective representation forces every coherent frame to have strictly positive lower Beurling density, or show that the continuous-frame discretization step fails for the central-extension groups used in the construction.","tokens_in":12122,"feed_emoji":"📐","tokens_out":876,"duration_ms":12876,"temperature":0.7,"pith_summary":"The paper shows that, for irreducible square-integrable projective representations of connected simply connected unimodular solvable Lie groups of exponential growth, one can build a coherent frame whose index set has lower Beurling density exactly zero. Earlier density theorems said that any such frame must have density at least the formal degree of the representation, but those theorems assumed the generating vector is localized. The construction removes that assumption and produces a counter-example, proving the localization hypothesis cannot be dropped. The same counter-example also shows that several companion results (stability under weak limits of translates, frame measure formulas) fail without localization. A sympathetic reader cares because density theorems are used to decide when discrete coherent systems can span a Hilbert space; knowing the precise hypotheses is essential for applications in harmonic analysis and signal processing on groups.","feed_headline":"Coherent frames can have zero Beurling density","feed_subtitle":"Localization is essential: without it the usual density lower bound fails on exponential solvable groups","key_machinery":"Central extension G_σ of G that converts the projective representation into an ordinary representation, followed by restriction to a closed simply-connected non-unimodular subgroup H isomorphic to the affine or Grélaud group; continuous Parseval frames of translates on H are discretized and transferred back to produce a zero-density frame on G.","core_discovery":"Let (π, H_π) be an irreducible square-integrable projective representation of a connected, simply connected unimodular solvable Lie group G of exponential growth. Then there exist a vector η in H_π and a discrete set Γ subset G with lower Beurling density D−(Γ) = 0 such that the coherent system π(Γ)η is a frame for H_π. Consequently the localization condition η ∈ B_π cannot be removed from the density lower bound for frames.","pith_inferences":["The same zero-density phenomenon may appear for other non-type-R solvable groups once continuous frames of translates are known for their non-unimodular subgroups.","Practical sampling schemes on exponential solvable groups cannot rely solely on Beurling density; some form of localization or decay on the generating vector must be verified separately.","The construction suggests that density theorems for projective representations should be restated in terms of the associated central extension, where ordinary representation theory applies."],"forward_implications":["The localization hypothesis η ∈ B_π is indispensable for the density lower bound D−(Γ) ≥ d_π on unimodular amenable groups.","Stability of frames under weak limits of translates need not hold without localization.","Frame-measure formulas that relate density to frame bounds fail once localization is dropped.","Zero-density coherent frames can be realized inside subgroups of dimension at most three."],"fun_headline_variants":["Axial form factor drives charged-current e-p cross sections","Spin asymmetries probe g2 in electron-proton scattering","e-p charged current data constrain axial vector form factor","Cross sections and polarizations test g1 and g2 parameterizations","Weak charged current e-p scattering for JLab and MAMI"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument needs non-unimodular subgroups (affine or Grélaud type) that admit continuous frames of translates which can be discretized, and that these frames survive the passage through the central extension.","fun_headline_variants_meta":{"raw":{"variants":["Axial form factor drives charged-current e-p cross sections","Spin asymmetries probe g2 in electron-proton scattering","e-p charged current data constrain axial vector form factor","Cross sections and polarizations test g1 and g2 parameterizations","Weak charged current e-p scattering for JLab and MAMI"]},"model":"grok-4.5","effort":"low","cost_usd":0.005792,"raw_usage":{"total_tokens":1700,"prompt_tokens":1006,"num_sources_used":0,"completion_tokens":89,"cost_in_usd_ticks":57920000,"prompt_tokens_details":{"text_tokens":1006,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":605,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1006,"tokens_out":89,"duration_ms":5146,"temperature":1.0,"reasoning_tokens":605,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T14:52:21.586819+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a connected simply-connected unimodular solvable Lie group of exponential growth whose every square-integrable projective representation forces every coherent frame to have strictly positive lower Beurling density, or show that the continuous-frame discretization step fails for the central-extension groups used in the construction.","supporting_citations":[],"review_version":1}