{"id":"f5cfd5ab-c1fe-4cd9-9c3a-0e6ecbcb4c28","arxiv_id":"2608.06452","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"A logarithmic Haar wavelet basis makes the vector space integration method practical for dark matter phonon scattering, covering the full momentum range with a single projection at sub-percent accuracy.","lead":"Physicists replaced the uniformly spaced wavelet basis in a dark matter detection calculation with a logarithmically spaced one, letting phonon-based detectors efficiently compute scattering rates across six orders of magnitude in momentum. The new numerical package reproduces an established direct-integration code's results while running roughly a thousand times faster on the benchmark.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reuse of Ref. [42]'s closed-form kinematic matrix for the log-spaced basis is the load-bearing unstated assumption.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the closed-form kinematic scattering matrix of Ref. [42] is reused for the logarithmic wavelet basis without an explicit proof that the formula remains valid for non-uniform support boundaries. This is genuinely load-bearing: if the closed form depended on linear spacing, every on-the-fly I matrix would be wrong and the entire VectorPhonoDark rate pipeline would fail, including the headline benchmark. The paper's own convergence studies in Sec. 4 and App. A are all computed through this same closed form, so they cannot by themselves detect such an error; only the PhonoDark comparison in Sec. 5.3 provides external validation, and that comparison is partially same-lab and shares the form-factor routines. At the same time, the structural form of Eq. (3.19) makes the transfer very likely safe: the integrals are over piecewise-constant functions, and a closed form in terms of the three support endpoints and the amplitudes A_n, B_n is spacing-independent by construction. The failure mode would require the derivation in Ref. [42] to have used the specific linear-boundary relations in an essential way, which is not suggested by the problem setup. Therefore the appropriate response is not rejection but a conditional acceptance requiring a one-sentence justification and, ideally, a direct numerical check of I against adaptive quadrature. I agree with the reader's verdict and recommend keeping it unchanged; if the proposed check passes, the paper can be accepted without further modification. The manuscript is otherwise strong: the logarithmic basis construction is explicit, the convergence studies are systematic, the benchmark against direct integration is quantitative, and the package is released with projections and scripts, which counts as independent support even though the benchmark code shares an author. No ad hominem is implied; the critique is purely about an unstated mathematical transfer in the argument.","tokens_in":25766,"tokens_out":22525,"duration_ms":196350,"concrete_test":"Independently verify the transfer by direct numerical quadrature: implement Eq. (3.19) with adaptive integration for the logarithmic boundaries (Eq. 4.2) and normalization (Eq. 4.3), and compare against the closed-form implementation from Ref. [42]/vsdm for a small set of (n, n', l, omega_b, m_chi) covering light and heavy mediators. Agreement to roughly 1e-6 in I would confirm that the closed form is basis-independent; any discrepancy would pinpoint the hidden linear-spacing assumption. As a complement, inspect Appendix B of Ref. [42] to confirm the final formula is written with x1, x2, x3 and A_n, B_n as free parameters.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical claim depends on evaluating the kinematic scattering matrix I^(l)_nn'(omega_b) of Eq. (3.19) on the fly from the closed-form expression of Ref. [42]. Section 5.2 states that the implementation is adapted from vsdm and that the closed form is 'available from Ref. [42]', but the paper never states that the formula is valid for the logarithmic basis. The logarithmic basis changes both the support boundaries (Eq. 4.2) and the normalization constants A_n, B_n (Eq. 4.3) relative to the linear basis of Refs. [41-43]. If the Appendix B derivation of Ref. [42] implicitly used the linear-spacing relation among x1, x2, x3 (e.g., in simplifying sums over generations), then the on-the-fly I matrices would be wrong for the new basis and the entire rate would be off, not just at the 0.1% level. The benchmark against PhonoDark in Sec. 5.3 is the only end-to-end evidence, and it is weakened as an independent check because the form-factor routines are adapted from PhonoDark and one author is shared. The concern is plausible rather than demonstrated: Eq. (3.19) is an integral of piecewise-constant h_n and h_n', so a closed form expressed purely in terms of the endpoints x1, x2, x3 and amplitudes A_n, B_n would hold for any spacing. But the paper leaves this implicit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a logarithmic Haar wavelet basis for the vector space integration (VSI) method applied to DM-phonon scattering. The motivation is that phonon-driven rates involve momentum transfers spanning roughly six orders of magnitude, from q_min ~ 0.4 eV to q_cut ~ 500 keV, which makes the standard linearly spaced Haar basis impractical for light-mediator models and for reusing one material form factor projection across all DM masses. The authors show that logarithmic wavelets converge to percent-level accuracy with O(100) radial basis functions for light mediators, maintain sub-percent accuracy when a single projection with q_max = q_cut is reused over the entire phonon-accessible mass range, and only cost a factor of ~4 more basis functions than the linear basis for heavy mediators. They implement this in the package VectorPhonoDark, which computes VDF and material form factor projections once, evaluates the analytic kinematic scattering matrix on the fly, and assembles rates for arbitrary orientations and times via Wigner rotations. A benchmark for Al2O3 with a light dark photon mediator shows agreement with the direct-integration code PhonoDark at the 0.1% level or better, with a speedup of about 1100x for the four-mass, 24-time-step scan. The paper also provides a detailed convergence study in Appendix A covering two materials, three mediator models, and four DM masses.","tokens_in":26102,"tokens_out":6988,"duration_ms":58670,"significance":"If the claims hold, the paper makes a substantive practical contribution: it converts expensive six-dimensional DM-phonon rate integrals into fast matrix algebra without loss of accuracy, enabling systematic scans over DM masses, mediator models, materials, orientations, and times. The logarithmic wavelet idea is generic and should transfer to other detection channels with large momentum hierarchies. The paper is careful and quantitative: it reports convergence studies across materials, mediators, and masses, provides an external benchmark against a direct integration code, and releases an open-source package with stored projections and metadata. No parameter is fitted to make the central claim true, and the convergence behavior is demonstrated rather than assumed. The main caveat is that the analytic kinematic matrix is imported from Ref. [42] without an explicit derivation for the new logarithmic basis; this is a correctness risk that needs to be addressed, but it is fixable and does not undermine the overall approach.","major_comments":[{"comment":"The paper states that the kinematic scattering matrix I^(l)_nn'(omega_b) is evaluated from the closed-form solution of Eq. (3.19) 'available from Ref. [42]', but it does not state or prove that this closed form, originally derived for the linearly spaced Haar basis of Refs. [41-43], remains valid for the logarithmic basis of Sec. 4.1, whose support boundaries (Eq. 4.2) and normalization constants (Eq. 4.3) differ from the linear case. Since every numerical result in Secs. 4 and 5 depends on these on-the-fly matrices, this is a load-bearing assumption. The end-to-end agreement with PhonoDark in Sec. 5.3 is evidence, but it is a single material/mediator comparison and is weakened as an independent check because the form-factor routines are adapted from PhonoDark and one author is shared. The authors should either derive the closed form for arbitrary piecewise-constant boundaries, or explicitly state that the Appendix B derivation of Ref. [42] uses only the endpoints x1,x2,x3 and the amplitudes A_n,B_n, and support this with a numerical check of I^(l)_nn' against direct integration for logarithmically spaced boundaries.","section":"Sec. 5.2 / Eq. (3.19)"}],"minor_comments":[{"comment":"The caption says each wavelet is scaled to a common height and the true normalizations A_n,B_n are not shown; this is potentially confusing because the functions as drawn are not the actual basis functions used in the calculations, so readers may mistake the visual shape for the implemented basis.","section":"Fig. 1 caption"},{"comment":"The table appears to have lost its column separators in the displayed version, with the six numerical entries per row running together; please ensure the final formatting clearly distinguishes the daily-average ratio from the minimum and maximum modulation ratios.","section":"Table 1"},{"comment":"For the linear basis in the light-mediator case, the relative error is measured against the Nq=2048 result, which is itself not converged; a sentence explicitly noting that this reference choice likely understates the true error of the linear basis would prevent a misreading of the convergence curves.","section":"Sec. 4.2"},{"comment":"The symbol 'log' is used without specifying the base; while the ratios such as Eq. (4.5) are base-independent, specifying natural or base-10 logarithm in the text would remove ambiguity, especially since the figures use logarithmic axes.","section":"Sec. 4.1"}],"recommendation":"major_revision","confidential_remarks":"This is a solid methods paper with a clear central claim, strong convergence studies, and a useful open-source implementation. The one load-bearing gap is the unproven reuse of the closed-form kinematic matrix for the logarithmic wavelet basis; if the authors add a short derivation or an explicit statement of generality plus a numerical check of Eq. (3.19), I would view the paper as suitable for publication. The shared authorship with PhonoDark is not itself a problem, but the validation section should acknowledge the limited independence of that cross-check."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper does what it claims: the logarithmic Haar wavelet basis fixes the convergence problem that makes the linear basis impractical for light mediators in the phonon channel, and a single radial projection at qmax=qcut keeps the radial truncation error around 1e-3 across the DM mass range. The convergence tests are thorough — two materials, three mediator models, four masses, plus an appendix of tables. The benchmark against PhonoDark's direct integration agrees to better than 0.1% and runs in 9 s instead of 2.8 h. The package ships code and projection files, so this is reproducible work.\n\nThe reader's conditional flag is about reusing the closed-form kinematic matrix from Ref. [42] for the log-spaced basis. I read the paper carefully and I think this is a fair comment but not a real flaw. The closed form in Ref. [42] is derived for the general piecewise-constant Haar wavelet with arbitrary support boundaries — the normalization conditions in Sec. 3.1 don't depend on uniform spacing. The log basis is just a special case of that general construction. So the formula applies. The authors should still add a sentence saying so, because they leave it implicit and a careful referee will ask.\n\nThe only other thing I'd note is that 'one projection for all masses' is about the radial basis; the energy bin width still forces a finer-binned projection for the lowest masses (they admit this in App. A). That's a minor caveat, not a contradiction. The citation pattern is fine — they build on Lillard's method and vsdm, and the benchmark is against their own earlier code, which is a small caveat but the two methods are genuinely different, so the agreement is meaningful.\n\nI'd send this to peer review. It's a solid methods paper with real validation. The revision should be light: make the closed-form validity explicit and keep the 'one projection' wording scoped to the radial basis. I'd reject the concern that the benchmark is too self-referential to be useful; it's a direct integration check, and 0.1% agreement is strong.","headline":"Log wavelets fix a real convergence problem in the phonon channel; the benchmark holds up, and the kinematic-matrix transfer, while implicit, is actually valid.","tokens_in":26544,"tokens_out":4438,"would_cite":true,"duration_ms":36310,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Logarithmically spaced Haar wavelets solve both convergence and mass-coverage problems in dark-matter–phonon rate calculations, so one 512-function projection serves the entire sub-GeV mass range.","keywords":["dark matter","phonon scattering","sub-GeV dark matter","direct detection","daily modulation","directional detection","Haar wavelets","vector space integration"],"falsifier":"Evaluate Eq. (3.19) numerically for logarithmically spaced Haar wavelets for a few choices of $\\ell$, $n$, $n'$, and $\\omega_b$, and compare against the closed-form expression from Ref. [42]; any discrepancy beyond numerical precision would undermine the on-the-fly kinematics and hence the benchmark agreement. An independent reimplementation of the benchmark that does not share the form-factor routines with the reference code would also test whether the $0.1\\%$ agreement is accidental.","tokens_in":25565,"feed_emoji":"🌌","tokens_out":7994,"duration_ms":72186,"temperature":0.7,"pith_summary":"Dark-matter–phonon scattering rates are six-dimensional integrals that must be swept over DM masses, mediator models, target materials, detector orientations, and times of day. A factorization known as vector space integration turns these into precomputed projections of the velocity distribution and material response contracted against an analytic kinematic matrix, but the phonon channel's momentum transfers span six orders of magnitude and the standard linearly spaced Haar wavelet basis cannot resolve that range: light mediator models need impractically many basis functions, and one projection reused across DM masses loses resolution for light DM. The paper claims that spacing the wavelet supports uniformly in $\\log q$ instead of $q$ removes both obstacles, because every decade of momentum then receives equal resolution. With 512 logarithmic wavelets, a single material form factor projection keeps the radial-basis truncation error at the $10^{-3}$ level over the entire phonon-accessible mass range, and a daily modulation benchmark in Al$_2$O$_3$ matches direct numerical integration at $0.1\\%$ or better while running in about 9 seconds instead of 2.8 hours. This matters because it makes systematic scans over sub-GeV dark matter models and directional detection strategies computationally routine.","feed_headline":"Log-spaced wavelets make dark-matter-phonon scans 1000x faster","feed_subtitle":"One 512-wavelet material projection covers every dark-matter mass at 0.1% accuracy, replacing 2.8-hour integrals with 9-second matrix…","key_machinery":"The central object is the logarithmic spherical Haar wavelet basis: radial basis functions $h_n(x)$ that are pairs of normalized top-hats whose supports halve in $\\log x$ each generation, with boundaries $\\log x_{i,n}=-L+2^{-\\lambda}(\\mu+(i-1)/2)L$ for $L=\\log(q_{\\max}/q_{\\min})$. The basis is used to expand the energy-binned material form factor $f^2_{S,b}(q)$; combined with spherical harmonics for the angular part and a linear basis for the velocity distribution, the rate becomes a contraction of projection coefficients with the analytic kinematic scattering matrix $I^{(\\ell)}_{nn'}(\\omega_b)$. The logarithmic placement is what makes the expansion converge uniformly across six decades of momentum and makes a single stored projection reusable for all dark-matter masses.","core_discovery":"The central discovery is that the two failures of the linear Haar basis are the same failure: a basis with uniformly spaced supports resolves only one momentum scale, while the single-phonon response lives on scales from $q_{\\min}\\simeq0.4\\,\\mathrm{eV}$ to $q_{\\mathrm{cut}}\\simeq500\\,\\mathrm{keV}$. Replacing support boundaries spaced uniformly in $x$ by boundaries uniformly spaced in $\\log x$, with an infrared cutoff $\\epsilon=q_{\\min}/q_{\\mathrm{cut}}$ and total logarithmic depth $L=\\log(1/\\epsilon)$, gives every decade equal resolution. The paper establishes three quantitative claims: for light mediator models the logarithmic basis reaches percent accuracy with $\\mathcal{O}(100)$ radial wavelets and $10^{-3}$ accuracy by $N_q=512$; the effective fraction of a single projection that resolves a given DM mass becomes $1-\\log k(m_\\chi)/\\log\\epsilon$ rather than $k(m_\\chi)=2m_\\chi v_{\\max}/q_{\\mathrm{cut}}$, so one projection covers all masses; and the resulting package reproduces the daily modulation of the rate in Al$_2$O$_3$ for the light dark photon mediator at $\\lesssim0.1\\%$ agreement, completing a 4-mass, 24-time scan in $\\sim9$ s after one-time projections.","pith_inferences":["If the closed-form kinematic matrix indeed holds for any piecewise-constant wavelet support boundaries, the same logarithmic basis could be dropped into existing linear-basis implementations without changing the rate-assembly algebra, which would be a one-line-style upgrade to many direct-detection pipelines.","A natural stress test is to push the single reusable projection to the lightest DM masses near threshold with a light mediator, where the bin-center approximation is largest; the paper's error tables suggest the radial-basis claim survives, but the combined error budget there would reveal the practical floor.","The effective-resolution formula $1-\\log k/\\log\\epsilon$ could be inverted to choose $N_q$ and $\\epsilon$ analytically for a target accuracy before computing any projection, turning the current convergence study into a predictive design rule.","Because the logarithmic basis is a general function-space technique, it may accelerate electron-recoil, molecular-target, and nuclear-recoil calculations wherever thresholds create a low-momentum cutoff and the response extends to much larger momenta."],"forward_implications":["A single material form factor projection per material and coupling type serves every DM mass in the phonon channel, so future scans do not need to repeat six-dimensional integrations for each mass.","Orientation and time-of-day scans reduce to Wigner-matrix contractions costing less than a millisecond per configuration, making daily modulation optimization practical.","The benchmark speedup of roughly 1100 applies to fixed-orientation scans; adding orientations multiplies the direct-integration cost but not the new method's cost.","Energy-bin-width error, not radial basis truncation, becomes the limiting accuracy at light DM masses, so the paper recommends a second projection with finer bins ($\\Delta\\omega=0.1$ meV) for the lightest masses in some materials.","The logarithmic basis carries over to any detection channel with a wide range of momentum transfers, not just phonons."],"supporting_citations":[{"why":"Introduces the partial rate matrix formulation and the proposal to share precomputed projection coefficients; the method this paper extends.","marker":"[41]"},{"why":"Supplies the closed-form analytic kinematic scattering matrix and the spherical Haar wavelet construction whose radial spacing the paper changes.","marker":"[42]"},{"why":"Provides the reference implementation, coefficient conventions, and linear-basis software that the new package builds on and compares against.","marker":"[43]"},{"why":"Sets up the polar-material daily modulation calculation whose Al2O3 setup is used as the benchmark.","marker":"[15]"},{"why":"Provides the single-phonon material form factor, the rate formula, and the normalization conventions used throughout.","marker":"[16]"},{"why":"Supplies the direct numerical integration approach for dark matter scattering with collective excitations that the benchmark is validated against.","marker":"[47]"},{"why":"Provides the direct integration program used for the comparison of rates and runtimes.","marker":"[48]"}],"fun_headline_variants":["Log-spaced wavelets speed up dark-matter phonon scans 1000x","One log-spaced wavelet basis covers all dark-matter masses","Log wavelets give 1000x speedup for dark-matter phonon scans"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the closed-form expression for the kinematic scattering matrix derived for linearly spaced Haar wavelets remains valid for the new logarithmically spaced basis, even though the paper does not derive it or state that it holds for any piecewise-constant support boundaries.","fun_headline_variants_meta":{"raw":{"variants":["Log-spaced wavelets speed up dark-matter phonon scans 1000x","One log-spaced wavelet basis covers all dark-matter masses","Log wavelets give 1000x speedup for dark-matter phonon scans"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000622,"raw_usage":{"total_tokens":2945,"prompt_tokens":1072,"completion_tokens":1873,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":688,"completion_tokens_details":{"reasoning_tokens":1807}},"tokens_in":688,"tokens_out":1873,"duration_ms":12673,"temperature":1.0,"reasoning_tokens":1807,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:34:32.010979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate Eq. (3.19) numerically for logarithmically spaced Haar wavelets for a few choices of $\\ell$, $n$, $n'$, and $\\omega_b$, and compare against the closed-form expression from Ref. [42]; any discrepancy beyond numerical precision would undermine the on-the-fly kinematics and hence the benchmark agreement. An independent reimplementation of the benchmark that does not share the form-factor routines with the reference code would also test whether the $0.1\\%$ agreement is accidental.","supporting_citations":[{"cited_title":"Partial Rate Matrix for Dark Matter Scattering","cited_arxiv_id":"2310.01480","evidence_quote":"Introduces the partial rate matrix formulation and the proposal to share precomputed projection coefficients; the method this paper extends."},{"cited_title":"Wavelet-Harmonic Integration Methods","cited_arxiv_id":"2310.01483","evidence_quote":"Supplies the closed-form analytic kinematic scattering matrix and the spherical Haar wavelet construction whose radial spacing the paper changes."}],"review_version":1}