{"id":"a2a7216a-4740-4cdc-a5ed-f2d81d205298","arxiv_id":"2607.10607","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Dark-matter abundance fixes the scalar peak area via its cubic moment, so the scalar-induced MHz gravitational-wave amplitude is predicted from the same peak without free normalization.","lead":"A small-scale bump in the primordial density spectrum can create both superheavy conformal-fermion dark matter and a MHz gravitational-wave background whose amplitude is fixed by the dark-matter abundance. Laboratory high-frequency searches would then turn a null result into a mass lower bound or a detection into a joint test of one inflationary feature.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged high-reheating assumption.","rationale":"The reader's strongest claim accurately restates the paper's central result: once Ω_χ h^{2} fixes A_ζ via the cubic moment, the SIGW peak is predicted without residual scalar normalization. The algebra (Secs. II–IV), numerical convergence (Table II), and consistency checks (PBH tail, dark radiation, crossover mass generation) are internally coherent. The high-reheating requirement is the single most load-bearing external assumption, already flagged by the reader; no stronger internal flaw (e.g., inconsistent moment definitions, unaccounted shape degeneracy that re-introduces free normalization, or contradiction with the quoted A_χ) is present. Therefore the CONDITIONAL verdict and HIGH confidence remain appropriate; no adjustment is warranted.","tokens_in":18752,"tokens_out":558,"duration_ms":4664,"concrete_test":"Recompute the radiation-era convolution (Eq. 15) for the numerical Mukhanov–Sasaki spectrum of Sec. VI after replacing the transfer function I^{2}(u,v) by a standard early-matter-era or gradual-reheating kernel (e.g., Inomata et al. 2019); if the resulting h^{2}Ω_pk_GW shifts by more than a factor of a few relative to the benchmark 5.33×10^{-12}, the mass-reach translation (Eq. 25) must be re-quoted with an explicit S_reh band.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Eq. 1 / Eq. 23) is an algebraic elimination of A_ζ between the cubic fermion moment (Eq. 12) and the quadratic SIGW peak (Eq. 20). Under the paper's stated assumptions—radiation-era kernel, fixed A_χ ≃ 0.015 from the cited production calculation, and a localized peak whose shape factors R_3 and C_GW are recomputed from the same P_ζ—the relation is tautological and holds. The reader's weakest assumption (T_reh ≳ 1.12×10^14 GeV so that S_reh = 1) is correctly identified as the dominant external condition; if delayed reheating applies, both amplitude and mass reach rescale by S_reh^{1/2} (Eq. 38). No deeper internal inconsistency, hidden free parameter, or algebraic error appears in the closure itself. The engineered single-field feature and external A_χ are acknowledged limitations but do not undermine the moment-matching logic once those inputs are granted.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper argues that a localized small-scale peak in the primordial curvature spectrum P_ζ simultaneously produces conformal-fermion dark matter through a cubic moment of P_ζ and a scalar-induced gravitational-wave (SIGW) background through a quadratic radiation-era convolution. By fixing the integrated scalar area A_ζ from the observed relic density Ω_χ h^{2} (Eq. 14), the free normalization of ordinary SIGW templates is eliminated, yielding the closed relation h^{2}Ω_pk_GW = C_V (Ω_χ h^{2})^{2}/(M_χ/GeV)^{2} (f_0/MHz)^{6} \times Δ^{-2} exp(9Δ^{2})/R_3^{2} (Eq. 1 / Eq. 23). A concrete single-field Mukhanov–Sasaki realization with a transient slow-roll dip generates a broad MHz peak (f_pk_GW ≈ 2.63 MHz, h^{2}Ω_pk_GW ≈ 5.3\times10^{-12}) that lies far below the Gaussian PBH threshold; null HFGW searches then become lower bounds on M_χ while a detection is overconstrained by abundance, frequency, amplitude and width.","tokens_in":19057,"tokens_out":1105,"duration_ms":9220,"significance":"If the radiation-era kernel and the adopted production coefficient A_χ hold, the result supplies a genuinely predictive link between superheavy conformal dark matter and laboratory MHz gravitational-wave searches. The algebraic elimination of A_ζ is clean, the numerical pipeline (Mukhanov–Sasaki spectrum, cubic shape factor R_3 = 1.6287, radiation-era convolution with documented convergence in Table II) is reproducible, and the closure converts existing and projected HFGW sensitivities into concrete mass reach (Eq. 25) and an inverted mass inference (Eq. 26). The construction therefore turns a free-normalization SIGW template into a falsifiable multi-observable test of one primordial feature.","major_comments":[{"comment":"Sec. VII.A, Eqs. (36)–(38): the entire numerical prediction and mass-reach translation rest on the radiation-era kernel, which requires T_reh ≳ 1.12\times10^{14} GeV. The paper correctly notes that delayed reheating replaces the amplitude by an unspecified factor S_reh, yet provides neither a concrete evaluation of S_reh for any standard reheating history nor a quantitative band on how large the correction can be. Because the central claim is a definite MHz amplitude and mass bound, this external condition should be either justified more tightly or accompanied by an explicit range of S_reh so that the predicted signal and M_lim_χ can be assessed under realistic post-inflationary evolution.","section":null},{"comment":"Sec. II.B, Eq. (11): the production coefficient A_χ ≃ 0.015 is taken from the external conformal-fermion calculation of Refs. [10,11] and is never recomputed or varied for the specific Mukhanov–Sasaki peak used here. Because A_χ enters C_χ and therefore C_V, any O(1) uncertainty in the kernel (spin sum, constraint normalization, or time integral) rescales the entire predicted h^{2}Ω_pk_GW and the inferred mass. A short sensitivity scan or an explicit statement of the uncertainty inherited from the cited production calculation is needed before the closure can be treated as quantitatively robust.","section":null}],"minor_comments":[{"comment":"Table I is useful but the final column is somewhat repetitive; a single sentence in the introduction already states the same point.","section":null},{"comment":"Fig. 2 (right panel) labels a “local quadratic support proxy” without defining the proxy function; a one-line formula would help the reader.","section":null},{"comment":"Several arXiv preprints in the reference list carry future dates (2026); these should be updated or flagged as “in preparation” if they remain unpublished.","section":null},{"comment":"Notation for the shape factors switches between R_3 and R3, and between C_GW and CGW; a uniform choice would improve readability.","section":null},{"comment":"Eq. (21) gives a convenient analytic fit for C_LN_GW(Δ), but the maximum fractional deviation of 1.6 % is stated without showing the underlying scan; a brief appendix plot would strengthen the claim.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid, self-contained application of existing conformal-production and SIGW kernels. The high-reheating assumption is the only load-bearing external condition; once it is quantified or bounded, the paper is suitable for a specialized cosmology journal. Novelty is incremental rather than transformative, but the closure relation itself is clean and the numerical checks are careful."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is the abundance-normalized closure (Eq. 23 / their Eq. 1). Once Ω_χ h² fixes A_ζ through the cubic moment of P_ζ, the quadratic SIGW peak is fixed—no free scalar normalization left. That is a real, usable step relative to the Garani et al. production papers and the usual SIGW literature. The algebra is clean, the Mukhanov–Sasaki benchmark is explicit (R_3 = 1.6287, C_GW, convergence table <7 %), and they give the mass-reach and mass-inference formulae that experimental groups can actually plug into. Consistency checks (Gaussian PBH tail, dark radiation, crossover mass generation) are done at the right level of seriousness.\n\nSoft spots are the ones the paper itself flags. The radiation-era kernel needs T_reh ≳ 10^14 GeV; delayed reheating multiplies everything by an S_reh factor they leave free. A_χ is taken from the external production calculation, the inflationary feature is engineered (Gaussian dip in ε), and there is no public code. None of that breaks the central relation under the stated assumptions; it just means the prediction is conditional on high-scale reheating and the production coefficient. Shape factors measured from the same spectrum introduce mild self-consistency, not circularity.\n\nThis is for people working on SIGW templates, non-thermal superheavy DM, or MHz detector planning. It does not solve a major open problem, but it removes a free parameter and gives a falsifiable joint prediction. I would send it to referees; the math and the checks are good enough to deserve that time. Worth a careful look if you care about the MHz band or conformal production.","headline":"Clean algebraic closure that removes free scalar amplitude from SIGW templates and ties MHz peaks to conformal-fermion mass; solid under its stated assumptions.","tokens_in":19713,"tokens_out":451,"would_cite":true,"duration_ms":4584,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Dark-matter abundance fixes the amplitude of a MHz gravitational-wave signal from the same primordial peak.","keywords":["conformal dark matter","scalar-induced gravitational waves","MHz gravitational waves","primordial curvature spectrum","superheavy fermions","inflationary small-scale features","relic abundance normalization"],"falsifier":"A laboratory MHz stochastic search that either (a) sets a limit stronger than the mass-reach curve implied by the observed dark-matter density at the measured peak frequency and width, or (b) detects a peak whose amplitude, frequency, and width cannot be reproduced by any single curvature spectrum that also yields the correct relic density.","tokens_in":19634,"feed_emoji":"📡","tokens_out":641,"duration_ms":6467,"temperature":0.7,"pith_summary":"A localized bump in the primordial curvature spectrum can create two relics at once: superheavy conformal-fermion dark matter produced gravitationally from the cubic moment of the spectrum, and a stochastic gravitational-wave background induced at second order from its quadratic convolution. The observed dark-matter density therefore fixes the overall scalar normalization, so the usual free amplitude in scalar-induced gravitational-wave templates is removed. The remaining peak height, frequency, and width of the MHz signal are then predicted from the dark-matter mass, the peak scale, and the spectral shape of one primordial feature. A concrete single-field inflationary evolution produces a broad laboratory-band signal that stays safely below the Gaussian primordial-black-hole threshold. A null high-frequency search becomes a lower bound on the fermion mass; a detection must simultaneously match the relic density, peak frequency, amplitude, and width.","feed_headline":"Dark matter locks the strength of a MHz gravity-wave signal","feed_subtitle":"One primordial bump sets both the fermion relic and the laboratory-band wave amplitude","key_machinery":"Abundance-normalized closure relation: the cubic moment that sets the conformal-fermion yield is eliminated against the quadratic radiation-era convolution that sets the induced tensor amplitude, yielding an explicit formula for the peak gravitational-wave density in terms of dark-matter mass, peak frequency, width, and shape factors only.","core_discovery":"Once the dark-matter abundance is used to fix the integrated area of a localized primordial curvature peak through its cubic moment, the peak amplitude of the scalar-induced gravitational-wave spectrum is completely determined by the same peak’s mass scale, frequency, width, and shape factors. There is no free scalar normalization left in the tensor prediction, so a MHz laboratory signal is tied directly to the conformal-fermion mass.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Conformal dark matter sets MHz gravitational-wave strength","Primordial peak fixes DM relic and lab MHz wave amplitude","DM cubic moment locks scalar-induced GW signal","One feature ties fermion mass to MHz gravity-wave peak","Null MHz search bounds conformal-fermion mass"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The calculation assumes the MHz modes re-enter the horizon only after the universe has already reheated into radiation, so the standard radiation-era transfer functions apply; a much later reheating would replace those kernels and change the predicted amplitude.","fun_headline_variants_meta":{"raw":{"variants":["Conformal dark matter sets MHz gravitational-wave strength","Primordial peak fixes DM relic and lab MHz wave amplitude","DM cubic moment locks scalar-induced GW signal","One feature ties fermion mass to MHz gravity-wave peak","Null MHz search bounds conformal-fermion mass"]},"model":"grok-4.5","effort":"low","cost_usd":0.006124,"raw_usage":{"total_tokens":1542,"prompt_tokens":734,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":61240000,"prompt_tokens_details":{"text_tokens":734,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":726,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":734,"tokens_out":82,"duration_ms":6449,"temperature":1.0,"reasoning_tokens":726,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T10:30:34.104561+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"A laboratory MHz stochastic search that either (a) sets a limit stronger than the mass-reach curve implied by the observed dark-matter density at the measured peak frequency and width, or (b) detects a peak whose amplitude, frequency, and width cannot be reproduced by any single curvature spectrum that also yields the correct relic density.","supporting_citations":[],"review_version":1}