{"id":"c430bcb5-c657-4504-97fa-265b4dcf9dcc","arxiv_id":"2511.00644","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The Principle of Minimal Heating fixes the otherwise-arbitrary smearing profile: Gaussian for GRW, a quartic-then-cutoff profile for CSL, and a parabola-then-cutoff profile for Diósi–Penrose and Tilloy–Diósi models.","lead":"This paper proposes a selection rule for the 'smearing' function that keeps spontaneous-collapse and classical-quantum-gravity models from heating matter infinitely fast: use the smearing that minimizes the heating rate. Applied to the standard models (GRW, CSL, Diósi–Penrose, Tilloy–Diósi), the rule returns a Gaussian only for GRW, and non-Gaussian profiles that cut predicted heating by up to 47% for the others.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"PMH's unique optimizers depend on the arbitrary constraint set Eq. (3); the paper concedes other constraints change optimizers, so the one-parameter TD claim is not fully determined.","rationale":"I verified the central derivations: the Gaussian optimizer for GRW, the quartic-cutoff for CSL, and the parabolic-cutoff for DP/TD are correct under the stated constraints, and the 47%/22% heating penalties are reproduced by direct integration. The reader's weakest assumption is indeed the load-bearing one: the entire optimization framework is conditional on the constraint set (3). The paper's own Discussion explicitly acknowledges that different constraints would change the optimizers and that a broader space of correlators is unexplored. This makes the concern genuine, but it is a limitation rather than an internal inconsistency or a calculational error. The refutation logic in the abstract is also slightly overstated because it omits the qualifier that refutation must be heating-based and that refuting the optimal variant only rules out variants with the same constraint choice; the Conclusions hedge with 'arguably.' However, these issues do not overturn the paper's main mathematical results; they justify the CONDITIONAL verdict already given. No new fatal flaw was found, so the verdict should remain UNCHANGED.","tokens_in":17584,"tokens_out":11104,"duration_ms":123472,"concrete_test":"Re-run the CSL and DP optimization problems (Eqs. S.2.1 and S.3.0.1) under alternative natural width constraints: (a) fixed full width at half maximum of g, (b) fixed support radius, (c) fixed variance of √g instead of g, and (d) signed g with fixed variance. Compare the resulting profiles and the Gaussian-vs-optimal heating penalties against Table I. If the optimizers remain proportional to the reported compact-support profiles up to a global rescaling of r_C, the PMH is robust; if the profiles or penalties change materially, the paper's unique-prediction claim is an artifact of the particular constraint set (3).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that the Principle of Minimal Heating uniquely fixes the smearing profiles and reduces the Tilloy–Diósi model to a single parameter r_C—is mathematically correct only for the specific constraint set in Eq. (3): g ≥ 0, ∫g = 1, ∫xg = 0, and fixed variance ∫x²g = 3r_C². The paper itself concedes in the Discussion that \"different constraints would, in most cases, change the optimizer(s).\" This is not a minor caveat: the PMH is a stipulated selection principle, and the constraint set is the only thing that makes the optimization well-posed. Equally natural choices—fixing the full width at half maximum, fixing the support radius, fixing the variance of √g rather than g, or allowing signed smearings—would generally yield different profiles and different heating-rate penalties. The Discussion even raises the larger space of directly shaping the correlator γ_C, which could make the smearing optimization evaporate. Thus the PMH does not fully resolve the original arbitrariness; it transfers that arbitrariness to the choice of constraints. The headline claims about the 47%/22% Gaussian penalties and the single-parameter TD model are all conditional on this unmotivated choice. The calculation of the optimizers themselves is sound; the weakness is at the level of the principle's uniqueness, not the algebra.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a 'Principle of Minimal Heating' (PMH): for a fixed smearing length r_C, among positive, normalized, centered smearing distributions with variance 3r_C^2, choose the distribution that minimizes the model's heating rate. The principle is applied to the GRW, CSL, and DP collapse models and to the Tilloy–Diósi (TD) hybrid classical-quantum gravity model. The reported optimizers are Gaussian for GRW, the quartic bubble g(r)∝(9r_C^2-r^2)^2 for CSL, and the parabolic bubble g(r)∝(7r_C^2-r^2) for DP and for the matched-smearing TD model. Replacing the optimal profile by a Gaussian increases the heating rate by 47% for CSL and 22% for DP at fixed variance. The paper argues that this removes an otherwise ad hoc modeling choice and reduces the TD model to a single free parameter r_C.","tokens_in":17890,"tokens_out":12381,"duration_ms":112719,"significance":"If the results are correct, the paper gives a concrete, state-independent variational prescription that removes one layer of arbitrariness in collapse and hybrid models, and it provides closed-form optimizers that can be used in future phenomenological and experimental work. The central functional evaluations (the CSL Euler–Lagrange solution, the DP identity I_DP[g]=π∫g^2, the radii R=3r_C and R=√7 r_C, and the 47%/22% Gaussian penalties) reproduce, and the calculations are transparent enough to be independently checked. The contribution is useful even though the PMH is a stipulated selection principle rather than a derived consequence of the models.","major_comments":[{"comment":"The paper's headline claims—'unique optimizer', 'entirely determined by only one free parameter', and 'if experimentally refuted, would strongly disfavor all variants'—are stated in the Abstract without the qualification that they hold only within the constraint set of Eq. (3): g≥0, ∫g=1, ∫xg=0, ∫x^2g=3r_C^2. The Discussion concedes that 'different constraints would, in most cases, change the optimizer(s)'. Since the PMH is a stipulated principle, the choice of constraints is part of the principle; it is not itself fixed by any physical argument. Other natural formulations (fixed support radius, fixed FWHM, fixed variance of √g, or signed smearings/correlator shaping raised in the Discussion) would generally lead to different optimizers and different heating-rate penalties. The Abstract and Conclusion should therefore say 'under constraints (3)' and 'disfavor any other version within thi","section":"Abstract; Discussion; Eq. (3)"},{"comment":"There is a concrete algebraic error in the DP derivation. With g(r)=μ/(2π)(R^2-r^2), the normalization integral gives 4π∫_0^R r^2 g(r) dr = 4μR^5/15, not μR^5/15, so the printed μ=15R^{-5} is incorrect. The correct value is μ=15/(4R^5). With the printed μ, the variance equation would give 12R^2/7, not 3R^2/7, so the printed equations are internally inconsistent. The final distribution in Table I is nevertheless correct when the proper μ is used, but the derivation as printed needs to be fixed.","section":"Appendix S3, normalization equation"},{"comment":"Equation (S.4.3.2) writes γ̃_C(k)=(Ṽ(k)/(2ℏ))|g̃_{r_G}|/|g̃_{r_C}|. Since V(x)=-G/|x|, Ṽ(k)=-4πG/k^2, this gives a negative γ̃_C, inconsistent with the positive correlator used for the DP model. The main-text Eq. (7) has the correct positive expression 2πG/(ℏk^2)|g̃_{r_G}|/|g̃_{r_C}|. Please correct the sign/absolute value in Appendix S4.3 so the two derivations agree.","section":"Appendix S4.3, Eq. (S.4.3.2); main-text Eq. (7)"}],"minor_comments":[{"comment":"The row labeled 'Gaussian increase' would be clearer as 'Percentage increase in heating rate when the Gaussian is used instead of the optimal profile'.","section":"Table I caption"},{"comment":"The sentence 'Different constraints would, in most cases, change the optimizer(s)' is important for the interpretation of the whole paper. It should appear earlier, at least in the Introduction after the statement of PMH, so readers are not misled by the unqualified uniqueness claims in the Abstract.","section":"Discussion"},{"comment":"When restoring physical units, the substitutions leading from R=√7 to the final expression in Table I are not shown. A one-line derivation would help readers verify the normalization and variance.","section":"Appendix S3"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a quantum-physics letters journal and the main mathematical results are reproducible. The required revisions are local: two concrete derivation errors in the supplementary material and a needed qualification of the constraint-dependence of the PMH in the Abstract/Conclusions. None of these appears to invalidate the final formulas, so I do not recommend rejection. The editor may wish to emphasize to the authors that the overstatement in the Abstract should be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short take: this paper makes a real contribution. It proposes a selection principle — pick the smearing profile that minimizes the heating rate for a fixed smearing length — and then carries out the variational problems correctly for GRW, CSL, DP, and the Tilloy–Diósi hybrid model. The finding that the Gaussian is optimal only for GRW, with the quartic (CSL) and parabolic (DP/TD) profiles, is new as far as I can tell, and the headline comparison numbers (47% and 22% heating penalties for using the Gaussian) are direct functional evaluations. I spot-checked the derivations in the appendices; they reproduce, including the radii R=3r_C and √7 r_C.\n\nWhat's genuinely new: the PMH closes a gap left by the Tilloy–Diósi principle of least decoherence, which fixed the noise correlator but left the smearing arbitrary. The paper's optimal profiles thus make the TD model effectively one-parameter. That is a useful step for people trying to bound or falsify these models.\n\nThe soft spots are real but not disqualifying. The optimization is only well-posed under the constraint set in Eq. (3): non-negativity, normalization, fixed variance, centering. The author explicitly concedes in the Discussion that different constraints would change the optimizers, so the uniqueness is conditional on that set. This is not a hidden flaw, but it does mean the PMH relocates rather than eliminates the original arbitrariness. If you consider signed smearings or a different width measure equally natural, the strong version of the claim is weaker than the abstract's phrasing. Relatedly, the refutation logic in the abstract — that ruling out the optimal TD variant disfavors all versions — only holds for heating-based refutation; the paper's own appendix shows rigid-body decoherence is smearing-independent. The Conclusions' 'arguably' is the honest phrasing.\n\nTwo smaller points. The state-independence of the heating rate is load-bearing; the paper notes it fails for Poissonian-type models, so the principle's scope is limited. And Appendix S3 contains a factor-of-4 inconsistency in the constraint algebra; the final profile is correct, so it looks like a typographical error, but it should be fixed.\n\nBottom line: a careful, honest paper with correct math and a useful new idea. The central caveat is the stipulated constraint set, which the author acknowledges. I'd send it to referees. A good referee should ask the author to clarify the status of the constraint choice and to align the abstract with the caveats, but the core results stand.","headline":"A genuine new selection principle with correct variational math; the uniqueness claim is conditional on the stated constraints, but the paper is honest about that and deserves refereeing.","tokens_in":18429,"tokens_out":3060,"would_cite":true,"duration_ms":31829,"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":"A new variational principle fixes the arbitrary smearing in collapse and hybrid gravitational models, selecting a unique optimal profile for each model.","keywords":["spontaneous collapse models","continuous spontaneous localization","Diósi-Penrose model","Tilloy-Diósi hybrid model","heating rate","smearing distribution","variational optimization","classical-quantum gravity"],"falsifier":"Find a positive, normalized, centered distribution with variance 3r_C² that yields a strictly lower value of the CSL or DP heating functional than the claimed optimizer — analytically or by numerical variational search — and the central claim is refuted.","tokens_in":17361,"feed_emoji":"⚛️","tokens_out":4201,"duration_ms":42127,"temperature":0.7,"pith_summary":"The paper introduces a 'Principle of Minimal Heating': among all possible smearings of the mass density operator with a given width, choose the one that makes the spontaneous heating rate smallest. This turns a previously arbitrary modeling choice into a solvable variational problem. The unique optimizers are a Gaussian for the GRW collapse model and compact-support polynomial profiles for the CSL and DP models, with the same DP profile applying to the Tilloy–Diósi hybrid classical-quantum model of Newtonian gravity. The optimal profiles cut predicted heating by 47% (CSL) and 22% (DP) relative to the usual Gaussian. For the hybrid model, combining this principle with the earlier least-decoherence principle leaves a single free parameter, the smearing length, which can be bounded from above and below.","feed_headline":"Optimal smearing found for collapse models by minimizing heating","feed_subtitle":"Gaussian wins only for GRW; new profiles cut predicted heating by 47% (CSL) and 22% (DP).","key_machinery":"The central device is a set of variational problems: minimize the functional I[√g] (GRW), I[g] (CSL), or I_DP[g] (DP/TD) under positivity, normalization, and fixed-variance constraints. The solution uses the Pólya–Szegő rearrangement inequality to show the optimum is radial and decreasing, then Lagrange multipliers to solve the one-dimensional Euler–Lagrange equations. The resulting profiles are explicit, compact-support, and unique.","core_discovery":"For collapse and hybrid master equations whose heating rate is a state-independent functional of the smearing distribution, the paper minimizes that functional under the constraints g ≥ 0, ∫g = 1, and fixed variance ∫x²g = 3r_C². The unique minimizers are: a Gaussian for GRW; g(x) = (105/(32π(3r_C)^7))(9r_C² − x²)²₊ for CSL; and g(x) = (15/(8π(√7 r_C)^5))(7r_C² − x²)₊ for DP and for the matched-smearing Tilloy–Diósi model. The Gaussian is optimal only for GRW. For the Tilloy–Diósi hybrid model with matched measurement and feedback smearings, the optimizer coincides with the DP one, so the model is determined by r_C alone and becomes experimentally falsifiable both from below (heating) and ab","pith_inferences":["If the principle is adopted as a model-selection criterion, the 47%/22% reductions in heating are modest enough that current experiments may not distinguish Gaussian from optimal smearings; higher precision is needed for a decisive test.","The result is sensitive to the constraint set: allowing signed smearings or directly shaping the noise correlator could change or eliminate the optimizers, a possibility the paper explicitly leaves open.","A natural extension is to test the same variational principle under different moment constraints (e.g., fixed fourth moment) to see whether the optimal profiles are robust.","For Poissonian-type models whose heating rate is state-dependent, the principle does not directly apply, so a generalized formulation would be needed."],"forward_implications":["For each model, experimental constraints on heating now map directly onto a single optimal smearing profile; if the optimized variant is ruled out, all smearing variants of that model are disfavored.","Published bounds based on Gaussian smearings overestimate heating by 47% for CSL and 22% for DP, so existing limits on r_C should be re-evaluated.","The Tilloy–Diósi hybrid model is reduced to one free parameter, r_C, with both upper and lower experimental bounds, making the entire model class testable.","The single-particle equivalence between GRW and CSL is an artifact of Gaussian smearing; with the optimal profiles the equivalence breaks.","For models with different measurement and feedback smearings, the heating optimization splits into two independent problems, one for each smearing."],"fun_headline_variants":["Heating minimization picks optimal smearing in collapse models","Gaussian only optimal for GRW; heating-minimized profiles for CSL, DP","Minimal heating selects unique smearing profiles for collapse models","Gaussian optimal only for GRW when heating is minimized","Heating-minimized smearing fixes Tilloy-Diosi to one parameter"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The unique optimizers and the one-parameter Tilloy–Diósi claim rest on the specific constraints of non-negativity, normalization, and fixed variance (plus centering); the paper itself notes that different constraints would, in most cases, change the optimizers.","fun_headline_variants_meta":{"raw":{"variants":["Heating minimization picks optimal smearing in collapse models","Gaussian only optimal for GRW; heating-minimized profiles for CSL, DP","Minimal heating selects unique smearing profiles for collapse models","Gaussian optimal only for GRW when heating is minimized","Heating-minimized smearing fixes Tilloy-Diosi to one parameter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000742,"raw_usage":{"total_tokens":3196,"prompt_tokens":844,"completion_tokens":2352,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":2260}},"tokens_in":588,"tokens_out":2352,"duration_ms":16682,"temperature":1.0,"reasoning_tokens":2260,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T00:31:12.668550+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a positive, normalized, centered distribution with variance 3r_C² that yields a strictly lower value of the CSL or DP heating functional than the claimed optimizer — analytically or by numerical variational search — and the central claim is refuted.","supporting_citations":[],"review_version":1}