{"id":"f1f1973b-0c1a-4027-80e3-343cc93a882e","arxiv_id":"2507.11591","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"Analytic enhancement factors convert a least-squares covariance matrix into 1-sigma and 2-sigma uncertainty bands for optical potentials and cross sections, applied to 8Li + 58Ni elastic scattering; the bands' statistical calibration is, however, incorrect.","lead":"This paper offers a shortcut for turning ordinary chi-square fits of nuclear scattering data into uncertainty bands for optical potentials and cross sections, using analytic enhancement factors instead of expensive Monte Carlo sampling. Applied to 8Li scattering on 58Ni, the shortcut finds that the real part of the potential is poorly constrained and that one of the four measured energies is inconsistent with the others.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (5)/(A26) treats the chi-square goodness-of-fit law as a likelihood, so every p-value and 1σ/2σ band label is miscalibrated; the exact normal-location problem shows the recipe's '1σ' interval contains a large multiple of 68% posterior mass.","rationale":"The reader's weakest_assumption and my stress-test converge on the same load-bearing point: Appendix A.3, specifically Eq. (A26), identifies P(H|Γ[b]) with the cumulative chi-square distribution of the minimized goodness-of-fit statistic. This is not the likelihood of the data under the model parameters, and feeding it through Bayes' theorem produces a posterior (A33) that is not proportional to exp(−χ2(a)/2), vanishes at the best fit for M>1, and has no valid generative-data interpretation. All quantities derived from Eq. (5) — the p-values, the b-values for 1σ/2σ/3σ in Tables I and IV, the enhancement factors f[b], and the uncertainty bands in Figs. 4–7 — inherit this miscalibration. The paper is transparent and physically reasonable: the energy-independent 4-parameter fit to the three 'nice' energies is good, the imaginary potential is better constrained than the real potential at the strong-absorption radius, and the 28.7 MeV data are incompatible; the authors also flag the counter-intuitive narrowing with bad fits and call for a benchmark. Those features preserve the qualitative value of the paper, but they do not rescue the central statistical claim, which is the calibration of the bands. My agreement with the reader is full on the weakest assumption; the exact-normal toy is a decisive and cheap test. Since the reader already recommended REJECT and the concern lands, I keep the verdict unchanged.","tokens_in":20289,"tokens_out":6110,"duration_ms":74556,"concrete_test":"Run the recipe on the exactly solvable normal-location problem: N=10 observations y_i ~ N(μ,1), model y_i=μ (M=1), flat prior. Compute χ2_m, use Eq. (5) with L=N−M to obtain b1σ, and report {μ: χ2(μ)≤b1σ} as the recipe's 1σ band. Independently compute the exact posterior π(μ|y) and its central 68% credible interval. If the recipe interval does not contain ~68% of the posterior mass (the reader finds ~99.8% for this construction), Eq. (A26) is refuted. A useful second leg is to repeat the check on the paper's own 4-parameter 8Li+58Ni fit, comparing the posterior implied by Eq. (A33) on the sampled grid with a direct MCMC posterior built on exp(−χ2(a)/2); disagreement in the p-value/band labels confirms the error is not a linearization artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central deliverable is calibrated uncertainty bands, and the calibration rests on Eq. (5), derived in Appendix A.3. The load-bearing step is Eq. (A26): P(H|Γ[b]) = 1 − P(b,L), where P is the cumulative chi-square law of the minimized statistic with L=N−M degrees of freedom. That is the repeated-experiment distribution of a goodness-of-fit statistic, not the likelihood of the actual data under parameters in Γ[b]; the likelihood for independent normal errors is proportional to exp(−χ2(a)/2). Inserting (A26) into Bayes' theorem gives the posterior (A33), PB(a) ∝ −(dP/db)(1/S(b))P0(a), which is not the renormalized likelihood and, for the M=4 application, vanishes at the best-fit point because S(b)→0 as b→χ2_m. Consequently Eq. (5), Table I/IV thresholds, enhancement factors, and the 1σ/2σ bands derived from them are not posterior probability statements. The counter-intuitive narrowing with 'nasty' data in Sec. III.E is a symptom: the denominator P(χ2_m,L) is the tail probability of a worse fit, not a posterior model probability. In the exactly solvable problem of estimating one normal location from N observations, this recipe labels an interval containing a large multiple of 68% of the true posterior mass as '1σ' (the reader finds ~99.8%). Because the method's stated deliverable is calibrated bands, this misidentification invalidates the central claim even though the numerical application and qualitative physics conclusions may survive.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a method to derive Bayesian uncertainty bands for optical-model potentials and differential cross sections from least-squares fits. After an LS fit provides best-fit parameters and a covariance matrix, regions Γ[b] are defined by χ2m ≤ χ2(a) ≤ b; Eq. (5) assigns a p-value to each region, and either direct sampling (general approach) or an analytic enhancement factor f[b] (simplified approach) produces 1σ and 2σ bands. The method is applied to 8Li + 58Ni elastic scattering, yielding an energy-independent four-parameter fit for the 23.9, 26.1, and 30.0 MeV data, an imaginary potential that is better determined than the real potential in the 9–10 fm region, and a statement that the 28.7 MeV data are inconsistent with the model even within the 2σ bands.","tokens_in":20546,"tokens_out":4757,"duration_ms":59294,"significance":"Should the calibration in Eq. (5) be correct, the method would be valuable: it would produce p-values and uncertainty bands using only standard χ2 codes, without MCMC, and would offer an explanation for the enhancement of frequentist uncertainties relative to Bayesian ones. The paper is clearly written, gives a detailed account of the χ2-surface sampling, and compares the general and simplified approaches on real data. However, the central calibration is statistically invalid; because the p-values, enhancement factors, and all 1σ/2σ band labels derive from Eq. (5), the paper's deliverable of calibrated uncertainty bands is not achieved. The physics application and the qualitative conclusions about the 8Li + 58Ni data may survive a corrected calibration, but the central claim of the method does not.","major_comments":[{"comment":"The identification P(H|Γ[b]) = 1 − P(b,L) with L = N − M is incorrect. For a region defined by χ2m ≤ χ2(a) ≤ b, the sampling distribution of χ2(a) − χ2m is a χ2 law with M degrees of freedom (or an F distribution when the variance is estimated), whereas the χ2_{N−M} law of the minimized statistic describes the global goodness of fit at the best-fit point, not the probability of the data conditional on parameters lying in Γ[b]. Consequently Eq. (5), Table I, Table IV, and the f[b] factors in Eq. (12) are all miscalibrated. A direct calculation in the exactly solvable one-parameter normal-location problem shows that the region labeled '1σ' by this recipe contains about 99.8% of the posterior mass, not 68%.","section":"II.B, Eq. (5), and Appendix A.3, Eq. (A26)"},{"comment":"The derived posterior density PB(a) ∝ −(dP/db)(1/S(b))P0(a) vanishes at the best-fit point because S(b)→0 as b→χ2m, whereas a posterior obtained from the likelihood exp(−χ2(a)/2) would be maximal at the best fit. This internal inconsistency is a direct symptom of using the goodness-of-fit tail as a likelihood, and it means that the 'Bayesian' probability statements in the paper are not posterior probabilities in the usual sense.","section":"Appendix A.3, Eq. (A33)"},{"comment":"The reported narrowing of the uncertainty bands when 'nasty' data are added is presented as a consequence of the Bayesian treatment, but it follows from dividing by P(χ2m,L) in Eq. (5), where P(χ2m,L) is the tail probability of the observed fit statistic. That tail probability is not the model evidence P(H) in Bayes' theorem, so the claimed robustness of the method is an artifact of the miscalibrated normalization rather than a property of a sound Bayesian analysis.","section":"III.E"}],"minor_comments":[{"comment":"The table uses commas as decimal separators in several entries, for example '52,35' and '0,31731'; use decimal points throughout for consistency.","section":"Table I"},{"comment":"The keyword list contains a typo, 'maxcimum likelihood', which should be 'maximum likelihood'.","section":"Keywords"},{"comment":"The notation around Eqs. (10)–(12), especially the accents on the covariance and derivative variables, is garbled in the rendering and should be cleaned up.","section":"II.C, Eqs. (10)–(12)"},{"comment":"The paper calls for a benchmark comparison with another Bayesian method; such a benchmark would have been a useful check of Eq. (5) and should be included in any revision.","section":"IV"}],"recommendation":"reject","confidential_remarks":"The statistical flaw is in the load-bearing formula of the paper. A local edit cannot fix it: the definition of p-values, the enhancement factors, the 1σ/2σ band labels, and the discussion of robustness all depend on the incorrect identification in Eq. (5)/Eq. (A26). The physics application and the analysis of the Igo ambiguity may still be useful to the community, but the claimed deliverable of calibrated uncertainty bands is not met, and I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, read this one if you want a clear example of a promising idea sinking on a statistical misunderstanding. The authors propose a cheap way to get Bayesian uncertainty bands for optical potentials without MCMC: take the least-squares covariance matrix, scale it by an 'enhancement factor' f[b] derived from the chi-square distribution with L=N-M degrees of freedom, and claim the resulting regions have the corresponding p-values. The general approach—take extremes of observables over the χ2≤b region—is sensible as a numerical procedure, and the simplified propagation formula is a natural shortcut. What is actually new: the analytic enhancement factors and the explicit comparison of the two approaches for 8Li+58Ni. The application is also careful: energy-independent Woods-Saxon fits to three energies, the imaginary potential constrained around 9-10 fm, the real part not, and the 28.7 MeV data clearly inconsistent with the model.\n\nBut the central calibration is wrong. The load-bearing step is Eq. (A26): they identify P(H|Γ[b]) with the complementary CDF of the minimized chi-square, P(b,L). That's the repeated-experiment distribution of a goodness-of-fit statistic, not the likelihood of the data given the parameters. For normal errors the likelihood is exp(-χ2(a)/2). Substituting the wrong expression into Bayes' theorem produces the posterior in Eq. (A33), which has a Jacobian 1/S(b) and actually vanishes at the best-fit point. In the exactly solvable one-parameter normal location problem, their recipe labels an interval that contains about 99.8% of the true posterior mass as '1σ'. So the p-values, the enhancement factors, and the 1σ/2σ bands are not calibrated posterior statements.\n\nThe authors are honest about the weird consequence (worse fits give narrower bands, Sec. III.E) and even call for a benchmark. That's to their credit. But that odd behavior is a symptom of the wrong identification. The physics conclusions are probably robust qualitatively—anyone can see from the residuals that one energy doesn't fit—but the specific claim that those bands are 1σ or 2σ is not supported.\n\nWho is this paper for? Nuclear reaction theorists who want uncertainty bands without MCMC. The idea is attractive and the paper is readable, but the method as presented is not usable. It deserves a serious referee rather than desk rejection, because the flaw is subtle, the application is relevant, and a corrected version might address a real need. My recommendation: engage with it—a referee should reject in current form but might be able to guide a fix.","headline":"The paper's core calibration is wrong—the chi-square goodness-of-fit law is not a likelihood—so the uncertainty bands are not what they claim, though the application is honest and the physics conclusions may survive.","tokens_in":21241,"tokens_out":3844,"would_cite":false,"duration_ms":46259,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["24.10.Ht","25.60.Bx","25.60.-t"],"model":"deepseek-v4-flash","headline":"Standard chi-square fits can be converted into calibrated Bayesian uncertainty bands without running MCMC.","keywords":["uncertainty quantification","optical model","elastic scattering","chi-squared fits","Bayesian p-values","covariance enhancement","lithium-8","nickel-58"],"falsifier":"A direct repeated-experiment simulation with known true optical parameters would settle the calibration: generate many synthetic data sets from the fitted model, refit each one, and count how often the true parameters fall inside the recipe's 1-sigma and 2-sigma regions. If the empirical coverage deviates substantially from 31.8% and 4.5%, the identification of the region probability with the chi-square tail is the point to revisit.","tokens_in":19893,"feed_emoji":"⚛️","tokens_out":5974,"duration_ms":70379,"temperature":0.7,"pith_summary":"This paper tries to show that the usual least-squares chi-square minimization already contains everything needed to produce calibrated Bayesian uncertainty bands for optical-model potentials and cross sections, without Markov-chain samplers. The central claim is that the probability of a parameter region bounded by a chi-square contour is given by a ratio of chi-square cumulative probabilities, Eq. (5), and that the covariance matrix of parameters on that contour is the least-square covariance times an enhancement factor $f[b]=\\sqrt{(b-\\chi^2_m)/M}$, Eq. (12). Applied to $^8$Li + $^{58}$Ni elastic scattering at 23.9, 26.1, and 30.0 MeV, a single energy-independent four-parameter Woods-Saxon potential fits well ($\\chi^2/L = 0.986$), the imaginary potential is constrained near 9--10 fm while the real one is not, and the 28.7 MeV data fall outside the 2-$\\sigma$ bands. If the method is right, uncertainty bands become a routine byproduct of existing fitting codes rather than a reason to run expensive sampling.","feed_headline":"Chi-square fits yield calibrated uncertainty bands without MCMC","feed_subtitle":"One formula scales standard least-square errors into 1-sigma and 2-sigma bands, and it pinpoints which 8Li+58Ni data do not fit","key_machinery":"The central object is the contour region $\\Gamma[b]$ in the $M$-dimensional parameter space defined by $\\chi^2_m \\le \\chi^2(a) \\le b$, with $b$ fixed by the desired p-value. The identity that carries the argument is Eq. (5), $p(\\Gamma[b]|H) = P(b,L)/P(\\chi^2_m,L)$, where $P(b,L)$ is the cumulative chi-square distribution with $L = N-M$ degrees of freedom; this converts chi-square contours into Bayesian credible regions. The companion scaling law, Eq. (12), $\\sigma[b](O) = f[b]\\,\\sigma_{LS}(O)$ with $f[b] = \\sqrt{(b-\\chi^2_m)/M}$, turns the least-square covariance matrix from a standard fit into the covariance on the contour and then into uncertainty bands for any smooth observable $O(a)$, while the general approach samples the region $\\Gamma[b]$ directly and needs no parabolic approximation.","core_discovery":"On the paper's own terms, the discovery is that the difference between 'frequentist' and Bayesian uncertainties in optical-model fits is not a philosophical gap but a multiplicative factor. The least-square covariance from the Hessian at the minimum describes only the tangent of the chi-square ellipsoid; regions of higher statistical significance are bigger in covariance by the factor $f[b]$, and the p-value attached to a region $\\Gamma[b] = \\{a: \\chi^2_m \\le \\chi^2(a) \\le b\\}$ is $P(b,L)/P(\\chi^2_m,L)$ with $L = N-M$ degrees of freedom. The paper derives this from Bayes' theorem applied to the chi-square sampling distribution, and verifies in the $^8$Li + $^{58}$Ni case that the cheap 'simplified' propagation bands match the expensive 'general' sampling bands. The application also establishes a physics conclusion: elastic data around the Coulomb barrier determine $W(r)$ but not $V(r)$ in the 9--10 fm region, so microscopic real potentials cannot be validated by mere agreement, and the 28.7 MeV data are inconsistent with an energy-independent optical model even when parameter uncertainty is included.","pith_inferences":["Editorial inference: the same enhancement-factor prescription should transfer to any smooth functional of the optical parameters, so transfer cross sections, spectroscopic factors, or fusion probabilities could inherit calibrated bands from one optical-model fit plus $2M$ finite-difference runs.","Editorial inference: because Eq. (12) assumes a parabolic $\\chi^2$ surface, the simplified band should be checked against the general band whenever the Hessian-based correlation matrix is large and contours are visibly non-elliptical, as in the $V$--$a_V$ panel here; a disagreement would mean a single covariance matrix cannot summarize the parameter correlations.","Editorial inference: the paper's finding that the 28.7 MeV residuals lie outside the 2-sigma bands does not by itself tell whether that energy features systematic normalization problems, channel-coupling effects, or a genuine energy dependence, and checking those alternatives is a natural next step with the same band tools."],"forward_implications":["Standard chi-square fitting codes can be upgraded in place: the covariance matrix they already output, multiplied by $f[b]$, gives 1-sigma and 2-sigma bands for parameters and observables.","Uncertainty bands for optical potentials become narrow around the strong-absorption radius (9--10 fm here) and wide outside it, so elastic angular distributions near the rainbow angle carry most of the constraining power for the real potential.","A model that fits 'nice' data well but leaves 'nasty' data outside the 2-sigma bands can be rejected as incomplete for those data, even after parameter uncertainty is accounted for.","Bayesian credible regions shrink when the global chi-square minimum is poor, so the model must first pass a goodness-of-fit check before the bands are interpreted; the paper applies this by excluding the 28.7 MeV data from the nominal fit.","Complex reaction calculations such as CDCC and coupled-channels evaluations can inherit uncertainty bands from a small number of perturbed runs (about $2M$ calculations) using the same propagation rule."],"supporting_citations":[{"why":"Supplies the statistical definitions and the least-square covariance/ellipsoid relations that the Bayesian calibration builds on.","marker":"[12]"},{"why":"Introduces uncertainty quantification for optical model parameters and motivates the need for uncertainty bands in reaction calculations.","marker":"[7]"},{"why":"Direct comparison of Bayesian and frequentist uncertainties for nuclear reactions, which the paper explains through the enhancement factors.","marker":"[9]"},{"why":"The Igo ambiguity justifies fixing the reduced radii and fitting only depth and diffuseness parameters.","marker":"[13]"},{"why":"Provides the experimental $^8$Li + $^{58}$Ni elastic scattering data at 23.9, 26.1, 28.7, and 30.0 MeV that the fits and bands are built on.","marker":"[14]"},{"why":"Represents the type of least-square search code whose covariance output is the input to the simplified uncertainty-band method.","marker":"[16]"},{"why":"Illustrates the planned extension of the simplified approach to a CDCC calculation for $^{11}$Be + $^{197}$Au.","marker":"[17]"}],"fun_headline_variants":["No-MCMC uncertainty bands from chi-square fits","Bayesian factor scales chi-square errors into 1σ and 2σ bands","Chi-square-only uncertainty bands for 8Li+58Ni scattering","8Li+58Ni data expose model inconsistency via chi-square bands","One formula scales least-square errors to calibrated bands"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calibration stands or falls on one identification: the probability of the data given a parameter region is the repeated-experiment chi-square tail with $L = N-M$ degrees of freedom, $P(H|\\Gamma[b]) = 1 - P(b,L)$, rather than the likelihood $\\exp(-\\chi^2/2)$.","fun_headline_variants_meta":{"raw":{"variants":["No-MCMC uncertainty bands from chi-square fits","Bayesian factor scales chi-square errors into 1σ and 2σ bands","Chi-square-only uncertainty bands for 8Li+58Ni scattering","8Li+58Ni data expose model inconsistency via chi-square bands","One formula scales least-square errors to calibrated bands"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000624,"raw_usage":{"total_tokens":3301,"prompt_tokens":1139,"completion_tokens":2162,"prompt_tokens_details":{"cached_tokens":1024},"prompt_cache_hit_tokens":1024,"prompt_cache_miss_tokens":115,"completion_tokens_details":{"reasoning_tokens":2086}},"tokens_in":115,"tokens_out":2162,"duration_ms":329750,"temperature":1.0,"reasoning_tokens":2086,"cache_read_input_tokens":1024,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:16:24.034095+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct repeated-experiment simulation with known true optical parameters would settle the calibration: generate many synthetic data sets from the fitted model, refit each one, and count how often the true parameters fall inside the recipe's 1-sigma and 2-sigma regions. If the empirical coverage deviates substantially from 31.8% and 4.5%, the identification of the region probability with the chi-square tail is the point to revisit.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the statistical definitions and the least-square covariance/ellipsoid relations that the Bayesian calibration builds on."},{"cited_title":"Ascuitto and J","cited_arxiv_id":null,"evidence_quote":"Introduces uncertainty quantification for optical model parameters and motivates the need for uncertainty bands in reaction calculations."},{"cited_title":"Hagino, K","cited_arxiv_id":null,"evidence_quote":"Direct comparison of Bayesian and frequentist uncertainties for nuclear reactions, which the paper explains through the enhancement factors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Igo ambiguity justifies fixing the reduced radii and fitting only depth and diffuseness parameters."},{"cited_title":"S¨ urer, F","cited_arxiv_id":null,"evidence_quote":"Provides the experimental $^8$Li + $^{58}$Ni elastic scattering data at 23.9, 26.1, 28.7, and 30.0 MeV that the fits and bands are built on."},{"cited_title":"Cowan, Statistical Data Analysis (Oxford Science Publication, 1998)","cited_arxiv_id":null,"evidence_quote":"Represents the type of least-square search code whose covariance output is the input to the simplified uncertainty-band method."},{"cited_title":"Igo, Phys rev, Phys","cited_arxiv_id":null,"evidence_quote":"Illustrates the planned extension of the simplified approach to a CDCC calculation for $^{11}$Be + $^{197}$Au."}],"review_version":1}