{"id":"01914d4f-3f2d-4da8-ab19-4bc127a9c340","arxiv_id":"2411.15817","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Closed-form entropy formulas, parameter monotonicity results, discrete-distribution entropy convergence limits, and a Gaussian-vector determinant bound are derived for common parametric distributions.","lead":"This paper derives and collects closed-form formulas for Shannon, Rényi, Tsallis, Sharma-Mittal and other entropy measures on five common probability distributions, and studies how these entropies depend on distribution parameters. It is a reference-style mathematics paper for anyone who needs exact entropy expressions rather than numerical estimates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sharma–Mittal closed forms in Prop. 3.2(6) and 3.6(6) invert the Gamma-function ratio; the reference-formula claim fails unless corrected.","rationale":"I verified the reader's numerical counterexample: as printed, the gamma Sharma–Mittal formula contains the Gamma ratio upside down, and the chi-squared formula inherits the same error. This is exactly the kind of simplification failure that undermines the paper's central claim to be a dependable formula reference. I also found a separate missing 2^{1−β} factor in the Laplace Sharma–Mittal formula, which reinforces the same concern rather than replacing it. The remaining analytical results—Poisson monotonicity, gamma parameter dependence, exponential entropy behaviour, dominated-convergence arguments in Section 5, and the Gaussian determinant extremal discussion—do not rely on the defective formulas and appear internally coherent. The errors are localized and correctable, so conditional acceptance remains the appropriate verdict; the reader's conclusion stands, although the specific gamma/chi-squared inversion is the primary load-bearing defect.","tokens_in":25266,"tokens_out":29154,"duration_ms":237459,"concrete_test":"Recompute the Sharma–Mittal entries directly from definition (2.5) using the displayed integrals of Section 3. For Proposition 3.2(6), test (µ, λ, α, β) = (2, 1, 2, 3): direct integration gives ∫p^α = 1/4, q = 2, hence H_SM = 15/32, whereas the printed expression gives 255/512; flipping the Gamma ratio to Γ^{(1−β)/(1−α)}(α(µ−1)+1) / Γ^{α(1−β)/(1−α)}(µ) restores agreement for all parameters. For Proposition 3.9(6), test (λ, α, β) = (2, 2, 3): direct integration gives J = 1/2 and H_SM = 3/8, while the printed expression gives 0; inserting the missing 2^{1−β} factor restores agreement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central value is its promise that every closed form in Section 3 correctly evaluates the defining integral. That promise fails for the Sharma–Mittal entries. Proposition 3.2(6), as printed, contains the ratio Γ^{α(1−β)/(1−α)}(µ) / Γ^{(1−β)/(1−α)}(α(µ−1)+1). Substituting the correct integral J = λ^{α−1} α^{−α(µ−1)−1} Γ(α(µ−1)+1) / Γ^α(µ), obtained in Proposition 3.2(2), into definition (2.5) with q = (1−β)/(1−α) gives the reciprocal placement: Γ(α(µ−1)+1)^q in the numerator and Γ(µ)^{αq} in the denominator. For µ=2, λ=1, α=2, β=3, the defining formula yields H_SM = 15/32, while the printed expression yields 255/512. Proposition 3.6(6) inherits the same inversion after substituting λ=1/2, µ=ν/2. A second independent simplification error appears in Proposition 3.9(6): since J^{(La)} = λ^{α−1}/(2^{α−1}α), the Sharma–Mittal expression must contain the factor 2^{1−β}, which the printed formula omits. These errors are localized and do not affect the main theorems of Sections 4–6, but they invalidate the paper's advertised role as a reliable closed-form reference until corrected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines six entropy functionals (Shannon, Rényi, generalized Rényi, Tsallis, generalized Rényi with two parameters, and Sharma–Mittal), the Kullback–Leibler divergence, and a modified Shannon entropy, and derives closed-form expressions for gamma, exponential, chi-squared, Laplace, and log-normal distributions. It then studies parameter monotonicity for Poisson, gamma, and exponential entropies, proves convergence of Shannon entropies under negative-binomial-to-logarithmic and binomial-to-Poisson distributional limits, and gives extremal results for the determinant of covariance matrices of Gaussian vectors with fixed variances. The intended contribution is a reliable reference table of entropy formulas together with new monotonicity, robustness, and extremal results.","tokens_in":25579,"tokens_out":8172,"duration_ms":68832,"significance":"If the Section 3 table were correct, the paper would serve as a useful formula reference for information-theoretic and stochastic-modelling applications. The paper does contain genuine self-contained contributions: a new proof that Poisson entropy increases in its intensity, including an exact evaluation of the limiting derivative; a monotonicity proof for the two-parameter gamma Shannon entropy; dominated-convergence arguments for entropy limits; and clean determinant-extremal arguments for Gaussian vectors in Section 6. The Poisson proof is not circular and goes beyond the earlier argument in [5]. However, the paper's central formula-table claim is currently undermined by three algebraic errors in the Sharma–Mittal entries in Section 3, and the proof of Proposition 5.4 is not complete as printed.","major_comments":[{"comment":"The printed Sharma–Mittal formula for the gamma distribution inverts the Gamma-function ratio. Using the formula for J = ∫ p^α(x)dx obtained in Proposition 3.2(2) and the definition (2.5) with q = (1−β)/(1−α), the correct expression is H_SM = (1/(1−β))[ λ^{β−1} α^{(α(1−μ)−1)(1−β)/(1−α)} Γ^{q}(α(μ−1)+1)/Γ^{αq}(μ) − 1 ]. The displayed formula in the paper has Γ^{αq}(μ) in the numerator and Γ^{q}(α(μ−1)+1) in the denominator. Direct evaluation of the defining integral for μ=2, λ=1, α=2, β=3 gives H_SM = 15/32, while the printed expression yields 255/512. This error directly contradicts the paper's advertised role as a reliable closed-form reference.","section":"Section 3.1, Proposition 3.2(6)"},{"comment":"The chi-squared Sharma–Mittal formula inherits the same inverted Gamma-function ratio after substituting λ=1/2 and μ=ν/2 into the gamma formula. The corrected expression should contain Γ^{q}(α(ν/2−1)+1) divided by Γ^{αq}(ν/2), not the reciprocal placement displayed in the paper. This entry is therefore also quantitatively incorrect for generic parameter choices.","section":"Section 3.3, Proposition 3.6(6)"},{"comment":"The Laplace Sharma–Mittal formula omits the factor 2^{1−β}. With J^{(La)} = λ^{α−1}/(2^{α−1}α) and q=(1−β)/(1−α), one has J^q = λ^{β−1} 2^{1−β} α^{(1−β)/(α−1)}. The printed formula lacks the factor 2^{1−β}, so it is wrong for every β ≠ 1, including the numerical check β=3.","section":"Section 3.4, Proposition 3.9(6)"},{"comment":"The proof of Proposition 5.4 is fragmented as printed. The opening paragraph states that both lower and upper bounds for P_{p,r}(k) are needed, but the displayed text after the first paragraph moves directly to an upper estimate, and the paragraph beginning 'Also, for 0<r<1...' together with the final domination series appears after the proof's QED marker. No explicit lower bound for P_{p,r}(k) is displayed. The proof should be rewritten as one continuous argument with the lower and upper bounds stated explicitly and the uniform-in-r domination shown in full.","section":"Section 5.1, Proposition 5.4"}],"minor_comments":[{"comment":"In the proof of Proposition 3.2, after part 3, the sentence 'The statements 4)–6) immediately follow from (3.1)' cannot be correct, because (3.1) is the Shannon entropy formula. The Tsallis and Sharma–Mittal formulas follow from the integral J computed in part 2 (or from the corresponding expressions obtained there), so this cross-reference should be corrected.","section":"Proof of Proposition 3.2"},{"comment":"Several indicator functions are rendered as '/BD(0,+∞)(x)' or similar artifacts, e.g., in Definition 3.1; the typesetting should be cleaned to use standard indicator notation such as 1_{(0,+∞)}(x).","section":"Typesetting"},{"comment":"Reference [17] contains corrupted text ('Witu/suppress la') and should be repaired; the correct name appears to be Wituła. A few other bibliographic entries also have minor capitalization or formatting issues.","section":"References"},{"comment":"The proof of Lemma 4.2 applies l'Hôpital's rule and term-by-term differentiation of a power series without explicitly justifying the interchange; this is standard for exponential series but should be stated for completeness.","section":"Lemma 4.2"}],"recommendation":"major_revision","confidential_remarks":"The errors in Section 3 are localized and do not appear to affect the main theorems of Sections 4–6, but they strike at the paper's core selling point as a formula reference. I recommend that the editor ask the authors to verify every closed form in Section 3, either by a direct substitution of the defining integrals or by a numerical validation table, and to rewrite the proof of Proposition 5.4 as a continuous argument. The Poisson monotonicity proof and the Gaussian determinant discussion are the strongest parts of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the paper is a useful, mostly correct reference for entropy formulas, but its advertised reliability currently fails in exactly the place the stress-test flags: the Sharma-Mittal closed forms for gamma, chi-squared, and Laplace are wrong as printed. Those are localized algebra slips, not load-bearing for Sections 4–6, but they invalidate the paper's central claim until fixed.\n\nWhat is genuinely new: explicit Tsallis, generalized Renyi, and Sharma-Mittal expressions for gamma, chi-squared, and lognormal that I haven't seen assembled elsewhere; a new, concise proof that Poisson entropy increases in λ; and two convergence results (binomial-to-Poisson, negative-binomial-to-logarithmic) proved by finding summable dominants. The proof of Poisson monotonicity is neat and the limit argument in Lemma 4.2 is correct. The lognormal derivations are clean and the KL divergence formulas are standard but correctly re-derived.\n\nThe soft spots are real but countable. I checked the stress-test's arithmetic: for gamma with µ=2, λ=1, α=2, β=3, the defining integral gives H_SM = 15/32 while Proposition 3.2(6) prints 255/512. The gamma-function ratio is inverted. Proposition 3.6(6) inherits the same inversion after the chi-squared substitution, and Proposition 3.9(6) drops a 2^{1-β} factor that J^{(La)} forces. Any reader using these formulas will get wrong numbers. Also, the proof of Proposition 5.4 is not complete as printed: after bounding 1/(Γ(r)(1-p^r)) it jumps to the final bound on P log(1/P) without showing the lower bound side; the displayed argument is a fragment. This is likely repairable but as it stands it is not a proof. The convergence theorem may still be true, but a referee should ask for the missing details.\n\nThe rest is solid. Section 4's dependence results on gamma entropy are correct, Section 6's determinant extremes are straightforward Hadamard applications, and the authors are honest about the unproved fGn hypothesis. The self-citations provide context rather than carrying the derivations.\n\nWho this is for: applied information theorists and statistical physicists who want a single formula sheet. It is not a breakthrough, but it is a service. With the three corrections and the completed proof, it deserves publication. I would send it to a serious referee rather than desk-reject; the referee's job is narrow and mechanical.","headline":"Useful entropy formula reference with correctable algebra slips in the Sharma-Mittal entries; send to referee after requiring the fixes.","tokens_in":26100,"tokens_out":2513,"would_cite":false,"duration_ms":22366,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A17","62B10","60E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives closed-form entropy formulas for six entropy families on five common distributions and proves their parameter monotonicity, convergence, and extremal properties.","keywords":["Shannon entropy","Rényi entropy","generalized Rényi entropy","Tsallis entropy","Sharma–Mittal entropy","Kullback–Leibler divergence","gamma distribution","Hadamard inequality"],"falsifier":"For the gamma distribution with $\\mu=2$, $\\lambda=1$, $\\alpha=2$, $\\beta=3$, compute the Sharma–Mittal entropy numerically from the defining integral and compare it with Proposition 3.2(6): the integral gives $15/32$ and the printed formula gives $255/512$, so this single substitution is enough to decide whether that closed form is correct.","tokens_in":25073,"feed_emoji":"📊","tokens_out":13509,"duration_ms":105537,"temperature":0.7,"pith_summary":"The paper sets out to turn entropy computation into a formula lookup: for the gamma, exponential, chi-squared, Laplace and log-normal distributions it derives closed-form expressions for the six entropy functionals (Shannon, Rényi, the two generalized Rényi versions, Tsallis and Sharma–Mittal), for the modified Shannon entropy, and for the Kullback–Leibler divergence, all written directly in terms of the distribution parameters. It then reads parameter dependence off those formulas, giving a new proof that Poisson entropy increases with intensity, a proof that gamma Shannon entropy decreases in the rate $\\lambda$ and increases in the shape $\\mu$, and a full $\\lambda$-profile for every exponential entropy. It also proves two convergence results: the conditional (zero-truncated) negative binomial entropy converges to the logarithmic entropy as $r\\to0$, and the binomial entropy converges to the Poisson entropy when $np_n\\to\\lambda$. Finally, for Gaussian vectors with fixed variances it uses Hadamard's determinant inequality to locate the maximum and minimum of the Shannon entropy and to say which covariance structures realize them. If the formulas are correct, the paper gives a reference table that replaces numerical integration in applications using these entropy measures.","feed_headline":"Six entropies, five distributions, all in closed form","feed_subtitle":"Explicit entropy formulas for common probability laws, with parameter trends and convergence limits.","key_machinery":"The load-bearing machinery is a chain of explicit integral evaluations: each density is substituted into the defining entropy integrals, and the resulting Gamma integrals $\\int_0^\\infty x^{a-1}e^{-x}\\,dx=\\Gamma(a)$ and their differentiated versions produce closed forms in $\\Gamma$, digamma $\\psi$, and trigamma $\\psi'$. Parameter monotonicity is then read from estimates such as $\\psi'(z)=\\sum_{n=0}^\\infty (z+n)^{-2}$ and the bound $\\psi'(z)<1/z^2+1/z$ used in Theorem 4.4. For the discrete convergence theorems, the mechanism is dominated convergence: the terms $p_n(k)\\log(1/p_n(k))$ are bounded by summable sequences built from elementary binomial and negative-binomial inequalities. For the Gaussian extremal problem, the mechanism is Hadamard's inequality $\\det A\\le\\prod_i a_{ii}$ for positive semidefinite matrices with fixed diagonal, with equality only for diagonal matrices; inserted into the Gaussian vector entropy formula $\\frac n2(1+\\log2\\pi)+\\frac12\\log\\det A$, it yields the maximum at independent components and the minimum at rank-one covariance.","core_discovery":"On the paper's own terms, the central discovery is that all six entropy functionals plus the modified Shannon entropy and the Kullback–Leibler divergence are explicitly evaluable for the five named distribution families, and that the resulting expressions support exact statements about monotonicity, limits and extremes. The gamma calculations are representative: they express the Shannon entropy as $H_{\\mathrm{SH}}^{(\\mathrm{gamma})}(\\lambda,\\mu)=-\\log\\lambda+\\log\\Gamma(\\mu)+\\mu-\\psi(\\mu)(\\mu-1)$, and the other gamma entropies through Gamma-function ratios; the exponential and chi-squared families then follow by specialization $\\mu=1$ and $\\lambda=1/2,\\mu=\\nu/2$, while the log-normal and Laplace cases are computed from moment formulas. The paper further claims that the gamma Shannon entropy decreases in $\\lambda$ and increases in $\\mu$ over the whole parameter range, that each exponential entropy has a precisely described $\\lambda$-behaviour with identified zero crossings, that Shannon entropies converge along the two discrete distribution limits, and that among centered Gaussian vectors with fixed variances the entropy is maximized exactly by uncorrelated (independent) components and minimized by perfectly correlated components.","pith_inferences":["The same Gamma-integral technique should extend to other densities of the form $x^{a-1}e^{-bx}$ times a polynomial in $\\log x$, such as Weibull or inverse-gamma laws; the paper does not compute those, but nothing in the method appears special to the five chosen families.","The two convergence theorems suggest Shannon entropy is continuous along the exhibited distributional limits, yet the paper does not prove a general continuity principle; finding the broadest class of discrete limits with entropy convergence is a natural next step.","The Gaussian vector extremal result, combined with the fractional Gaussian noise discussion, suggests a finite-dimensional sandwich for the entropy of Gaussian processes with prescribed marginal variances, though the paper only treats fixed-dimension vectors."],"forward_implications":["For any of the five distributions, Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropies can be evaluated by substituting parameters into a closed formula, eliminating numerical quadrature in those cases.","The gamma monotonicity theorem fixes the sign of every partial derivative: increasing the rate $\\lambda$ always lowers entropy and increasing the shape $\\mu$ always raises it, for all $\\lambda,\\mu>0$.","The exponential results establish a strict ordering of entropy scales, with $H_R>H_{SH}$ for $\\alpha<1$ and $H_R<H_{SH}$ for $\\alpha>1$, and locate the exact zero crossings of Tsallis and Sharma–Mittal entropies.","The convergence theorems justify approximating binomial entropy by Poisson entropy and zero-truncated negative binomial entropy by logarithmic entropy, with explicit dominated-convergence bounds rather than numerical observation.","For Gaussian vectors with fixed variances, the entropy range is exactly the interval from the rank-one correlation case (determinant zero) to the diagonal case (determinant $\\prod_i a_{ii}$), so every covariance matrix's entropy lies between these two computable extremes."],"supporting_citations":[{"why":"Supplies the earlier Gaussian entropy calculations and the parameter-dependence approach that this paper extends to other distribution families.","marker":"[12]"},{"why":"States the Poisson entropy monotonicity theorem for which the paper gives a new, more concise proof via series differentiation.","marker":"[5]"},{"why":"Gives the Gaussian vector entropy formula $\\frac n2(1+\\log2\\pi)+\\frac12\\log\\det A$ that defines the extremal problem in Section 6.","marker":"[19]"},{"why":"Provides Hadamard's determinant inequality with its equality condition, the core mechanism of the Gaussian vector extremes.","marker":"[17]"},{"why":"Supplies the digamma and trigamma series representations and Gamma-function identities used in the gamma and chi-squared computations.","marker":"[9]"},{"why":"Proves non-degeneracy of fractional Gaussian noise covariance matrices, used to assert uniqueness of the determinant minimum in Remark 6.3.","marker":"[2]"},{"why":"Defines the fractional Gaussian noise entropy problem and the unresolved monotonicity hypothesis that motivates the Gaussian vector determinant analysis.","marker":"[13]"}],"fun_headline_variants":["Six entropies, five distributions, all in closed form","Explicit entropy formulas for gamma, chi-squared, and more","Shannon entropy extremes for Gaussian vectors identified","Parameter dependence of six entropy measures mapped","Closed-form entropy expressions for five probability laws"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything in Sections 4 and 5 that is built on the Section 3 formulas assumes every displayed closed-form expression is a correct simplification of its defining integral; if any one formula mis-simplifies a Gamma-function ratio, the monotonicity and convergence statements that rely on it inherit the error.","fun_headline_variants_meta":{"raw":{"variants":["Six entropies, five distributions, all in closed form","Explicit entropy formulas for gamma, chi-squared, and more","Shannon entropy extremes for Gaussian vectors identified","Parameter dependence of six entropy measures mapped","Closed-form entropy expressions for five probability laws"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1365,"prompt_tokens":892,"completion_tokens":473,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":414}},"tokens_in":508,"tokens_out":473,"duration_ms":5294,"temperature":1.0,"reasoning_tokens":414,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:53:20.416176+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the gamma distribution with $\\mu=2$, $\\lambda=1$, $\\alpha=2$, $\\beta=3$, compute the Sharma–Mittal entropy numerically from the defining integral and compare it with Proposition 3.2(6): the integral gives $15/32$ and the printed formula gives $255/512$, so this single substitution is enough to decide whether that closed form is correct.","supporting_citations":[{"cited_title":"Malyarenko, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the earlier Gaussian entropy calculations and the parameter-dependence approach that this paper extends to other distribution families."},{"cited_title":"Braiman, A","cited_arxiv_id":null,"evidence_quote":"States the Poisson entropy monotonicity theorem for which the paper gives a new, more concise proof via series differentiation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Gaussian vector entropy formula $\\frac n2(1+\\log2\\pi)+\\frac12\\log\\det A$ that defines the extremal problem in Section 6."},{"cited_title":"R´ o˙ za´ nski, R","cited_arxiv_id":null,"evidence_quote":"Provides Hadamard's determinant inequality with its equality condition, the core mechanism of the Gaussian vector extremes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the digamma and trigamma series representations and Gamma-function identities used in the gamma and chi-squared computations."},{"cited_title":"Banna, Y","cited_arxiv_id":null,"evidence_quote":"Proves non-degeneracy of fractional Gaussian noise covariance matrices, used to assert uniqueness of the determinant minimum in Remark 6.3."},{"cited_title":"Malyarenko, Y","cited_arxiv_id":null,"evidence_quote":"Defines the fractional Gaussian noise entropy problem and the unresolved monotonicity hypothesis that motivates the Gaussian vector determinant analysis."}],"review_version":1}