{"id":"85673cc3-b7c7-4fc1-8d0f-98000c711943","arxiv_id":"2608.06232","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A common telescoping identity yields closed forms, complete asymptotics, and enveloping series for the mean absolute deviation of the four classical discrete laws.","lead":"The mean absolute deviation for the four classical discrete distributions is shown to collapse to a single point mass, and full asymptotic expansions with exact lattice corrections are derived for the Poisson, negative binomial, and hypergeometric laws. The paper gives a common telescoping proof, interprets it via size biasing, and proves that at integer means the expansions become sign-alternating series whose partial sums bracket the true value.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central asymptotic expansions (15), (21), (28) all pass through the unproved black-box Lemma 2.1 and Eq. (4) from [9]; their uniformity in the lattice displacement h is the key condition that could fail.","rationale":"After reconstructing the proofs, I find the collapse identities and the asymptotic machinery internally coherent. The four telescoping cases check out; the size-bias identities in Theorem 3.3 are correct; the prefactor/tail cancellations in Theorems 5.1 and 6.1 are exact, including the two diagonal factors in the hypergeometric case; and Proposition 4.3 is a valid Binet-kernel argument. The Poisson theorem also checks: using m=floor λ and t=1-θ gives the stated cancellation and reflection identity. The single place where the derivation depends on something not proved in this paper is the imported Lemma 2.1 and Eq. (4) in Section 2. Because the expansions are 'complete' in the strong uniform sense quoted there, and because h is not fixed but moves with r or N, the uniformity part of Lemma 2.1 is genuinely load-bearing. I agree with the reader that this is the weakest assumption. The paper's own remark that the matching across degenerate limits is left open is a scope limitation, not a defect. My verdict is unchanged: conditional acceptance pending verification of the imported lemma.","tokens_in":17272,"tokens_out":36862,"duration_ms":322271,"concrete_test":"Independently derive Lemma 2.1 from Eq. (4), checking that the remainder bound is uniform when the shifts are allowed to depend on x but stay in a fixed compact set and the scalings stay in a compact subset of (0,∞). Then test (21) numerically: take p=0.3, r=100,101,...,200 (so h takes many values), compute the exact E|X-μ| and compare with the right-hand truncations M=1,2,3; the normalized error times r^{M+1} should be bounded uniformly across the h sequence. If the bound diverges, the uniformity hypothesis behind the black-box lemma is the failure point.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper is honest that Section 2 imports Eq. (4) and Lemma 2.1 from the companion preprints [9,10]. Every complete expansion in the paper (Theorems 4.1, 5.1, 6.1) is obtained by applying Lemma 2.1 to a gamma quotient and then cancelling an elementary tail against the collapse prefactor. The delicate point is not the algebraic assembly, which I checked and it is consistent, but the uniformity claim in Lemma 2.1: the remainder after M terms is O(x^{-M-1}) uniformly for shifts in compact sets and scalings in compact subsets of (0,∞). This uniformity is essential because h=h_r (negative binomial) and h=h_N (hypergeometric) vary with the large parameter while staying in [0,1); the expansion treats h as an arbitrary compact shift. If Lemma 2.1 fails to be uniform in that sense, the Bernoulli-polynomial coefficients B_{m+1}(h) would not be legitimate to all orders and the cancellation steps in §5.1 and §6.1 would not yield the stated O(r^{-M-1}) or O(N^{-M-1}) errors. The paper gives no proof of Lemma 2.1, and [9] is not included, so the central claim is conditional on that lemma. This is a missing support, not an identified contradiction; the rest of the proof chain, including the telescoping identities, size-bias cancellations and the Binet-kernel enveloping argument, appears internally consistent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a unified treatment of the mean absolute deviation about the mean for the binomial, Poisson, negative binomial and hypergeometric laws. Section 3 proves a common telescoping identity, derives the collapse formula E|X−µ| = 2g(ν)P(ν), and recasts it in the universal size-biased form (13). Sections 4–6 derive complete asymptotic expansions for the three non-binomial laws, with Bernoulli-polynomial coefficients carrying the lattice displacement of the mean, and prove enveloping sign-alternating series at integer means through a single Binet-kernel argument. Section 8 gives careful historical attribution. The main results are Theorems 4.1, 5.1, 6.1 and the enveloping Theorems 4.4, 5.3, 6.3.","tokens_in":17529,"tokens_out":10508,"duration_ms":94366,"significance":"If the asymptotic results hold, they are a genuine advance: all-order expansions with closed Bernoulli-polynomial coefficients, exact lattice oscillation, no fitted parameters, and a common size-bias mechanism for the cancellation of elementary tails. The telescoping proof of the classical collapse formulas is clean, the universal form (13) is illuminating, and Proposition 4.3 is an elegant self-contained kernel argument that yields bracketing for all three laws at once. The paper is honest about the classical provenance of the closed forms. The main weakness is that the complete expansions all pass through Lemma 2.1 and Eq. (4), imported as black boxes from the companion preprints [9,10]; the central asymptotic claims are therefore conditional on an unverifiable supporting lemma.","major_comments":[{"comment":"The complete expansions of Theorems 4.1, 5.1 and 6.1 are all obtained by applying the imported gamma-quotient expansion Lemma 2.1 and the shifted Stirling series (4), and neither is proved in this manuscript. This is load-bearing because the required uniformity in the lattice displacement h (which varies with r or N in [0,1)) and in the scalings governed by p or (κ,η) is precisely what legitimizes the Bernoulli-polynomial coefficients to all orders. Please supply a proof of Lemma 2.1 and of the uniform remainder in Eq. (4), or replace the black-box reference by a complete and verifiable derivation; as it stands, the main asymptotic theorems remain conditional.","section":"§2 (Lemma 2.1 and Eq. (4))"},{"comment":"In the tail-cancellation step, the paper truncates the exponent of P(ν) after M series terms, replaces the truncated elementary tail by the exact log-factor via Lemma 2.2, and then multiplies by the growing prefactor 2ν/p (respectively 2g(ν)). The argument asserts that after cancellation the relative error remains O(r^{-M-1}) (respectively O(N^{-M-1})) uniformly in h, but the interaction of a prefactor of order r with the uniform remainder and with the boundedness of the factor pν/(qr) is not shown in detail. This uniformity is exactly what Lemma 2.1 must deliver, so the check should be made explicit.","section":"§5.1, Step 4 (and §6.1, Step 4)"}],"minor_comments":[{"comment":"The displayed equality 'p√2πrq' should read 'p/√(2πrq)' or 'p(2πrq)^{-1/2}'; as printed the equality to 1/√(2πσ²) is false.","section":"§5.1, proof of Theorem 5.1, Step 2"},{"comment":"The lemma is stated for shifts in compact sets, but the applications require h ∈ [0,1); to apply the lemma verbatim one should note that [0,1] is compact and that the uniformity extends to the endpoint by the stated remainder estimates.","section":"§2 (Lemma 2.1)"},{"comment":"For the binomial, write the weight with explicit parentheses as p^{-m}+(-1)^{m+1}q^{-m} to avoid a possible parse error; the text already notes that the negative-binomial entry (p/q)^m-p^m is the same bracket in reverse order, which is helpful and should remain.","section":"§7 (family table)"},{"comment":"The uniformity statement 'uniform in θ∈[0,1)' should explicitly tie θ to λ through θ={λ}; otherwise the statement can be misread as uniformity over an arbitrary independent parameter θ, which is not what the proof establishes.","section":"Theorem 4.1"},{"comment":"Since [9] and [10] are companion preprints that are not yet published, the paper should state their availability or review status; this would help readers verify the imported Lemma 2.1 and Eq. (4).","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is worth publishing if Lemma 2.1 is supplied or a fully verifiable reference is provided. The core derivation is transparent, the classical attributions are careful, and I found no circularity or fitted parameters. The main risk is the uniformity of the gamma-quotient expansion in the lattice displacement; the rest of the proof chain appears internally consistent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper completes a natural programme, and the new asymptotic expansions look right. The only real question is the black-box Lemma 2.1 imported from the author's companion preprints; nothing in the paper proves it. If that lemma is solid, the results stand.\n\nWhat is new: the complete asymptotic expansions for Poisson (15), negative binomial (21), and hypergeometric (28), with closed Bernoulli-polynomial coefficients carrying the lattice displacement exactly. The enveloping theorems at integer means are also new beyond the binomial case, and the single Binet-kernel argument (Proposition 4.3) is clean and sign-consistent. The collapse identities are classical, and the author says so; the telescoping is known in the Pearson/Ord w-function literature. The genuinely useful addition there is the uniform statement (Theorem 3.3) and the size-bias interpretation, which explains why the hypergeometric needs two tail cancellations. The proof assembly is detailed, and I checked the algebraic steps: the cancellation between the elementary tails and the collapse prefactors works, and the coefficients in (21), (22), (28), (29) are consistent.\n\nThe soft spot is real but narrow. Section 2 states Lemma 2.1 and the shifted Stirling form (4) as black boxes from [9]. Every complete expansion in the paper goes through Lemma 2.1, so the central theorems are conditional on that lemma's uniformity in the shifts. The author states the uniformity condition explicitly, and the shifts h remain in [0,1), which is a compact set, so the stated condition is exactly what is needed. But there is no proof in this paper, and [9] is a preprint. That is a missing support, not an error I can point to. The rest of the proof chain -- the telescoping, the size-bias cancellations, the Binet-kernel enveloping -- is internally consistent, and I found no sign of circularity or fitted parameters.\n\nThe paper also says clearly that matching across the degenerate limits (rq -> lambda, etc.) is left open; that is a scope limitation, not a flaw.\n\nBottom line: this deserves a serious referee. I would ask the author to either include a proof of Lemma 2.1 or make the companion preprint impossible to miss in the review package; the referee should verify the uniformity claim. The paper is not a desk reject. I would cite it if I worked on asymptotics of discrete laws.","headline":"A carefully built extension of the binomial mean-deviation expansions to the other three classical laws; the results stand if the imported Stirling-type lemma from the companion papers holds up.","tokens_in":18101,"tokens_out":2438,"would_cite":true,"duration_ms":22958,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60E05","62E20","41A60","60C05","11B68","33B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The mean absolute deviation of each classical discrete law collapses to a single point mass; the three non-binomial cases have complete asymptotics with Bernoulli-polynomial coefficients.","keywords":["mean absolute deviation","Poisson distribution","negative binomial distribution","hypergeometric distribution","Bernoulli polynomials","size biasing","Stirling series","enveloping series"],"falsifier":"Take an integer-mean case, say $X\\sim\\mathrm{Poi}(10)$, and compute $\\log E|X-\\lambda|/\\sqrt{2\\lambda/\\pi}$ numerically or through the exact identity $J(\\lambda)$; then check whether the partial sums of $\\sum_{j\\ge1}a_{2j-1}(0)\\lambda^{1-2j}$ with $a_{2j-1}(0)=-B_{2j}/((2j-1)2j)$ alternate around the value with the stated strict signs, since a single violation would falsify the enveloping theorem. For a non-integer check, compare $\\lambda=10.3$ against the first two multiplicative coefficients $d_1,d_2$ of Theorem 4.1.","tokens_in":17028,"feed_emoji":"📊","tokens_out":8440,"duration_ms":72336,"temperature":0.7,"pith_summary":"The paper proves that the mean absolute deviation about the mean of any of the four classical discrete laws—binomial, Poisson, negative binomial, and hypergeometric—is exactly twice a single point mass of the distribution, located just above the mean. It then shows for the three non-binomial laws that this point mass has a complete asymptotic expansion in inverse powers of the natural parameter, with coefficients given in closed form by Bernoulli polynomials evaluated at the fractional part of the mean. The key point is that the coefficients carry the lattice displacement of the mean exactly, so the expansions remain valid at every parameter value, not only along arithmetic subsequences. At integer means the expansions reduce to sign-alternating series in odd powers, and a single kernel argument shows those series envelope the logarithm of the normalized mean absolute deviation: successive partial sums bracket the true value. If true, this gives uniform, arbitrary-order asymptotics for a central measure of spread in the Poisson, negative-binomial, and hypergeometric regimes, with explicit oscillating corrections.","feed_headline":"Mean deviation collapses to one term for all four classical laws","feed_subtitle":"One telescoping proof for all four laws; complete expansions carry the lattice oscillation exactly.","key_machinery":"The argument runs on two black-box asymptotics imported from [9]: the shifted Appell form of Stirling's series, $\\log\\Gamma(x+t)\\sim(x+t-\\tfrac12)\\log x-x+\\tfrac12\\log2\\pi+\\sum_{n\\ge1}(-1)^{n+1}B_{n+1}(t)/(n(n+1))x^{-n}$, and Lemma 2.1, a complete expansion for a quotient of gamma functions with unequal scalings. On top of these, the paper uses a common telescoping identity $(k-\\mu)P(k)=g(k)P(k)-g(k+1)P(k+1)$ whose $g$ has the degree of the Katz–Ord ratio, and the size-bias absorption identities that turn the collapse prefactor $2g(\\nu)$ into $2\\sigma^2$ times a single mass of a shifted law. For the enveloping theorems the load-bearing object is the kernel $\\varphi(t)=\\frac1t(\\frac1{e^t-1}-\\frac1t+\\frac12)$, whose positive partial-fraction expansion (via the zeta identity $\\sum_{k\\ge1}2(4\\pi^2k^2)^{-n}=(-1)^{n-1}B_{2n}/(2n)!$) gives Proposition 4.3: any exponentially decaying strictly negative $\\Delta$ produces a series with partial sums bracketing $\\int\\varphi\\,\\Delta$. The Bernoulli polynomials $B_{n+1}(\\cdot)$ at the lattice displacement of the mean carry the oscillation in all complete expansions.","core_discovery":"For $X$ among $\\mathrm{Bin}(N,p)$, $\\mathrm{Poi}(\\lambda)$, $\\mathrm{NB}(r,p)$, and $\\mathrm{Hyp}(N,K,n)$, with mean $\\mu$ and $\\nu=\\lceil\\mu\\rceil$, the paper establishes a common telescoping identity $(k-\\mu)P(k)=g(k)P(k)-g(k+1)P(k+1)$ with $g(k)$ given by (8), from which $E|X-\\mu|=2g(\\nu)P(\\nu)$. Through size-biasing this takes the uniform form $E|X-\\mu|=2\\sigma^2P^{\\downarrow}\\{\\nu-1\\}$, where $P^{\\downarrow}$ is the mass function of the corresponding size-biased law. For the Poisson, negative binomial, and hypergeometric laws, the paper derives complete asymptotic expansions: for the Poisson, $E|X-\\lambda|\\sim\\sqrt{2\\lambda/\\pi}\\,\\exp(-\\sum_{n\\ge1}B_{n+1}(\\{\\lambda\\})/(n(n+1))\\,\\lambda^{-n})$, with analogous Bernoulli-polynomial expansions in the other two cases. The elementary, non-Bernoulli tail of the underlying Stirling expansion is removed exactly by the collapse prefactor $g(\\nu)$, so the complete series is pure Bernoulli-polynomial at every order. When the mean is an integer, even-index coefficients vanish and the series becomes a sign-alternating odd series; a single kernel identity shows the series is enveloping, so each partial sum is a strict two-sided bound on the normalization.","pith_inferences":["One could press the same telescoping scheme on other members of the Pearson/Ord family whose ratio has degree two; the logarithmic distribution, which the paper notes lies outside its framework, is a natural first candidate.","The three open matchings (binomial-to-Poisson as $p=\\lambda/N$, negative-binomial-to-Poisson as $rq\\to\\lambda$, and hypergeometric-to-binomial as $\\eta\\to0$) could plausibly be resolved by uniform expansions built from the same gamma-quotient lemma, yielding one transition-valid formula instead of the current degradation of the effective parameter.","Because the enveloping series carry strict sign information, they can be used as certified interval arithmetic for the mean absolute deviation: truncating at an odd or even order gives a deterministic upper or lower bound over a whole parameter range.","The continuous Cesàro vanishing of each Poisson coefficient suggests that for generic $\\lambda$ the oscillating corrections randomize over a period, so the leading Gaussian term is the honest average; this is a testable high-precision prediction."],"forward_implications":["At integer means each law has a sign-alternating enveloping series, so partial sums provide strict two-sided bounds on the mean absolute deviation.","The elementary tail of the Stirling expansion is cancelled exactly by the collapse prefactor, so no separate non-Bernoulli terms appear at any order in the three non-binomial expansions.","The negative-binomial expansion's effective parameter is $1/(rq)$, equivalently $1/(p\\mu)$, so the approximation degrades as $q\\to0$; the matching to the Poisson case as $rq\\to\\lambda$ is left open.","The hypergeometric expansion is naturally written in terms of $Nad$, the independence-table cell mass, rather than the variance $\\sigma^2$, and the finite-population factor $N/(N-1)$ never enters the coefficients.","The universal form $E|X-\\mu|=2\\sigma^2P^{\\downarrow}\\{\\nu-1\\}$ gives an exact lattice version of the continuous formula ‘mean deviation equals twice the variance times a density value at the mean’ for all four laws."],"supporting_citations":[{"why":"Supplies Lemma 2.1, the gamma-quotient expansion with unequal scalings, and the shifted Appell form of Stirling's series on which the three complete expansions are built.","marker":"[9]"},{"why":"Gives the binomial mean-deviation expansion and the size-bias cancellation mechanism that the paper extends to the other three laws.","marker":"[10]"},{"why":"Supplies the integral representation of the log-gamma remainder and the partial-fraction expansion of its integrand that drive the enveloping proposition.","marker":"[21]"},{"why":"Records the hypergeometric closed form with quadratic $g$ that Theorem 3.1 reproduces as part of a uniform statement.","marker":"[19]"},{"why":"Defines the enveloping-series sense used to state the bracketing theorems.","marker":"[18]"},{"why":"Provides the historical and structural context for the classical collapse identity that the paper unifies.","marker":"[8]"}],"fun_headline_variants":["One telescoping identity collapses MAD for binomial, Poisson, NB, hypergeometric","Full asymptotic expansions for MAD of Poisson, negative binomial, hypergeometric","Enveloping series for normalized MAD: partial sums bracket it"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central assumption is that an imported asymptotic formula for ratios of gamma functions with unequal scalings (Lemma 2.1) is valid uniformly over the compact ranges of shifts and scalings used here; if that formula fails or lacks the stated uniformity, the complete expansions (15), (21), and (28) do not follow.","fun_headline_variants_meta":{"raw":{"variants":["One telescoping identity collapses MAD for binomial, Poisson, NB, hypergeometric","Full asymptotic expansions for MAD of Poisson, negative binomial, hypergeometric","Enveloping series for normalized MAD: partial sums bracket it"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001424,"raw_usage":{"total_tokens":5778,"prompt_tokens":1009,"completion_tokens":4769,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":4707}},"tokens_in":625,"tokens_out":4769,"duration_ms":34065,"temperature":1.0,"reasoning_tokens":4707,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:56:36.058329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an integer-mean case, say $X\\sim\\mathrm{Poi}(10)$, and compute $\\log E|X-\\lambda|/\\sqrt{2\\lambda/\\pi}$ numerically or through the exact identity $J(\\lambda)$; then check whether the partial sums of $\\sum_{j\\ge1}a_{2j-1}(0)\\lambda^{1-2j}$ with $a_{2j-1}(0)=-B_{2j}/((2j-1)2j)$ alternate around the value with the stated strict signs, since a single violation would falsify the enveloping theorem. For a non-integer check, compare $\\lambda=10.3$ against the first two multiplicative coefficients $d_1,d_2$ of Theorem 4.1.","supporting_citations":[{"cited_title":"Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution","cited_arxiv_id":"2607.18494","evidence_quote":"Supplies Lemma 2.1, the gamma-quotient expansion with unequal scalings, and the shifted Appell form of Stirling's series on which the three complete expansions are built."},{"cited_title":"Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion","cited_arxiv_id":"2607.19844","evidence_quote":"Gives the binomial mean-deviation expansion and the size-bias cancellation mechanism that the paper extends to the other three laws."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the integral representation of the log-gamma remainder and the partial-fraction expansion of its integrand that drive the enveloping proposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Records the hypergeometric closed form with quadratic $g$ that Theorem 3.1 reproduces as part of a uniform statement."},{"cited_title":"Pólya, G","cited_arxiv_id":null,"evidence_quote":"Defines the enveloping-series sense used to state the bracketing theorems."},{"cited_title":"Diaconis, S","cited_arxiv_id":null,"evidence_quote":"Provides the historical and structural context for the classical collapse identity that the paper unifies."}],"review_version":1}