{"id":"cffe6e08-0ada-4bde-a715-17040b12da25","arxiv_id":"2607.19844","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Binomial masses a fixed distance from the upper mode have an all-order expansion whose elementary tail is exactly the size-bias factor, leaving pure Appell coefficients.","lead":"This paper derives complete asymptotic expansions for binomial probabilities at a fixed integer distance from the mode, keeping the oscillating fractional part of the mean explicit. The structural cleanup—identifying the logarithmic tail as a size-bias factor—makes local binomial approximations easier to trust and to extend.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem inherits unproved base expansion from companion [7]; paper explicitly quotes rather than reproves Theorem 2.1 (Remark 5.2).","rationale":"The reader's weakest assumption identifies exactly the load-bearing dependency on [7, Theorem 3.1]. The manuscript itself flags this in Remark 5.2, so the concern is not manufactured. The internal derivations are transparent and algebraically consistent; the size-bias factorization (Prop. 2.3, Thm 2.4), the Appell normalisations (Thm 3.3), and the Euler–Maclaurin splitting (Thm 4.2) are all derived carefully and verified numerically. But without an independent proof of the quoted base expansion, the main theorem is conditional. The proposed test—re-deriving [7, Theorem 3.1]—directly settles whether the concern lands. If the base theorem holds, the paper is a solid contribution; if not, it collapses. Hence the reader's CONDITIONAL verdict is appropriate and should be unchanged.","tokens_in":18899,"tokens_out":5107,"duration_ms":53113,"concrete_test":"Obtain [7, Theorem 3.1] and independently re-derive it from the gamma-function quotient (e.g., via Stirling's series), verifying that the relative remainder is indeed O(N^{-M-1}) uniformly for p in compact K and t in compact H, with a constant independent of p and t. If the re-derivation succeeds, the concern is resolved; if it fails or requires additional restrictions on t, Theorem 5.1 lacks support and the verdict should be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.1, the paper's central contribution, rests entirely on Theorem 2.1, quoted from [7, Theorem 3.1] without proof. The paper states this openly in §2 ('We take as given...') and in Remark 5.2 ('Theorem 2.1 is quoted, not reproved'). Theorems 2.4, 3.3, and 5.1 all inherit the existence, uniformity, and remainder O(N^{-M-1}) from this external result. If [7, Theorem 3.1] is unproven, has a hidden hypothesis, or its uniformity in t on compact sets fails, the main expansion collapses—the internal structural work (size-bias factorization, Appell sequences, Euler–Maclaurin splitting) is sound but cannot replace the missing base. The paper's own numerical checks (§9) verify the final expansion in specific cases, but they do not establish the general uniform remainder on which the theorem depends. This is a genuine premise distinct from the paper's contributions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper considers X ~ Bin(N,p), nu = ceil(Np), h_N = nu - Np, and gives asymptotic expansions for Pr{X = nu + r} as N tends to infinity, uniformly for p in compact subsets of (0,1) and |r| <= R. The starting point is a quoted theorem ([7, Thm 3.1], stated as Theorem 2.1) providing a complete local expansion with coefficients A_k(t;p). The paper's contribution is structural: it identifies the non-Bernoulli tail in A_k with log(m/(Np)) and hence with the binomial size-bias factor; removes it to obtain a 'pure Appell' normalization; introduces a second, p-to-q symmetric normalization using the even Appell sequence hat B_n; proves that the integer shift r enters only through finite power sums S_k or T_k (the Euler-Maclaurin correspondence); and assembles these into Theorem 5.1. It also derives the exact mode rule from the first-order coefficient, verifies the expansions by exact ratio identities and by numerical checks to about 60 digits, and computes Cesaro averages of the oscillating coefficients.","tokens_in":19199,"tokens_out":16950,"duration_ms":167175,"significance":"If the base expansion is valid, the paper delivers a genuinely complete, uniformly valid expansion for fixed-distance modal binomial probabilities in the regime where the fractional part h_N oscillates. The identification of the tail with the size-bias factor is elegant and explains the earlier cancellation in the companion paper on mean absolute deviation. The Euler-Maclaurin split shows that the r-dependence is elementary, and the two normalizations clarify the trade-off between Appell purity and p-to-q symmetry. The paper is transparent about its dependence on [7]; the internal algebra is coherent, and the claimed agreement with special cases and with exact ratio identities is reassuring. These structural results are likely to be useful for lattice local limit problems and for the associated mean-deviation literature.","major_comments":[{"comment":"The main expansion is not self-contained. Theorem 5.1 (Eqs. (29)-(30)) inherits its existence, uniformity in p and t, and the O(N^{-M-1}) relative remainder entirely from the quoted companion theorem [7, Thm 3.1], stated here as Theorem 2.1. The paper explicitly says in Remark 5.2 that Theorem 2.1 is quoted, not reproved. This is load-bearing: if [7, Thm 3.1] is not available, has a hidden hypothesis, or fails to be uniform in t, then Theorems 2.4, 3.3 and 5.1 collapse. The numerical checks in Section 9 verify specific rates and identities but do not establish the uniform remainder. I request that a proof of Theorem 2.1 be included in an appendix or provided through a published reference; without this, the central claim remains conditional on an external same-author preprint.","section":"Section 2, Theorem 2.1; Remark 5.2; Theorem 5.1"},{"comment":"The sentence 'Theorem 5.1 must reproduce this identically' overstates the logical status. Theorem 5.1 is an asymptotic statement with a formal infinite exponent, not a theorem that yields exact finite-N identities. The displayed derivation shows that the formal series for the exponent difference telescopes to the logarithm of the exact ratio, which is a useful algebraic consistency check; however it is not a consequence of the remainder theorem. Please rephrase (and perhaps note that the matching is exact only in the formal or convergent-series sense) to avoid implying that an asymptotic theorem proves an exact identity.","section":"Section 7.1, Eq. (34)"}],"minor_comments":[{"comment":"The claim that [3, Theorem 3.1] contains a sign misprint in P_1 would be easier to check if the relevant formula from [3] were reproduced alongside the present derivation.","section":"Section 7.2"},{"comment":"The proof of equidistribution of h_N for irrational p is compressed; since h_N is 1 - {Np} except at atoms, a one-line reduction to Weyl's theorem would be clearer.","section":"Section 8, Proposition 8.1"},{"comment":"The numerical checks are extensive, but the paper does not include code or exact evaluation details. A short reproducibility note would be useful, though this is not required for the mathematics.","section":"Section 9"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the reliance on the companion preprint [7]. If [7] is already accepted, a published citation may be enough; otherwise the author should append the proof. I do not see an internal error in the algebraic development, and the paper is transparent about the dependency, so I waver between major_revision and reject; I choose major_revision because the gap is fixable within the scope of a revision. Please also consider whether the overstatement in Section 7.1 needs correction before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is a genuinely useful companion, but its main theorem inherits existence, uniformity, and remainder from the author's own [7, Theorem 3.1], which is quoted, not reproved. Everything else is internal, transparent, and supported by checks.\n\nWhat is actually new: the identification of the logarithmic tail in the naive local expansion with the binomial size-bias factor -log(m/(Np)); removing it leaves a pure Appell expansion. The symmetric normalization uses the even Appell sequence bB_n and recovers the classical Stirling prefactor. The entire r-dependence reduces to power sums — left endpoint versus trapezoidal — via an Euler-Maclaurin correspondence. Those structural claims are proven cleanly and consistently. The mode-rule consistency check, the exact telescoping of the ratio b(ν+r)/b(ν), the recovery of the central-binomial coefficients of [3], and the Cesàro averaging are all nice and appear correct. Numerical checks at 50–60 digits behave as predicted.\n\nThe soft spot is genuine: Theorem 5.1 rests on Theorem 2.1, imported from [7], and the paper openly says so in §2 and Remark 5.2. If that base expansion or its uniformity fails for oscillating t = h+r with hN a fractional part, the main theorem collapses. That concern is load-bearing in principle but not fatal in practice: the quoted expansion has been independently checked here to high order numerically, and all the internal structural derivations stand on their own. Still, the dependency must be resolved before accepting — either by proving Theorem 2.1 in an appendix or by having [7] refereed in tandem. The self-citation is not itself a problem; the issue is that the cited result is not formally verified or published.\n\nWho this is for: probabilists and asymptotic analysts working on lattice expansions, binomial modes, or local limit refinements. It deserves a serious referee, but the referee should see [7] as part of the package.","headline":"Solid structural companion that turns a known all-order expansion into clean Appell form, but the load-bearing base expansion is imported from an unrefereed companion preprint.","tokens_in":19616,"tokens_out":2341,"would_cite":true,"duration_ms":27050,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60C05","41A60","11B68","33B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that binomial probabilities at a fixed integer distance from the mode have a complete asymptotic expansion in powers of 1/N, uniform in p and in the shift, with the oscillating fractional part of the mean encoded in Bernou","keywords":["binomial distribution","local limit theorem","mode","asymptotic expansion","Bernoulli polynomials","Appell sequence","size-biasing","Euler–Maclaurin formula"],"falsifier":"Compute the exact probability Pr{X = ⌈Np⌉+r} for p = 1/√7 and r = 3 for N = 10^4, 10^5, 10^6 using the gamma-function representation of the binomial coefficient, and compare with Theorem 5.1 truncated after M = 2 terms. The relative error should decrease by factor ~10^3 for each factor of 10 in N, uniformly in the oscillating h_N; any slower decay refutes the uniformity claim. Alternatively, verify the exact identity (28) at a non-integer h (say h=1/3, r=2, p=1/3): both sides are rational polynomials in h and p, so the equality can be checked symbolically; a single mismatch refutes the Bernoul","tokens_in":18838,"feed_emoji":"🎯","tokens_out":7095,"duration_ms":63404,"temperature":0.7,"pith_summary":"The paper establishes that the binomial probability at any fixed integer distance r from the upper mode has a complete asymptotic expansion in powers of 1/N, uniform in the success probability p on compact subintervals of (0,1) and in |r| ≤ R. The expansion keeps the oscillating fractional part h_N = ⌈Np⌉ − Np exactly in the coefficients, which are built from Bernoulli polynomials and finite power sums. A single elementary mechanism drives the whole structure: the logarithmic tail of the standard local expansion is exactly the logarithm of the size-bias factor m/(Np), and removing it leaves an expansion governed by a pure Appell sequence. A second, symmetrized normalisation recovers the classical Stirling prefactor and an even Appell sequence. From these coefficients the paper derives the exact binomial mode rule, recovers the central-binomial expansions of the literature as a special case, and gives closed-form Cesàro averages of the oscillating coefficients.","feed_headline":"Complete expansion found for binomial probabilities at the mode","feed_subtitle":"Keeping the fractional part of the mean in the coefficients reproduces the exact mode rule and the central-binomial expansions.","key_machinery":"The size-bias identity m b(m;N,p) = Np b(m−1;N−1,p) is the load-bearing identity: it converts the logarithmic tail of the naive expansion into the exact prefactor Np/m. The two Appell sequences — the Bernoulli polynomials B_n(t) for the size-biased normalisation and the even sequence bB_n(t) = (B_n(t)+B_n(t+1))/2, generated by (z/2)coth(z/2)e^{tz}, for the symmetric normalisation — carry the coefficients. The difference identity B_{k+1}(h+r)−B_{k+1}(h) = (k+1) Σ_{j=0}^{r-1}(h+j)^k is what reduces the shift r to finite power sums.","core_discovery":"The paper's central claim is Theorem 5.1: as N→∞, uniformly for p in a compact subset of (0,1) and integers r with |r|≤R, Pr{X=⌈Np⌉+r} ∼ (Np/(Np+h+r)) (2πNpq)^{-1/2} exp( Σ_{k≥1} [eA_k(h;p) − Ξ_k(p) S_k(h,r)/k] / N^k ), where h = ⌈Np⌉ − Np, eA_k are built from Bernoulli numbers and Bernoulli polynomials, Ξ_k(p) = (−1)^{k+1}p^{-k}+q^{-k}, and S_k(h,r) is a finite power sum along h, h+1, ..., h+r−1. Equivalently, after a symmetric normalisation the prefactor becomes the classical Stirling factor and the power sum becomes trapezoidal. The paper shows that the elementary tail of the previously known expansion is exactly the size-bias factor, so the 'pure' coefficients are Appell; that the intege","pith_inferences":["The same size-bias-plus-Appell mechanism should transfer to other lattice distributions possessing a size-bias identity — Poisson, negative binomial, hypergeometric — replacing the Bernoulli sequence by the appropriate Appell generator; the paper suggests this but does not carry it out.","In the crossover region r ≍ √N, where the fixed-distance expansion loses its ordering, one could interpolate between the Bernoulli-polynomial coefficients and the Hermite-polynomial Edgeworth expansion; the paper identifies the regime but leaves the transition unexplored.","The exact Cesàro-average coefficients (Proposition 8.1) imply a practical smoothing: for a single large N and unknown fractional part, one can use the averaged coefficients, with the error controlled by the discrepancy of the sequence {Np}; this is a testable approximation that the paper does not formulate.","Because Corollary 5.5 gives the modal window sum in closed form as a cubic in R, it offers a fast way to compute local coverage probabilities without summing the individual probabilities; this may be useful in confidence-interval calculations, though the paper does not pursue that application."],"forward_implications":["Every fixed-distance local mass of a binomial law is now known to all orders in 1/N, with the fractional part of the mean preserved in the coefficients; this is directly useful for likelihood approximations and continuity corrections near the mode.","The exact mode rule — upper mode at ⌈Np⌉ when h≤p, at ⌈Np⌉−1 when h>p — is reproduced by the first asymptotic coefficient, so the expansion correctly encodes the lattice rounding, not just the smooth skewness correction.","The central-binomial expansions of the literature (p=1/2, N even) are recovered as the special case h=0, and the vanishing of even-order coefficients at p=q explains the alternating structure there.","The mean absolute deviation of the binomial, via De Moivre's identity, inherits a pure Appell expansion, resolving the term-by-term cancellation observed in the companion paper.","Averaged over N, the oscillating coefficients have closed-form Cesàro limits: zero for irrational p (by equidistribution) and the explicit periodic mean b^{-j}B_j for rational p=a/b, which gives a smooth all-purpose approximation to the modal expansion."],"fun_headline_variants":["Binomial at mode: full expansion with size-bias tail","Size-bias factor exposed in binomial mode asymptotics","All-order expansion for binomial mode probabilities","Binomial mode rule derived from Appell coefficients","Exact mode asymptotics: Stirling prefactor and Appell terms"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire expansion rests on the un-reproved companion theorem [7, Theorem 3.1] asserting a complete uniform asymptotic expansion of the binomial local mass at arbitrary real displacement t, with relative remainder O(N^{-M-1}) on compacta; if that quoted expansion fails in uniformity, the present Theorem 5.1 loses its error term as well.","fun_headline_variants_meta":{"raw":{"variants":["Binomial at mode: full expansion with size-bias tail","Size-bias factor exposed in binomial mode asymptotics","All-order expansion for binomial mode probabilities","Binomial mode rule derived from Appell coefficients","Exact mode asymptotics: Stirling prefactor and Appell terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000294,"raw_usage":{"total_tokens":1553,"prompt_tokens":756,"completion_tokens":797,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":722}},"tokens_in":500,"tokens_out":797,"duration_ms":9350,"temperature":1.0,"reasoning_tokens":722,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T01:50:10.761313+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact probability Pr{X = ⌈Np⌉+r} for p = 1/√7 and r = 3 for N = 10^4, 10^5, 10^6 using the gamma-function representation of the binomial coefficient, and compare with Theorem 5.1 truncated after M = 2 terms. The relative error should decrease by factor ~10^3 for each factor of 10 in N, uniformly in the oscillating h_N; any slower decay refutes the uniformity claim. Alternatively, verify the exact identity (28) at a non-integer h (say h=1/3, r=2, p=1/3): both sides are rational polynomials in h and p, so the equality can be checked symbolically; a single mismatch refutes the Bernoul","supporting_citations":[],"review_version":2}