{"id":"4f9c7c15-dd79-4487-8562-73f1f7bf7e7b","arxiv_id":"2506.08503","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The Erdős-Kac theorem is transformed with the delta method, yielding variance-stabilising square-root and width-optimal three-quarter-power interval estimates for omega(m), with a score interval also derived.","lead":"A number theorist applies the delta method to the Erdős-Kac theorem to build interval estimates for the number of distinct prime factors of an integer, showing that a square-root transform stabilises the variance and a three-quarter power gives the narrowest intervals. The paper then tests these intervals on actual integers and finds that a Poisson-based interval is the most reliable for small numbers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Trained interval estimates in Sec. 7 rest on fitted SDs that appear roughly 40% smaller than the Erdős–Kac variance even inside the training range; the claimed reliability of the trained score/Poisson intervals is therefore unverified.","rationale":"The theoretical core of the paper—Theorem 1.1, the Box-Cox form in Theorem 3.1, variance stabilization at λ = 1/2, and width optimality at λ = 3/4—is mathematically sound; the delta-method derivations are correct, and the second-order width expansion checks out. The reader's conditional verdict is appropriate. However, the most load-bearing weakness is more specific than generic extrapolation risk: the fitted standard deviations in Table 5 are inconsistent with the asymptotic variance even within the training range, by a nearly uniform factor of about 0.6 across all three λ values. This suggests that the training algorithm as described, or its numerical implementation, is not estimating the intended quantity. Since the abstract and discussion present the trained score and Poisson intervals as the practically reliable outcome, this calibration issue directly undermines a central applied claim. The concern is testable by reproduction, and it does not affect the main theoretical results, so the verdict remains CONDITIONAL rather than REJECT. The reader's weakest assumption captured the extrapolation risk but not the more fundamental in-range miscalibration, hence partial agreement.","tokens_in":24000,"tokens_out":31764,"duration_ms":339254,"concrete_test":"Independently reimplement Section 7's training algorithm exactly as written: for m ∈ [10^4, 10^6], smooth ω^λ with j = 2000, fit f̂_{μ,λ} by nonlinear least squares as in Table 5, then compute the moving-average RMS residual from Step 4. Compare the resulting σ̃_λ(m) to Table 5 and to the theoretical SD λ·ℓ2^{λ−1/2}. As a minimal check, directly compute the empirical SD of ω(t) over a window of length 4001 around m = 10^5 and compare it to f̂_{σ,1}(10^5): if the empirical SD is close to √ℓ2 ≈ 1.56 while the fitted value is ≈ 0.95, the trained intervals are miscalibrated and the reliability claims in Section 7 collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 7 estimates the standard deviation of the normalized residual (ω^λ − f̂_{μ,λ}^λ)/λ and fits it as (γ0 + γ1·ℓ2)^λ. For λ=1, Table 5 gives f̂_{σ,1}(m) ≈ 0.05 + 0.368·ℓ2(m); at m = 10^5 (ℓ2 ≈ 2.44), this is ≈ 0.95, whereas the Erdős–Kac variance of ω in a window of that size is ℓ2(m) ≈ 2.44, so the empirical SD should be ≈ 1.56. For λ=1/2, the variance-stabilized residual should have SD ≈ 1, but Table 5 gives (γ0 + γ1ℓ2)^(1/2) ≈ √(0.355) ≈ 0.60 at the same m. The plotted values of f̂_{σ,λ}^{1/λ} in Figure 7 (0.32–0.38 for λ=1/2 and 0.90–1.00 for λ=1) are consistently about 60% of the theoretical delta-method SD. This discrepancy is unlikely to be an extrapolation failure; it appears to be a calibration problem inside the training range itself. If the fitted SDs are miscalibrated, the trained interval widths used in Figure 8 are wrong, and the paper's conclusion that trained score and Poisson intervals are reliable is unsupported even in-sample, not merely out-of-sample.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies the delta method to Billingsley's probabilistic version of the Erdős–Kac theorem and obtains a central limit theorem for g(ω/ℓ2) when g'(1)≠0. Choosing g as a Box–Cox power gives a transformed Erdős–Kac theorem; the paper identifies λ=1/2 as the variance-stabilizing transformation and λ=3/4 as the power that asymptotically minimizes the width of the two-sided interval estimate for ω. It then introduces local adjustment functions to refine the intervals for finite m, develops fuzzy coverage probabilities to handle the discreteness of ω, and compares five interval estimates (λ=1/2, 3/4, 1, Poisson, and score) for m∈[10^5,10^14]. After fitting mean and standard deviation functions on m∈[10^4,10^6], it claims that the trained score and Poisson intervals are reliable. The paper also states analogous transformed Erdős–Pomerance theorems.","tokens_in":24343,"tokens_out":13592,"duration_ms":151202,"significance":"If the main claims hold, the theoretical part is a clean and useful contribution: it gives a principled family of Erdős–Kac-based intervals, and the variance-stabilization and width-optimality results are natural and well-motivated. The asymptotic expansion behind Theorem 4.1 is correct, and the fuzzy-coverage framework is an appropriate response to the discreteness of ω. The claimed numerical advantage of the trained score and Poisson intervals, however, is not supported by the calibration evidence as presented: the fitted standard deviations in Table 5 are markedly smaller than the Erdős–Kac variance even inside the training range, so the reliability conclusion rests on unverified numerics. The paper would also be strengthened by providing the omitted proofs and by making the numerical evaluation reproducible.","major_comments":[{"comment":"The fitted scale parameters in Table 5 appear inconsistent with the Erdős–Kac variance on the training range. At m=10^5 (ℓ2≈2.44), the λ=1 value gives f̂σ,1≈0.0499+0.3677·2.44≈0.95, whereas the Erdős–Kac variance of ω at that scale is ℓ2≈2.44, so the empirical standard deviation should be about 1.56, not 0.95. For λ=1/2, the fitted value is about 0.60 instead of the variance-stabilized value 1. This is not a small finite-sample correction. Since the trained interval widths in Figure 8 are built from these f̂σ,λ values, the reported reliability of the trained Box-Cox, score, and Poisson intervals is not established. Please reconcile Table 5 with a direct computation of the residual standard deviations on the training range, or supply the code/data so that the reader can verify the calibration.","section":"Section 7, Table 5, Figure 8"},{"comment":"Theorem 3.1 is the central result used for all subsequent interval constructions, and Theorems 8.1–8.3 are stated in Section 8, but their proofs are omitted 'for brevity.' At least for Theorem 3.1, a short derivation from Theorem 1.1 should be included; for Section 8, either provide the analogous delta-method derivations or state explicitly that they follow from (25) by the same argument. As written, the reader cannot verify the λ=0 cases or the rates without reconstructing the calculations.","section":"Theorems 3.1, 8.1–8.3"},{"comment":"The displayed denominator in Theorem 8.1, 2g'(1)/(√3 ℓ2), is not consistent with Theorem 8.2 or with (25). Substituting g(x)=yλ(x) into that denominator and into the numerator gives a ratio with a denominator containing ℓ2^{2λ−1}, not ℓ2^{2λ−1/2} as in Theorem 8.2; the λ=1 case would not reduce to (25). The correct denominator should be 2g'(1)/(√3√ℓ2). The same notational correction is needed in the λ=0 statement of Theorem 8.2 and in the following sentence, so that the variance-stabilizing power λ=1/4 is derived from the correct rate.","section":"Theorem 8.1 and Theorem 8.2"}],"minor_comments":[{"comment":"There are several typographical slips, including 'the the' in the introduction and 'the the vicinity' in Section 4; a careful proofreading pass is needed.","section":"Section 1"},{"comment":"The sentence beginning 'when ⌈Lλ,α(m)⌉=⌊Uλ,α(m)⌋+1, we may further assume that at least one of ... is included' is not reflected in the fuzzy coverage formula that follows; the intended convention should be stated precisely or removed.","section":"Section 6"},{"comment":"The numerical results in Tables 4–5 and Figures 4–8 are not reproducible from the text alone because no code or detailed data-processing pipeline is given; providing the code or a detailed pseudocode would substantially help the reader check the calibration issue raised above.","section":"Section 7"},{"comment":"The claim that 'all the theoretical results follow even if we replace ω with Ω' is stated without comment; since Ω is also an additive function satisfying the relevant Billingsley conditions, a one-sentence justification would make the remark self-contained.","section":"Section 9"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core of the paper is sound and the topic is suitable for a number theory audience, but the empirical claims about trained interval estimates are currently not supported. The calibration discrepancy in Section 7 is the main obstacle; if the authors can correct or explain the fitted standard deviations and provide reproducible numerics, the paper would be acceptable. I would not reject on the basis of the theoretical content alone."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the theoretical section is solid, and the width-optimal lambda=3/4 result is a real addition to the Erdős–Kac toolbox. The training section is the soft underbelly, and the stress-test concern about the fitted SDs is on target.\n\nWhat's new: applying the delta method to omega/log log m and then the Box-Cox family gives a clean family of EK-type CLTs. The variance stabilization at lambda=1/2 and width minimization at lambda=3/4 are simple consequences but appear new in that exact form, and the score-interval analogue is a neat extension. The Erdős-Pomerance variants are a bonus. The paper is careful about the P_n convergence framework and the local adjustment functions. All of that is worth keeping.\n\nSoft spots: Section 7 is where I part ways. The fitted SDs in Table 5 and Figure 7 are roughly 60% of the EK scale: for lambda=1 at m=10^5, f-hat_sigma is about 0.95 versus sqrt(log log m) ~1.56. The paper just plots them and moves on. Either the variance in a short window really is that much smaller than the global EK variance, which would need an explanation, or the computation is off. This matters because the trained score interval uses those SDs directly. The fuzzy coverage numbers in Figure 8 look good, but without code/data or sensitivity analysis I cannot tell whether the intervals are correctly calibrated. The out-of-sample extrapolation from [10^4,10^6] to 10^14 is a separate worry, but the in-sample mismatch is the more immediate problem.\n\nAlso: proofs for Theorem 3.1 and Theorems 8.1-8.3 are omitted, but that is fine because they follow directly from earlier results. The paper is honest that the training is heuristic.\n\nBottom line: the theoretical half is legitimate and deserves a serious referee. The training half needs substantial revision before I would trust the reliability claims. Send it to review, but expect the SD calibration to be the main issue.","headline":"The delta-method/Box-Cox part of this paper is correct and genuinely useful, but the trained-interval reliability claims sit on a calibration that the paper never explains.","tokens_in":24917,"tokens_out":17722,"would_cite":true,"duration_ms":188542,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N40","62E20","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper transforms the Erdős–Kac theorem to build interval estimates for the prime-divisor count, with a square-root variance stabilizer, a three-quarters width optimizer, and a trained score interval for small integers.","keywords":["Box-Cox transformation","fuzzy interval estimate","Poisson interval estimate","score interval estimate","variance-stabilizing transformation","Erdős–Kac theorem","prime omega function","delta method"],"falsifier":"Take the actual values of $\\omega(m)$ for $m$ near $10^8$ and $10^{12}$, compute the smoothed mean and standard deviation of $\\omega^\\lambda$ with the same window and powers as Section 7, and compare them with the fitted functions in Table 5; a deviation of more than a few percent would show that the power-of-a-linear-function model does not extrapolate, and the trained interval estimates' claimed out-of-sample reliability would fail. An even more direct check is to compute the exact fuzzy coverage probabilities of the trained score and Poisson intervals at $m=10^{13}$ and $m=10^{14}$ and see whether they remain inside Bradley's liberal band around the nominal $0.6319$.","tokens_in":23736,"feed_emoji":"🔢","tokens_out":17010,"duration_ms":165771,"temperature":0.7,"pith_summary":"This paper extends the Erdős–Kac theorem, which says that the number $\\omega(m)$ of distinct prime divisors of $m$ is asymptotically normal with mean and variance $\\log\\log m$, by showing that the theorem survives a nonlinear change of scale. Provided a transformation $g$ is differentiable at $1$ with nonzero derivative, the ratio $g(\\omega/\\ell_2)$ is also asymptotically standard normal after the usual centering and scaling, where $\\ell_2(m)=\\log\\log m$. Applying the Box-Cox family of transformations singles out two useful members: the square root, which makes the limiting variance a constant independent of $m$, and the three-quarters power, which asymptotically minimizes the width of a two-sided interval estimate for $\\omega$. The paper then constructs a score-type interval estimate from the transformed theorem and evaluates all these intervals for small integers using fuzzy coverage probabilities, finding that a Poisson-based interval is the most reliable without training and that the score and Poisson intervals become accurate after a simple training step. If these claims hold, statisticians and number theorists get practical, computable interval estimates for how many distinct prime factors an integer has, including for numbers far smaller than the regime where the original theorem is usually trusted.","feed_headline":"Square-root transform stabilizes prime-divisor counts","feed_subtitle":"The 3/4 power minimizes interval width; a trained score interval covers even small integers reliably.","key_machinery":"The argument is carried by the delta method fitted into the probabilistic framework of additive arithmetic functions. Starting from the central limit theorem for additive functions, the paper introduces the remainder function $r(x) = (g(x)-g(1))/(x-1) - g'(1)$, which is continuous at $1$, and uses Slutsky's theorem under the discrete uniform measure to prove that $g(f_n/A_n)$ is asymptotically normal whenever the additive function $f_n$ is (Theorem 2.2). Specializing $f_n=\\omega$, $A_n=\\ell_2 + O(1)$, $B_n^2=\\ell_2 + O(1)$ gives Theorem 1.1. The Box-Cox transformation $y_\\lambda(x) = (x^\\lambda-1)/\\lambda$ (or $\\log x$ at $\\lambda=0$) turns this into an explicit $\\lambda$-family of normal approximations; the asymptotic expansion of the interval width, whose first correction term contains the factor $(\\lambda-1)(2\\lambda-1)$, identifies $\\lambda=3/4$ as the width-minimizer. For small integers, the paper replaces ordinary coverage probability with a fuzzy coverage probability that partially credits the fractional upper and lower endpoints of an interval, and it estimates local adjustment functions $f_{\\mu,\\lambda}$ and $f_{\\sigma,\\lambda}$ by fitting power functions of linear functions of $\\ell_2$ to smoothed $\\omega$. The score interval estimate comes from solving $(\\omega-\\ell_2)/\\sqrt{\\omega} = \\pm z$ for $\\omega$, in the manner of Wilson's interval for a binomial proportion.","core_discovery":"The central discovery is Theorem 1.1: if $g$ is differentiable at $1$ and $g'(1)\\neq 0$, then under the discrete uniform measure on $\\{1,\\dots,n\\}$, the transformed ratio $(g(\\omega/\\ell_2)-g(1))/(g'(1)/\\sqrt{\\ell_2})$ converges in distribution to the standard normal $\\Phi$. With $g$ equal to the Box-Cox power $y_\\lambda$, this becomes $(\\omega^\\lambda - \\ell_2^\\lambda)/(\\lambda \\ell_2^{\\lambda-1/2}) \\Rightarrow \\Phi$, and the same argument gives versions with local adjustment functions $f_\\mu$ and $f_\\sigma$ in place of $\\ell_2$ and $\\sqrt{\\ell_2}$. The two distinguished powers are $\\lambda=1/2$, where the denominator no longer depends on $m$ so the variance is stabilized, and $\\lambda=3/4$, which minimizes the asymptotic width of the two-sided interval because the first nonconstant term in the width's expansion is proportional to $(\\lambda-1)(2\\lambda-1)$. Solving the quadratic obtained by standardizing with $\\omega$ in the denominator gives a score interval estimate in the spirit of Wilson, and numerical work with fuzzy coverage probabilities shows that the score interval, and the Poisson interval based on Landau's formula, keep their coverage closest to nominal for $m$ between $10^5$ and $10^{14}$ once their means and standard deviations are estimated from $\\omega$ values on $[10^4,10^6]$. The same transformation machinery is applied to the Erdős–Pomerance theorem, where it yields variance stabilization at $\\lambda=1/4$ while the optimal width remains at $\\lambda=3/4$.","pith_inferences":["Because variance stabilization at $\\lambda=1/2$ makes deviations comparable across different $m$, one could define a universal rarity score $z = (\\sqrt{\\omega} - \\sqrt{\\ell_2})/0.5$ and rank integers of vastly different sizes on one scale; the paper does not propose such a score but its own Table 1 is the seed of it.","The training procedure is only validated on $m$ up to $10^{14}$; a natural robustness test is to retrain on shifted windows (e.g., $10^6$–$10^8$) and see whether the fitted power-law forms in Table 5 drift, which would indicate that the choice of training range, rather than the asymptotic theory, drives the reported coverage.","The relative success of the Poisson interval estimate supports a shifted-Poisson model as a better finite-sample description of $\\omega$ than the normal law; the paper's discussion of Landau's formula and the shifted-Poisson mass function marks the Poisson approximation as a natural target for a rate-of-convergence theorem.","The fuzzy-coverage device is transferable: any interval for a lattice-valued statistic whose endpoints fall between integers suffers the same jump problem, so the same fractional-credit definition could be applied to binomial, Poisson, or hypergeometric intervals, not just to $\\omega$."],"forward_implications":["The square-root transformation yields a simple rarity rule: integers with $|\\sqrt{\\omega(m)} - \\sqrt{\\log\\log m}| > 1.5$ make up roughly 1 in 400 of all integers, and >2.0 roughly 1 in 16,000, giving an easily remembered scale for interpreting $\\omega$.","The three-quarters power gives the asymptotically narrowest two-sided $100(1-\\alpha)\\%$ interval for $\\omega$ near large $m$: $[\\ell_2(1 - 3z/(4\\sqrt{\\ell_2}))^{4/3}, \\ell_2(1 + 3z/(4\\sqrt{\\ell_2}))^{4/3}]$, a direct improvement over the usual $\\ell_2 \\pm z\\sqrt{\\ell_2}$ interval.","The score interval estimate of $\\omega$, obtained by solving the quadratic $(\\omega-\\ell_2)^2/\\omega = z^2$, is the most reliable of the normal-based intervals after training, with fuzzy coverage probabilities close to nominal for $m \\in [10^5, 10^{14}]$.","The Poisson interval estimate, centered at $\\ell_2(m)+1$ with width calibrated to the nominal level, is relatively reliable even without training, and it tends to be narrower than the transformed Erdős–Kac intervals (e.g., at $m\\approx 10^{70}$ it gives $[4.52, 7.65]$ versus Billingsley's $[3.05, 7.11]$).","The transformation results transfer to the Erdős–Pomerance theorem: $\\omega(\\varphi(m))$ is normal after a Box-Cox-type transform with variance stabilization at $\\lambda=1/4$ and asymptotically optimal interval width at $\\lambda=3/4$."],"supporting_citations":[{"why":"supplies the original Gaussian law for additive functions, the statement that the paper generalizes through transformation.","marker":"Erdős and Kac, 1939, 1940"},{"why":"gives the normal order of $\\omega(m)$, i.e., $\\omega/\\ell_2 \\Rightarrow 1$, which justifies transforming the ratio $\\omega/\\ell_2$.","marker":"Hardy and Ramanujan, 1917"},{"why":"furnishes the discrete-uniform probabilistic view, Slutsky's theorem under $P_n$, and the additive-function central limit theorems that Theorem 2.2 builds on.","marker":"Billingsley, 1969, 1974"},{"why":"defines the Box-Cox transformation whose $\\lambda$-family generates the variance-stabilizing and width-optimal interval estimates.","marker":"Box and Cox, 1964"},{"why":"originates the score-interval construction that the paper adapts to $\\omega$ by solving a quadratic equation.","marker":"Wilson, 1927"},{"why":"provides the confidence-region/score-interval strategy and the comparison of studentization versus variance stabilization that frames the interval estimates.","marker":"Politis, 2024"},{"why":"yields the asymptotic formula for the number of integers with exactly $d$ distinct prime divisors, which underlies the Poisson interval estimate and the asymptotic coverage calculation.","marker":"Landau, 1900"},{"why":"establishes the normal law for $\\omega(\\varphi(m))$ that is transformed in Section 8.","marker":"Erdős and Pomerance, 1985"},{"why":"motivates the fuzzy interval interpretation and fuzzy coverage probabilities used to handle jumps at integer endpoints.","marker":"Geyer and Meeden, 2005"},{"why":"supplies the delta-method theorem whose proof structure is adapted to the $P_n$ setting in Lemma 2.1 and Theorem 2.2.","marker":"Serfling, 2009"}],"fun_headline_variants":["Prime divisor intervals: square-root stabilizes, 3/4 minimizes","Variance-stabilizing and width-optimal transforms for prime omega","Score interval reliable for small integers in prime divisor counts","Optimal power transforms for Erdős–Kac interval estimates","Transform prime omega: square-root variance, 3/4 width"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise outside the asymptotic theory is the Section 7 working assumption that the mean and standard deviation of the transformed prime-divisor count can be represented as fixed powers of linear functions of $\\log\\log m$, with coefficients fitted on $m\\in[10^4,10^6]$ and trusted to extrapolate to $m=10^{14}$ and beyond; if that empirical model drifts, the claimed reliability of the trained score and Poisson intervals for out-of-sample $m$ does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Prime divisor intervals: square-root stabilizes, 3/4 minimizes","Variance-stabilizing and width-optimal transforms for prime omega","Score interval reliable for small integers in prime divisor counts","Optimal power transforms for Erdős–Kac interval estimates","Transform prime omega: square-root variance, 3/4 width"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000685,"raw_usage":{"total_tokens":3164,"prompt_tokens":1062,"completion_tokens":2102,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":678,"completion_tokens_details":{"reasoning_tokens":2014}},"tokens_in":678,"tokens_out":2102,"duration_ms":20145,"temperature":1.0,"reasoning_tokens":2014,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:09:00.991919+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the actual values of $\\omega(m)$ for $m$ near $10^8$ and $10^{12}$, compute the smoothed mean and standard deviation of $\\omega^\\lambda$ with the same window and powers as Section 7, and compare them with the fitted functions in Table 5; a deviation of more than a few percent would show that the power-of-a-linear-function model does not extrapolate, and the trained interval estimates' claimed out-of-sample reliability would fail. An even more direct check is to compute the exact fuzzy coverage probabilities of the trained score and Poisson intervals at $m=10^{13}$ and $m=10^{14}$ and see whether they remain inside Bradley's liberal band around the nominal $0.6319$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the normal order of $\\omega(m)$, i.e., $\\omega/\\ell_2 \\Rightarrow 1$, which justifies transforming the ratio $\\omega/\\ell_2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"furnishes the discrete-uniform probabilistic view, Slutsky's theorem under $P_n$, and the additive-function central limit theorems that Theorem 2.2 builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the Box-Cox transformation whose $\\lambda$-family generates the variance-stabilizing and width-optimal interval estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"originates the score-interval construction that the paper adapts to $\\omega$ by solving a quadratic equation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the confidence-region/score-interval strategy and the comparison of studentization versus variance stabilization that frames the interval estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"yields the asymptotic formula for the number of integers with exactly $d$ distinct prime divisors, which underlies the Poisson interval estimate and the asymptotic coverage calculation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"motivates the fuzzy interval interpretation and fuzzy coverage probabilities used to handle jumps at integer endpoints."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the delta-method theorem whose proof structure is adapted to the $P_n$ setting in Lemma 2.1 and Theorem 2.2."}],"review_version":1}