{"id":"f21ad222-9060-45e9-97f6-6e2dc9557853","arxiv_id":"2608.00445","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The packing test detects FvML deviations only when the concentration kappa is of order p^{3/4}/(log n)^{1/4}, and Watson deviations only when p-2*kappa is of order sqrt(p log n), with a discontinuous phase transition in the second-order scaling.","lead":"This paper gives the exact detection thresholds of the packing test for spherical uniformity under high-dimensional Fisher-von Mises-Langevin and Watson alternatives. It confirms rigorously that the packing test is suboptimal in both models and identifies a discontinuous phase transition in the Watson model.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact Watson threshold is not fully proven: the size/power-one claims outside fixed finite rho are left to a sketched appendix and unproved uniformity in rho.","rationale":"The paper is a serious and largely self-contained contribution; the fixed-rho regimes are argued in detail, and the reader's flagged Chen-Stein b2 bounds appear to be handled under the stated hypotheses. I do not see an internal inconsistency in Theorems 2-4 themselves. However, the abstract and conclusion present an exact threshold as a Theta statement, which logically requires controlling alternatives that scale away from the critical window: rho_n -> 0, rho_n -> infinity, and rho_n -> 1 at rates other than the critical one. The paper does not provide uniform-in-rho Poisson approximations, and Appendix B is explicitly a sketch for the rho_n -> 1 case. This is more load-bearing than the b2 estimate inside the fixed-rho regimes, because a failure there would leave the claimed threshold without a lower or upper boundary, not merely change a constant in a proved limit. The concern is concrete and testable: one can repeat the saddle-point computation for non-critical rates and check whether the joint exceedance probability is o(n^{-3}). I therefore recommend CONDITIONAL acceptance: the central theorems are credible, but the exact-threshold wording should be reduced to the proved regimes, or the boundary uniformity in rho should be supplied.","tokens_in":41501,"tokens_out":35297,"duration_ms":323206,"concrete_test":"Directly compute, by the Laplace/saddle estimates of Section 9, the order of n^3 P(I12=I13=1) for rho_n = 1 +- (log n)^{-alpha} with alpha in (0,1/2) and alpha in (1/2,infinity). If for any of these rates the correlation term fails to be o(n^{-3}), the Chen-Stein approximation at rho=1 is not established and the -1/2 log-log coefficient in Theorem 4 is not universal. As a second check, derive the fixed-x limit of P(P_n <= x) when rho_n -> infinity and verify it equals the null Gumbel exp{-(8pi)^{-1/2} e^{-x/2}}; a different limit would contradict the claimed 'only when Delta_n = Theta(sqrt(p log n))' interpretation.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is an exact detection threshold, p-2*kappa = Theta(sqrt(p log n)), together with the power formula (12). But the proofs establish limiting distributions only for fixed rho in (0,infinity) (Theorems 2 and 3) and for the critical scale sqrt(log n)(rho_n-1) -> Delta (Theorem 4). The boundary cases needed for the Theta statement are not fully proved. Appendix B sketches power one for rho_n -> 1 at arbitrary rates, but it asserts without proof that 'Step 2 of the proof of Theorem 4 still yields n^3 P(I12=I13=1) -> 0'. That step relies on the special y,z change of variables and the critical-scale condition; when sqrt(log n)(rho_n-1) -> +-infinity the saddle structure changes, and the b2 correlation estimate is not supplied. Similarly, the paper infers from the fixed-rho formula that rho_n -> infinity forces power to tend to the size, but no theorem covers that regime; one must show the equatorial perturbation eT_i = O_P(1/sqrt(Delta_n)) does not alter the null Gumbel limit. Since the abstract and Sections 4 and 5 state the exact threshold without these qualifications, the boundary behavior is load-bearing: if the missing b2 bounds fail, the Poisson approximation at rho=1 breaks down and the claimed discontinuous phase transition may not be universal.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the packing test for uniformity on the high-dimensional sphere under Fisher--von Mises--Langevin (FvML) and Watson alternatives. Using Poisson approximation, the author derives non-null limiting distributions for the largest squared inner product. Theorem 1 gives a shifted Gumbel limit under FvML alternatives with concentration κ=τ p^{3/4}/(log n)^{1/4}. Theorems 2--4 give Gumbel limits under Watson alternatives in three regimes: local (ρ>1), non-local (ρ∈(0,1)), and critical (ρ=1), where ρ is the limiting value of Δ_n/√(p log n) with Δ_n=(p−2κ_n)/2. The paper concludes that the FvML detection threshold is Θ(p^{3/4}/(log n)^{1/4}) and the Watson threshold is p−2κ=Θ(√(p log n)), with a discontinuous second-order phase transition at ρ=1. Proofs are built on tangent--normal decompositions, Chen--Stein Poisson approximation, Laplace's method, and large-deviation estimates.","tokens_in":41662,"tokens_out":14089,"duration_ms":117610,"significance":"If the boundary gaps are closed, the paper would provide the first exact non-null extreme-value analysis of the packing test under two standard high-dimensional directional models. The results would rigorously confirm the empirical suboptimality of the packing test and reveal a genuinely novel second-order phase transition in the Watson model. The mathematical work is substantial: the tail asymptotics in Propositions 3, 5, 6, and 7 are derived with explicit error rates; the Chen--Stein b2 term is addressed via truncation and saddle-point analysis; and the limiting constants are explicit and parameter-free. The paper is not circular: prior work is cited only for context and for comparison rates, not as an input to the proofs. However, as detailed below, the exact-threshold claims in the abstract and Section 5 go beyond what the stated theorems and the sketched Appendix B actually prove.","major_comments":[{"comment":"The exact-threshold claim 'p−2κ=Θ(√(p log n))' is not established by the theorems as stated. Theorem 2 covers only ρ_n→ρ∈(1,∞), Theorem 3 only ρ_n→ρ∈(0,1), and Theorem 4 only √(log n)(ρ_n−1)→Δ∈R. The inference from Theorem 2 that the limiting power tends to α as ρ→∞ is a statement about fixed-ρ limits, not about sequences ρ_n→∞; likewise, Theorem 3 gives no control for ρ_n→0. Since the Θ notation requires both Δ_n/√(p log n)→∞ (no detection) and Δ_n/√(p log n)→0 (detection with probability tending to one), the stated threshold is stronger than the proved convergence results. Please either add proofs for these boundary regimes or rephrase the abstract and Section 5 to state the threshold only as a consequence of Theorems 2--4 for the regimes they cover.","section":"§4.1, Eq. (12)"},{"comment":"The proof of power one for ρ_n→1 at arbitrary rates is not complete. The assertion 'Step 2 of the proof of Theorem 4 still yields n^3 P(I12=I13=1)→0' is made without proof, but Step 2 in Section 10.2 relies on the critical-scale change of variables y=L^{1/4}(α1+α2)/√2, z=√(L/2)(α1−α2) and on the limit √(L)(ρ_n−1)→Δ. When √(L)(ρ_n−1)→±∞ the saddle structure and the associated rate (62) are different, so the b2 bound cannot be taken as a black box. Because Eq. (12) and the threshold claim use power one at ρ=1, this missing correlation estimate is load-bearing; if the estimate fails, the Poisson approximation at the boundary would break down and the phase-transition conclusion would not follow.","section":"Appendix B"},{"comment":"The power formula (12) is stated for all ρ∈[0,∞), but the proof of the ρ∈[0,1) part relies on Theorem 3 with ρ∈(0,1) and on Appendix B for ρ=1; the endpoint ρ=0 is not covered by either. If the intended definition of detection threshold does not require the ρ_n→0 case, please state this explicitly; otherwise supply a proof for Δ_n=o(√(p log n)). The same issue affects the interpretation of the non-local regime as 'asymptotically powerful throughout this regime', since ρ_n→0 is part of that regime by the notation Δ_n/√(p log n)→ρ∈[0,1).","section":"§4.2, Theorem 3"}],"minor_comments":[{"comment":"The proof omits the minimization ('straightforward minimization of the quadratic function'). Since Lemma 6 supplies the constant (3+c)/2 used in the b2 estimate of Theorem 4, please include the explicit computation.","section":"Appendix A, Lemma 6"},{"comment":"The abstract and Section 5 state the Watson threshold without the technical conditions p/(log n)^2→∞ (Theorems 1--2) and p/(log n)^3→∞ (Theorems 3--4); these qualifications should appear wherever the threshold is asserted.","section":"Abstract and §5"},{"comment":"The symbol L is defined both as log n and as a generic sequence diverging to infinity; this dual use is confusing in Propositions 5--7 and should be disambiguated.","section":"§2, Notation"},{"comment":"In the displayed power formulas, the exponent on (1−α) is not visible in the manuscript rendering; please verify that β_FvML=1−(1−α)^{cosh(2τ^2)} and β_Wat=1−(1−α)^{(1−ρ^{-2})^{-1/2}} appear correctly in the final version.","section":"After Theorem 1 and Eq. (12)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's fixed-regime derivations are careful and appear sound; the main obstacle is the unproved boundary regimes behind the Θ statement. If the author can supply the missing boundary-case proofs, or explicitly narrow the claimed threshold to the regimes covered by Theorems 2--4, the paper would be a strong contribution. I do not see evidence of circularity; prior work is used only for context and comparison rates."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Tuan Pham's paper gives exact detection thresholds and non-null Gumbel limits for the packing test under FvML and Watson alternatives. This is genuinely new: prior work showed blindness at the minimax scales, but not the packing test's own thresholds, and the discontinuous second-order phase transition at rho=1 is a real discovery. The proofs are largely self-contained, the Poisson approximation structure is clear, and the constants check out (e.g., the shift 2 log cosh(2 tau^2), the factor (1-rho^{-2})^{-1/2}). The paper is also honest about its technical conditions, e.g., p/(log n)^3 -> infinity in Theorems 3-4.\n\nThe soft spot is exactly where the stress-test note points. Theorems 2 and 3 give fixed rho in (1,infinity) and (0,1), and Theorem 4 gives rho_n at the critical scale. But the abstract and Section 4 claim the full threshold Theta(sqrt(p log n)) and the power formula (12), including power one for rho_n -> 1 at arbitrary rates and power tending to size for rho_n -> infinity. Those boundary cases are not actually proved. Appendix B is a sketch: it asserts without proof that Step 2 of Theorem 4 still yields n^3 P(I12=I13=1) -> 0 when rho_n -> 1. That step relied on the y,z change of variables at critical scale; away from that scale the saddle structure changes and the b2 bound is not supplied. Similarly, no theorem covers rho_n -> infinity; you need uniformity in rho to pass from the fixed-rho limit to the boundary, and it isn't there.\n\nI don't think this is fatal—the missing pieces look patchable, and the main phenomenon is probably right. But as it stands, the exact-threshold claim is slightly ahead of the proof. The paper would be stronger if the boundary cases were either proved or explicitly stated as conjectures/conditions. The sketched minimization in Lemma 6 and the p/(log n)^3 condition are minor by comparison.\n\nWho is this for? People working on high-dimensional uniformity testing and extreme-value statistics on the sphere. It deserves a serious referee: the core argument is substantive and mostly rigorous, and the gap is in the boundary of the phase transition, not in the main mechanism. I would send it to review, with the instruction to push for a complete Appendix B or a scaled-back claim.","headline":"Sharp non-null asymptotics for the packing test, with the Watson boundary regime sketched rather than proved.","tokens_in":42289,"tokens_out":1823,"would_cite":true,"duration_ms":16257,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H15","62E20","60G70","60F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives the exact detection thresholds and non-null limiting distributions of the packing test for spherical uniformity under FvML and Watson alternatives.","keywords":["packing test","spherical uniformity","detection threshold","Fisher-von Mises-Langevin distribution","Watson distribution","Poisson approximation","Chen-Stein method","phase transition"],"falsifier":"Simulate the Watson model at the critical scale $\\Delta_n/\\sqrt{p\\log n}\\to 1$ with large $p$ and $n$, and estimate $n^3\\mathbb{P}(|X_1^\\top X_2|>t_n,\\,|X_1^\\top X_3|>t_n)$: if this quantity fails to tend to zero, the Chen--Stein correlation bound in Sections 8.2--9.2 is false and Theorem 4's limiting coefficient is not $-1/2$.","tokens_in":41172,"feed_emoji":"🎯","tokens_out":11220,"duration_ms":83883,"temperature":0.7,"pith_summary":"The paper pins down the exact detection thresholds and non-null limiting laws of the packing test for spherical uniformity, in the two standard high-dimensional parametric alternatives. Under Fisher--von Mises--Langevin alternatives, the threshold is $\\kappa = \\Theta(p^{3/4}/(\\log n)^{1/4})$; under Watson alternatives, it is $p-2\\kappa = \\Theta(\\sqrt{p\\log n})$. Because these thresholds are strict, the packing test is suboptimal in sample size compared with the known minimax rates in both models. The paper also shows that, in the Watson model, the asymptotic scaling of the largest squared inner product undergoes a discontinuous second-order phase transition at the critical regime, while no analogous phenomenon occurs in the FvML model.","feed_headline":"Exact thresholds show packing test pays a log n penalty","feed_subtitle":"Under FvML and Watson alternatives, the test is suboptimal in sample size and has a new phase transition.","key_machinery":"The carrying mechanism is Poisson (Chen--Stein) approximation for the exceedance indicators $I_{ij} = \\mathbf{1}\\{|X_i^\\top X_j|>t_n\\}$, viewed through the tangent--normal decomposition of each observation. The mean of the approximating Poisson variable comes from sharp tail asymptotics of a single inner product, with a Laplace-method saddle-point analysis of the rate function; the variance is controlled by the dependency-graph term $b_2$, whose negligibility requires truncating to small neighborhoods of the saddle points and using uniform large-deviation bounds. The named objects that carry the argument are the rate-function minimizers $A(\\rho)=\\sqrt{2(1+\\rho^2)}$, $a(\\rho)=2(\\sqrt{2(1+\\rho^2)}-\\rho)$, $K(\\rho)$, and the critical kernel $K_{\\rm cr}(\\Delta)$; the change in saddle-point structure at $\\rho=1$ is what produces the phase transition.","core_discovery":"The central claim is that the packing test statistic $P_n = p\\max_{i<j}(X_i^\\top X_j)^2 - 4\\log n + \\log\\log n$ has an exact, fully quantified non-null behaviour. Under FvML alternatives with $\\kappa_n = \\tau p^{3/4}/(\\log n)^{1/4}$, $P_n$ converges to a shifted Gumbel law with shift $2\\log\\cosh(2\\tau^2)$, giving the detection threshold $\\Theta(p^{3/4}/(\\log n)^{1/4})$. Under Watson alternatives with $\\Delta_n = (p-2\\kappa_n)/2$ and $\\rho_n = \\Delta_n/\\sqrt{p\\log n}\\to\\rho$, the limiting law of $P_n$ is Gumbel in all regimes, but the scaling of the largest squared inner product changes: for $\\rho>1$ the null scaling persists and the threshold is $p-2\\kappa = \\Theta(\\sqrt{p\\log n})$; for $\\rho=1$ the coefficient of $\\log\\log n/p$ jumps from $-1$ to $-1/2$; for $0<\\rho<1$ both the leading and second-order coefficients depend on $\\rho$. These results rigorously confirm the empirical observation that the packing test is strictly suboptimal in sample size for both models.","pith_inferences":["A consequence the paper leaves implicit is that at fixed concentration the FvML threshold forces an exponentially large sample: $\\log n \\sim p^3/\\kappa^4$ is necessary for detection.","The critical scale restriction $\\sqrt{\\log n}\\,(\\rho_n-1)\\to\\Delta$ suggests an uncovered family of intermediate regimes with $\\rho_n-1$ of order $(\\log n)^{-\\gamma}$ for $\\gamma\\ne 1/2$, where the second-order coefficient may interpolate between $-1$ and $-1/2$.","A direct way to test the phase transition in simulations is to fit the slope of $pM_n^2$ versus $\\log\\log n$ at $\\rho_n\\approx 1$; Theorem 4 predicts the slope $-1/2$, whereas both adjacent regimes give $-1$."],"forward_implications":["At the FvML minimax scale $\\kappa\\asymp p^{3/4}/\\sqrt n$, the packing test has asymptotic power equal to its size, so it cannot detect at the optimal rate.","At its own thresholds, the packing test has explicit asymptotic power: $1-(1-\\alpha)^{\\cosh(2\\tau^2)}$ under FvML and the piecewise formula (12) under Watson.","Under Watson alternatives with $\\rho\\le 1$, the packing test has asymptotic power one; for $\\rho>1$, its power is strictly between size and one.","The largest squared inner product scales like $4\\log n/p$ whenever $\\rho\\ge 1$, with the $\\log\\log n$ coefficient jumping from $-1$ to $-1/2$ at $\\rho=1$, and acquires $\\rho$-dependent coefficients for $\\rho<1$."],"supporting_citations":[{"why":"Introduces the packing test statistic and its null Gumbel distribution, the baseline to which the paper's non-null laws reduce at $\\tau=0$.","marker":"Cai et al. [2013]"},{"why":"Provides the Chen--Stein Poisson approximation lemma with dependency-graph bounds $b_1,b_2$ used in every theorem.","marker":"Arratia et al. [1989]"},{"why":"Establishes the FvML minimax detection rate $\\kappa\\sim p^{3/4}/\\sqrt n$ that the packing threshold is compared against.","marker":"Cutting et al. [2017]"},{"why":"Supplies the Bingham test's Watson non-null theory and the low-dimensional Watson minimax rate used in the comparison.","marker":"Cutting et al. [2022]"},{"why":"Supplies the high-dimensional Watson minimax rate $p/2-\\Theta(\\sqrt{pn})$ that defines suboptimality in that regime.","marker":"Jiang and Pham [2025b]"},{"why":"Shows the packing test is asymptotically blind at the FvML optimal scale, the motivation for computing its actual threshold.","marker":"Jiang and Pham [2025a]"},{"why":"Gives the self-normalized Cramér-type large-deviation theorem used in conditional tail-probability asymptotics.","marker":"Jing et al. [2003]"}],"fun_headline_variants":["Packing test pays log n penalty in both uniformity models","Exact thresholds confirm packing test's log n weakness","Packing test suboptimal: exact thresholds reveal log n tax","Packing test's exact threshold shows log n penalty and phase jump","Packing test is suboptimal: log n penalty quantified exactly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the correlation term $b_2$, specifically the estimate $\\mathbb{P}(I_{12}=1,I_{13}=1)=o(n^{-3})$ in the Watson analysis, is negligible after truncation to saddle-point neighborhoods; without that estimate the Poisson approximation and the four limiting laws do not go through.","fun_headline_variants_meta":{"raw":{"variants":["Packing test pays log n penalty in both uniformity models","Exact thresholds confirm packing test's log n weakness","Packing test suboptimal: exact thresholds reveal log n tax","Packing test's exact threshold shows log n penalty and phase jump","Packing test is suboptimal: log n penalty quantified exactly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000739,"raw_usage":{"total_tokens":3322,"prompt_tokens":991,"completion_tokens":2331,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":2247}},"tokens_in":607,"tokens_out":2331,"duration_ms":14265,"temperature":1.0,"reasoning_tokens":2247,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:21:22.460916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the Watson model at the critical scale $\\Delta_n/\\sqrt{p\\log n}\\to 1$ with large $p$ and $n$, and estimate $n^3\\mathbb{P}(|X_1^\\top X_2|>t_n,\\,|X_1^\\top X_3|>t_n)$: if this quantity fails to tend to zero, the Chen--Stein correlation bound in Sections 8.2--9.2 is false and Theorem 4's limiting coefficient is not $-1/2$.","supporting_citations":[],"review_version":2}