{"id":"0efb5f51-8c7e-4070-ad92-7212b63aaf06","arxiv_id":"2608.08112","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A distribution-free CUSUM test for angular mean-direction changepoints is proposed, with a Kolmogorov limiting null distribution and financial applications.","lead":"A new test uses a torus-based 'square of an angle' transform to detect shifts in the mean direction of circular data with a CUSUM statistic. The authors apply it to the daily timing of extreme prices in Bitcoin, Ethereum, and Gold.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Consistency claim (Theorem 5/Corollary 7) rests on unproven 'Since Δ≠0' in Appendix 8.4; for continuous periodic h(μ)=E[sgn(θ)A_C^(0)(θ)], Δ=0 occurs for distinct mean directions, so the test is not consistent for all stated alternatives.","rationale":"The reader's weakest assumption exactly identifies the gap; my agreement is 'agree'. The null lemma is not the issue: for i.i.d. a_i with finite variance, the CUSUM statistic converges to sup|B0|. The central claim is the H1 behavior. In Appendix 8.4, Eqs. (8.12)-(8.13) decompose T_n into a Brownian bridge plus √n c*(u), where c*(u) is proportional to Δ. The proof then states 'Since Δ≠0' without derivation. This is load-bearing. The transformed score is bounded, and the map μ↦E[a] is continuous and periodic on the circle. Such a map cannot be one-to-one unless it is constant; therefore there exist mean-direction changes with zero drift, and for those alternatives the CUSUM statistic has no diverging term. Consequently the claimed consistency and changepoint-estimator consistency fail for the stated H1. The paper's simulations only cover parameter pairs where Δ empirically differs from zero, so they cannot detect this gap. The i.i.d. working approximation in the application is secondary and is explicitly flagged by the authors. Since the central theoretical claim is unsupported as stated, the reader's REJECT verdict is appropriate, so no adjustment is needed.","tokens_in":25901,"tokens_out":6785,"duration_ms":73807,"concrete_test":"Compute h(μ)=∫_0^{2π} sgn(θ)A_C^(0)(θ) f_vM(θ;μ,κ=3)dθ by quadrature on a fine grid μ∈[0,2π); locate two distinct μ1,μ2 with |h(μ1)−h(μ2)|<1e−12. Then simulate n=10^4 von Mises observations with mean μ1 before k*=n/2 and μ2 after, run the SAMC test at the 5% level over 10^3 replications. If empirical power stays near 0.05 instead of tending to 1, the Δ≠0 premise fails and Theorem 5/Corollary 7 are false as stated. If no equal-h pair exists (possible only if h is constant), the concern is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's H1 consistency (Theorem 5 and Corollary 7) depends entirely on the assertion in Appendix 8.4 that 'Since Δ≠0', where Δ=m1−m2 and m_i=E[a_i]=E[sgn(Θ)A_C^(0)(Θ)]. This is asserted, not proved, and is generally false. For fixed concentration, h(μ)=m(μ) is a continuous 2π-periodic function on the circle. A continuous periodic nonconstant map from S^1 to R cannot be injective, so there exist μ1≠μ2 with h(μ1)=h(μ2); if h is constant, this holds for every pair. In either case the alternative in (3.1) does not guarantee Δ≠0. When Δ=0, the drift term √n c*(u) in Eq. (8.13) vanishes, T_n(u) remains O_p(1) (a Brownian bridge plus negligible term), M_n does not diverge, and Type-II error does not go to zero. Thus the consistency and changepoint-estimator claims are unsupported as stated; the theorem would need an explicit identifiability condition Δ≠0, which the paper neither states nor verifies. Lemma 4 (null distribution) is a standard FCLT application and is not affected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a distribution-free CUSUM test for a single changepoint in the mean direction of independent circular observations. Each angle θ_i is transformed to a_i = sgn(θ_i) A_C^{(0)}(θ_i), where A_C^{(0)} is the 'square of an angle' derived from the proportionate area on a torus. The test statistic is the supremum of the studentized CUSUM process of the a_i. Under H_0 the paper claims M_n converges to the Kolmogorov distribution (Lemma 4); under H_1 it claims the test is consistent and the estimated changepoint fraction converges to u* (Theorem 5 and Corollary 7). The method is compared by simulation with a rank-based test (Lombard 1986), a graph-based test (Chen and Zhang 2015), and an arc-length baseline, and is applied to daily timestamps of extreme prices in Bitcoin, Ethereum, and gold. Section 6 acknowledges that the i.i.d. assumption is only a working approximation for the financial data.","tokens_in":26139,"tokens_out":8254,"duration_ms":78049,"significance":"Lemma 4 is a standard functional central limit theorem argument and is essentially correct, and the simulation study is extensive, covering von Mises and wrapped Cauchy families, local alternatives, and a comparison with the von Mises GLRT. However, the paper's principal theoretical claims beyond the null limit—consistency under H_1 and convergence of the changepoint estimator—are not established because the proof assumes Δ≠0 without proof, and this assumption is generally false for smooth circular distributions. The empirical application reports p-values under an i.i.d. assumption the paper itself describes as a working approximation. The result, if properly repaired, would be a useful addition to the circular changepoint literature, but in its current form the central claims are unsupported. No working code is provided; the availability statement contains a placeholder.","major_comments":[{"comment":"The consistency proof of Theorem 5 and Corollary 7 depends entirely on the assertion that Δ = m_1 - m_2 is nonzero whenever μ_1 ≠ μ_2 in (3.1). This is never proved and is not generally true. Writing h(μ) = E[sgn(Θ) A_C^{(0)}(Θ)] with Θ ~ F(·; μ), the integrand is bounded and piecewise continuous, so h is continuous and 2π-periodic. A continuous periodic map from S^1 to R cannot be injective unless it is constant; hence for any nonconstant h there exist μ_1 ≠ μ_2 with h(μ_1)=h(μ_2), and if h is constant the equality holds for all pairs. For such alternatives the drift term √n c*(u) in (8.13) vanishes, T_n(u) remains O_p(1), M_n does not diverge, and the probability of Type-II error does not tend to zero. Thus Theorem 5 and Corollary 7 fail for the alternative class stated in (3.1). The identifiability condition Δ≠0 would need to be added as an explicit assumption and verified, not merely asserted, for the distributions used in the simulations and the data application.","section":"Appendix 8.4, Eq. (8.13) and the sentence 'Since Δ≠0'"},{"comment":"The paper explicitly acknowledges that the i.i.d. assumption underlying Lemma 4 is only a 'working approximation' for the financial timestamp sequences, and that the runs test 'cannot establish independence, stationarity, or all probabilistic conditions required by the asymptotic theory.' Under serial dependence the studentized CUSUM need not converge to the Brownian bridge, so the p-values reported in Tables 5-8 are not justified. Because these p-values are the quantitative support for the paper's empirical claims, the application section should be recast as exploratory, or a dependence-robust calibration (e.g., block bootstrap or a long-run variance estimator) should be supplied before the market-event alignments are presented as evidence.","section":"Section 6.A and Section 5, Tables 5-8"}],"minor_comments":[{"comment":"The title contains the typo 'APPLICA TION' and should read 'APPLICATION'.","section":"Title, page 1"},{"comment":"The y-axis label 'lavel' appears in all four figures and should be spelled 'level'.","section":"Figures 6, 7, 8, and 9"},{"comment":"The sentence introducing the test statistic contains the garbled fragment '[h tan^{-1*}(E(sin Θ)/E(cos Θ))]'; the mean-direction formula is not properly displayed and the symbol h is used without definition in that context.","section":"Section 3.1, paragraph preceding Eq. (3.4)"},{"comment":"The post-changepoint mean direction is reported as '87.7083(135.00)'; 87.7083 radians is not a valid circular mean and does not equal 135 degrees, indicating a unit or transcription error that should be corrected.","section":"Table 7, last row"},{"comment":"The null data are generated as N(μ, σ²) without specifying μ and σ; because the limiting distribution is distribution-free this is acceptable in principle, but the parameter values and the number of replications N used for Tables 3 and 4 should be stated.","section":"Algorithm 1"},{"comment":"The repository link is given as 'INSERT-PERMANENT-REPOSITORY-LINK'; the reproducibility claim is therefore not currently verifiable.","section":"Data and Code Availability Statement"}],"recommendation":"reject","confidential_remarks":"The manuscript's central consistency theorem is not a local gap: the proof assumes an identifiability condition that is not implied by the stated alternative class, and the empirical conclusions in Section 5 depend on an assumption the authors themselves admit is only approximate. Adding a Δ≠0 assumption would change the paper's target from 'change in mean direction' to 'change in E[a_i]' and would require reworking the simulations and the financial interpretation. In view of the scope change this would entail, I do not see this as a fixable-by-revision issue. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: the null-distribution part is fine and the geometric transformation is genuinely new, but the consistency theorem rests on an unproven \"Δ≠0\" and as stated the test is not consistent for all alternatives in H1. Expect heavy revision, not desk reject.\n\nWhat's new: applying the square-of-angle transform (from their earlier torus geometry paper) to build a CUSUM statistic for circular mean changepoints. That is a real addition to the angular changepoint toolkit. The proof of Lemma 4 is a standard FCLT on the i.i.d. transformed sequence; the simulations are extensive, including von Mises and wrapped Cauchy, comparisons to Lombard, gSeg, arc-length baseline, plus a local-power study against the von Mises GLRT. The financial application to daily timestamps of extreme prices is a sensible use case and the runs tests are reported. The authors also explicitly list limitations: i.i.d. as a working approximation, no multiple-changepoint machinery.\n\nSoft spots: the load-bearing one is Theorem 5 and Corollary 7. Appendix 8.4 defines Δ=m1−m2 and then simply says \"Since Δ≠0\". That condition is not derived and is generally false for smooth circular families: h(μ)=E[sgn(Θ)A_C^(0)(Θ)] is continuous and 2π-periodic, hence non-injective, so there are distinct mean directions with identical expected score. For those alternatives the drift term vanishes, M_n doesn't diverge, and consistency fails. The theorem needs an explicit identifiability assumption, e.g. Δ(μ1,μ2)≠0, which may hold for many pairs but needs to be stated and checked. This is not a minor gap; it's the core of the H1 claim. Also, the code availability statement is a placeholder (\"INSERT-PERMANENT-REPOSITORY-LINK\"), so reproducibility is limited. The p-values in the data tables are also a bit sloppy (e.g., a subsegment p-value of 0.0004 for a segment of length 970 with no further changepoint seems inconsistent with the stated calibration, and Table 7 has a suspicious entry). These are minor relative to the Δ issue.\n\nWho this is for: researchers in circular statistics and applied changepoint work who might use the transformation or the null result. It deserves a serious referee: the idea is worthwhile and the null part is correct, but the consistency claim needs repair before publication. My recommendation: send to review, insist the authors either prove Δ≠0 under stated conditions or restrict H1 accordingly, and require the code link to be completed.","headline":"The null-distribution result and the geometry-based transformation are worthwhile, but the consistency proof rests on an unproven and generally false 'Δ≠0', so the central H1 claim needs major repair.","tokens_in":26700,"tokens_out":2128,"would_cite":false,"duration_ms":21625,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H11","62G10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A torus-geometry “square of an angle” turns circular observations into signed real scores, so a studentized CUSUM has a Kolmogorov limit under the null and a consistent changepoint estimator under the alternative.","keywords":["angular data","changepoint detection","circular mean direction","distribution-free test","CUSUM","Kolmogorov distribution","square of an angle","finance"],"falsifier":"Compute the function $h(\\mu)=\\mathbb{E}[\\operatorname{sgn}(\\Theta)A_C^{(0)}(\\Theta)]$ for a smooth circular density $f(\\theta-\\mu)$, such as the von Mises with $\\kappa=3$, and use periodicity and continuity of $h$ to find two mean directions $\\mu_1\\ne\\mu_2$ with $h(\\mu_1)=h(\\mu_2)$; simulate $n=500$ observations with a changepoint at $n/2$ between those means and measure the rejection rate of $M_n$. If the rejection rate stays at the nominal level while the mean direction has plainly shifted, the consistency claim does not cover that alternative.","tokens_in":25652,"feed_emoji":"📈","tokens_out":14577,"duration_ms":126722,"temperature":0.7,"pith_summary":"The paper sets out to establish a distribution-free test for a changepoint in the mean direction of angular data, along with an estimator of the changepoint location; the procedure is called the Square Angle Mean Change (SAMC) test. The key move is to map each angle $\\theta_i$ to a signed “square of an angle” $a_i=\\operatorname{sgn}(\\theta_i)A_C^{(0)}(\\theta_i)$, a quadratic measure of separation derived from the area element of a torus, and then run a studentized cumulative-sum (CUSUM) statistic on the real-valued sequence $a_i$. The paper claims that under the null hypothesis of no change the maximum of this CUSUM converges in distribution to the Kolmogorov law $\\sup_{0<u<1}|B_0(u)|$, making critical values pivotal across circular distributions; under the alternative the test is consistent and the estimated changepoint fraction $\\hat{k}^*/n$ converges in probability to the true fraction $u^*$. The practical payoff is that analysts can test for regime shifts in directional data without specifying a parametric circular model, and the paper demonstrates this with simulations against rank-based, graph-based, likelihood-based, and arc-length benchmarks and with applications to the daily timing of extreme prices in Bitcoin, Ethereum, and gold.","feed_headline":"Torus score turns circular data into a Kolmogorov CUSUM test","feed_subtitle":"No parametric assumption is needed to detect shifts in the daily timing of price extremes.","key_machinery":"The load-bearing object is the “square of an angle,” defined as $A_C^{(0)}(\\theta)=A_T[(0,0),(\\theta,\\theta)]$, the smallest of the four torus-surface areas determined by the rectangle from $(0,0)$ to $(\\theta,\\theta)$, normalized by $4\\pi^2 rR$ with $r/R=1$. Lemma 3 gives the closed form $A_C^{(0)}(\\theta)=(2\\pi)^{-2}\\theta(\\theta+\\sin\\theta)$ on $[0,\\pi]$ and $(2\\pi)^{-2}(2\\pi-\\theta)(2\\pi-(\\theta+\\sin\\theta))$ on $(\\pi,2\\pi]$. The signed scores $a_i=\\operatorname{sgn}(\\theta_i)A_C^{(0)}(\\theta_i)$ are what carry the argument: they convert the circular mean-direction problem into a real-valued sequence with finite second moments, so the studentized CUSUM $T(k)$ inherits a Brownian-bridge limit and hence the Kolmogorov null distribution. The construction is one-to-one almost everywhere and independent of the mean direction, so the complete transformed sample preserves Fisher information; the efficiency trade-off in the paper comes from using moment-based CUSUM aggregation rather than the full transformed likelihood.","core_discovery":"The central claim is that changepoint detection in angular mean direction reduces to a one-dimensional CUSUM problem through the transformation $a_i=\\operatorname{sgn}(\\theta_i)A_C^{(0)}(\\theta_i)$, where the “square of an angle” $A_C^{(0)}(\\theta)$ is the proportionate minimum area cut out by the point $(\\theta,\\theta)$ on a torus. On the i.i.d. transformed sequence, the statistic $M_n=\\max_{1\\le k<n}|T(k)|$ converges in distribution to $\\sup_{0<u<1}|B_0(u)|$, the Kolmogorov distribution, under $H_0$ (Lemma 4). Under $H_1$, Theorem 5 states that $\\hat{k}^*/n \\to u^*$ in probability, and Corollary 7 states that the type-II error probability decays to zero, provided the change in mean direction induces a nonzero drift $\\Delta$ in the expected transformed score. The proof route is the functional central limit theorem on the $a_i$ sequence, whose finite second moments and stationary variance structure are guaranteed by the torus-geometric construction.","pith_inferences":["Editorial inference: the expected transformed score $h(\\mu)=\\mathbb{E}[\\operatorname{sgn}(\\Theta)A_C^{(0)}(\\Theta)]$ under a smooth circular density $f(\\theta-\\mu)$ is continuous and periodic in $\\mu$, so it cannot be one-to-one on the circle; for mean-direction pairs with $h(\\mu_1)=h(\\mu_2)$ the CUSUM drift $\\Delta$ vanishes, and the consistency proof of Theorem 5 would not apply to such alternat","Editorial inference: the i.i.d. assumption is the entry point for the Brownian-bridge limit; for high-frequency financial timestamps, a dependence-robust version with a long-run variance estimator would be the natural extension, and the paper explicitly flags this as open.","Editorial inference: although the paper stops at a single changepoint, the same signed square-of-angle scores could be fed into a recursive binary-segmentation wrapper to detect multiple changes; the paper notes that such an extension needs separate penalization and error control."],"forward_implications":["Critical values for $M_n$ can be taken from the Kolmogorov distribution (or its finite-grid version $K_\\infty^{(n)}$) regardless of the underlying circular law, so practitioners need no distributional fit and no tuning parameter beyond the significance level.","A single scan of the sequence, of order $O(n)$ operations, yields both a decision and an estimated changepoint location, and the estimated fraction $\\hat{k}^*/n$ converges to the true changepoint fraction when a change is present.","For mean-direction shifts in $[-\\pi/2,\\pi/2]$, the simulations show the SAMC test at or above the power of Lombard's rank-based test, the graph-based gSeg test, and an arc-length CUSUM, with the largest gains for concentrated von Mises data.","On the financial data, the test detects a changepoint in the daily timing of the lowest price for Bitcoin (August 2019), Ethereum (October 2018), and gold (March 2023), and in the highest-price timing for gold (March 2023), while finding no change in the highest-price timing for Bitcoin or Ethereum."],"supporting_citations":[{"why":"It supplies the torus area element and the definition of the proportionate area behind the square of an angle.","marker":"Biswas and Banerjee (2025)"},{"why":"It supplies Donsker's theorem and the Skorohod-topology embedding used to get the Brownian-bridge limit in Lemma 4.","marker":"Billingsley (2013)"},{"why":"It supplies Slutsky's theorem used in the proof of Lemma 4.","marker":"Athreya and Lahiri (2006)"},{"why":"It supplies the argmax continuous mapping theorem used to prove consistency of the changepoint estimator in Theorem 5.","marker":"Ferger (2004)"},{"why":"It provides the rank-based angular changepoint test that SAMC is compared against in the power simulations.","marker":"Lombard (1986)"},{"why":"It provides the graph-based gSeg changepoint method that SAMC is compared against.","marker":"Chen and Zhang (2015)"},{"why":"It provides the von Mises likelihood-ratio changepoint benchmark used in the local-power study.","marker":"Ghosh et al. (1999)"},{"why":"It supplies the circular arc-length distance used to define the arc-length benchmark test $C_n$.","marker":"Mardia (2000)"}],"fun_headline_variants":["Angular CUSUM test finds shifts in timing of price extremes","Torus geometry yields distribution-free changepoint test for angles","Kolmogorov CUSUM for angular data detects changepoints in finance","No assumptions: shift in timing of price extremes caught by angular CUSUM","Angular changepoint test works on Bitcoin, Ethereum, and Gold spikes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a change in circular mean direction always changes the average of the signed square-of-angle values, because the consistency proof assumes this change $\\Delta$ is nonzero without deriving it; if two different mean directions share the same average, the CUSUM drift disappears and consistency is not established.","fun_headline_variants_meta":{"raw":{"variants":["Angular CUSUM test finds shifts in timing of price extremes","Torus geometry yields distribution-free changepoint test for angles","Kolmogorov CUSUM for angular data detects changepoints in finance","No assumptions: shift in timing of price extremes caught by angular CUSUM","Angular changepoint test works on Bitcoin, Ethereum, and Gold spikes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000621,"raw_usage":{"total_tokens":2870,"prompt_tokens":929,"completion_tokens":1941,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":1844}},"tokens_in":545,"tokens_out":1941,"duration_ms":15342,"temperature":1.0,"reasoning_tokens":1844,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:25:14.521231+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the function $h(\\mu)=\\mathbb{E}[\\operatorname{sgn}(\\Theta)A_C^{(0)}(\\Theta)]$ for a smooth circular density $f(\\theta-\\mu)$, such as the von Mises with $\\kappa=3$, and use periodicity and continuity of $h$ to find two mean directions $\\mu_1\\ne\\mu_2$ with $h(\\mu_1)=h(\\mu_2)$; simulate $n=500$ observations with a changepoint at $n/2$ between those means and measure the rejection rate of $M_n$. If the rejection rate stays at the nominal level while the mean direction has plainly shifted, the consistency claim does not cover that alternative.","supporting_citations":[],"review_version":1}