{"id":"d6704177-427f-404b-b6b7-0e85bb8277c0","arxiv_id":"1908.04006","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves, via conformal invariance of Brownian motion, that the hyperbolic secant law is invariant under g(x)=(2/π)log|sinh(πx/2)|, and gives a general transfer principle for exit distributions of symmetric domains.","lead":"This math note shows how Brownian motion in symmetric regions can be used to build transformations that leave probability distributions fixed. It proves a new map that preserves the hyperbolic secant distribution, and re-derives some known results with short martingale proofs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The example map f(z)=Log cosh z used to prove Corollary 9 is not a conformal automorphism of W, so Proposition 8 does not apply as stated; a direct check is needed.","rationale":"Read in good faith, the paper's new result is Corollary 9. Proposition 8 is the transfer principle, and the example is intended to instantiate it. The reader flagged the independence and scaling assumptions; my stress test found a more direct gap in the instantiation of the theorem. The composed map f is not a conformal automorphism of the stated W: the derivative vanishes at 0, so the Carathéodory/bijection step in Proposition 8 cannot be invoked. With the rescaled W needed for the corollary's constants, the map has a pole at 0. These are not mere typos: they mean the printed proof does not establish the central claim. However, the conclusion is independently checkable, and the direct change-of-variables computation confirms it, so the mathematics is likely sound but the argument needs repair. The missing independence assertion in Proposition 3 is standard and externally supported, so I do not treat it as the most load-bearing issue; the non-automorphism gap is more central to the claimed novelty. The support for the final result through direct verification means the reader's conditional verdict remains appropriate.","tokens_in":6050,"tokens_out":13871,"duration_ms":147263,"concrete_test":"Perform the direct pushforward computation for μ(dx)=½sech(πx/2)dx under g(x)=(2/π)log|sinh(πx/2)|: for each y, solve g(x)=y to get x=±(2/π)asinh(e^{πy/2}), and check that μ(dx) pushes forward to μ(dy) using the derivative g'(x)=coth(πx/2). If the densities match, Corollary 9 is true and the flaw is confined to the stated proof of the example; if they do not, the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 8 is stated for a conformal automorphism f of W, but the example f(z)=Log cosh z on W={-π/2<Im z<π/2} has f'(0)=tanh 0=0, so it is not locally injective and cannot be an automorphism. If one instead rescales W to height 2 so that the density constants match the corollary, the analogous map is f(z)=(2/π)Log sinh(πz/2), which has a singularity at z=0 and is not a self-map of W. Thus Corollary 9 does not follow from Proposition 8 as printed; an additional argument or a direct calculation is needed. The scaling mismatch between the stated W and the corollary constants compounds this. I verified the final invariance by direct change of variables, so the claim itself appears true, but the proof is incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies invariance of probability distributions under analytic maps via planar Brownian motion. Section 1 uses optional stopping to compute exit distributions of Brownian motion, giving short proofs of the Cauchy law for half-plane exit, a hyperbolic secant law for strip exit, and a characteristic-function identity for log|Cauchy|. Section 2 proves that Newton's method applied to x^2+1 yields, for almost every starting point, empirical measures converging weakly to the standard Cauchy distribution, using the ergodicity of the Boole transformation. Section 3 proposes a general transfer principle (Prop. 8) for symmetric simply connected domains and applies it to find a nonlinear map g(x)=(2/\\pi)\\log|\\sinh(\\pi x/2)| that supposedly preserves the distribution (1/2)\\sech(\\pi x/2)dx. The central new assertion is Corollary 9.","tokens_in":6126,"tokens_out":16859,"duration_ms":187094,"significance":"The intended result, if properly proved, is a nice addition to the short list of exactly invariant distributions for explicit nonlinear maps, and the method connecting conformal invariance, Brownian exit laws, and distribution preservation is attractive. Section 2's ergodic-theoretic proof of the Cauchy empirical law for Newton's method is clean and self-contained. The paper's claims appear to be true: Corollary 9 can be verified directly by a change of variables, and the optional stopping arguments are essentially correct. However, the proof of the main new result as printed does not follow from the stated Proposition 8, because the map used is not a conformal automorphism, and there is a scaling mismatch between the domain and the corollary's constants.","major_comments":[{"comment":"The map f(z)=Log cosh z is not a conformal automorphism of W={-\\pi/2<Im z<\\pi/2}. Its derivative is f'(z)=tanh z, so f'(0)=0 and f is not locally injective at 0; moreover f(-z)=f(z), so f is not injective globally. The composition f=\\psi^{-1}\\circ \\phi\\circ\\psi inherits the failure of univalence from \\phi, whose derivative vanishes at i. Therefore Proposition 8 cannot be applied to this f, and Corollary 9 does not follow from the printed argument. Since a direct change-of-variables calculation does establish the invariant law, the gap is fixable, but the manuscript must either supply that calculation or prove a suitable extension of Proposition 8 to non-injective analytic self-maps with the required boundary behavior.","section":"Section 3 (construction of f)"},{"comment":"There is a scaling mismatch between the domain W and the constants in Corollary 9. For W={-\\pi/2<Im z<\\pi/2}, the upper boundary point is \\pi(x)=x+i\\pi/2, and the calculation in the text gives g(x)=Re f(\\pi(x))=Log|\\sinh x|, not g(x)=(2/\\pi)Log|\\sinh(\\pi x/2)|. Moreover, after rotating and scaling the exit law from Proposition 3, the distribution of Re(B_\\tau) for this W is (1/\\pi)\\sech x dx, not (1/2)\\sech(\\pi x/2)dx. The corollary corresponds instead to the strip {|Im z|<1} (or to a scaled version of the calculation), and this rescaling must be stated explicitly and carried through consistently.","section":"Section 3 (scaling)"},{"comment":"The proof asserts without justification that E[e^{\\theta Re(B_T)}]=cosh\\theta and that Re(B_T) and Im(B_T) are independent. Both facts are true here: Re(B_T) takes the values \\pm 1 with equal probability, and the sign is independent of |Im(B_T)| by reflection symmetry across the imaginary axis. But no argument is supplied, and this factorization is the only derivation of the sech law used in Section 3. The proof should include the symmetry argument explicitly.","section":"Section 1, Proof of Proposition 3"}],"minor_comments":[{"comment":"The phrase 'Arg(B_T) is uniform on {-\\pi/2,\\pi/2}' should say that Arg(B_T) takes the two values \\pm\\pi/2 with probability 1/2 each; as written it could suggest a continuous uniform distribution on the interval.","section":"Section 1, Proposition 4 proof"},{"comment":"The independence of ln|B_T| and Arg(B_T) is asserted rather than proved; a one-sentence reflection-symmetry argument across the imaginary axis would suffice.","section":"Section 1, Proposition 4 proof"},{"comment":"The reference to 'Theorem 1' in the proof of Proposition 4 should be 'Proposition 1'.","section":"Section 1, Proposition 4 proof"},{"comment":"The notation \\sinh(\\pi/2 x) is ambiguous; \\sinh(\\pi x/2) would be clearer.","section":"Section 3, Corollary 9"},{"comment":"The reference [Hor] lacks a year and venue information; if it is an online source, a URL and access date would be helpful.","section":"References"},{"comment":"The proof implicitly uses that F sends the standard Cauchy distribution to normalized Lebesgue measure on the circle; stating this fact explicitly would make the ergodicity transfer fully transparent.","section":"Section 2, Lemma 7"}],"recommendation":"major_revision","confidential_remarks":"The Section 3 gap is real but fixable: a direct change-of-variables proof of Corollary 9 is short, and the paper's main idea remains sound. The scaling inconsistency suggests the normalization used in the corollary was not the one carried through the calculation, and the authors should be asked to rewrite Section 3 carefully. The unproved symmetry facts in Section 1 are also easily supplied. I do not see grounds for doubting the truth of the results, only for doubting the completeness of the written proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: Sections 1 and 2 are clean and worth reading; Corollary 9 is true, but Section 3's proof as printed has a real gap. I checked the invariant map directly by change of variables, so the result is right, but it does not follow from the stated Proposition 8.\n\nThe transfer principle (Prop 8) is a nice observation and is likely correct for genuine automorphisms. The optional stopping proofs in Section 1 are neat, and the ergodic proof for the Boole transformation in Section 2 is elegant. The paper is honest about re-proving known results, and the new sech-law map is a legitimate extension.\n\nNow the soft spots, concentrated in Section 3. First, W is defined as {−π/2 < Im z < π/2}, but the exit distribution of Re(B_τ) from that strip is (1/π)sech(x), not the (1/2)sech(πx/2) used in Corollary 9. That scaling mismatch is concrete and easy to fix. Second, the map f(z)=Log cosh z is not a conformal automorphism of W — f'(0)=0 — so Proposition 8 does not apply. The composite f = ψ^{-1}∘ϕ∘ψ maps W into itself, but it is not one-to-one. The corollary therefore does not follow from the stated proposition; it is saved only by a direct calculation that the text does not provide.\n\nThere are smaller gaps too: the independence of Re(B_T) and Im(B_T) in Prop 3, and of ln|B_T| and Arg(B_T) in Prop 4, is asserted without proof. These are standard facts and can be supplied, but they are load-bearing.\n\nAll of this is fixable. The authors should either replace the example with one where f is a genuine automorphism, or prove a more general version of Prop 8 for non-injective conformal maps with the needed boundary behavior, or simply include the direct change-of-variables proof. As is, I would not accept without revision, but I would send it to a referee rather than desk reject: the central theorem is true and the earlier sections are solid.","headline":"Corollary 9 is true, but the proof as written doesn't establish it — the example doesn't satisfy the transfer principle's hypotheses, and the strip is scaled wrong.","tokens_in":6694,"tokens_out":9204,"would_cite":false,"duration_ms":89394,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J65","60E10","37A05","30C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A log-sinh map leaves the hyperbolic secant law exactly invariant.","keywords":["Cauchy distribution","hyperbolic secant distribution","Boole transformation","Newton's method","Brownian motion","conformal invariance","optional stopping","invariant distributions"],"falsifier":"Simulate planar Brownian motion started at $0$ and stopped on leaving the strip $\\{-1<\\Re z<1\\}$, then estimate $\\mathbb{E}[e^{i\\theta \\Im B_T}]$ and the independence of $\\Re B_T$ and $\\Im B_T$. If the characteristic function is not $\\operatorname{sech}\\theta$, or if the product of $\\mathbb{E}[e^{\\theta\\Re B_T}]$ and $\\mathbb{E}[e^{i\\theta\\Im B_T}]$ does not equal $1$, then Proposition 3 fails. Directly, sample $X$ from $\\frac{1}{2}\\operatorname{sech}(\\frac{\\pi}{2}x)\\,dx$, apply $g(x)=\\frac{2}{\\pi}\\log|\\sinh(\\frac{\\pi}{2}x)|$, and compare the empirical law with the same density.","tokens_in":5783,"feed_emoji":"📐","tokens_out":10003,"duration_ms":100369,"temperature":0.7,"pith_summary":"This note establishes a transfer principle for invariant distributions: if a symmetric domain has a known Brownian-exit law and a conformal automorphism fixing the real axis, that automorphism produces an explicit transformation preserving the law. Applying the principle to a strip yields a new exact invariant law: the density $\\frac{1}{2}\\operatorname{sech}(\\frac{\\pi}{2}x)\\,dx$ is invariant under $g(x)=\\frac{2}{\\pi}\\log|\\sinh(\\frac{\\pi}{2}x)|$. The same methods give Brownian proofs of the Cauchy exit law, a proof that Newton's method on $x^2+1$ asymptotically yields Cauchy-distributed iterates, and a Brownian proof of a known identity for the log-modulus of a Cauchy variable. A sympathetic reader would care because exactly invariant laws under nonlinear maps are rare, and the transfer principle suggests a general way to build them.","feed_headline":"Hyperbolic secant distribution survives a log-sinh transformation","feed_subtitle":"Brownian exit from a strip yields an explicit map g that keeps the sech density exactly invariant.","key_machinery":"The load-bearing object is Proposition 8, a transfer principle for exit laws: for a symmetric simply connected domain $W$, the distribution $\\Delta_a$ of the real part of Brownian motion's exit point is pushed forward by the real part of any conformal automorphism $f$ fixing the real axis, with $f$ applied to the starting point. The proof uses conformal invariance of Brownian motion, Schwarz reflection ($f(\\bar z)=\\overline{f(z)}$), and Carathéodory's theorem to extend $f$ to the boundary. The other engine is optional stopping for martingales $e^{\\theta B_t}$, which computes the Cauchy exit law from a half-plane, the hyperbolic secant law from a strip, and the identity $\\mathbb{E}[e^{i\\lambda \\frac{2}{\\pi}\\ln|C|}]= \\operatorname{sech}\\lambda$ for standard Cauchy $C$.","core_discovery":"The central new assertion is Corollary 9: the distribution $\\frac{1}{2}\\operatorname{sech}(\\frac{\\pi}{2}x)\\,dx$ is invariant under $g(x)=\\frac{2}{\\pi}\\log|\\sinh(\\frac{\\pi}{2}x)|$. This follows from Proposition 8, which states that for a simply connected domain $W$ symmetric about $\\mathbb{R}$, if $X$ has the law of $\\operatorname{Re}(B_\\tau)$ for Brownian motion starting at $a$ and $f$ is a conformal automorphism of $W$ sending $\\mathbb{R}$ into itself, then $\\operatorname{Re}(f(\\pi(X)))$ has the corresponding law for $f(a)$; here $\\pi(x)$ is the upper boundary point above $x$. With $W$ a strip, the exit law is the hyperbolic secant law, and the automorphism built from $z\\mapsto \\frac{1}{2}(z-z^{-1})$ conjugated by $z\\mapsto i e^z$ gives the stated invariant map.","pith_inferences":["The example's horizontal strip of width $\\pi$ is connected to the corollary's density by the change of variable $x\\mapsto \\frac{\\pi}{2}x$; the unscaled invariant statement is that $\\frac{1}{\\pi}\\operatorname{sech} x$ is preserved by $\\ln|\\sinh x|$, and Corollary 9 is its rescaling, though the paper does not spell out this step.","If the unproved independence assertion in Proposition 3 is supplied, the same transfer principle likely extends to other one-parameter families of exit distributions, giving invariant laws for other special functions.","The map $g$ is an explicit nonlinear dynamical system preserving a probability measure; the paper does not analyze its mixing or periodic points, but the invariance alone invites a dynamical study."],"forward_implications":["The hyperbolic secant law now has a new, explicitly written invariant map, so iterates of $g$ preserve the law exactly rather than approximately.","Any other symmetric domain with a computable Brownian-exit law and a conformal automorphism fixing the real axis yields an invariant map by the same Proposition 8, enlarging the known catalogue of invariant distributions.","Newton's method applied to $x^2+1$ has Cauchy-distributed empirical iterates for almost every starting point, with the Boole transformation ergodic under the Cauchy law.","The optional-stopping method reproduces the Cauchy and hyperbolic secant exit laws and the $\\log|C|$ identity with short martingale proofs, suggesting the technique may work for other holomorphic exponentials."],"supporting_citations":[{"why":"Supplies the idea that a distribution can be realized as the Brownian-exit law from a symmetric simply connected domain, which Proposition 8 builds on.","marker":"[Gro19]"},{"why":"Provides Carathéodory's theorem, used to extend the conformal automorphism $f$ continuously to the boundary of $W$.","marker":"[Gol69]"},{"why":"Gives the uniqueness principle for analytic functions used to prove $f(\\bar z)=\\overline{f(z)}$, the Schwarz-reflection step in Proposition 8.","marker":"[Rud06]"},{"why":"Background reference for Lévy's conformal-invariance theorem for Brownian motion, which carries the exit law through the conformal map.","marker":"[Bas95]"},{"why":"Source for conformal invariance and for the standard Cauchy exit law of the half-plane, used in Proposition 1 and Remark 2.","marker":"[Dur84]"},{"why":"The known characterization of Cauchy-preserving maps that motivates the general invariance framework.","marker":"[Let77]"},{"why":"Gives the original direct calculation of Cauchy-distributed functions of Cauchy variates that the paper generalizes.","marker":"[PW67]"},{"why":"Names the Boole transformation and is referenced for its ergodic properties used in Section 2.","marker":"[A W73]"}],"fun_headline_variants":["Sech density invariant under log-sinh map","Brownian strip exit fixes hyperbolic secant law","Explicit transformation keeps sech distribution invariant","Log-sinh map preserves sech distribution exactly","New invariant: sech law under log-sinh transformation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation of the hyperbolic secant exit law assumes, without proof, that the real and imaginary parts of Brownian motion's exit point from a strip are independent and that $\\mathbb{E}[e^{\\theta \\operatorname{Re}(B_T)}]=\\cosh\\theta$; if this factorization fails, Proposition 3 and hence the new invariant map in Corollary 9 are not established, and the example's stated width must also be rescaled to match the corollary.","fun_headline_variants_meta":{"raw":{"variants":["Sech density invariant under log-sinh map","Brownian strip exit fixes hyperbolic secant law","Explicit transformation keeps sech distribution invariant","Log-sinh map preserves sech distribution exactly","New invariant: sech law under log-sinh transformation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001675,"raw_usage":{"total_tokens":6566,"prompt_tokens":789,"completion_tokens":5777,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":5705}},"tokens_in":405,"tokens_out":5777,"duration_ms":45233,"temperature":1.0,"reasoning_tokens":5705,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:56:34.004345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate planar Brownian motion started at $0$ and stopped on leaving the strip $\\{-1<\\Re z<1\\}$, then estimate $\\mathbb{E}[e^{i\\theta \\Im B_T}]$ and the independence of $\\Re B_T$ and $\\Im B_T$. If the characteristic function is not $\\operatorname{sech}\\theta$, or if the product of $\\mathbb{E}[e^{\\theta\\Re B_T}]$ and $\\mathbb{E}[e^{i\\theta\\Im B_T}]$ does not equal $1$, then Proposition 3 fails. Directly, sample $X$ from $\\frac{1}{2}\\operatorname{sech}(\\frac{\\pi}{2}x)\\,dx$, apply $g(x)=\\frac{2}{\\pi}\\log|\\sinh(\\frac{\\pi}{2}x)|$, and compare the empirical law with the same density.","supporting_citations":[],"review_version":1}