{"id":"8247697e-68ba-4528-bfad-8eef1e5e3a07","arxiv_id":"2608.22568","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every entrywise preserver of one-positive-eigenvalue Gram matrices of hyperbolic type has the form f(t)=1+γ 1_{(1,∞)}(t)+β(t^α-1)/α+∫(1-t^{-s}) dν(s), with α∈[0,1] and β large enough relative to γ and ν, completing Schoenberg's program.","lead":"This paper classifies all functions that, when applied entrywise to Lorentz-Gram matrices, keep exactly one positive eigenvalue. It closes the negative-curvature case of a classification begun by Schoenberg in Euclidean and spherical geometries.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central characterization depends on Krein's screw-line trichotomy, imported without proof; no internal gap found, but this external step should be verified.","rationale":"The reader's weakest assumption identifies exactly the load-bearing external step: Krein's classification of screw lines. I agree this is the least securely supported link in the necessity argument. However, this is a standard, clearly cited classical theorem, and the paper nowhere misstates or overuses it; the remaining proof is internally coherent. I checked the key transitions: Theorem 3.2 correctly transfers M_1+-preservation to an embedding, Section 4.1 correctly converts characteristics (α,C) into elliptic/parabolic/hyperbolic alternatives, Section 5.1's positivity argument leading to Widder's theorem is sound, and Section 5.2's use of conditional negative definiteness and the Bernstein representation is valid modulo the usual integrability details. The self-citation [5] for the 3-PMP matrix N is a published auxiliary result and is not circular. The only genuine risk is external: if Krein's trichotomy has a boundary case not covered by the statement in Section 4, the characterization would fail. A verification of the original theorem settles this risk. Since the cited theorem is standard and the paper is explicit about relying on it, the concern does not change the accept verdict; it is a verification note rather than a flaw.","tokens_in":36606,"tokens_out":18816,"duration_ms":172790,"concrete_test":"Obtain Iokhvidov–Krein [15, Sections 26–27] (or an independent modern exposition of Krein's screw-line classification) and check the statement in Section 4 line by line: (i) in the elliptic case, σ is a positive Radon measure with total mass a−1 and no mass at 0; (ii) in the hyperbolic case, σ has total mass a−1; (iii) in the parabolic case, an atom at 0 is allowed and the measure conditions in (4.1) are exactly ∫_{(0,1]} dμ(s)/s²=∞ and ∫_{[1,∞)} dμ(s)/s²<∞; (iv) there is no fourth case. If all four points match, re-run the Section 4.1 trichotomy table; if any boundary condition differs, trace its effect on the measure representations in Sections 5.1 and 5.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The necessity direction of the main theorem has the chain: M_1+-preservation ⇒ (by Theorem 3.2 and Krein's embedding theorem) a map Φ with Φ∘ℓ0 a screw line whose metric function is φ; then Krein's screw-line classification forces f∘cosh = cosh∘φ into exactly one of the elliptic, hyperbolic, or parabolic integral forms; Sections 5.1–5.2 use the hyperbolic form for α>0 and the parabolic form for α=0, C=∞ to finish the additive representation. The link in this chain that is not proved or even reproduced in the paper is the screw-line trichotomy imported from Krein [17] and Iokhvidov–Krein [15, Sections 26–27], stated in Section 4 as the assertion that Λ=cosh∘λ has exactly one of three forms. This is genuinely load-bearing: if the trichotomy were incomplete, or if a boundary case in the parabolic class (e.g., the atom at 0 and the integrability conditions in (4.1)) were mis-stated, the necessity argument would inherit the error. This is not an internal inconsistency: the surrounding proofs are careful, the external theorem is classical, and the citation is explicit. The residual risk is purely that the exact hypotheses and boundary conventions of the imported theorem have not been verified by the authors, so a reader must consult the original sources to confirm the classification applies verbatim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies entrywise maps f:[1,∞)→[1,∞) that send the class of Lorentz–Gram matrices (Gram matrices of points in infinite-dimensional Lobachevsky space) to itself. The main result, stated in display (1.1), characterizes such distance preservers as functions of the form f(t)=1+γ 1_{(1,∞)}(t)+β(t^α−1)/α+∫_{(0,∞)}(1−t^{−s}) dν(s), where α∈[0,1], β,γ≥0, and ν is a positive Radon measure subject to β≥α(1+γ+ν((0,∞))) for α>0 and ∫ min{s,1} dν(s)<∞ for α=0. The necessity proof combines Krein's embedding theorem with Krein's classification of screw lines; the sufficiency is shown through explicit matrix decompositions, Bernstein-function theory, and closure properties. The paper also establishes closure under composition and pointwise limits, uniqueness of the representation, a multiplicative representation, and a negative answer to a question of Cohen and Lifshits on fractional Ornstein–Uhlenbeck fields.","tokens_in":36872,"tokens_out":14152,"duration_ms":113896,"significance":"If the main theorem is correct, it completes Schoenberg's classification of entrywise preservers in the constant-curvature trichotomy by filling the negative-curvature case. The representation is explicit and yields concrete consequences: it unifies and extends the Monod–Py power preservation result, resolves the Cohen–Lifshits question, and connects to complete Nevanlinna–Pick kernels and Bochner subordination. The proof is detailed and largely self-contained, with careful handling of boundary cases, pointwise limits, and the regularisation of discontinuous preservers. The main residual risk is the unproved importation of Krein's screw-line classification, which is load-bearing for the necessity direction.","major_comments":[{"comment":"The necessity direction of the main theorem (display (1.1)) depends crucially on the trichotomy for metric functions of screw lines stated in Section 4. This theorem is imported from Krein [17] and Iokhvidov–Krein [15, Sections 26 and 27] without proof. The trichotomy is the bridge that forces f∘cosh to be elliptic, hyperbolic, or parabolic, and any mismatch in hypotheses or boundary conventions (e.g., the integrability conditions in (4.1), the support condition on σ, or the no-mass-at-0 statement) would propagate into the characterization. Since the paper claims to be self-contained, I request that the authors either prove the classification or provide a precise statement with all hypotheses and a verification that the classical sources establish exactly this version. At a minimum, a remark detailing the boundary cases would substantially reduce the residual risk.","section":"Section 4 (Screw lines) and Section 5 (Characterisation of distance preservers)"}],"minor_comments":[{"comment":"The main characterization is stated in the abstract and in the introduction but is not labelled as a theorem. Numbering this result, e.g., as Theorem 1.1, and restating it in Section 5 would improve citability and readability.","section":"Section 1, display (1.1)"},{"comment":"The uniqueness of the representing measure ν is attributed to 'standard Laplace-transform arguments'. Given that the rest of the paper is very careful, a brief proof or a precise reference for this uniqueness would be welcome.","section":"Section 3.8"},{"comment":"The screw-line trichotomy would benefit from a uniqueness statement for the representing measures in each of the three cases, since the later proofs identify the parameters a and b from asymptotic limits.","section":"Section 4"},{"comment":"The sentence 'To ensure our presentation is self contained ... and full proofs of auxiliary results in Appendix A' is too strong: the proofs of Krein's embedding theorem (Theorem 3.1) and the screw-line classification (Section 4) are not provided in the appendix. Please tone down this claim or add the missing material.","section":"Introduction, Section 1"},{"comment":"In the display involving the second difference, the notation uses forward increments defined just after; this is fine, but the definition of Δ_δ should be placed before its first use for readability.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is of high quality and the main result is likely correct. My only substantive concern is the unproved reliance on Krein's screw-line classification. If the authors can confirm (e.g., by providing the precise statement from Iokhvidov–Krein or a proof sketch) that the trichotomy applies verbatim, the paper should be accepted. I have no concerns about novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the real thing. It closes the negative-curvature case that Schoenberg left open, and the main characterization (1.1) is a clean L\\'evy\\-Khintchine type representation. The paper proves both directions in detail, recovers the Monod\\-Py power result as a special case, and answers the Cohen\\-Lifshits question. The exposition is careful, and the closure arguments under composition and pointwise limits are handled honestly and without hand\\-waving. The one self\\-citation, to the authors' 3\\-PMP theorem, is used for exactly what it proves and is not circular.\n\nThe soft spot is the necessity direction's reliance on Krein's screw\\-line classification of metric functions, stated in Section 4 but not proved or reproduced. That trichotomy is genuinely load\\-bearing: it is the bridge from entrywise preservation to the three integral forms. It is classical and precisely cited, so this is not a fatal gap, but a referee should verify that the hypotheses and boundary conventions\\—especially the parabolic integrability conditions in (4.1)\\—match the paper's usage. The paper calls itself self\\-contained, which is slightly overstated on this one point. I do not see an internal error; the risk is inherited from the external literature.\n\nThere is also an acknowledgment of AI assistance in the proofs. That is unusual, but the authors state they verified everything, and the mathematics itself shows no sign of the sloppiness that would make me worry. The reader's soundness score of 8 feels right; the external dependence is the only thing keeping it from a 9.\n\nWho is this for? Anyone working on matrix entrywise preservers, hyperbolic geometry, or kernels of real hyperbolic type. It deserves a serious referee and, assuming the Krein step checks out, publication. I would bring it to a reading group and would cite it.","headline":"Completes Schoenberg's classification for negative curvature; the only real soft spot is a load-bearing external theorem (Krein's screw-line trichotomy) imported without proof.","tokens_in":37400,"tokens_out":1161,"would_cite":true,"duration_ms":12810,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["15A45","51M10","46E22","42A82"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper classifies every entrywise transform that maps Lorentz–Gram matrices to Lorentz–Gram matrices, completing the negative-curvature counterpart of the classical zero- and positive-curvature results.","keywords":["Lobachevsky space","Lorentz–Gram matrices","entrywise preservers","hyperbolic distance","Bernstein functions","screw lines","kernels of real hyperbolic type","infinitely divisible matrices"],"falsifier":"Take the parameter inequality seriously: choose $\\alpha=1$, $\\gamma=0$, $\\nu=0$, and $\\beta<1$, so $f(t)=1+\\beta(t-1)$ is increasing but violates (1.2); if the $3\\times3$ Lorentz–Gram matrix $\\begin{bmatrix}1&a&b\\\\a&1&a\\\\b&a&1\\end{bmatrix}$ with $b=2a^2-1$ is mapped by $f$ to a matrix with two positive eigenvalues for some $a>1$, the necessity direction is refuted. Conversely, any preserver not representable in form (1.1), for example a flat bounded function with $f(1)=1$ and $f(t)-1$ decaying faster than $t^{-s}$ for all $s>0$, would refute sufficiency if it preserves the class $M_{1+}$.","tokens_in":36424,"feed_emoji":"📐","tokens_out":9299,"duration_ms":85252,"temperature":0.7,"pith_summary":"This paper asks which functions, applied entrywise to a Lorentz–Gram matrix, always produce another Lorentz–Gram matrix. For Lobachevsky space, the answer is exactly the parameterized family in display (1.1): a baseline power term, a possible jump at the identity, and an integral term, subject to one inequality coupling the drift to the jump and measure. If correct, this closes the last curvature case left open by the classical Euclidean and spherical classifications and gives a complete description of distance preservers of hyperbolic space.","feed_headline":"One formula classifies all hyperbolic distance preservers","feed_subtitle":"Entrywise transforms of Lorentz–Gram matrices reduce to one integral family, completing the curvature trilogy.","key_machinery":"The load-bearing tool is the screw-line trichotomy: every metric function of a continuous helical path in Lobachevsky space is elliptic, parabolic, or hyperbolic, meaning that $\\Lambda=\\cosh\\circ\\lambda$ has one of three integral representations involving cosine transforms. Combined with the Hankel-kernel theorem and the fact that the Hadamard inverse of a Lorentz–Gram matrix is positive semidefinite and infinitely divisible, this trichotomy forces any preserver's transform $f\\circ\\cosh$ into the three integral forms from which the additive representation follows.","core_discovery":"A function $f:[1,\\infty)\\to[1,\\infty)$ with $f(t)=1$ only at $t=1$ preserves the class of Lorentz–Gram matrices under entrywise application if and only if $f$ has the form $$f(t)=1+\\gamma\\mathbf{1}_{(1,\\infty)}(t)+\\$\\beta$\\frac{t^\\$\\alpha$-1}{\\$\\alpha$}+\\int_{(0,\\infty)}(1-$t^{{-s}}$)\\,d\\nu(s),$$ with $\\alpha\\in[0,1]$, $\\beta,\\gamma\\ge 0$, and $\\nu$ a positive Radon measure satisfying $\\beta\\ge\\alpha(1+\\gamma+\\nu((0,\\infty)))$ when $\\alpha>0$ and $\\int\\min\\{s,1\\}\\,d\\nu(s)<\\infty$ when $\\alpha=0$, with not all of $\\beta,\\gamma,\\nu$ zero. The same class is exactly the set of functions for which the inner transform $\\varphi=\\operatorname{arcosh}\\circ f\\circ\\cosh$ sends the hyperbolic metric into another metric on Lobachevsky space. Continuous preservers also admit a multiplicative exponential-Bernstein representation, which the paper relates to subordinators.","pith_inferences":["Inference: The explicit family gives a direct route to data-driven hyperbolic embeddings: one can fit the parameters $(\\alpha,\\beta,\\gamma,\\nu)$ rather than fix a single kernel transform, because every admissible parameter choice provably preserves the hyperbolic geometry constraints.","Inference: The theorem suggests a precise analogy between Euclidean conditionally negative definite kernels and Lorentz–Gram kernels; the condition $\\beta\\ge\\alpha(1+\\gamma+\\nu((0,\\infty)))$ is the hyperbolic analogue of Bernstein-function monotonicity, so algorithms developed for Euclidean distance transforms may have direct hyperbolic counterparts.","Inference: A natural testable extension is to Gram matrices with more than one negative eigenvalue; the screw-line trichotomy likely resolves into a finite number of regimes indexed by the signature, and the same additive-integral method could apply."],"forward_implications":["Every power map $t\\mapsto t^\\alpha$ with $\\alpha\\in[0,1]$ is a preserver, so the known result that such powers preserve kernels of real hyperbolic type appears as a special case.","The class is closed under composition and pointwise limits, so the family (1.1) forms a complete semigroup of nonlinear transforms on Lorentz–Gram matrices.","Each continuous preserver admits a multiplicative representation $f(t)=t^\\alpha\\exp\\!\\left(\\int_{(0,\\infty)}(1-t^{-s})\\,d\\mu(s)\\right)$, connecting the classification to Bernstein-function theory and to subordination of stochastic processes.","The fractional Ornstein–Uhlenbeck covariance $\\exp(-a\\,d(x,y)^\\alpha)$ on hyperbolic space is not positive semidefinite for $\\alpha>1$ in all sufficiently large dimensions, settling an open question about such fields."],"supporting_citations":[{"why":"Supplies the screw-line trichotomy: every metric function of a helical path in Lobachevsky space is elliptic, parabolic, or hyperbolic.","marker":"[17]"},{"why":"Provides the detailed spectral-theoretic statement of the screw-line classification and its integral representations.","marker":"[15]"},{"why":"Gives the Hankel-kernel theorem used to pass from positive semidefinite kernels to Laplace-integral representations.","marker":"[33]"},{"why":"Shows the Hadamard inverse of a one-positive-eigenvalue matrix is positive semidefinite, a key step for reciprocal transforms.","marker":"[2]"},{"why":"Supplies the corollary on Hadamard inverses of infinitely divisible matrices used in the same step.","marker":"[23]"},{"why":"Establishes the zero-curvature classification that this paper extends to constant negative curvature.","marker":"[30]"},{"why":"Gives the positive-definite kernel integral representation used to prove real analyticity and the multiplicative form.","marker":"[9]"},{"why":"Defines kernels of real hyperbolic type, identifying the objects whose entrywise preservers are classified here.","marker":"[24]"}],"fun_headline_variants":["All hyperbolic distance preservers: one formula","Hyperbolic distance preservers fully classified","One integral family completes curvature trilogy","Entrywise Lorentz-Gram preservers solved"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The characterization inherits its structure from a previously established trichotomy for the distance functions of helical paths in Lobachevsky space; if that trichotomy has a missing boundary case, the whole characterization would inherit the gap.","fun_headline_variants_meta":{"raw":{"variants":["All hyperbolic distance preservers: one formula","Hyperbolic distance preservers fully classified","One integral family completes curvature trilogy","Entrywise Lorentz-Gram preservers solved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000141,"raw_usage":{"total_tokens":1096,"prompt_tokens":812,"completion_tokens":284,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":232}},"tokens_in":428,"tokens_out":284,"duration_ms":3318,"temperature":1.0,"reasoning_tokens":232,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-27T18:24:42.291759+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the parameter inequality seriously: choose $\\alpha=1$, $\\gamma=0$, $\\nu=0$, and $\\beta<1$, so $f(t)=1+\\beta(t-1)$ is increasing but violates (1.2); if the $3\\times3$ Lorentz–Gram matrix $\\begin{bmatrix}1&a&b\\\\a&1&a\\\\b&a&1\\end{bmatrix}$ with $b=2a^2-1$ is mapped by $f$ to a matrix with two positive eigenvalues for some $a>1$, the necessity direction is refuted. Conversely, any preserver not representable in form (1.1), for example a flat bounded function with $f(1)=1$ and $f(t)-1$ decaying faster than $t^{-s}$ for all $s>0$, would refute sufficiency if it preserves the class $M_{1+}$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the screw-line trichotomy: every metric function of a helical path in Lobachevsky space is elliptic, parabolic, or hyperbolic."},{"cited_title":"Iokhvidov and M.G","cited_arxiv_id":null,"evidence_quote":"Provides the detailed spectral-theoretic statement of the screw-line classification and its integral representations."},{"cited_title":"Widder.The Laplace Transform","cited_arxiv_id":null,"evidence_quote":"Gives the Hankel-kernel theorem used to pass from positive semidefinite kernels to Laplace-integral representations."},{"cited_title":"Bapat, Multinomial probabilities, permanents and a conjecture of Karlin and Rinott.Proc","cited_arxiv_id":null,"evidence_quote":"Shows the Hadamard inverse of a one-positive-eigenvalue matrix is positive semidefinite, a key step for reciprocal transforms."},{"cited_title":"Micchelli, Interpolation of scattered data: distance matrices and conditionally positive def- inite functions.Constr","cited_arxiv_id":null,"evidence_quote":"Supplies the corollary on Hadamard inverses of infinitely divisible matrices used in the same step."},{"cited_title":"Schoenberg","cited_arxiv_id":null,"evidence_quote":"Establishes the zero-curvature classification that this paper extends to constant negative curvature."},{"cited_title":"Faraut and K","cited_arxiv_id":null,"evidence_quote":"Gives the positive-definite kernel integral representation used to prove real analyticity and the multiplicative form."},{"cited_title":"Monod and P","cited_arxiv_id":null,"evidence_quote":"Defines kernels of real hyperbolic type, identifying the objects whose entrywise preservers are classified here."}],"review_version":1}