{"id":"9c9841a7-b59e-46dc-a678-713d81fcc83d","arxiv_id":"1909.00559","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Over local fields, the maximum likelihood Gaussian is the span of the data, conditional independence is an F_q-realizable matroid, and in dimension 2 the tropicalized Gaussian is exactly a supermodular tropical polynomial.","lead":"This paper starts a statistical theory for data over p-adic numbers and other local fields, where Gaussian distributions are uniform measures on lattices. It shows how to fit such Gaussians to data, characterizes when coordinates are conditionally independent, and links the resulting valuation patterns to tropical polynomials.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 omits integrality of the supermodular coefficients; with real coefficients the 'if' direction is false, e.g. P(v)=max(0,v1-1/2,v2-1/2,v1+v2-1) is supermodular but not integer-valued and so cannot be a Gaussian tail.","rationale":"I read the manuscript in good faith. The reader's specific concern about the conditioning step in Section 3.2 does not actually land: with the lower-triangular HNF read as X1=pi^a Y1 and X2=pi^c x Y1 + pi^b Y2, the event V1>=v1 only constrains Y1, so the conditional law is indeed uniform on the stated sublattice L_{v1}, and the tail formula follows. The reader is also right that Theorem 1.2 is stated for all Gaussian vectors while its proof invokes a full-rank HNF; that is a genuine gap, though the statement may be salvageable by a degenerate-case argument. The more direct flaw in the paper's central claim is the missing integrality in Theorem 1.3: the bijection with supermodular functions is only true for integer supermodular coefficients. Since this is a one-word correction that does not alter the intended construction, I do not change the reader's CONDITIONAL verdict, but I flag it as the most load-bearing textual issue.","tokens_in":13991,"tokens_out":44147,"duration_ms":584480,"concrete_test":"Evaluate the restriction of P0(v)=max(0, v1-1/2, v2-1/2, v1+v2-1) at v=(1,0): the value is 1/2. Since Lemma 3.2 forces phi_L to be integer-valued for every lattice L, P0 cannot be the tail function of any K-Gaussian. This settles that Theorem 1.3's 'if' direction is false without an integrality hypothesis. A minimal fix is to add c10,c01,c11 in Z to the statement and re-verify the converse reparametrization; no other part of the proof needs to change.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bijection in Theorem 1.3 is stated as: a function phi on Z^2 equals phi_L for some lattice L iff phi is the restriction of a tropical polynomial P(v)=max(c00, v1-c10, v2-c01, v1+v2-c11) with c00=0 and c00+c11 >= c01+c10. No integrality condition is imposed on c10,c01,c11. The proof, however, only works for integer coefficients: Lemma 3.2 shows phi_L is integer-valued, and the converse constructs a lattice from the integer reparametrization a=c10, c=c01, b=c11-c10. With real coefficients the 'if' direction is false. For example, P0(v)=max(0, v1-1/2, v2-1/2, v1+v2-1) satisfies c00=0 and c00+c11=1 >= 1 = c01+c10, but P0(1,0)=1/2, so its restriction to Z^2 is not integer-valued and cannot equal -log_q P(V>=v) for any Gaussian lattice L. Thus the theorem as printed overstates the bijection: the correct statement is with integer supermodular coefficients, as Conjecture 5.1 explicitly says. This is a statement-level error in the paper's headline result, not merely a missing proof step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops foundations for statistical inference for Gaussian measures over non-Archimedean local fields, using lattices as the analogue of covariance matrices. Its main results are: (1) uniqueness of the maximum likelihood lattice for full-rank data (Theorem 1.1); (2) representation of conditional independence among coordinates of a local-field Gaussian vector by an F_q-realizable matroid (Theorem 1.2); (3) a bijection, in dimension two, between tropicalizations of Gaussian measures and supermodular functions on the discrete cube (Theorem 1.3); and (4) a discussion of the Bruhat-Tits building as a parameter space for Gaussian measures, with conditional-independence models formulated as an open problem. The proofs rely on Evans' characterization of local-field Gaussians, van Rooij's orthonormality criterion, and Smith normal form / Hermite normal form decompositions of lattices.","tokens_in":14264,"tokens_out":5059,"duration_ms":49067,"significance":"The paper addresses a novel and worthwhile topic: statistics over local fields has not been systematically developed, and the lattice-based perspective is natural and potentially influential. The MLE and conditional-independence characterizations are clean and, modulo the qualification discussed below, largely supported by the proofs. The proposed bijection between tropicalized two-dimensional Gaussians and supermodular functions is an appealing bridge between probability and tropical geometry, and the explicit use of Hermite normal form and Bruhat-Tits buildings gives the paper concrete computational content. However, the statement of Theorem 1.3 is currently false without an integrality condition, which is a load-bearing error in the headline result. The conditional-distribution step in its proof also needs to be justified explicitly. With these corrections and a suitable non-degeneracy assumption for Theorem 1.2, the paper would make a solid contribution.","major_comments":[{"comment":"Theorem 1.3 as stated asserts that every real supermodular tropical polynomial of the form (1.1) with c00=0 and c00+c11 >= c01+c10 restricts to phi_L for some lattice L. This is false. Lemma 3.2 shows that phi_L is integer-valued, but P_0(v)=max(0, v1-1/2, v2-1/2, v1+v2-1) satisfies the stated inequalities while P_0(1,0)=1/2, so its restriction to Z^2 is not integer-valued and cannot equal -log_q P(V>=v) for any Gaussian lattice. The correct statement requires integer coefficients, exactly as given in Conjecture 5.1. The theorem as printed overstates the bijection and must be corrected.","section":"Section 3.2, Theorem 1.3"},{"comment":"The proof depends on the assertion that the conditional distribution of X given V1>=v1 is the Gaussian distribution on the sublattice L_{v1} displayed in the proof. This step is asserted without proof, and it is the basis for the tail formula for V2 and hence for the full expression of phi_L. The assertion is in fact plausible: the event V1>=v1 forces the first lattice coordinate into pi^{max(v1-a,0)} O_K, producing L_{v1}; however, the paper should include this argument explicitly, since the bijection in Theorem 1.3 fails if this conditional law is not exact.","section":"Section 3.2, proof of Theorem 1.3"},{"comment":"Theorem 1.2 is stated for an arbitrary Gaussian vector X in K^d, but the proof invokes Lemma 2.2 to obtain a Hermite normal form for the support lattice L, and Lemma 2.2 requires L to have full rank d. For a degenerate Gaussian, the theorem needs either an explicit non-degeneracy assumption or a separate reduction to the support subspace. As written, the statement and proof do not match on this point.","section":"Theorem 1.2"}],"minor_comments":[{"comment":"The text reads 'm01 = c' instead of 'c01 = c' when reparametrizing the coefficients of the tropical polynomial.","section":"Section 3.2, proof of Theorem 1.3"},{"comment":"The displayed conditional independence statements list 'X3 independence X4 | X1' twice; the third statement should presumably be 'X2 independence X3 | X1'.","section":"Example 3.1"},{"comment":"The condition 'not all epsilon_i are 0 or 1' is ambiguous; it should read 'not all epsilon_i are 0 and not all epsilon_i are 1'.","section":"Section 4, Proposition 4.2"},{"comment":"There are several typographical errors, including 'semidefinite', 'simlimarity', and 'rader', which should be corrected in a final revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The integrality gap between Theorem 1.3 and Conjecture 5.1 should be reconciled prominently; as it stands, the main theorem is false exactly as printed, although the intended integer-coefficient statement appears defensible. The paper is within the journal's scope and, after the required corrections, would be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper deserves a serious referee. It opens a new subarea — statistics for local-field Gaussians — and the main results are mostly right. But Theorem 1.3, the headline characterization, is misstated: the coefficients c10, c01, c11 in (1.1) have to be integers for the converse to hold. With real coefficients the 'if' direction is false; a tropical polynomial like max(0, v1-1/2, v2-1/2, v1+v2-1) is supermodular but its restriction to Z^2 is not integer-valued and no lattice tail distribution takes non-integer exponents. The proof only constructs a lattice when the coefficients are integers, and the authors' own Conjecture 5.1 says 'integer supermodular coefficients'. So this is a statement-level bug, not a deep flaw: fix Theorem 1.3 to require integer coefficients (or integer-valued phi) and the bijection goes through.\n\nWhat is actually new: Theorem 1.1 gives the MLE lattice as span_{O_K}(X), with a clean proof using the minimal Haar measure; Theorem 1.2 shows conditional independence is encoded by an F_q-realizable matroid, which is a genuinely non-archimedean phenomenon (there is no analogue over R); the tropicalization bijection in dimension 2, once corrected, is a nice exact result. The paper also introduces the Bruhat-Tits building as a parameter space, with explicit adjacency computations. The proofs of 1.1 and 1.2 are complete; the SVD/Smith normal form background is used properly.\n\nSoft spots: Theorem 1.2 is stated without the full-rank assumption; the proof requires Hermite normal form for a lattice of full rank d. The degenerate case may be salvageable by conditioning on the support subspace, but the theorem as printed promises more than is proved. The conditioning step in the proof of Theorem 1.3 — 'the conditional distribution of X given V1 >= v1 is Gaussian on the sublattice L_{v1}' — is asserted in one sentence. It is actually true; you can verify it by writing X = A Y and conditioning on Y1. But it should be spelled out; it's the most subtle step in the paper. The paper's citation of [Tra18] for d=1 is fair; the d=2 case is new. I don't see a circularity problem.\n\nWho is this for? Anyone working in p-adic probability, ultrametric analysis, or tropical statistics. It is a foundation paper; the significance is contained but real. It should go to peer review, and the referee report will need to insist on the integrality fix and the full-rank qualification. After that, I'd take Theorem 1.3 as correct.","headline":"Genuinely opens a subfield of p-adic statistics; the tropicalization bijection in Theorem 1.3 is misstated without an integrality condition, but the intended result and most of the paper are solid.","tokens_in":14853,"tokens_out":4326,"would_cite":true,"duration_ms":36575,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H05","60E05","12J25","14T90"],"pacs":[],"model":"deepseek-v4-flash","headline":"Local-field Gaussians admit a complete statistical theory: closed-form MLE, matroidal conditional independence, and a supermodular tropical bijection in dimension two.","keywords":["local fields","p-adic Gaussian measures","tropicalization","maximum likelihood","conditional independence","Bruhat-Tits building","supermodular functions","matroids"],"falsifier":"Take $K=\\mathbb{Q}_2$ and the lattice $L$ generated by $(1,1)$ and $(0,2)$ over $\\mathbb{Z}_2$ (Hermite form with $a=0,b=1,c=0$). Enumerate the points of $L$ with $\\mathrm{val}(X_1)\\ge 0$ and count the fraction with $\\mathrm{val}(X_2)\\ge v_2$ for $v_2=0,1,2$; if the fractions are not $q^{-\\max(v_2-\\min(b,c+\\max(v_1-a,0)),0)}$ with $v_1=0$, the asserted conditional law is false.","tokens_in":13764,"feed_emoji":"🌴","tokens_out":9339,"duration_ms":78965,"temperature":0.7,"pith_summary":"This paper lays foundations for statistics over local fields, such as the p-adic numbers, where Gaussian measures are uniform probability measures on lattices. It establishes that maximum likelihood estimation has a closed-form solution, that conditional independence among coordinates is governed by a matroid over the residue field, and that in dimension two the valuation, or tropicalization, of a Gaussian is exactly a tropical polynomial with supermodular coefficients on the unit square. These results matter because they reduce statistical questions over exotic fields to concrete discrete geometry: lattices, matroids, and piecewise-linear functions.","feed_headline":"2-D p-adic Gaussians tropicalize to supermodular polynomials","feed_subtitle":"In dimension two, the valuation image of a Gaussian is a supermodular tropical polynomial, and vice versa.","key_machinery":"The load-bearing object is the lattice $L \\subset K^d$, the analogue of the covariance matrix: every non-degenerate Gaussian is uniform Haar measure on $L$. Three tools carry the argument. The Hermite normal form writes any lattice as the image of $O_K^d$ under a lower-triangular matrix with diagonal entries $\\pi^a, \\pi^b$ and off-diagonal entry $\\pi^c x$, giving canonical integer parameters. The valuation map $\\mathrm{val}: K^d \\to \\mathbb{Z}^d$ tropicalizes the measure, and the tail probability $P(V \\ge v)$ counts lattice points in $L \\cap \\pi^v O_K^d$. The proof of Theorem 1.3 conditions on $V_1 \\ge v_1$, uses the conditional law on an explicit sublattice $L_{v_1}$, and reads off a tail formula that matches the tropical polynomial; the Bruhat-Tits building of $\\mathrm{SL}_d(K)$ is introduced as the parameter space counterpart of the positive-semidefinite cone.","core_discovery":"The central claim is a bijection in dimension two. For a non-degenerate Gaussian $X$ in $K^2$ with lattice $L$, define $\\phi_L(v) = -\\log_q P(V \\ge v)$ for $v \\in \\mathbb{Z}^2$. The paper proves that $\\phi_L$ equals the restriction to $\\mathbb{Z}^2$ of the tropical polynomial $P(v) = \\max(0, v_1-a, v_2-c, v_1+v_2-a-b)$, and that every polynomial of this form with $c_{00}=0$ and $c_{00}+c_{11} \\ge c_{01}+c_{10}$ arises from a lattice. The same framework yields Theorem 1.1, that the MLE lattice is the $O_K$-span of the data, and Theorem 1.2, that conditional independence statements are exactly bases of an $\\mathbb{F}_q$-realizable matroid.","pith_inferences":["A direct corollary of the d=2 bijection that the paper leaves implicit: the tropicalized Gaussian family has exactly three integer parameters $(a,b,c)$ with $c \\le b$, so fitting to valuation-only data is a finite combinatorial search rather than a continuous optimization problem.","Because the Bruhat-Tits building $B_2(K)$ is a tree, the Gaussian models in $K^2$ sit on a tree; asking whether each conditional-independence model $M_{I,J}$ is a geodesically convex subtree would connect Theorems 1.2 and 1.3 and could lead to fast model-selection algorithms.","The proof idea of conditioning on one coordinate at a time suggests a recursive algorithm for $\\phi_L$ in higher dimensions based on the independence lattice $\\mathrm{ind}(L)$; testing Conjecture 5.1 by computing $\\phi_L$ on random lattices for $d=3$ would be a direct computational check."],"forward_implications":["Maximum likelihood estimation over local fields is solved in closed form: the fitted lattice is the $O_K$-span of the data, and when the data has full rank it is the unique full-dimensional maximizer.","Conditional independence among coordinates of a $K$-Gaussian is a matroidal property: the maximal conditionally independent sets form the bases of a matroid representable over the residue field $\\mathbb{F}_q$, so checking independence reduces to linear algebra over a finite field.","In dimension two the family of tropicalized Gaussians coincides with the family of tropical polynomials supported on the unit square with supermodular coefficients; the case $c=b$ gives independent coordinates and a product of shifted geometric distributions for the valuation vector.","The Bruhat-Tits building gives a geometric parameter space for non-degenerate Gaussians, with adjacency described by explicit matrix conditions, so statistical models such as conditional-independence families can be studied as subsets of the building.","If the paper's Conjecture 5.1 is correct, the same supermodular-polynomial picture holds in every dimension $d$, giving a finite parametrization of $K$-Gaussians by $2^d - 1$ integer coefficients."],"supporting_citations":[{"why":"Gives the foundational theorem that K-Gaussians are exactly uniform Haar measures on lattices, used as the starting point for all three main theorems.","marker":"[Eva01b]"},{"why":"Provides the non-archimedean singular value decomposition used to prove the orthonormal form of a lattice and the measure formula λ(L)=q^{-val(det A)}.","marker":"[Eva02]"},{"why":"Supplies the Gram-Schmidt-type theorem over local fields used to construct the orthonormal form of lattices.","marker":"[Sch84]"},{"why":"Supplies the modulo-π orthonormality criterion used to translate conditional independence into linear independence over the residue field.","marker":"[vR78]"},{"why":"Sets up the tropical Gaussian program and the one-dimensional result that tropicalized Gaussians are shifted geometric distributions, which the paper extends to dimension two.","marker":"[Tra18]"}],"fun_headline_variants":["2D local field Gaussians tropicalize to supermodular polynomials","Tropicalization of 2D local field Gaussians is supermodular","Supermodular tropical polynomials biject with 2D local field Gaussians","p-adic Gaussian tropicalization: supermodular in 2D","Dimension 2: p-adic Gaussian tropicalization is exactly supermodular"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bijection in Theorem 1.3 rests on the unproved assertion that, conditional on $V_1 \\ge v_1$, the Gaussian is uniform on the explicit sublattice $L_{v_1}$; if that conditional law were not uniform, the tail formula and the bijection would fail.","fun_headline_variants_meta":{"raw":{"variants":["2D local field Gaussians tropicalize to supermodular polynomials","Tropicalization of 2D local field Gaussians is supermodular","Supermodular tropical polynomials biject with 2D local field Gaussians","p-adic Gaussian tropicalization: supermodular in 2D","Dimension 2: p-adic Gaussian tropicalization is exactly supermodular"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001693,"raw_usage":{"total_tokens":6644,"prompt_tokens":816,"completion_tokens":5828,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":432,"completion_tokens_details":{"reasoning_tokens":5730}},"tokens_in":432,"tokens_out":5828,"duration_ms":102383,"temperature":1.0,"reasoning_tokens":5730,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:47:09.118801+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $K=\\mathbb{Q}_2$ and the lattice $L$ generated by $(1,1)$ and $(0,2)$ over $\\mathbb{Z}_2$ (Hermite form with $a=0,b=1,c=0$). Enumerate the points of $L$ with $\\mathrm{val}(X_1)\\ge 0$ and count the fraction with $\\mathrm{val}(X_2)\\ge v_2$ for $v_2=0,1,2$; if the fractions are not $q^{-\\max(v_2-\\min(b,c+\\max(v_1-a,0)),0)}$ with $v_1=0$, the asserted conditional law is false.","supporting_citations":[],"review_version":1}