{"id":"82b30999-1f7e-4a59-80e9-a505cee18407","arxiv_id":"2511.15827","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For matrices over PID rings of integers, local triangularizability/diagonalizability at all places implies global triangularizability/diagonalizability; a stratified Brauer–Manin obstruction governs some non-PID failures.","lead":"This paper proves a local-global principle: over principal ideal domains, an integral matrix that can be triangularized or diagonalized at every prime can also be triangularized or diagonalized over the ring itself. It also introduces a stratified Brauer–Manin obstruction that explains failures in more general number rings, and verifies it in several special cases.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.12's fibration target is mis-specified: the image lies in ΣXY=1, not merely ΣXY≠0, so the good-fibration assertions carrying Theorem 5.4 are not proved as written.","rationale":"The reader's weakest assumption identifies the same load-bearing point: Lemma 5.12's 'good fibration' assertions are insufficiently proved, and the displayed codomain of the fibration is inconsistent with the actual image. My reading confirms this and makes it more precise: the target should be the hypersurface ΣXY=1, not the localization at ΣXY≠0; otherwise the morphism is not even surjective on k-points, let alone on O_v-points. The concern is localized to the proof of Theorem 5.4, not to the PID local-global results in Section 3, which are independently argued and appear sound. The diagonalization results in Theorem 4.2 are also well-supported. Because the defect is a concrete, identifiable specification error in an otherwise elaborate induction, a conditional verdict is appropriate rather than rejection: the statement may be true and repairable, but the proof as written has a genuine gap. Hence no change to the reader's CONDITIONAL verdict.","tokens_in":21677,"tokens_out":10296,"duration_ms":97262,"concrete_test":"Take the smallest case m=1, n=2, s=(1,1) in Lemma 5.12. Then W_s ⊂ A^9 is given by T_{1,1}=0, T_{3,1}=0, det(T)=1, and f sends T to (X,Y)=(T_{2,1}, C_{2,1}). Direct computation gives det(T)=XY on W_s. (i) Pick a number-field point (X,Y)=(2,1) in the stated codomain Spec k[X,Y,1/(XY)]: the fiber equations force det(T)=2, so the fiber is empty; hence f is not surjective onto the stated target. (ii) Redo the proof with the corrected target B'=Spec k[X,Y]/(XY−1). Verify explicitly whether the fiber over a point (x,y) with xy=1 is SL_{2,0}×A^2 and whether W_s(O_v)→B'(O_v) is surjective for almost all v. If the fiber computation fails, Lemma 5.12 and Theorem 5.4 require a new argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.4 is the advertised 'special cases' result, and its proof is carried by Lemma 5.12 together with Lemma 5.15. In Lemma 5.12 the induction step considers f: W_s → Spec k[X_{s_{n−2}+1},...,X_{s_{n−1}}; Y_{...}; 1/(Σ X_α Y_α)], with X_α=T_{m+n−1,α} and Y_α=C_{m+n−1,α}. On W_s the determinant is 1, and the preceding expansion gives det(T)=Σ X_α Y_α. Therefore the image of f is contained in the hypersurface Σ X_α Y_α=1. The stated codomain is only the localization at Σ X_α Y_α≠0; it contains points with Σ X_α Y_α=c≠1, over which the fiber is empty. Hence f is not surjective on rational or integral points of the stated target, contradicting the second condition of Definition 5.5. This is not a notational quibble: the subsequent 'Therefore' asserts, without coordinate verification, that each fiber is SL_{m+1,s_2−1}×A^{...} or W'_{s'}×A^{...}; those fiber descriptions are exactly what make f a good fibration and yield W_s(k)=W_s(A_k)_• by induction. If the target is corrected to Spec k[X,Y]/(ΣXY−1), the proof may be repairable, but the lemma as written is not proved. Remark 5.16 additionally asserts order≤5 cases without proof, which does not fill this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a local-global principle for triangularizability and diagonalizability of an integral matrix over a number field. It proves that when the ring of integers is a PID, local triangularizability/diagonalizability over completions or residue rings implies global. It also constructs counterexamples over non-PID rings of integers. The paper then translates the problem into rational and integral points on triangularization and diagonalization varieties. For diagonalization, Theorem 4.2 shows that every connected component of the reduced diagonalization variety is a product of general linear groups, so the Brauer–Manin obstruction is the only obstruction to rational and integral points. For triangularization, the paper introduces a stratified Brauer–Manin obstruction and claims in Theorem 5.4 that for M similar to diag(λ I_m, J_n(λ)), every irreducible component of the triangularization variety is smooth and k-rational, and the stratified obstruction is the only obstruction.","tokens_in":22103,"tokens_out":10035,"duration_ms":92902,"significance":"The local-global statements for PIDs (Propositions 3.3 and 3.5) and the geometric description of the diagonalization variety (Theorem 4.2) are clean, elementary, and appear correct. These parts have independent value. The stratified Brauer–Manin obstruction is a novel notion, and Theorem 5.4, if fully proved, would be a meaningful advance toward Conjectures 5.2 and 5.3. However, the proof of Theorem 5.4 as written is not rigorous: it depends on several asserted good-fibration properties, an apparently incorrect codomain in Lemma 5.12, and a sketched ideal equality in Lemma 5.14. The advertised special-case result is therefore not yet established. The manuscript is likely repairable, but a substantial revision is needed.","major_comments":[{"comment":"The morphism f is defined with target Spec k[X_{s_{n-2}+1},...,X_{s_{n-1}}; Y_{...}; 1/(Σ X_α Y_α)]. On W_s the determinant is 1 and the displayed expansion gives det(T) = Σ X_α Y_α, so the image of f lies in the hypersurface Σ X_α Y_α = 1. The stated target contains points with Σ X_α Y_α = c ≠ 1, over which the fiber is empty. Thus the second condition of Definition 5.5 (surjectivity on O_v-points for almost all v) fails for the stated codomain, and the induction proving W_s(k)=W_s(A_k)_• collapses. The target should presumably be Spec k[X,Y]/(ΣXY−1); with that correction, the surjectivity and fiber-description arguments still need to be supplied.","section":"Section 5, Lemma 5.12"},{"comment":"Several load-bearing good-fibration claims are merely asserted. Corollary 5.7 says 'One can verify that f is a good fibration' and 'surjectivity on integral points arises naturally from the construction', without proof; the target V also contains a notational typo (T instead of Z). Lemma 5.11 says 'One checks that f is a good fibration whose fibers are always isomorphic to A^{s-r}' with no coordinate verification. Lemma 5.12 asserts that the fibers are SL_{m+1,s_2−1}×A^{m+2−s_1} (n=2) or W'_{s'}×A^{m+n−2−s_{n−1}} (n≥3) without proof. These fiber identities are exactly what makes f good and drives the induction; they must be proved explicitly.","section":"Section 5, Corollary 5.7, Lemma 5.11, Lemma 5.12"},{"comment":"The proof of the claimed ideal equality is not complete. The cofactor-expansion argument relies on informal statements such as 'we observe', 'if there is any surviving expansion term', and 'the expansion process must stop', without a precise induction invariant or termination argument. Since Lemma 5.14 is used to replace X_M by X'_M and Theorem 5.4 concerns the irreducible components of X_M, this equality is load-bearing. Moreover, the assertion that 'same k-points and thus same reduced subscheme structure' needs justification: for non-reduced schemes of finite type over k, the set of k-points does not determine the reduced structure in general.","section":"Section 5, Lemma 5.14"},{"comment":"The final step of Theorem 5.4 states 'one can easily check that V_r ≃ GL_{m+1}×G_m×A^{m+1}_k' and 'V_r ≃ V_{r'}×G_m×A^{m+n−1}_k' without giving the isomorphisms; these claims carry the induction. Remark 5.16 also asserts that the authors have established Conjectures 5.2 and 5.3 for all M of order ≤5, but no proof or reference is provided. Such an unsupported result should either be proved or explicitly labelled as work in progress.","section":"Section 5, proof of Theorem 5.4 and Remark 5.16"}],"minor_comments":[{"comment":"The citation [SZ14, Thmeorem C] contains a typo: 'Thmeorem' should be 'Theorem'.","section":"Section 2, Lemma 2.1"},{"comment":"'enries' should be 'entries'.","section":"Section 3, Proposition 3.5"},{"comment":"In part (1), the closed subscheme □ is defined using T_{i,j} but the ambient ring is in variables Z_{i,j}. This should be Z_{i,j}.","section":"Section 5, Corollary 5.7"},{"comment":"W_s is defined modulo det(T_{ij})_{n×n}−1; the notation should be det(T)−1 with T the (m+n)×(m+n) matrix. Also, 'W'_s is isomorphic to W_s×G^m' should read W_s×G_m.","section":"Section 5, Lemma 5.12"},{"comment":"The claim about order ≤5 needs either a proof or a clear statement that it is work in progress.","section":"Section 5, Remark 5.16"}],"recommendation":"major_revision","confidential_remarks":"The core PID results and Theorem 4.2 are solid and worth publishing. The main issue is the proof of Theorem 5.4, which is not in publishable shape: the codomain error in Lemma 5.12 is a clear mathematical mistake, and the numerous 'one checks' in the fibration arguments must be filled in. These gaps appear repairable, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has two real accomplishments. The PID local-global statements (Propositions 3.3 and 3.5) are clean, correctly proved, and new, and the non-PID counterexamples in Remarks 3.4 and 3.7 look right. Theorem 4.2, showing that diagonalization varieties break into connected components each isomorphic to a product of general linear groups, is a genuinely nice structural result and the proof holds up. The introduction of the stratified Brauer–Manin obstruction is also a natural idea and worth developing.\n\nThe trouble is concentrated in the proof of the advertised special-case theorem, Theorem 5.4. That proof is carried by Lemma 5.12, and the stress-test concern is accurate. The morphism f: W_s -> Spec k[X,Y,1/(ΣXY)] is not well-posed as a good fibration because on W_s the determinant is 1, so ΣXY = 1 identically. The stated target contains points with ΣXY ≠ 1, and those points have empty fiber. The second condition of Definition 5.5 therefore fails for the morphism as written. The fix is plausibly just to change the target to Spec k[X,Y]/(ΣXY−1), and the rest of the fiber computation might then go through, but the lemma as written is not proved, and Theorem 5.4 currently rests on it.\n\nThis is a load-bearing gap, not a cosmetic typo. The same proof also leans on Corollary 5.7 and Lemma 5.11, where the crucial fiber descriptions and surjectivity on integral points are asserted with 'one checks' and 'arises naturally'; those need real arguments too. Lemma 5.14, by contrast, is intricate but is at least actually argued in the text, so I would not put it in the same category. Remark 5.16's claim of order-≤5 results without proof should be removed or supported.\n\nI want to be clear: the earlier sections are good, and the special-case theorem is plausible and probably repairable. But the paper should not be accepted with Theorem 5.4 in its current form. The right move is to send it to a serious referee with a request to focus on Lemma 5.12 and the fibration lemmas; the author can then fix the target and fill in the missing verifications.\n\nWho is this for? Arithmetic geometers working on Hasse principles for matrix varieties, and linear algebraists interested in local-global questions. A careful reader gets real value from Sections 3–4 even if Theorem 5.4 needs repair. I would bring it to a reading group and would cite the PID results if I worked in the area.","headline":"Sections 3–4 are solid and genuinely new, but Theorem 5.4 is carried by a good-fibration lemma whose target is mis-specified as written; worth refereeing, not acceptable as is.","tokens_in":22547,"tokens_out":6142,"would_cite":true,"duration_ms":61357,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G12","15A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that when the ring of integers of a number field is a principal ideal domain, a matrix that is triangularizable or diagonalizable over every local completion is already triangularizable or diagonalizable over the ring of i","keywords":["local-global principle","Hasse principle","triangularizability","diagonalizability","Brauer–Manin obstruction","stratified Brauer–Manin obstruction","Jordan block","principal ideal domain"],"falsifier":"Compute the morphism in Lemma 5.12 for the smallest nontrivial case m=1, n=2: check whether the map f: W_s → Spec k[X,Y]/(ΣXY−1) is smooth and surjective on O_v-points for almost all v, and whether each fiber is isomorphic to SL_{2,s_2−1}×A^{3−s_1}; if the fiber structure or the codomain is wrong, the induction in Theorem 5.4 loses its foundation.","tokens_in":21569,"feed_emoji":"🧮","tokens_out":7659,"duration_ms":69154,"temperature":0.7,"pith_summary":"The paper establishes a local-global principle for two basic matrix properties over number fields: if the ring of integers is a principal ideal domain, then a matrix that is triangularizable (or diagonalizable) over every completion—or even over almost all residue fields—is triangularizable (resp. diagonalizable) over the number field and over its ring of integers. The proof is elementary and uses unimodular completion plus the Chebotarev density theorem to pass from local splitting of the characteristic polynomial to global splitting. When the ring of integers is not a PID, the principle can fail, and the paper translates the problem into algebraic geometry by associating to each matrix a triangularization variety X_M and a diagonalization variety Y_M. For diagonalization, Y_M is shown to be a disjoint union of products of general linear groups, so the classical Brauer–Manin obstruction is the only obstruction to rational and integral points. For triangularization, the paper introduces a stratified Brauer–Manin obstruction and proves that it is the only obstruction for matrices similar to a diagonal block plus a single Jordan block, with every irreducible component of X_M smooth and k-rational.","feed_headline":"Local diagonalizability is global over PID integer rings","feed_subtitle":"Same for triangularizability; failures over non-PID rings are traced through Brauer–Manin.","key_machinery":"The central mechanism is the translation of a linear-algebra question into the arithmetic of an auxiliary variety. For a fixed M, the triangularization variety X_M ⊂ GL_n is defined by w·det(T)=1 and (T^*MT)_{ij}=0 for i>j, so its rational or integral points are exactly the transition matrices that achieve triangularization; the diagonalization variety Y_M is defined analogously by off-diagonal entries vanishing. For diagonalization, the proof uses the fact that Y_{M,red} splits as a disjoint union of products of general linear groups, reducing the Brauer–Manin statement to known theorems for linear algebraic groups. For triangularization, the load-bearing tool is the 'good fibration' techni","core_discovery":"The central claim is that the local-global principle holds for triangularizability and diagonalizability of matrices over rings of integers that are PIDs: if M is triangularizable (resp. diagonalizable) over O_v for every place v—or even over the residue fields O_v/𝔭O_v for almost all v—then M is triangularizable (resp. diagonalizable) over k and over O_k. The key reduction is Lemma 3.1: over a PID, a matrix is integrally triangularizable if and only if it is triangularizable over the fraction field. To explain failures over non-PIDs, the authors define the triangularization variety X_M and diagonalization variety Y_M as closed subvarieties of GL_n cut out by requiring T^*MT to be triangular","pith_inferences":["The paper's approach suggests a general recipe for proving local-global principles for matrix properties: encode the property as a variety, decompose it via eigenvalue permutations, and use the fibration method; the Jordan-block case appears to be the hardest core, so proving the conjectures for a matrix with several Jordan blocks of the same eigenvalue would be a natural next test.","The 'good fibration' assertions in Lemma 5.12 are the point most worth scrutinizing: the codomain is written as a localization at ΣXY, but the image lies in the hypersurface ΣXY=1, so a reader extending this work should first verify that the morphism to Spec k[X,Y]/(ΣXY−1) is indeed smooth and has the claimed fibers SL_{s,r}×A^d.","Because the diagonalization variety is a union of algebraic groups, the Brauer group of Y_M is computable in explicit examples; one could use the non-PID counterexamples from Section 3 to compare the classical Brauer–Manin set with the actual rational points and see exactly which Brauer classes account for the failures.","The reduction to a single eigenvalue in Theorem 5.4 means that matrices with multiple Jordan blocks for the same eigenvalue form the immediate test case for Conjectures 5.2 and 5.3, and a concrete computation there would either confirm the conjectural picture or provide a counterexample."],"forward_implications":["Over principal ideal domains such as the integers, a matrix triangularizable modulo almost every prime is triangularizable over the ring itself, giving an effective integral Hasse principle for triangularizability.","The local-global principle for integral diagonalizability and triangularizability holds exactly when the ring of integers is a PID; for every non-PID ring of integers, the paper constructs explicit counterexamples.","For diagonalizability, the Brauer–Manin obstruction is always the only obstruction, so any failure of the local-global principle for diagonalizability is explained by the Brauer group of the associated variety.","For triangularizability, the stratified Brauer–Manin obstruction is the only obstruction for matrices with a single nontrivial Jordan block, and according to the paper's final remark also for all matrices of size at most 5.","If the paper's Conjectures 5.2 and 5.3 hold in full generality, the stratified Brauer–Manin obstruction would be the only obstruction to rational and integral points for every triangularization variety X_M."],"fun_headline_variants":["PID rings: local triangularizability implies global","Over PID integer rings, local diagonalizability is global","Local-global principle for matrix triangularizability proven over PIDs","PID rings settle local-global for triangularization and diagonalization","For PID rings, local-to-global holds; non-PIDs need Brauer–Manin"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of Theorem 5.4 rests on the claim that certain morphisms are 'good fibrations'—smooth, surjective on integral points for almost all places, with fibers isomorphic to affine spaces or special linear groups—and these claims are verified only by 'one checks' rather than by detailed proof; if any of those fibrations is not good (or if the apparent codomain error in Lemma 5.12 changes the fiber structure), the induction showing that the stratified Brauer–Manin set is cov","fun_headline_variants_meta":{"raw":{"variants":["PID rings: local triangularizability implies global","Over PID integer rings, local diagonalizability is global","Local-global principle for matrix triangularizability proven over PIDs","PID rings settle local-global for triangularization and diagonalization","For PID rings, local-to-global holds; non-PIDs need Brauer–Manin"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001308,"raw_usage":{"total_tokens":5109,"prompt_tokens":627,"completion_tokens":4482,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":371,"completion_tokens_details":{"reasoning_tokens":4395}},"tokens_in":371,"tokens_out":4482,"duration_ms":31090,"temperature":1.0,"reasoning_tokens":4395,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T21:18:35.257242+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the morphism in Lemma 5.12 for the smallest nontrivial case m=1, n=2: check whether the map f: W_s → Spec k[X,Y]/(ΣXY−1) is smooth and surjective on O_v-points for almost all v, and whether each fiber is isomorphic to SL_{2,s_2−1}×A^{3−s_1}; if the fiber structure or the codomain is wrong, the induction in Theorem 5.4 loses its foundation.","supporting_citations":[],"review_version":1}