{"id":"b9a3923a-a6e0-443f-9a8b-353b14967e7f","arxiv_id":"2606.28976","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Over reduced rings and quotients by integrally closed ideals, a matrix with all principal minors 1 has all powers with the same property; over arbitrary rings the right invariant is the stronger 1-nullcyclic condition.","lead":"This paper proves that if a square matrix has every principal minor equal to 1, then all of its integer powers do too, whenever the entries live in a 'well-behaved' ring such as a reduced ring or a quotient of the integers. It also identifies the stronger combinatorial condition, 1-nullcyclicity, that is preserved by powers over every commutative ring.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Lemma 4.3 is the most delicate combinatorial input, but its proof is sound and the induction in Theorem 4.6 holds together.","rationale":"The reader correctly identifies Lemma 4.3 as the most load-bearing assumption, and the proof of Theorem 1.2 does rely on it structurally. However, on careful re-derivation the lemma is correct: the two constructed cycles are short, arc-disjoint in the multigraph, and the leftover balanced digraph gives short cycles that avoid the chosen vertex. The subsequent Lemma 4.5 and the induction in Theorem 4.6 are also sound. Thus I do not find a mathematical flaw that would change the verdict. The reader's CONDITIONAL verdict is reasonable given the lack of formal verification and the AI-generated provenance, but my stress test does not expose a concrete error. I therefore leave the verdict unchanged.","tokens_in":15044,"tokens_out":16552,"duration_ms":132445,"concrete_test":"Formalize Lemma 4.3 in a proof assistant (e.g., Lean), or at minimum brute-force verify it for all pairs of distinct Hamilton cycles for |S| ≤ 8 by enumerating cycles and checking that the proof's construction yields a partition into at least three cycles of length < |S|. Because Lemma 4.3 is finite and purely combinatorial, any counterexample would immediately invalidate the proof of Theorem 4.6 and hence Theorem 1.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem (Theorem 1.2) depends on Theorem 4.6, whose key combinatorial engine is Lemma 4.3. I examined Lemma 4.3 closely. The construction of C and C′ from a vertex with distinct outgoing arcs is valid: each has length at most |S|−1, they are arc-disjoint in the multigraph sense (counting multiplicities when H and H′ share arcs), and the remaining balanced multigraph yields cycles avoiding v, so the whole union partitions into at least three cycles of length < |S|. Lemma 4.5 then correctly promotes this to products in K_{<r}(A)^t, and the symmetric-polynomial/Viète step in Theorem 4.6 shows each cycle weight is integral over J(A). The only blemish is an apparent typo in the invocation of Lemma 4.2(d): the conclusion should be \\overline{L} = \\overline{J(A)}, not L = \\overline{J(A)}. Since the text later uses w_A(H) ∈ \\overline{L}, the intended argument is unaffected. The absence of a machine-checked proof is a verification gap, not a detected flaw. No load-bearing mathematical objection identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies matrices over commutative rings whose principal minors are all 1 (called 1-principled) and asks whether this property is preserved under taking powers. The main theorem (Theorem 1.2) proves it for quotients D/I where I is integrally closed, yielding the cases Z/d, quotients of Prüfer domains, principal quotients of normal domains, and all reduced rings (Corollaries 4.9, 5.4, 5.8, 5.10). The proof introduces stronger conditions — principled, nullcyclic, 1-nullcyclic — proves they are stable under powers (Propositions 2.3, 3.5 and Lemma 3.4), and then bridges from principled to nullcyclic through a universal integrality theorem (Theorem 4.6): every nontrivial cycle weight is integral over the ideal generated by principal-minor defects. The engine of Theorem 4.6 is a combinatorial lemma (Lemma 4.3) on partitioning the multiset union of two distinct Hamilton cycles into short cycles, together with an induction using elementary symmetric polynomials and Viète's formulas.","tokens_in":15326,"tokens_out":16953,"duration_ms":134854,"significance":"If the results hold, this is a clean and substantive generalization of Putnam 2021 B5, moving from Z/2 to a broad class of rings. The paper identifies 1-nullcyclicity as the ring-independent property that explains stability under powers, and the universal integrality statement of Theorem 4.6 is a valuable result in its own right. The exposition is detailed, largely self-contained, and gives explicit examples and a counterexample over a non-reduced ring. I examined the most delicate input, Lemma 4.3; its proof is sound (for |S|=2 the hypothesis is vacuous because the two Hamilton cycles coincide up to rotation). The walk-sum arguments and the Vieta induction in Theorem 4.6 are coherent. I found no load-bearing mathematical error.","major_comments":[],"minor_comments":[{"comment":"After equation (4.4), the text invokes Lemma 4.2(d) and concludes \"L = \\overline{J(A)}\". The correct conclusion is \"\\overline{L} = \\overline{J(A)}\", or one can directly use the already-established inclusion L ⊆ \\overline{J(A)} to get \\overline{L} ⊆ \\overline{J(A)}. As printed, the equality is generally false and the cited lemma concerns integral closures. This is a typographical slip and does not affect the proof's conclusion.","section":"§4, Theorem 4.6"},{"comment":"The statement that the 4×4 counterexample is \"minimal in the sense that there are no matrices of size ≤ 3 that are 1-principled but have non-1-principled powers\" is asserted without proof or reference. This is not load-bearing, but it should be justified or explicitly marked as an empirical observation.","section":"Footnote 6"},{"comment":"The notation \"\\overline{I}\" for integral closure is introduced in Definition 4.1, and subsequently the same overline is used for ideals such as \\overline{L} in Theorem 4.6. The proof would be easier to read if the overline notation were also explained in a sentence just before its first use in Lemma 4.2, or if the authors consistently say \"integral closure\" at the first few occurrences.","section":"§1–§4"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript carries an unusual provenance statement, but I judged the mathematics on its own merits. The central theorem is defensible and the proof structure is sound; the only substantive correction is the missing overline in Theorem 4.6, which is clearly a typographical issue. The paper is suitable for publication after the minor revisions are made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper proves a genuinely new generalization of Putnam 2021 B5, and I think the math is sound. The headline result is Theorem 1.2: if a matrix over D/I has all principal minors equal to 1, and I is integrally closed, then every integer power has the same property. That covers reduced rings and Z/d, which is exactly the right scope after [2] showed the naive generalization fails over arbitrary finite rings. The key new object is the 1-nullcyclic condition, which is stable under all integer powers over any commutative ring, and the universal integrality theorem (4.6) saying every nontrivial cycle weight lies in the integral closure of the principal-minor-defect ideal. That is a nice unifying statement.\n\nWhat the paper does well: the proofs are detailed and internally consistent. I checked the delicate combinatorial Lemma 4.3—partitioning the union of two Hamilton cycles into at least three shorter cycles—and the construction is valid. The induction in Theorem 4.6 using Vieta's formulas holds together. The self-citation to [2] is not circular: Remark 3.3 gives an alternative proof, and Theorem 4.6 is proven directly. The counterexample from [2] is correctly used to show why the integrally closed condition is needed.\n\nSoft spots: first, the paper discloses in a footnote that it was drafted by GPT-5.5 with strategic prompting and then edited. That is not a mathematical flaw, but it does mean independent verification, ideally formalization, is advisable before treating the result as fully established. I found no load-bearing errors, so this is a verification gap, not a detected defect. Second, there is a small typo in the proof of Theorem 4.6: Lemma 4.2(d) should give the equality of the integral closures, \\overline{L} = \\overline{J(A)}, not L = \\overline{J(A)}. The intended conclusion w_A(H) ∈ \\overline{J(A)} still follows. Third, Lemma 4.2 is quoted from [6] without proof; that is standard but worth noting.\n\nWho should read this: commutative algebraists and algebraic combinatorists, especially anyone interested in principal minors, integral closure, and matrix powers; also people who track Putnam problems. It deserves a serious referee. I would send it to peer review, with a note asking the referee to pay special attention to Lemma 4.3 and Theorem 4.6, and to confirm the typo fix.","headline":"A solid, genuinely new generalization of Putnam 2021 B5 with a delicate but sound combinatorial core; the AI provenance disclosure is the only real concern, and it argues for independent verification rather than distrust.","tokens_in":15828,"tokens_out":3103,"would_cite":true,"duration_ms":26703,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13B22","15A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a matrix over any quotient of a commutative ring by an integrally closed ideal — including every reduced ring and every ring Z/d — has all its powers (positive and negative) sharing the property that all principal mino","keywords":["1-principled matrix","principal minors","nullcyclic matrix","integral closure of ideals","powers of matrices","Putnam problem B5","Hamilton cycles","cycle weights"],"falsifier":"Enumerate all pairs of distinct Hamilton cycles on a finite set (e.g., size 5) and check whether their multiset union always admits a partition into at least three cycles of length < |S|; one counterexample would falsify Lemma 4.3 and therefore the main theorem. On the ring side, a finite-ring search (e.g., over Z/4) for a 1-principled matrix whose square is not 1-principled would directly contradict Corollary 5.4 and the main theorem.","tokens_in":14906,"feed_emoji":"🧮","tokens_out":7916,"duration_ms":70407,"temperature":0.7,"pith_summary":"The paper establishes that the property of having all principal minors equal to 1 is preserved under taking any integer power of a matrix, whenever the ground ring is a quotient of a commutative ring by an integrally closed ideal. This covers every reduced ring and every ring Z/d, and it recovers a Putnam competition problem as a special case. The proof isolates a stronger condition — nullcyclicity, the vanishing of all products along nontrivial cycles — that passes to all powers over any ring, and shows that over these well-behaved rings every 1-principled matrix is 1-nullcyclic. A universal theorem shows that cycle weights are always integral over the ideal generated by principal-minor defects, which is the engine that makes the special cases work.","feed_headline":"All powers keep a matrix's all-1 principal minors","feed_subtitle":"Holds for reduced rings and for Z/d, and more generally whenever the kernel ideal is integrally closed.","key_machinery":"Two components carry the argument. The universal integrality theorem (Theorem 4.6) uses Viète's formulas and a determinant expansion: for each size r, the elementary symmetric functions of the weights of Hamilton cycles on an r-element set lie in the ideal generated by shorter cycle weights and principal-minor defects, forcing each cycle weight to be integral over J(A). To establish the needed ideal-membership bounds, the paper proves a purely combinatorial lemma (Lemma 4.3): the multiset union of two distinct Hamilton cycles on a finite set can be partitioned into at least three cycles, each having length strictly smaller than the set's size. This lemma is what lets the author place product","core_discovery":"The paper proves Theorem 1.2: if I is an integrally closed ideal of a commutative ring D and A is a matrix over D/I whose principal minors are all 1, then every power A^m (m any integer) also has all principal minors 1. The key upstream result is Theorem 4.6: for any n×n matrix A over any commutative ring, the weight of every nontrivial cycle — a product of entries along a cycle — lies in the integral closure of the ideal J(A) generated by the differences between principal minors and the corresponding products of diagonal entries. When J(A) lies in an integrally closed ideal I, reducing modulo I forces all cycle weights to vanish, so A is nullcyclic; nullcyclicity is inherited by powers via","pith_inferences":["The main theorem suggests classifying rings over which the Putnam property holds in full; the paper's evidence points to quotients by integrally closed ideals, and a natural testable question is whether every ring with the property admits such a presentation.","Lemma 4.3 may be useful beyond matrices: any weight function on arcs of a complete digraph that vanishes on short cycles forces products over Hamilton cycles to lie in powers of the ideal generated by short-cycle weights, suggesting a general graph-theoretic integrality principle.","The walk-expansion proof of nullcyclicity inheritance works over any ring, so 1-nullcyclicity is a genuinely ring-independent sufficient condition; one could explore a 'nullcyclic up to nilpotents' variant that might extend the result beyond integrally closed quotients."],"forward_implications":["Over every reduced ring, a 1-principled matrix has all integer powers 1-principled (Corollary 4.9).","Over Z/d for any integer d, including the original Putnam setting d=2, every power of a 1-principled matrix is 1-principled (Corollary 5.4).","The same conclusion holds over quotients of Prüfer domains by arbitrary ideals and over principal quotients of normal domains (Corollaries 5.8 and 5.10).","Without the 1's: a principled matrix over such rings has all nonnegative powers principled (Theorem 1.9).","The universal integrality theorem gives a new ideal-theoretic constraint: even when cycle weights do not vanish, they are always integral over J(A), so their behavior is controlled by integral closure rather than by ad hoc computations."],"fun_headline_variants":["Powers keep 1-principled matrices 1-principled","If principal minors are 1, so are all powers' minors","Cyclic-product ideal proof: powers stay nullcyclic","Integrally closed ideals yield power-closed 1-principality"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument's load-bearing premise is Lemma 4.3, the combinatorial assertion that the multiset union of two distinct Hamilton cycles on a finite set can be partitioned into at least three cycles each shorter than the set; the main theorem follows only if this decomposition always exists.","fun_headline_variants_meta":{"raw":{"variants":["Powers keep 1-principled matrices 1-principled","If principal minors are 1, so are all powers' minors","Cyclic-product ideal proof: powers stay nullcyclic","Integrally closed ideals yield power-closed 1-principality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000237,"raw_usage":{"total_tokens":1507,"prompt_tokens":1070,"completion_tokens":437,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":814,"completion_tokens_details":{"reasoning_tokens":363}},"tokens_in":814,"tokens_out":437,"duration_ms":4636,"temperature":1.0,"reasoning_tokens":363,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T04:34:11.643671+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all pairs of distinct Hamilton cycles on a finite set (e.g., size 5) and check whether their multiset union always admits a partition into at least three cycles of length < |S|; one counterexample would falsify Lemma 4.3 and therefore the main theorem. On the ring side, a finite-ring search (e.g., over Z/4) for a 1-principled matrix whose square is not 1-principled would directly contradict Corollary 5.4 and the main theorem.","supporting_citations":[],"review_version":3}