{"id":"13e7f8bf-7f42-4e73-bc53-8cb854b554e4","arxiv_id":"2607.19416","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"An explicit 5×5 integer matrix is constructed whose transpose-times-self is scalar, whose row/column and both diagonal square-sums are that scalar, and whose 25 squared entries are all distinct.","lead":"This paper constructs the first proper Euler magic matrix of order 5—a 5×5 integer matrix whose rows, columns, and both main diagonals agree in their sum of squares, with all 25 squared entries distinct. The trick is to rotate a known near-miss with mirror-symmetric planar rotations, reducing the two diagonal conditions to one rational equation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The construction is internally consistent. Theorem 3.1's invariance algebra checks out, the rational point lies on F2=0 via the displayed identities, and the remaining burden is exact finite verification, which the appendix discharges with a complete script rather than an unverifiable assertion. The reader's weakest-assumption identification is reasonable—the unexpanded identity and the script output are the least-hand-checkable parts—but I do not consider this a load-bearing objection because the computational certificate is concrete and reproducible. An independent rerun or symbolic re-expansion would settle the matter, but nothing in the current text suggests a substantive gap. Therefore the ACCEPT verdict should stand unchanged.","tokens_in":9411,"tokens_out":13536,"duration_ms":116121,"concrete_test":"Run the Appendix A script in a fresh Python/SymPy session (or independently in Sage/Mathematica): verify symbolically that (D(N)-A(N))*(1+x^2)^2*(1+y^2)^2 + 256*F1*F2 == 0 for N=G(x)M0G(y), then verify d*N(755/547,671/631)==M5, M5*M5.T == (143*d)^2*I5, det(M5)==(143*d)^5, and that all 300 pairwise entry-square differences are nonzero. If all checks pass, Theorem 4.1 is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—existence of a proper Euler magic matrix of order 5—is supported by an explicit matrix and an elementary reduction. The only step not expanded by hand is the polynomial identity for D(N)-A(N) in Section 4, together with the finite integer checks. These are exactly what the Appendix A exact-arithmetic script verifies symbolically and over Z, with printed PASS output in Appendix B. I spot-checked M0's row and diagonal sums and the substitution F2(755/547, 671/631)=0; both are correct. The proof has no circularity and no unstated mathematical assumption beyond the computational certificate, which is fully specified and reproducible. Section 5's speculative remarks do not affect Theorem 4.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs an explicit 5x5 integer matrix M5 and proves that it is a proper Euler magic matrix of order 5: M5 M5^t = gamma I with gamma = (143 d)^2, the sums of squares along both main diagonals equal gamma, and all 25 entry-squares are pairwise distinct. The construction starts from a 5x5 near-miss M0 due to Müller and applies Givens rotations supported on a mirror-symmetric row/column pair, which leaves D+A invariant and reduces the two diagonal conditions to the single equation F1 F2 = 0. The proof reduces all remaining claims to exact polynomial identities and finite integer checks, and the paper supplies a complete SymPy verification script and its PASS output in appendices.","tokens_in":9628,"tokens_out":13656,"duration_ms":120169,"significance":"If correct, this resolves the smallest open order for proper Euler magic matrices, complementing Euler's order-4 example and Müller's order-8 construction. The proof is an explicit existence proof with a reproducible machine-checked certificate: the full matrix is displayed, the key polynomial identity is verified symbolically, properness is checked as 25 distinct integer squares, and the determinant/gcd claims are verified exactly. The exploratory remarks in Section 5 are clearly separated from the main theorem and do not affect its validity.","major_comments":[],"minor_comments":[{"comment":"The identity (D(N)-A(N))*((1+x^2)^2(1+y^2)^2) = -256 F1 F2 is the computational heart of the proof. It is verified by the exact SymPy script in Appendix A, but the main text gives no human-readable derivation. Since the theorem depends on this identity, a short expansion or an explicit statement that this is an exact rational-function identity checked by the attached code would improve readability. This is a presentation issue, not a correctness issue.","section":"Section 4"},{"comment":"The text says the repeated entries equal to 20 in M0 are 'indicated in bold', but the bold formatting is not visible in the plain-text reproduction. Please mark the two positions (3,2) and (5,3) explicitly in the published version so the near-miss property is immediately visible.","section":"Section 4, displayed M0"},{"comment":"The phrase 'the image of the rotation curve F2(x,y)=0' is slightly imprecise because the map (x,y) -> N(x,y) may have fibers' issues. I suggest writing 'the image of the curve F2=0 under (x,y) -> N(x,y)' to avoid ambiguity. This does not affect the theorem.","section":"Remark 4.2"},{"comment":"The verification script uses 0-based indices for the Givens rotations, while the main text uses 1-based mirror pairs {2,4}. The comments in the script make the correspondence clear, but a sentence in the main text explicitly connecting the 0-based indices (1,3) to the mirror pair {2,4} would help readers who inspect the code.","section":"Appendix A"}],"recommendation":"accept","confidential_remarks":"This is a clean, self-contained computational construction. The main theorem is verified by exact arithmetic in the included script, and I found no mathematical gap. The minor comments are purely presentational and do not require re-review after revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing to know: this paper settles the smallest open order for proper Euler magic matrices. The explicit 5×5 matrix M5 is printed in full, and the proof that it works is essentially exact computation. I read the argument through and I think the central claim holds up.\n\nWhat is genuinely new is the mirror-pair Givens invariance: left- or right-multiplying by a rotation on a mirror pair {a, n+1−a} preserves D+A but not D−A, so the two diagonal conditions collapse to a single equation after rotating a near-miss seed. That is a real organizing observation, and it explains why Müller's near-miss could be fixed by one rational rotation pair. The paper also gives a clean hand-checkable rational point on the curve F2=0, with the messy polynomial identity verified symbolically in the appendix. The SymPy script is self-contained, covers all 15 checks, and the printed output shows all passing. For this kind of existence proof, that is acceptable and even good practice.\n\nThe soft spots are real but minor. The identity D(N)−A(N) = −256 F1 F2 / den is asserted as \"a direct computation\" and not expanded by hand; the appendix is the actual evidence. That is fine for a referee who trusts a concrete script, but it does mean the theorem's certificate is the code, not the prose. The paper is also a witness construction, not a method: the search for the rational point is not described beyond a brief height-bounded remark, and Section 5 explicitly leaves orders 7 and beyond open. The Kronecker-product remark and the Spin(5)/Sp(2) speculation are standard or speculative; they do not affect Theorem 4.1. The reader's report and the stress-test note both correctly find no load-bearing flaw. I agree.\n\nCitation pattern looks clean: Müller's reduction and near-miss seed are credited and re-verified, not just cited. No circularity.\n\nWho is this for? Anyone working on magic squares of squares, rational orthogonal matrices, or small Diophantine existence problems. It is short, explicit, and checkable. It deserves a serious referee: an expert should run the script and confirm the matrix, but I see no reason to desk-reject. Send it out.","headline":"Order 5 is no longer open: the paper gives an explicit proper Euler magic matrix with a sound proof and a reproducible exact-arithmetic certificate, even though the construction is more a clever search than a general method.","tokens_in":9998,"tokens_out":2076,"would_cite":true,"duration_ms":21347,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11C20","15B36","05B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"An explicit 5×5 integer matrix is Euler magic and proper.","keywords":["Euler magic matrix","magic square of squares","order 5","Givens rotation","rational orthogonal matrix","SO(5)","Diophantine equation","integer matrix"],"falsifier":"Independently recompute D(N(x,y))−A(N(x,y)) symbolically for the seed and rotations used, substitute (755/547,671/631), and directly verify all 300 pairwise differences of the squared entries of the displayed M5, along with all row and column sums; a single repeated entry square or a row sum different from γ would disprove the construction.","tokens_in":9320,"feed_emoji":"🧮","tokens_out":7122,"duration_ms":66419,"temperature":0.7,"pith_summary":"This paper resolves the last small open case in the theory of Euler magic matrices: order 5. It exhibits a concrete 5×5 integer matrix M5 whose row dot products vanish off the diagonal, whose squared entries all sum to the same number along every row, column, and both main diagonals, and whose 25 entry squares are pairwise distinct. The construction starts from a known near-miss whose entry squares take only 24 distinct values, then applies two rational rotations on a mirror-symmetric pair of rows and columns. A key invariant shows that the two diagonal conditions collapse to one rational equation after such rotations; a rational point on that equation is found, and clearing denominators yields the integer matrix. Because a proper Euler magic matrix yields a magic square of squares, the result supplies an explicit order-5 magic square of squares.","feed_headline":"Explicit 5x5 matrix closes Euler magic order 5","feed_subtitle":"The integer matrix has 25 distinct entry squares and equal square sums on both diagonals, ending the smallest open case.","key_machinery":"The load-bearing object is the mirror-pair Givens rotation: a rational planar rotation acting on rows and columns a and n+1−a, with cosine and sine parameterized by (1−t²)/(1+t²) and 2t/(1+t²). Any such rotation, applied on the left or right of an Euler magic matrix, leaves D+A unchanged, so the two diagonal conditions become the single condition D=A. Together with the explicit factorization D−A = −256F1F2/((1+x²)²(1+y²)²), this reduces the construction to finding a rational point on F2=0, which the paper does by a height-bounded search; the rational point, after clearing denominators, becomes the integer matrix M5.","core_discovery":"For the starting near-miss seed and two mirror-pair Givens rotations G(x),G(y), the sum D+A of the two diagonal square-sums is preserved, so the rotated matrix is Euler magic exactly when D=A. Symbolically D−A = −256F1F2/((1+x²)²(1+y²)²), so the two diagonal conditions collapse to one rational equation F1F2=0. The rational point (755/547,671/631) lies on F2=0; multiplying the resulting rational orthogonal matrix by d=434617·424201 gives the integer matrix M5 with γ=(143d)². Exact arithmetic confirms M5M5ᵀ=γI, both diagonal sums, 25 distinct entry squares, and entry gcd 1.","pith_inferences":["A natural next step would be to search for rational points on the curve F2=0 other than the one used; if that curve is rational, the construction might yield an infinite family of order-5 examples rather than a single matrix.","The same two-sided mirror-rotation template could be tried at orders 7 and 9, starting from any seed with a single repeated pair of entry squares; success would extend the result to all odd orders.","The paper leaves open whether the exceptional isomorphism Spin(5) ≅ Sp(2) gives a more conceptual parametrization; working that out could replace the height-bounded search with a structural derivation.","Because the decisive identities are delegated to an exact-arithmetic script, an independent reimplementation of the fifteen printed checks would be a cheap and decisive way to strengthen confidence in the theorem."],"forward_implications":["A proper Euler magic matrix of order 5 exists; the smallest open case is closed.","Squaring the entries of M5 yields a 5×5 magic square of squares with common sum γ=(143d)².","The rescaled matrix lies in SO(5,Q) and cuts the variety V5 smoothly of dimension 8 near it, so M5 sits in a positive-dimensional real family of Euler magic matrices with the same constant.","Because the Kronecker product of two Euler magic matrices is again Euler magic, the order-5 example combines with other orders to produce Euler magic matrices of composite order.","The same mirror-pair mechanism works in every odd order; for larger odd orders only a suitable near-miss seed and a rational solution of the resulting equation are missing."],"fun_headline_variants":["Explicit 5x5 Euler magic matrix constructed","Order-5 Euler magic solved explicitly","5x5 matrix ends smallest open Euler magic case","Distinct-square 5x5 achieves Euler magic","Euler magic order 5 falls to explicit construction"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof rests on the computer-verified polynomial identity D−A = −256F1F2/((1+x²)²(1+y²)²) and on the printed output of an exact-arithmetic script checking properness; if either contains an unnoticed transcription or execution error, the constructed matrix may not be Euler magic or proper.","fun_headline_variants_meta":{"raw":{"variants":["Explicit 5x5 Euler magic matrix constructed","Order-5 Euler magic solved explicitly","5x5 matrix ends smallest open Euler magic case","Distinct-square 5x5 achieves Euler magic","Euler magic order 5 falls to explicit construction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000669,"raw_usage":{"total_tokens":2846,"prompt_tokens":666,"completion_tokens":2180,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":2107}},"tokens_in":410,"tokens_out":2180,"duration_ms":15837,"temperature":1.0,"reasoning_tokens":2107,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:23:29.042398+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently recompute D(N(x,y))−A(N(x,y)) symbolically for the seed and rotations used, substitute (755/547,671/631), and directly verify all 300 pairwise differences of the squared entries of the displayed M5, along with all row and column sums; a single repeated entry square or a row sum different from γ would disprove the construction.","supporting_citations":[],"review_version":1}