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A proper Euler magic matrix of order $5$

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read An explicit 5×5 integer matrix is Euler magic and proper.

desk verdict 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. read the letter →

arxiv 2607.19416 v1 pith:WCSXYOH2 submitted 2026-07-18 math.GM

Scott Duke Kominers This is my paper · ORCID
classification math.GM MSC 11C2015B3605B20
keywords Eulermagicmatrixsquareofsquaresorder5GivensrotationrationalorthogonalSO(5)Diophantineequationinteger
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

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.

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.

minor comments (4)
  1. [Section 4] 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.
  2. [Section 4, displayed M0] 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.
  3. [Remark 4.2] 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.
  4. [Appendix A] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is an explicit witness with independent verification.

full rationale

The paper's central claim — existence of a proper Euler magic matrix of order 5 — is supported by an explicit integer matrix M5 and a self-contained derivation. The only input taken from prior work (Müller's reduction and the seed M0) is both stated in full and re-verified inside the paper: Lemma 2.1 is proved, M0 is displayed and checked by exact arithmetic, and the single-equation reduction is justified by Theorem 3.1 with a symbolic proof. The key algebraic identity (D(N)-A(N))(1+x^2)^2(1+y^2)^2 = -256 F1 F2 is stated as a direct computation and independently certified by the exact-arithmetic script in Appendix A, whose printed PASS output covers all 15 checks. The rational point (755/547, 671/631) is explicitly exhibited and its membership in F2=0 is verified by hand with displayed intermediate values. Properness and gcd are checked as 300 exact integer differences, not assumed from any fitted quantity. No prediction is derived from a parameter fitted to the target result; no load-bearing claim reduces to a self-citation. The speculative remarks in Section 5 do not affect Theorem 4.1. Thus no circular step is present.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The construction is an explicit witness, so the free-parameter count is small: the chosen rational rotation parameters make D=A and avoid the seed's single square coincidence. No new entities are postulated. The main load-bearing premises are the unexpanded polynomial identity and the correctness of the exact-arithmetic appendix, plus the faithful transcription of Müller's seed.

free parameters (1)
  • (x0, y0) = (755/547, 671/631) = 755/547, 671/631
    Rational point on the curve F2=0 found by a height-bounded search; chosen so the rotated matrix is Euler magic and avoids the coincidence locus. It is a witness, not a fit to external data.
assumptions (3)
  • ad hoc to paper The direct-computation identity (D(N)-A(N))·((1+x^2)^2(1+y^2)^2) = -256 F1 F2 is correct.
    Stated in Section 4 as 'a direct computation'; verified by the Appendix A symbolic script rather than by hand expansion. This is the load-bearing algebraic reduction.
  • domain assumption Müller's near-miss M0 is correctly transcribed and satisfies MM^t = 143^2 I, D=A=143^2, det = 143^5, and exactly 24 distinct entry-squares.
    The property is re-verified in Appendix A, but the theorem inherits M0's structure; a transcription error would break the construction.
  • domain assumption The Appendix A script, as printed, executes in Python 3.12/SymPy 1.14 and its Appendix B output truthfully reports 15/15 PASS.
    No external verifier or commit hash is provided; the computational existence proof depends on this exact-arithmetic certificate.

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Cite this review

Pith. "Pith review of A proper Euler magic matrix of order $5$." pith.science (2026). https://pith.science/paper/WCSXYOH2

@misc{pith2026260719416,
  author       = {Pith},
  title        = {Pith review of: A proper Euler magic matrix of order $5$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCSXYOH2}},
  note         = {Machine review of arXiv:2607.19416}
}
abstract

An Euler magic matrix is an integer matrix $M$ with $MM^{t}=\gamma I$ whose squared entries sum to $\gamma$ along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler constructed an order-$4$ proper example, and M\"{u}ller settled orders $3$ (none exist) and $8$, leaving order $5$ as the smallest open case. We construct such a matrix, by rotating one of M\"{u}ller's "near-misses" under a mirror-symmetric coordinate pair so that the two diagonal conditions collapse to a single rational equation; the same invariant suggests a uniform approach to the odd orders.

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Reference graph

Works this paper leans on

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