Theory
Theory contents
The material is divided into self-contained series. The main line runs from the linear form through arithmetic to partial square configurations, while matrix multiplication algebra develops in parallel.
Foundations
Definitions, the complete linear classification, and the main Diophantine problem.
The general form of a 3×3 magic square
Proofs of M=3E and the existence and uniqueness of m(E,x,y), with exact versions over groups, rings, fields, ℤ, ℚ, and ℝ.
The magic square of squares problem
The 9/9 and 7/9 problems, the known example, and the transition from the linear normal form to Diophantine constraints.
Arithmetic restrictions
Local congruences, sums of two squares, and restrictions on prime divisors.
Residues and quadratic residues
Coordinate congruence, restrictions modulo 3, 4, 5, and 24, and an application of the Legendre symbol to prime divisors of entries.
Prime divisors in a minimal 9/9 square
Gaussian factorizations of sums of two squares, the central-root factorization, and restrictions on prime divisors of the other eight roots.
Partial square configurations
From the classification of 4/9 and 5/9 patterns to progressions of squares, 6/9 configurations, and the tfmn method.
General theory of the 4/9 and 5/9 patterns
D₄ orbits, elimination of E,x,y, quadratic systems, colored supports, and criteria for completeness of parametrizations.
ABCDG: a globally complete algorithm
The entire rational surface is decomposed into smooth conics over P¹; exact fiber solving, projection, and fair dovetailing give a complete algorithm with an explicit inverse.
Arithmetic progressions of squares and the dir function
The complete rational and integral parametrization of three squares in progression, the eight lines of a magic square, and the normalized difference dir(m,n).
6/9 patterns: the sixteen positional types
A proof of the orbit count, the three independent quadrics of every pattern, the eight progressions in a magic square, and the exact scope of the two parallel tfmn classes.
fmn and tfmn in the 6/9 problem
Two parallel progressions in a 6/9 pattern, the function f, its squarefree part tf, the difference-matching criterion, and the elliptic curve of a fixed tf value.
The early F1–F8 families
Complete elementary parametrizations of F1–F4, the F7 identity, the F8 quartic lift, and the exact scope of the outdated simplified classification.
7/9 patterns: a complete trigonometric atlas
Eight D4 orbits, a complete basis of four quadrics for every pattern, and a reversible reduction of every system to one equation in three rational angles.
Elliptic geometry of tfmn
Complete coordinate descriptions of equal square classes through rational points on elliptic curves and surfaces.
Genus-one curves, Jacobians, and elliptic surfaces
The common language of the series: Jacobians of genus-one quartics, fibers and sections, Weierstrass models, Kodaira types, rational and K3 surfaces, the Shioda–Tate formula, and three distinct notions of rank.
F7+: parameter pairs and the congruent-number surface
The exact bijection between nondegenerate projective pairs [m:n] of fixed tf and nontrivial rational points on y²=x³−T²x, up to the sign of y.
Integral tfmn forms: containers, Kummer coordinates, and the group law
The four containers as a harmonic rank-two frame, their Kummer signature, exact square corrections under addition, doubling and tripling formulas, and the link with the Monsky matrix.
F4+: an elliptic surface for pairs of Pythagorean areas
Complete normalization of equal area square classes, a birational Weierstrass model, the resulting rational elliptic surface, and its quadratic K3 base change.
F7+ and F4+: generating equal tf values
F7+ turns known points on a fixed curve into parameter pairs, while F4+ produces the square class together with two seed points; the group law extends them to a lattice of solutions.
F9+: quadratic substitutions and the cancellation theorem
A complete classification of homogeneous quadratic substitutions with exact polynomial cancellation, 204 nondegenerate projective branches, and the residual quartic as an exact criterion for equal tf values.
The elliptic layers of F9+
The basic curve y²=x³−2x, genuine one-parameter families, recorded quadratic layers, and their embedding into F4+ followed by the passage to F7+.
The main line next returns to the individual 6/9 patterns: the F-series has now supplied the required language of elliptic curves and surfaces.
Elliptic surfaces for 6/9 patterns
Detailed derivations of chosen elliptic charts for 6/9 patterns: from three quadratic conditions through genus-one quartics or the shared equal-difference surface to K3 geometry and explicit solution families.
ABCDEH: two progressions and a K3 surface
A simultaneous parametrization of BEH and CDH, the residual quartic, its split Jacobian with configuration 2I₄+8I₂, and the proved rank bound 2≤rank≤4.
ABCDEJ: two progressions with a common endpoint
The gluing of AEJ and BDJ, the residual quartic, a 2I₄+8I₂ K3 surface, a non-torsion section, and an explicit polynomial family from 2P.
ABCDFH: two progressions and a palindromic quartic
The AFH and CDH progressions glued by BDFH, a split 2I₄+8I₂ K3 surface, an exact non-torsion certificate, and an explicit polynomial family.
ABCDHJ: two progressions sharing the entry D
The BDJ and CDH progressions glued by ABHJ, a palindromic quartic, a split 2I₄+8I₂ K3 surface, and a non-torsion section derived from a tangent parabola.
ABCEGH and ABCEGJ: one surface, two readings
Two patterns with shared center E lead to the same quartic and the same split 2I₄+8I₂ K3 surface; two explicit sections are independent, and one polynomial family yields solutions to both patterns.
ABDEFH: three progressions of squares
Two progressions sharing the center E and a third relation between their endpoints lead to a correctly twisted quartic, a split 4I₄+4I₂ K3 surface, and an explicit non-torsion family.
ABDEFJ: a chain of three progressions
The DEF and AEJ progressions sharing the center E, linked by the BJD progression, lead to a palindromic quartic, a split 4I₄+4I₂ K3 surface, and an explicit non-torsion family.
ABEFGH: a triangle of pairwise means
Three pairwise progressions reduce to r(x)r(z)=r(y), a pullback of the Legendre family, a 4I₄+4I₂ K3 surface, and an explicit polynomial family.
ABCDEF: an even quartic and a 12I₂ surface
The DEF progression and two norms on the CDE block reduce to an even quartic, a split 12I₂ K3 surface, a non-torsion section, and an explicit polynomial family.
ABEFGJ and ABDFHJ: tfmn as a shared Kummer K3 surface
Parametrizing the two parallel progressions turns the yellow quadric exactly into equality of tf; the two patterns are linearly equivalent and define Km(E×E) for E:y²=x³−x.
ABCDEG: two Gaussian gluings and a 12I₂ surface
The central CEG progression and two yellow quadrics give an even genus-one quartic, a split K3 Jacobian, a non-torsion section, and an explicit degree-18 family.
ABCDFG: red, yellow, and blue quadrics
The BFG progression, a Gaussian norm, and an x²+2y² norm again give an even quartic, a split 12I₂ K3 Jacobian, a non-torsion section, and a degree-18 family.
ABCDGJ: one progression and two Gaussian norms
The BDJ progression and the two yellow norms ACGJ/BCDG reduce to an even quartic, a split 12I₂ K3 Jacobian, a non-torsion section, and an explicit degree-18 family.
ABCGHJ: a K3 surface without a red progression
Two Gaussian norms and one blue norm define a smooth intersection of three quadrics, a split 4I₄+4I₂ fibration, a non-torsion section, and an explicit degree-14 family.
The general geometry of 6/9: an expanded atlas
All 16 types are ordered by condition complexity and distributed among six combinatorial groups and four model classes; the article explains the geometric-rank ladder, native sections, and the precise meaning of F4+ and F9+ analogues.
The series covers all 16 positional 6/9 types and concludes with a general classification of their K3 geometry.
Matrix multiplication algebra
A separate branch on products of magic matrices and the five-dimensional algebra of semimagic squares.
Magic, charming, and semimagic squares
The standard associated and balanced components, four product laws, the complete five-dimensional form, and exact decompositions.
Block structure and split quaternions
The isomorphism SM₃(K)≅K⊕M₂(K), two central idempotents, an explicit basis 1,i,j,k, and the exact distinction from Hamilton's quaternions.
Next: the adjugate, the spectrum, and the exact structure of the integral lattice.