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Elliptic surfaces for 6/9 patterns · 5.10

The ABEFGJ and ABDFHJ Patterns: tfmn as a Shared Kummer K3 Surface

The two parallel rectangular types are not exceptions to the general theory of three quadrics. Parametrizing the two red conics turns the yellow quadric exactly into equality of their differences, hence equality of tf. The two patterns are linear readings of the same Kummer K3 surface.

1. Two exact systems

Let lowercase letters denote rational square roots of the selected entries. The preferred basis of quadrics for ABEFGJ is

ABEFGJ:{a2+j2=2e2,b2+f2=2g2,a2+b2=f2+j2. \mathrm{ABEFGJ}:\qquad \begin{cases} a^2+j^2=2e^2,\\ b^2+f^2=2g^2,\\ a^2+b^2=f^2+j^2. \end{cases}

The first two equations are the AEJ and BGF progressions. The third is the yellow Gaussian relation ABFJ. For ABDFHJ the corresponding system is

ABDFHJ:{f2+h2=2a2,b2+d2=2j2,b2+h2=d2+f2. \mathrm{ABDFHJ}:\qquad \begin{cases} f^2+h^2=2a^2,\\ b^2+d^2=2j^2,\\ b^2+h^2=d^2+f^2. \end{cases}

Here the red progressions are HAF and DJB, and the yellow relation is supported on BDFH. By the general pattern theory, each three-equation system is necessary and sufficient for recovering the coordinates E,x,y of its magic square.

2. The universal red conic

For rational m,n put

R(m,n)=−m2+2mn+n2,S(m,n)=m2+n2,W(m,n)=m2+2mn−n2,f(m,n)=mn(m−n)(m+n). \begin{aligned} R(m,n)&=-m^2+2mn+n^2,\\ S(m,n)&=m^2+n^2,\\ W(m,n)&=m^2+2mn-n^2,\\ f(m,n)&=mn(m-n)(m+n). \end{aligned}

The two identities

S2−R2=4f(m,n),W2−S2=4f(m,n) S^2-R^2=4f(m,n),\qquad W^2-S^2=4f(m,n)

show that R²,S²,W² form a progression of squares with oriented difference 4f(m,n). Multiplying the roots by λ multiplies the entries by λ² and changes the difference to 4λ²f(m,n). Away from constant progressions this parametrization of the conic is complete, up to the usual projective symmetries of the pair (m,n).

3. ABEFGJ: the yellow quadric is equality of differences

Take two independent parametrized progressions and mark their forms by subscripts 1 and 2:

(j,e,a)=λ(R1,S1,W1),(b,g,f)=μ(R2,S2,W2). (j,e,a)=\lambda(R_1,S_1,W_1),\qquad (b,g,f)=\mu(R_2,S_2,W_2).

Both red quadrics now hold identically. The residual yellow quadric factors without introducing a new curve:

a2+b2−f2−j2=(a2−j2)−(f2−b2)=8(λ2f(m,n)−μ2f(p,q)). \begin{aligned} a^2+b^2-f^2-j^2 &=(a^2-j^2)-(f^2-b^2)\\ &=8\bigl(\lambda^2f(m,n)-\mu^2f(p,q)\bigr). \end{aligned}

Thus ABFJ adds no independent geometry after the red conics have been parametrized: it requires the JEA and BGF progressions to have the same oriented difference.

4. ABDFHJ: the same equation in another placement

For the second pattern, place the same two progressions differently:

(h,a,f)=λ(R1,S1,W1),(d,j,b)=μ(R2,S2,W2). (h,a,f)=\lambda(R_1,S_1,W_1),\qquad (d,j,b)=\mu(R_2,S_2,W_2).

Then

b2+h2−d2−f2=(b2−d2)−(f2−h2)=−8(λ2f(m,n)−μ2f(p,q)). \begin{aligned} b^2+h^2-d^2-f^2 &=(b^2-d^2)-(f^2-h^2)\\ &=-8\bigl(\lambda^2f(m,n)-\mu^2f(p,q)\bigr). \end{aligned}

Only the sign changed because of the chosen ordering. The geometric condition is unchanged: the two parallel progressions must have the same difference.

5. The exact tfmn theorem for the two patterns

Necessity and sufficiency

Suppose both parametrized progressions are nondegenerate. The following conditions are equivalent:

there exist λ,μ∈Q×:λ2f(m,n)=μ2f(p,q);f(m,n)f(p,q)∈(Q×)2;tf⁡(m,n)=tf⁡(p,q);the pairs give a rational ABEFGJ solution;the pairs give a rational ABDFHJ solution. \begin{aligned} &\text{there exist }\lambda,\mu\in\mathbb Q^\times: \quad \lambda^2f(m,n)=\mu^2f(p,q);\\ &\frac{f(m,n)}{f(p,q)}\in(\mathbb Q^\times)^2;\\ &\operatorname{tf}(m,n)=\operatorname{tf}(p,q);\\ &\text{the pairs give a rational ABEFGJ solution;}\\ &\text{the pairs give a rational ABDFHJ solution.} \end{aligned}

The last two implications are constructive. For ABEFGJ put

E=λ2S12,x=4λ2f1,y=μ2S22−E. E=\lambda^2S_1^2,\qquad x=4\lambda^2f_1,\qquad y=\mu^2S_2^2-E.

Then A,E,J become the first progression and B,G,F the second. For ABDFHJ first put

A=λ2S12,J=μ2S22,d0=4λ2f1=4μ2f2, \mathsf A=\lambda^2S_1^2,\qquad \mathsf J=\mu^2S_2^2,\qquad d_0=4\lambda^2f_1=4\mu^2f_2,

and then

E=A+J2,x=A−J2,y=d0. E=\frac{\mathsf A+\mathsf J}{2},\qquad x=\frac{\mathsf A-\mathsf J}{2},\qquad y=d_0.

These formulas recover all six selected square entries. Hence no additional placement equations remain for ABEFGJ or ABDFHJ after equality of tf has been imposed.

6. Two linear readings

The relation between the patterns is stronger than sharing an equation. Let (E,x,y) define an ABEFGJ square. Apply the linear substitution

(E,x,y)⟼(E+y2, −y2, x)=(E′,x′,y′). (E,x,y)\longmapsto \left(E+\frac y2,\,-\frac y2,\,x\right) =(E',x',y').

In the new square, the ABDFHJ entries receive exactly the old selected values:

A′=E,B′=F,D′=B,F′=A,H′=J,J′=G. A'=E,\qquad B'=F,\qquad D'=B,\qquad F'=A,\qquad H'=J,\qquad J'=G.

The inverse substitution is E=E′+x′, x=y′, y=−2x′. Thus the projectivized solution spaces of the two patterns are rationally and linearly isomorphic. This is not a D₄ board symmetry: it changes the positional type while preserving the Diophantine surface.

7. The shared Kummer K3 surface

Remove the common scale of each pair and on the affine chart put r=m/n, s=p/q, and ρ=μ/λ. Equality of differences becomes

S:r(r2−1)=ρ2s(s2−1). \mathcal S:\qquad r(r^2-1)=\rho^2s(s^2-1).

Consider the elliptic curve

C:U2=r(r2−1)=r3−r. \mathcal C:\qquad U^2=r(r^2-1)=r^3-r.

On the product of two copies, U²=r(r²−1) and V²=s(s²−1). The ratio ρ=U/V is invariant under the simultaneous sign change (U,V)↦(−U,−V) and satisfies the equation of 𝒮. Conversely, that equation gives the same function field. Hence

S∼(C×C)/{±1}. \mathcal S\sim (\mathcal C\times\mathcal C)/\{\pm1\}.

The double cover of ℙ¹×ℙ¹ is branched over four vertical and four horizontal lines corresponding to r,s∈{0,1,−1,∞}. Their sixteen intersections give ordinary double points. The minimal resolution is the Kummer surface Km(𝒞×𝒞), hence a K3 surface.

8. Where F4+, F7+, and F9+ fit

The surface equation is the algebraic form of tfmn. The function tf chooses the canonical squarefree representative and turns the existence of rational ρ into a discrete equality of square classes.

LanguageRole on the common surface
F4+A general affine chart: both projective pairs are normalized to a common first parameter.
F7+Each pair becomes a point on a congruent-number curve E_T; a tfmn solution is a pair of points on one fiber.
F9+Explicit curves and one-parameter layers inside the common surface.

Thus the general pattern theory, tfmn, and the F-series are not competing methods; they are successive coordinate layers of one object.

9. Exact scope of the result

Constant progressions f(m,n)=0, namely m=0, n=0, or m=±n, are excluded. Swapping or changing signs of parameters may reverse the orientation; this is handled by the ordering of the entries and the sign of tf.

The theorem covers every nondegenerate rational solution of the three quadrics for both patterns, not merely the known formula families. Positivity, pairwise distinctness, and the absence of extra square entries in the complement are separate conditions on a specialization and are not part of the surface definition.

Conclusion

ABEFGJ and ABDFHJ are distinct positional types but the same Diophantine surface. The tfmn relation arises naturally from the general three-quadric system and gives its complete rational description after the two red conics are parametrized.

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