A Ramanujan Pell Elliptic K3 Surface and Primitive Five-Cube Near Misses

We construct a trace-16 Ramanujan--Pell family of primitive positive five-cube near misses $$a_n^3+b_n^3+c_n^3+d_n^3+e_n^3=t_n^3+(-1)^{n+1},$$ obtained from a six-cube identity of quadratic forms and the negative Pell orbit generated by $8+\sqrt{65}$. The six coefficient sequences have rational recurrence generating functions with common reciprocal denominator $$R(q)=1-257q-257q^2+q^3=(1+q)(1-258q+q^2).$$ The quadratic identity is derived from a conic source, explaining the constants, while the Pell mechanism accounts for both the alternating error term and the denominator. The same construction yields the Mordell curve $$E_K:\ y^2=x^3-432K(u)^2,\qquad K(u)=(1-13u+26u^2)^3+(6+182u^2)^3.$$ We prove that its minimal smooth projective model is an elliptic K3 surface with geometric fibre configuration $6IV$; over $\mathbb Q$, the singular-fibre divisor is supported at one degree-two and one degree-four closed point. The section induced by the displayed decomposition of $K$ has canonical height $4/3$. The torsion groups over $\mathbb Q(u)$, $\mathbb Q(\sqrt{-3})(u)$, and $\overline{\mathbb Q}(u)$ are respectively $0$, $\mathbb Z/3\mathbb Z$, and $\mathbb Z/3\mathbb Z$. Over $\mathbb Q(\sqrt{-3})(u)$, the complex multiplication orbit of the section gives an Eisenstein Mordell--Weil sublattice and a visible generated rank-16 sublattice of the geometric Neron--Severi group of discriminant $-108$; no fullness or primitivity is claimed. We identify the remaining free Mordell--Weil problem with an explicit equivariant Hom module, exhibit an anti-symplectic reciprocal involution, and show that $w^3=K(u)$ has a Fermat-cubic quotient. No modularity assertion is made for the recurrence functions.