Theta–Minima Coupling Hypothesis: Lattice-Theta Invariant from Experimental Solid-Electrolyte Crystallography

This manuscript proposes a new-looking multiplicative lower-bound principle linking the Gaussian theta series of a Euclidean lattice to its successive minima, and introduces a computable coupling invariant branded as the Sakib Index SI. The central object is the Theta–Minima Coupling Hypothesis, which asserts that for any lattice Λ ⊂ Rn and any t > 0, the theta mass ΘΛ(t) is bounded below by a product of one-dimensional theta factors evaluated at the successive minima. While a full proof is open, we support the hypothesis using real-world, open experimental crystallography data: the OBELiX dataset of ∼600 synthesized lithium solid-state electrolyte materials with experimentally measured ionic conductivity and unit-cell parameters [1, 2]. From 562 entries with complete unit-cell parameters and con- ductivity, we compute SI after volume-normalization to determinant 1 (a purely mathematical transformation of experimental lattice parameters), and find: (i) SI is numerically near 0 for orthogonal (right-angled) cells and positive for strongly non-orthogonal cells (notably γ ≈ 120◦ NASICON-type entries), (ii) SI correlates strongly with an orthogonality-defect proxy (ρ ≈ 0.88), and (iii) within the NASI- CON family (n = 133), SI shows a moderate positive correlation with log10 ionic conductivity (ρ ≈ 0.41). We present ten data-based figures and six conceptual diagrams, and outline testable physical interpretations of SI as a geometry-coupling descriptor for transport and stability.