Arithmetic Uniqueness and Modular Factors in Calabi–Yau Fourfolds: A reproducible finite-data framework and a worked literature-based calculation

Summary This research note studies when a finite set of Fourier coefficients determines a unique modular form inside a modular-form space fixed in advance. A valid Sturm bound gives uniqueness, while existence is tested separately by exact consistency of the finite coefficient map. Scope and computations The note formulates and proves a finite-data existence–uniqueness criterion and gives its exact linear-algebra test. It reproduces two published local-factor polynomial expansions for the Z6-symmetric Hulek–Verrill Calabi–Yau fourfold quotient at phi = 1, deriving b_17 = -14 and b_19 = -22 from the distinguished factors. It also expands the cited weight-three, level-fifteen eta-product through q^19 and provides Python and SageMath code for reproducibility. Limitations The geometric factors and candidate modular form are drawn from Jockers, Kotlewski, and Kuusela (2024). The calculations here are literature-based arithmetic checks; they do not provide new point counts, prove agreement at every prime, or establish a new geometric modularity theorem.