Sphere Volume Identities and the Algebraic Proximity of α⁻¹ to I = 4π³ + π² + π

We record a family of exact algebraic identities expressing the invariant I ≡ 4π³ + π² + π ≈ 137.036 as a linear combination of unit-sphere volumes Vol(S¹), Vol(S³), and of the unit 6-ball volume Vol(B⁶), with small-integer rational coefficients. The three equivalent forms—polynomial, Horner factorisation, and sphere-and-ball-volume decomposition—are all algebraic identities in π, not numerical approximations. We compare I with the CODATA 2022 recommended value of the inverse fine-structure constant α⁻¹ = 137.035999084(21), observing a relative deviation of +2.22 × 10⁻⁶ (with I exceeding α⁻¹_exp). All numerical comparisons are performed at 50-digit precision using mpmath. The purely algebraic factorisation I = π²H + π with H ≡ 4π + 1 establishes H as a geometrically anchored object—the formal sum of the 0-cell count and the area of the unit 2-sphere in a minimal CW decomposition—without importing dynamical claims. We perform a Monte-Carlo look-elsewhere estimate within a restricted family of integer-coefficient π-polynomials of degree ≤ 3, finding that the proximity of I to α⁻¹ is not generic at the 10⁻⁵ level. We make no claim that I derives α⁻¹; the identification is classified as a conjectural numerical proximity. This preprint is self-contained and does not depend on any specific model of physics beyond the Standard Model.