Physical and Mathematical Constants as Projectives of a Common Foundation: A General Solution from M3(C) Structure

The standard treatment of fundamental constants rests on two separate domains: physical constants such as α⁻¹ = 137. 036 are measured experimentally and classified by CODATA, while mathematical constants such as π and e are defined through analytic or geometric necessity. No structural criterion distinguishes derivable constants from contingent ones, and the total number of fundamental constants has no theoretical bound. This separation creates a deep asymmetry. Physicists treat α⁻¹ as a contingent empirical fact with no known reason for its specific value; mathematicians treat π as logically necessary. Yet both are dimensionless, scale-invariant, and protocol-independent. The question of why they belong to different ontological categories has no answer within the standard paradigm. The present paper resolves this within Operatiology by introducing Operational Invariants (操作遍数, Sousa Hensuu), dimensionless constants derivable from the common foundation 𝒞⁽³⁾_Πd realised as M₃ (ℂ). An ontological trichotomy partitions all quantities into Class I (Operational Invariants), Class II (dimensional constants encoding unit system choice), and Class III (running coupling constants). A Class I Completeness Theorem is proved without circularity. The Euler-Mascheroni constant γE is established as M₃ (ℂ) -grounded, enabling identification of the BCS gap ratio κBCS = 2π/e^γE = 3. 52775. . . as a new Operational Invariant. The distinction between physical and mathematical constants is dissolved: π, e, √2, √3, γE, and α⁻¹ are ontologically equivalent Class I Operational Invariants, all projectives of the common foundation 𝒞⁽³⁾_Πd.