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 γ_E is established as M₃(ℂ)-grounded as the unique structural invariant measuring the discrepancy between the discrete factorial structure of U(n) (Tier 2) and its continuous analytic extension (Tier 3), and as the compression residue of the relational closure of M₃(ℂ) into finite algebraic invariants. This enables identification of the BCS gap ratio κ_BCS = 2π/e^γ_E = 3.52775... as a new Operational Invariant. Version 2 extended the framework by defining the set S of Operational Invariants without restriction to the candidate space F_ext: S = {κ | M₃(ℂ)-grounded and A4-irreducible}. S properly contains F_ext; confirmed elements of S \ F_ext include e, ln 2, α⁻¹, μ, and ξ². Each element of S is assigned a structural category: (A) M₁-algebraic; (B) M₁-condition/M₃-value (πᵏ, e); (C) projection-connection invariant (γ_E); (D) M₃-operation on M₁-element (ln 2); (E) composite. The constants √2 = γ⁻¹/² and √3 = K = √2·δ satisfy Class I observational conditions but are A4-redundant and excluded from S. The finiteness of S was proved in full, covering both S ∩ F_ext and S \ F_ext. Version 3 completes the grounding structure. M₃(ℂ)-groundedness is now established via three structurally independent and jointly exhaustive conditions: (G1) volumetric grounding, governing πᵏ; (G2) normalization grounding, governing γ_E; and (G3) exponential unit grounding, newly introduced, governing the natural logarithm base e as the value of the Lie algebra exponential map at the multiplicative unit of the A2-selected field ℝ. The uniqueness of this evaluation point and the exhaustiveness of the three conditions are each proved as independent results. A new section establishes, paper by paper, the citation integrity of the companion derivations of e, μ, π, and α⁻¹ against the present version, resolving a prior citation gap with the companion paper on e and confirming the absence of impact elsewhere. The distinction between physical and mathematical constants is dissolved: π, e, √2, √3, γ_E, and α⁻¹ are ontologically equivalent Class I Operational Invariants, all projectives of the common foundation C⁽³⁾_Πd. This Version 3 supersedes Version 2 (DOI: 10.5281/zenodo.20647095) in all respects. Published on June 18, 2026 DOI: 10.5281/zenodo.20747691