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This work introduces and develops the family of Alpha functions on primes: where the sum traverses all primes. The family is simultaneously parameterized in the complex variable s and in the real exponent k, generating a two-dimensional theory with new analytical and arithmetic properties. The main results of this paper are: (1) the fundamental differential system and its extension to mixed derivatives (§§5–6); (2) the complete classification of singularities at s = 1 and the family of generalized Mertens constants for (§7); (3) integral representations via Mellin , Riemann – Liouville and Laplace (§§ 8 –10); (4) logarithmic convexity and the constant (rigorous bound, primes; §11); heuristic estimation ; honest interval as a canonical minimizer of the inverse family ( § 11); (5) the numerically verified quotient functional equations for the linear fitting of H(ω) in the real regime ω (see Note 12.26); the value of Result 12.37 corresponds to the fitting of the quotient Q within the family of four symmetric operations in a different experiment, the value of Result 12.37 corresponds to the fitting of the quotient Q within the family of four symmetric operations , calculated in a different regime; (6) the family of deformed Alpha constants with its minimizer (§23); (7) the vertical differential system inherited by the generalized Brun families , the exhaustive classification of 41 admissible constellations within the universe , (37 of length ≤ 4 and 4 quintuples of length 5) into two branches and , and the spectral quotient with its exact differential equation, and, as a preliminary block, the Alpha family over mirror primes with unconditional differential inheritance and convergence under the Emirp Density Hypothesis (EDH; §19.8). The continuation of this program—including the spectral equivalence conjecture of with the zeros of , the complete theory of the spectral ratio, and the extensions to function composition and analytic continuation beyond remains as the author's later work. Main result: We establish the existence of quotient functional equations for the smoothed/truncated version —the exact object of Theorem 12.10—with numerical adjustment in the regime for the coefficients of . The unsmoothed function does not admit a complete functional equation in the classical sense (§ 13). It constitutes an example of quotient-type functional equations for exclusive sums over primes; no precedent has been found in the consulted references (Apostol [1976], Finch [2003], OEIS).