Primodial Duality Theory-Foundation to G^RCD(QAP)

Abstract. Primordial Duality Theory (PDT) derives all physical structure from a single axiom Ω: the irreducible duality of Potential P and Actualization A. A central
result of the framework is that the cascade branching factor must equal Euler’s number, b = e. Existing derivations of this result invoke the single-valuedness condition of quantum-mechanical phase, ϕcycle = S/ℏ = 2πn, thereby importing the physical Planck constant ℏ at a stage prior to the cascade whose output is supposed to derive ℏ. We identify this as a genuine logical circularity (not merely an open theorem), diagnose its dimensional structure, and provide a fully non-circular proof. The proof rests solely on Axiom Ω, the definition of ontic depth in natural units (nats), the homogeneity of the cascade (a theorem of Ω), and a variational principle—the Actualization Efficiency Principle—that is the direct operational content of Ω. The main result, the Cascade Efficiency Theorem, states that b = e is the unique real number greater than one that maximises the function η(b) := b^1/b. We show that this condition is equivalent to the self-consistency requirement δD = 1 nat per cascade step, both reducing to the equation ln b = 1. All three proposed non-circular routes (efficiency maximisation, depth-rate self-consistency, and Quine-atom information closure) are proven equivalent. We trace the logical dependencies explicitly, give the corrected derivation chain from Ω to b = e to ℏ as a downstream calibration, and derive immediate corollaries: the Bethe lattice Te, the Kesten–McKay spectral density ρe, spectral moments m2 = e + 1 and m4 ≈ 23.933, and the non-Gaussian fixed point g∗ ≈ 0.14099.