Unified Primal Fractal Theory (UPFT) — A Minimal Discrete Algebraic Framework

Unified Primal Fractal Theory (UPFT) — A Minimal Discrete Algebraic Framework We present the Unified Primal Fractal Theory (UPFT), a minimal discrete algebraic framework based on the primordial matrix \(E_n = n^2\) for \(n = 1\) to \(5\). This matrix yields a single pivotal integer \(Z = 41\) and trace \(Tr = 42.025\), together with the torsion gear \(T = 13/12\) and Doppler flux slippage \(R = 626/624\). From these five integers alone, the theory derives — without any free parameters — the vacuum impedance \(Z_0 ≈ 376.73\,Ω\), the inverse fine-structure constant \(α⁻¹ ≈ 137.04\), the primordial frequency \(ν = 1.375\,Hz\), the Hubble constant \(H_0 ≈ 68.73\,km/s/Mpc\), the Rydberg constant, the Bohr radius, and several other fundamental constants with phenomenological precision often exceeding 99.99%. The same matrix naturally accounts for the 41 degenerate codons in the genetic code, the 13 protofilaments of microtubules, and the Delta-band resonance (1.375 Hz) associated with DNA repair and quantum coherence in Orch-OR theory. The framework also exhibits deep compatibility with the Dirac equation and echoes Ramanujan’s rapidly convergent series for \(π\) through the 14th Fibonacci number \(F₁₄ = 377\). UPFT proposes that the universe is not a continuous spacetime but a self-resonant discrete circuit whose fundamental “heartbeat” is the phase slippage \(R = 626/624\). This minimal grammar appears sufficient to unify vacuum physics, quantum biology, and consciousness within a single fractal architecture.