The Big Bang as the Unique Quantum White Fountain: A Fractal Mechanics Derivation

White fountains (white holes) — the time-reverse of black holes — have been theorized since Novikov (1964) but lack a formation mechanism. Loop Quantum Gravity (LQG) independently predicts a quantum bounce replacing the Big Bang singularity. We propose, within the Fractal Mechanics (FM) framework, that the Big Bang was a unique quantum white fountain, occurring exactly once in cosmic history and unrepeatable by any subsequent physical process. The uniqueness follows from a critical mass argument: the bounce condition requires 𝑀 ≥ 𝑀_crit(𝑡) = (4𝜋/3)𝑀_𝑃 × (𝑡/𝑡_𝑃 )³. At 𝑡 = 𝑡𝑃 , 𝑀_crit ≈ 𝑀_𝑃 — satisfied by the primordial universe. Today, 𝑀_crit(𝑡₀ ) ≈ 10¹⁷⁶ kg, exceeding the observable universe’s mass by 10¹²³ orders of magnitude. No black hole or cosmic structure can trigger a second bounce. The initial geometry, derived from catastrophe theory, is a cardioid — the unique stable singularity of a quasi-spherical wavefront. This cardioid provides a natural mechanism for inflation (geometric curvature relaxation), a lower bound on baryogenesis (𝜂 ≥ 10⁻¹²), and specific CMB predictions (𝑛𝑠 ≈ 0.96–0.97, hemispherical asymmetry aligned with the cardioid axis) testable with Simons Observatory.