√2 as the Self-Consistency Constant of Fractal-Temporal Spacetime: Non-Collision Stability, Scale Isolation, and the Tsirelson Bound

Why √2 and not the golden ratio, not π, not e? This paper provides the mathematical answer. We prove that √2 is the unique scaling ratio in a base-2 system that achieves exactly zero non-trivial intermodulation collisions between fractal scale levels. The mechanism is a parity separation lemma: for r = √2, the power rᵃ is rational when a is even and irrational when a is odd. This rigid partition eliminates every dangerous intermodulation resonance (krᵃ ± ℓrᵇ = 1 has no solution unless both exponents are even), leaving only the harmless dyadic sector. We test this analytically and numerically against six alternative constants (φ, 3/2, √3, π/2, e^1/2, 4/3). At all tested thresholds, √2 achieves exactly zero non-trivial collisions while every other constant — including the golden ratio, the "most irrational" number by the Hurwitz theorem — retains dangerous collisions. The zero count is not a finite-precision artifact: it is an exact analytical result from the three-case parity argument. The paper provides three independent physical motivations for why nature would select for noncollision: energetic stability (no resonant energy transfer between scales), long-time survival (colliding configurations thermalize, only noncolliding hierarchies persist), and curvature regularity (collisions produce metric singularities in the fractal-temporal framework). The √2 scaling is then derived from quadratic energy composition of orthogonal modes: two independent binary degrees of freedom combine without interference to produce an effective amplitude scaling of exactly √2. Additional results include: the derivation of the Tsirelson bound 2√2 from commutator structure, the √2-cascade of correlation bounds across entanglement levels, the prohibition of Popescu-Rohrlich correlations via deterministic setting reconstruction, and the phase-space isotropy argument yielding α = p² = 2 (connecting the kinetic parameter of the Lagrangian to the potential exponent). This paper serves as the algebraic foundation for the fractal-temporal framework and is referenced by the companion papers on gravity, dark energy, quantum entanglement, and the spectral program for the Riemann Hypothesis