Flow-Time Theory v5.0: From GPS to Black Holes — Gravitational Modulation of the Entropy Gradient

Flow-Time Theory v5 extends the framework from the existence of the U-axis (cosmological entropy gradient direction) to the gravitational modulation of the local entropy gradient. Starting from a purely entropic scalar field action with no a priori assumption of gravity, we derive the weak-field linear approximation ∇ρ = (∇ρ)_0 (1 + α φ_g) from first principles after minimally coupling to the metric. We then present a unified function that connects weak-field and strong-field regimes without discontinuity or divergence. The entropy rate formula dS/dλ = k |∇ρ_U| cosθ · f(r) · sgn(ρ) is obtained, where f(r) smoothly interpolates between the weak-field limit (1 + α φ_g) and the strong-field limit (→0 at the horizon). This unified formulation preserves the GPS clock offset prediction (+45 μs/day gravitational part), deep-sea organism longevity, spaceflight accelerated aging, and black hole horizon freezing, all within a single mathematical framework. The sign function sgn(ρ) ensures dS/dλ ≥ 0 always. A comparison table clarifies the relationship between FTT and standard General Relativity. Experimental predictions include P1-U measurements at different altitudes and depths. The magnitude of each predicted effect is explicitly stated, with most being extremely small — detection requires high-precision measurements or long integration times.