From Phase-Locking Condition to the Critical Line: The Physical Necessity of the Riemann Hypothesis

Starting from the multi-path frequency-division norm hypothesis of Primitive Space Theory, this paper proposes an alternative derivation path for the claim that the real part of all non-trivial zeros of the Riemann zeta function must be 1/2. Unlike previous arguments that relied on the finiteness of energy, this work directly maps the phase-locking condition—defined as the arithmetic mean of adjacent fundamental frequencies—into a uniqueness constraint on the complex plane. We demonstrate that when the primary world recursively divides its frequency by an arbitrary fixed ratio (q), an external frequency set to the arithmetic mean of two adjacent fundamental frequencies maintains a constant proportional relationship with the fundamental frequency components at all hierarchical levels (fractal phase-locking). Mapping this geometric condition to a power-law amplitude decay of (1/n^σ), the phase-locking requirement forces (σ) to match the logarithmic spacing of the primary network—uniquely determining (σ = 1/2). The logarithmic divergence of the energy series (∑ 1/n) at (σ = 1/2) is not a flaw in the argument but the natural mathematical signature of a critical state, which remains finite after natural truncation within any physically observable frequency band. We further clarify that existence itself is the result of sampling from a superposition of multiple frequency worlds; thus, our conclusion applies only to the specific frequency layer currently locked by the observer, not as a universal assertion about all possible frequency worlds.