The Riemann Hypothesis as a Derived Theorem in Pure Time Theory

We demonstrate that the Riemann Hypothesis (RH) is not an isolated conjecture of pure mathematics, but a necessary theorem within a physically grounded formal system. Based on the Pure Time Theory (PTT), we construct a canonical scalar field Trelax(xμ) whose spectral decomposition reflects the complex structure of the non-trivial zeros of the Riemann zeta function. We show that enforcing internal consistency —specifically, the injectivity of the associated phase map — imposes a strict constraint on the real parts of the spectral modes: βk = 1/2 for all k. This constraint is equivalent to the classical RH, but derived without assuming it. The proof proceeds axiomatically within a formal system T defined entirely in first-order logic and logically independent of RH. All five axioms (T1–T5) are shown to be RH-free, and the truth of RH emerges solely from the requirement of phase coherence under time evolution. The spectral sum f (x) constructed from all known non-trivial zeros ρk of ζ(s) reveals that any deviation from the critical line leads to violations of injectivity, thereby collapsing the system’s coherence. We further demonstrate a formal bijection between the spectral indices of f (x) and the zeros of ζ(s), establishing a logical equivalence between internal consistency in T and the classical truth of RH. This mechanism — the “consistency transfer” — anchors the truth of RH in a physical ontology: the scalar architecture of time. Thus, the Riemann Hypothesis cannot be fully resolved within traditional mathematics alone. It requires a foundational reconstruction where time, not abstract geometry, dictates the admissibility of number-theoretic structures. Our work establishes RH as a physical theorem — irreducible to pure conjecture — and PTT as its necessary framework.