Universal Spectral Machinery: A Shared Framework for Number Theory and Quantum Field Theory via √2-Fractal Operators

What if the same mathematics that governs prime numbers also governs particle physics — not as a metaphor, but as two specializations of a single spectral operator? We construct a universal self-adjoint operator on a fractal-weighted Hilbert space with scaling parameter √2, governed by three analytic pillars: a uniform Mourre estimate (ensuring absence of singular continuous spectrum), trace-class bounds (connecting eigenvalues to geometric data), and a parametrix expansion (controlling perturbative corrections with geometric decay (√2) ⁻¹ per tower level). These pillars are proved rigorously. From this operator, a Universal Trace Formula is derived that admits two distinct specializations. Under the arithmetic specialization — where the comb potential encodes primes as Vₐrith = Σₚ Σₖ (log p / p^k/2) δ (x − k log p) — the trace formula recovers the Weil explicit formula relating prime numbers to zeta zeros. Under the physics specialization — where the comb potential encodes particle masses — the same trace formula produces the QFT trace formula relating particles to resonances. The structural parallel is captured by a precise duality dictionary: primes correspond to elementary particles, non-trivial zeros to resonances, the critical line Re (s) = 1/2 to unitarity of the S-matrix, the Möbius function μ (n) to fermion statistics (−1) F, the Euler product to the partition function, the Selberg trace formula to the Gutzwiller trace formula, and the functional equation to CPT symmetry. This dictionary is presented as a structural observation — the same mathematics governs both domains — not as a claim of physical causation. The framework does not prove the Riemann Hypothesis (which requires a spectral completeness result not yet established) nor compute exact particle masses (which requires solving the full eigenvalue problem). These are identified as the central open problems — and the paper's conclusion suggests they may be the same hard problem viewed from different angles. The Hilbert space weights (√2) ⁻ⁿ are not postulated but derived from the fractal-temporal Lagrangian, where the kinetic parameter α = 2 = (√2) ² is fixed by the Newtonian limit. This grounds the spectral machinery in a physical action principle rather than abstract operator theory.