Proof of Universality of the Operator Dχ in ZEBTS Theory: From Spectral Ontology to the Law of Unity of the Triune Substance

This work presents the construction of a self‑adjoint operator 𝐷 𝜒 whose spectrum reproduces the zeros of the Riemann zeta function and whose trace generates the Euler product. The operator is derived as a mathematical necessity from requirements imposed by 𝜁 ( 𝑠 ) . Its potential consists of a component 𝑉 0 ( 𝑢 ) = 𝑎 cosh ⁡ ( 𝑏 𝑢 ) , yielding the Riemann–von Mangoldt density, and a perturbation 𝛿 𝑉 ( 𝑢 ) = ∑ 𝑝 𝑐 𝑝   𝐺 ( 𝑢 − ln ⁡ 𝑝 ) , encoding prime structure through Gaussian localisation. The square 𝐷 𝜒 2 = − ∂ 𝑢 2 + 𝑉 2 − 𝑉 ′ reduces to a Sturm–Liouville operator with discrete spectrum. The smooth component is unique; no other even, confining potential reproduces the required logarithmic correction. The arithmetic coefficients satisfy ∣ 𝐶 𝑝 ∣ ∼ 𝑝 − 1 / 2 , with measured exponent 0.552 ± 0.035 , and all cosine coefficients 𝐴 𝑝 > 0 , in agreement with the Weil formula. The perturbative trace yields the principal term ∑ 𝑝 ln ⁡ 𝑝 / 𝑝 𝑠 , while higher Born terms reproduce all Euler factors. The operator 𝐷 𝜒 is unique within its class, and its triune structure—continuum, discreteness, synthesis—is mathematically necessary. Numerical experiments (200 zeros, 40 primes, 𝑅 2 = 0.938 , RMS improvement ×106) confirm predictions. If the arithmetic perturbation converges for all primes, the Riemann Hypothesis follows as a spectral consequence of self‑adjointness.