Proof of Universality of the Operator Dχ in ZEBTS Theory: From Spectral Ontology to the Law of Unity of the Triune Substance
Abstract
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.
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BibTeX
@misc{mukha2026proof,
title = {{Proof of Universality of the Operator Dχ in ZEBTS Theory: From Spectral Ontology to the Law of Unity of the Triune Substance}},
author = {Anatolii Mukha},
year = {2026},
month = sep,
doi = {10.33774/coe-2026-gwf17},
url = {https://doi.org/10.33774/coe-2026-gwf17},
note = {CC BY},
}
RIS
TY - UNPB
TI - Proof of Universality of the Operator Dχ in ZEBTS Theory: From Spectral Ontology to the Law of Unity of the Triune Substance
AU - Anatolii Mukha
PY - 2026
DA - 2026/09/20
DO - 10.33774/coe-2026-gwf17
UR - https://doi.org/10.33774/coe-2026-gwf17
AB - 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.
ER -