Zero Operator Theory: Ab Initio Derivation of 41 Physical Quantities as a Proposed Resolution to the Naturalness and Hierarchy Problems

Abstract

Zero Operator Theory (ZOT v3.6): Over-Constrained Architecture and Global Interlocking of 41 Physical QuantitiesDescription: The Zero Operator Theory (ZOT v3.6) proposes a resolution to the long-standing Naturalness and Hierarchy problems in modern theoretical physics by replacing the classical assumption of a continuous space-time continuum with a discrete, causal-informational quantum vacuum termed the Emergent Compressed Vacuum (ECV). The model operates under an over-constrained framework with zero free degrees of freedom ($\text{DOF} \equiv 0$). Unlike conventional theories that depend on the empirical fine-tuning of dozens of arbitrary free parameters, ZOT establishes a strictly irreversible, unidirectional causal sequence—governed by a Directed Acyclic Graph (DAG)—derived from only **two irreducible native seeds**: the Constant of Motion ($C_{\text{ZOT}} = 518500.0$) and the Thermodynamic Exponent ($\gamma = 0.383002$). The cornerstone of the architecture is its Global Interlocking Index ($IIG = 316.2$), wherein the Universal Vacuum Dressing Formula (CILP-ECV) serves as an elastic impedance response bridge between the sub-Planckian informational mesh and observable physical phenomena. Through this mechanism, the theory achieves the ab initio derivation of 41 fundamental constants and physical quantities without post-hoc tuning or slack variables. This single causal chain unifies the SI metrological sector ($c, \hbar, G, e$), the Higgs boson mass ($125.28\text{ GeV}$), the dressed masses of the electroweak ($W/Z$) gauge bosons, the anomalous magnetic moment of the muon ($a_\mu \approx 0.00116592$), the active neutrino mass scale ($m_\nu \approx 0.0804\text{ eV}$), and the cosmological dark energy density ($rho_{\text{DE}} = 6.4 \times 10^{-11}\text{ eV}^4$). Because the system is strictly over-constrained, each of the 41 derivations acts simultaneously as a boundary condition and a consistency test for all others. This mathematical rigidity turns the model's extreme Popperian falsifiability into its primary epistemological strength: minor shifts in baseline empirical data would collapse the interlocking chain, whereas increasing metrological precision continuously confirms the model without requiring manual recalibration. This technical manuscript was translated from Brazilian Portuguese into English with the assistance of Google AI technology (Gemini), operating within a Human-in-the-Loop (HITL) framework. All mathematical derivations, LaTeX syntax, unit mappings, and canonical terms of the Zero Operator Theory (ZOT v3.6) were rigorously verified and audited to ensure full technical accuracy and alignment with international publication standards. List of the 41 Master Observables Derived *Ab Initio* by ZOT v3.6 Sector 1: Metrology and Fundamental SI Constants 1. Base Speed of Light ( c {bare}) 2. Projected Speed of Light (c) 3. Reduced Planck Constant (hbar) 4. Planck Constant (h) 5. Gravitational Constant (G) 6. Boltzmann Constant (k_B) 7. Vacuum Permeability (mu_0) 8. Vacuum Permittivity (varepsilon_0) 9. Characteristic Vacuum Impedance (Z_0) 10. Rydberg Constant (R_\infty) 11. Avogadro Constant (N_A) 12. Faraday Constant (F) 13. Stefan-Boltzmann Constant (sigma_{SB}) Sector 2: Fundamental Couplings, Angles, and Phase Parameters 14. Inverse Fine-Structure Constant (alpha^{-1}) 15. Fine-Structure Constant (alpha) 16. Strong Coupling Constant at (M_Z) Scale (alpha_s(M_Z)) 17. Fermi Coupling Constant (G_F) 18. Sine Squared of the Weinberg Angle (sin^2\theta_W) 19. Electrospherical Golden Angle (psi) Sector 3: Leptonic Spectrum, Baryons, and Flavor Physics 20. Yang-Mills Mass Gap / QCD Scale (mu_{vac}}) 21. Proton Rest Mass (M_p\) 22. Neutron Rest Mass (m_n) 23. Muon/Electron Mass Ratio (m_\mu/m_e) 24. Tau/Electron Mass Ratio (m_\tau/m_e) 25. Muon Magnetic Anomaly (a_\mu) 26. Electron Magnetic Anomaly (a_e) 27. Active Neutrino Mass (m_\nu) Sector 4: Electroweak Sector, Higgs Boson, and Heavy Quarks 28. Higgs Vacuum Expectation Value (v_{ZOT}) 29. Higgs Boson Mass (m_H) 30. \(W^\pm\) Gauge Boson Mass (m_W) 31. \(Z^0\) Gauge Boson Mass (m_Z) 32. Top Quark Mass (m_t) Sector 5: Dark Sector, Cosmology, and Boundary Invariants 33. Dark Matter Mass Density Parameter (Omega_{DM}) 34. Dark Energy Energy Density (rho_{DE}) 35. Dark Energy Density Parameter (Omega_{DE}) 36. MOND Critical Acceleration (a_0\) 37. Bare Cosmological Constant (Lambda_{bare}) 38. Geometric Ceiling of the Periodic Table (Z_{limit}) 39. Euler's Number (e) 40. Critical Geometric Threshold (c_{top}) 41. Temporal Lattice Spacing / Static Mesh Impedance (lambda_{lat}, \zeta) Note: This document is the official English translation of the original article in Brazilian Portuguese, available at DOI: https://zenodo.org/uploads/22832916 "This study is part of the overarching compendium of the Zero Operator Theory (ZOT) framework. For further details regarding canonical terminology, analytical derivations, and an in-depth exploration of the theoretical structure, readers are invited to consult the complete repository of works deposited under the author's name: Bartolome, Ricardo on Zenodo."