Geometric Origin of the One-Half Factor: Thermal Potential Energy Floor and Zero-Point Energy in the 0-Sphere Model: A Conceptual Clarification

#46 A Conceptual Clarification The 0-Sphere Model energy identity E0 = cos⁴θ + sin⁴θ + ½sin²(2θ) contains a geometric constraint that has not previously been stated explicitly: the combined thermal potential energy of the two kernels, Te1 + Te2, is bounded below by E0/2 for all values of the internal phase θ. This lower bound is tight — it is achieved exactly at θ = π/4 and θ = 3π/4, where the photon sphere simultaneously attains its maximum kinetic energy E0/2. The bound is a mathematical identity, not a dynamical assumption. We propose that this geometric floor is the possible geometric analogue of the one-half factor that appears in the zero-point energy ½ℏω of a quantum harmonic oscillator: the kernel thermal potential energy cannot be removed from the system because it is geometrically prevented from falling below E0/2. This complements the prior dissolution of the vacuum energy problem in the 0-Sphere framework (Paper #32), which showed that the enormous QFT vacuum energy is an artifact of local field ontology. The present paper provides the positive counterpart: within the 0-Sphere ontology, the one-half factor has a concrete geometric origin in the structure of the energy identity itself. Main Result (Theorem 1 — Kernel Energy Floor) For the 0-Sphere energy identity, the combined kernel thermal potential energy satisfies Te1(θ) + Te2(θ) = cos⁴θ + sin⁴θ ≥ ½ = E0/2 for all θ ∈ ℝ, with equality if and only if θ = π/4 + nπ/2 for integer n. The proof uses the AM–GM inequality applied to (u, v) = (cos²θ, sin²θ) with u + v = 1, giving uv ≤ 1/4 and hence u² + v² ≥ 1/2. No Lagrangian, no Hamiltonian operator, and no uncertainty principle is required. Key Contributions Exact geometric lower bound E0/2 on kernel thermal potential energy, derived via the AM–GM inequality from the quartic (Stefan–Boltzmann) thermodynamic structure Complementary upper bound E0/2 on photon sphere kinetic energy — the floor and ceiling are equivalent statements of the same AM–GM constraint Proposed correspondence between the geometric floor and the quantum mechanical zero-point energy ½ℏω, connected through the Zitterbewegung frequency identification E0 = ℏω Explanation of why the Stefan–Boltzmann exponent n = 2 uniquely produces the factor 1/2 (generalised analysis: for power 2n, the floor is 21−n; only n = 2 gives 1/2) Numerical prediction: E0 ≈ 20.7 keV (approximately 1/24.7 of the electron rest-mass energy), compared with the Dirac Zitterbewegung quantum 2mec² ≈ 1022 keV; the factor of ~50 between the two predictions is entirely accounted for by the ratio 2/β ≈ 49.4 Completion of a two-part account of zero-point energy in the 0-Sphere framework: Paper #32 dissolved the enormous QFT vacuum energy sum; the present paper provides the geometric origin of the residual factor 1/2 Relation to Prior Work This paper is the positive counterpart to Paper #32 (Dissolution of the Vacuum Energy Problem). A recurring pattern in the series is documented: Paper #11 derived the geodesic path structure stated qualitatively in Paper #1; Paper #25 derived the conservation-law origin of spin stated in Paper #20; Paper #35 established the confinement mechanism introduced structurally in Paper #33. The present paper occupies this successor role with respect to Paper #32. Open Questions First-principles derivation of E0 = ℏω (currently used as an identification; the Hamiltonian framework of Paper #18 is the natural starting point) Casimir effect within the worldline/history-based ontology Relationship between E0 ≈ 20.7 keV and the full electron rest-mass energy mec² = 511 keV Generalised virial theorem for the 0-Sphere oscillator Series Context This is Paper #46 in the 0-Sphere Model series (2018–2026). The series develops a deterministic geometric framework in which the electron is modelled as two thermally coupled kernels (perfect black bodies) exchanging energy via a photon sphere. The foundational energy identity was introduced in Paper #1 (2018).