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 \( E_0 = \cos^4\theta + \sin^4\theta + \tfrac{1}{2}\sin^2(2\theta) \) contains a geometric constraint that has not previously been stated explicitly: the combined thermal potential energy of the two kernels, \( Te_1 + Te_2 \), is bounded below by \( E_0/2 \) for all values of the internal phase \( \theta \). This lower bound is tight — it is achieved exactly at \( \theta = \tfrac{\pi}{4} \) and \( \theta = \tfrac{3\pi}{4} \), where the photon sphere simultaneously attains its maximum kinetic energy \( E_0/2 \). 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 \( \tfrac{1}{2}\hbar\omega \) of a quantum harmonic oscillator: the kernel thermal potential energy cannot be removed from the system because it is geometrically prevented from falling below \( E_0/2 \). 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 \[ Te_1(\theta) + Te_2(\theta) = \cos^4\theta + \sin^4\theta \ge \tfrac{1}{2} = \frac{E_0}{2} \quad \text{for all } \theta \in \mathbb{R}, \] with equality if and only if \( \theta = \tfrac{\pi}{4} + n\tfrac{\pi}{2} \) for integer \( n \). The proof uses the AM–GM inequality applied to \( (u, v) = (\cos^2\theta, \sin^2\theta) \) with \( u + v = 1 \), giving \( uv \le \tfrac{1}{4} \) and hence \( u^2 + v^2 \ge \tfrac{1}{2} \). No Lagrangian, no Hamiltonian operator, and no uncertainty principle is required. Key Contributions Exact geometric lower bound \( E_0/2 \) on kernel thermal potential energy, derived via the AM–GM inequality from the quartic (Stefan–Boltzmann) thermodynamic structure Complementary upper bound \( E_0/2 \) 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 \( \tfrac{1}{2}\hbar\omega \), connected through the Zitterbewegung frequency identification \( E_0 = \hbar\omega \) Explanation of why the Stefan–Boltzmann exponent \( n = 2 \) uniquely produces the factor \( \tfrac{1}{2} \) (generalised analysis: for power \( 2n \), the floor is \( 2^{1-n} \); only \( n = 2 \) gives \( \tfrac{1}{2} \)) Numerical prediction: \( E_0 \approx 20.7\,\mathrm{keV} \) (approximately \( 1/24.7 \) of the electron rest-mass energy), compared with the Dirac Zitterbewegung quantum \( 2m_e c^2 \approx 1022\,\mathrm{keV} \); the factor of \( \sim 50 \) between the two predictions is entirely accounted for by the ratio \( \tfrac{2}{\beta} \approx 49.4 \) 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 \( \tfrac{1}{2} \) 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 \( E_0 = \hbar\omega \) (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 \( E_0 \approx 20.7\,\mathrm{keV} \) and the full electron rest-mass energy \( m_e c^2 = 511\,\mathrm{keV} \) 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).