Reflection Positivity and a Matter-Sector Spectral Gap on the Euclidean Substratum of Absolute Frame Theory

Abstract

Version of August 21st, 2026, following the development within the Absolute Frame Theory. Absolute Frame Theory (AFT) realizes its substratum 𝒜 as a flat Euclidean space. The Osterwalder–Schrader (OS) axioms and reflection positivity (RP) are therefore the native language for passing from the substratum to a Lorentzian quantum theory on the observable manifold. We isolate, as a first-stage target, the bounded-routing sector of the embedding action. That sector is a free (Gaussian) tension term plus the bounded pointwise routing perturbation Re[λΨ(X)], with λΨ ∈ L∞ and Ψ a bounded plane wave fixed by the companion analysis. We state the stage-one target as a conjecture and organize it into four proof obligations. They are the lower-boundedness of the Euclidean action, RP under the reflection-symmetric pointwise routing weight, the finite channel capacity N_crit as a Lorentz-invariant regulator, and the thermodynamic-continuum limit. The first two are proved here, jointly, as a finite-volume reflection-positivity theorem (lower-boundedness of the action and reflection positivity of the bounded-routing measure under the single Euclidean-time reflection, yielding an Osterwalder–Schrader physical Hilbert space at finite volume). The third is a plausibility claim. The fourth, together with the spectral mass gap, is placed explicitly out of scope. We are explicit about the limitation: this sector is a Gaussian theory dressed by a bounded multiplicative perturbation, not a nontrivial interacting four-dimensional quantum field theory. The existence of the latter with a mass gap is the universal Yang–Mills/Millennium debt rather than a defect specific to AFT. As a second, independent Euclidean result we prove a strictly positive spectral gap for the linearized Jacobi operator of the embedding's normal (matter-sector) fluctuations. Its finite bound-state count is the family number (a finite count; its value is background data, not derived here). We stress that this one-particle gap is not the non-perturbative Yang–Mills mass gap, which remains out of scope. The nontrivial content of AFT lives instead in the gauge topology of the embedding, which sidesteps the triviality of quartic (φ⁴) scalar self-interaction in four dimensions while leaving confinement and clustering open.