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

Absolute Frame Theory (AFT) realizes its substratum A as a flat Euclidean space, whereby the Osterwalder--Schrader (OS) axioms and reflection positivity (RP) become 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---a free (Gaussian) tension term plus the bounded pointwise routing perturbation Re (X), with L^ and 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: lower-boundedness of the Euclidean action, RP under the reflection-symmetric pointwise routing weight, the finite channel capacity N₂ₑ₈ₓ 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, and 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, whose 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 (⁴) scalar self-interaction in four dimensions while leaving confinement and clustering open.