Reflection Positivity, Loop Space, and OS-Positivity Obstructions to a Class of Yang-Mills Mass-Gap Constructions

We study whether a reflection-positive renormalization-group construction of Yang-Mills theory can proceed through a specific, narrow mechanism -- projection-sandwich fluctuation pieces C - Pi C Pi with Pi an orthogonal projection (distinct from, and not a negative result about, the Brydges finite-range decomposition C = sum_j Gamma_j with each Gamma_j positive and finite-range) -- and we map where the difficulty actually lies. We do NOT prove, and do NOT approach, the Yang-Mills mass gap; that statement is deliberately outside the certified profile and is never self-declared. For a reflection Theta with positive-time subspace H_+, an OS-positive covariance C, and a reflection-symmetric coarse-graining Pi preserving H_+ ("Case A"), we prove (Theorem 3.2) that the projection-sandwich piece is OS-positive if and only if Pi commutes with the OS Gram operator W. Corollaries: (a) fine-translation-invariant local blockings are trivial (Lemma 3.4, a trigonometric-polynomial rigidity -- the lattice analogue of the non-existence of an exact FIR ideal band-pass filter), so such a reflection-positive RG cannot even be started; (b) a concrete block-harmonic block-spin is a certified finite-volume OS-positivity failure (exact rational -5194733/685440000 < 0); (c) the general infinite-volume block-spin case is left OPEN, obstructed by the spectral multiplicity (degeneracy) of the free-field dispersion for spatial dimension >= 2. A general null-escape obstruction and the structure of the OS-null cone are also proved (Propositions 4.1, 4.2). We then translate the resulting locality-versus-positivity tension into the loop-space / master-loop-equation formulation (heat-kernel master field, Lemoine), with the loop machinery grounded from source: its convergence requirement is the same Kotecky-Preiss "decay beats entropy" threshold rho = A exp(-T/4) < 1. The loop route genuinely bypasses gauge fixing (the Gribov problem) and makes positivity an exact central-charge selection rule, yet inherits the identical continuum obstruction. Finite quantitative probes (discovery tier) support a "pick-two-of-three" incompatibility between reducing loop entropy, preserving the exact selection rule, and Kotecky-Preiss convergence to the continuum; and they show that the strongest measure-deformation tool, the log-Sobolev inequality, is gap-equivalent (its constant equals the mass gap for the free field, by Gross), routing through an unsolved large-field regulator, which we restate in functional-analytic language as a uniform Bakry-Emery CD(rho>0, infinity) curvature / strong-convexity / support-of-invariant-measure condition. The observation that all examined routes reduce to one such a-priori estimate ("conservation of difficulty") is a conjecture-grade synthesis over the proved theorems and the grounded literature, NOT a theorem. This is an honest reconnaissance producing proved structural results plus a tiered map, not a construction.