Structural Audit of Mixed-Dimensional Reductions on a Thales Constraint Manifold

This technical note performs a structural audit of mixed-dimensional reductions on a Thales right-triangle manifold. Beginning with the general Pythagorean constraint a^2 + b^2 = H^2, we analyze the projection identities, altitude relation, and geometric-mean properties as consequences of a single defining equation. A mixed-dimensional reduction R_{\text{mix}} = a + b^2 is introduced, lowering one leg from quadratic (area-level projection) to linear (geometric-mean level) participation while retaining the other at quadratic level. An embedding correction factor C_d = d/(d-1) is examined as an external dimensional uplift. A rigorous degree-of-freedom count shows that the Thales construction constitutes a two-parameter family: one equation in three unknowns leaves two independent degrees of freedom. Neither the projection structure nor the mixed reduction reduces this freedom. The choice of which leg to reduce, the requirement of dimensionless normalization, and the embedding dimension d are all external inputs rather than geometric necessities. When specific constants from independent domains are supplied—\phi from algebraic recursion, e from analysis, and d = 3 from the crystallographic restriction theorem—the construction yields a unique numerical output R = e + \phi + 1 \approx 5.336, which is empirically close to the Planck dark-to-baryonic matter ratio. The manuscript demonstrates that this result is a compatibility outcome rather than a derivation: the framework organizes relationships among independently defined constants but does not generate them. The boundary between structural necessity, optimization choice, and parameter insertion is made explicit throughout. The Thales manifold is therefore classified as a compatibility geometry: a structure capable of coherently organizing externally supplied constants with high precision, but not forcing them from first principles.