Paper 24: Scale-Space Signatures in FLRW Consistency Diagnostics: A Third Class of Cosmological Model

The Scale Space framework treats logarithmic scale as a genuine fourth spatial coordinate, producing a five-dimensional metric with AdS4 geometry in which matter geodesics include a scale-direction component. A photon propagating through this geometry acquires a scale-direction momentum (scale drift) generated by the Christoffel coupling Γ^s xx, even when emitted with zero scale velocity. This modifies the angular diameter distance dA(z) and therefore all FLRW-consistency diagnostics. We derive the null geodesic equations, obtain the closed-form analytic solution for the scale drift, compute the observed redshift from the matter congruence, and prove an exact algebraic theorem: the ratio d^SS A /d^FLRW A (zobs) = sqrt[1 +β^2 s], where β^2 s = α^2H^2L^2 is the dimensionless scale kinetic energy of co-moving matter. This ratio is proved invariant under any constant-F correction to gtt. The resulting FLRW consistency corrections are ∆O > 0 (apparent spatial openness), ∆C < 0 (CBL test violation), and ∆A > 0 (KSS eff < 0, open-curvature-like), all organising naturally as polynomials in u = ln(1 + z) rather than power laws in (1+ z). This log-scale functional form distinguishes Scale Space as a third class of cosmological model in the Heinesen–Clifton taxonomy, alongside but distinct from Dyer–Roeder optical effects and Buchert backreaction. With the local gravity parameter identification β2 s = 8, the corrections are observationally too large; quantitative viability requires β2 s ≪ 1, which depends on the cosmological value of α not yet determined by the current framework. The qualitative signatures and the log-scale functional form are robust predictions
of the toy model.