The Fine-Structure Constant and the Matter Fraction of the Universe from the General Mittermeier Flow Equation: Vacuum-to-Cosmology Closure, Holographic Matter Projection, Backreaction Geometry, a Geometric Unification Architecture and a Higher-Dimensional Lift Target

The Mittermeier Attractor Theory (MAT) proposes that the observed universe is the late-time readout of a single self-organizing attractor equation rather than a patchwork of separately fitted constants and cosmological parameters. The theory starts from a closed ultraviolet datum for the dimensionless vacuum product, κM = e^ (-2) / ρ, and from this datum selects an ideal metrological branch, αM = κM / 14, so that the fine-structure sector is not inserted phenomenologically but generated from the attractor skeleton itself. The same architecture is then rewritten through a coherent π-dual manifold layer, where the chart slope, curvature bridge, and infrared plateau lock exactly into one another. In this layer MAT yields an exact late-time matter-vacuum partition, Ωₘ, π = 1/π, Ω_Λ, π = 1 - 1/π, turning the matter sector from an external fit parameter into a direct geometric output. The dynamical heart of the framework is the transcendental General Mittermeier Flow Equation, an exact two-threshold beta law rather than a downstream summary of fitted outputs, β_λ, π (u) = 3π/10- 2 e^ (-1) / √π / 1 + ( (28e / √π) αM) u+ e^ (-2) / (98 αM) / 1 + (28e αM) u. Its integrated form fixes the vacuum chart, and inversion at the observed vacuum benchmark produces a single chart backreaction residue, δαchart, π = 0. 015190721131109, which is then transported into both metrology and cosmology. On the cosmology side the same residue supports two exact readout depths: a chart-side Planck-facing lane and a fully FRW-completed DESI-facing lane. Including the universal complete-closure invariant εcl = αM² / (4π) = 4. 237495891 ppm, MAT yields the refined late-time matter fractions Ωₘ, Planckʳefined = 0. 315140915118918, Ωₘ, DESIʳefined = 0. 298051353846421, placing the theory directly in the Planck 2018 and DESI DR2 observational corridors without introducing a free late-time fitting sector. On the metrology side the theory is equally structured. MAT does not pretend that the ideal branch αM and the realized laboratory value α₀ are identical by decree. Instead, it derives a bridge-healing chain and distinguishes between the universal architectural invariant εcl and the exact realized bridge closure εₘet: = κM α₀ (xₛtar - xₕeal) = 4. 23810288164383 × 10^ (-6), with only a minute residual sliver δ₃: = εₘet - εcl = 6. 06990271 × 10^ (-10). This distinction matters because it shows that MAT is not built like a numerology exercise. It first derives the ideal skeleton, then the realized readout, and finally the small but explicit difference between them. The broader scientific claim is therefore not merely that MAT lands near known numbers. It is that one closed attractor architecture appears to generate, in a single auditable chain, an ultraviolet vacuum normalization, a fine-structure branch, a curvature-transfer law, an exact matter key, observation-facing Planck and DESI readouts, a dynamic FRW completion, and a refined complete closure. In this sense MAT reverses the usual explanatory order of fundamental physics: instead of treating constants and cosmological fractions as inputs and asking for interpretation later, it treats them as structured outputs of one vacuum-to-cosmology flow. If that chain survives expert scrutiny, the theory would not be an incremental extension of existing phenomenology but a candidate reorganization of how metrology, geometry, and cosmology are related.