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, and a Candidate Unification Architecture

We present the Mittermeier Attractor programme as a unified vacuum-to-cosmology architecture in which the fine-structure constant α, the observed vacuum scale, and the late-time matter fraction Ωm arise as correlated projections of a single geometric flow. Central to the framework is the General Mittermeier Flow Equation, a transcendental attractor normal form that starts from a closed ultraviolet boundary anchor fixed by the plastic constant (ρ³ = ρ + 1) and the 14-lock principle, so that the metrological branch is not inserted phenomenologically but selected from an internal vacuum skeleton. The decisive structural step is the coherent π-dual manifold recoding. Under the single deformation factor λπ = 25/(8π), the chart gate, curvature bridge, RG slope, and infrared plateau reorganize into an exact locked manifold layer. This layer generates a distinguished holographic matter–vacuum partition, with the ideal late-time matter and vacuum keys anchored at 1/π and 1 − 1/π, respectively. The recoding also preserves nontrivial integer locks of the construction, including the canonical lock αRG,π · β∞,π = 2 and the one-eighth bridge ck,π = αRG,π/8, which tie the manifold slope to the curvature-transfer scale. On the cosmology side, the same recoding fixes an explicit retained-manifold FRW embedding law: the RG scale variable is advanced by a baseline channel blended with a k ~ H manifold channel through an equality-gated weight whose amplitude is fixed at Aπ = 1/(2π). This yields a closed-form vacuum-to-FRW closure rule that introduces no new phenomenological couplings and, by construction, localizes the correction to the matter–vacuum transition corridor. A further central result is the quantitative coupling of the π-partition to the fine-structure backreaction. The chart backreaction extracted from the healed α bridge is δαchart,π = 0.015190721131 and induces an exact Mittermeier Ω-healing law, yielding the healed late-time densities Ωm,heal = 0.315145152615 and ΩΛ,heal = 0.684854847385. These values are provided with 12-decimal-place precision and are in rigorous consonance with the Planck 2018 matter fraction (Ωm ≈ 0.315 ± 0.007). Within the present framework, the observed departure from the ideal π-partition is therefore reinterpreted as a late-time geometric healing effect tied to the chart backreaction, rather than as an anthropic accident. We also resolve the apparent gate tension inside the FRW embedding. The exact manifold gate remains the primary 1/(2π) activation amplitude, while the successful cosmological completion is governed by a derived quarter-gate that is shown to be the plateau-renormalized image of the same manifold algebra rather than an external fit parameter. With this exact derived gate lock and a tightly constrained projected residue near 1/50, the low-redshift BAO-plus-growth demonstrator improves from 88.33 to 3.44 while preserving the FRW construction equations at machine precision. The cosmological coincidence problem is thereby recast as a controlled relaxation toward geometric resonance. The resulting picture is not yet a completed theory of everything or a full proof of quantum gravity. It is, however, a mathematically organized and numerically nontrivial unification architecture: a single constrained vacuum programme simultaneously structures metrology, curvature transfer, and late-time cosmology, and it delineates concrete work packages in which the same projection logic could be extended toward gauge-sector embedding, high-curvature closure, and a broader unification outlook.