A Closed Vacuum-to-Cosmology Readout of the Late-Time Vacuum Scale in Mittermeier Attractor Theory: Dimensionless Resolution of the Gravitational Vacuum Hierarchy, Calibrated Closure of Lambda₀, and Transported Matter-Vacuum Predictions

The vacuum catastrophe is the problem that local quantum field theory naturally suggests a gravitational vacuum sector of order unity in Planck units, whereas the late Universe is governed by a vacuum scale of order 10^-122. This companion addresses the gravity-facing part of that hierarchy problem in a form that can be reconstructed from one synchronized source of truth. The central running quantity is λ̂ (u) = Λ (u) G (u), where Λ (u) is the running vacuum term, G (u) is the running gravitational coupling, and u is the chart coordinate that measures position along the exact vacuum flow. The problem is therefore posed as a forward calculation: start from a fixed ultraviolet boundary datum, integrate one fixed flow, and determine which late-time dimensionless vacuum scale that flow actually reads out. In synchronized Mittermeier Attractor Theory (MAT), the ultraviolet boundary condition is fixed internally by the Mittermeier Planck-boundary constant κM = e^-2 / ρ, with ρ defined by ρ³ = ρ + 1. Here ρ is the plastic constant. The same boundary seed selects the theory-internal metrological fine-structure branch αM = κM / 14. In the synchronized lane this already fixes the threshold pair through qM = 28 e αM = 2 e^-1 / ρ and pM = qM / √π, so the plastic-constant lock and the 14-lock are not appended later as interpretive decorations but propagated directly into the exact vacuum flow itself. The corresponding π-manifold chart is fixed by an exact infrared plateau β_∞, π = 3π / 10 and an exact chart slope αRG, π = 20 / (3π), with αRG, π β_∞, π = 2. The vacuum lane is then governed by one exact two-threshold flow and its exact back-integration. In that sense, the late-time vacuum value is not inserted as an independent fit parameter after the fact. It is the endpoint of a rigid ultraviolet-to-infrared transport law. The same synchronized architecture also contains a coherent π-dual manifold layer in which the gate, curvature bridge, chart slope, plateau, and late-time projection depths are algebraically linked rather than introduced one by one. This linked closure structure is essential for interpretation: the numerically sharp outputs reported here do not arise as isolated endpoint coincidences, but as repeated readouts of one locked architecture. At the observation-facing benchmark Λ₀ = 2. 9 × 10^-122, the synchronized readout yields ucosmo, π = 299. 05635044704695, δαchart, π = 0. 015190721131109086, and δ₃ = 6. 069902711019906 × 10^-10. Here δαchart, π is the realized chart backreaction residue produced by the exact two-threshold flow, and δ₃ is the remaining realized-versus-ideal metrological closure sliver at the observation-facing benchmark. The same rigid architecture then defines a calibrated closure point Λ₀* = 2. 816998637863968 × 10^-122, selected by the exact internal condition δ₃ (Λ₀*) = 0. In addition, the paper records the explicit structured near-lock approximation Λ₀, NL = 2. 816997989820800 × 10^-122, whose mantissa differs from the calibrated value by only M* − MNL = 6. 480431684607879 × 10^-7. The main analytic result is the exact threshold-balance law at the calibrated point together with its controlled large-u expansion. Truncating that expansion at O (u^-2) reproduces the exact balance with residual 5. 074873694610460 × 10^-7, showing that the same asymptotic tail that governs the ultraviolet-to-infrared vacuum flow also governs the closure-point analysis. The late-time vacuum hierarchy is therefore no longer treated here as a bare unexplained target. Within the synchronized MAT lane it becomes a reproducible boundary-to-readout calculation with explicit internal audits, explicit claim classes, and direct falsifiability. For standard late-time cosmology, the significance is immediate: if this structure continues to match sharper future determinations of the vacuum scale, the fine-structure constant, and the matter-vacuum sector, then Λ₀ is no longer merely an empirical input but the macroscopic readout of a rigid microscopic-to-cosmological attractor. If it does not, the same many-digit structure makes the proposal decisively testable.