The General Mittermeier Flow Equation: A Closed Two-Threshold RG Normal Form from the Planck–Boundary Constant to an α–π Vacuum–Particle Bridge

We extract a rigid, reproducible two-threshold RG skeleton from the Mittermeier RG Attractor and state it in a form that makes an explicit particle–vacuum bridge visible. The central object is the General Mittermeier Flow Equation: an algebraically closed rational beta function for β_λ (u): = -∂ᵤ ln λ̂ (u) in a monotone, dimensionless flow coordinate u ≥ 0, whose parameters depend only on αM, π, and e. Closed SSOT chain The attractor fixes the plastic constant ρ by ρ³ = ρ + 1 and the Mittermeier Planck–Boundary normalization: κM: = λ̂ (0) = e^- (2 + ln ρ) = e⁻² / ρ αM: = κM / 14 = e⁻² / (14ρ) Perfect closure in αM, π, e and the integers 28 and 98 Defining the refined thresholds by qM = 2e⁻¹ / ρ and pM = qM / √π, the flow admits the exact ρ-free rewrite: β_λ (u) = 15/16 - 2e⁻¹ / √π / 1 + (28e / √π · αM) u + e⁻² / (98αM) / 1 + (28e · αM) u Here, the integer 28 = 2 × 14 is forced by the threshold sector after substituting ρ = e⁻² / (14αM), and 98 = 7 × 14 is forced by the fixed residue ratio 1/7 in the IR pole. This "particle bridge" is a change of variables, not a fit: the paper verifies pointwise identity between the canonical refined normal form and the fine-structure constant αM form. Analytic invariants Working directly with the general αM–π–e form, we derive a compact invariant set: An exact u-plateau of 15/16. Strict monotonicity β'_λ (u) > 0 for all u ≥ 0 (no overshoot). A universal deep-IR approach law: β_λ (u) = 15/16 - C · u⁻¹ + O (u⁻²) with a closed coefficient C (αM, π, e). A unique analytic inflection point uᵢnf. Numerical values at the closed constant: β_λ (0) ≈ 0. 711638, C ≈ 0. 983986, and uᵢnf ≈ 0. 629160. Scale-chart implication (deep IR) With the SSOT chart, the plateau β_λ (u) → 15/16 implies λ̂ (u) ~ e^- (15/16) u and therefore λ̂ (k) ~ (k / MPl) ^ ( (15/16) αRG). For the QG-closed/FRW-closure choice αRG = 32/15, this exponent equals 2, i. e. , λ̂ (k) ~ (k / MPl) ² as k → 0 (infrared). Metrology interface Using CODATA-2022 α₀⁻¹ = 137. 035999177 (21), the measured coupling implies: κM / (14α₀) - 1 = -13. 245 ppm This ppm proximity is slightly larger than the CODATA uncertainty on α₀ and is reported as a deterministic numerical pattern; deriving α₀ dynamically would require an action-level completion beyond this companion.