The exact surplus acceleration from the geodesic equation in the recursion metric with the exact R(D_st) solution: g_surplus = cH₀R(1 + 2R⁴)/6. Setting g_surplus = a₀ gives the unique solution R_trans = 0.748, from which a₀ is derived from c and H₀ alone — not postulated.

Source: Research Calculator V5 (10.5281/zenodo.19970405); Identifying Dgal from the MOND Transition (10.5281/zenodo.20010274); Rotation-Curve Law and RAR (10.5281/zenodo.19683339). Observed a₀ = 1.380×10⁻¹⁰ m/s². The density ratio D_trans/D_GUT = 16.79 follows from a₀, c, and H₀ with no free parameters.

The exact solution derived from the classical recursion Lagrangian: R/(1 + 2R⁴)^1/4 = C · D_st^1/6, with the single constant C = 3^−1/4 fixed by the normalisation R(D_GUT) = 1. The exponent n = 1/6 is derived, not fitted. There are no free parameters. The asymptote is R → 0 as D_st → 0 (c_eff → ∞: inflation is the D_st → 0 limit); the finite observed R₀ is set by the minimum density of the recursion medium, not by a parameter.

Source: The Exact Solution for R(D_st) from the Classical Recursion Field (10.5281/zenodo.19968963); Research Calculator V5 (10.5281/zenodo.19970405). Verification values from the paper: R = 0.0353 at 10⁻⁸, 0.1638 at 10⁻⁴ (electroweak), 0.3555 at 10⁻² (QCD), 1.0000 at D_GUT. The equation of state w = (1−2R⁴)/(1+2R⁴) reaches w = −1/3 at the GUT scale.

Forward chain (derived). From the pressure axiom: golden angle θ_φ = 360/φ², hexagonal curvature κ₆ = 6/π², spin coupling C = (S_e/S_γ)²·π²·κ₆ = 3/2, torsion density Ī = 3α/2, torsion slope g = 6κ₆Ī^3/2, slip operator Δθ = 2g·(180/π), and the self-consistency condition 1/α = θ_φ − Δθ. This is a transcendental equation for α with no free parameters and no α as input to the geometric chain. Solved live below by fixed-point iteration.

Source: The Fine-Structure Constant from the Cohesion UFT: A Complete Derivation (10.5281/zenodo.19739815); The Photon-Electron Coupling Derivation (10.5281/zenodo.19965731). The +0.007 between the geometric prediction and CODATA is identified, not fitted: viewed from inside the cascade it is the axial t²/2 kinematic term in the pair-production spectral density; viewed from outside it is the pressure signature of the containing scale at cascade level k = −1. Both accounts describe the same correction (The Fine-Structure Constant Is the Coupling Between Scales, 10.5281/zenodo.19966284).

Self-consistency check (slip-time form). The same condition read in time: 1/α = θ_φ − 2g/ω, where 2g/ω is the slip operator. The value g/ω = 0.004117 rad was determined by working backward from CODATA; its forward evaluation from the field equations is the stated open problem of the Golden Angle paper. The forward chain above is the derived result; this card verifies the identity numerically.

Sources: The Fine-Structure Constant and the Golden Angle, V5 (10.5281/zenodo.19676781); Slip Acceleration (10.5281/zenodo.21222605) — the formalisation of this card. Empirical comparison: CODATA 1/α = 137.035999.

The scaling factors are derived, not fitted: λ₁ = 3/(2α) and λ₂ = 3/(8πα). The muon and tau masses follow from the electron anchor: m_μ = λ₁m_e, m_τ = λ₂m_μ. The ratio λ₁/λ₂ = 4π is the solid-angle ratio.

Source: The Mass Spectrum in the Cohesion UFT (10.5281/zenodo.20008508); Torsion Interval Scaling Factors (10.5281/zenodo.21222258). Observed: m_μ = 105.658 MeV, m_τ = 1.77686 GeV.

The toggle threshold Φ = 32/(3π² − 4) separates the two stable free-field polarities. Below Φ the recursion is hexapolar; above it, bipolar.

Source: The Binary Recursion Toggle (10.5281/zenodo.19747782); Hexagonal Recursion and the Six-Peak Slip Structure (10.5281/zenodo.19720151).

The foundational energy identity: energy equals pressure times recursion volume.

Source: E = pr (10.5281/zenodo.19759691); Cohesion UFT: Inertia = Pressure (10.5281/zenodo.20184123).