This page implements the Cohesion UFT's closed-form expressions as a working calculator. Every result can be compared against measured values without adopting the framework's theoretical foundations. Each module states the paper its expressions come from. The full computational companion — nineteen modules, including the closure count of one generation, the targets the framework's own running must meet, the perihelion chain from the toggle threshold, and the open programme carried as the series' seven problems — is the Research Calculator on Zenodo; the link resolves to the latest version.
The MOND acceleration scale — closure
The exact surplus acceleration from the geodesic equation in the recursion metric with the exact R(Dst) solution: gsurplus = cH₀R(1 + 2R⁴)/6. Setting gsurplus = a₀ gives the unique solution Rtrans = 0.748. As Rtrans is fixed by a₀ itself, a₀ is a closure; the derived content is the scale cH₀/6 and the transition location.
Source: Research Calculator (10.5281/zenodo.19970404, resolving to the latest version); 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. Inverting the exact R(Dst) solution at Rtrans gives Dtrans/DGUT = 0.439: the transition sits at 44% of the GUT density, as any R below 1 requires. Version 6.1 corrects the 16.79 printed in the Dgal paper, whose evaluation misplaced the factor (1 + 2R⁴); Rtrans and a₀ are unchanged.
The density-dependent propagation function R(Dst) — exact solution
The exact solution derived from the classical recursion Lagrangian: R/(1 + 2R⁴)1/4 = C · Dst1/6, with the single constant C = 3−1/4 fixed by the normalisation R(DGUT) = 1. The exponent n = 1/6 is derived, not fitted. There are no free parameters. The asymptote is R → 0 as Dst → 0 (ceff → ∞: inflation is the Dst → 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(Dst) from the Classical Recursion Field (10.5281/zenodo.19968963); Research Calculator (10.5281/zenodo.19970404, resolving to the latest version). 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 DGUT. The equation of state w = (1−2R⁴)/(1+2R⁴) reaches w = −1/3 at the GUT scale.
Fine-structure constant — geometric derivation and slip operator
Forward chain (derived). From the pressure axiom: golden angle θφ = 360/φ², hexagonal curvature κ₆ = 6/π², spin coupling C = (Se/Sγ)²·π²·κ₆ = 3/2, torsion density Ī = 3α/2, torsion slope g = 6κ₆Ī3/2, slip operator Δθ = 2g·(180/π), and the self-consistency condition 1/α = θφ − 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 = θφ − 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.
Lepton mass spectrum — torsion interval scaling
The scaling factors are derived, not fitted: λ₁ = 3/(2α) and λ₂ = 3/(8πα). The muon and tau masses follow from the electron anchor: mμ = λ₁me, mτ = λ₂mμ. 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: mμ = 105.658 MeV, mτ = 1.77686 GeV.
Binary recursion toggle
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).
Energy identity — E = pr
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).
Every expression on this page traces to a specific paper. For the complete catalog, see the Series page; for the conceptual foundations, see the Framework page; for the derived quantities stated as design constraints, see the Engineering page.