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UIDT Formalism v3.7.2

PURPOSE: Central repository for UIDT equations
RULE: Math (LaTeX) MUST be separated from interpretation

Lagrangian

Full UIDT Lagrangian

$$\mathcal{L}_{\text{UIDT}} = \mathcal{L}_{\text{YM}} + \mathcal{L}_S + \mathcal{L}_{\text{int}}$$

Yang-Mills Sector

$$\mathcal{L}_{\text{YM}} = -\frac{1}{4} F^a_{\mu\nu} F^{a\mu\nu}$$

Scalar Sector

$$\mathcal{L}_S = \frac{1}{2} \partial_\mu S \partial^\mu S - V(S)$$

Scalar Potential

$$V(S) = \frac{\lambda_S}{4} (S^2 - v^2)^2$$

Interaction (Non-Minimal Coupling)

$$\mathcal{L}_{\text{int}} = -\frac{\kappa}{4} S^2 F^a_{\mu\nu} F^{a\mu\nu}$$

Core Equations

Vacuum Equation

$$\left\langle S \right\rangle = v = 47.7 \text{ MeV}$$

Schwinger-Dyson Equation

$$\Box S + \lambda_S S (S^2 - v^2) + \frac{\kappa}{2} S F^2 = 0$$

Renormalization Group Equation

$$\mu \frac{d\lambda_S}{d\mu} = \beta_{\lambda_S}(\lambda_S, \kappa, g)$$

Fixed-Point Condition

RG Fixed Point Constraint

$$5\kappa^2 = 3\lambda_S$$

Verification: $$5 \times (0.500)^2 = 1.250$$ $$3 \times 0.417 = 1.251$$ $$|\Delta| = 0.001 < 0.01 \checkmark$$

Mass Gap Derivation

Spectral Gap

$$\Delta = \gamma \cdot \Lambda_{\text{QCD}}$$

With:

  • $\gamma = 16.339$ (kinetic VEV) [A-]
  • $\Lambda_{\text{QCD}} \approx 0.1046$ GeV
  • $\Delta = 1.710 \pm 0.015$ GeV [A]

Scalar Mass

$$m_S^2 = 2\lambda_S v^2$$ $$m_S = 1.705 \pm 0.015 \text{ GeV}$$

Stability Conditions

Perturbative Stability

$$\lambda_S < 1 \quad \Rightarrow \quad 0.417 < 1 \checkmark$$

Vacuum Stability

$$V''(v) = 2\lambda_S v^2 > 0 \quad \Rightarrow \quad 2.907 > 0 \checkmark$$

Cosmological Equations

Hubble Parameter (Calibrated)

$$H_0 = 70.4 \pm 0.16 \text{ km/s/Mpc}$$

Category C: Calibrated to DESI DR2, NOT derived.

Vacuum Energy Suppression

$$\rho_{\text{vac}}^{\text{obs}} = \rho_{\text{vac}}^{\text{QFT}} \times \pi^{-2} \times \prod_{n=1}^{99} f_n(g)$$

Category C: Phenomenological 99-step cascade.

UIDT Wavelength

$$\lambda_{\text{UIDT}} = 0.660 \pm 0.005 \text{ nm}$$

Pillar II-CSF (Covariant Scalar-Field)

Conformal Density Mapping

$$ \gamma_{CSF} = \frac{1}{2 \sqrt{\pi \ln(\gamma_{UIDT})}} $$

Planck-Singularity Regularization

$$ \rho_{max} = \Delta^4 \cdot \gamma^{99} $$

Equation of State (Placeholders)

$$ w_0 = -0.99, \quad w_a = +0.03 $$

Strict Caveat: The CSF extensions are strictly evaluated under Evidence Category [C]. They map phenomenologically upon the [A-] calibrated lattice invariant $\gamma = 16.339$. Limitation L4: $\gamma$ is calibrated, not fundamentally derived. Limitation L5: The $N=99$ RG step limit remains empirical.

Casimir Prediction

Force Anomaly

$$\frac{\Delta F}{F} = +0.59% \quad \text{at} \quad d = 0.66 \text{ nm}$$

Category D: Unverified prediction.

Optimal Distance (v3.7.1 corrected)

$$d_{\text{opt}} = 0.854 \text{ nm}$$

Numerical Precision

Residual Thresholds

Equation System Residual Status
Three-Equation Closure < 10⁻⁴⁰ ✅
Branch 1 3.2×10⁻¹⁴ ✅
Branch 2 (excluded) 1.8×10⁻¹² ❌
Verification Tolerance < 10⁻¹⁴ ✅

Constraint Summary

Constraint Expression Value Status
RG Fixed Point 5κ² = 3λ_S 1.250 ≈ 1.251 ✅
Perturbative λ_S < 1 0.417 ✅
Vacuum V''(v) > 0 2.907 ✅
Gamma γ_kinetic ≈ γ_MC 16.339 ≈ 16.374 ✅

Reference Implementation

See WORKSPACE/derivations/ for:

  • uidt_proof_core.py — Core proof implementation
  • rg_flow_analysis.py — RG flow calculations
  • error_propagation.py — Uncertainty analysis

CITATION: Rietz, P. (2025). UIDT v3.7.2. DOI: 10.5281/zenodo.17835200