APPENDIX A: MATHEMATICAL FORMALISM
A.1 Full Lagrangian Expansion
$$\mathcal{L}{\text{total}} = \mathcal{L}\chi + \mathcal{L}\Phi + \mathcal{L}{\text{int}} + \mathcal{L}_{\text{matter}}$$
χ-Field Term: $$\mathcal{L}\chi = \frac{1}{2}(\partial\mu \chi)(\partial^\mu \chi) - \frac{m_\chi^2}{2}\chi^2 - \frac{\lambda_\chi}{4!}\chi^4$$
Φ-Field Term: $$\mathcal{L}\Phi = \frac{1}{2}(\partial\mu \Phi)(\partial^\mu \Phi) - V(\Phi)$$
Interaction Term:
Mathematical Equation
Visual: $$\mathcal{L}_{\text{int}} = \kappa \chi \Phi + \lambda C[\chi, \Phi] - g \chi \bar{\psi}\psi$$
Spoken: When we read this, it is telling us that kappa in a more natural way.
Matter Coupling:
Mathematical Equation
Visual: $$\mathcal{L}{\text{matter}} = \bar{\psi}(i\gamma^\mu D\mu - m)\psi$$
Spoken: When we read this, it is telling us that gamma in a more natural way.
A.2 Field Equations
χ-Field Equation:
Mathematical Equation
Visual: $$\Box \chi + m_\chi^2 \chi + \frac{\lambda_\chi}{6}\chi^3 = -\kappa \Phi - g\bar{\psi}\psi$$
Spoken: When we read this, it is telling us that kappa in a more natural way.
Φ-Field Equation:
Mathematical Equation
Visual: $$\Box \Phi + \frac{dV}{d\Phi} = -\kappa \chi$$
Spoken: When we read this, it is telling us that kappa in a more natural way.
A.3 Stress-Energy Tensors
χ-Field Stress-Energy:
Mathematical Equation
Visual: $$T_{\mu\nu}^\chi = \partial_\mu \chi \partial_\nu \chi - g_{\mu\nu}\mathcal{L}_\chi$$
Spoken: When we read this, it is telling us that T_{munu} in a more natural way.
Φ-Field Stress-Energy:
Mathematical Equation
Visual: $$T_{\mu\nu}^\Phi = \partial_\mu \Phi \partial_\nu \Phi - g_{\mu\nu}\mathcal{L}_\Phi$$
Spoken: When we read this, it is telling us that T_{munu} in a more natural way.
Ring 2 — Canonical Grounding
Ring 3 — Framework Connections
Canonical Hub: CANONICAL_INDEX