APPENDIX H: THE EMERGENT LAGRANGIAN AND GRANULAR DICTIONARY

(Rigorous Proof of Correspondence and Physical Mapping)

H.1 Theorem of Convergence

We postulate that the discrete mechanical evolution of the vacuum, governed by the Fundamental Granular Function ($F_{\text{Planck}}$), converges to the continuous Standard Model Action ($S_{\text{SM}}$) under the following limits:

1. Geometric Limit: Grain size $L_P \to 0$.

2. Temporal Limit: Time step $\Delta t_P \to 0$.

3. Elastic Limit: The local yield function satisfies $Y < 0$ everywhere (stress remains below the Mohr-Coulomb failure threshold).

H.2 The Discrete Granular Action

In SBF, the "Action" is the summation of the difference between Kinetic and Potential energies of the discrete grains over time.

$$S_{\text{SBF}} = \sum_{t=0}^{T} \sum_{k \in \text{lattice}} \left[ T_k(t) - \mathcal{E}_{k,\text{Local}}(t) \right] \Delta V_k \Delta t$$

H.2.1 The Kinetic Term (The Emergence of Time)

The kinetic term $T_k$ represents the inertial resistance of the vacuum grains to rearrangement.

$$T_k = \frac{1}{2} \rho \left( \frac{\Delta \mathbf{u}_k}{\Delta t} \right)^2 + \frac{1}{2} \rho_{\text{rot}} \left( \frac{\Delta \boldsymbol{\phi}_k}{\Delta t} \right)^2$$

• Physical Interpretation: In SBF, "Time" is not a fundamental coordinate. It is a measure of the lattice's inertia. The finite mass density $\rho$ of the grains ensures that interactions are not instantaneous, creating the "speed of light" limit ($c = \sqrt{K/\rho}$).

• Note on Gauge Choice: In this derivation, we adopt the Weyl (Temporal) Gauge ($A_0 = 0, \boldsymbol{\phi} \neq 0$). This is consistent with the SBF's treatment of time as an emergent property of lattice inertia rather than a fundamental geometric dimension. By setting the scalar potential to zero, the electric field arises purely from the time-evolution of the vector potential ($\mathbf{E} = -\partial_t \mathbf{A}$), mapping directly to the inertial term of the granular update rule.

H.2.2 The Potential Term (Stored Energy)

Derived from the stored elastic energy (Section E.2.1):

$$\mathcal{E}_{k,\text{Local}} = \frac{1}{2} K (\nabla \cdot \mathbf{u})^2 + \frac{1}{2} G (\nabla \times \mathbf{u})^2 + \frac{1}{2} G (\nabla \boldsymbol{\phi})^2 + \kappa_{\text{coup}} (\nabla \mathbf{u} \cdot \nabla \boldsymbol{\phi})$$

H.3 The Continuum Limit

We apply the continuum limit where the grain size vanishes relative to the observation scale.

Step A: Geometry to Calculus

The finite differences become partial derivatives:

$$\lim_{L_P \to 0} \frac{\mathbf{u}(x+L_P) - \mathbf{u}(x)}{L_P} = \nabla \mathbf{u}$$

The Riemann sum transitions to a spacetime volume integral:

$$\sum_{k} \Delta V_k \Delta t \to \int d^3x \, dt$$

Step B: Scaling of Constants

To recover the correct dimensions for Field Theory, the mechanical constants must scale with the Planck units. As the scale changes, the effective stiffness "runs" according to the Renormalization Group flow:ShutterstockExplore

• Stiffness ($K, G$): $K \approx \frac{E_{\text{Planck}}}{L_P^3}$ (Energy Density).

• Mass Density ($\rho$): $\rho \approx \frac{M_{\text{Planck}}}{L_P^3}$.

Step C: The Integral Form

Substituting these limits into $S_{\text{SBF}}$:

$$S_{\text{SBF}} \to \int dt \int d^3x \left[ \underbrace{\frac{1}{2}\rho (\partial_t \mathbf{u})^2}_{\text{Kinetic Density } \mathcal{T}} - \underbrace{\mathcal{E}_{\text{Local}}(\nabla \mathbf{u})}_{\text{Potential Density } \mathcal{V}} \right]$$

H.4 The Emergent Lagrangian Density

This yields the effective continuum Lagrangian density $\mathcal{L}_{\text{Effective}} = \mathcal{T} - \mathcal{V}$, recovering the Standard Model sectors as effective field theories:

1. Gravity Sector: The scalar compression terms ($\nabla \cdot \mathbf{u}$) and translational inertia ($\partial_t \mathbf{u}$) recover the scalar-tensor gravitational Lagrangian.

2. Electromagnetic Sector: The rotational terms ($\partial_t \boldsymbol{\phi}$ and $\nabla \boldsymbol{\phi}$) recover the Proca/Maxwell Lagrangian structure ($\partial_\mu A_\nu \partial^\mu A^\nu$).

3. Interaction Sector: The coupling term $\nabla \mathbf{u} \cdot \nabla \boldsymbol{\phi}$ generates the interaction Lagrangian $\mathcal{L}_{\text{int}} = -J^\mu A_\mu$.

H.5 The Granular Dictionary

(Electrodynamics from Mechanics)

Having established that the Maxwell Lagrangian emerges from the granular rotational stiffness, we can now map the abstract constants of QED to the concrete mechanical properties of the vacuum grains.

H.5.1 Derivation of Electromagnetic Constants from Granular Mechanics

We derive the vacuum permittivity ($\epsilon_0$) and permeability ($\mu_0$) not as fundamental constants, but as the emergent mechanical properties—Rotational Inertia and Torsional Stiffness—of the granular lattice.

1. The Geometric Inputs

We model the vacuum substrate as a Random Close Packing (RCP) of Planck grains with:

• Grain Mass: $m_{grain} = m_P$ (Planck Mass)

• Grain Radius: $r = L_P/2$ (Planck Length diameter)

• Packing Fraction: $\phi_{RCP} \approx 0.64$

• Coordination Number: $Z \approx 14.4$

2. Deriving Permittivity ($\epsilon_0$) as Rotational Inertia Density

The "Electric Field" energy density $\frac{1}{2}\epsilon_0 E^2$ corresponds to the kinetic energy of the grains' microrotation ($\dot{\theta}$).

• Single Grain Inertia: For a solid sphere, $I = \frac{2}{5} m_P r^2 = \frac{1}{10} m_P L_P^2$.

• Inertia Density ($i$): The rotational inertia per unit volume, corrected by the packing fraction $\phi_{RCP}$:
  $$i_{rot} = \phi_{RCP} \cdot \frac{I_{grain}}{Vol_{grain}} = \phi_{RCP} \cdot \frac{\frac{1}{10}m_P L_P^2}{\frac{4}{3}\pi (L_P/2)^3}$$
  Simplifying:
  $$i_{rot} \approx \frac{m_P}{L_P} \cdot \zeta_{geom}$$
  (Where $\zeta_{geom}$ is a dimensionless geometric factor of order $\approx 0.1$).

Identification: In the mechanical Lagrangian, the coefficient of the kinetic term $\frac{1}{2} i_{rot} \dot{\theta}^2$ maps directly to the electric permittivity.

$$\epsilon_0 \equiv i_{rot} \quad (\text{Rotational Inertia Density})$$

3. Deriving Permeability ($\mu_0$) as Torsional Stiffness

The "Magnetic Field" energy density $\frac{1}{2\mu_0} B^2$ corresponds to the potential energy of the lattice twisting strain $(\nabla \theta)^2$.

• Planck Stiffness: The yield force of a single grain contact is the Planck Force $F_P = E_P / L_P$.

• Shear Modulus ($G$): The energy density required to deform the packing is $G = F_P / L_P^2$.

• Torsional Stiffness ($J$): The torque response to a twist gradient. For a Cosserat medium, this scales with the Shear Modulus and the cross-sectional area of the grain interaction ($L_P^2$):
  $$J_{torsion} \approx G \cdot L_P^2 = \left(\frac{F_P}{L_P^2}\right) L_P^2 = F_P = \frac{m_P c^2}{L_P}$$

Identification: In the mechanical Lagrangian, the coefficient of the potential term $\frac{1}{2} J_{torsion} (\nabla \theta)^2$ maps to the inverse magnetic permeability.

$$\frac{1}{\mu_0} \equiv J_{torsion} \implies \mu_0 \equiv \frac{1}{J_{torsion}} \quad (\text{Inverse Stiffness})$$

4. The Speed of Light Check

Standard electrodynamics requires $c = 1/\sqrt{\epsilon_0 \mu_0}$. We test if our mechanical derivation recovers this limit naturally.

$$c_{SBF} = \frac{1}{\sqrt{\epsilon_0 \mu_0}} = \sqrt{\frac{1/\mu_0}{\epsilon_0}} = \sqrt{\frac{J_{torsion}}{i_{rot}}}$$

Substituting the granular values:

$$c_{SBF} = \sqrt{\frac{m_P c^2 / L_P}{m_P / L_P}} = \sqrt{c^2} = c$$

Conclusion:

The speed of light is not arbitrary; it is the shear wave velocity of the granular vacuum.

• $\epsilon_0$ is the inertia resisting the spin-up of vacuum grains (Electric Field).

• $\mu_0$ is the lattice stiffness resisting the twisting of vacuum grains (Magnetic Field).

• The product $\epsilon_0 \mu_0$ is fixed by the ratio of Mass ($m_P$) to Tension ($F_P$) in the Planck substrate.

H.5.2 Mechanical Interpretation of Fields

This mapping demystifies electromagnetic fields:

H.5.3 Prediction: The Magnetic Yield Limit

A unique granular prediction: magnetic fields cannot be infinite. Magnetic energy density $B^2/(2\mu_0)$ corresponds to physical shear stress on the lattice.