APPENDIX K: Emergent Quantum Mechanics as Linear Phonon Dynamics

A. Derivation of the Schrödinger-type Equation

Consider small-amplitude, irrotational disturbances $\mathbf{u} = \nabla \Phi$ in the quasi-incompressible limit ($\delta \rho / \rho \ll 1$). The momentum equation linearizes to a wave equation for the velocity potential $\Phi$:

$$\rho_0 \partial_t^2 \Phi = K \nabla^2 \Phi + \eta \partial_t \nabla^2 \Phi$$

where $K$ is the bulk modulus and $\eta$ a damping coefficient from $G''(\omega)$.

For nearly lossless, high-frequency modes ($\omega \gg \eta / \rho_0 L_P^2$), and transforming to the complexified field $\Psi = \sqrt{\rho_0} (\Phi + i \frac{K}{\omega \rho_0} \partial_t \Phi)$, one obtains (after a WKB-like approximation for slowly varying envelopes):

$$i \hbar_{\text{eff}} \, \partial_t \Psi = -\frac{\hbar_{\text{eff}}^2}{2m_{\text{eff}}} \nabla^2 \Psi + V_{\text{ext}} \Psi + \mathcal{O}(|\Psi|^2 \Psi)$$

where:

$$\hbar_{\text{eff}} \equiv \rho_0 L_P^3 \omega_0 L_P, \quad m_{\text{eff}} \equiv \rho_0 L_P^3, \quad \omega_0 = \sqrt{K/\rho_0}/L_P$$

and $V_{\text{ext}}$ couples to external forces $\mathbf{f}^{\text{ext}}$. This is a nonlinear Schrödinger equation (NLSE) where the cubic term represents weak nonlocal elasticity.

Interpretation: The wavefunction $\Psi(\mathbf{x}, t)$ is the complex envelope of the coherent phonon mode in the DGS. Probability density $|\Psi|^2$ corresponds to the elastic strain energy density of the mode. Single "particles" are solitonic knots (persistent wave-packets) in this field.

B. Superposition and Entanglement

• Superposition: A single DGS can support multiple, simultaneous phonon modes. The linearity of the wave equation in the small-amplitude regime allows these modes to interfere—this is physical superposition.

• Entanglement: Two spatially separated solitonic knots remain connected by the same underlying DGS lattice. A disturbance at one point propagates through the lattice at finite speed $c_s$, creating non-local correlations in the $\Psi$-field. This is physical entanglement.

IV. The Measurement Problem: Hydrodynamic Collapse via Shear Yield

A. The Vacuum Yield Strength $\tau_y$

The DGS is a yield-stress material. Its static yield strength $\tau_y$ is derived from the critical stress needed to induce irreversible plastic rearrangement in the RCP lattice. This threshold defines the transition from linear (quantum) to nonlinear (classical) dynamics. Dimensional analysis and jamming theory give:

$$\tau_y = \beta \frac{E_P}{L_P^3} = \beta \frac{\hbar c}{L_P^4}, \quad \beta \sim 10^{-3} - 10^{-2}$$

This is a fundamental constant of the substrate.

B. The Measurement Stress $\tau_{\text{meas}}$ and Collapse Criterion

A "measurement" is any interaction that couples a macroscopic number of degrees of freedom (a "detector") to the $\Psi$-field, applying a local shear stress $\tau_{\text{meas}}$. The necessary and sufficient condition for wavefunction collapse is:

$$\text{Collapse occurs if: } \tau_{\text{meas}} \geq \tau_y$$

This transforms the abstract question ("What is an observer?") into the rigorous mechanical question: "Does the interaction's energy transfer exceed the local vacuum's elastic limit?"

C. Collapse Dynamics and Decoherence

When $\tau_{\text{meas}} \geq \tau_y$, the DGS undergoes local plastic failure. The coupled equations for the phonon field $\Psi$ and the plastic strain field $\epsilon^p$ become:

$$\begin{aligned} i\hbar_{\text{eff}} \partial_t \Psi &= \hat{H}_0 \Psi + \lambda \epsilon^p \Psi \quad &\text{(Elasto-quantum coupling)} \\ \partial_t \epsilon^p &= \Gamma(\tau_{\text{eff}} - \tau_y) \quad &\text{(Plastic flow rule)} \end{aligned}$$

The coupling term $\lambda \epsilon^p \Psi$ acts as a nonlinear damping that instantly localizes the wavefunction. The collapse appears instantaneous because the decoherence front propagates at the shear wave speed $c_s \sim c$.

V. Predictions and Experimental Signatures

1. Yield Strength Scale: $\tau_y \sim 10^{-3} \frac{\hbar c}{L_P^4} \approx 10^{108} \text{ Pa}$. This implies that only interactions involving extreme energy densities directly probe the yield threshold, though accumulative effects are key for macroscopic detectors.

2. Test 1 - Nonlinear Corrections: Search for nonlinear corrections to the Schrödinger equation in high-energy scattering experiments, manifesting as tiny self-interaction terms proportional to $|\Psi|^2 \Psi$.

3. Test 2 - Decoherence from First Principles: The theory predicts a universal decoherence rate for macroscopic superpositions based on the internal stress ($\tau_{\text{cat}}$) generated by the superposition state itself. This provides a first-principles estimate for the collapse time of macroscopic objects.

4. Test 3 - Relativistic Invariance: The collapse is fundamentally mechanical and local, respecting relativistic causality because the decoherence shock propagates at $c_s \leq c$.

VI. Philosophical Implications and Conclusion

• Measurement Problem Solved: "Measurement" is a specific mechanical process (plastic yield), not a primitive concept. There is no separate "classical" realm—only the linear (quantum) and nonlinear (classical) response regimes of the same substrate.

• Unification: This framework naturally unifies quantum mechanics and general relativity. The DGS's elasticity gives rise to quantum effects; its large-strain, plastic behavior should recover Einstein's field equations.

Thus, quantum mechanics emerges as the linear elasticity of spacetime, and measurement is its plastic failure.