TFP Section 10 (v9)

SECTION 10 — EMERGENT QUANTUM OBSERVABLES (v9.1)

Quantization from Substrate Eigenmodes and Causal Coherence
By John Gavel

10.0 Overview

Quantum observables emerge from the stability spectrum of the substrate’s linearized dynamics (Section 2.2) and the causal structure of correlations (Section 3). No wavefunctions or complex Hilbert spaces are postulated — all quantum behavior derives from:

• Eigenmodes of the response matrix \( L \) (Section 2.2)
• Operational distance \( d_{ij} \) (Section 3.2)
• Calibration to physical units (Section 5)

Phase, superposition, and uncertainty arise as coarse-grained descriptions of constrained binary dynamics (Section 3.1.2).

10.1 Substrate Response Matrix and Eigenmodes

Linearizing the stochastic update rule (Section 2.2.2) around a stable motif yields:

\[ \delta \mathbf{F}(t + \tau_0) = L \cdot \delta \mathbf{F}(t) \tag{10.1} \]

where \( L \) is the real-valued response matrix with elements:

\[ L_{ij} = \alpha(i,j) \left( 1 - \frac{\Delta T_i}{T_0} \right) \tag{10.2} \]

and \( \alpha(i,j) \) is the adaptive coupling (Section 2.1.1). The eigenvalue problem is:

\[ L \mathbf{v}_n = \lambda_n \mathbf{v}_n \tag{10.3} \]

Eigenmodes \( \mathbf{v}_n \) define stable collective oscillations; eigenvalues \( \lambda_n = \lambda_n^{\text{re}} + i \lambda_n^{\text{im}} \) determine relaxation and frequency.

10.2 Energy Quantization

From Section 5.3.1, the physical action unit is \( \hbar_c = M_c L_c^2 / T_c \). The energy of eigenmode \( n \) is:

\[ E_n = \hbar_c \cdot \frac{|\lambda_n^{\text{im}}|}{\tau_0} \tag{10.4} \]

since \( |\lambda_n^{\text{im}}| / \tau_0 \) is the angular frequency (Section 4.3.2). In the rest frame (\( \mathbf{p} = 0 \)), the effective mass is:

\[ m_n = \frac{E_n}{c_{\text{pred}}^2} = \frac{\hbar_c |\lambda_n^{\text{im}}|}{c_{\text{pred}}^2 \tau_0} \tag{10.5} \]

where \( c_{\text{pred}} = L_c / T_c \) (Section 5.2.3). This links mass to the **temporal oscillation rate** of the eigenmode.

10.3 Spatial Localization and Mass Hierarchy

Eigenmode spatial extent is measured by the inverse participation ratio:

\[ \text{IPR}_n = \frac{ \left( \sum_i |v_n(i)|^2 \right)^2 }{ \sum_i |v_n(i)|^4 } \tag{10.6} \]

Large IPR → extended mode → small \( |\lambda_n^{\text{re}}| \) → light particle Small IPR → localized mode → large \( |\lambda_n^{\text{re}}| \) → heavy particle

Thus the mass hierarchy is:

\[ \frac{m_n}{m_1} = \frac{|\lambda_n^{\text{im}}|}{|\lambda_1^{\text{im}}|} \tag{10.7} \]

with no free parameters — ratios are fixed by eigenmode structure (Section 4.7).

10.4 Emergent Phase and Superposition

Although \( F_i \in \{±1\} \) has no phase, the **correlation lag** \( \tau_{i\to j} \) (Section 3.1.2) defines a **discrete phase difference**:

\[ \theta_{ij} = \omega \tau_{i\to j} \tag{10.8} \]

where \( \omega = |\lambda^{\text{im}}| / \tau_0 \) is the eigenmode frequency. For a region \( R \), define the coarse-grained field:

\[ \Psi_R(t) = \frac{1}{|R|} \sum_{i \in R} F_i(t) e^{i \theta_i(t)} \tag{10.9} \]

where \( \theta_i(t) \) is the accumulated lag from a reference site. This is **not fundamental** — it is a derived observable encoding causal history (Section 3.1.2). Superposition arises because \( \Psi_R \) is a linear combination of eigenmodes:

\[ \Psi_R(t) = \sum_n c_n \mathbf{v}_n e^{-i \omega_n t} \tag{10.10} \]

10.5 Uncertainty Principle

From Section 3.2.2, the minimum resolvable distance is \( \Delta x_{\text{min}} = L_c \). From Section 4.3.2, the momentum scale is \( P_c = M_c L_c / T_c \). Thus:

\[ \Delta x \geq L_c, \quad \Delta p \geq P_c \tag{10.11} \]

Multiplying:

\[ \Delta x \Delta p \geq L_c P_c = \frac{M_c L_c^2}{T_c} = \hbar_c \tag{10.12} \]

For energy and time, \( \Delta E \geq \hbar_c / T_c \) and \( \Delta t \geq T_c / |\lambda^{\text{im}}| \), so:

\[ \Delta E \Delta t \geq \hbar_c \tag{10.13} \]

Uncertainty arises from **finite node capacity** (Section 11.4), not from non-commuting operators.

10.6 Spin and Angular Momentum

From Section 3.5, spin is the topological winding number:

\[ S_p = \sum_{(u,v) \in \partial p} F_u F_v \cdot \mathrm{sgn}(u \to v) \tag{10.14} \]

For a stable eigenmode, the net spin is quantized. After calibration (Section 5):

\[ S_z = \frac{\hbar_c}{2} S_p \tag{10.15} \]

Orbital angular momentum arises from spatial winding of eigenmodes:

\[ L_z = i \hbar_c \sum_i x_i \partial_y v_n(i) - y_i \partial_x v_n(i) \tag{10.16} \]

which yields \( L^2 = \ell(\ell+1) \hbar_c^2 \) for integer \( \ell \).

10.7 Relativistic Dispersion

For a motif with momentum \( \mathbf{p} = M_n \mathbf{v}_C \) (Section 6.3.2), the energy is:

\[ E^2 = (p c_{\text{pred}})^2 + (m_n c_{\text{pred}}^2)^2 \tag{10.17} \]

where \( c_{\text{pred}} = L_c / T_c \) (Section 5.2.3). Lorentz invariance emerges from the statistical homogeneity of correlations (Section 6.6.3):

\[ \langle F_i F_j \rangle = f(|\mathbf{x}_i - \mathbf{x}_j|) \tag{10.18} \]

10.8 Wave-Particle Duality

Localized eigenmodes (\( \text{IPR}_n \ll N \)) behave as particles; extended modes (\( \text{IPR}_n \sim N \)) exhibit interference. The transition scale is the coherence length:

\[ \xi_{\text{quantum}} = \text{average } d_{ij} \text{ over correlated sites} \tag{10.19} \]

If the measurement scale \( \Delta x \ll \xi_{\text{quantum}} L_c \), particle-like behavior dominates; if \( \Delta x \gg \xi_{\text{quantum}} L_c \), wave-like interference appears.

10.9 Axiomatic Closure

Quantum Observable	Substrate Origin	Axiom
Energy Quantization	L-matrix eigenvalues	A2, A6, A9
Uncertainty	Finite node capacity	A2, A3
Spin	Flow circulation (Section 3.5)	A2, A3
Phase	Causal lag accumulation (Section 3.1.2)	A6, A9

10.10 Bridge to Section 11 — Measurement

Section 10 provides:

• Quantized energy levels from eigenmodes
• Uncertainty from finite resolution
• Phase as derived from causal history
• Spin from topological circulation

Section 11 shows how measurement collapse emerges from phase synchronization in stable motifs, with no external observer required.