Section 10 — Emergent Quantum Observables (TFP, v11.1)

Section 10 — Emergent Quantum Observables (TFP, v11.1)

Quantization from Substrate Eigenmodes and Causal Coherence

10.0 Overview

Quantum observables in TFP emerge entirely from substrate constraints: linear response, causal correlations, finite handshake capacity, adjacency, and timing produce eigenmodes, frequencies, phase, uncertainty, spin, and relativistic dispersion. No external Hilbert space or operator postulates are required. Physical units are calibrated using anchors from Section 5.

10.1 Substrate Response Matrix and Eigenmodes

Linearize stochastic updates about a stable motif: \[ \delta F(t + \tau_0) = L \cdot \delta F(t) \] where \(L\) is the real substrate response matrix with elements: \[ L_{ij} = \alpha(i,j) \cdot \big(1 - \Delta T_i / T_0\big) \] encoding adjacency weights and adaptive coupling.

Eigenproblem: \[ L \cdot v_n = \lambda_n \, v_n, \quad \lambda_n = \lambda_n^\text{re} + i \lambda_n^\text{im} \] Real part → relaxation; Imaginary part → oscillation frequency.

10.2 Energy Quantization (Mode Frequency → Energy)

Define substrate action unit: \[ \hbar_c = \frac{M_c L_c^2}{T_c} \] Angular frequency of mode \(n\): \[ \omega_n = \frac{|\lambda_n^\text{im}|}{\tau_0} \] Mode energy: \[ E_n = \hbar_c \, \omega_n \] Rest-frame effective mass: \[ m_n = \frac{E_n}{c_\text{pred}^2} = \frac{\hbar_c |\lambda_n^\text{im}|}{c_\text{pred}^2 \tau_0}, \quad c_\text{pred} = \frac{L_c}{T_c} \] Interpretation: Mass measures temporal oscillation cost of an eigenmode in substrate action units.

10.3 Spatial Localization and Mass Hierarchy

Inverse Participation Ratio (IPR): \[ \text{IPR}_n = \frac{\big(\sum_i |v_n(i)|^2\big)^2}{\sum_i |v_n(i)|^4} \] Large IPR → extended mode → light mass; Small IPR → localized mode → heavy mass. Mass ratios: \[ \frac{m_n}{m_1} = \frac{|\lambda_n^\text{im}|}{|\lambda_1^\text{im}|} \] determined entirely by network topology, J/L structure, and K=12 embedding.

10.4 Emergent Phase and Superposition

Discrete causal lag: \[ \theta_{ij} = \omega \, \tau_{i \to j}, \quad \tau_{i \to j} = d_{ij} \, \tau_0 \] Coarse-grained complex observable: \[ \Psi_R(t) = \frac{1}{|R|} \sum_{i \in R} F_i(t) \, e^{i \theta_i(t)}, \quad \theta_i(t) = \text{accumulated lag from reference site} \] Superposition: \[ \Psi_R(t) = \sum_n c_n \, v_n \, e^{-i \omega_n t} \] Valid where coarse-graining is meaningful (Sections 8–9).

10.5 Uncertainty Principle from Finite Capacity

Minimal resolvable scales: \[ \Delta x_\text{min} = L_c, \quad P_c = \frac{M_c L_c}{T_c} \] Action product: \[ \Delta x \, \Delta p \ge L_c \, P_c = \hbar_c \] Energy-time bound: \[ \Delta E \, \Delta t \ge \hbar_c \] Origin: finite node capacity and address collisions. Interpretation: uncertainty = hardware jitter in the 132-bit handshake substrate.

10.6 Spin and Angular Momentum as Topological Winding

Winding number for plaquette \(p\): \[ S_p = \sum_{(u,v) \in \partial p} F_u F_v \, \text{sgn}(u \to v) \] Calibrated spin component: \[ S_z = \frac{\hbar_c}{2} S_p \] Orbital angular momentum: \[ L_z = i \hbar_c \sum_i \left( x_i \frac{\partial}{\partial y} v_n(i) - y_i \frac{\partial}{\partial x} v_n(i) \right) \] Quantization: \(L^2 = \ell(\ell+1) \hbar_c^2\). Spin emerges from scheduling/topology of updates; Pauli exclusion arises from handshake constraints.

10.7 Relativistic Dispersion from Correlation Homogeneity

Momentum from centroid motion: \[ p = m_n \, v_C \] Emergent Lorentz-invariant dispersion: \[ E^2 = (p \, c_\text{pred})^2 + (m_n \, c_\text{pred}^2)^2 \] with \(c_\text{pred} = L_c / T_c\). Lorentz symmetry emerges statistically from isotropic correlations: \(\langle F_i F_j \rangle = f(|x_i - x_j|)\).

10.8 Wave–Particle Duality and Coherence Length

Coherence length: \[ \xi_\text{quantum} = \text{average } d_{ij} \text{ over correlated sites} \] Measurement scale: \[ \Delta x \ll \xi_\text{quantum} \, L_c \Rightarrow \text{particle-like} \] \[ \Delta x \gg \xi_\text{quantum} \, L_c \Rightarrow \text{wave interference emerges} \] IPR determines particle vs. wave behavior.

10.9 Axiomatic Closure (Quantum Observables)

Observable	Substrate Origin	Axiom
Energy quantization	L eigenvalues	A2, A6, A9
Uncertainty	Finite node capacity	A2, A3
Spin	Flow circulation / winding	A2, A3
Phase	Causal lag accumulation	A6, A9

10.10 Hardware Limit, Remainder Resolution, and Physical Constants

Key substrate states:

• 132 — active handshake budget (12 × 11)
• 137.036 — folding resultant → α⁻¹ (Section 6.5)
• 140 — total functional bits (132 + 8)
• 144 — K² total logical adjacency (12²)

Temporal remainder ("the Gap") → gravitational leakage. Perfect resonance (zero remainder) → ideal Riemann-zero-like conditions. Dimensional transference: 3D emerges from process scaling; unresolved remainders produce curvature when not cleared in one tick (Section 7.4.1).

10.11 Bridge to Section 11 — Measurement

Section 10 establishes discrete spectra, eigenmode structure, and hardware constants (132, 137, 140, 144). Section 11 describes measurement as locking a handshake address to resolve a temporal remainder: collapse is a causal synchronization event where the 132-bit budget commits to a routing outcome.