Section 5: Emergence of Fundamental Concepts and Structure

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Section 5: Emergence of Fundamental Concepts and Structure — Defining Dimensions

5.1 Internal Speed Limit as an Emergent Causal Limit

5.1.1 Emergent Causality from Flow Asymmetry

Causality and directional propagation naturally arise from local temporal asymmetry and irreversibility in the discrete flow network. This is expressed by the condition:

The discrete second difference of flow at node \( i \) and step \( n \), denoted \( A_i(n) \), is nonzero if and only if the temporal asymmetry parameter \( \Psi_i(n) \) is nonzero, which also corresponds to a nonzero local time-averaged reflection rate \( M_i \) (mass). In other words:

\( A_i(n) \neq 0 \iff \Psi_i(n) \neq 0 \iff M_i > 0 \)

Here:

  • \( A_i(n) \): discrete second difference of the flow (dimensionless)
  • \( \Psi_i(n) \): temporal asymmetry or directional bias (dimensionless)
  • \( M_i \): time-averaged reflection rate at node \( i \) (dimensionless)

This shows that a nonzero asymmetry is necessary for coherent oscillations and information propagation, aligning local directional bias with energy flow and breaking time-reversal symmetry — fundamental for emergent causality.

5.1.2 Deriving Internal Speed Scale from Coherence Constraints

Stable and coherent network dynamics require that local differences in flow velocity between neighboring nodes be bounded.

Define local flow velocity update: \( u_i(n) = F_i(n+1) - F_i(n) \)

The coherence constraint requires: \( |u_i(n) - u_j(n)| \leq u_{\text{critical}} \) for all neighboring nodes \( i, j \)

Where \( u_{\text{critical}} \) depends on local coupling \( \alpha(i,j,n) \), nodal capacities \( C_i \), and asymmetry \( \Psi_i(n) \).

With discrete minimums:

  • \( \Delta x = 1 \)
  • \( \Delta t = 1 \)
so: \( c_{\text{internal}} \leq \frac{\Delta x}{\Delta t} = 1 \)

Therefore, the emergent internal causal speed is: \( c_{\text{internal}} = 1 \)

5.1.3 Physical Interpretation

This speed limit defines the maximum rate of coherent information propagation. Causal order emerges from:

  • \( A_i(n) \neq 0 \): directional bias
  • \( |u_i - u_j| \leq u_{\text{critical}} \): bounded neighbor mismatch
  • Saturation from nodal capacity \( C_i \)
Together forming a discrete analogue of relativistic causality.

5.2 Emergent Energy and Mass

5.2.1 Energy from Flow Activity

Define local energy: \( E_i(n) = [u_i(n)]^2 \)

Total network energy: \( E_{\text{total}}(n) = \sum_i E_i(n) \)

Physical energy recovered by: \( E_{i,\text{phys}} = E_i(n) \cdot E_c \)

5.2.2 Mass from Reflection and Time-Averaging

Mass as time-averaged reflection: \( M_i = \langle R_i(n) \rangle \), where \( R_i(n) \in \{0,1\} \)

Time-averaged energy: \( \langle E \rangle \propto M_i \cdot \langle \omega_i^2 \rangle \)

\( \omega_i \): internal oscillation frequency (dimensionless), with: \( \omega_{i,\text{phys}} = \frac{\omega_i}{T_c} \)

5.2.3 Inertial Mass and Resistance to Flow Change

Excitation mass difference: \( m_{\text{excitation}} = E_{\text{disrupted}} - E_{\text{stable}} \)

Effective inertial mass: \( m_{\text{eff}} = \eta(\gamma) \cdot m_{\text{excitation}} \)

Physical mass: \( m_{\text{eff,phys}} = m_{\text{eff}} \cdot M_c \)

5.3 Mass-Energy Relation and Relativistic Analogue

From flow dynamics:

  • \( E_i = u_i^2 \)
  • \( M_i = \langle R_i(n) \rangle \)
  • \( c_{\text{internal}} = 1 \)
Leading to: \( E \propto M \cdot c_{\text{internal}}^2 \Rightarrow E \propto M \)

5.4 Emergent Spatial Geometry

5.4.1 Flow-Based Distance Metric

Distance between nodes \( i, j \) defined by: \[ d_{ij}^2 \propto \left( \sum_k \alpha(i,k,n) - \sum_k \alpha(j,k,n) \right)^2 \] with physical distance: \[ x_{\text{phys}} = x_{\text{internal}} \cdot L_c \]

5.5 Testable Deviations and Predictive Structure

5.5.1 Modified Dispersion Relations

Discrete corrections yield: \[ E^2 \approx p^2 + m^2 + \delta E_{\text{discrete}}^2 \] from:

  • Nodal capacity saturation
  • Soliton deformation near capacity
  • Topological quantization

5.5.2 Stability Transitions

Observable near breakdown:

  • Deviations from \( E \propto M \)
  • Fluctuations in \( c_{\text{internal}} \)
  • Soliton fragmentation from excessive coupling

5.6 Logical Coherence and Non-Circular Foundations

All quantities derived internally from:

  • \( F_i(n) \)
  • \( \alpha(i,j,n) \)
  • \( \Psi_i(n) \)
  • \( C_i \)
No external assumptions or circular logic required.

5.7 Summary

The emergent notions of mass, energy, motion, causality, and space arise cohesively from asymmetric temporal flow dynamics on a discrete network substrate. Dimensional consistency is maintained by applying physical scales — energy \( E_c \), length \( L_c \), and time \( T_c \) — only when mapping internal quantities to observables.

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