Section 6: Forces as Emergent Flow Couplings and Interactions

Section 6: Forces as Emergent Flow Couplings and Interactions — Defining Dimensions

6.0 Preliminaries on Physical Units

Physical units in this framework arise exclusively through mapping from underlying dimensionless network quantities using characteristic physical scales introduced in Section 1. These characteristic scales are:

• Characteristic length, denoted \( L_c \), with units of length [L].
• Characteristic time, denoted \( T_c \), with units of time [T].
• Characteristic energy, denoted \( E_c \), with units of mass times length squared divided by time squared \([M L^2 T^{-2}]\).
• Characteristic speed, \( c_{\text{char}} = \frac{L_c}{T_c} \), with units length per time \([L T^{-1}]\).
• Characteristic mass, \( M_c = E_c \times \frac{T_c^2}{L_c^2} \), units of mass \([M]\).
• Characteristic action, \( \hbar_c = E_c \times T_c \), units of mass times length squared divided by time \([M L^2 T^{-1}]\).

These scales provide the essential bridge between dimensionless internal variables and measurable physical quantities.

6.1 Fundamental Flow Quantities

The fundamental dynamical variable is the local flow \( F_i(n) \) at node \( i \) and discrete step \( n \), which is dimensionless by construction. This flow encodes both instantaneous magnitude and directional phase of information processing at the node.

Each node has an intrinsic processing capacity \( C_i \), also dimensionless, representing the maximum allowed number of flow exchanges per discrete time step. This capacity sets saturation thresholds critical for emergent mass formation and dynamic stability.

Reflection events at node \( i \) are modeled by a binary indicator \( R_i(n) \) taking values 0 or 1, marking bounce-back occurrences when saturation or capacity limits are reached. Persistent reflection patterns form the basis of inertial mass emergence.

6.2 Coherent Structures and Solitons

Emergent quasi-particles arise as coherent, localized flow structures denoted \( \Phi_a(n) \), dimensionless quantities representing soliton-like modes sustained by recursive, phase-aligned flow motifs.

Physically, these amplitudes map to \( \Phi_a \times \sqrt{E_c} \), giving units of the square root of mass times length divided by time, consistent with oscillatory amplitude in physical units.

The dimensionless coupling strength \( g \) governs interaction amplitudes between solitons.

The misalignment angle \( \theta_{ab}(n) \), dimensionless, encodes the relative phase or orientation difference between solitons \( a \) and \( b \). Variation in \( \theta_{ab}(n) \) generates emergent forces by inducing decoherence and energy penalties.

6.3 Interaction Geometry, Coupling, and Informational Friction

The internal causal signal speed \( c_{\text{internal}} \) is dimensionless (defined as unity in network units). When mapped to physical units, it corresponds to \( c_{\text{internal}} \times \frac{L_c}{T_c} \), thus carrying units of length per time.

The characteristic interaction range \( \lambda \) is dimensionless and physically corresponds to \( \lambda \times L_c \), specifying the spatial extent over which forces and couplings operate.

The vacuum expectation value \( v \) represents the coherent condensate-level flow amplitude, physically scaled as \( v \times \sqrt{E_c} \), bearing units of square root of mass times length over time.

Emergent mass parameters \( m \) are dimensionless within the network and map to physical mass by multiplication with \( M_c \).

6.3.2 Informational Friction ( \( \delta \) ): Formal Derivation and Multi-Scale Interpretation

Informational friction \( \delta \) quantifies the network’s intrinsic resistance to sustaining recursive phase coherence, thereby governing the damping rates of oscillatory feedback loops.

Starting with local flow entropy at node \( i \):

Define entropy \[ S_i = -\sum_k p_k(i) \log p_k(i) \] where \[ p_k(i) = \frac{\alpha(i,k)}{\sum_m \alpha(i,m)} \] with \( \alpha(i,k) \) representing flow magnitudes from node \( i \) to neighbor \( k \).

For each neighbor pair \( (i,j) \), define the discrete informational conflict \[ \xi(i,j) = (F_i - F_j)^2 \times (S_j - S_i) \] capturing mismatch between flow magnitudes and local entropy gradients.

The local magnitude-based friction at node \( i \) is computed as the average over neighbors \( j \) of \[ (F_i - F_j)^2 \times |S_j - S_i| \]

The local directionally-aware friction is similarly averaged but uses the signed difference \( S_j - S_i \), reflecting directional bias.

At a larger spatial scale \( l \), coarse-grained friction values \( \delta_{\text{abs}}(l) \) and \( \delta_{\text{dir}}(l) \) are defined as averages over nodes in the region.

While dimensionless, \( \delta \) acts as a fundamental parameter controlling damping of coherence amplitudes and limits on coherence length when combined with characteristic physical scales.

6.4 Topology Factor — Structural Origin and Observables

The topology factor at node \( i \) is a dimensionless scalar quantifying how richly looped and feedback-supportive the local network is, enhancing recursive coherence capabilities.

It can be modeled as proportional to the loop density at node \( i \) divided by node degree, multiplied by the square of the dominant eigenvalue of the adjacency matrix restricted to the neighborhood of node \( i \).

Higher topology factor values correspond to longer coherence lengths and more stable recursive processing domains.

6.5 Coupling Action and Energy Cost

The coupling action \( S_{\text{coupling}} \) is dimensionless within network units. When mapped to physical units, it corresponds to \[ S_{\text{coupling}} \times \hbar_c \] with physical units of mass times length squared divided by time.

This action quantifies the cost of coupling solitons, determining the strength and stability of their interactions.

The soliton potential energy \( V(\Phi) \) is dimensionless internally and physically scales as \[ V(\Phi) \times E_c \] carrying units of energy.

6.6 Flow Saturation, Misalignment Effects, and Emergent Mass

Misalignment between soliton phases imposes an energy penalty \( \Delta E \), dimensionless within the network, which maps to physical energy \[ \Delta E \times E_c \]

The emergent mass of force carriers, denoted \( m_{\text{carrier}} \), is dimensionless and scales to physical mass by multiplication with \( M_c \).

The interplay between informational friction \( \delta \) and the topology factor determines whether force carriers remain effectively massless or develop a mass gap, reflecting the stability and damping of coherence within the network.

6.7 Summary: Role of \( \delta \) and Topology Factor in Emergent Forces

Informational friction \( \delta \) arises fundamentally from local entropy gradients and turbulent mismatches in flow, quantifying resistance to maintaining recursive coherence.

Topology factor measures the local richness of loops and feedback capacity, supporting recursive flow structures.

Together, these dimensionless scalars underpin emergent force carriers, soliton stability, interaction coupling strength, and the formation of mass gaps. This forms a unified framework connecting discrete network flow dynamics directly to continuous physical interactions without requiring predefined Lagrangian formulations.

Integration with Manuscript

The parameters \( \delta \) and topology_factor were introduced in Sections 1 and 3 as central elements limiting flow coherence and enabling emergent geometric and physical structure.

Experimental and Theoretical Implications

Their scale-dependent behavior predicts testable phenomena such as varying coherence lengths and running coupling constants. These have relevance for condensed matter analogs and cosmological-scale observations.

Effective Coherence Depth

The maximum recursive coherence depth, \( N_{\text{rec}} \), is approximately given by the reciprocal of the product \[ N_{\text{rec}} \approx \frac{1}{\delta \times \text{topology_factor}} \] This quantity sets the effective coherence lifetime and spatial interaction range within the network substrate.