Section 1: First Principles and Foundations of Temporal Flow Physics (Revised)

Section 1 — First Principles and Foundations of Temporal Flow Physics

by John Gavel

1.1 Purpose and Framing (Constructive Program)

Temporal Flow Physics (TFP) is a constructive program aimed at deriving spacetime, fields, and matter-like structures from a minimal discrete dynamical substrate. The fundamental premise is that time and information flow are primary; geometric notions like metric, connection, or continuum fields emerge from patterns in the evolution of discrete complex variables on a network. This section states the core axioms, variables, canonical dynamics, and the construction protocol that avoids circular assumptions.

1.2 Ontology and Primitives (Axioms and Definitions)

Postulate 1 — Discrete Dynamical Substrate

The fundamental structure is a directed graph G = (V, E), where:

• V = {1, ..., N} is a finite set of nodes (events or informational units).
• E ⊆ V × V is a set of directed edges connecting nodes.
• Time is discrete: t ∈ ℕ (non-negative integers), with a fundamental unit time step Δt = 1.

Definition 1 — Complex Flow Variable

Each node i ∈ V at time t carries a single complex state variable F_i(t) ∈ ℂ, called the flow, expressed as:

$$ F_i(t) = A_i(t) \times \exp(i \times \varphi_i(t)) $$

where:

• A_i(t) = |F_i(t)| ≥ 0 is the amplitude,
• φ_i(t) ∈ [0, 2π) is the phase angle,
• i = \sqrt{-1} is the imaginary unit.

The flow variable is the minimal representation encoding amplitude and phase transport at each node.

Axiom A — Locality of Dynamics

For each node i, define the incoming neighbor set N(i) = {j | (j → i) ∈ E}. The update of F_i(t) depends only on the current states of nodes in N(i) and a fixed finite set of global parameters P. No external geometric or coordinate fields are assumed.

Formally, for each i:

$$ F_i(t + 1) = G_i(\{F_j(t) \mid j \in N(i)\}; P) $$

where G_i is the update map, possibly node-dependent but local.

Axiom B — Discrete Time and Finite Memory

The update map G_i depends only on the current time t (minimal choice), not on an unbounded past history.

Axiom C — Minimal Parameter Set

The global parameters P include at least:

• Coupling strength μ ∈ [0,1], controlling update weighting.
• Complex edge weights w_{ij} = r_{ij} \times \exp(i \times θ_{ij}) for edges j → i, where r_{ij} ≥ 0 modulates amplitude transmission and θ_{ij} encodes phase bias and temporal asymmetry.
• Optional nonlinear parameter α ∈ ℝ for self-phase modulation.
• Temporal asymmetry parameter γ_{asym}, encoded in θ_{ij} or dissipative choices of μ, r_{ij}.

1.3 Canonical Update Laws (Minimal Explicit Dynamics)

1.3.1 Linear Minimal Update

The simplest local update for node i is a weighted average of its own current state and the states of its incoming neighbors:

$$ F_i(t + 1) = (1 - \mu) \times F_i(t) + \frac{\mu}{|N(i)|} \times \sum_{j \in N(i)} \left[w_{ij} \times F_j(t)\right] $$

where |N(i)| is the number of incoming neighbors of node i.

1.3.2 Nonlinear Coherence Coupling Update

Incorporating nonlinear self-phase modulation:

$$ F_i(t + 1) = (1 - \mu) \times F_i(t) + \frac{\mu}{|N(i)|} \times \sum_{j \in N(i)} \left[w_{ij} \times F_j(t) \times \exp\left(i \times \alpha \times |F_j(t)|^2\right)\right] $$

This allows amplitude-dependent phase evolution, supporting richer dynamical coherence.

1.4 Operational Observables (Derived from Flow Data)

Without assuming geometry, we define operational quantities from the time series {F_i(t)} and the fixed graph topology.

1.4.1 Time-Averaged Correlations

For a large sampling time T, define the lag-τ correlation between nodes i and j:

$$ C_{ij}(\tau) = \frac{1}{T} \sum_{t=0}^{T-1} \left[F_i(t) \times \overline{F_j(t + \tau)}\right] $$

This complex function captures amplitude and phase correlations at time lag τ.

1.4.2 Normalized Transport Estimator

Estimate directional influence from j to i at lag τ by normalizing correlations:

$$ T_{j \to i}(\tau) = \frac{C_{ij}(\tau)}{\sqrt{C_{ii}(0) \times C_{jj}(0)}} $$

The lag τ^* maximizing |T_{j \to i}(\tau)| is chosen as optimal, or τ^* = 1 is used as a default.

1.4.3 Edge Phase — Discrete Connection

The operational phase associated with edge j \to i is:

$$ a_{j \to i} = \arg\left(T_{j \to i}(\tau^*)\right) $$

This is a discrete precursor to a gauge connection encoding phase transport bias.

1.4.4 Closed-Loop Holonomy

For any simple directed cycle C = (v_0 \to v_1 \to \cdots \to v_{m-1} \to v_0), the holonomy is the net phase accumulated:

$$ H_C = \arg\left(\prod_{k=0}^{m-1} T_{v_k \to v_{k+1}}(\tau^*)\right) $$

This quantifies discrete curvature-like effects in the network.

1.5 Derived Control Quantities (Network-Level Metrics)

Informational Friction δ

Linearize the update map near steady state, obtaining Jacobian matrix J. Let ρ(J) be its spectral radius (max eigenvalue magnitude). Then:

$$ \delta = 1 - \rho(J) $$

This measures decay rate of perturbations, controlling system responsiveness and coherence stability.

Topology Factor TF

The topology factor measures the abundance of closed loops in the network, essential for emergent curvature and coherence phenomena.

Formally, it is defined as a weighted sum over cycle lengths from 3 up to a maximum length L_{max}:

$$ TF = \sum_{L=3}^{L_{max}} c_L \times \mathrm{trace}(A^L) $$

where:

• A is the adjacency matrix of the directed graph,
• trace(A^L) counts the total number of closed walks of length L (including cycles),
• c_L are weighting coefficients to emphasize or de-emphasize cycles of different lengths.

This factor captures how the presence and distribution of loops influence the network’s capacity to support geometric structures and stable flow coherence.

Correlation-Derived Distance

Define an operational distance between nodes i and j as:

$$ d_{ij} = -\log\left(|T_{i \leftrightarrow j}|\right) $$

where T_{i \leftrightarrow j} is the symmetric combination of T_{i \to j} and T_{j \to i}.

This distance supports embedding local patches in an emergent geometric space.

1.6 Emergence of Geometry and Dynamics

Postulate 3 — Flow-Defined Metric

Local coherent alignment of flows defines an emergent metric tensor:

$$ g_{\mu\nu} = \eta_{\mu\nu} + \epsilon_{\mu\nu}(F) $$

where \eta_{\mu\nu} is the Minkowski background metric, and \epsilon_{\mu\nu}(F) is a symmetric tensor perturbation induced by local flow patterns derived from correlations and phase alignments.

Postulate 4 — Flow Conservation

Total flow density \rho_F is conserved within closed regions of the discrete substrate, analogous to a discrete Gauss's law.

Curvature and Einstein Equation Emergence

From g_{\mu\nu}, compute Levi-Civita connection \Gamma^\lambda_{\mu\nu}, Ricci tensor R_{\mu\nu}, and scalar curvature R using standard geometric formulas adapted to the discrete setting.

Curvature arises from flow divergence and holonomies around cycles.

Flow energy-momentum conservation links curvature to stress-energy T_{\mu\nu} derived from flow densities and fluxes.

In a continuum limit, this yields the Einstein field equation:

$$ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8 \pi G_{eff}}{c^4} \times T_{\mu\nu} $$

where G_{eff} depends on microscopic parameters \gamma_{asym}, \delta, and TF.

1.7 Non-Circular Construction Protocol

To avoid assuming geometry a priori, the construction proceeds strictly:

1. Specify graph G, parameters P, and initial flow states F_i(0).
2. Evolve dynamics collecting {F_i(t)} over time.
3. Compute correlation C_{ij}(\tau) and transport T_{j \to i}(\tau).
4. Extract edge phases a_{j \to i} and holonomies H_C.
5. Define correlation distances d_{ij} and embed patches geometrically.
6. Construct metric g_{\mu\nu} and curvature quantities from embeddings and holonomies.

1.8 Worked Example: 3-Node Directed Triangle

Consider nodes V = {1, 2, 3} with edges 1 \to 2, 2 \to 3, 3 \to 1.

Parameters: \mu = 0.3; complex weights w_{ij} have specific phases.

Initial states:

• F_1(0) = 1.0
• F_2(0) = 0.5 \times \exp(i \times \pi/4)
• F_3(0) = 0.8 \times \exp(i \times \pi/2)

One update step using linear rule:

$$ F_1(1) = (1 - \mu) F_1(0) + \mu w_{3 \to 1} F_3(0) $$

$$ F_2(1) = (1 - \mu) F_2(0) + \mu w_{1 \to 2} F_1(0) $$

$$ F_3(1) = (1 - \mu) F_3(0) + \mu w_{2 \to 3} F_2(0) $$

This demonstrates local flow evolution incorporating phase biases, seeding complex network behavior from simple rules.