From Discrete to Continuous: How TFP Bridges the Gap

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From Discrete to Continuous: How TFP Bridges the Gap A blog post explaining how Temporal Flow Physics reveals continuity as emergent from discrete relational processes The False Dichotomy That Has Haunted Physics For decades, physics has been caught in what seems like an irreconcilable tension: quantum mechanics suggests reality is fundamentally discrete and probabilistic, while general relativity describes smooth, continuous spacetime. String theory, loop quantum gravity, and other approaches have tried to resolve this by choosing one side or the other. But what if this isn't actually a choice we need to make? What if continuous spacetime and discrete quantum processes are two perspectives on the same underlying reality? Through my work on Temporal Flow Physics (TFP), I've discovered that continuity doesn't compete with discreteness — it emerges from it . Let me show you how this works, both logically and mathematically. The Key Insight: Continuity as Cohere...
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Emergent Space from Temporal Flow: Gauge Invariance and Coherence in Action

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Emergent Space from Temporal Flow: Gauge Invariance and Coherence in Action

By John Gavel
Note: I use AI-assisted tools to help write and organize my research.

Introduction

What if space isn’t fundamental?
In Temporal Flow Physics (TFP), I explore a radical proposition: space and geometry don’t come first—they emerge. Beneath them lies a discrete network of one-dimensional temporal flows evolving causally. These flows don’t move in space; instead, space is what forms when flows align, synchronize, and sustain structure over time.

This post shares a major theoretical breakthrough and new simulation results demonstrating a key feature of my model: gauge invariance. Despite varying internal settings, the coherent patterns that build space remain unchanged—mirroring the kind of invariance we expect from fundamental physical symmetries.

Core Dynamics: Causal Flows on a Discrete Network

Each node i in the network carries a scalar temporal flow F_i(t) evolving in discrete time steps. The flow’s update depends on four primary forces:

  1. Potential Force:
    Pulls each node toward a preferred value F_A, via a stabilizing potential.
    Example: V(F) = α × (F_i − F_A)²

  2. Continuity Force:
    Aligns each node with the average of its causal neighbors j in N(i).

  3. Asymmetry Force (Causal Frustration):
    Applies restoring pressure when a node’s causal exchange capacity is underutilized, driving it back toward the attractor.

  4. Inter-grid Coupling:
    Ensures phase coherence between grids of differing dimensionality or causal depth. Links across these grids enforce alignment without assuming a shared background space.

Unified Dynamics Equation

The complete evolution for each node i is:

dF_i/dt =
  − α × (F_i − F_A)
  + β × (average over neighbors F_j − F_i)
  − γ × Frustration_i × (F_i − F_A)
  + κ × (average over inter-grid partners F_j − F_i)

Where:

  • F_A: attractor value (gauge offset)

  • Frustration_i = 1 − min(exchange_budget, available_neighbors) / available_neighbors

  • Parameters α, β, γ, κ tune attraction, continuity, asymmetry, and cross-grid coupling

Gauge Invariance: Why the Attractor Doesn’t Matter

Suppose we define a shifted field: G_i(t) = F_i(t) − F_A.
Substituting into the dynamics, all explicit dependence on F_A disappears. The updated evolution becomes:

dG_i/dt =
  − α × G_i
  + β × (average over neighbors G_j − G_i)
  − γ × Frustration_i × G_i
  + κ × (average over inter-grid partners G_j − G_i)

This proves that the physical behavior of the system—coherence, tension, emergent geometry—remains unchanged under global shifts of F_A. That’s gauge invariance in action.

Observable Quantities: Coherence and Frustration

Two key observables define the system’s emergent behavior:

  • Intra-system Coherence:
    Measures uniformity within each grid.
    Defined as: 1 / standard deviation of neighbor flows

  • Inter-system Coherence:
    Measures how well flows align between grids (e.g., 2D vs 3D).
    Defined as: 1 / average absolute difference across inter-grid links

Both metrics are gauge invariant—they depend only on relative values, not on absolute field magnitude.

Frustration is a persistent property of each grid, measuring internal mismatch between causal velocity and connection density. It behaves like curvature: an intrinsic resistance to uniform alignment.

Simulation Results: Emergent Coherence and Dimensional Effects

I tested the system with two configurations:

  • Scenario 1: F_A = 0.0

  • Scenario 2: F_A = 0.5

Each simulation used coupled lattices:

  • One 2D grid (20×20)

  • One 3D grid (10×10×10)

  • 100 inter-grid links between them

  • Variable causal exchange budgets to simulate causal tension

Results:

  • Both simulations converged to high coherence within and across grids—independent of F_A

  • All inter-grid couplings stabilized, confirming emergent stitching of space-like structure

  • Frustration remained constant per grid, revealing each grid’s intrinsic causal properties

  • The 3D grid consistently reached higher coherence than the 2D grid, supporting the notion that dimensionality emerges from deeper recursion and oscillatory stability

Interpretation: Space from Flow, Not the Other Way Around

These results demonstrate that space is not a background stage. Instead, it forms when discrete causal flows stabilize into coherent, phase-locked patterns. The dynamics are self-contained: no pre-built space, no imposed metric—just evolution, tension, and correlation.

Gauge invariance shows us something deeper: what matters is not what value a field takes, but how it interacts. Shifting the background doesn’t change the structure. This echoes foundational principles of modern physics—and here, it arises from scratch in a temporal network.

Next Steps and Extensions

This simulation validates key TFP principles. Upcoming work will extend these results to full multiplet flows (SU(2), SU(3)), explore quantized oscillation spectra in high-curvature regions, and develop real-time visualizations of causal stitching.


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