3. Emergent Spatial Distance and Temporal Neutrality

Section 3: Emergent Spatial Distance and Temporal Neutrality

3.1 Postulate: Space Emerges from Relational Flow Misalignment

In Temporal Flow Physics (TFP), space is not fundamental. It emerges from relative differences between discrete, bidirectional temporal flows \(F_i(t)\) defined on lattice sites \(i\).

Each flow \(F_i(t)\) is a local, dimensionless oscillator. There is no built-in direction of time; space and time both emerge from coherence patterns and causal exchange between flows.

3.2 The Temporal Index \(t\) Is Not Time

The index \(t\) is a discrete update label, not physical time. It indexes steps in causal evolution but has no intrinsic temporal meaning.

Physical time arises from patterns in flow synchronization, not from \(t\) itself (see Section 7).

3.3 Temporal Neutrality of the Substrate

3.3.1 Reversible Evolution Rules

Flows evolve through time-symmetric update rules:

\[ \text{Forward evolution:} \quad F_i(t + \Delta t) = F_i(t) + O(u_i) \] \[ \text{Backward evolution:} \quad F_i(t - \Delta t) = F_i(t) - O(u_i) \]

These rules are invariant under \(t \to -t\) when flows are isolated or not causally coupled.

3.3.2 No Global Arrow of Time

The substrate is temporally neutral. Only when flows become causally entangled or saturated with exchanges does local time direction emerge.

Analogy: Think of a lattice of clocks that can tick forward or backward. Only when they begin interacting do consistent ticking directions (causal arrows) appear.

3.4 Emergence of Local Time Directionality

When flows begin exchanging values—due to misalignment forces, continuity coupling, or potential energy—they form local patterns of synchronization and directionality.

This interaction breaks the time symmetry locally, giving rise to a preferred temporal direction along causal chains, while the rest of the substrate remains neutral.

3.5 Spatial Distance from Flow Misalignment

Spatial distance is not defined a priori. It emerges from differences between neighboring flows. A minimal measure is:

\[ |x_i - x_j|^2 = \ell_P^2 \times (F_i - F_j)^2 \]

Here, \(\ell_P\) is the Planck length.

This expression is dimensionally consistent: length squared equals length squared times (dimensionless)\(^2\).

It respects bidirectional symmetry: it’s invariant under \(F_i \to -F_i\).

Once flows are causally coupled, this measure acquires directional structure.

3.6 Dimensionality Is Emergent, Not Fixed

Recent simulations show:

• Spatial dimension (e.g., 1D vs 3D) is not fundamental.
• It emerges statistically from how flows organize into coherent solitons.
• When three flows coordinate their phases and amplitudes (as in triplet balancing), local 3D-like behavior emerges.
• In other regions, simpler 1D or 2D dynamics dominate.

Dimensionality becomes a local function of:

• Soliton saturation
• Coherence density
• Causal frustration
• Flow exchange capacity

Geometry, including curvature and confinement, emerges from local flow structure, not from a fixed background.

3.7 Figure: From Flow Symmetry to Emergent Space

Panel A: Uncoupled flows freely oscillating
Caption: "Temporally symmetric substrate: flows evolve bidirectionally."

Panel B: Causally coupled flows forming directional coherence solitons
Caption: "Emergent local time direction and spatial separation from interaction."

Panel C: Map of coherence density or soliton dimension
Caption: "Regions with high triplet locking show 3D structure; others remain 1D or 2D."

3.8 Dimensional Consistency

All distance and space-related quantities are defined using Planck units. The causal index \(t\) is dimensionless.

Physical time is recovered from:

\[ t_{\text{phys}} = t \times \Delta t_P \]

Where \(\Delta t_P\) is the Planck time (time per update step).

3.9 Summary and Implications

Space and time are emergent, not built-in. They arise from interacting flows.

The substrate is symmetric in time and dimension; direction and geometry emerge from interactions.

Effective dimension varies by region, depending on soliton structure and exchange saturation.

This supports curvature, mass-dependence of space, and localized time asymmetry (see Sections 4, 9, and 13).

3.10 Local Emergence of Time Directionality

Temporal Flow Physics begins with a substrate evolving through fundamentally time-symmetric, reversible update rules. In isolated nodes \(i\), the flows obey symmetric forward and backward discrete dynamics, expressed as: \(F_i\) at time \(t + \Delta t\) equals \(F_i\) at time \(t\) plus a function of the local flow rate \(u_i\); similarly, \(F_i\) at time \(t - \Delta t\) equals \(F_i\) at time \(t\) minus that function. The local flow rate \(u_i\) at time \(t\) is defined as the difference quotient \(\frac{F_i(t + \Delta t) - F_i(t)}{\Delta t}\), which has units of inverse time. These rules remain invariant under time reversal, \(t \mapsto -t\), in the absence of interactions.

However, local interactions between flows break this perfect symmetry by generating irreversible behavior and causal directionality within regions of coupled flows. To characterize this local time asymmetry, we define the local time asymmetry measure \(A_i(t)\) as the discrete second temporal difference:

\[ A_i(t) = F_i(t + \Delta t) - 2 F_i(t) + F_i(t - \Delta t) \]

This quantity \(A_i(t)\) is dimensionless, consistent with the dimensionless nature of the flow variables \(F_i(t)\). When \(A_i(t) \approx 0\), the evolution at node \(i\) is locally time-symmetric, indicating reversible, non-directional flow dynamics. Conversely, when \(|A_i(t)|\) becomes significantly different from zero, it signals a local breaking of time-reversal symmetry, corresponding to an emergent preferred direction of causal evolution. Physically, \(A_i(t)\) captures the curvature or acceleration of the flow in discrete time and acts as an indicator of local causal arrow formation.

The asymmetry arises naturally from flow coupling—interactions encoded in discrete evolution equations that incorporate the flows of neighboring nodes—yielding irreversible dynamics locally. Additionally, soliton saturation in stable, phase-locked flow structures restricts reversibility by locking internal phases and amplitudes. Nonlinear potentials acting on the flows enforce preferred configurations and bias forward evolution.

The discrete evolution equation governing flow dynamics can be written as:

\[ \frac{k}{\Delta t^2} \left( F_i(t + \Delta t) - 2 F_i(t) + F_i(t - \Delta t) \right) - \frac{\lambda k}{\Delta t^2} \sum_{j \in N(i)} \left( F_i(t) - F_j(t) \right) - \frac{\partial V}{\partial F_i} = 0 \]

This shows that the local acceleration \(A_i(t)\) is directly linked to spatial coupling and potential gradients. These terms break time-reversal invariance locally while maintaining global temporal neutrality when averaged over the entire network.

Importantly, the substrate remains globally temporally neutral because isolated or weakly coupled flows retain near-zero \(A_i(t)\), while time asymmetry arises only in regions of strong causal entanglement and flow soliton coherence. This reconciles the seemingly conflicting descriptions: Section 3’s reversible, temporally neutral substrate forms the foundation of evolution, whereas Section 5’s causal speed limits and directional flow gradients describe emergent local time arrows resulting from flow interactions. Thus, time directionality is not fundamental but an emergent, local property of interacting temporal flows, captured quantitatively by nonzero values of \(A_i(t)\).

In summary, the quantity

\[ A_i(t) = F_i(t + \Delta t) - 2 F_i(t) + F_i(t - \Delta t) \]

quantifies local time asymmetry. When this is near zero, the evolution is reversible and temporally neutral; when significantly nonzero, it indicates emergent causal directionality arising from flow interactions. This formalism provides a dimensionally consistent and physically meaningful bridge between time-symmetric discrete flow evolution and the causal time observed macroscopically.