Causal Flow and CPT Inversion: A Geometric Foundation for Quantum Phenomena

by John Gavel

Core Thesis

In Temporal Flow Physics (TFP), all physical phenomena—including quantum behavior, entanglement, CPT symmetry, and spacetime geometry—emerge from the dynamic interactions of quantized, one-dimensional temporal flows. These flows are fundamental entities; no additional quantum formalism is required. The complete structure of physical reality arises from the geometric, statistical, and causal relations between these temporal flows, as defined by the core principles of TFP.

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I. Metric Emergence from Flow Correlations and Relations

Principle 1 asserts that time and temporal flows are fundamental: the universe consists of discrete, complex-valued 1D temporal flows $F_i(t)$, each associated with a node $i$.
Principle 2 states that space and geometry emerge from comparisons between flows: spatial relations and metric structure are not primitive but arise from discrete differences

$$\Delta F_{ij}(t) = F_j(t) - F_i(t)$$

and from gradients of the entropy field $S_i(t)$, associated with flow misalignments.

To bridge the discrete flow network to continuous spacetime, we apply a coarse-graining or embedding process. Define a coarse-grained field $F(x,t)$ that represents the local average of nearby flows. The fluctuations around this average are:

$$\delta F(x,t) = F(x,t) - \bar{F}(x,t)$$

where $\bar{F}(x,t)$ is the local background flow.

We define the emergent metric as the expectation value of gradient correlations in these fluctuations:

$$g_{\mu\nu}(x) = \langle \partial_\mu \delta F(x) \, \partial_\nu \delta F(x) \rangle$$

This symmetric, positive semi-definite rank-2 tensor arises directly from local flow dynamics and defines the metric structure of emergent spacetime.

Proof Sketch

• If $\delta F(x)$ transforms as a scalar under coordinate transformations, $\partial_\mu \delta F$ transforms as a covector.

• The product $\partial_\mu \delta F \otimes \partial_\nu \delta F$ thus defines a second-rank tensor.

• Taking the expectation value preserves covariance:

$$\langle \partial_\mu \delta F \, \partial_\nu \delta F \rangle \rightarrow \frac{\partial x^{\mu'}}{\partial x^\mu} \frac{\partial x^{\nu'}}{\partial x^\nu} \langle \partial_{\mu'} \delta F \, \partial_{\nu'} \delta F \rangle$$

Thus, $g_{\mu\nu}(x)$ is a legitimate emergent metric.

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II. Flow Norm and Emergent Lorentz Invariance

With the emergent metric $g_{\mu\nu}$, we define the flow norm:

$$\|F\|^2 = g_{\mu\nu} \, \partial_\mu F \, \partial_\nu F$$

This scalar quantity is invariant under coordinate transformations and serves as a natural kinetic term for $F$. It encodes the "geometric inertia" of the flow. The Lorentzian structure of the metric, and hence local Lorentz invariance, emerges as a consequence of the underlying flow symmetry.

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III. CPT Symmetry as Flow Inversion at Causal Saturation

Principle 5 posits that CPT symmetry emerges from discrete inversion symmetries of the fundamental flows, expressed as:

$$F_i(t) \rightarrow -F_i(-t)$$

This inversion is triggered at the causal saturation point—the limit at which flow rate reaches its maximal allowed value, as described by Principle 3:

$$|\partial_t F_i| \leq c_t$$

In the continuum limit, causal saturation is reached when:

$$\|\partial_\mu F(x)\| \rightarrow c$$

At this boundary, a CPT inversion is activated:

$$F(x) \rightarrow -F(-x), \quad x^\mu \rightarrow -x^\mu, \quad t \rightarrow -t$$

Since $g_{\mu\nu}(x)$ is built from symmetric products of derivatives (odd under $x^\mu \rightarrow -x^\mu$), the metric is even under CPT inversion:

$$g_{\mu\nu}(x) = g_{\mu\nu}(-x)$$

Thus, CPT symmetry acts as a geometric boundary reflection, preserving action and causal structure across the inversion surface.

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IV. Entanglement from Flow Proximity in Configuration Space

In TFP, entanglement arises from flow adjacency in configuration space, not from abstract Hilbert space correlations.

Given two localized flows $F_1$ and $F_2$, define their geodesic separation in the emergent configuration manifold:

$$d(F_1, F_2) = \inf_\gamma \int_\gamma \sqrt{g_{\mu\nu} \, \frac{dF^\mu}{d\lambda} \, \frac{dF^\nu}{d\lambda}} \, d\lambda$$

Postulate: Two flows are entangled if and only if their geodesic distance satisfies

$$d(F_1, F_2) \leq \ell_P$$

where $\ell_P$ is the Planck length. Entanglement is thus a geometric adjacency relation in flow space.

Justification

Flows that originate from a shared causal event and undergo synchronized CPT inversion satisfy:

$$F_1(x) = -F_2(-x)$$

This symmetry ensures perfect correlation. The non-local interaction terms in the TFP action (e.g. $\sum_{i,j} W_{ij} \cdot I_{ij}$) maintain this alignment dynamically.

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V. Decoherence as Metric Deformation

Let two entangled flows $(F_1,F_2)\; evolve\; within\; a\; shared\; geometry$$g_{\mu\nu}^{\text{ent}}$. If an external perturbation induces a local deformation:

$$\delta g_{\mu\nu}(x) = g_{\mu\nu}^{\text{env}}(x) - g_{\mu\nu}^{\text{ent}}(x)$$

then coherence is lost when this deformation exceeds a critical threshold:

$$|\delta g_{\mu\nu}(x)| > \epsilon_c$$

Here, $\epsilon_c$ is the decoherence threshold. Entanglement in TFP is thus lost when metric compatibility between flow paths is broken.

Interpretation: Decoherence is not probabilistic collapse but a geometric phase misalignment—the loss of adjacency in the configuration space of flows, consistent with Principle 4 and the segmentation dynamics of the entropy field.

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VI. Causal Flow Bound and CPT Trigger (Recap)

The Causal Flow Limit from Principle 3 becomes, in proper time:

$$\frac{dF}{d\tau} = u^\mu \partial_\mu F \rightarrow c$$

with $u^\mu = dx^\mu/d\tau$. When this condition is saturated:

$$\frac{dF}{d\tau} = c \quad \Rightarrow \quad \text{CPT inversion event}$$

This CPT reflection regulates divergences and prevents classical singularities, acting as a Planck-scale geometric regulator.

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VII. Conserved Flow Current Across CPT Boundaries

For a complex coarse-grained flow field $F(x)$, define the conserved current:

$$J^\mu = i(F^* \partial_\mu F - F \partial_\mu F^*)$$

Under smooth CPT boundary conditions, this current satisfies:

$$\nabla_\mu J^\mu = 0$$

This conservation ensures that flow amplitude and phase remain continuous across inversion boundaries, sustaining quantum correlations over mirrored domains.

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VIII. Singularities Regularized by CPT Inversion

In standard physics, singularities (e.g., $r \rightarrow 0$ in black holes) involve diverging gradients. In TFP, these are dynamically avoided via CPT inversion:

$$F \rightarrow -F, \quad r \rightarrow -r$$

This inverts the flow curvature and keeps it finite:

$$\|\nabla \delta F\|^2 = g_{\mu\nu} \, \partial_\mu \delta F \, \partial_\nu \delta F \rightarrow \text{finite}$$

Hence, geodesics remain complete, and the would-be singularity is replaced by a smooth inversion surface, preserving causal continuity.

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Conclusion

Temporal Flow Physics recasts spacetime, quantum mechanics, and gravitational singularities as emergent phenomena rooted in the causal and statistical structure of 1D temporal flows. The entire apparatus of modern physics—metric geometry, CPT invariance, quantum entanglement, and black hole regularity—arises without added postulates, from a unified flow-based ontology. The causal organization of flow is not merely a feature of the universe—it is the universe.