Causality from Topological Persistence in Scalar Fields
2026-01-20
One-Sentence Summary. Causality emerges in the PNP framework because a topologically non-trivial scalar field configuration cannot remain static without violating stress–energy conservation, forcing ordered evolution.
Abstract. We derive causality from first principles within the Point–Not–Point (PNP) framework. At its core lies the topological irreducibility of the fundamental \((1)\) mode: the simplest closed oscillation of a scalar field \(U\) exhibiting a \(\pi\) phase inversion, or “bounce.” This \(\mathbb{Z}_2\) invariant enforces loop persistence and forbids extinction without a phase slip. We explicitly ground this mode in the discrete solution space of source-free Maxwell dynamics (the toroidal hydrogenic spectrum). From this physically motivated topology, we prove that such a mode cannot remain static, formalizing cause–effect not as a postulate, but as the inevitable action of the field propagator on a persistent topological sector.
Keywords. PNP Framework, Topological Persistence, Causal Geometry, Scalar Field Theory, Z2 Invariant, Emergent Time
In standard formulations of physics, causality is assumed as a primitive ordering of events—time exists, and things move through it. In the Point–Not–Point (PNP) framework, we invert this relationship. We propose that causality emerges from Topology: specifically, from the requirement that a non-trivial field configuration must evolve to maintain its structural integrity.
We show that a minimally nontrivial loop of the scalar field \(U\) (the fundamental “Entity”) persists under evolution. We prove that such a mode cannot remain static without violating local momentum conservation. Time here is not assumed as a background ordering, but emerges as the parameter labeling successive configurations required to preserve topology. Thus, cause–effect is the temporal manifestation of topological persistence.
Let \(U: \mathcal{M} \to \mathbb{R}\) be a real scalar energy field. We define the complex envelope \(A(\mathbf{x},t)\) as a local phase–amplitude decomposition of the oscillatory solutions of \(U\):
\[ A(\mathbf{x},t) = \rho(\mathbf{x},t)\,e^{i\phi(\mathbf{x},t)}, \qquad \rho \ge 0, \ \phi \in \mathbb{R} \pmod{2\pi} \]
Note: The complex envelope \(A\) is a bookkeeping device for local oscillatory structure in a real scalar field; no additional \(U(1)\) degrees of freedom are introduced.
The topological object central to this theory—the \((1)\) mode—is not an arbitrary mathematical postulate. It is the abstraction of the fundamental standing-wave solution to Maxwell’s equations on a toroidal manifold.
As demonstrated in the derivation of the Schrödinger equation from source-free electromagnetism [1], the imposition of single-valuedness on a toroidal field configuration yields a discrete spectrum of modes labeled by integer winding numbers \((m, n)\). For the symmetric case (\(m = n\)), the energy of these modes scales as \(E \propto 1/n^2\), reproducing the Rydberg series characteristic of bound atomic states (Hydrogen) without invoking point charges.
The \((1)\) mode corresponds to the ground state (\(n=1\)) of this physical hierarchy. It represents the “simplest knot” compatible with the wave equation—a closed loop of energy with a \(\pi\) phase twist. While higher \(n\) modes describe excited states, the \((1)\) mode represents the irreducible topological obstruction that defines the entity’s existence. By focusing on the \((1)\) mode, we are not inventing a shape; we are analyzing the topological properties of the most fundamental stable structure allowed by classical field dynamics.
We denote topological sectors by \((n)\) with \(n \in \mathbb{N}\), representing the winding number of the phase around the core.
The \((1)\) mode is defined geometrically as a closed loop \(C\) encircling a core such that one traversal advances the phase \(\phi\) by \(\pi\) (a Möbius-like twist). This requires two traversals to return to the initial state.
The holonomy along \(C\) is:
\[ H(C) = \exp\!\left(i\oint_C \nabla\phi \cdot d\mathbf{l}\right) \in \{+1, -1\} \]
This defines the discrete \(\mathbb{Z}_2\) index \(\nu\) (Parity):
\[ \nu = \frac{1 - H(C)}{2} = n \pmod 2 \]
Physically, the \((1)\) mode traps the essence of a “continuous bounce.” The field flows through the core, inverts phase, and re-emerges, effectively reflecting off its own nodal structure without ever encountering a hard boundary; a self-referential flow.
Crucially, \(\nu\) is a topological invariant. It cannot change continuously; it can only change via a Phase Slip (where \(\rho \to 0\) at a point on \(C\)), effectively breaking the loop.
The source-free PNP equation of motion is given by the vanishing of the exterior derivative of the dual:
\[ d(\star dU) = 0 \]
With a Lagrangian density \(\mathcal{L}(U, \nabla U)\), the stress–energy tensor is:
\[ T_{\mu\nu} = \nabla_\mu U\,\nabla_\nu U - g_{\mu\nu}\,\mathcal{L}, \qquad \nabla_\mu T^{\mu\nu} = 0 \]
Note: No specific form of \(\mathcal{L}\) is required for this argument beyond locality, positivity of energy density, and the existence of a conserved stress–energy tensor.
We define the Energy Density (\(u\)) and Flux (\(J^\mu\)) relative to a local time vector \(t^\nu\):
\[ u = T^{00}, \qquad J^\mu = T^\mu{}_\nu\,t^\nu \]
We now prove that “Time” is the byproduct of the \((1)\) mode’s necessary self-perpetuation.
The configuration space decomposes into disjoint sectors labeled by \(\nu\). The evolution generated by \(d(\star dU)=0\) preserves sector labels except at singularities (Phase Slips). Therefore, a persistent entity satisfies:
\[ \nu(t+\Delta t) = \nu(t) = 1 \]
Assume, for the sake of contradiction, that the field is static: \(\Phi(t+\Delta t) = \Phi(t)\) for all \(t\). This implies \(\partial_t U = 0\) everywhere on the loop, which means the momentum flux density (energy flow) \(T^{0i}\) must vanish.
However, for a loop with \(\pi\)-twist topology (the \((1)\) mode), the phase gradient \(\nabla \phi\) is non-zero and twisted. This implies nonzero spatial stress components (\(T^{ij} \neq 0\)). A static configuration with non-zero internal stress requires external support to maintain force balance (\(\partial_j T^{ij} \neq 0\) without flow).
In a source-free vacuum, no such external force exists. Therefore, a static \((1)\) mode violates local momentum balance. Topology alone does not generate motion; rather, the incompatibility between nontrivial topology and static force balance in a source-free field enforces evolution.
Conclusion: To maintain the \((1)\) mode (Persistence), the field cannot be static.
\[ \Phi(t+\Delta t) \neq \Phi(t) \]
Unlike conventional instabilities which depend on parameters, the instability of a static \((1)\) mode is topologically protected.
Let \(\mathcal{P}_{\Delta t}\) be the evolution operator. On the persistent sector:
\[ \Phi(t+\Delta t) = \mathcal{P}_{\Delta t}\,\Phi(t) \]
“Cause” is the state \(\Phi(t)\). “Effect” is the state \(\Phi(t+\Delta t)\). The link between them is not an axiom, but the Propagator of Topological Persistence. The effect is simply the next necessary configuration to prevent the loop from breaking.
We can extend this to interactions. From \(\nabla_\mu T^{\mu\nu}=0\) in a stationary, spherically symmetric flow:
\[ \partial_r T^{rr} + \frac{2}{r}\big(T^{rr} - T^{\theta}{}_{\theta}\big) = 0 \]
For tangentially dominated energy transport (a spinning torus), \(T^{rr} \approx -u(r)\). The induced radial acceleration on test configurations is:
\[ a_r(r) \propto -\partial_r T^{rr}(r) \approx \partial_r u(r) \]
For configurations whose energy density exhibits vortex-like decay (\(u(r) \sim 1/r\)), this yields \(a_r \sim -1/r^2\). Thus, this framework suggests a gravitation-like interaction arising from the Organization of Energy Flow, without the need to postulate intrinsic mass.
In the PNP framework, we do not need to postulate that “Time Flows” or “Gravity Attracts.”
Reality is a self-driving machine: it moves because it is topologically forbidden from standing still.