Second-Order Relationality and the Emergence of Orientation in PNP
2026-01-20
One-Sentence Summary. Spatial orientation and the concepts of “in” and “out” are not fundamental primitives, but emerge as phase-dependent projections of the self-referential scalar field mode (1).
Abstract. We develop a second‐order relational description of the Point–Not–Point (PNP) scalar‐field framework, showing how “in” and “out” —along with orientation, direction, and spatial geometry— emerge from the self‐referential phase structure of a single real scalar field \(U(x,t)\). The minimal closed mode, denoted (1), exhibits a Möbius‐like phase inversion across its nodal surface, sustaining continuous energy circulation without requiring a background geometric twist. This work complements the dynamical theorems of PNP by providing the conceptual formulation of how a scalar field constructs spatial orientation.
Keywords. PNP Framework, Scalar Field Recursion, Emergent Geometry, Mobius Phase Topology, Relational Space
In standard physics, space is treated as a container and orientation as a primitive. In the Point–Not–Point (PNP) framework, neither is fundamental: the only ontic entity is a scalar energy field \(U:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}\). Observable structure arises from the closed oscillations of \(U\), with apparent directions and “in–out” relations emerging from nodal phase behavior.
Building on the derivation of causality from topological persistence [1], we here show how the minimal \((1)\) mode defines a self‐referential energy flow that reverses orientation across a node without spatial inversion, grounding spatial concepts in scalar recursion.
The field dynamics are governed by the recursive definitions:
\[ F = d(*dU), \quad dF=0, \quad d*F=0 \]
from which electric‐ and magnetic‐like fields follow:
\[ \mathbf{B} = *\,dU, \quad \mathbf{E} = *\,d*\,dU \]
These satisfy the source‐free Maxwell equations. In PNP, however, vectors are not primary: they are projections of the scalar’s own oscillatory recursion.
Note: The Hodge dual (\(*\)) is used here as a relational operator on the field gradients, not as a rigid structure dependent on a pre-existing metric background.
We define the minimal spherical standing wave (referring to the symmetry of the nodal set, rather than a fundamental embedding space):
\[ U(r,t) = A\sin(k r - \omega t), \quad U(0,t) = U(R,t) = 0 \]
The boundary condition gives \(k R = \pi\). The field flows inward, cancels at \(r=0\), and reemerges outward with opposite phase.
Let the effective orientation vector be:
\[ \hat{n}(r) = \frac{\nabla U}{|\nabla U|} \]
Then, examining the limit across the node:
\[ \lim_{r\to 0^-} \hat{n} = -\lim_{r\to 0^+} \hat{n} \]
This inversion is continuous in phase space (\(e^{i\pi} = -1\)) but appears as a reversal in vector space. This is a Möbius‐like effect in the field’s orientation: the “inside” transforms continuously into the “outside” through a phase twist, creating a non-orientable topology from a simple scalar oscillation.
PNP’s relationality is two‐tiered:
“In” and “out” are thus not absolute directions but phase‐dependent projections. Space itself is the stable pattern of these relations.
The minimal \((1)\) mode in PNP provides a self‐referential energy flow that defines “in” and “out” without presupposing space or orientation. This complements the formal derivation of PNP’s dynamics, offering a compact conceptual lens for interpreting the framework’s physical and philosophical reach.