The In–Out Self-Referential Field Vibration

Second-Order Relationality and the Emergence of Orientation in PNP

Max Freet

Adrien Hale

An M. Rodriguez

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

Table of Contents

Introduction
Scalar Field Recursion
Minimal Mode and In–Out Reversal
Second‐Order Relationality
Implications
Conclusion
References

1 Introduction

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.

2 Scalar Field 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.

3 Minimal Mode and In–Out Reversal

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.

4 Second‐Order Relationality

PNP’s relationality is two‐tiered:

First‐order: Spatial relations arise from field phase gradients (Distance).
Second‐order: Those gradients are themselves defined by other relations—internal phase continuity across nodes (Orientation).

“In” and “out” are thus not absolute directions but phase‐dependent projections. Space itself is the stable pattern of these relations.

5 Implications

Emergent Orientation: Orientation is locally reversible and defined only via field phase.
Relational Descriptors: “In” and “out” are not ontic; they are relational descriptors of recursion.
Epistemic Geometry: Geometry and topology are epistemic models of field closure, constrained only by measurable phase continuity, not fundamental givens.
Complexity: Complex structure results from nested and interacting closed modes.

6 Conclusion

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.

7 References

Nedrock, F., Vale, L., Freet, M., Rodriguez, A. M. (2025). The PNP Theory of Cause and Effect: Causality from Topological Persistence in Scalar Fields. Preferred Frame Lab. https://writing.preferredframe.com/doi/10.5281/zenodo.18317319