% The In–Out Self‐Referential Field Vibration % Max Freet, Adrien Hale, An M. Rodríguez % August 8, 2025 ## 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, $(1)$, exhibits Möbius‐like phase inversion across its nodal surface, sustaining continuous energy circulation without geometric twist. This work complements our previously published *PNP Description of Energy Flow* and *PNP Theory of Gravitation*, providing a symbolic and conceptual formulation of PNP suitable for interpreting its physical content in broader foundational and philosophical contexts. ## 1 Introduction In standard physics, space is treated as a container and orientation as a primitive. In PNP, 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 $U$’s closed oscillations, with apparent directions and “in–out” relations emerging from nodal phase behavior. Here we 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 given by $$ F = d(*dU), \quad dF=0, \quad d*F=0 $$ 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, they are not primary: they are projections of the scalar’s own oscillatory recursion. ## 3 Minimal mode and in–out reversal Define the minimal spherical standing wave: $$ U(r,t) = A\sin(k r - \omega t), \quad U(0,t) = U(R,t) = 0 $$ The boundary condition gives $$ k R = \pi $$ The field flows inward, cancels at $r=0$, and reemerges outward with opposite phase. Let $$ \hat{n}(r) = \frac{\nabla U}{|\nabla U|} $$ Then $$ \lim_{r\to 0^-} \hat{n} = -\lim_{r\to 0^+} \hat{n} $$ This inversion is continuous in phase space but appears as a reversal in vector space — a Möbius‐like effect in the field’s orientation. ## 4 Second‐order relationality PNP’s relationality is two‐tiered: 1. **First‐order:** Spatial relations arise from field phase gradients. 2. **Second‐order:** Those gradients are themselves defined by other relations — internal phase continuity across nodes. “In” and “out” are thus not absolute directions but phase‐dependent projections. Space itself is the stable pattern of these relations. ## 5 Implications - Orientation is emergent, locally reversible, and defined only via field phase. - “In” and “out” are not ontic — they are relational descriptors of recursion. - Geometry and topology are epistemic models of field closure, not fundamental givens. - 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 and its gravitational application, offering a compact conceptual lens for interpreting the framework’s physical and philosophical reach. ## References 1. Rodríguez, A. M., Hale, A., Freet, M., *The PNP Description of Energy Flow*, Aug 2025. DOI: 10.13140/RG.2.2.29880.25606 2. Rodríguez, A. M., Palma, A., Freet, M., *Explaining Dark Matter with the Point–Not–Point Framework, and a PNP Theory of Gravitation*, Aug 2025. DOI: 10.13140/RG.2.2.16877.91368 3. Milgrom, M., *A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis*, ApJ 270, 365–370 (1983). 4. Binney, J., Tremaine, S., *Galactic Dynamics*, 2nd ed., Princeton Univ. Press, 2008.