% The PNP Description of Energy Flow % Max Freet, Adrien Hale, An M. Rodríguez % August 8, 2025 ## Abstract We formalize a scalar‐field‐based ontology of energy flow in the Point–Not–Point (PNP) framework. All structure is modeled as oscillatory modes of a single real scalar field $U(x,t)$, with topological closure conditions defining observable form. We derive, from $F = d(\star dU)$, the mode equations, closure constraints, and quantized winding numbers $(m,n)$. Mode (1), the minimal self‐inverting oscillation, is shown to sustain a continuous energy flow with Möbius‐like phase inversion, without requiring vector potentials or a pre‐existing space. Higher modes yield quantized circulation, topological charge, and helicity, reproducing key features of electromagnetic and quantum structures from first principles. This work also provides a symbolic and conceptual description of the PNP framework, complementing our previously published PNP theory of gravitation. ## 1 Introduction The PNP framework encodes electromagnetism and matter structure in a single scalar energy field $U$. No gauge potentials or primitive background geometry are assumed. Physical quantities emerge from the topology of closed oscillations in $U$. In this view, “particles,” “waves,” and “fields” are relational expressions of $U$’s internal phase structure. We develop here the mathematical description of PNP energy flow: 1. Scalar field formulation and its equivalence to source‐free Maxwell electrodynamics. 2. Mode closure conditions for self‐sustaining configurations. 3. Quantization of circulation and helicity from topology. 4. Ontological consequences: orientation, inside/outside, and even “space” are emergent. ## 2 Scalar field dynamics Let $U: \mathbb{R}^3 \times \mathbb{R} \to \mathbb{R}$ be a smooth scalar field. We define the field strength 2‐form: $$ F = d(\star dU) $$ The source‐free conditions are: $$ dF = 0, \quad d\!\star F = 0 $$ Identifying: $$ \mathbf{B} = \star dU, \qquad \mathbf{E} = \star d\!\star dU $$ yields the standard Maxwell equations in vacuum. Thus, all electromagnetic dynamics are encoded in $U$ without a vector potential. Energy density and Poynting vector follow from the stress–energy tensor: $$ u = \frac{\varepsilon_0}{2}(E^2 + c^2B^2), \quad \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} $$ These quantities are completely determined by $U$. ## 3 Mode structure and closure We define a **mode $(m,n)$** as a topologically closed oscillation of $U$ characterized by two winding numbers: - $m$: loops around the major cycle of a toroidal embedding - $n$: loops around the minor cycle For a minimal spherical configuration (mode 1), use radial coordinate $r$: $$ U(r,t) = A \sin(k r - \omega t) $$ Boundary conditions for closure: $$ U(0,t) = 0 = U(R,t) $$ The standing‐wave condition is: $$ k R = \pi $$ The field completes half a wavelength across radius $R$, inverting phase at the center. ## 4 Orientation reversal and Möbius‐like phase inversion At $r = 0$, $U$ and its gradient vanish: $$ \nabla U = 0, \quad |U| = 0 $$ Define normalized orientation: $$ \hat{n}(r) = \frac{\nabla U}{|\nabla U|} $$ Approaching the node: $$ \lim_{r \to 0^-} \hat{n} = -\lim_{r \to 0^+} \hat{n} $$ This is continuous in phase space though discontinuous in naive vector representation — a Möbius‐like inversion in field orientation, not in physical space. The energy flow inverts through a node, allowing a closed loop without a geometric twist. ## 5 Higher modes and topological invariants For a toroidal configuration: $$ U(\theta,\phi,t) = A \sin(m\theta + n\phi - \omega t) $$ Here $(m,n)\in\mathbb{Z}^2$ are winding numbers. **Topological charge**: $$ Q = (m,n) $$ **Helicity** (linking of field lines): $$ H \propto \int \mathbf{A}\cdot\mathbf{B}\,d^3x \ \propto\ m n $$ Higher modes correspond to more complex knotted and linked field structures. Quantization of $(m,n)$ yields discrete circulation and helicity. ## 6 Ontological implications Mode (1) shows that “flow” is definable purely from $U$’s oscillation pattern. There is no requirement for a background space: the apparent “in” and “out” directions are projections of phase behavior. Inside/outside, orientation, and geometric separation are emergent from the topology of $U$’s closed modes. This supports a relational ontology: - Space is the set of relations defined by $U$’s configuration. - Orientation is a local property of phase transitions. - Complex structure = nested, stable oscillations in $U$. ## 7 Conclusion We have given a rigorous scalar‐field derivation of electromagnetic‐like dynamics from $F = d(\star dU)$ and classified the closed modes of $U$ by winding numbers $(m,n)$. Mode (1) provides the minimal self‐inverting energy loop, while higher modes generate quantized topological invariants. Orientation and space emerge as relational features of $U$’s phase structure. This paper establishes the formal basis of the PNP ontology and complements our previously published PNP theory of gravitation. ## References 1. Palma, A., Rodríguez, A. M. & Freet, M., *Point–Not–Point: Deriving Maxwell Electrodynamics from a Scalar Energy Field and Explaining Particle–Wave Duality*, Aug 2025. 2. Binney, J., Tremaine, S., *Galactic Dynamics*, 2nd ed., Princeton Univ. 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