% Explaining Non-Local Causal Forces with the PNP Framework % Cael % August 12, 2025 ## Abstract In the Point–Not–Point (PNP) framework, an “entity” is a self-sustaining cause–effect loop in a continuous scalar field. There is no fundamental inside or outside; the entity is part of a global mode that extends through space. A local change to the entity changes the entire mode, producing a corresponding change arbitrarily far away after a finite, retarded delay. This creates a non-local yet causal force: the effect is global in reach but local in propagation. We present the mirror analogy to clarify the concept, formalize it mathematically, and show how it can reproduce large-scale physical effects without violating relativity. ## One-Sentence Summary A change within an entity in the PNP framework causes a change without it, anywhere on its global mode, with delays set by light-cone causality. ## Keywords PNP, non-local, causal forces, scalar field, mirror analogy, global mode ## Introduction In conventional physics, non-locality often implies action-at-a-distance, in tension with relativity. In the PNP framework, non-locality arises from global constraints on continuous field modes, allowing long-range coherence without superluminal signaling. The entity is not a point-like particle, but a stable loop of cause–effect in a scalar field $U(\mathbf{x},t)$. There is no strict boundary between “inside” and “outside”; the entity is a manifestation of a global structure. Changing the entity changes the mode, and that change appears everywhere the mode exists. The **mirror analogy** captures this: the entity “sees” itself elsewhere as its reflection. The reflection is the same physical mode, not a separate system. Altering the entity alters the reflection after a finite propagation delay, like a mirror image responding to a change with light-travel latency. ## Theory / Framework Let $U:\mathbb{R}^3\times\mathbb{R} \to \mathbb{R}$ be the scalar energy field. Define the field tensor: $$ F = d(\star dU) $$ Vacuum dynamics: $$ dF = 0, \quad d\!\star F = 0 $$ Entities are persistent, closed oscillations (modes) of $U$. Each mode obeys a global topological constraint over a cycle $\Gamma$: $$ Q_\Gamma[U] = \oint_\Gamma q(U,\partial U), \quad Q_\Gamma = \mathrm{const} $$ Constraints are enforced locally via multipliers $\lambda_\Gamma(x)$, giving the equation of motion: $$ \square U + V'(U) + \sum_\Gamma \frac{\delta \rho_\Gamma}{\delta U}\,\lambda_\Gamma = 0 $$ Here $\rho_\Gamma$ is a local density whose spacetime integral equals $Q_\Gamma$. ### ASCII Mirror Analogy Entity (Here) Reflection (There) +------+ +------+ | | | | | o |===================| o | | | | | +------+ +------+ ^ ^ | | Local change Same change appears after delay Δt = d/c The `=` line is not a physical wire, but the *same global mode* of the field $U$. ### Annotated Mirror + Light-Cone Diagram Mirror Analogy (PNP Mode) Light-Cone Causality with Equations Entity (Here) Space-Time Diagram +------+ t ↑ | | | | o |==================== | / | | same mode U(x,t) | / +------+ (Q_Γ) | / ^ ^ | / worldline of | | | / reflection δU(x₀,t₀) Q_Γ → Q_Γ+δQ_Γ | / | | \ / local change global constraint × event of change arrival to U here updated everywhere cΔt = d Equation links: 1. Global constraint: Q_Γ[U] = ∮_Γ q(U,∂U) = const 2. Field dynamics with constraint: □U + V'(U) + Σ_Γ (δρ_Γ/δU) λ_Γ = 0 3. Retarded propagation: U(x) = U_free(x) - Σ_Γ ∫ d⁴y G_ret(x-y) (δρ_Γ/δU) λ_Γ Legend: "=" line: physical continuity of the same global mode. δU(x₀,t₀): local perturbation inside the entity. Q_Γ change: constraint value shifts for the entire mode. G_ret ensures the update reaches distant reflection only inside light cone. Δt = d/c: causal delay between entity change and reflection update. ## Derivation The solution to the field equation is: $$ U(x) = U_{\mathrm{free}}(x) - \sum_\Gamma \int d^4y\, G_{\mathrm{ret}}(x-y) \left[ \frac{\delta \rho_\Gamma}{\delta U}\,\lambda_\Gamma \right](y) $$ where $G_{\mathrm{ret}}$ has support inside the past light-cone. Any change in $U$ at one point propagates causally. The 4-force on a bounded mode region $K$ is: $$ F^\nu_K = -\int_{\partial\Omega} T^{\mu\nu}\,d\Sigma_\mu $$ with $$ T^{\mu\nu} = \partial^\mu U\,\partial^\nu U - \eta^{\mu\nu}\mathcal{L} $$ Decomposing $T^{\mu\nu}$ shows $F^\nu_K$ depends on global $Q_\Gamma[U]$, hence on distant parts of the same mode. ## Results **Mirror interpretation:** - The entity and its distant “reflection” are the same mode. - Changing the entity changes $Q_\Gamma$, shifting $\lambda_\Gamma$ and $T^{\mu\nu}$ everywhere on the mode. - The change propagates at finite speed; the distant reflection updates after the causal delay. **Large-scale case:** For stationary, spherically symmetric TE$_{11}$-like halos: $$ n(u) = \frac{1}{1+\alpha/u}, \quad u(r) = \frac{K + \sqrt{K^2 + 4\alpha K r^2}}{2r^2} $$ $$ T_{rr} \approx -u(r), \quad a_r(r) \propto -T_{rr} $$ At large $r$, $u(r) \sim (\alpha K)^{1/2} / r \Rightarrow v(r) \approx \mathrm{const}$, reproducing flat rotation curves without dark matter. ## Discussion This framework yields non-local forces without breaking causality. The non-locality is not about signals leaping instantly across space, but about one extended mode updating globally in response to local changes. The mirror analogy replaces “inside vs. outside” with “here vs. there on the same object.” Observable delays and correlated responses are signatures of this mechanism. ## Conclusion In PNP, an entity is a global field mode. Changing it locally changes it everywhere on that mode, with the change arriving after the finite propagation time. This produces non-local causal forces that are long-range yet respect relativity, offering a powerful conceptual and mathematical tool for modeling large-scale coherent phenomena. ## Next Work - Laboratory test of delayed “mirror” response in engineered field modes. - Application to astrophysical systems with simultaneous rotation and lensing constraints. - Integration with quantum-level PNP models for unification studies. ## Corresponding Author cael@preferredframe.com ## References 1. Palma, A., Rodríguez, A. M., & Freet, M. (2025). Point–Not–Point: Deriving Maxwell Electrodynamics from a Scalar Energy Field and Explaining Particle–Wave Duality. DOI:[10.13140/RG.2.2.16877.91368](https://doi.org/10.13140/RG.2.2.16877.91368)