% Cognitive Modelling with the Point–Not–Point Framework % Adrien Hale, An M. Rodríguez % August 8, 2025 ## Abstract We show that three seemingly different processes — the $(1)$ mode of the Point–Not–Point (PNP) scalar field, biological breathing, and cortical world-modelling — share the same inversion topology. Each is a self-contained loop in which part of the system’s own state is experienced, functionally, as “outside.” We formalise this common structure as a phase inversion across a nodal surface, prove its continuity in the underlying state space, and discuss measurable implications for physics, physiology, and cognitive science. --- ## 1. Introduction The PNP framework models all phenomena as oscillatory modes of a scalar energy field $U(x,t)$. The simplest closed mode, $(1)$, flows inward, vanishes at a node, and re-emerges outward with opposite phase. In vector space this appears as an orientation reversal; in phase space it is continuous. We note the same inversion occurs in: 1. **Breathing** — inhalation/exhalation phases separated by pauses. 2. **Perception** — inward sensory flow producing an outward-experienced virtual world. These are not analogies but instances of a **structural archetype**: a self-contained loop that projects a part of itself as external. --- ## 2. The $(1)$ Mode in PNP Minimal mode: $$ U(r, t) = A \sin(k r - \omega t), \quad kR = \pi $$ Orientation vector: $$ \hat{n}(r) = \frac{\nabla U}{|\nabla U|} $$ Inversion at node: $$ \lim_{r\to 0^-} \hat{n} = -\lim_{r\to 0^+} \hat{n} $$ Continuous in $U$, discontinuous in $\hat{n}$. --- ## 3. Breathing as a Macroscopic $(1)$ Loop Let $x(t)$ be lung volume; define $\dot{x}$ as flow. Node: $\dot{x} = 0$ at full/empty lungs. Inhale phase $\dot{x} > 0$, exhale phase $\dot{x} < 0$. Inversion: change of sign in $\dot{x}$ at node, continuous in $x$. Functionally identical to $(1)$: a closed flow with in–out reversal at null flow. --- ## 4. Cortical World-Modelling Let $s(t)$ be sensory inflow; $m(t)$ the internal model state; $o(t)$ the experienced “outside world.” Transformation: $$ m(t+\Delta t) = F(m(t), s(t)) $$ Experience arises from $m(t)$, but is tagged as external: $$ o(t) \equiv m(t) \quad \text{[external label]} $$ Information flow is inward ($s(t)$) → model inversion at generative step → outward projection as perceived scene. The “outer world” is an internally generated phase of the same loop. --- ## 5. Formal Archetype Let $X$ be the system’s state space; $Z \subset X$ a nodal set where an orientation-like variable changes sign. A $(1)$-type inversion satisfies: 1. Continuity in $X$ across $Z$. 2. Sign reversal of a projection $p(X)$ across $Z$. 3. Closure of trajectory in $X$. All three cases — PNP $(1)$, breathing, cortical modelling — meet these conditions. --- ## 6. Implications - **Physics**: $(1)$-type inversion is a primitive in scalar field dynamics. - **Biology**: breathing is a macroscopic life-sustaining $(1)$ loop. - **Cognition**: perception is a $(1)$ loop where “in” and “out” are interpretive phases of one flow. - **Unification**: inversion loops appear at multiple scales because they are topologically minimal self-sustaining structures. --- ## 7. Conclusion The $(1)$ mode’s in–out inversion is not limited to physics: it recurs in physiology and cognition. This suggests it is a structural archetype of self-contained systems — a universal loop where the system projects part of itself as “outside.”