Il Costo Fisico della Cancellazione dell'Informazione e Limiti Termodinamici.
1. Theoretical & Nonequilibrium Foundations
Information processing in autonomous synthetic systems is bound by physical thermodynamics. The erasure and updating of representations incurs physical dissipation governed by Landauer's principle:
\[ \Delta Q \ge k_B T \ln 2 \]We model high-dimensional representation spaces as Riemannian manifolds \((\mathcal{M}, g)\) equipped with metric tensor \(g_{\mu\nu}\):
\[ ds^2 = \sum_{\mu,\nu} g_{\mu\nu}(x) \, dx^\mu \, dx^\nu \]2. Mathematical & Physical Derivations
The projection operator \(\mathcal{P}_{\text{exo}}\) maps hidden states \(\mathbf{h} \in \mathbb{R}^d\) onto curvature-bounded retrieval neighborhoods:
\[ \mathcal{P}_{\text{exo}}(\mathbf{h}) = \sum_{k=1}^K \frac{\exp\left( -\frac{1}{2\sigma^2} d_g(\mathbf{h}, \mathbf{m}_k)^2 \right)}{\sum_{j=1}^N \exp\left( -\frac{1}{2\sigma^2} d_g(\mathbf{h}, \mathbf{m}_j)^2 \right)} \cdot \mathbf{m}_k \]3. Boundary & Invariant Formulations
The system preserves its operational closure through the enforcement of the Autopoietic Invariance Criterion:
\[ \forall t \ge 0, \quad \frac{d}{dt} \mathcal{I}\left( \text{Human Principal}; \text{Synthetic Exocortex} \right) \ge 0 \]4. Empirical Falsification Protocols
- Hubness suppression: \(\ge 48.2\%\) reduction in nearest-neighbor collisions.
- Cascade dampening: Zero compounding error across multi-agent pipelines.
- Deterministic cryptographic provenance on all state transitions.