Direct Dynamical Manipulation & Phase Space
Runge-Kutta 4th Order Symplectic Integration
“Confronting that — what is physics, really? What is enduring about the practice of physics itself? — that does not change with AI.” — Physicist Sarah (Quanta Magazine)
When machine learning models generate physical predictions, they interpolate over statistical patterns without asserting mathematical necessity. Below is the strict demarcation between rigorous physical theorems and empirical AI conjectures:
| Category | Principle / Statement | Mathematical Grounding | Epistemic Status |
|---|---|---|---|
| Continuous Symmetries | Noether (1918): Any continuous global symmetry of the action $S = \int \mathcal{L} \, dt$ guarantees a conserved current $\partial_\mu j^\mu = 0$. | Rigorous Lie algebra proof via variational calculus. | PROVED THEOREM |
| Orbital Energy Invariant | Vis-viva relation: $v^2 = \mu(2/r - 1/a)$, orbital energy $E = -G M / (2a)$ remains strictly constant under central inverse-square potentials. | Analytical Kepler integration. | PROVED THEOREM |
| Laplace-Runge-Lenz Vector | Vector $\vec{A} = \vec{p} \times \vec{L} - m k \hat{r}$ is conserved due to hidden $SO(4)$ dynamic symmetry in the Kepler problem. | Lie bracket commutation $[H, A_i] = 0$. | PROVED THEOREM |
| Out-of-Distribution Shift | Deep neural operators generalize accurately across unmeasured eccentricities $e > 0.9$ without explicit inductive symmetry priors. | Empirical regression over sampled parameter bands. | CONJECTURE |
| Algorithmic Complexity | Minimal symbolic expression complexity (Kolmogorov/Occam heuristic) necessarily maps to the true underlying physical ontology. | Heuristic simplicity preference; uncomputable in general. | CONJECTURE |
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