This table states the cross-domain claim directly: the manifold-constrained architecture is not only a theorem-library pattern.
| Architectural role | Math | AI | Control | Physics | Systems |
|---|---|---|---|---|---|
| Admissible class | legal object/state family | legal latent/state manifold | safe operating region | allowed state sector | valid configuration envelope |
| Projected core | coercive / invariant sector | constrained representation core | stabilizing internal model | protected mode / symmetry sector | trusted service core |
| Transport / continuation | deformation or persistence law | routed update / training or inference flow | evolution law / observer flow | conserved or constrained propagation | dataflow / request propagation |
| Restart | normalization / surgery / re-entry | reset, checkpoint, fallback, retokenized re-entry | controller reset / receding-horizon refresh | renormalization / chart switch | failover / restart / rehydration |
| Remainder | explicit residual / complement | residual channel / off-manifold part | disturbance / model error | fluctuation / uncontrolled mode | backpressure / error budget / residual load |
| Rigidity | bad-limit exclusion | collapse / degenerate-mode exclusion | instability-mode exclusion | forbidden phase / singular limit exclusion | outage / degenerate-state exclusion |
| Certificate | theorem constant / strict margin | safety or capability certificate | stability certificate | conservation / admissibility certificate | service or policy certificate |
| Endpoint identification | bridge to intended theorem object | map to intended behavior or task object | target equilibrium / reference behavior | physical observable identification | external service-level semantics |
The point is not that these domains are identical. The point is that the same causal jobs recur across them and can be named publicly instead of rediscovered under new vocabulary each time.