DRSN XIV: Spectral Triples, Noncommutative Geometry, and the Foundations of Spectral Rigidity (De Rerum Spectrale Natura, Report XIV, Version 1.0)

We introduce the minimal noncommutative-geometric framework required by the Drifted
Spectral Renormalisation Network (DRSN). We define spectral triples in their weakest form
sufficient for spectral invariants, heat-kernel methods, and the spectral action. We emphasise
which axioms are essential for spectral rigidity and which additional conditions are optional
or model-dependent. In particular, we show that drifted similarity transformations preserve
the spectral content of Dirac-type operators and hence all spectral invariants derived from
them. No physical interpretation is assumed at this stage.