DRSN XII: Heat Kernel, Heat Trace, and Spectral Invariants (De Rerum Spectrale Natura, Report XII, Version 1.0)

We develop the heat-kernel layer of the Drifted Spectral Renormalisation Network (DRSN).
For nonnegative self-adjoint operators with trace-class heat semigroup we prove that the
heat trace is a spectral invariant and therefore rigid under bounded similarity drift. We then
recall the small-time heat-trace asymptotics for Laplace-type operators on compact manifolds
and identify the resulting Seeley–DeWitt coefficients as spectrally rigid objects. Finally, we
formalise where and how effective (scale-dependent) quantities arise only through truncation,
projection, and other information-loss operations, in accordance with the DRSN Separation
Principle fixed in DRSN XI.