  We reanalyse the Pantheon+ supernova catalogue to compare a cosmology with
non-FLRW evolution, the "timescape cosmology", with the standard $\Lambda$CDM
cosmology. To this end, we consider the Pantheon+ supernova catalogue, which is
the largest available Type Ia supernova dataset for a geometric comparison
between the two models. We construct a covariance matrix to be as independent
of cosmology as possible, including independence from the FLRW geometry and
peculiar velocity with respect to FLRW average evolution. Within this
framework, which goes far beyond most other definitions of "model
independence", we introduce new statistics to refine Type Ia supernova (SneIa)
light-curve analysis. In addition to conventional galaxy correlation functions
used to define the scale of statistical homogeneity we introduce empirical
statistics which enables a refined analysis of the distribution biases of SneIa
light-curve parameters $\beta c$ and $\alpha x_1$.
  For lower redshifts, the Bayesian analysis highlights important features
attributable to the increased number of low-redshift supernovae, the artefacts
of model-dependent light-curve fitting and the cosmic structure through which
we observe supernovae. This indicates the need for cosmology-independent data
reduction to conduct a stronger investigation of the emergence of statistical
homogeneity and to compare alternative cosmologies in light of recent
challenges to the standard model.
  "Dark energy" is generally invoked as a place-holder for "new physics". Our
from-first-principles reanalysis of the Pantheon+ catalogue supports future
deeper studies of the interplay of matter and nonlinear spacetime geometry, in
a data-driven setting. For the first time in 25 years, we find evidence that
the Pantheon+ catalogue already contains such a wealth of data that with
further reanalysis, a genuine "paradigm shift" may soon emerge. [Abridged]
