Appendix C. Comparison with Standard Frameworks

C.1 Relation to Gleason's Theorem

Gleason's theorem establishes that any measure $\mu$ assigning probabilities to projection operators on a Hilbert space of dimension $\geq 3$, subject to non-contextuality and additivity, must take the form

where $\rho$ is a density operator. In particular, for pure states $\rho = |\psi\rangle\langle\psi|$, this yields the Born rule $\mu(P_i)=|\langle i|\psi\rangle|^2$.

The present framework is compatible with Gleason's result but conceptually distinct. Gleason's theorem is an axiomatic characterization: it assumes the existence of a probability measure satisfying specific consistency conditions and derives its functional form. By contrast, the structural stability approach does not postulate probabilities a priori. Instead, it derives the squared-norm measure as the unique structurally stable assignment under composition, perturbation, and large-$N$ limits.

In this sense, Gleason's theorem can be viewed as a consistency check on admissible measures, while the present work provides an explanatory principle for why the conditions of Gleason's theorem should be physically privileged. The Born rule emerges here not merely as a mathematically allowed measure, but as the only one compatible with stability, typicality, and empirical robustness.

C.2 Comparison with Decoherence and Envariance

Decoherence theory explains the suppression of interference between branches of the wavefunction through environmental entanglement, thereby accounting for the emergence of classical pointer states. However, decoherence alone does not fix the numerical weights of outcomes; it explains why certain bases are stable, but not why outcomes occur with Born weights.

Envariance-based approaches (notably those of Zurek) attempt to derive the Born rule by exploiting symmetry properties of entangled states under environment-assisted transformations. While powerful, these derivations rely on specific symmetry assumptions and often presuppose some notion of probability or equiprobability at an intermediate step.

The structural stability framework complements and extends these ideas. Decoherence kernels appear naturally as dynamical filters that enforce stability of certain subspaces, while envariance can be interpreted as a special case of symmetry under perturbations. Crucially, however, the squared-norm weights arise here as stable fixed points of both dynamical decoherence and geometric measure concentration, without invoking decision theory, subjective probability, or symmetry postulates beyond invariance under small perturbations.

Thus, decoherence and envariance are recovered as mechanisms within a broader stability-based picture, rather than serving as foundational explanations on their own.

C.3 Relation to General Relativity

In classical General Relativity (GR), the fundamental organizing principle is dynamical: spacetime geometry evolves according to the Einstein field equations derived from an action principle. Singularities arise as generic solutions under reasonable energy conditions, signaling a breakdown of the classical description.

The present work differs in emphasis by elevating structural stability to a foundational role. In the gravitational context, this leads naturally to the no-singularity principle: geometries with divergent curvature invariants are structurally unstable and therefore physically inadmissible. Regular black hole models and geodesically complete spacetimes are favored because they remain robust under perturbations.

Conceptually, this mirrors the quantum case. Just as probability assignments are selected by stability and typicality rather than postulated, spacetime geometries are selected by their robustness against pathological behavior rather than by dynamical equations alone. General Relativity is recovered as the stable, low-curvature limit of this selection, in which the Einstein equations provide an excellent effective description.

From this perspective, GR and quantum theory are unified at the level of principles rather than equations: both emerge as effective theories selected by structural stability in their respective domains. Singularities in gravity and ad hoc probability axioms in quantum mechanics appear as analogous structural pathologies, resolved by the same underlying criterion.