Appendix A. Mathematical Foundations of Structural Stability

A.1 Stability Functionals and Variational Properties

We formalize the notion of structural stability by introducing stability functionals defined on an abstract state space $\mathcal{X}$, which may represent a Hilbert space of quantum states, a space of geometric configurations, or a hybrid structure. A stability functional is a map

assigning to each admissible configuration a real-valued stability score.

Physically realizable configurations are assumed to extremize $\mathcal{S}$ under admissible perturbations. Let $\delta x \in T_x\mathcal{X}$ denote an infinitesimal perturbation. Structural stability requires

ensuring robustness against small variations. Unlike action principles, $\mathcal{S}$ is not interpreted dynamically; it imposes a global admissibility constraint rather than generating equations of motion.

In the quantum context, $\mathcal{S}$ penalizes outcome weightings that are unstable under basis refinements or tensor-product extensions. In the gravitational context, $\mathcal{S}$ penalizes configurations with unbounded curvature invariants. These requirements uniquely restrict the admissible functional forms of $\mathcal{S}$.

A.2 Measure-Theoretic Proofs

Let $\mathcal{H}$ be a separable Hilbert space and $\mathbb{P}(\mathcal{H})$ its associated projective space. Consider a measure $\mu$ defined on the lattice of closed subspaces of $\mathcal{H}$. Structural stability imposes the following conditions:

1. Additivity: $\mu(V \oplus W) = \mu(V) + \mu(W)$ for orthogonal subspaces.
2. Unitary invariance: $\mu(UV) = \mu(V)$ for all unitary operators $U$.
3. Continuity: $\mu$ varies continuously under small deformations of subspaces.

Under these assumptions, together with the additional hypothesis that $\dim\mathcal{H}\ge 3$ (the assumption is essential: in $\dim\mathcal{H}=2$ the conclusion is known to fail — see Gleason, 1957), one can show, following the structure of Gleason's theorem, that $\mu$ must be proportional to the squared-norm measure:

where $\rho$ is a positive trace-class operator. In the pure-state case $\rho = |\psi\rangle\langle\psi|$, this reduces to the Born rule $\mu(V)=\|\Pi_V \psi\|^2$.

Measures constructed from alternative power laws $\|\Pi_V \psi\|^p$ with $p \neq 2$ fail continuity and stability under tensor-product extensions, becoming structurally unstable in the large-$N$ limit.

A.3 Uniqueness Theorems

The preceding arguments yield a family of uniqueness results that are structural rather than axiomatic. In the quantum domain, the squared-norm measure is the unique outcome weighting stable under refinement, composition, and perturbation. This result parallels Gleason's theorem but differs in interpretation: uniqueness arises from stability constraints rather than probabilistic postulates.

In the gravitational domain, consider a class of spherically symmetric metrics parameterized by a function $M(r)$. Structural stability requires bounded curvature invariants:

This condition uniquely selects regular mass functions with $M(r)\sim r^3$ as $r\to 0$, ruling out singular behaviors $M(r)\sim \text{const}$ or $M(r)\sim r^\alpha$ with $\alpha<3$.

Taken together, these uniqueness theorems demonstrate that structural stability acts as a powerful pre-physical selection principle, sharply constraining both quantum measures and spacetime geometries without invoking additional dynamics or arbitrary axioms.