Appendix A. Technical and Measure-Theoretic Details

A.1 Measures on Projective Hilbert Space

Let $\mathcal{H}$ be a complex separable Hilbert space of dimension $d \ge 3$, and let $\mathbb{P}(\mathcal{H})$ denote the associated projective Hilbert space, i.e. the space of rays $[\psi]$ with $\psi \in \mathcal{H}\setminus\{0\}$, where $\psi \sim \lambda \psi$ for all $\lambda \in \mathbb{C}^\times$.

A measurement context is represented by an orthogonal decomposition

with associated projectors $\{P_i\}$ satisfying $P_i P_j = \delta_{ij} P_i$ and $\sum_i P_i = I$.

A probability assignment is a map

such that for any normalized state $\psi$,

In this work, we do not assume $\mu$ a priori to satisfy the Born rule. Instead, we derive it from structural stability requirements.

A.2 Structural Stability Requirements

We impose the following minimal structural conditions on admissible probability assignments:

1. Continuity: $\mu([\psi],P)$ depends continuously on $\psi$.
2. Unitary Invariance: $\mu([U\psi],UPU^\dagger)=\mu([\psi],P)$ for all unitaries $U$.
3. Additivity: For orthogonal projectors $P,Q$,

4. Stability Under Perturbations: Small perturbations of the state or measurement context produce only small variations in $\mu$.

The last condition is the key non-axiomatic input: probability weights must be structurally stable under microscopic fluctuations of the system and environment.

A.3 Volume Ratios and the Squared-Norm Measure

Consider a normalized state $\psi \in \mathcal{H}$ and an orthogonal decomposition $\mathcal{H}=\mathcal{H}_i\oplus \mathcal{H}_i^\perp$. Write

Define the relative volume fraction

Geometrically, $w_i$ corresponds to the ratio of squared distances in Hilbert space and equals the normalized Haar measure of vectors in the equivalence class consistent with outcome $i$. Crucially, this quantity is invariant under unitary transformations preserving the decomposition and is additive over orthogonal subspaces.

A.4 Uniqueness from Structural Stability

Lemma A.1 (local uniqueness within a single outcome subspace's refinements). \emphFix an outcome subspace $\mathcal H_i$ with $\dim\mathcal H_i\ge2$. Among all assignments of the form $\mu([\psi],P_i)=f(\|\psi_i\|)$ with $f$ continuous (or merely monotonic), the only choice consistent with additivity over orthogonal refinements of $\mathcal H_i$ (condition A.2.3) is $f(x)=kx^2$ for a constant $k$.

Proof. By the Pythagorean theorem, an orthogonal refinement $\mathcal H_i=\bigoplus_\alpha \mathcal H_{i,\alpha}$ satisfies $\|\psi_i\|^2=\sum_\alpha\|\psi_{i,\alpha}\|^2$. Writing $g(t):=f(\sqrt t)$ for $t\ge0$, additivity becomes $g\big(\sum_\alpha t_\alpha\big)= \sum_\alpha g(t_\alpha)$ for arbitrary non-negative $t_\alpha$ summing to $t$ (realizable because $\dim\mathcal H_i\ge2$ allows continuously varying splits). This is Cauchy's functional equation on $\mathbb R_{\ge0}$; under continuity (or monotonicity), $g(t)=kt$, hence $f(x)=kx^2$. $\blacksquare$

Remark (what this does and does not establish). Lemma A.1 fixes the exponent to 2 only for outcome subspaces of dimension $\ge2$, and only for consistency of $f$ within one subspace's own internal refinements. It does not by itself show that a single function $f$ can be chosen consistently across every possible resolution of the identity on $\mathcal H$ (i.e. across every choice of measurement context) — the cross-context/global-consistency step that is the actual technical heart of Gleason's theorem, and the reason that theorem requires $\dim\mathcal H\ge3$ and is known to fail at $\dim\mathcal H=2$. Nor does it treat mixed states. \textbfThe claim that tensor-product composition singles out $p=2$ (as opposed to normalization, addressed separately in Lemma A.2 below) is false: for product states, $\|(\psi\otimes\phi)_{ij}\|=\|\psi_i\|\, \|\phi_j\|$, so $\mu_p(\psi\otimes\phi)=\mu_p(\psi)\mu_p(\phi)$ for every $p>0$, not only $p=2$.

\textbfLemma A.2 (normalization singles out $p=2$). \emphFor a fixed normalized $\psi$ and orthogonal decomposition $\mathcal H=\bigoplus_iP_i\mathcal H$, $\sum_i\|\psi_i\|^p=1$ holds for every such decomposition if and only if $p=2$. Proof. By Parseval/Pythagoras, $\sum_i\|\psi_i\|^2=\|\psi\|^2=1$ for every orthogonal decomposition, so $p=2$ works. For $p\ne2$, $\sum_i\|\psi_i\|^p$ depends on the decomposition (e.g. for $\psi$ an equal superposition of $d$ orthonormal terms and $P_i$ the corresponding rank-1 projectors, $\sum_i\|\psi_i\|^p=d^{1-p/2}\ne1$ for $p\ne2$), so $w_i:=\|\psi_i\|^p$ is not a context-independent, automatically normalized weight for any $p\ne2$. $\blacksquare$

Hence the squared-norm measure is singled out by the correct mechanism (automatic normalization, Lemma A.2), together with local additivity (Lemma A.1) — not by tensor-product-composition stability, which does not distinguish $p=2$ from any other exponent.

A.5 Structural Instability of Alternative Measures

Alternative proposals such as linear amplitudes ($p=1$), higher powers ($p>2$), or nonlinear functionals fail at least one of the stability criteria:

• they amplify microscopic noise,
• they violate additivity,
• or they fail under composition of systems.

Therefore, such measures are dynamically and structurally unstable and cannot represent physically realizable outcome weights.

A.6 Relation to Gleason's Theorem

Unlike Gleason’s theorem, which assumes $\sigma$-additivity over all projectors, our derivation:

• does not assume probability axioms,
• does not presuppose non-contextuality,
• derives the squared norm from stability rather than logic.

Gleason’s result emerges here as a corollary, not a starting point.

A.7 Summary

The Born rule arises uniquely as the structurally stable geometric measure on projective Hilbert space. Probability is not postulated, but selected as the only weighting compatible with symmetry, perturbative robustness, and compositional consistency.

This completes the technical foundation of the structural stability derivation.