Appendix B. Large-N Analysis and Measure Concentration

B.1 Concentration Inequalities

Consider a finite-dimensional Hilbert space $\mathcal{H}$ of dimension $d$ and the $N$-fold tensor product space $\mathcal{H}^{\otimes N}$. Let $|\Psi\rangle = |\psi\rangle^{\otimes N}$ be a product state representing $N$ identical preparations of the same quantum system. Observables corresponding to empirical frequencies act naturally on this space.

A key mathematical fact is the concentration of measure phenomenon: for large $N$, functions that depend smoothly on many degrees of freedom become sharply peaked around their expectation values. More precisely, let $f:\mathbb{P}(\mathcal{H}^{\otimes N})\to\mathbb{R}$ be a Lipschitz-continuous function with Lipschitz constant $L$. Then Lévy-type inequalities imply

where $c>0$ is a universal constant.

In the context of measurement outcomes, $f$ may be taken as the empirical frequency of a given projector. The exponential suppression in $N$ ensures that deviations from the mean become overwhelmingly unlikely as $N\to\infty$.

B.2 Typicality Proofs

Let $\{ |i\rangle \}$ be an orthonormal basis associated with a measurement, and let

be the squared-norm weights. Define the frequency operator $\hat{F}_i^{(N)}$ acting on $\mathcal{H}^{\otimes N}$ as

where the projector appears in the $k$-th slot.

One can show that the product state $|\Psi\rangle$ is sharply concentrated around eigenstates of $\hat{F}_i^{(N)}$ with eigenvalues close to $p_i$. More precisely,

which vanishes as $N\to\infty$.

Thus, in the large-$N$ limit, almost all branches (in the sense of Hilbert-space measure) exhibit empirical frequencies arbitrarily close to the squared-norm weights. This establishes typicality: the Born weights emerge as the overwhelmingly dominant behavior without invoking stochastic postulates.

B.3 Relation to Empirical Frequencies

The above results connect directly to observed experimental frequencies. Consider repeated measurements of the same quantum state prepared $N$ times. The empirical frequency $f_i^{(N)}$ of outcome $i$ converges almost surely to $p_i$ as $N\to\infty$, not by assumption but as a consequence of geometric concentration in $\mathcal{H}^{\otimes N}$.

Importantly, this convergence does not rely on interpreting $p_i$ as a primitive probability. Instead, $p_i$ characterizes the measure-theoretic weight of stable outcome sectors in state space. Frequencies are emergent properties of typical states under tensor-product extension.

Alternative weightings, such as $|\langle i|\psi\rangle|^p$ with $p\neq 2$, fail this test: the corresponding frequency operators do not concentrate, and empirical frequencies become unstable under composition. Hence such assignments are structurally inadmissible in the large-$N$ limit.

In summary, large-$N$ analysis shows that the Born rule is uniquely compatible with measure concentration, typicality, and empirical stability. What appears as probabilistic behavior is, in this framework, a deterministic consequence of high-dimensional geometry and structural robustness.