Appendix B. Large-N Limit and Concentration of Measure

B.1 Many-Copy Quantum States

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and consider $N$ identical copies of a quantum system prepared in the same normalized state $\psi\in\mathcal{H}$. The composite system is described by the tensor product space

Measurements are assumed to act independently on each subsystem, corresponding to projectors of the form

where $\{P_i\}$ is an orthogonal resolution of the identity on $\mathcal{H}$.

An outcome sequence is characterized by empirical frequencies

where $n_i$ is the number of occurrences of outcome $i$ in the sequence.

B.2 Empirical Frequencies Without Probability

Crucially, we do not assume any probabilistic interpretation at this stage. The state $\Psi^{(N)}$ induces a geometric weight on the subspace of $\mathcal{H}^{(N)}$ corresponding to a given frequency vector $\mathbf{f}=(f_1,\dots,f_k)$.

The squared norm of the projection of $\Psi^{(N)}$ onto the subspace with frequencies $\mathbf{f}$ is given by

where $\psi_i = P_i \psi$.

This quantity is purely geometric and combinatorial; no probability postulate is invoked.

B.3 Concentration of Measure

Define

As $N\to\infty$, the multinomial weights above exhibit a sharp concentration around

More precisely, for any $\epsilon>0$,

This follows from standard concentration results (e.g. Chernoff or Sanov-type bounds), but here the interpretation is geometric: \emphasymptotically all of the Hilbert-space norm of $\Psi^{(N)}$ lies in frequency sectors consistent with $f_i=w_i$.

B.4 Typicality and Structural Selection

In the large-$N$ limit, the set of frequency vectors $\mathbf{f}$ satisfying

forms an overwhelmingly dominant sector of $\mathcal{H}^{(N)}$.

Any alternative weighting scheme $\mu([\psi],P_i)\neq \|\psi_i\|^2$ generically leads to:

• dispersion rather than concentration,
• failure of automatic normalization across decompositions (Appendix A, Lemma A.2 – not tensor-product-composition instability, which does not distinguish $p=2$ from any other exponent),
• non-typical frequency behavior.

Thus, squared-norm weights are selected not as probabilities, but as typicality weights in high-dimensional Hilbert space.

B.5 Emergence of the Born Rule Without Probability

The Born rule emerges as the statement

where “typical” means occupying asymptotically all of the Hilbert-space norm in the large-$N$ limit.

No stochastic assumptions, decision theory, or subjective probability enters the argument. Probability appears only as a derived effective concept summarizing typical frequency behavior.

B.6 Structural Instability of Non-Born Weights

Suppose one postulates alternative weights $\mu_i = f(\|\psi_i\|^2)$ with $f \neq \mathrm{id}$. Then, generically:

• the large-$N$ state does not concentrate on a single frequency vector,
• frequency fluctuations remain macroscopic,
• empirical regularity is lost.

Such weightings are therefore structurally unstable and observationally inconsistent.

B.7 Summary

In the large-$N$ limit, the squared-norm measure is uniquely selected by:

• concentration of Hilbert-space measure,
• compositional stability,
• emergence of empirical frequencies.

The Born rule is thus not an axiom but a law of large-dimensional geometry.