The quantum harmonic oscillator from binary sequences
The quantum harmonic oscillator (QHO) is normally built on a structure that contains a Hilbert space, ladder operators, and a Born rule. Here we derive it from counting. We adopt three information-theoretic postulates: information is carried by binary sequences of length $n$; only symbol counts, not the sequences themselves, are observable; and the transition measure must account for exactly the $2^n$-configuration capacity of the register. The first two postulates force the counting problem into the Hamming/Delsarte association scheme and endow it with an exact symmetry under relabeling of the two symbols. Given an exchange-symmetric bilinear transition measure we prove a rigidity theorem: relabeling symmetry and normalization together force the alternating sign $(-1)^t$ which drives interference. The variable that carries the sign, i.e. the overlap between input and output sequences, or equivalently their Hamming distance $d_{ab}$, is a hidden quantum number of the transition as a whole, defined only by the two endpoints jointly and unknowable from either alone. Interference thus enters as alternating-sign bookkeeping over a nonlocal, relational variable $d_{ab}$, rather than as a separate dynamical ingredient. This forced weighting makes the transition probability proportional to the square of a Krawtchouk polynomial, whose difference equation has an exactly equally spaced spectrum and which converges to the QHO as $n\to\infty$. At fixed excitation, deviations from the QHO are $\mathcal{O}(1/n)$, and are testable in laboratory realizations of the oscillator, since any real system has finite information capacity. In this framework, interference, quadratic (Born-rule-like) probabilities, and the particle-hole symmetric oscillator spectrum are consequences of counting; only the bilinear form of the measure is assumed (though well motivated) rather than derived.