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.
We construct an information theory framework in which the fundamental objects are binary sequences of length $n$, equipped with the bitwise XOR operation. The only physical observables are counts of XOR-generated symbol classes, while the exact locations of symbols are inaccessible. Averaging over those locations ultimately yields the Minkowski interval as an invariant object that maximizes (information) entropy. Correlations between two binary sequences are base-4 sequences that we label as ``events'', and events are connected with maps. The entire kinematic structure of Special Relativity is recovered under minimal assumptions, i.e. counts that represent space and time increments carry equal informational weight. The central claim is that Lorentz symmetry is the typical large-n behaviour of XOR counting. At finite $n$ the framework yields a discrete rapidity spectrum, a bound $\gamma_{\max} = O(\sqrt{n})$, and interval fluctuations of relative size $O(n^{-1/2})$, with standard special relativity recovered as $n \to \infty$. However, the light cone and one null coordinate are exact for every microscopic configuration, so the symmetry group is undeformed and dispersion relations are unmodified. Thus, the theory, though discrete, implies no Lorentz violation of the standard phenomenological kind. Ultra-high-energy cosmic rays already require $n \gtrsim 10^{23}$; interferometry excludes the variant in which $n$ scales linearly with system size, leaving a holographic area law. The most striking prediction of the framework concerns systems at the maximum of their information capacity, where $n$ is necessarily finite: black holes and de Sitter space. There the corrections are of order one within a Planck proper length of the horizon, regardless of the horizon's size, and the spacetime description fails altogether at the endpoint of black-hole evaporation, where $n$ itself is of order unity.
Sam Powers, D. Stojkovic· 0 citations
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