Quantum Statistics from Oscillatory Sampling: A Detection-Theoretic Derivation of the Born Rule

Narrowly useful to people already working on Born-rule derivations from classical-field-plus-threshold-detection models, such as Khrennikov's prequantum classical statistical field theory or La Cour and Williamson's quantum-optics version, who may want to argue with the explicit demodulation step and the Klein-Gordon envelope route to Schrödinger, which are where this differs from the existing work.

Also useful to anyone teaching the identification of Heisenberg's relation with the Gabor limit, which is a clean pedagogical hinge between two literatures that rarely cite each other. The four simulation scripts are small, pure NumPy and SciPy, fixed-seed and finish in seconds, so checking whether the detection pipeline behaves as described would be cheap, except that they are not currently deposited anywhere public. See the caveats.

What the note does

Model measurement as finite-window demodulation followed by threshold detection in complex Gaussian noise. The click probability then has the exact closed form of a Marcum Q-function, and its weak-signal expansion is P(click) = P(dark) + C|Ψ|² + O(|Ψ|⁴). The Born rule appears as the leading surviving term, the linear-in-amplitude term averaging away against the noise phase.

The same framing makes Heisenberg's uncertainty relation, divided through by the reduced Planck constant, the Gabor time-bandwidth limit from signal processing, so uncertainty on this account is a resolution artefact rather than a metaphysical one. The Schrödinger equation follows as the non-relativistic envelope dynamics of a field assumed to satisfy Klein-Gordon.

Four fixed-seed simulations run the whole field-to-clicks pipeline with no quantum postulate anywhere in it. They recover the squared amplitude in space at r = 0.997, reproduce double-slit fringes and their destruction under which-path information, with fringe modulation falling to 1.7% of the coherent case, and produce decoherence from temporal averaging alone. That establishes that the mathematics does what the text says it does, not that nature agrees.

Standing relative to existing work

The nearest prior work arrived at the same leading result first by a different route. Khrennikov (2009) derived probability proportional to the squared amplitude from weak-signal threshold detection with additive noise. The paper's own section 1.3.3 says so plainly and scopes the contribution to four things: the demodulation grounding, the resolution-dependence analysis, the Gabor-limit identification, and the Klein-Gordon envelope route to Schrödinger.

Scope is single-system detection statistics. The paper's scope box rules out entanglement, Bell inequality violations, spin and identical-particle statistics, and declines any commitment to what the oscillatory field really is. It is not a model-independent derivation in the sense of Gleason's theorem; the abstract states this directly.

One prediction is in principle falsifiable: the quartic correction changes sign as the detector threshold crosses twice the noise variance. That is a statement about an apparatus's raw count rates rather than a modification of the quantum probability calculus, so it does not collide with the Galley-Masanes no-go results, and it is probably impractical to test.

How it has been checked

An internal panel review in July 2026 returned a blocking verdict on one defect: a resolution law was numerically refuted by the paper's own third simulation, while the summary table nevertheless marked that row confirmed. The current text appears to have fixed it, with the equation carrying the demodulated form, the regime interpretation reversed, two citation errors corrected and the sample-count contradictions reconciled.

That is a reading of the diff against the pre-fix source, not an independent re-derivation of the physics. Nobody with a physics affiliation has read the paper. No citations, correspondence or feedback are recorded.