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Emergence of Born Statistics from the Equidistribution of the Observer-System Beat: A Conjecture on the Localized-Kernel Model

Year: 2026
Type: preprint
Abstract: CONJECTURE PAPER. The preceding localized-kernel paper derived the FORM of the Born distribution |psi|^2 from the reproducing-kernel property of a constant-amplitude odd-harmonic isolated peak wave S_N, leaving the 'existence of randomness' as an open postulate. Here we conjecture a mechanism for that origin: the scan variable phi0 is not free but the OBSERVER'S phase; since the observer is a physical object of finite frequency with no absolute phase reference (subjective-space premise), the measurement sequence scans the observer-system relative phase (beat/heterodyne) (nu-psi)t, which equidistributes (Weyl) for an irrational ratio. The logic is two-stage. (I) Conditionally proven core: reproducing kernel (proven) gives a deterministic read (pi/2)psi(phi0) at each phi0; granting equidistribution (assumption O1), the position-resolved intensity-profile SHAPE = |psi_base|^2. Role separation: Weyl supplies only a uniform measure (no bias); the |psi|^2 weight is the deterministic kernel read; the intensity->single-click step is the bridge. (II) Conjectured bridge (intensity -> single-click probability via threshold detection, lineage A), NOT derived. Both input cases (localized vs spread) are derived from the same kernel (output difference = input difference only), with a falsification battery that all four consistency conditions pass. The conjecture sits at the intersection of Khrennikov's PCSFT + threshold detection and the method of arbitrary functions (Poincare-Feintzeig), with a falsifiable finite-N (O(1/N)~l/R) Born-rule deviation. Verification code reproduces the numerical claims to machine precision. Not a derivation of the Born rule nor a solution of the measurement problem.
DOI: 10.5281/zenodo.20967081
Language: English
Open Access status: green
OpenAlex ID: W7166453014
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