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Deriving the Form of the Born Distribution from the Reproducing-Kernel Property of a Localized Odd-Harmonic Wave: Reducing the Remaining Postulates to the Squaring Rule and Randomness

Noriaki Kihara

Published Jun 27, 2026• 0 citations
Publisher

Abstract

We regard the constant-amplitude odd-harmonic sum S_N (isolated peak wave) on the half-wavelength phase interval as the localized kernel of observation, and adopt two premises: (i) observation is the convolution of the kernel with phase differences and the probability is the square of the observed amplitude; (ii) phase differences finer than the band are ignored (finite-N truncation). Because S_N is the truncated reproducing kernel of the odd-harmonic basis on this interval, the observed amplitude reproduces any band-limited base wave without distortion, and the observed distribution coincides exactly with |psi|^2 (after normalization). For a complex base wave the squaring becomes a genuine modulus |Z|^2 = Z conj(Z) (not Z^2), reaching the real-imaginary cross structure of the Born square. The paper derives the FORM of the distribution (faithful reproduction by the reproducing-kernel property); the squaring rule and the probability interpretation remain postulates. It does not claim to derive the Born rule or solve the measurement problem; rather it reduces the remaining postulates to the squaring rule and the existence of randomness. The reproducing-kernel identity itself is a classical fact of Fourier analysis (Dirichlet kernel / RKHS); the novelty is the mapping between the physical localized wave, its reproducing kernel, and the observation model. All results are reproduced to machine precision by the accompanying verification code.

DOI
10.5281/zenodo.20965526

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

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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.

•Jun 27, 2026•1 citation

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One of the most successful, yet most puzzling, structures of quantum mechanics is Born's rule:P(\mathbf{x},t)=|\Psi(\mathbf{x},t)|^2.It establishes, with extraordinarily high experimental accuracy, the connection between the quantum wave function and the statistical distribution of actual measurement outcomes. Yet a more fundamental question remains worth asking: Why is probability given precisely by the modulus squared of the complex wave amplitude? Standard quantum mechanics uses Born's rule as a fundamental rule. This paper instead poses an open research question: Could Born's rule be not an irreducible probabilistic axiom, but a statistical emergence resulting from the combined action of continuous wave-amplitude superposition, quantized energy exchange, and local interaction dynamics? We propose a candidate physical picture:wave-amplitude superposition->spatial interference structure->local interaction->quantized energy exchange->discrete events}->probability distribution The central mathematical question is \Gamma(\mathbf{x})\stackrel{?}{\propto}|\Psi(\mathbf{x})|^2 where $\Gamma(\mathbf{x})$ denotes the occurrence rate of localized discrete detection events. This paper does not claim to have completed such a derivation. Instead, it explicitly formulates the problem as an \textbf{open, computable, testable, and falsifiable research challenge}. We particularly invite young physicists, mathematicians, and researchers skilled in AI and modern computational tools to explore this problem.

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Born's statistical interpretation of quantum mechanics transformed the wave function from a dynamical object into a quantity whose squared modulus determines the probability distribution of measurement outcomes. Since the formulation of the Born rule in 1926, the relation between a continuously evolving quantum state and discrete detection events has remained one of the central conceptual questions of quantum theory. This paper proposes a physical reconsideration of that relation. Rather than treating the appearance of a localized detection event as an immediate consequence of wave-function collapse, we distinguish two physically different stages: continuous wave-packet propagation and discrete detector readout. In the first stage, a localized wave packet evolves through an interaction potential and undergoes phase reconstruction, deflection, focusing, or redistribution. In the second stage, the continuously distributed wave amplitude interacts locally with microscopic detector degrees of freedom, producing discrete detection events. The central hypothesis explored here is that the experimentally observed distribution of individual events may arise from a local detector-response functional acting on the incident wave amplitude. In an appropriate regime, a detector response of the form \[R(\mathbf{x}) \propto |\psi(\mathbf{x})|^2\]could emerge from the microscopic interaction between the wave packet and the detector, rather than being imposed as a primitive probabilistic postulate. The paper does not claim a completed derivation of Born's rule. Instead, it identifies the physical assumptions that such a derivation would require, relates them to historical electron-scattering experiments, and formulates a concrete research program for deriving the observed quadratic detection law from wave amplitude, energy transfer, phase structure, and microscopic detector dynamics.

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A Thought Experiment on Double-Slit Interference from a Localized Odd-Harmonic Source: Shape Preservation Is Conditional and Fragile, and the Single-Wavelength N=1 Is the Robust Special Case

Noriaki Kihara

Thought experiment (model calculation), not a measurement. Extends the position-readout push-forward (companion paper, isSupplementTo) to a localized source: localization is represented by a constant-amplitude odd-harmonic sum (isolated peak wave S_N). Although S_N is a sum of odd harmonics, it localizes in the far field WITHOUT changing shape only when the source-slit diagonal r_k=sqrt(L^2+(W/2)^2) is an integer/half-integer multiple of the fundamental wavelength: sqrt(L^2+(W/2)^2)=(m/2) lambda0. Alignment is a band of width ~1/(2N) (narrower for larger N); a ~3% change in lambda0 decides interference vs scattering. The central-alignment tolerance is W/L-independent; larger W gives greater off-axis fragility. An aligned localized source inherits the cos^2 fringe-shift readout under positional fluctuation but its peaks fall below cos^2 off-axis (extra scattering loss). The single wavelength N=1 has none of this fragility and reproduces the companion paper to machine precision. Net result is negative: localization adds an alignment constraint and off-axis fragility; shape preservation is conditional and fragile, with N=1 the robust special case. Distinguished from spatial coherence / van Cittert-Zernike visibility loss.

•Jun 29, 2026•1 citation
We regard the constant-amplitude odd-harmonic sum S_N (isolated peak wave) on the half-wavelength phase interval as the localized kernel of observation, and adopt two premises: (i) observation is the convolution of the kernel with phase differences and the probability is the square of the observed amplitude; (ii) phase differences finer than the band are ignored (finite-N truncation). Because S_N is the truncated reproducing kernel of the odd-harmonic basis on this interval, the observed amplitude reproduces any band-limited base wave without distortion, and the observed distribution coincides exactly with |psi|^2 (after normalization). For a complex base wave the squaring becomes a genuine modulus |Z|^2 = Z conj(Z) (not Z^2), reaching the real-imaginary cross structure of the Born square. The paper derives the FORM of the distribution (faithful reproduction by the reproducing-kernel property); the squaring rule and the probability interpretation remain postulates. It does not claim to derive the Born rule or solve the measurement problem; rather it reduces the remaining postulates to the squaring rule and the existence of randomness. The reproducing-kernel identity itself is a classical fact of Fourier analysis (Dirichlet kernel / RKHS); the novelty is the mapping between the physical localized wave, its reproducing kernel, and the observation model. All results are reproduced to machine precision by the accompanying verification code.
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.
Observational note. For the isolated peak wave formed by superposing constant-amplitude odd harmonics on the half-wavelength phase interval, giving two copies a common relative phase 2a applied identically to all harmonics yields, by the product-to-sum identity, psi = 2 cos(a) S_N, hence I = 4 cos^2(a) I_N. Two elementary consequences follow: (i) waveform invariance - the normalized waveform is independent of the relative phase and coincides with that of a single copy; (ii) a contrast law - the relative phase scales the squared amplitude by cos^2(a) without changing the waveform. No physical interpretation is given.
This paper shows how the L2 norm underlying the Born rule arises as a phase-invariant quadratic observable when linear spectral representations are applied to discrete systems governed by L1 structure. Rather than assuming Hilbert space geometry, the result demonstrates how quadratic probability emerges from a combination of discrete combinatorics, linear representation, and phase invariance. The central result establishes that, given an underlying L1 causal structure, a linear spectral representation of states, and invariance under phase rotations, the expected observable magnitude under uniform phase averaging reduces to a sum of squared amplitudes. This recovers the L2 norm and yields the Born rule after normalization. Within this framework: Discrete systems are modeled using L1-based combinatorial structure Observers introduce linear complex representations of these systems Phase variables act as unobservable degrees of freedom and are averaged using a uniform invariant measure Cross-terms cancel under phase averaging, leaving a purely quadratic observable This provides a structural explanation for why quantum probability takes a quadratic form. The result can be interpreted as a consequence of Parseval-type identities applied under phase invariance, rather than as an independent postulate. The paper includes: A formal derivation of the quadratic observable from phase averaging A proof that only the quadratic (p = 2) measure preserves underlying combinatorial volume Numerical validation demonstrating consistency with Parseval identity Simulations reproducing standard interference behavior and norm preservation A construction showing compatibility between defect-based dynamics and Schrödinger-type evolution under additional assumptions The work does not attempt a full derivation of quantum mechanics. Instead, it establishes how L2 structure and Born-rule probabilities arise naturally from discrete L1 systems when represented in a linear spectral form with phase invariance. This result connects discrete geometry, harmonic analysis, and quantum probability, and provides a bridge between combinatorial models and continuous Hilbert space representations.
One of the most successful, yet most puzzling, structures of quantum mechanics is Born's rule:P(\mathbf{x},t)=|\Psi(\mathbf{x},t)|^2.It establishes, with extraordinarily high experimental accuracy, the connection between the quantum wave function and the statistical distribution of actual measurement outcomes. Yet a more fundamental question remains worth asking: Why is probability given precisely by the modulus squared of the complex wave amplitude? Standard quantum mechanics uses Born's rule as a fundamental rule. This paper instead poses an open research question: Could Born's rule be not an irreducible probabilistic axiom, but a statistical emergence resulting from the combined action of continuous wave-amplitude superposition, quantized energy exchange, and local interaction dynamics? We propose a candidate physical picture:wave-amplitude superposition->spatial interference structure->local interaction->quantized energy exchange->discrete events}->probability distribution The central mathematical question is \Gamma(\mathbf{x})\stackrel{?}{\propto}|\Psi(\mathbf{x})|^2 where Γ(x)\Gamma(\mathbf{x})Γ(x) denotes the occurrence rate of localized discrete detection events. This paper does not claim to have completed such a derivation. Instead, it explicitly formulates the problem as an \textbf{open, computable, testable, and falsifiable research challenge}. We particularly invite young physicists, mathematicians, and researchers skilled in AI and modern computational tools to explore this problem.
Born's statistical interpretation of quantum mechanics transformed the wave function from a dynamical object into a quantity whose squared modulus determines the probability distribution of measurement outcomes. Since the formulation of the Born rule in 1926, the relation between a continuously evolving quantum state and discrete detection events has remained one of the central conceptual questions of quantum theory. This paper proposes a physical reconsideration of that relation. Rather than treating the appearance of a localized detection event as an immediate consequence of wave-function collapse, we distinguish two physically different stages: continuous wave-packet propagation and discrete detector readout. In the first stage, a localized wave packet evolves through an interaction potential and undergoes phase reconstruction, deflection, focusing, or redistribution. In the second stage, the continuously distributed wave amplitude interacts locally with microscopic detector degrees of freedom, producing discrete detection events. The central hypothesis explored here is that the experimentally observed distribution of individual events may arise from a local detector-response functional acting on the incident wave amplitude. In an appropriate regime, a detector response of the form R(x)∝∣ψ(x)∣2R(\mathbf{x}) \propto |\psi(\mathbf{x})|^2R(x)∝∣ψ(x)∣2could emerge from the microscopic interaction between the wave packet and the detector, rather than being imposed as a primitive probabilistic postulate. The paper does not claim a completed derivation of Born's rule. Instead, it identifies the physical assumptions that such a derivation would require, relates them to historical electron-scattering experiments, and formulates a concrete research program for deriving the observed quadratic detection law from wave amplitude, energy transfer, phase structure, and microscopic detector dynamics.
Thought experiment (model calculation), not a measurement. Extends the position-readout push-forward (companion paper, isSupplementTo) to a localized source: localization is represented by a constant-amplitude odd-harmonic sum (isolated peak wave S_N). Although S_N is a sum of odd harmonics, it localizes in the far field WITHOUT changing shape only when the source-slit diagonal r_k=sqrt(L^2+(W/2)^2) is an integer/half-integer multiple of the fundamental wavelength: sqrt(L^2+(W/2)^2)=(m/2) lambda0. Alignment is a band of width ~1/(2N) (narrower for larger N); a ~3% change in lambda0 decides interference vs scattering. The central-alignment tolerance is W/L-independent; larger W gives greater off-axis fragility. An aligned localized source inherits the cos^2 fringe-shift readout under positional fluctuation but its peaks fall below cos^2 off-axis (extra scattering loss). The single wavelength N=1 has none of this fragility and reproduces the companion paper to machine precision. Net result is negative: localization adds an alignment constraint and off-axis fragility; shape preservation is conditional and fragile, with N=1 the robust special case. Distinguished from spatial coherence / van Cittert-Zernike visibility loss.