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.
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.
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.
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.
A numerical experiment that simulates and observes the de Broglie relation lambda=h/p inside a single-source double-slit classical wave model. An electron is modeled as a spatially localized wave built from an equal-amplitude sum of odd harmonics (N>1). The central result is NOT the value of h (which is a near-tautological self-consistency, since lambda0=h/p is the input and the resonance lattice is dense ~1e-19 m): it is (i) that a highly localized odd-harmonic classical wave can survive two-slit interference as a sharp single peak only under an inter-harmonic alignment condition, and (ii) that the observed central fringe spacing DeltaX = lambda' D/W is set by the fundamental wavelength and is independent of the harmonic count N (N controls only the localization sharpness ~1/(N+1)). Scope is stated explicitly: this is neither a first-principles derivation of p*lambda=h nor a derivation of a Delta_x*Delta_p ~ h uncertainty relation. Each electron is given independent Monte Carlo draws of source position (cos^2, +/-lambda0/2) and initial wavelength (cos^2, +/-1%), with interference, alignment and exclusion recomputed exactly per trial. Bilingual (Japanese + English) drafts with source programs.
Observational note (thought-experiment style; not a proof or claim paper). A sequel to the preceding note, which defined a single particle-like state as a phase-bearing local mode on the subjective geometry S^4(R_U) with spread W_a = 2 R_a and deferred the multi-particle case. Here we treat the minimal case: two particle-like states with the same spread phase, on a closed one-dimensional relational setting (period-1 phase line, a single great circle of S^4). Representing each particle as a rectangular standing wave (odd harmonics only), the n-th harmonic amplitude of the superposition is A_n = (2/n) cos(n pi f), where f is the relative position in fundamental wavelengths, and the total squared norm (energy-like quantity) is the triangle wave E(f) = (pi^2/4)(1-2f) for 0 <= f <= 1/2 (the autocorrelation of a square wave). Within the continuous f, discrete feature points appear: fusion (f=0), orthogonality / independent coexistence (f=1/4), and complete cancellation (f=1/2). Depending on whether the superposition is symmetrized or antisymmetrized, complete overlap (f=0) is allowed (boson-like) or makes the field vanish, i.e. exclusive (fermion-like). For unequal frequencies, interference arises only from shared odd harmonics (the irreducible ratio r/s with both r,s odd, i.e. matching 2-adic valuation); without sharing, interference vanishes and placement is free. All results are presented as observations of structural correspondence (with standing-wave quantization and the exchange symmetry of quantum statistics); no derivation of quantum statistics or the spin-statistics theorem is claimed. Notably the existence of the three feature points is universal to any half-wave-antisymmetric (odd-harmonic) profile, while only the linear (triangle-wave) envelope is specific to the square-wave (1/n) choice. This record contains Japanese and English versions (Markdown, LaTeX, PDF) and one figure.
As we saw in the previous chapter, a sinusoidal wave of amplitude ψ0 > 0, wavenumber k > 0, and angular frequency ω > 0, propagating in the positive x-direction, can be represented in terms of a wavefunction of the form ψ(x, t) = ψ0 cos(ω t − k x). (7.1) This type of wave is conventionally termed a one-dimensional plane wave. It is onedimensional because its associated wavefunction depends on only a single Cartesian coordinate. Furthermore, it is a plane wave because the wave maxima, which are located at ω t − k x = j 2π, (7.2) where j is an integer, consist of a series of parallel planes, normal to the x-axis, that are equally spaced a distance λ = 2π/k apart and propagate along the x-axis at the fixed phase velocity ω/k = v, where v is the characteristic wave speed. These conclusions follow because Equation (7.2) can be rewritten in the form x = d, (7.3) where d = v t − j λ. Moreover, (7.3) is the equation of a plane, normal to the x-axis, whose distance of closest approach to the origin is d.
In a previous note (1) we argued that the free particle classical action A (relativistic or nonrelativistic) may be written in terms of v=x/t and then varied in x and t independently. One finds that dA/dx partial = p and dA/dt partial = -E implying A= -Et+px with x and t independent. This is the opposite of the classical mechanical idea of x(t). Creating an eigenvalue equation yields -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = Eexp(iA). exp(iA) plays the role of an unusual type of probability we argue which describes two dimensional periodicity (cos(px) and sin(px) ). We have argued that the quantum particle is the medium for a two dimensional wavelike motion of an energy-momentum wave. A wave type structure is needed to describe dynamics (like a classical wave), but unlike a classical wave two dimensions are required because there is no preference of any x point for a free particle. Both cos(px) and sin(px) both treat x in a preferential manner as their values vary with x. For a classical wave this is fine because the creation point of the wave imposes boundary conditions, but there is no such boundary condition imposed on a free particle. exp(ipx-iEt0 must somehow then link to P(x)=constant. One may note that exp(-iA)exp(iA) = 1 for the free particle. This may be argued to be linked to spatial invariance i.e. P(x)=constant where P(x) is not exp(ipx). Here we argue that exp(-iA)exp(iA) represents a product of two unusual probabilities , one for motion in a certain direction and the other for the time reversed scenario. If the quantum particle is the medium of an energy-momentum wave exp(-iEt+ipx), then the product of the regular and time reversed unusual probabilities cancel or undo each other. In other words one does not see any change in space as long as one has a noninteracting exp(-iEt+ipx) . Given that -id/dx is a linear operator this product yields a momentum of 0. We consider this idea and its implications on a momentum distribution Sum over p a(p)exp(ipx) which may represent a bound state or other states. In particular, it seems that P(x)=W*(x,t)W(x,t) which yields a classical type of probability (P(x) real, positive and normalized to 1) is linked to a product (merging) of forward and time reversed situations so that any spatial density which remain due to the quantum particle acting as a medium for an energy-momentum wave (as well as the time reversed part) would represent an actual change in spatial density. exp(ipx) already contains two periodic spatial functions (cos(px) and sin(px)), but does not manifest it unless there is an interaction.