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
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.
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.
Thought experiment (model calculation), not a measurement. For a fixed geometry (L=10, W=5, lambda0=1) a stationary point source with positional fluctuation P(y)=cos^2(pi y/lambda) is passed through a double slit. Each trial shifts an identically shaped far-field fringe by the exact geometric path difference u(y), a nearly linear map of the source position. Over repeated trials the fringe-shift distribution is the push-forward rho(u)=P(y(u))|dy/du| of P, preserving the cos^2 shape because the map is paraxially linear; the ~3% departure is the non-paraxial nonlinearity (1/2 tan^2 theta). This is a change-of-variables (push-forward), not a derivation of a probability law. Distinguished from intensity-accumulation visibility loss (spatial coherence / van Cittert-Zernike).
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.
We ask whether the impact point of an individual electron in an interference experimentcan be predicted from its launch point, and what precision this would require. Particlesare described deterministically, guided by the ux of the wave as in the de BroglieBohmformulation, and wave sources are placed according to the base table of a directional-productmodel, xk,n = s ck + Cn with eight levels c = (0, 1, 1.5, 2.5, 3, 4, 5.5, 6). A simulation of thecontrolled electron double-slit experiment of Bach et al., with the electron wavelength writtenas the directional product (pc)2 = K(K + 2mec2), reproduces the interference pattern fromdeterministic, non-crossing paths (correlation 0.98), but shows that launch points spreadover one slit (54 nm) reach six fringes: predicting the fringe of a single electron requires itsposition in the slit plane to be known to about 5 nm. Propagating this requirement back toa nanotip source with the measured eective radius of 0.550.80 nm gives a required launchprecision of 0.48 pm for source-to-object distances of 0.11 mm at 600 eV, far below atomicdimensions. A review of single-electron experiments shows that sources already localise theemission point to below 1 nm without destroying coherence, and that single electrons fromsuch sources still arrive at dierent fringes; no experiment records the in-wave launch positionthat a deterministic prediction would need. A repeat-launch simulation shows why: xing thelaunch position by a measurement of 5 nm (1 nm) precision spreads the impacts over 4.3 (22)fringes and lowers the correlation with the interference pattern from 0.996 to 0.46 (−0.05).Within pilot-wave theory an individual prediction is therefore possible only for an ensembleout of quantum equilibrium, and the proposed test is a test of quantum non-equilibrium.We identify which existing data (per-pulse emission times and arrival positions) can testtiming-based prediction rules, and summarise a pre-registration protocol for a decisive test,in which the prediction rule is to be xed before any data are taken.
The double-slit experiment is commonly interpreted as evidence that a single electron must be described by a spatially extended wavefunction whose path amplitudes interfere. Here, we present an alternative geometric formulation within the Bivector Standard Model, in which the observed far-field interference pattern is reproduced while the phase is attributed to an internal bivector clock carried by the electron. In this approach, each electron remains localized and produces a single detection event, while the familiar fringe pattern emerges statistically from the accumulation of many impacts. Interference arises from the comparison of clock phases associated with geometrically distinct paths, rather than from the superposition of spatial waves. The resulting probability distribution recovers the standard two-slit interference factor, the single-slit diffraction envelope, and the usual fringe-spacing relation in the Fraunhofer regime. The de Broglie wavelength emerges as the spatial manifestation of this transported phase. This formulation provides a geometric account of the origin of phase in single-electron interference, consistent with standard results while offering a distinct physical interpretation.
A benchmark, not a new theory of diffraction. The propagation mathematics used here is established scalar wave optics. What is tested is an architectural question: which information must be present in a solver before the slit pattern can exist? The generator receives a device class, one invariant free-space evolution operator, a binary geometric admissibility mask, and one square-law detector. It does not receive the single-slit minimum formula, the double-slit path-difference relation, the fringe spacing, or a target intensity profile. Only the aperture mask changes between runs. Analytic diffraction formulas are isolated in a separate process that executes after the raw detector profiles have been written to disk. The feature detector that reads those profiles is scale-free and uses one threshold pair, unchanged across every aperture; its isolation from the device parameters is enforced by test rather than asserted in prose. Writing the cascade as R_B = D[U2 C_B U1 psi_P], the observed pattern is the detector-visible residuum of a state space restricted by device geometry and evolved by an invariant free-space law. Results Single slit, 30 µm: first four blind minima agree with post-hoc finite-angle Fraunhofer controls to a maximum relative position error of 0.00953 %. Blind width change to 17 µm, with propagator, detector, feature detector and thresholds all frozen: maximum error 0.00176 %. The detected first-minimum scaling ratio is 1.765138 against an inverse-width prediction of 30/17 = 1.764706. Double slit, 30/150 µm: complex-field propagation is linear to 4.17e-16 relative error, while square-law detection yields a centre coherent-to-incoherent intensity ratio of 1.99999984 and a full normalised profile matching the post-hoc control over ±20 mm with RMSE 6.14e-5. An independent direct-space Rayleigh-Sommerfeld solver, sharing no code path with the angular-spectrum generator, reproduces the same minima to within 1.4 µm. This supplies the validator that the projector-illuminated case lacks, since that case has no closed form: the 66 µm shift of the first minimum under non-uniform upstream illumination exceeds the inter-method disagreement by a factor of 48. Eleven automated tests pass in the reference run. The claim, and its failure condition The contribution is a separation rather than a physical result. The generative content of scalar diffraction admits an exhaustive partition into four individually phenomenon-independent components, and the analytic result laws occupy none of them. That completeness is the claim. It fails visibly if a new aperture geometry within the scalar regime requires a phenomenon-specific change to the evolution operator or the detector. The paper deliberately does not claim that quantum mechanics is unnecessary for interference: in the scalar regime tested, that statement would be uncontroversial rather than bold. The generator does contain physics, and the square-law detector is a postulate, not a theorem. The broader hypothesis about constrained joint state spaces is stated as a research programme and reserved for a subsequent tensor-space test. Contents Complete Python package, both executable stages plus the cross-implementation control, raw profiles, JSON results, all figures as PDF and PNG, eleven pytest acceptance tests, the LaTeX source and compiled PDF, a prior-art note, and a SHA-256 manifest of every file. Reproduce with: python scripts/run_all.py Software in the archive is released under the MIT License; the paper and figures are released under CC BY 4.0.
This thought experiment begins with a naive question about classifying wave-packet collapse and decoherence as separate phenomena. Using experience with laser holography as a guide, it asks whether coherence is not an absolute attribute of a single wave but depends on the partner wave, the harmonic structures of both, their phases and the readout conditions. A localized wave built from aligned high harmonics forms a sharp peak, but its spatiotemporal alignment with another localized wave is extremely restrictive. When a localized wave scatters on a monochromatic background wave, its high harmonics can be redistributed to the background. If a particle acquires a mass-like response through interaction with the background, why does it not keep losing harmonics and dissipate? Re-examining the author's earlier closed two-wave scattering experiment shows that localization is not lost in one direction but is redistributed back and forth between the two waves. The original program was re-run as a byte-identical copy (all data and figures identical to the original output): the two channels are orthogonal, so the per-collision normalization is a no-op and the update is the pure exchange operator U_R; no dissipation occurs, each wave keeps unit power while its peak height changes by a factor of about 30, and the effective harmonic orders satisfy N_A = 1 + 31 sin^2(k omega/2) with N_A + N_B = 33 exactly. If such circulation closes stably, finite periodicity U^n = I may be observed as a result, not imposed as a rule. The paper asks whether periodic closure, phase locking, quasi-stationarity and lifetimes appear as results, and whether the temporal and spatial readouts of the circulation correspond to internal clock delay and Lorentz contraction. The 'background' is treated not as a container with places, times and a number of waves, but as the anonymous part of the total state not yet distinguished by interaction readouts; the paper ends by asking whether particle species are names for conserved quantities of the interaction. This is a thought experiment that formulates verification tasks arising from the reinterpretation of existing numerical experiments; it does not complete derivations of mass, the Lorentz metric, quantum energy levels or real particle lifetimes. Main text in Japanese with an English translation; the verification package of the re-run is included as a zip archive.