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.
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 paper presents the 2013 double-slit experiment "Particle Solution to the Double Slit Experiment" (University of Arizona), together with a more rigorous 2026 reconstruction of the same test. Part I reproduces the original study verbatim. A slit was cut at a dark fringe on Screen 1 and the light was measured on Screen 2 behind it. The detector current behind the dark fringe was 1.4 mA, equal to the 1.4 mA laser-off noise. The result was interpreted as supporting the particle view. Part II specifies a high-precision modular protocol designed to discriminate between dark-fringe occupancy and interference: a common baseline (Module 0), the Screen-1 dark-fringe profile (A), an aperture-width series (B), Screen-2 profiles behind dark and bright apertures (C), an offset scan (D), and a closure map (E). The standard optical-field prediction is frozen before acquisition. If no light is observed on Screen 2 behind the dark-fringe aperture while the bright control transmits, the particle view is supported. If two fringes appear, matching the frozen template, the wave view is supported. Status: Part I includes the original 2013 detector readings. Part II is a prospective protocol; no data have been acquired under it. Files: manuscript (PDF), LaTeX source, supplementary forward model (double_slit_field_model.py), and example configuration (asbuilt_example.json), which reproduces the illustrative values in Section II.12.
A frozen numerical record of a two-dimensional electron double slit in which the idealized impenetrable wall is replaced by a barrier of finite height and finite thickness, and the response is measured against a finite low-transmission control rather than against an infinite wall. Three artifacts. The primary record contains a preregistration of five cases, an independent split-operator solver, the closure evidence and a gate ledger; it was cryptographically committed before a blind exchange with an independent worker who had frozen his own calculation. A detector-position supplement varies only the plane at which the current is recorded, 1–14 nm, and identifies which planes are physically usable rather than forcing them. A wall-height record varies only the barrier height, 2.35–24 eV on both sides of the control, and both thicknesses. What the records establish: the finite-minus-control response separates into a through-thickness term, calibrated independently by the no-slit geometry, and a face and edge term that survives when the thickness kills tunnelling; a hard spatial partition into "material" and "aperture" contributions is not well defined, since shifting the mask by one or two grid cells reverses the sign of the attributed bulk contribution; the residual vanishes only when the two walls are identical and reverses sign when the test wall is the harder one, so it is the difference between two standard responses and not a property of one wall; and its form is one scalar function of the wall height times one geometry constant. What they do not establish, and prove they cannot establish in this geometry: which property of the wall carries that function, since a null regressor with no derivation behind it fits equally well and the measurement that would separate the two is smaller than the numerical systematics. No additional quantum path, no Born-rule violation and no physics beyond the time-dependent Schrödinger equation is claimed or supported. Withdrawn constructions, corrections and errata are retained in the records rather than removed. Each archive carries a manifest, a verifier that re-derives its summary from the raw output and rebuilds its report byte-identically, and a set of mutations that must each be caught — six, five and seven negative controls respectively. The two supplements reproduce values of the primary record's gate ledger bit for bit. Every claim carries its register: exact result, computed here, conditional, constitutive, open, or prediction.
The double-slit experiment combines three observational facts: interference fringes in the coherent regime, position-resolved detections, and continuous fringe suppression under which-path resolution. Standard quantum mechanics accommodates all three through postulates without deriving them from deeper structure. This paper shows that the standard far-field interference law and the coherence-to-which-path transition follow as structural corollaries of results established in Quantum Mechanics as the Boundary Algebra of a Non-Invertible Projection (doi:10.5281/zenodo.19535588) and its companion (doi:10.5281/zenodo.19571321). Two results suffice for the algebraic derivation: the Born rule as the canonical quadratic measure on Im(Π), and the kernel spinor as the Hilbert quadrature of the boundary field. A third result — measurement irreversibility as a kinematic consequence of the projection geometry — belongs to the physical interpretation of measurement and temporal ordering within the same framework. The paper derives the Fraunhofer intensity law I(x) ∝ |Ψ₁ + Ψ₂|² and the detector-overlap formula with fringe visibility V = |γ|, within the standard Fraunhofer approximation and without additional postulates. It does not claim a first-principles derivation of the full boundary propagator or of single-event arrival statistics. A companion numerical simulation implements all three coherence regimes. A self-contained appendix summarises the three imported results in compact form.
The double-slit experiment is often framed as evidence that microscopic objects are “both waves and particles,” or that a wavefunction “collapses” upon measurement. This paper gives a QMU-native, apparatus-centric explanation in the Aether Physics Model (APM) by separating the experiment into three operational layers: (1) Transport layer (space): Light is treated as a conserved transport content, denoted here as ligt, distributed over expanding photon fronts. The measurable quantity in space is the delivered areal density (intensity) lint(ν;r). Interference is asserted at this transport layer: two outgoing channels can superpose and form stable bright and dark regions in lint at the screen plane before any detection event occurs. (2) Boundary layer (aperture): The slit-mask is modeled as a transfer operator T acting on the incident delivery density:lintout(ν;r)=Tlintin(ν;r).For a double slit, the outgoing density is produced by two transfer channels, T1 and T2, whose superposition yields an intensity-level interference formlinttot(P)=lint1(P)+lint2(P)+2lint1(P)lint2(P)cosΔϕ(P).The phase difference is written in QMU-normalized form using the chronovibration identity and the dimensionless cadence:c=FqλC,ν∗=Fqν,λ=ν∗λC,so thatΔϕ(P)=2πλΔℓ(P)=2πν∗λCΔℓ(P).In the far-field approximation with slit separation d, screen distance L, and transverse coordinate x, Δℓ(x)≈dx/L, givingΔϕ(x)≈2πν∗LdλCx.This emphasizes that the fringe map is a geometric count in units of λC scaled by the dimensionless cadence ν∗. (3) Detection layer (matter): Detection is a discrete resonance gate event in matter, not a continuous “field readout.” The receiver is represented by a binary acceptance functionG(ν∗,alignment)∈{0,1},and the event-rate density follows the gated effective drivelinteff(r)=∫G(ν∗,alignment)R(ν∗)lint(ν;r)dν,N˙e(r)∝linteff(r).Single-particle hits therefore build the fringe pattern because discrete events sample a pre-existing transport interference pattern in lint. Coherence window and a falsifiable fringe condition The paper introduces a receiver coherence window τg and its QMU-normalized form τg∗=Fqτg. This produces a detector gate coherence lengthLg=cτg=λCτg∗,and a direct, falsifiable interference conditionΔℓ(P)≲λCτg∗.A key operational refinement is the \emph{minimum law} for the observed visibility rolloff length:Lrolloff≈min(Lc,Lg),where Lc is a transport coherence length set by source bandwidth / wavefront quality, and Lg is the detector’s gate coherence length. This predicts two distinct regimes and a detector-dependent saturation plateau when Lc is increased beyond Lg. Which-path marking as controlled timing-jitter injection Which-path devices are treated as measurable degradations (timing/phase jitter, alignment scrambling, or mode partitioning), not metaphysical collapse. A minimal quantitative model treats marking as injecting an RMS delay σt between channels, with QMU-normalized jitter σt∗=Fqσt. For Gaussian jitter, the visibility obeysV=V0exp[−2π2(ν∗)2(σt∗)2],yielding a clean functional form for tunable weak-to-strong marking protocols. Slit-edge participation and QMU surface knobs (including qsfo) To make boundary participation experimentally reportable, the slit transfer operator is decomposed asT=Tgeom+Tsurf,where Tsurf captures mask material / coating / roughness / polarization-sensitive surface channels. The paper introduces QMU-native surface oscillation knobs:qsfo:=λCc3=λC2Fq3,qsfi:=c3λC=qsfo1,and defines a dimensionless surface-participation ratioΠsurf:=Ωsurfqsfi,so that “edge effects” are reported as measured QMU parameters rather than qualitative caveats. What this upload contains This Zenodo upload includes the full paper (PDF) written in the QADI template style, with: (i) a transport/boundary/detection decomposition, (ii) QMU-normalized fringe geometry, (iii) a detector-dependent coherence-window prediction (τg∗ and Lg), (iv) the minimum-law saturation test, (v) a timing-jitter which-path reduction law, and (vi) QMU surface-mode reporting via qsfo/qsfi. The appendix provides SI bridges and explicitly states that numerical anchors are reported using the MKS (SI) version of the QMU conversion tables. Suggested citation Thomson III, D. W. The Double-Slit Experiment in the Aether Physics Model (APM): Transfer Operators, Gate Coherence, and QMU Predictions. DOI: 10.5281/zenodo.18446424 Zenodo (Version 1.0).
## Record metadata - **Title:** One FIN Operator, Two Temporal Dynamics: Unitary Propagation, Reversible Diffusion, a Finite Double-Slit Experiment, and Operational Completion- **Creator:** Żuchowski, Krzysztof- **Affiliation:** Independent Researcher — Fractal Information Theory Project- **ORCID:** 0009-0002-0909-3613- **Resource type:** Publication — Preprint - **Version:** 1.0.0- **Publication date:** 2026-07-19- **Language:** English- **Publisher:** Zenodo- **Access:** Open access, subject to the license confirmation below- **Reserved DOI:** [10.5281/zenodo.21435332](https://doi.org/10.5281/zenodo.21435332) ## Abstract This technical note studies a precisely defined twelve-site FIN matrix and the two canonical temporal calculi generated by its weighted graph Laplacian. Let (W) be the zero-diagonal real symmetric cyclic-distance matrix on (mathbb Z_{12}) sampled from K(d)=1+d1.8cos(0.18575d+0.16250), using cyclic distances (d=1,ldots,6). Its common row sum is s=1.660307278766099 to fifteen decimal places, and A=sI−W. All off-diagonal weights used at (N=12) are positive. Therefore (A) is a connected weighted graph Laplacian and ⟨f,Af⟩=21∑x,yWxy∣fx−fy∣2≥0. The same positive self-adjoint operator generates the unitary group Ut=e−itA and the irreducible, symmetric, uniform-reversible Markov semigroup Pt=e−tA. Although the two evolutions share Fourier eigenvectors and eigenvalues, their operational predictions are inequivalent. Markov diffusion has linear short-time escape, exact Chapman–Kolmogorov composition and exponential relaxation to the uniform law. Unitary position probabilities have quadratic short-time escape, interference, failure of population-level Chapman–Kolmogorov composition and no pointwise equilibrium. The release includes a reproducible finite double-slit analogue. A coherent preparation at vertices (-2) and (+2) is compared with a perfectly distinguishing which-path record and with classical diffusion. The interference term is displayed across all twelve detectors and through time. At the first local maximum of total interference, (t_*=1.5525), the symmetry detector has unit ideal visibility; reducing the overlap of the two environment/apparatus records continuously suppresses the fringe. The note also identifies the missing structure needed to convert the operator into an experimental model. The smallest established-style candidate is a **calibrated process–tester model**: a causal process tensor (quantum comb), an admissible tester containing preparations and instruments with classical records, and a clock/reference calibration. This object carries state, multitime dynamics, preparation, instruments, environmental memory, apparatus and records up to operational equivalence. The operator alone does not select a physical clock, dimensional scale, environment, apparatus, Born rule, orientation or temporal semantics. No cosmological, Theory-of-Everything or continuum-physics claim is made. ## Plain-language significance One finite operator can generate both wave-like and diffusion-like mathematics. That fact does not mean an observer turns one physical law into another. It means the static operator underdetermines the experiment. State preparation, phase reference, clock, environment, measurement instrument and stored record must be supplied before either family becomes a testable physical model. The included double-slit analogue makes the distinction visible: coherent alternatives interfere; a which-path record removes the cross term; classical diffusion produces a different distribution. These are three operational protocols built around the same finite matrix. ## Main results and confidence labels ### Proven - (A=sI-W) is positive semidefinite with (ker A=\operatorname{span}\{\mathbf 1\}) under the declared cyclic-distance convention.- The spectral gap is (0.754121154207080) to displayed precision.- (U_t=e^{-itA}) is unitary.- (P_t=e^{-tA}) is an entrywise positive, symmetric, doubly stochastic Markov semigroup for (t>0).- The scalar shift (s) is the unique scalar shift of this regular (W) that makes (A\mathbf1=0).- Born-position matrices (|U_t|^{\circ2}) do not form a nontrivial differentiable Markov semigroup.- No nontrivial continuous finite evolution can be both unitary and entrywise nonnegative stochastic.- The two-slit probability law follows from the stated coherent preparation, record overlap and detector POVM.- A dimensionful time, energy, mass, length or action scale cannot be selected from the dimensionless matrix alone. ### Computed and independently checkable - All six weights, the complete spectrum, the spectral gap and the Dirichlet-form residual.- Markov mixing times and the nonuniform unitary Cesàro position law.- Unitarity, stochasticity and Chapman–Kolmogorov residuals.- The first two-slit interference maximum (t_*=1.5525) and total-variation interference (0.1232806696).- Ideal phase-fringe visibility (1) at detector (x=0) for record overlap (eta=1). ### Candidate / strong structural evidence - A calibrated process–tester model is the smallest standard mathematical structure found that simultaneously carries the requested operational roles. ### Not established - That the FIN matrix is a measured Hamiltonian or measured stochastic generator.- That unitary or diffusive time is selected by FIN itself.- That the displayed parameter (t) is measured in seconds.- That Shannon entropy in the plots is thermodynamic entropy.- That the (N=12) Markov interpretation survives a continuum limit.- That a microscopic apparatus, environment, selector, orientation or physical constants are derived from (A). ## Double-slit simulation The simulated slit modes are (a=10\equiv-2\pmod{12}) and (b=2). The coherent system–record state is ∣Ψϕ⟩=2∣a⟩∣Ea⟩+eiϕ∣b⟩∣Eb⟩,η=⟨Eb∣Ea⟩. With (u_j(x,t)=\langle x|U_t|j\rangle), the detector law is pϕ,η(x,t)=2∣ua∣2+∣ub∣2+Re(ηe−iϕuaub). The figure compares: 1. coherent propagation, (eta=1);2. a which-path record / incoherent mixture, (eta=0);3. classical diffusion (P_t(\delta_a+\delta_b)/2). This is a finite two-path interferometer on (mathbb Z_{12}), not a continuum optical diffraction calculation. ## Important finite-size limitation The Markov conclusion uses positivity of all off-diagonal weights present at (N=12). The fixed profile becomes negative at distance (8): K(8)=−0.0017958784. A naive cyclic extension first includes that distance at (N=16), where the resulting off-diagonal matrix entry cannot be a Markov jump rate. Any truncation, rectification, absolute-value prescription or parameter running would be an additional axiom. The release therefore makes an exact (N=12) claim and no automatic continuum-limit claim. ## Files in this release | File | Purpose | SHA-256 ||---|---|---|| `FIN_Dual_Dynamics_Operational_Completion.pdf` | Archival English research note | `53bad7dfc10446211d9bc6bc28da53470ee3e46b4b4084d23f2301a439f2456c` || `FIN_Dual_Dynamics_Operational_Completion.tex` | LaTeX source | `be38e241f3b4b7ba73230845cc584806c1b5102c18de6dbcc1dde0818b614765` || `fin_dual_dynamics_release.py` | Deterministic numerical audit and figure generator | `671dca9f2ac0b40f17bad3cda8e253e934db825503f76e1efa787b47c262ad47` || `FIN_Dual_Dynamics_reference_results.json` | Machine-readable expected results | `3917ad393abe286d1f39caeac6596d218b56b369cffdfe405a3aa6f79156e5ab` || `FIN_One_Spectrum_Two_Dynamics.png` | Unitary/Markov comparison figure | `5e37f1ff1e49cce4855247c03f4bd298ade73e3da1ecd4136ad77875c43d2f42` || `FIN_Double_Slit_Operational_Simulation.png` | Four-panel double-slit simulation | `a6a1f59b9c6d5b4df17df6580679b18f9706421ca5a0aedeb916da84ac27488d` || `FIN_Dual_Dynamics_MANIFEST.sha256` | Checksums for the archival files | Generated after the final build | The Markdown release description is intentionally omitted from its own checksum table to avoid a self-referential digest. ## Reproducibility Reference environment used for this release: - Linux 7.0.0-28-generic, x86_64- Python 3.12.3- NumPy 1.26.4- Matplotlib 3.10.8- pdfTeX 3.141592653-2.6-1.40.25 / TeX Live 2023- Floating-point arithmetic: IEEE-754 binary64- Random seed used only for the Dirichlet identity spot-check: `20260719` Run: ```bashpython3 fin_dual_dynamics_release.pypdflatex -interaction=nonstopmode -halt-on-error FIN_Dual_Dynamics_Operational_Completion.texpdflatex -interaction=nonstopmode -halt-on-error FIN_Dual_Dynamics_Operational_Completion.tex``` The script reconstructs the operator from the displayed analytic parameters, generates both figures and writes the complete numerical audit to JSON. It tests symmetry, the constant mode, Dirichlet positivity, unitarity, stochasticity, minimum transition entries, Chapman–Kolmogorov composition, mixing, the raw-(W) heat failure, the first negative extended-profile weight and normalization of all double-slit distributions. ## Keywords FIN kernel; spectral operator; weighted graph Laplacian; unitary group; heat semigroup; reversible Markov chain; continuous-time quantum walk; double-slit interference; which-path information; decoherence; quantum instrument; process tensor; quantum comb; calibrated process–tester model; observer dependence; operational equivalence; finite operator theory; spectral dynamics. ## Suggested citation > Żuchowski, K. (2026). *One FIN Operator, Two Temporal Dynamics: Unitary Propagation, Reversible Diffusion, a Finite Double-Slit Experiment, and Operational Completion* (Version 1.0.0) [Technical note]. Zenodo. https://doi.org/10.5281/zenodo.21435332 ```bibtex@misc{zuchowski_fin_dual_dynamics_2026, author = {Krzysztof Żuchowski}, title = {One FIN Operator, Two Temporal Dynamics: Unitary Pro