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One FIN Operator, Two Temporal Dynamics: Unitary Propagation, Reversible Diffusion, a Finite Double-Slit Experiment, and Operational Completion

Krzysztof Żuchowski

Published Jul 19, 2026• 0 citations
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Abstract

## 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 \[K(d)=\frac{\cos(0.18575d+0.16250)}{1+d^{1.8}},\] 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 \[\langle f,Af\rangle=\frac12\sum_{x,y}W_{xy}|f_x-f_y|^2\ge 0.\] The same positive self-adjoint operator generates the unitary group \[U_t=e^{-itA}\] and the irreducible, symmetric, uniform-reversible Markov semigroup \[P_t=e^{-tA}.\] 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 \[|\Psi_\phi\rangle=\frac{|a\rangle|E_a\rangle+e^{i\phi}|b\rangle|E_b\rangle}{\sqrt2},\qquad \eta=\langle E_b|E_a\rangle.\] With (u_j(x,t)=\langle x|U_t|j\rangle), the detector law is \[p_{\phi,\eta}(x,t)=\frac{|u_a|^2+|u_b|^2}{2}+\operatorname{Re}\!\left(\eta e^{-i\phi}u_a\overline{u_b}\right).\] 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

DOI
10.5281/zenodo.21435331

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Chapter 19: "Fractional Dynamics of Open Quantum Systems"

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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 ## 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)=\frac{\cos(0.18575d+0.16250)}{1+d^{1.8}},] 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 [\langle f,Af\rangle=\frac12\sum_{x,y}W_{xy}|f_x-f_y|^2\ge 0.] The same positive self-adjoint operator generates the unitary group [U_t=e^{-itA}] and the irreducible, symmetric, uniform-reversible Markov semigroup [P_t=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 [|\Psi_\phi\rangle=\frac{|a\rangle|E_a\rangle+e^{i\phi}|b\rangle|E_b\rangle}{\sqrt2},\qquad \eta=\langle E_b|E_a\rangle.] With (u_j(x,t)=\langle x|U_t|j\rangle), the detector law is [p_{\phi,\eta}(x,t)=\frac{|u_a|^2+|u_b|^2}{2}+\operatorname{Re}!\left(\eta e^{-i\phi}u_a\overline{u_b}\right).] 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

# Release 10.21 — FIN Laboratory Transfer Package ## P240 Optimal Tomography + P241 Blind-Custody Validator + P242 One-Shot Pipeline ### A scientific transfer package and executable specification for theoretical and experimental physicists ## Record metadata - **Creator:** Żuchowski, Krzysztof- **Affiliation:** Independent Researcher — Fractal Information Theory Project- **ORCID:** 0009-0002-0909-3613- **Resource type:** Publication — Preprint and executable specification- **Version:** 1.0.0- **Publication date:** 2026-07-27- **Language:** English- **Publisher:** Zenodo- **Access:** Open access- **License:** Creative Commons Attribution 4.0 International (CC BY 4.0)- **Repository:** https://github.com/hyconiek/Fractal-Nadsoliton-Theory- **Related project DOI:** https://doi.org/10.5281/zenodo.21435332 --- ## Abstract This release transfers the experimentally actionable part of the FIN research programme to an independent team of theoretical and experimental physicists. It begins from the finite strict twelve-node kernel K_strict on C_12, not from cosmological or Theory-of-Everything claims. With W = K_strict, s ≈ 1.660307, and A = sI - W, the exact Dirichlet identity ⟨f,Af⟩=12∑x,yWxy∣fx−fy∣2≥0\langle f, Af \rangle = \frac{1}{2} \sum_{x,y} W_{xy} |f_x - f_y|^2 \geq 0⟨f,Af⟩=21​∑x,y​Wxy​∣fx​−fy​∣2≥0 places the same self-adjoint generator under both the coherent group U_t = e^{-itA} and the reversible heat/Markov semigroup P_t = e^{-tA}. Program 240 delivers the optimal spectral tomography protocol. The inverse-log noise factor e^x/x yields the unique time minimizer at x = τ λ_max = 1. Equal allocation across all twelve basis preparations is proved minimax. Matrix-Bernstein spectral bounds supply a distribution-free concentration guarantee, accompanied by synthetic shot-plan analyses (50 000 shots per preparation). Program 241 supplies an executable blind-custody validator checking eleven mandatory measurement, data, calibration, control, chronology, hash, and role-separation fields. It enforces strict provider–registrar–analyst role separation with detached GPG signatures. Program 242 is a fail-closed one-shot analysis pipeline. It evaluates holdout prediction P_{2τ}^pred = P̂_τ^2 against the sealed P̂_{2τ} data and reports binary pass/fail without model repair. No external physical bundle is supplied in this release, keeping Program 242 locked until genuine laboratory data are admitted. --- ## Key Features & Main Results 1. Dual Semantics of One Generator: Proves that coherent quantum walk dynamics U_t = e^{-itA} and diffusive Markov heat flow P_t = e^{-tA} share the same spatial generator A = sI - W on C_12, while clarifying that temporal semantics require operational instrument definition.2. Exact Tomographic Time Minimizer (P240): Proves x = τ λ_max = 1 is the unique minimizer of local absolute-noise amplification e^x/x for matrix logarithm rate extraction.3. Minimax Equal Allocation (P240): Proves cyclic symmetrization makes equal shot allocation across all 12 basis preparations minimax within the declared convex design class.4. Production Custody Validator (P241): Provides an 11-field schema and hash validator (fin_lab_p241_validator.py) enforcing provider ≠ registrar ≠ analyst role separation.5. Fail-Closed One-Shot Pipeline (P242): Implements an atomic ledger-locked analysis pipeline (fin_lab_p242_pipeline.py) that evaluates time-homogeneity P_{2τ} = P_τ^2 and the 0.02 projective spectral fingerprint without post-hoc threshold adjustment or model repair.6. Fully Validated Test Suite: Includes an 18-test regression suite (test_fin_lab_p240_242.py), all passing cleanly. --- ## Included Files in FIN_Laboratory_Transfer_Package_P240_242.zip - FIN_Laboratory_Transfer_Package_P240_242.pdf — Archival English preprint and executable laboratory transfer specification.- fin_lab_p240_optimal_tomography.py — Target construction, time/allocation optimization, matrix Bernstein bounds, and synthetic shot planner.- FIN_Lab_P240_Design_Lock.json — Canonical registered design lock for Program 240.- FIN_Lab_P240_Optimal_Tomography_Results.json — Executed numerical results for P240.- fin_lab_p241_validator.py — Production 11-field schema, hash, chronology, and role-separation validator.- FIN_Lab_P241_Transfer_Template/ — Complete directory of raw data schemas, manifest templates, and calibration headers for laboratories.- fin_lab_p242_pipeline.py — Fail-closed one-run external analysis pipeline.- FIN_Lab_P242_Analysis_Lock.json — Canonical frozen analysis lock for Program 242.- test_fin_lab_p240_242.py — Complete test suite (18 unit/falsification tests).- FIN_Lab_P240_242_MANIFEST.sha256 — SHA-256 integrity manifest of all package files. --- ## Keywords quantum process tomography; Markov semigroups; continuous-time quantum walk; matrix logarithm; matrix Bernstein bounds; blind custody; preregistration; fail-closed pipeline; operational physics; spectral graph theory; experimental quantum optics; integrated photonics. --- ## Suggested Citation Żuchowski, K. (2026). FIN Laboratory Transfer Package — Release 10.21: P240 Optimal Tomography + P241 Blind-Custody Validator + P242 One-Shot Pipeline (FIN Research Monograph & Transfer Package; Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.21435332
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.
We reconstruct nonrelativistic quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. The single-run physical state is a finite topological core, its actual causal history, the substrate state, its sector, and the associated historical disturbance. A wave function is not identified with that ontic tuple. It is a ray in a Hilbert representation of admissible preparations and response amplitudes; a density operator is a statistical or reduced representation; and a record is a stable macroscopic event. The logical construction is acyclic. Finite-energy sector spaces and their continuous symmetry representations lead, under explicit representation conditions, to a complex Hilbert direct sum. A closed semibounded quadratic form defines the Hamiltonian and its self-adjoint realization. The standard Schrödinger action, unitary propagation, probability current, spectral observables, uncertainty relations, composite-system tensor products, spin representations, exchange sectors, instruments, and completely positive open dynamics then follow on their declared domains. Gleason-type reasoning fixes the Born form only after additivity, noncontextuality, continuity, and preparation–detector calibration are stated. Those operational conditions are not hidden physical axioms. The distribution of unresolved ontic initial data that realizes quantum equilibrium remains a realization certificate, not a theorem of topology alone. The Heisenberg representation is derived as a controlled transport of observables after Hilbert space and the self-adjoint generator have been constructed. Strongly continuous time translation, two-parameter nonautonomous propagation, common-domain Heisenberg equations, spectral matrix mechanics, regular Weyl relations, open-system adjoints, and instrument effects are connected by one certificate. Schrödinger and Heisenberg predictions are proved equivalent for one-time and finite instrument records, while finite-core, historical, truncation, domain, and detector errors remain separately bounded. Three frozen matrix benchmarks verify low-energy oscillator truncation and its state-weighted CCR defect, second-order unitary convergence for a driven doublet, Schrödinger–Heisenberg equality for a sequential instrument, finite auxiliary-history truncation, and quadratic leakage scaling for a conditional Dirac–Pauli interface. Independent reproduction, including the extended-precision audit, is closed for the frozen packet; physical coefficient calibration remains open. The next interface connects an owner-resolved Dirac–Pauli Hamiltonian to spin records through a Ward-compatible vertex, a causal passive history kernel, an independently audited doublet projection, SPAM-aware forward probabilities, nuisance-profiled identifiability, and a noncompensatory coefficient-to-record error theorem. The DPHC identifiability coordinate is executed on a frozen synthetic response family. A single-scale design is rank deficient, whereas an eighteen-setting frequency–gradient–order–time design has full profiled rank, controlled conditioning, near-nominal finite-sample coverage, and a calibrated held-out statistic. A robust twenty-setting design raises the worst registered information eigenvalue by 20.12%20.12\%20.12%. A ten-parameter nonlinear audit exhibits an exact radiative–detector equivalence curve. An independent dark record breaks that symmetry and restores full local rank. Continuous-set robustness, validated global exclusion, and independent implementation are executed, while physical response identification and empirical calibration remain open. Measurement is treated as a dynamics of system, apparatus, environment, and actual history. Unitary premeasurement and decoherence select robust record sectors but do not by themselves select one record. The three-axiom ontology supplies one actual branch history, while a measurable response map from the complete initial state and instrument to the final record supplies the missing dynamical content. Bell correlations require contextual and nonfactorizable ontic response while parameter independence and operational no-signalling are retained; no controllable superluminal channel is introduced. Historical response produces causal non-Markov Hamiltonians and memory-kernel master equations. Their normalization, positivity, complete positivity, and energy ledger are derived from a unitary enlarged realization whenever such a dilation exists. Finite-core form factors and causal memory yield explicit departures from point-core Markov quantum mechanics. Standard quantum mechanics is recovered as the unresolved-core, sector-fixed, quantum-equilibrium, short-memory limit. Quantum field theory is retained as a successful local, spectral, and asymptotic calculus, but field operators, Fock vectors, and vacuum functionals remain effective representations until a noncircular map from finite carriers and their complete causal histories has been certified. The manuscript proves a sequence of previously open analytic, mathematical, measurement, topological, statistical, and computational obligations on explicitly declared domains. These include non-Markov diffusion, graph-norm stability, distorted-wave and pole variation, finite passive-memory realization, Fisher-information identifiability, detector pushforwards, unique-record basins, Born-representation bounds, measurement independence, topology–Dirac charge conjugation, Stern–Gerlach records, action-scale matching, rank-two spin projection, Finkelstein–Rubinstein topology, Clifford reconstruction, and parent-to-record realization maps. Eighteen groups of numerical obligations are executed on frozen reference models. They cover spectral and scattering calculations, oscillator and two-level dynamics, Schrödinger–Heisenberg equivalence, finite-history truncation, Dirac–Pauli and Foldy–Wouthuysen reduction, gauge covariance, interference, Bell and GHZ correlations, open-system memory, Stern–Gerlach records, identifiability, global searches, topological invariants, path products, semiclassical limits, partial-wave optical closure, the relativistic/QFT interface, and conservative-parent two-level dynamics. Every environment explicitly typed as a computational obligation now has an executed frozen-model certificate; empirical identification and separately typed experimental inputs remain open. The present manuscript differs so substantially from the earlier version in scope, logical architecture, proofs, and executable certificates that it is issued as a new upload rather than as a minor revision. Because the subject is both foundational and technically consequential, readers professionally engaged in quantum foundations, mathematical physics, quantum measurement, or related fields are encouraged to consult both versions: the earlier version preserves the developmental route, whereas the present version gives the substantially expanded and reorganized reconstruction. ## Keywords Axiomatic quantum mechanics; finite particle histories; retarded Green functions; Schrödinger equation; Dirac equation; Born rule; topological spin; quantum measurement; non-Markovian dynamics; classical limit.