Hoffman’s conscious-agent formalism defines agents by Markov kernels on finite sets, anda programme has sought to obtain quantum mechanics—complex amplitudes, Hilbert space,unitary evolution and the Born rule—as outputs of that stochastic substrate. Papers I–III ofthis series closed the passive-readout, witness-inequality and Born-rule-inversion routes. Herewe examine the Eulerian control principles on the continuous-time nearest-neighbour jumprelative of the programme’s N -cycle substrate, with the phase S posited as an agent-carriedvariable and the rates depending on the population density, and ask which of them are exactlythe lattice Schrödinger pair at finite lattice spacing a. (i) No member with a kinetic termquadratic in the rates is exact, for any N , wherever its stationary drift exists (a degree count);by Legendre duality, (ii) no globally optimising member, with any cost, over an S-independentadmissible rate set, is exact on the whole configuration space, because the reduced Hamiltonianof a minimiser (maximiser) is convex (concave) in the phases while the lattice Schrödingerenergy is neither; (iii) an optimising member with bothprates free nevertheless exists—the2 —whose minimiser isconvex bond cost cQ (J) = ℏJ arcsin(J/Jmax ) + ℏJmax 1 − J 2 /Jmaxexactly the lattice Schrödinger pair, density dependence included, on the open winding-zerosector where every real bond phase satisfies |∆x | < πℏ/2, and which fails on the outer half ofthe zone, nodes included, and on every configuration with nonzero winding; with an activitycost selecting the symmetric part of the rates, that optimiser is the Bell–Vink process; (iv)exact stationary-point members exist beyond that sector, Bell–Vink among them, determinedup to the split of each bond energy and a uniform circulating current, with the branchselection as the rate law. The cost cQ is the entropic (Schrödinger-bridge) bond cost with thenoise scale continued to imaginary values, cE |Jmax →iJmax = −i cQ on principal branches, aformal correspondence. As a diagnostic, the fixed-noise Nelson–Yasue member with κ = +1carries the Euclidean sign of the quantum potential, κQ = −κ; the κ = −1 (Guerra–Morato)member is exact in the continuum and, where its stationary drift exists, departs from thelattice target at O(a2 ). Nelson’s law of motion, imposed on the fixed-total-rate kinematicswithin the rate bound |bx | ≤ 2νa for odd N , has the real-time sign, is exact for plane wavesand pure-potential configurations, and departs at O(a2 ) generically, with an exact vanishingfamily and in-regime counterexamples at N = 5, 7. A programme-status section recordsthat the substrate, read as a one-time operational probabilistic theory, lies in the classicalbranch of the Hardy and Chiribella–D’Ariano–Perinotti reconstructions, and corrects onestatement of Paper III (Barandes’ precondition is met by every finite-alphabet process; thesentence was true but empty). These are bounded no-go theorems for the reading constructedhere; no claim is made about quantum mechanics emerging from stochastic dynamics ingeneral. All machine results are exact-arithmetic and rerun bit-identically. The Round-1lattice results and the reconstruction-axiom checks were preregistered with blind priors; thetheorems added in revision were prompted by review and are machine-checked by successorverifiers, not preregistered.
Hoffman’s conscious-agent formalism defines agents by Markov kernels on finite sets, anda programme has sought to obtain quantum mechanics—complex amplitudes, Hilbert space,unitary evolution and the Born rule—as outputs of that stochastic substrate. Papers I–III ofthis series closed the passive-readout, witness-inequality and Born-rule-inversion routes. Herewe examine the Eulerian control principles on the continuous-time nearest-neighbour jumprelative of the programme’s N -cycle substrate, with the phase S posited as an agent-carriedvariable and the rates depending on the population density, and ask which of them are exactlythe lattice Schrödinger pair at finite lattice spacing a. (i) No member with a kinetic termquadratic in the rates is exact, for any N , wherever its stationary drift exists (a degree count);by Legendre duality, (ii) no globally optimising member, with any cost, over an S-independentadmissible rate set, is exact on the whole configuration space, because the reduced Hamiltonianof a minimiser (maximiser) is convex (concave) in the phases while the lattice Schrödingerenergy is neither; (iii) an optimising member with bothprates free nevertheless exists—the2 —whose minimiser isconvex bond cost cQ (J) = ℏJ arcsin(J/Jmax ) + ℏJmax 1 − J 2 /Jmaxexactly the lattice Schrödinger pair, density dependence included, on the open winding-zerosector where every real bond phase satisfies |∆x | < πℏ/2, and which fails on the outer half ofthe zone, nodes included, and on every configuration with nonzero winding; with an activitycost selecting the symmetric part of the rates, that optimiser is the Bell–Vink process; (iv)exact stationary-point members exist beyond that sector, Bell–Vink among them, determinedup to the split of each bond energy and a uniform circulating current, with the branchselection as the rate law. The cost cQ is the entropic (Schrödinger-bridge) bond cost with thenoise scale continued to imaginary values, cE |Jmax →iJmax = −i cQ on principal branches, aformal correspondence. As a diagnostic, the fixed-noise Nelson–Yasue member with κ = +1carries the Euclidean sign of the quantum potential, κQ = −κ; the κ = −1 (Guerra–Morato)member is exact in the continuum and, where its stationary drift exists, departs from thelattice target at O(a2 ). Nelson’s law of motion, imposed on the fixed-total-rate kinematicswithin the rate bound |bx | ≤ 2νa for odd N , has the real-time sign, is exact for plane wavesand pure-potential configurations, and departs at O(a2 ) generically, with an exact vanishingfamily and in-regime counterexamples at N = 5, 7. A programme-status section recordsthat the substrate, read as a one-time operational probabilistic theory, lies in the classicalbranch of the Hardy and Chiribella–D’Ariano–Perinotti reconstructions, and corrects onestatement of Paper III (Barandes’ precondition is met by every finite-alphabet process; thesentence was true but empty). These are bounded no-go theorems for the reading constructedhere; no claim is made about quantum mechanics emerging from stochastic dynamics ingeneral. All machine results are exact-arithmetic and rerun bit-identically. The Round-1lattice results and the reconstruction-axiom checks were preregistered with blind priors; thetheorems added in revision were prompted by review and are machine-checked by successorverifiers, not preregistered.
Systèmes quantiques ouverts et processus stochastiques quantiques De nombreux phénomènes de physique quantique ne peuvent être compris que par l'analyse des systèmes ouverts. Un appareil de mesure, par exemple, est un système macroscopique en contact avec un système quantique. Ainsi, tout modèle d'expérience doit prendre en compte les dynamiques propres aux systèmes ouverts. Ces dynamiques peuvent être complexes : l'interaction du système avec son environnement peut modifier ses propriétés, l'interaction peu créer des effets de mémoire dans l'évolution du système, . . . Ces dynamiques sont particulièrement importantes dans l'étude des expériences d'optique quantique. Nous sommes aujourd'hui capables de manipuler individuellement des particules. Pour cela la compréhension et le contrôle de l'influence de l'environnement est crucial. Dans cette thèse nous étudions d'un point de vue théorique quelques procédures communément utilisées en optique quantique. Avant la présentation de nos résultats, nous introduisons et motivons l'utilisation de la description markovienne des systèmes quantiques ouverts. Nous présentons a la fois les équations maîtresses et le calcul stochastique quantique. Nous introduisons ensuite la notion de trajectoire quantique pour la description des mesures indirectes continues. C'est dans ce contexte que l'on présente les résultats obtenus au cours de cette thèse. Dans un premier temps, nous étudions la convergence des mesures non destructives. Nous montrons qu'elles reproduisent la réduction du paquet d'onde du système mesuré. Nous montrons que cette convergence est exponentielle avec un taux fixe. Nous bornons le temps moyen de convergence. Dans ce cadre, en utilisant les techniques de changement de mesure par martingale, nous obtenons la limite continue des trajectoires quantiques discrètes. Dans un second temps, nous étudions l'influence de l'enregistrement des résultats de mesure sur la préparation d'état par ingénierie de réservoir. Nous montrons que l'enregistrement des résultats de mesure n'a pas d'influence sur la convergence proprement dite. Cependant, nous trouvons que l'enregistrement des résultats de mesure modifie le comportement du système avant la convergence. Nous retrouvons une convergence exponentielle avec un taux équivalent au taux sans enregistrement. Mais nous trouvons aussi un nouveau taux de convergence correspondant a une stabilité asymptotique. Ce dernier taux est interprété comme une mesure non destructive ajoutée. Ainsi l'état du système ne converge qu'après un temps aléatoire. A partir de ce temps la convergence peut être bien plus rapide. Nous obtenons aussi une borne sur le temps moyen de convergence.
We present the foundational layer of a research program that derives quantum-mechanical, relativistic, and thermodynamic structure from a single ontology: a ledger of sealed records. The ledger is governed by three axioms: R, laws are laws of whole sealed histories; S, no distinction without a record; and C, couplings are fixed by self-consistency under refinement. The program builds on the stochastic-quantum correspondence of Barandes' indivisible processes, replacing configuration trajectories by committed records as the primitive. The main results, each at stated scope and with machine-verified receipts, are as follows. First, the Born layer: record weights compose through square roots, and tame-class reconstruction recovers the Weyl algebra and the Schrodinger representation, via Stone-von Neumann, from record towers, with ladder operators, selection rules, and exact continuum targets approached at machine precision. Second, Lorentz signature: the commitment structure of records forces a Lorentzian (1,d) split with derived orientation classes. The spatial dimension d = 3 is selected by a minimality principle graded as input in the text, not derived. The operational identification lapse = local clock rate is verified to 3.7 x 10^{-8} in curved record geometries as a model-internal consistency receipt, with first 3+1-dimensional cosmological checks presented qualitatively and no cosmological number claimed in this batch. Third, the arrow of time: reflection positivity is a theorem for eventless sectors with finite primitive Markov presentations, the entropy-production functional is the path-measure relative entropy sigma = D(P_AB || P_BA), the classical identity of stochastic thermodynamics in record-pair form, and the presentation hypothesis is sharp. An explicit reversible, valid, finite-rank process, the "record clock", fails reflection positivity, separating record laws from general reversible processes. Fourth, thermality: horizon temperature T = kappa / 2 pi is recovered in exact lattice constructions, presented in full in the companion horizon paper and cited here only as a consistency credential. We state precisely what is axiom, what is theorem, what is named identification, and what is imported mathematics; the full dependency audit is part of the paper. We also delimit the program's frontier: the tame class, the regularity stratum of the emergent continuum, and continuous sector families.
De nombreux phénomènes de physique quantique ne peuvent être compris que par l'analyse des systèmes ouverts. Un appareil de mesure, par exemple, est un système macroscopique en contact avec un système quantique. Ainsi, tout modèle d'expérience doit prendre en compte les dynamiques propres aux systèmes ouverts. Ces dynamiques peuvent être complexes : l'interaction du système avec son environnement peut modifier ses propriétés, l'interaction peu créer des effets de mémoire dans l'évolution du système, . . . Ces dynamiques sont particulièrement importantes dans l'étude des expériences d'optique quantique. Nous sommes aujourd'hui capables de manipuler individuellement des particules. Pour cela la compréhension et le contrôle de l'influence de l'environnement est crucial. Dans cette thèse nous étudions d'un point de vue théorique quelques procédures communément utilisées en optique quantique. Avant la présentation de nos résultats, nous introduisons et motivons l'utilisation de la description markovienne des systèmes quantiques ouverts. Nous présentons a la fois les équations maîtresses et le calcul stochastique quantique. Nous introduisons ensuite la notion de trajectoire quantique pour la description des mesures indirectes continues. C'est dans ce contexte que l'on présente les résultats obtenus au cours de cette thèse. Dans un premier temps, nous étudions la convergence des mesures non destructives. Nous montrons qu'elles reproduisent la réduction du paquet d'onde du système mesuré. Nous montrons que cette convergence est exponentielle avec un taux fixe. Nous bornons le temps moyen de convergence. Dans ce cadre, en utilisant les techniques de changement de mesure par martingale, nous obtenons la limite continue des trajectoires quantiques discrètes. Dans un second temps, nous étudions l'influence de l'enregistrement des résultats de mesure sur la préparation d'état par ingénierie de réservoir. Nous montrons que l'enregistrement des résultats de mesure n'a pas d'influence sur la convergence proprement dite. Cependant, nous trouvons que l'enregistrement des résultats de mesure modifie le comportement du système avant la convergence. Nous retrouvons une convergence exponentielle avec un taux équivalent au taux sans enregistrement. Mais nous trouvons aussi un nouveau taux de convergence correspondant a une stabilité asymptotique. Ce dernier taux est interprété comme une mesure non destructive ajoutée. Ainsi l'état du système ne converge qu'après un temps aléatoire. A partir de ce temps la convergence peut être bien plus rapide. Nous obtenons aussi une borne sur le temps moyen de convergence.
### Abstract (Part One: Crystalline Time → Quantum Kinematics) We present Part One of a two-part programme that connects a discrete, non-split “crystalline time” framework to the safest core of quantum kinematics, with every nontrivial statement backed by simulation certificates. The aim of Part One is intentionally narrow: we build (i) a geometric lexicon for phases and symplectic structure, (ii) a minimal and assumption-transparent route to the Schrödinger equation via unitary time translation (Stone’s theorem), and (iii) the Heisenberg uncertainty principle via symplectic commutators and the Robertson inequality. Born weights and the Dirac operator are deferred to Part Two, where the structural and interpretive inputs are heavier. Section 1 introduces the geometric dictionary: “phases” are treated as holonomies of a U(1) connection on a complex line bundle, and classical mechanics is formulated on phase space as a symplectic manifold (T*Q, ω = dp ∧ dq). On the framework side, crystalline time is encoded by a non-split extension 0 → Z24 → Z48 → Z2 → 0 with the decomposition K = 2k + p (k ∈ Z24, p ∈ Z2). The fine tick K → K+1 necessarily toggles p, while the coarse tick K → K+2 acts as k → k+1 and preserves p. The leaf observer is p-blind; we model this by a projection/coarse-graining Πp over the Z2 fiber, yielding an effective open-system dynamics on the leaf. Time-kinetic support is regulated by bridge kernels S and J through Kt = Zt(S + εJ), with ε controlling whether the leaf sector is hyperbolic (ε > 0), constrained/elliptic (ε = 0), or ghost-like (ε < 0). All of these statements are verified by discrete operator checks at machine precision. Section 2 derives Schrödinger from one minimal physical principle: strongly continuous unitary time translations U(t). By Stone’s theorem, U(t) = exp(−iHt/ħ), giving iħ ∂t|ψ(t)⟩ = H|ψ(t)⟩. We then show how this “safe tower” is realized inside the framework: define the leaf embedding E+ (pure-even state), the leaf projection P+ (extract ψ+), and the fine-step unitary U on Z48. The cadence-dependent leaf amplitude map A_eff(m) = P+ U^m E+ exhibits an exact even/odd dichotomy (confirmed by closed-form derivation and machine-precision verification). For even cadences m = 2r (i.e., N|48 with m = 48/N even), A_eff(m) is a unitary kernel shift S^r: purity remains 1 and entropy remains 0. For odd cadences m = 2r+1, A_eff(m) = S^r (I+S)/2 is contractive: average retention Tr(A†A)/24 = 0.5, a near-zero singular mode appears, and entropy is produced under the projected density-matrix channel. Numerically, starting from a pure-even state, one fine tick generates substantial odd support (||ψ−|| = 0.6146). A divisor-of-48 scan shows that N=48 (m=1) and N=16 (m=3) produce identical entropy/purity trajectories (up to translation), while all even cadences (e.g., N=24,12,8,6) remain unitary and entropy-free. We further connect the odd-tick projected map to a continuum generator: in the long-wavelength, diffusive scaling limit the comoving odd-tick dynamics converges to ∂t φ = D ∂x^2 φ, with D = Δx^2/(8Δt). This diffusion limit is validated numerically with relative L2 errors ≈ 9.3×10^−6 (smooth profile) and ≈ 1.0×10^−5 (delta profile) at t ≈ 0.01. Finally, the ε-controlled cone speed c^2 = Zx/(2εZt) is confirmed by right-moving packet propagation with v_meas matching c_pred to machine precision for ε ∈ {1, 0.3, 0.1, 0.03}. Section 3 derives uncertainty from symplectic structure. The symplectic backbone motivates the canonical commutator [x, p] = iħ, with discrete phase-space realizations through Weyl pairs. The Robertson inequality ΔAΔB ≥ (1/2)|⟨[A, B]⟩| then yields the Heisenberg bound ΔxΔp ≥ ħ/2. On the leaf kernel Hilbert space, a Gaussian wavepacket saturates the bound numerically: ⟨[Q, P]⟩ ≈ (−9.37×10^−17) + 0.9999999998486 i, ΔQΔP ≈ 0.4999999999337, and (1/2)|⟨[Q, P]⟩| ≈ 0.4999999999243. In parallel, the unitary translation group built from the kernel momentum operator satisfies ||U(t+s) − U(t)U(s)|| ≈ 2.06×10^−14 and ||U(Δx) − S|| ≈ 2.17×10^−14, and a finite-difference Schrödinger check yields a relative residual ≈ 1.09×10^−6 at dt = 10^−6. Part One therefore closes a reproducible bridge: (i) crystalline time plus a p-blind projection yields an exact cadence-dependent split into unitary (even) and contractive/entropic (odd) leaf dynamics, (ii) the unitary kernel sector realizes Stone’s theorem and Schrödinger evolution explicitly, and (iii) the symplectic backbone yields commutators and uncertainty with quantitative saturation. Part Two will address Born weights and the Dirac operator, where the required axioms/structures are heavier and must be handled with the same “Court” discipline.
In a finite, timeless configuration space where probability is counting and the population of an outcome class is proved to equal N|α|², we construct a refinement chain from local devices—nearest-neighbour beamsplitters and per-cell phase increments, all on the 3,600-value phase lattice of the companion models—and evolve wavepacket classes by literal enumeration. The small-angle brickwork realizes the lattice Hamiltonian H = −2η cos k + V(x): the effective mass m* = 1/(2η) is the inverse hopping rate—mass as resistance to record restructuring, obtained from the move set rather than posited—and the potential is a cell-dependent phase cost per closure step. Counted populations reproduce the Schrödinger spreading law σ²(t) = σ₀² + (t/2m*σ₀)², the group velocity to seven significant figures against the exact device chain, and harmonic oscillation at ω = √(2ηκ), with Σₓ nₓ = N exact at every stage—the counting form of unitarity, tested exactly. Two findings go beyond agreement. First, the counted packet has a literal edge: populations fall to one configuration and then to zero exactly where the Born value crosses 1/N, so a counting world's particle has finite support where the continuum Gaussian merely thins—the rational-probability prediction rendered in position space. Second, the ontology's record-relative definition of classes is load-bearing for dynamics: counting once at the measurement stage (the chain as one composite device) yields residues of O(1/N), measured log–log slope −0.94, whereas imputing an integer census to unrecorded intermediate stages—partitions where no records are—degrades the scaling to slope −0.62 through the amplitude granularity of few-configuration tail classes. A definitional clause changes dynamical accuracy by a factor of order √N. What is not shown is the formal continuum limit: given [CS] the discrete effective dynamics is derived, with the same conditional status as the companion theorems, and the passage to the continuum equation remains the framework's open problem. The cell structure is a named input [CS]—but not a foreign one: its three clauses are the discrete shadow of the framework's own emergent-space program, and we locate each, finding that a preferred partition is what the stability criterion already selects, that adjacency is what the elementary refinement move set already supplies, and that only the graph's large-scale regularity—together with an assumed uniformity of hopping rate, which is a flatness assumption—remains open. Reading the cells as emergent space has an immediate consequence: the construction's maximal group velocity is one instance of the framework's conjectured invariant speed, computed rather than posited, since a single elementary move defines both the refinement step and the spatial adjacency. Relaxing uniformity—a position-dependent hopping rate, hence a position-dependent effective mass—is the discrete form of curvature, and is the natural next model.
Analogues classique de la dynamique quantique Cette thèse se concentre sur une formulation stochastique de la mécanique quantique connue sous le nom de mécanique de Nelson, que l’on a utilisée pour étudier la règle de Born. Cette théorie offre un cadre unique pour comprendre la dynamique quantique à travers des analogies classiques, en établissant notamment des parallèles entre l’évolution stochastique des particules browniennes et les systèmes quantiques. Nous avons appliqué cette approche pour optimiser les transitions des systèmes quantiques, tels que l’oscillateur harmonique quantique, en minimisant les coûts énergétiques et les temps de transition. A partir de ces résultats, la thèse explore l’extension de ces méthodes à des systèmes quantiques plus complexes, y compris les systèmes ouverts soumis à la décohérence et les systèmes multi-corps tels que les condensats de Bose-Einstein. Ce travail instaure les bases du développement de nouvelles techniques d’optimisation et de stratégies de contrôle pour les systèmes quantiques, pouvant potentiellement conduire à des applications innovantes en thermodynamique quantique et en contrôle de systèmes.