Reversible Finite-Frame Dynamics for Rational Quantum Grids: Preparation Laws and Matching-Dependent Statistical Predictions This paper develops a finite, reversible model of quantum control motivated by Tim Palmer’s rational quantum mechanics (RaQM). Rational restrictions on quantum states do not by themselves determine how nominal quantum gates should act when their exact outputs lie outside the admissible state set. The paper addresses this problem by constructing finite catalogues of quantum frames on which commands act as permutations, together with explicit preparation laws that assign statistical predictions to complete control sequences. The analysis distinguishes a conditional rational grid from its occupation-compatible subset, identifies their common projective monomial gate group, and proves an obstruction to uniformly accurate fixed-grid repair by local gates. Equal-measure partitions and perfect matchings are then used to construct reversible approximations of local and entangling controls on enlarged finite ray sets. A compensated preparation law yields a common random-unitary channel for each control word, while a finite consistency criterion determines whether a proposed preparation law works for all words. For fixed circuit depth, the resulting statistics converge uniformly to the nominal quantum predictions as the catalogue is refined. At sufficiently large depths, finite recurrence forces disagreement for a repeated entangling circuit. An explicit rematching of the CNOT control removes a seven-block discrepancy without increasing the catalogue size or the stated error bounds, showing that those coarse conditions alone do not determine the earliest observable deviation. The construction is therefore a mathematically explicit candidate completion of finite-state quantum kinematics, while leaving the physical selection of a particular matching, preparation law and reduction dynamics as additional assumptions. Exact Python and Wolfram Language checks accompany the finite examples.
Reversible Finite-Frame Dynamics for Rational Quantum Grids: Preparation Laws and Matching-Dependent Statistical Predictions This paper develops a finite, reversible model of quantum control motivated by Tim Palmer’s rational quantum mechanics (RaQM). Rational restrictions on quantum states do not by themselves determine how nominal quantum gates should act when their exact outputs lie outside the admissible state set. The paper addresses this problem by constructing finite catalogues of quantum frames on which commands act as permutations, together with explicit preparation laws that assign statistical predictions to complete control sequences. The analysis distinguishes a conditional rational grid from its occupation-compatible subset, identifies their common projective monomial gate group, and proves an obstruction to uniformly accurate fixed-grid repair by local gates. Equal-measure partitions and perfect matchings are then used to construct reversible approximations of local and entangling controls on enlarged finite ray sets. A compensated preparation law yields a common random-unitary channel for each control word, while a finite consistency criterion determines whether a proposed preparation law works for all words. For fixed circuit depth, the resulting statistics converge uniformly to the nominal quantum predictions as the catalogue is refined. At sufficiently large depths, finite recurrence forces disagreement for a repeated entangling circuit. An explicit rematching of the CNOT control removes a seven-block discrepancy without increasing the catalogue size or the stated error bounds, showing that those coarse conditions alone do not determine the earliest observable deviation. The construction is therefore a mathematically explicit candidate completion of finite-state quantum kinematics, while leaving the physical selection of a particular matching, preparation law and reduction dynamics as additional assumptions. Exact Python and Wolfram Language checks accompany the finite examples.
A finite reversible model can approximate individual quantum gates while retaining freedom in its transition tables and preparation contexts. We formulate the simultaneous certification problem in which one stationary controller and one preparation law must reproduce an entire collection of circuit probabilities. For a specified frame catalogue, accuracy graph and calibration constraints, we give an exact shared-table feasibility formulation and distinguish certificates for a fixed table from certificates for the complete model class. A four-frame qubit example exhibits a strict separation between fitting two circuits individually and fitting them jointly. A larger exact benchmark uses 464 frames and a phase symmetry that removes preparation-prior dependence. We independently reconstruct its admissible graph and certify a sharp repeated-circuit horizon: one controller stays within probability error 1/3 through 192 blocks, whereas every allowed controller violates that threshold by block 193. A separate controller reproduces all 8,190 nonempty Hadamard-phase words of length at most twelve within the same tolerance. We also derive a finite-window spectral obstruction and finite-shot confidence tests, illustrated by a reproducible sampling experiment. The conclusions concern the stated finite reversible model classes; they do not establish a universal memory bound or a discrepancy for physical quantum hardware.
AbstractStandard quantum mechanics is formulated using complex Hilbert spaces, real-valued probabilities, continuous one-parameter dynamics, and—where particles and fields have continuous degrees of freedom —function spaces constructed over continua such as R3 . In that precise mathematical sense, the real and complex continua are deeply embedded in the standard theory. It does not follow, however, that experiments have demonstrated the physical existence of an uncountable set of exact real numbers, still less that individual quantum systems physically “store” infinitely many digits. Every preparation, control operation, and measurement has finite precision, and many quantum predictions can be approximated arbitrarily closely using countable or finite descriptions.This distinction matters because the continuum is sometimes blamed for quantum theory's conceptual strangeness. That diagnosis is mostly unsupported. Superposition, interference, uncertainty, noncommutativity, entanglement, contextuality, Bell-inequality violations, and no-cloning already occur in finite-dimensional quantum systems and do not depend in any straightforward way on continuous position space or arbitrary irrational numbers. Restricting a Hilbert space to rational, algebraic, computable, or finitely specified states therefore does not automatically produce classical physics. Indeed, dense restricted subsets can approximate ordinary quantum predictions to arbitrary finite accuracy, although turning such a subset into a complete physical theory requires much more: closed dynamics, a composition rule, a measurement theory, robustness, and new empirical predictions.A particularly developed contemporary alternative is Tim Palmer's Invariant Set Theory and its recent formulation as Rational Quantum Mechanics (RaQM). RaQM is not accurately described as simply “quantum mechanics with irrational numbers banned.” It restricts the bases in which states are physically defined through number-theoretic rationality conditions, represents admissible states using finite bit strings, invokes a fractal invariant state space equipped with a non-Euclidean, p-adic-like metric, and violates Bell measurement independence at the level of exact counterfactual settings. Palmer's 2026 proposal further predicts a finite qubit-information capacity and an eventual saturation of quantum computational advantage. These are genuine, potentially falsifiable proposals, but they remain speculative and have not displaced standard quantum theory.The strongest conclusion is consequently asymmetric. There is a serious foundational question about whether the continuum is an ontological feature of nature or an extraordinarily efficient mathematical completion. Finite precision prevents existing experiments from proving a literal continuum. But there is presently little evidence that continuum mathematics causes quantum phenomena. Any discrete replacement must still account quantitatively for interference, contextuality, Bell correlations, entanglement, spectroscopy, atomic structure, and quantum computation. Discreteness can change quantum physics only when it is accompanied by additional physical postulates—and those postulates, rather than discreteness alone, must ultimately face experiment.
Relational quantum causal processes formulate finite operational contexts as normal positive functionals on local completely positive maps. Response differences generate an influence algebra, and its central projections define jointly readable Boolean events. We develop this starting point through a sequence of exact and controlled models. Fresh-environment unitary collision circuits produce dephasing-exchange kinetics with an exact charge-center fixed algebra, a uniform finite-step limit at fixed response order, and graph-controlled metastable Markov dynamics. An absorbing-state model exhibits a sharp transition between non-Abelian quantum memory and Boolean records. A reversal-covariant defect dynamics generates a locally finite partial order on a restricted graph family without assuming a Lyapunov time. Conditional on a certified order, a positive additive record measure, compactness, and identifiability, we prove subsequential convergence to a Lorentzian metric-measure space, finite reconstruction bounds, and uniqueness of admissible smooth limits. Complementary finite regulators provide controlled tests of modular-to-boost response, null tomography, same-update variational identities, induced quadratic gravity, and compatible common-refinement limits. These results are exact or controlled within their stated models, but they do not yet constitute a single background-independent microscopic law that jointly generates adjacency, time, volume normalization, dimension, signature, nonlinear Einstein constraints, and quantum matter. We therefore present RQCP-QG as a theorem-indexed framework that separates established mechanisms, conditional compositions, and open assumptions.
Deterministic Quantum Mechanics (DQM) is a research program investigating whether the probabilistic content of quantum mechanics can be understood as an emergent statistical description of an underlying deterministic physical theory. This paper consolidates the program’s cumulative results into a single account.The central proposal is that a complete microscopic state evolves deterministically, that the quantum state is a coarse-grained ensemble descriptor rather than the complete physical state, and that measurement outcomes are selected by which deterministic basin of an enlarged phase space a microscopic trajectory occupies. Under this proposal, Born probabilities are basin measures rather than fundamental stochastic laws.For a qubit, this program is realized exactly: a hidden variable λ on the unit sphere, a preparation-dependent density μ_r(λ) = [r·λ]₊/π, and the deterministic response A(n,λ) = sgn(n·λ) reproduce the qubit Born rule P(±|r,n) = (1 ± r·n)/2 exactly, together with sequential-measurement statistics and the associated expectation values. This preparation measure is itself shown to be the deterministic pushforward of a uniformly distributed microscopic preparation coordinate, removing any appeal to fundamental preparation-level stochasticity.Measurement is then developed as deterministic basin selection within a finite-time Hamiltonian architecture, with a finite-time stability theorem governing when microscopic perturbations cannot dislodge a trajectory from its basin. Entangled correlations are reproduced by an autonomous, rotationally covariant Bell Hamiltonian containing dynamical setting registers and a conserved-momentum phase shear; this mechanism reproduces the singlet correlation E(a,b) = −a·b and the CHSH value 2√2 in its ideal limit, while preserving exactly uniform local marginals at finite setting inertia and yielding an explicit, quantitatively bounded finite-inertia correction. A finite Hamiltonian bath (Model-C) supplies a deterministic amplification and record-retention mechanism; its analytic structure — including an exact completed-square cancellation of the static bath backreaction and an exact finite-bath spectral equation — is derived independently of, and is shown to survive, a documented numerical implementation failure (the C14 program), which is retained in the record as a diagnosed, non-Hamiltonian right-hand side rather than as evidence against the underlying construction.Extending the program beyond the qubit exposes a genuine mathematical obstruction: no deterministic basin defined directly on the qutrit pure-state manifold ℂP² can reproduce an arbitrary rank-one Born probability, because positivity forces any exact basin onto a measure-zero point. This is proved rigorously. The resolution is not the abandonment of determinism but the enlargement of the hidden state beyond the quantum ray itself, to a genuine mechanical phase space such as the cotangent bundle T*SU(3), on which the quantum state re-emerges as a coarse-grained moment and on which contextual, Kochen–Specker-compatible measurement dynamics can in principle be realized.Throughout, the paper enforces a strict evidentiary hierarchy — mathematical construction, analytic theorem, numerical validation, and experimental confirmation are never conflated — and closes with an explicit account of what has been proved, what remains open, and what would be required to render the program experimentally falsifiable.
No-signalling constraints on isolated outcome probabilities do not determine whether an apparatus preserves the joint law of its accessible records under repeated readout, network composition, or reuse. We study these extensions through a conditional-record criterion and four explicit models. For a wrapped-Cauchy retrocausal source, protection under two conditionally independent continuous readouts forces an angle-independent response; three readouts suffice without continuity. A finite angular-bandwidth assumption gives a quantitative leakage bound, whereas continuity alone gives no uniform stability. At a specified two-source Bell junction, the required correction matrix has full nonnegative rank on every even uniform angular grid. A Gaussian controller can cancel branch-mass dependence while leaving a distinguishable joint record, for which we derive exact noise and law-change tradeoffs. Finally, a fixed singlet with shared classical noise reproduces the variable-width isolated outcome family, but retained and refreshed controllers generate different temporal statistics. We prove a finite-resource obstruction and construct positive finite-controller approximations. Numerical studies comprising 8.3 million independent simulated histories support the model-specific predictions. Together, the results identify conditions that must accompany marginal no-signalling when extending an apparatus to richer records and interventions.
Can reversible quantum dynamics prepare a subsystem for measurement while retaining the physical archives of its earlier history? This paper provides an explicit mathematical construction and identifies the conditions under which such preparation succeeds. An exact, smooth split–copy–return protocol couples a Gaussian writer to a finite bank of position archives. The analysis shows why preparation of the writer alone is insufficient: singular archive populations and exact retained copies can preserve its fine information. A positive preparation theorem instead uses regularity and moment conditions on the complete original conditional law, retaining every archive coordinate and its correlations without imposing an upper bound on the probability density.A worked example with 102 position coordinates gives a joint conditional preparation error below 0.000010052. Under the stated assumptions, the same bound applies to a subsequent one-use labelled projective measurement with an arbitrary finite internal reference system. A separate physical receiver copies the writer’s earlier outcome and preserves it during the specified holding dynamics. The paper includes complete proofs, counterexamples, robustness estimates, and an analysis of reference systems and postselection. Its results concern conditional subsystem preparation and a one-use measurement within a controlled canonical-current model. Physical realization of the controls and independent justification of the admitted initial statistical laws remain separate requirements.