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.
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.
SCQG Paper 12 of 28 This paper sets out experiments for testing Semantic Commitment Quantum Gravity. It connects the theory’s physical equations to detector records, matched control interventions and decision rules fixed before data collection. The main laboratory proposal uses light coupled to a mechanical oscillator. It compares three conditions. Arm D adds measured noise independently of the record. Arm C measures the system and uses the result for feedback control. Arm S keeps that physical control matched and changes a stored instruction that selects a later action. The contrast C − D tests active record stabilization. The contrast S − C tests whether instruction content changes later choices on new task cases excluded from training and tuning. The paper gives equations for the cavity and oscillator, continuous measurement, feedback, noise, covariance reconstruction, work and heat. The calculations include the physical bath response, detector bandwidth, control latency and approximation error. Calibration and statistical procedures connect predicted changes in physical variability to quantities estimated from repeated runs. A source–receiver protocol distinguishes when a record becomes physically available from when the receiver can reliably identify its source. A separate clock protocol tests whether retained physical records can distinguish different cycles of a repeating clock. Gravity protocols combine causal order, discrete curvature, screen response, energy flux and tensor reconstruction to test the Einstein equation at specified scales. Each protocol specifies the controls, calibration, uncertainty, held-out evaluation, stopping rules and retained data needed to reproduce the analysis. These conditions determine when a result supports the tested prediction, reveals a physical discrepancy or remains inconclusive. This is Paper 12 of the Semantic Commitment / Quantum Gravity series.