This case study applies the Variety Dynamics (VD) analytical framework to the quantum computational ceiling proposed by Rational Quantum Mechanics (RaQM; Palmer, PNAS 2026). VD provides a structural explanation of the predicted ceiling that is independent of RaQM's gravitational discretisation argument, arriving at the same limit via the exponential and combinatorial scaling of transaction costs associated with maintaining quantum variety distributions in a physical substrate. In the limit, these transaction costs resolve to thermodynamic and geometric costs — the configuration-dependent stored resource associated with space-time curvature induced by the matter-energy arrangement of the substrate. The convergence of two independent analytical routes on the same structural prediction constitutes cross-framework support for the quantum computational ceiling. The analysis also provides evidence for Axiom 47 of the VD framework, which proposes that VD operates at a more foundational analytical level than physical theories. Axioms applied: 9, 25, 26, 28, 35, 36, 47, 48. This case study is part of a collection of case studies at https://variety-dynamics.org demonstrating the practicaL and theoretical application of Variety Dynamics to real world situations.
This case study applies the Variety Dynamics (VD) analytical framework to the quantum computational ceiling proposed by Rational Quantum Mechanics (RaQM; Palmer, PNAS 2026). VD provides a structural explanation of the predicted ceiling that is independent of RaQM's gravitational discretisation argument, arriving at the same limit via the exponential and combinatorial scaling of transaction costs associated with maintaining quantum variety distributions in a physical substrate. In the limit, these transaction costs resolve to thermodynamic and geometric costs — the configuration-dependent stored resource associated with space-time curvature induced by the matter-energy arrangement of the substrate. The convergence of two independent analytical routes on the same structural prediction constitutes cross-framework support for the quantum computational ceiling. The analysis also provides evidence for Axiom 47 of the VD framework, which proposes that VD operates at a more foundational analytical level than physical theories. Axioms applied: 9, 25, 26, 28, 35, 36, 47, 48. This case study is part of a collection of case studies at https://variety-dynamics.org demonstrating the practicaL and theoretical application of Variety Dynamics to real world situations.
This case study applies the Variety Dynamics (VD) analytical framework to foundational questions in physics and mathematics. It develops three related claims. First, varieties are informatic descriptions of reality — distinguishable options, states, decisions, or events — not the physical things they describe, and this distinction resolves persistent confusions in the foundations of physics including the quantum measurement problem. Second, quantum measurement, prediction, and physical theory are all informatic variety space operations; the wave function exhibits three modes of existence of which one — independent ontological status — is false; and the measurement problem dissolves when the informatic character of varieties is consistently maintained. Third, VD's variety space concept operates at the same foundational level as set theory and is arguably prior to it, since set theory's foundational operation of element distinguishability presupposes variety as an unnamed pre-formal primitive, and all specific physical state spaces (Hilbert space, phase space, Fock space) are special cases of variety space characterised by forgetful functors in the categorical sense. The case study also develops the connection between variety space and pre-sheaf structure in higher topos theory, where the variety-to-actualisation transition corresponds to sheafification. The analysis draws on the RaQM transaction cost argument established in Love (2026a) and provides evidence for Axiom 47's claim that VD operates at a more foundational level than physical theories. Axioms applied: 9, 25, 26, 32, 47, 48, 57 (in development).
This work is Part II of a two-part series on the Modular Superselection Framework. Part I (DOI: 10.5281/zenodo.17834017) established the theoretical foundation; Part II provides computational validation through AlexMax-Q discrete quantum dynamics. Building upon the Modular Superselection Framework established in Part I (Lazzari, 2025; DOI: 10.5281/zenodo.17834017), we present a concrete computational implementation showing that discrete ℤ_M-graded state spaces can support robust quantum interference. Using AlexMax fractal-arithmetic encoding to construct natural-number states organised into modular sectors, we systematically explore the emergence of quantum behaviour in a fully discrete setting. We identify a sharp distinction between purely geometric models and dynamically driven ones. In Framework v1.x, where overlap is determined solely by a fractal metric and discrete phase assignments, the resulting quantum–classical deviations never exceed ≈3%, placing the model firmly in a quasi-classical regime. In Framework v2.0, by contrast, we introduce explicit unitary dynamics generated by fractal-weighted Hamiltonians. Numerical simulations reveal strong interference, with quantum–classical deviations ranging from 18% to 43% across a broad parameter range of fractal decay rates, path asymmetries, and coupling geometries (fractal, modular, hybrid). These results provide the first explicit computational realisation of the postulates formulated in Part I: modular-arithmetic structures, when equipped with sector-preserving unitary evolution, can generate genuinely non-classical correlation patterns while maintaining a deterministic, ℤ_M-graded description of measurement. The AlexMax-Q framework thus connects number-theoretic fractal geometry with discrete quantum dynamics and supports an epistemic interpretation of quantum measurement within ℤ_M-graded structures.
Papers I–III derived specific physical structures from three operational axioms: an action gap, finite accessibility, and intersubjective stability. This paper inverts the question: what structures are forbidden? Seven exclusion theorems are proved, showing that large classes of alternative physics are inadmissible under the operational axioms. Each theorem includes explicit assumptions and rigorous mathematical proof. The excluded structures are: 1. **Arbitrarily precise extended synchronisation** — no protocol can certify synchronisation across a domain exceeding the coherence radius R_coh 2. **Higher-than-second-order local dynamics** — via Ostrogradsky instability and certification cost arguments 3. **Non-Born probability assignments** — via the Cauchy functional equation; the Born rule is the unique continuous, non-contextual invariant measure 4. **Non-three-dimensional geometry** — via flux invariance; only d = 3 permits radius-independent total distinguishability 5. **Non-quartic high-frequency dispersion** — via complex analysis of the EFT derivative expansion 6. **Stable relativistic gravitationally-coupled matter** above the accessibility ceiling 7. **Protocol-dependent physical constants** — universality is forced by intersubjective stability The exclusions collectively define an admissibility corridor: the narrow set of physical structures compatible with bounded intersubjective certification. The paper shows that most logically possible alternatives violate the axioms, and that observed physics lies on structural boundaries of the corridor. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** no-go theorems, admissibility corridor, finite accessibility, exclusion theorems, Born rule uniqueness, spatial dimensionality, Ostrogradsky instability, higher-order dynamics, extended synchronisation, intersubjective stability, certification cost, dispersion constraints, protocol invariance, operational axioms, action gap, quantum gravity constraints
This paper explores the potential of utilizing quantum state dynamics to construct a computational model for simulating complex systems. We propose a novel framework predicated on the continuous evolution of quantum states, aiming to unveil inherent patterns within these systems. The core mechanism involves establishing a dynamic geometric topology based on these quantum state changes, offering a pathway to more accurate and insightful simulations. This research introduces a framework that moves beyond static models, offering a new perspective on system dynamics and potentially unlocking significant insights into complex phenomena.
Our research meticulously navigates the realms of Fractional Quantum Mechanics (FQM), focusing on a critical examination of both numerical and analytical methods that harness the potential of fractional calculus to illuminate the quantum world's complexities. By embarking on this scholarly journey, we aim to decode the intricate dynamics that fractional equations reveal about quantum systems, pushing the boundaries of conventional quantum mechanics. This comparative study meticulously evaluates the efficacy and insights provided by these two distinct approaches, highlighting their contributions to a deeper understanding of quantum phenomena. As we traverse through the layers of quantum dynamics, our work seeks to contribute significantly to the theoretical framework of FQM, offering innovative perspectives and methodologies. The essence of this research lies in its potential to forge new theoretical pathways and inspire further exploration in the quantum domain, leveraging the unique capabilities of fractional calculus. By enriching the scientific community with our findings, we aspire to open new horizons in the study of quantum mechanics, marking a step forward in the ongoing quest to unravel the mysteries of the quantum universe.
Summary. A Comment on T. Palmer, Rational quantum mechanics: Testing quantum theory with quantum computers, PNAS 123(12), e2523350123 (2026), and a companion experimental proposal extending RaQM to optical continuous-variable and GKP quantum computers. Model-independent results. RaQM's capacity is an operational Schmidt-rank ceiling; circuit-width reduction (semiclassical QFT, Beauregard 2n+3, single-control phase estimation) cannot evade it, by Jozsa-Linden (2003) and Vidal (2003). The capacity constant is an analytic identity, K = log2(E_P c^2 / G h^2) = 341.42, with Diosi-Penrose prescription dependence bounded by an additive +/-30 bits. Conditional on RaQM. The absolute N_max, the CV/GKP scaling, and the shot-noise functional form. The simplest DP collapse is already experimentally constrained (Donadi et al., Nat. Phys. 17, 74, 2021). The premises are critically reviewed in the Comment. Falsification design. Lowest-N_max platform, ~100-bit Shor, bisection onset scan, decisive 2D (d_code x p_phys) invariance discriminator, GHZ cross-check; ~1e4-1e5 trials for 5-sigma at 90% power. Reproducibility. raqm_verify.py independently reproduces all of Palmer's anchor estimates (regression-tested); all figures are generated from the verified numerics.