An Alternative Structural Explanation of the Quantum Computational Ceiling Proposed by Rational Quantum Mechanics

Author: Terence Love

Date: March 2026 Framework: Variety Dynamics (VD)

DOI: https://doi.org/10.5281/zenodo.19140950

Summary

Variety Dynamics (VD) provides a structural explanation for the fundamental limit on quantum computational scaling proposed by Rational Quantum Mechanics (RaQM; Palmer, 2026). This explanation is independent of RaQM's gravitational discretisation argument and arrives at the same predicted ceiling via a structurally distinct route: the exponential and combinatorial scaling of transaction costs associated with maintaining quantum variety distributions in a physical substrate. This convergence of two independent analytical routes on the same structural limit constitutes significant cross-framework support for the predicted ceiling and illustrates VD's capacity to operate at the foundational level of physical theory.

The Phenomenon Under Analysis

In March 2026, Tim Palmer of the University of Oxford published a paper in Proceedings of the National Academy of Sciences introducing Rational Quantum Mechanics (RaQM). Palmer proposes that the continuous infinite-dimensional Hilbert space of standard quantum mechanics is an idealisation of a fundamentally discrete, information-bounded structure. The key claim is that the information available to describe an N-qubit system grows only linearly with N, while the quantum degrees of freedom requiring description grow exponentially with N. Beyond a threshold estimated at between 200 and 1,000 physical qubits, the available information is insufficient to fully specify the quantum state space. At that point, quantum algorithms requiring maximal entanglement — including Shor's algorithm for factoring large integers — lose their exponential advantage over classical computation. Palmer links this discretisation to gravity, proposing that gravitational effects determine the scale at which the continuous approximation of Hilbert space breaks down.

The VD analysis presented here does not dispute Palmer's gravitational argument. It demonstrates that VD arrives at the same structural ceiling via an independent route grounded in the transaction cost scaling properties of physical variety distributions.

VD Analytical Framing

VD analyses situations in terms of variety distributions — the availability and distribution of options across a situation — rather than through causal mechanisms. The central VD question for any physical situation is: what variety is available, to what entities, and what are the structural costs of maintaining or expanding that variety?

For a quantum computational situation, the relevant variety is the quantum state space: the set of distinguishable states the system can in principle occupy. In standard quantum mechanics, this variety grows exponentially with qubit count N. The physical substrate instantiating the quantum computation must maintain that variety distribution throughout the computation. VD analysis focuses on the structural costs of maintaining that variety in a physical substrate, and what happens when those costs exceed the capacity of any possible substrate.

Axiom-Grounded Variety Dynamics Analysis

Variety Requires Physical Instantiation

Axiom 28 establishes that all variety, regardless of how abstract, requires a physical substrate. A quantum computation does not occur in an abstract mathematical space — it is instantiated in physical qubits, cryogenic systems, electromagnetic fields, and the matter-energy configurations that sustain coherent quantum states. The variety of the quantum state space is not free. It must be physically carried.

Axiom 25 establishes that representing variety in physical information media always involves informatic constraints. The translation from abstract quantum state space to physical instantiation is not lossless. Physical media impose representational limits.

Physical Information Media Are Thermodynamically Bounded

Axiom 26 establishes that all physical information media are thermodynamic systems, and therefore all variety processing is subject to thermodynamic constraints. Maintaining coherent quantum superposition across N entangled qubits requires continuous expenditure of thermodynamic resources — cooling, shielding, error management, and the energy cost of sustaining the physical configuration that carries the quantum information state.

These thermodynamic costs are not merely engineering challenges. They are structural features of the relationship between variety and physical instantiation. In the limit, Coasian transaction costs — the overhead of generating, maintaining, and managing variety — resolve to thermodynamic and energy costs. This is the physical floor of the transaction cost concept: the minimum resource expenditure required to sustain a variety distribution in any possible physical substrate.

Transaction Costs Scale Super-Linearly with Variety

Axiom 35 establishes that transaction costs increase as variety increases. Axiom 36 sharpens this: transaction costs associated with variety increase exponentially or combinatorially with increases in variety, not linearly.

For a quantum computational system, the variety of the state space grows exponentially with qubit count N (as 2^N). The transaction costs of maintaining that variety in a physical substrate therefore scale exponentially or combinatorially with N. The physical resources required to sustain the quantum variety distribution grow far faster than the system itself.

This creates a structural diseconomy of scale. For small N, the transaction costs of maintaining quantum coherence are manageable and the exponential variety of the state space delivers genuine computational advantage. As N increases, the transaction costs of maintaining the full variety distribution grow exponentially, while the physical substrate's capacity to meet those costs grows at most linearly with the physical resources available.

The Geometric Cost of Substrate Maintenance

The thermodynamic cost of maintaining a physical substrate has a deeper structural form. In general relativity, gravity is not an acceleration or force in the Newtonian sense — it is the effect of space-time curvature induced by the matter-energy configuration of the physical system. That curvature constitutes a 'potential' — in the sense of a stored capacity, a possibility of doing work — and an 'energy' — in the sense of a stored but physically unspecified resource — associated with maintaining that particular configuration.¹ This geometric cost is the physical limit toward which transaction costs converge as variety distributions grow in complexity.

Maintaining N entangled qubits requires a physical configuration whose matter-energy arrangement curves local space-time in a configuration-dependent way. The geometric 'potential energy' of that curvature — in the structural sense above, not as a specific GR quantity — is the deepest physical form of the transaction cost identified by Axioms 35 and 36. As N increases and the variety of the quantum state space grows exponentially, the physical configuration required to instantiate that variety grows correspondingly, the matter-energy arrangement of the substrate grows in complexity, and the associated geometric cost grows super-linearly with qubit count.

Palmer's argument that gravity determines the discretisation scale of RaQM is, in VD terms, the observation that this geometric transaction cost of substrate maintenance becomes structurally prohibitive at a specific scale. VD identifies this as the point at which the exponentially scaling transaction costs of maintaining the quantum variety distribution exceed the capacity of any physically realisable substrate.

¹ The terms 'potential' and 'energy' are used here in their structural senses — 'potential' as possibility or stored capacity, 'energy' as an unspecified but real physical resource — rather than in the technical sense of Newtonian gravitational potential energy or any specific quantity in general relativity. The precise physical formalisation of the geometric cost is a matter for physics to determine. The structural claim is prior to that specific formalisation: space-time curvature induced by a matter-energy configuration constitutes a real configuration-dependent stored resource that must be accounted for in any complete description of the costs of physical instantiation.

The Variety Ceiling as Structural Discontinuity

Axiom 48 establishes that variety distributions can exhibit discontinuities — boundaries where small continuous changes in input variety produce discontinuous changes in the variety distribution structure. These discontinuities mark points of irreversibility.

The RaQM ceiling is precisely such a discontinuity. Below the threshold (estimated at 200–1,000 qubits), the physical substrate can sustain the full quantum variety distribution and the exponential advantage of quantum algorithms is realisable. At the threshold, the transaction costs of substrate maintenance become structurally prohibitive and the full variety distribution can no longer be instantiated. The system cannot represent all possible quantum states. The discontinuity is not a smooth asymptote — it is a hard structural boundary in the variety distribution landscape, beyond which the variety of the abstract quantum state space is physically unrealisable.

Axiom 9 frames this precisely: variety is the number of different options that are possible. Beyond the threshold, the exponential variety of the abstract quantum state space is not possible in any physical instantiation. The variety ceiling is real, not merely practically difficult to reach.

The Independent Route to Palmer's Prediction

Palmer arrives at the quantum computational ceiling via gravitational discretisation of Hilbert space. The VD argument arrives at the same ceiling via a structurally independent route:

Quantum state space variety grows exponentially with qubit count (Axiom 9)
All variety requires physical instantiation (Axiom 28)
Physical instantiation is subject to thermodynamic constraints (Axioms 25, 26)
In the limit, transaction costs resolve to thermodynamic costs and the geometric 'potential energy' of space-time curvature (in the structural sense: a configuration-dependent stored resource)
Transaction costs of maintaining quantum variety distributions scale exponentially/combinatorially with variety (Axioms 35, 36)
Therefore, the physical cost of instantiating the full quantum state space grows super-linearly with qubit count, producing a structural diseconomy of scale
At the threshold where substrate capacity is exhausted, the variety distribution exhibits a hard discontinuity (Axiom 48)
The discontinuity marks the point at which the abstract quantum variety space is physically unrealisable

This route does not require Palmer's gravitational discretisation argument. It requires only that substrate maintenance costs scale super-linearly with qubit count — which is empirically supported independently of RaQM, and which follows from first principles via Axioms 35 and 36.

The convergence of two independent analytical routes on the same structural prediction constitutes cross-framework support for the ceiling. If VD's transaction cost argument and Palmer's gravitational discretisation argument are structurally independent and predict the same limit, the probability that the limit is a genuine structural feature of quantum computation — rather than an artefact of either theoretical framework — is substantially increased.

What VD Reveals additional to Conventional Analysis

Conventional analysis of quantum computational scaling focuses on engineering challenges: decoherence, error rates, fault tolerance, and the physical difficulty of maintaining quantum coherence at scale. These are real challenges, but they are, in principle, addressable through improved engineering. They do not constitute a fundamental ceiling.

RaQM proposes a fundamental ceiling via a specific theoretical mechanism (gravitational discretisation) that, while compelling, depends on RaQM being correct — which remains to be experimentally verified.

The VD analysis reveals that the ceiling is a structural feature of the relationship between variety, physical instantiation, and transaction cost scaling that is independent of both the engineering challenges and the specific mechanism proposed by RaQM. The ceiling is not primarily an engineering problem or a gravitational effect. It is the point at which the structural cost of maintaining exponentially growing variety in a linearly scaling physical substrate becomes prohibitive. This structural characterisation holds regardless of the specific physical mechanism that enforces it.

Relationship to Axiom 47

Axiom 47 (tentative) proposes that VD operates at a more foundational level than physical theories through its grounding in set theory and its capacity to address discontinuities intrinsically. Physical phenomena represent variety distributions and transformations in state spaces.

This case study provides evidence for Axiom 47's claim. VD's analysis of the quantum computational ceiling does not apply VD concepts as analogies to quantum phenomena. It identifies the quantum state space as a variety distribution, the physical substrate as the instantiation medium, and the scaling mismatch as a structural feature of the variety-instantiation relationship. The physical theory (RaQM) describes one specific mechanism by which the structural limit is enforced. VD describes the structural limit itself, at a level that is independent of the specific physical mechanism.

This relationship — VD identifying the structural limit, physical theory identifying the enforcement mechanism — is consistent with VD operating at a more foundational analytical level than the physical theory.

Limitations and Open Questions

This analysis is theoretical and speculative. The following limitations apply:

The VD transaction cost argument predicts a structural ceiling but does not independently quantify where that ceiling falls. Palmer's estimate of 200–1,000 qubits comes from his gravitational discretisation argument, not from VD. VD confirms the structural form of the ceiling but not its specific location.

The connection between ‘Coasian’ transaction costs and thermodynamic/geometric costs is argued structurally here but has not been formally derived. That derivation is a task for the full mathematical treatment.

RaQM itself remains speculative and awaits experimental validation. Palmer notes it may be falsifiable within five years as qubit systems approach the predicted threshold. If RaQM is falsified — if quantum computers demonstrate sustained exponential advantage beyond 1,000 qubits — the VD analysis would need to examine whether the transaction cost scaling argument was misapplied or whether the predicted ceiling simply falls at a higher N than Palmer estimates.

Conclusion

VD provides a structural explanation of the quantum computational ceiling proposed by RaQM that is independent of RaQM's gravitational discretisation argument. The explanation is grounded in the exponential and combinatorial scaling of transaction costs associated with maintaining quantum variety distributions in a physical substrate, resolving in the limit to thermodynamic and geometric potential energy costs. The ceiling is identified as a variety distribution discontinuity — a hard structural boundary beyond which the abstract quantum variety space is physically unrealisable.

The convergence of the VD transaction cost argument with Palmer's gravitational discretisation argument on the same structural prediction is significant. It suggests the quantum computational ceiling is a robust structural feature of the relationship between variety and physical instantiation, not an artefact of any single theoretical framework.

This case study also provides concrete evidence for Axiom 47's claim that VD operates at a more foundational analytical level than physical theories. A full formal treatment of the transaction cost–thermodynamic–geometric cost connection is identified as a priority for subsequent mathematical development.

Axioms Applied

Axiom 9 (Variety definition), Axiom 25 (Variety dynamics and information systems), Axiom 26 (Variety dynamics, information systems, and thermodynamic constraints), Axiom 28 (All variety depends on a physical substrate), Axiom 35 (Transaction costs increase with variety), Axiom 36 (Exponential and combinatorial transaction cost scaling), Axiom 47 (Variety dynamics and fundamental physics — tentative), Axiom 48 (Discontinuity and irreversibility in variety distributions).

All axioms: Love, T. (2025). Variety Dynamics: Formal Statements of Axioms 1–50. Love Services Pty Ltd. https://doi.org/10.5281/zenodo.17571975

References

Palmer, T. (2026). Rational quantum mechanics: Testing quantum theory with quantum computers. Proceedings of the National Academy of Sciences, 123(12), e2523350123. https://doi.org/10.1073/pnas.2523350123

Love, T. (2025). Variety Dynamics: Formal Statements of Axioms 1–50. Love Services Pty Ltd. https://doi.org/10.5281/zenodo.17571975

Methodology Note

This analysis applies the Variety Dynamics framework through iterative human-AI collaboration. VD axioms and analytical framework specified by human expert (T. Love); variety enumeration, structural mapping, and initial drafting generated by Claude Sonnet 4.6 (Anthropic); reviewed, verified, edited and refined by T. Love through multiple iterations.