Variety Dynamics Case Study: Variety Space as Foundational to Physics and Mathematics

Author: Terence Love
Date: March 2026 Framework: Variety Dynamics (VD)
DOI: https://doi.org/10.5281/zenodo.19141638
© 2026 Terence Love, Love Services Pty Ltd

Summary

Variety Dynamics (VD) proposes that variety space, the potential variety (option)s to a situation , operates at a more foundational level than either physical theory or formal mathematics. This case study develops that claim in three directions. First, varieties (options) are informatic descriptions, not physical things, and this distinction resolves persistent confusions in the foundations of physics. Second, quantum measurement, prediction, and physical theory are all variety as informatic variety space operations, and the quantum measurement problem dissolves when this is recognised. Third, VD's variety space concept operates at the same foundational level as set theory and potentially prior to it, because sets are actualisations from variety space rather than variety space itself. These claims are developed through analysis of quantum mechanics, Rational Quantum Mechanics (RaQM; Palmer, 2026), and the structural relationship between variety and set-theoretic foundations.

This case study builds directly on the transaction cost analysis of the RaQM ceiling established in Love (2026a) and should be read in conjunction with it.

The Analytical Problem

Physical theories describe how the world behaves. Mathematical structures provide the formal language in which those descriptions are expressed. Neither physical theory nor formal mathematics typically asks what kind of thing a state space is before any state has been selected or actualised. That question — what is the structured totality of potential informatic descriptions prior to selection? — is precisely the question variety space addresses.

VD defines variety as the possibility of a variable to have different values: the set of options available prior to any particular option being taken. Variety space is the full distribution of those possibilities across a situation. It is pre-enumerative in the sense that it characterises what could be described before any determination of what is described has occurred.

This pre-enumerative character places variety space at a foundational level that physical theories do not reach and that set theory approaches but does not fully occupy. Physical theories begin after states have been identified. Set theory begins during and after elements are distinguished and collected. Variety space is the informatic structure that makes the processes of identification and collection possible.

Varieties as Informatic Descriptions and Quantum States

A foundational distinction is necessary before analysing quantum states through VD. Varieties are in essence informatic entities, descriptions of reality in terms of distinguishable choices, states, decisions, commitments, or events. A variety is not the physical thing it describes. The variety "red cup" is the informatic description of red cup as a distinguishable option in a choice space, not a red cup itself. The red cup is the physical realisation; the variety is the description of that option as distinct from other options. They are practically linked — the description is grounded in physical reality — but they are not ontologically identical.

This distinction is not merely terminological. It is the source of persistent confusion in physics, where mathematical descriptions of physical states are routinely treated as identical to the physical objects being described. The wave function is not the electron. It is an informatic description of the electron's possible states in terms of distinguishable measurement outcomes. Varieties are always in the informatic realm; physical objects are always in the physical realm. The relationship between them is the relationship between a description and what it describes.

With this distinction in place, the VD account of quantum states is precise. A quantum system prior to measurement is a physical object. The wave function is an informatic description of that system in terms of a variety distribution — a structured set of distinguishable possible measurement outcomes with associated probability amplitudes. The wave function does not describe a definite state because the variety distribution it represents does not yet contain a definite selection — the description is of the full range of possible outcomes, none yet actualised.

Axiom 9 defines variety as the possibility of a variable to have different values. The wave function description of a pre-measurement quantum system exhibits exactly this structure: the variable "measurement outcome" has multiple possible values, none yet determined. The variety distribution is the informatic description of those possibilities.

Axiom 32 establishes that counterfactuals and varieties correspond exactly — both represent multiple possible states of a situation. This equivalence is fundamental rather than analogical: counterfactuals are varieties, not merely similar to them. The non-actualised outcomes of a quantum measurement are counterfactuals in precisely Axiom 32's sense: informatic descriptions of what the system could have exhibited but did not. Their correspondence with the variety distribution is structural, not metaphorical. Varieties and counterfactuals are the same informatic entity approached from different directions — variety from the perspective of available options, counterfactual from the perspective of non-selected options.

The wave function therefore exhibits three distinct modes of existence, of which one is false. In the first mode it is a real informatic description — a variety distribution of possible measurement outcomes instantiated in physical media at thermodynamic cost, grounded in physical reality and practically linked to it. In the second mode it is a real physical theory with causal efficacy in the world: instantiated in physicists' brains, published papers, laboratory instruments, and computational systems, the wave function description directs experiments, organises research programmes, and has genuine downstream physical consequences through its instantiation in those media. Both of these modes are correct and consistent with the VD account. The third mode — in which the wave function is assumed to exist as an independent physical entity with direct causal effects on reality, superposition as a real physical condition, and branches as real physical worlds — is false on the VD account. It is the conflation of the informatic description with the physical thing it describes, elevated to ontological status and granted independent causal powers. This third mode generates the measurement problem as an apparent paradox and motivates the many-worlds interpretation. The VD position is that the third mode is not required and not justified: the wave function is real in modes one and two, and mode three is an ontological inflation and conflation of mode one that the informatic account does not support.

Measurement and Prediction as Informatic Variety Spaces

Measurement in quantum mechanics is not a physical event that stands outside the variety space and selects from it. Measurement is itself an informatic variety space — a structured description of possible interactions between a measuring apparatus and a physical situation, characterised by distinguishable apparatus states, distinguishable interaction types, distinguishable recording events, and distinguishable outcomes. The measurement process itself has its own variety distribution, distinct from but related to the variety distribution of the system being measured.

The quantum measurement problem — why and how superposition gives way to a definite outcome — has resisted resolution for a century across numerous interpretations (Copenhagen, many-worlds, pilot wave, relational, decoherence-based). VD's informatic framing suggests that a significant part of this resistance is an artefact of conflating the informatic description with the physical object. When the wave function is identified with the physical system rather than recognised as an informatic description of it, the transition from superposition to definite outcome appears paradoxical — a physical object cannot be in multiple states simultaneously and then suddenly be in one. But if the wave function is correctly identified as an informatic variety description, the transition is the replacement of one informatic description (variety distribution of possible outcomes) by a more specific informatic description (definite actualised outcome) as the interaction between system and apparatus is recorded. The paradox resides in the description, not in the physics.

The measurement interaction is itself described by a variety space: the possible states of the apparatus, the possible interaction types, the possible recording configurations. This generates the von Neumann chain — each stage of the measurement process can be described as a variety space, and the non-selected states at each stage are counterfactuals in precisely Axiom 32's sense, retaining their status as variety even as one path is actualised. The question of where the chain terminates is the question of where the variety description becomes fixed as a definite recorded outcome. On the VD account this is not a paradox but the natural structure of nested informatic variety descriptions, each grounded in a physical substrate that carries the description at thermodynamic cost.

Prediction is also an informatic variety space operation (Axiom 32). A prediction is an informatic description of a subset of the variety space of possible future states — a counterfactual specification of what could occur, weighted by whatever criteria — theoretical, empirical, structural — the predictor applies. The prediction selects from and assigns relative weights to elements of the variety space of possible outcomes. It does not reach outside the informatic realm into physical reality. The relationship between a prediction and the physical outcome it anticipates is the relationship between an informatic description and the physical situation it describes — practically linked but not identical.

David Deutsch's Beginning of Infinity (2011) offers a sophisticated account of explanation and prediction that converges with VD at several points, particularly in its treatment of good explanations as hard to vary — a claim that maps onto VD's observation that constrained variety distributions characterise stable and powerful analytical positions. Deutsch's framework is illuminating and has informed thinking about the relationship between explanation and physical reality in ways that are broadly consistent with the VD position. Where the frameworks diverge is in the treatment of the wave function: Deutsch's commitment to the many-worlds interpretation treats the wave function as ontologically real — as the physical object rather than the informatic description of it. On the VD account, this is the conflation that generates the measurement problem as an apparent paradox rather than a structural relationship between informatic descriptions at different levels of specificity. The divergence is a consequence of a foundational framing difference rather than an error on either side, and both frameworks continue to develop.

The structure of measurement involves two distinct and sequentially constituted variety spaces. The first is the measuring variety space: a prior informatic commitment to a set of measurable categories — lengths, charges, vector paths, light intensities, masses, continuous or discrete scales, valid measurement bounds, and so on. This is a choice of what kinds of distinctions will be made and what range of values will be treated as representable. It is constituted before any apparatus is designed or any physical situation approached, and it determines what aspects of physical reality will be rendered visible and what will remain outside the scope of description regardless of what instruments detect. The second is the variety space of the measuring process, establishing the mapping of measurements to measurable states in the situations being ’measured’. I.e. the variety set of means, instruments, protocols, and interpretive conventions developed to apply the measuring variety space to physical reality and to obtain measurements from specific situations. This includes calibration procedures, data quality criteria, and crucially the conventions for deciding what counts as a valid result and what is classified as instrument error.

The Antarctic ozone hole discovery illustrates the structural consequence of these prior informatic commitments with precision. NASA had been collecting satellite ozone data since 1979 using Total Ozone Mapping Spectrometer instruments. The measuring variety space had been constituted with a lower bound of 180 Dobson Units — values below that threshold were outside the defined valid measurement range. The mapping variety space — the quality control algorithm applied to the data — was designed to flag any reading below 180 DU as instrument error and exclude it from analysis. When ozone readings over Antarctica fell below this threshold from the late 1970s onwards, they were systematically excluded as invalid. After the British Antarctic Survey published ground-based measurements of the ozone hole in 1985, NASA scientists reprocessed the satellite data with the quality control flag removed and found that the ozone hole had been visible in the data as far back as 1976. The physical phenomenon was present in the measurements for nearly a decade. The prior informatic commitment to a bounded measuring variety space, enforced through the mapping variety space's quality control conventions, rendered it unrepresentable as a valid result. The ozone hole was invisible not because instruments failed to detect it but because the constituted measuring variety space had no room for it.

The VD position — that all varieties are informatic descriptions grounded in but not identical to physical reality, and that measurement and prediction are themselves informatic variety space operations — dissolves rather than solves the measurement problem. It relocates the question from "what physical mechanism causes collapse" to "what determines the transition between levels of informatic description specificity" — a question that is structurally tractable within the VD framework and does not require a specific physical mechanism to be identified.

RaQM and the Physical Limits of Variety Description Instantiation

Case Study 1 (Love, 2026a) established that RaQM's predicted quantum computational ceiling is a structural consequence of the exponential and combinatorial scaling of transaction costs associated with maintaining quantum variety distributions in a physical substrate. That analysis applies directly here with an additional dimension that the informatic framing makes precise.

RaQM proposes that the continuous infinite-dimensional Hilbert space of standard quantum mechanics is an idealisation of a discrete, information-bounded structure. In VD terms, this is the claim that the informatic description of a quantum system's possible states cannot grow without bound — that the physical substrate carrying the informatic description can support only a finite variety distribution, and that this finite bound grows only linearly with qubit count while the abstract variety description of standard quantum mechanics grows exponentially.

The physical substrate carries the informatic description at thermodynamic cost — this is the content of Axioms 25 and 26. The variety distribution is not free: representing it requires physical resources. As the variety description grows exponentially with qubit count, the physical resources required to carry that description grow super-linearly, producing the structural diseconomy of scale identified in Case Study 1. RaQM's ceiling marks the scale at which the physical substrate can no longer carry the full informatic description — where the thermodynamic cost of representing the variety distribution exceeds any physically realisable substrate capacity.

Axiom 32 establishes that VD operates at higher abstraction levels than either conventional physics or constructor theory — examining what variety distributions enable different theoretical representations themselves, and what could alter the space of possible representations. The RaQM analysis illustrates this: VD does not merely describe what quantum mechanics predicts within its current representational framework. It identifies the structural condition — the mismatch between the variety distribution of the informatic description and the physical substrate's capacity to carry it — that determines what quantum computational descriptions are physically realisable at all. RaQM is one specific theoretical representation of that structural condition. VD describes the meta-level variety constraint that makes the ceiling inevitable regardless of which specific representation is used to identify it.

Axiom 47 proposes that VD operates at a more foundational level than physical theories. The RaQM analysis illustrates this directly: VD identifies the structural mismatch between the informatic variety description and the physical substrate's capacity to carry it as the foundational source of the ceiling. The physical theory (RaQM) describes one specific mechanism — gravitational discretisation — by which that mismatch becomes operative. VD describes the structural condition that makes the ceiling inevitable regardless of the specific mechanism.

Axiom 48 identifies the ceiling as a variety distribution discontinuity — a hard structural boundary beyond which the informatic variety description of the abstract quantum system cannot be physically carried. This is not a smooth degradation of performance but a structural threshold, consistent with RaQM's prediction that the exponential advantage of quantum algorithms saturates abruptly rather than declining gradually.

Variety Space and the Foundations of Mathematics

Set theory is the standard foundation of mathematics. A set is a collection of distinct elements. The axioms of Zermelo-Fraenkel set theory with Choice (ZFC) provide the formal basis from which the bulk of mathematics is derived.

VD's variety space concept stands in a specific foundational relationship to set theory that is worth examining precisely. A set is determined by its members — this is the axiom of extensionality. But to specify members, elements must be distinguishable from one another. Distinguishability — the capacity for difference between elements — is not itself defined within ZFC. It is imported silently as a pre-formal primitive.

That primitive is variety. Distinguishability between elements is precisely the variety of the collection: the capacity of the variable "which element" to take different values. Without variety — without the structural possibility of difference — no two elements could be distinguished, and no set could be formed. Variety space is the pre-formal structure that ZFC's extensionality axiom presupposes but does not name.

This means variety space is not merely co-foundational with set theory. It is the unnamed pre-formal ground on which extensional set theory's most basic axiom depends. Sets are actualisations from variety space: the operation of forming a set is precisely the operation of selecting and fixing a collection of distinguishable elements from the variety space of possible elements.

The ZFC axiom of separation makes this concrete. Separation states that for any set and any property, there exists a subset containing exactly those elements satisfying the property. In VD terms, separation is a variety-selection operation: it carves an actualised subset from the variety space of the original set by applying a selection criterion. The subset is an actualisation; the original variety space of possible elements remains intact and available for further selection operations.

Axiom 9 defines variety as the possibility of a variable to have different values. The foundational operation of set theory — distinguishing elements and collecting them — presupposes exactly this: that the variable "which element" has the possibility of different values. Without that possibility, the operation cannot be performed.

The Pre-Enumerative Character of Variety Space and Its Categorical Structure

The critical distinction between variety space and sets is temporal and ontological, not merely formal. Variety space is pre-enumerative: it characterises the structured totality of potential distinctions before any enumeration has occurred. A set is post-enumerative: it is the result of an enumeration operation that has already occurred.

This distinction maps directly onto the quantum mechanical situation. The pre-measurement quantum state is pre-enumerative — the possible outcomes exist as a structured distribution before any measurement selects among them. The post-measurement outcome is post-enumerative — a specific element has been selected and the result is a definite, actualised state.

It also maps onto the foundational mathematical situation. Before a set is formed, there is a variety space of possible elements from which the set's members will be drawn. The set is the post-enumerative result of the selection operation. ZFC begins at the post-enumerative stage and provides axioms for operating on already-formed sets. It does not provide an account of the pre-enumerative variety space from which sets are formed.

VD operates at the pre-enumerative stage. This is what Axiom 47 means when it proposes that VD operates at a more foundational level than physical theories: VD characterises the structure of possibility prior to the selection operations that physical theories describe. Physical theories describe how selection operations occur and what their outcomes are. VD describes the variety space within which those operations take place. As Axiom 32 establishes, this means VD occupies a higher abstraction level than both conventional physics and constructor theory — examining what variety distributions enable different theoretical representations themselves, not merely what those representations describe within their current frameworks.

The relationship between variety space and specific physical state spaces is most precisely characterised in the language of category theory and higher topos theory rather than set-theoretic containment. Hilbert space — the state space of quantum mechanics — is a variety space with additional structure imposed: an inner product defining geometry, a norm defining distance, and completeness ensuring no limit points are absent. The relationship between Hilbert space and variety space is characterised by a forgetful functor in the categorical sense: a structure-preserving map from the category of Hilbert spaces to the category of variety spaces that forgets the inner product, norm, and completeness while retaining the underlying variety structure. Hilbert space is variety space with extra structure; variety space is Hilbert space with that structure forgotten.

This categorical framing extends to all physical state spaces. Phase space in classical mechanics, Fock space in quantum field theory, and configuration space in Lagrangian mechanics are each variety spaces with different additional structures imposed. Each admits a forgetful functor to the category of variety spaces. The direction of those forgetful functors establishes the generality relationship precisely: variety space is the most general structure, and specific physical state spaces are special cases obtained by imposing additional constraints. No set-theoretic containment claim is required and no foundational priority dispute arises — the categorical relationship is structural and unambiguous.

In higher topos theory, this relationship becomes still more precise. A higher topos is a category rich enough to internalise logic, homotopy theory, and the full machinery of mathematics. Within a higher topos, variety space corresponds to the pre-sheaf structure — the totality of potential local data distributed across a base of possible contexts, prior to any globalisation. The variety-to-actualisation transition corresponds to sheafification: the operation by which local potential data is assembled into a globally coherent actualised structure. The pre-measurement quantum state is a pre-sheaf of potential measurement outcomes; measurement is the sheafification operation that produces a globally consistent actualised result. This is not an analogy — it is the same structural operation characterised at different levels of abstraction.

Axiom 57 of VD (in development), supported by Axiom 32, proposes that VD subsumes constructor theory, knot theory, tessellation, topology, and related mathematical structures as special cases. In categorical terms, this is the claim that each of these structures admits a forgetful functor to the category of variety spaces — that each is a variety space with additional constraints. The higher topos framework provides the mathematical setting in which this subsuming relationship can be formally established.

Limitations and Open Questions

The claim that variety space is foundational to both physics and mathematics is a strong philosophical claim that requires formal development beyond what this case study provides.

The connection between variety space and the pre-formal primitive of distinguishability in ZFC is argued structurally here but has not been formally derived. A rigorous treatment requires either a formal demonstration that variety can be defined without presupposing set-theoretic structures, or a precise characterisation of the mutual dependency between variety and sets that establishes their co-foundational status without circularity. That formal treatment is identified as the primary task for the standalone mathematical paper in development.

The VD informatic framing of the measurement problem — as a transition between levels of informatic description specificity rather than a physical collapse event — dissolves rather than solves the measurement problem in the conventional sense. It does not specify a physical mechanism. This is intentional: the physical mechanism is a matter for physics to determine, and VD's contribution is to identify the structural condition — the relationship between informatic variety descriptions at different levels of specificity — that any such mechanism must satisfy.

The categorical and higher topos framing of Section 6 is stated at a level of precision appropriate for a website case study. The full formal treatment — establishing the forgetful functors between specific physical state space categories and the category of variety spaces, and characterising variety space as a pre-sheaf structure within a higher topos — requires mathematical development beyond the scope of this case study. That development is identified as a priority for the standalone mathematical paper in preparation. The claim that Axiom 57 subsumes constructor theory, knot theory, topology, and related structures as special cases of VD is similarly identified as requiring formal categorical demonstration in that paper.

Two related transitions at the boundary between the informatic and physical realms remain underspecified in the current VD framework and in constructor theory: the transition from variety selection to design — how an informatic selection from a variety space becomes a coherent, implementable specification sufficient to guide manufacture — and the transition from design to manufactured physical entity — how a constructor uses a design to produce a specific physical instantiation. Both transitions involve informatic-to-physical crossings that are structurally distinct from the variety distribution dynamics VD currently formalises. The variety-to-design transition in particular involves a substantial increase in informatic specificity — from a selection among options to a fully specified, internally coherent design — whose structural conditions are not yet axiomatised in VD. Engineering design theory and design process research have addressed both transitions in practical and theoretical terms (REFS), and integrating those accounts into the VD formal framework is identified as a development priority.

Conclusion

Varieties are informatic descriptions of reality in terms of distinguishable options, states, decisions, or events — not the physical things they describe. This distinction, consistently maintained, reframes several foundational problems in physics and mathematics.

The wave function is an informatic variety description of a quantum system's possible measurement outcomes, not the physical system itself. It is real in two modes — as an informatic description instantiated in physical media, and as a theory with causal efficacy through its instantiation — but not in the third, false mode of an independent physical entity with direct causal powers. Measurement is itself an informatic variety space — a structured description of possible apparatus-system interactions and their distinguishable outcomes, with non-selected outcomes retaining their status as counterfactual varieties (Axiom 32). Prediction is an informatic variety space operation selecting from and weighting possible future states. The quantum measurement problem is the structural question of how informatic descriptions at one level of specificity transition to descriptions at a more specific level — a question tractable within VD that does not require identifying a specific physical collapse mechanism.

VD operates at higher abstraction levels than conventional physics and constructor theory, examining what variety distributions enable different theoretical representations themselves (Axiom 32). RaQM's predicted ceiling marks the scale at which the physical substrate can no longer carry the full informatic variety description of a quantum system — where thermodynamic costs of representing the variety distribution exceed any realisable substrate capacity. Set theory's foundational operation of element distinguishability presupposes variety as its unnamed pre-formal primitive. Variety space is pre-enumerative; sets are post-enumerative results of selection operations on variety space.

All specific physical state spaces — Hilbert space, phase space, Fock space, configuration space — are variety spaces with additional structure imposed, related to the category of variety spaces by forgetful functors. In higher topos theory, variety space corresponds to the pre-sheaf structure from which actualised states emerge through sheafification. These categorical relationships establish the generality of variety space over specific physical state spaces without requiring absolute foundational priority claims.

Together these observations support Axiom 47's claim that VD operates at a more foundational analytical level than physical theories, and extend that claim to the foundations of mathematics. The formal development of the forgetful functor relationships, the higher topos characterisation of variety space, and the informatic grounding of the variety-set foundational relationship are identified as priorities for subsequent mathematical work.

Axioms Applied

Axiom 9 (Variety definition), Axiom 25 (Variety dynamics and information systems), Axiom 26 (Variety dynamics, information systems, and thermodynamic constraints), Axiom 32 (Variety dynamics, counterfactuals, and constructors), Axiom 47 (Variety dynamics and fundamental physics — tentative), Axiom 48 (Discontinuity and irreversibility in variety distributions), Axiom 57 (VD subsuming special cases — in development).

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

Deutsch, D. (2011). The Beginning of Infinity: Explanations that Transform the World. Allen Lane.

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

Love, T. (2026a). Variety Dynamics Case Study: An Alternative Structural Explanation of the Quantum Computational Ceiling Proposed by Rational Quantum Mechanics. Zenodo. https://doi.org/10.5281/zenodo.19140950

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

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); structural mapping, argument development, and initial drafting generated by Claude Sonnet 4.6 (Anthropic); reviewed, verified, edited and refined by T. Love through multiple iterations. Final analysis reflects human expert judgment of variety distributions, structural dynamics, and analytical sufficiency.