Universal Representations of Matter

• Universal representations of matter are frameworks that mathematically and physically model diverse forms of reality, from elementary particles to cosmic structures.
• They integrate methods such as relational ontology, algebraic symmetry, quantum entanglement, and machine learning to bridge classical and quantum descriptions.
• These approaches generate unifying analytic tools that inform interpretations of standard models, dark matter profiles, and emergent gauge structures.

Universal representations of matter are mathematical, physical, or computational schemes that aim to describe all forms of material reality—elementary particles, fields, condensed phases, and even dark sectors—within a shared, rigorously defined framework. These approaches seek universality either by imposing structural parsimony, manifest symmetry, information-theoretic primitives, or algebraic unification, often extending or superseding standard models of particle physics, quantum mechanics, and cosmology.

1. Atomistic–Structural Realist Ontology and Relational Foundations

Esfeld, Deckert & Oldofredi’s universal relational ontology defines matter as a collection of $N$ structureless matter points, fundamentally individuated only by their spatial relations. Neither absolute space, intrinsic properties, nor a substratum exist—only the instantiational network of spatial relations remains. Changes in physical systems are entirely shifts within this web. Mathematically, the instantaneous configuration is a point in the set $\Omega$ of all relational graphs: each $\Delta \in \Omega$ encodes the web of pairwise distances or relations across $N$ points.

The dynamical law is a first-order evolution $dQ_t/dt = v_t(Q_t)$ in $\mathbb{R}^{3(N-1)}$, where $Q_t$ parametrizes relative positions. The velocity field $v_t$ is chosen to recover classical Newtonian motion (via a potential $V(Q)$) or Bohmian quantum mechanics (via a wavefunction $\Psi_t(Q)$ evolving under a Hamiltonian $H$), simply by different choices of dynamical parameters. Extensions to GRW collapse, QFT, and quantum gravity fit naturally, with all field-theoretic ingredients relegated to non-ontological, dynamical roles (Esfeld et al., 2015).

This scheme achieves universality via parsimony, relational invariance, and the capacity to derive both classical and quantum histories from a single law applied to N-point configurations.

2. Algebraic and Group-Theoretic Universal Classifications

Meng’s construction replaces Lorentz structure in quantum theory with Euclidean Jordan algebra (EJA), associating each simple EJA $J$ with a conformal group $G = \mathrm{Conf}(J)$ whose unitary lowest-weight representations yield “abstract fundamental matter particles.” All quantum states of such a particle form a bound-state Hilbert space $\mathcal{H}$, and ground states decompose into irreducible $K$-modules (compact subgroup $K$ of $G$).

The search for universality proceeds by classifying abstract “universes” via the dimension of their charged-particle spaces. Singularly, $\mathrm{SO}(11,2)$ delivers exactly 32 charged particle types—matching the Standard Model’s matter content when decomposed at low temperature—alongside dark-matter candidates and an infinite tower of generations. Phenomenologically, constraints from conformal symmetry enforce an $11$-dimensional Lorentzian structure at high temperature, while a “fifth force” emerges via a broken $\mathrm{SO}(4,2)$, furnishing CP violation and quantitative agreement with CKM mixing parameters (Meng, 2012).

This formalism unifies matter content, symmetry breaking, and generational structure via group-theoretic and algebraic data, offering universality grounded in representation theory.

3. Information-Theoretic Models: Qubit Oceans and String-Net Condensates

The quantum information paradigm posits matter (and forces) as emergent collective excitations of an underlying lattice of entangled qubits: a “qubit ocean.” String-net models (Levin–Wen) render photons and gauge bosons as transverse flux fluctuations, while fermionic statistics emerge through the topological braiding of open string endpoints, even when all degrees of freedom are fundamentally bosonic.

The string-net Hamiltonian aggregates local branching (vertex) and loop-creation (plaquette) terms, and continuum limits naturally recover $U(1)$ and $SU(N)$ gauge actions. Fermionic creation operators are realized as nonlocal strings of Pauli operators, with their anticommutation enforced by global entanglement structure. This framework generalizes to encompass all observed gauge and matter content, with the unifying principle supplied by long-range entanglement. Model extensions can theoretically accommodate gravity and exotic hidden sectors (Wen, 2017).

Thus, universal representation is achieved by identifying the informational structure—the pattern of entanglement—as the substrate of all observed material phenomena.

4. Universal Hilbert-Space and Operator Representations

Quantum foundations and condensed matter physics admit a universal mixed-space operator basis $Y(u,v)$, for two conjugate operators $Q,P$. Any operator $\hat{O}$ can be expanded over $Y(u,v)=\exp[-i(u\cdot P - v\cdot Q)/\hbar]$, with expansion coefficients given as Fourier transforms of the Wigner quantum distribution. This enables a single representational kernel for bosonic, fermionic, and spin systems that encompasses all familiar mappings (Jordan–Wigner, Holstein–Primakoff, Bogoliubov), as well as coherent states and quantum optics formulations.

Functional extensions generate universal representations for field-theoretic quantum transport, treating operator evolution as a Moyal-type partial difference equation over quantum characteristic functionals. These representations are manifestly general, supporting open-system evolution and time-reversal breaking in nonequilibrium settings (Buot et al., 2022).

5. Universal Machine Learning Representations in Scientific Foundation Models

Using large-scale foundational scientific neural networks, recent empirical work quantifies the convergence of learned representations across diverse models (string-, graph-, atomistic-, protein-based) by direct comparison of their latent embedding spaces. Centered Kernel Alignment (CKA), Distance Correlation (dCor), intrinsic dimension estimation, and asymmetric information imbalance expose a robust alignment among top-performing models when evaluated on shared chemical inputs.

Key findings:

• Representations become increasingly aligned as model performance improves (low energy-MAE), demonstrating monotonic convergence.
• Two regimes emerge: tight alignment in-distribution (ID) for high-performance models, local sub-optimality for weaker models, and collapse to low-information manifold structure out-of-distribution (OOD), reflecting architecture-induced bias.
• Intrinsic dimension bottlenecks and mutual information scores indicate that foundation models encode a common, low-dimensional manifold for matter, spanning modalities and datasets.

True universality, however, remains contingent on coverage of diverse matter domains and robust OOD generalization. Nevertheless, representational alignment offers a quantitative benchmark and selection tool for universal scientific models (Edamadaka et al., 3 Dec 2025).

6. Unified Gauge Frameworks and Representation Invariance

Universal representation is realized in the Principle of Representation Invariance (PRI), requiring physical gauge laws to be independent of basis choices in the internal $SU(N)$ algebra. The total Lagrangian for gravity, electromagnetism, weak, and strong interactions is constructed so that all matter fields couple via exactly the same minimal pattern:

$$\mathcal{L} = \mathcal{L}_{\text{GR}} + \mathcal{L}_{\text{QED}} + \mathcal{L}_{\text{Weak}} + \mathcal{L}_{\text{QCD}},$$

with each covariant derivative and coupling invariant under arbitrary basis transformation. This enforces a universal treatment of all matter fields and interactions. Modified field equations result from the Principle of Interaction Dynamics (PID), and dual gauge fields (adjoints) emerge naturally—a conceptual source for Higgs-like bosons and the dark sector (Ma et al., 2012).

7. Algebraic Universal Models: Division Algebras and Emergent Gauge Structure

Algebraic constructions employing division algebras (complex numbers $\mathbb{C}$, quaternions $\mathbb{H}$, octonions $\mathbb{O}$) generate universal representations for matter and antimatter. Projectors associated with octonionic units split the full tensor algebra $$T = \mathbb{C} \otimes \mathbb{H} \otimes \mathbb{O}$$ into seeable and unseeable subspaces, directly embedding one family of quarks and leptons, and their anti-family, within 128D spinor representations. Extra octonionic dimensions are responsible for SU(3) color charge and the hiding of antimatter, while left- and right-action projectors extract Standard Model gauge symmetries naturally. This not only accounts for matter–antimatter asymmetry but also points to an unseeable antimatter universe connected by higher-dimensional spaces (Dixon, 2014).

8. Quantum Supreme Matter and Holographic Emergence

Holographic duality provides a universal representation for quantum-supreme matter—systems of maximal, long-range entanglement, inaccessible to conventional or weakly-coupled field theories. The Ryu–Takayanagi formula computes entanglement entropy via bulk geometric minimal surfaces, while AdS/CFT correspondence determines spectral and transport observables through scaling geometries. General classes of scaling exponents (dynamical $z$, hyperscaling violation $\theta$) map directly onto universal thermodynamic and response features in strongly correlated materials. This approach has predicted linear-T resistivity, Planckian dissipation, and scaling spectra compatible with "strange metal" regimes (Zaanen, 2021).

9. Unified and Cogenerative GUT Multiplets for Baryonic and Dark Matter

Grand Unified Theories (GUT) with gauge groups such as SU(6) embed all visible (Standard Model) and dark-matter fermions into unified multiplets. Accidental U(1) global symmetries naturally stabilize dark-matter candidates, while higher-dimensional couplings and sphaleron processes cogenerate matching baryon and dark-matter asymmetries. The relative cosmic abundances, dark-matter mass scale, and stability emerge from representation content and symmetry architecture (Barr, 2011).

10. Universal Dark Matter Halo Representation

Perturbative solutions of the Einstein equations for galactic halos yield an analytic universal mass-density profile for dark matter: $\rho(r) = a_0/(2\pi G r)$, with $a_0$ fixed by rotation curve and Tully–Fisher data. This profile underpins observed constant surface densities and scaling relations (Burkert, NFW), and the inclusion of a baryonic mass removes the central cusp analytically—solving the core–cusp problem. The singular universality of $a_0$ provides a single analytic form for all halos, connecting dark-matter phenomenology to fundamental gravitational dynamics (Haddad et al., 28 Jun 2024).

11. Quantum-Phase-Field Formalism and Emergent Gravity

Steinbach’s monistic quantum-phase-field formulation represents all energy states by normalized scalar fields $\phi_i$, from which mass and spatial energies emerge as bulk and gradient terms. Gravitational-like attractions arise from Casimir energy fluctuations in finite intervals between particle-domains, leading to effective Newtonian forces and scale-dependent repulsion that predict void sizes matching cosmic structures. This universal representation links mass and space causally, entropically, and dynamically in a background-independent manner (Steinbach, 2017).

12. Geometric Matter: Space Without Time

The Matter: Space Without Time (MST) paradigm posits material regions as geometric degeneracy loci of a single four-manifold metric $g_{ab}$. Matter is defined as "space without time," with the metric degenerating in its temporal direction. Field equations and particles are derived solely from the metric structure, encoding all charges and topological invariants in the local geometry. This framework unifies particles, fields, space, and time algebraically but lacks explicit gauge and quantum completions (Ghazi-Tabatabai, 2012).

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The concept of universal representations of matter spans relational, algebraic, quantum-informational, gauge-theoretic, machine-learning, and geometric domains. Each approach proposes a unifying template that not only subsumes the Standard Model and its extensions, but also provides analytic or computational tools for emergent fields, quantum phases, cosmological backgrounds, and dark matter. Despite substantial progress, full mathematical and physical universality—including robust out-of-distribution generalization, dynamic and topological quantization, and complete gauge–quantum unification—remains an enduring challenge and research frontier.