GLOSSARY — docs/theories/repos/

TriadicFrameworks Canonical Documentation Scope: Quantum-theoretic module registry — docs/theories/repos/ Status: Canonical · Living Document Generated: 2026-07-13

Purpose#

This glossary defines the formal vocabulary in use across the three quantum-theoretic modules registered in docs/theories/repos/. It is the single source of truth for term definitions within this triadic cluster. All terms are canonical: they carry specific architectural meaning within TriadicFrameworks and must not be used loosely or interchangeably.

Entries are organized alphabetically. Each entry identifies the module(s) to which the term is native or across which it is materially relevant: [QB] = quantum_basis, [QL] = quantum-lattice, [LQ] = LattiQ, [ALL] = all three modules, [META] = triadic meta-term governing the cluster as a whole.

Cross-references are indicated with → See also.

Terms#

Abstraction Boundary [LQ] [META]: The enforced separation between the internal structure of a triadic cluster and the interface it exposes to external consumers. LattiQ is the abstraction boundary of this cluster. Callers of LattiQ are explicitly prohibited from reasoning about lattice topology or basis primitives directly; all such complexity is mediated through the interface. Abstraction boundaries are the mechanism by which triadic clusters achieve compositional stability. → See also: Integration Seam, Interface (Layer III), LattiQ.

Algebraic Composition Rules [QB]: The formal rules governing how basis vectors and operators combine within the Hilbert space. These rules are defined in quantum_basis and constitute part of the canonical unit type contract that all higher modules must respect. → See also: Canonical Unit Type, Formal Postulate, State-Space Primitive.

Alpha (α) [LQ]: A configurable hyperparameter in LattiQ controlling the fraction of the probability distribution used in the CVaR loss function. Tuning α trades off noise robustness against solution fidelity in VQE and QAOA solvers. → See also: CVaR, QAOA, VQE.

ARPACK-NG [QB]: An open-source library of Arnoldi/Lanczos eigenvalue routines used by quantum_basis as a backend eigensolver for large sparse Hamiltonians. → See also: Eigensolver, Lanczos Method.

Basis Vector [QB]: An element of the many-body Hilbert-space basis as constructed by quantum_basis. Basis vectors are the primitive objects from which all quantum-state representations in this cluster are formed. A basis vector encodes a specific occupancy configuration of the degrees of freedom present in the model. → See also: Hilbert Space, Many-Body Basis, State-Space Primitive.

Band Structure [QL]: The energy dispersion relation E(k) of a lattice model as a function of crystal momentum k, computed by quantum-lattice. Band structures are a primary output of tight-binding calculations and serve as phenomenological fingerprints that feed into both ED verification and VQA encoding. → See also: Hamiltonian, Tight-Binding Model.

Bogoliubov–de Gennes (BdG) Formalism [QL]: A mean-field framework implemented in quantum-lattice for describing superconducting systems via a Nambu spinor basis. BdG Hamiltonians capture particle-hole symmetry and are used to model unconventional superconductivity on lattice geometries. → See also: Nambu Basis, Superconductivity (Unconventional).

Bosonic Degree of Freedom [QB]: A quantum degree of freedom obeying Bose–Einstein statistics, characterized by commutation relations rather than anti-commutation. quantum_basis supports mixed bosonic and fermionic configurations within the same many-body basis. → See also: Fermionic Degree of Freedom, Many-Body Basis.

Canonical Review [META]: The governance process required before any module is added to, or any breaking change is made to, the docs/theories/repos/ directory. Canonical review assesses whether a proposed change preserves triadic slot integrity and does not destabilize the cluster's interface contract. → See also: Governance, Triadic Slot.

Canonical Unit Type [QB] [ALL] [META]: The formally defined primitive types — basis vectors, operators, state representations — that quantum_basis establishes and that all modules in this cluster must use without alteration. Canonical unit types are the mechanism of type coherence across the cluster: they are enforced at the substrate level rather than retrofitted at integration seams. → See also: Algebraic Composition Rules, Formal Postulate, Integration Seam, Type Coherence.

Chern Number [QL]: An integer-valued topological invariant computed by quantum-lattice that classifies the topology of energy bands in two-dimensional lattice systems. A non-zero Chern number is the signature of a quantum Hall phase. → See also: Topological Invariant, Z₂ Invariant.

Cluster [ALL] [META]: The collective term for the three modules — quantum_basis, quantum-lattice, LattiQ — treated as a single composable unit within TriadicFrameworks. The cluster is the atom of canonical composition: external theories interact with the cluster through its Layer III interface, not with individual modules. → See also: Quantum-Theoretic Primitive Cluster, Triadic Unit.

Cluster Versioning [META]: The practice of incrementing the version of all three modules simultaneously when a breaking change is made to quantum_basis. Because quantum_basis is the dependency terminus, changes to it propagate to the entire cluster; versioning must reflect that propagation. → See also: Dependency Terminus, quantum_basis.

Compositional Direction [META]: The architectural principle that triadic clusters compose upward through their Layer III interface modules. Cross-cluster dependencies are always Layer-III–to–Layer-III bindings; no cluster reaches into another cluster's Layer I or Layer II internals. → See also: Abstraction Boundary, Integration Seam, Interface (Layer III).

Conditional Value-at-Risk (CVaR) [LQ]: A loss function used in LattiQ in place of the standard expectation-value objective. CVaR averages only the lowest-energy fraction α of measurement outcomes, providing robustness against quantum hardware noise. → See also: Alpha (α), QAOA, VQE.

Cross-Canon Impact Assessment [META]: A mandatory governance step triggered whenever LattiQ's interface contract is modified. The assessment evaluates which other triadic clusters or higher-order TriadicFrameworks theories depend on LattiQ as a boundary and whether those dependencies remain satisfiable. → See also: Canonical Review, Interface (Layer III), LattiQ.

Dependency Terminus [QB] [META]: The module in a triadic cluster that has no upstream dependencies within the cluster. In this cluster, quantum_basis is the dependency terminus — the floor. All other modules depend on it; it depends on none of them. → See also: quantum_basis, Cluster.

Downward Grounding [ALL] [META]: One of two integration directions of the quantum-theoretic cluster within the broader TriadicFrameworks canon. Downward grounding means this cluster provides the mathematical and structural foundation for any canon module that makes claims about quantum state, superposition, entanglement, or measurement. Such claims must be grounded through LattiQ's interface. → See also: Upward Composition, Interface (Layer III).

Eigensolver [QB]: A numerical routine that computes eigenvalues and eigenvectors of a sparse Hamiltonian matrix. quantum_basis exposes two eigensolver backends: Lanczos and FEAST. The output of an eigensolver constitutes the exact diagonalization (ED) spectrum, which serves as the benchmark against which higher-level approximations are measured. → See also: Exact Diagonalization, FEAST Eigensolver, Lanczos Method.

Entanglement Geometry [QL] [META]: The structural encoding, within quantum-lattice, of entanglement relationships between quantum states as explicit first-class objects. Entanglement geometries are not emergent side-effects in this canon; they are declared structural elements of the lattice. → See also: Lattice Topology, Superposition Arrangement.

Exact Diagonalization (ED) [QB]: The numerical paradigm implemented by quantum_basis, in which the full Hamiltonian matrix is constructed in the many-body Hilbert-space basis and diagonalized to obtain exact eigenvalues and eigenvectors. ED produces ground-truth spectra against which tight-binding and variational results are benchmarked. → See also: Eigensolver, Hamiltonian, Many-Body Basis.

Extension Hook [LQ] [META]: A documented integration point in LattiQ through which the quantum-theoretic cluster can be composed with other triadic clusters in the TriadicFrameworks canon. Extension hooks are the canonical mechanism for cross-cluster composition; they must not expose internal lattice topology or basis primitives. → See also: Integration Seam, Compositional Direction.

FastVQA [LQ]: The open-source variational quantum algorithm framework on which LattiQ is built. FastVQA provides the circuit execution and optimization substrate; LattiQ layers lattice-problem encoding and SVP-specific configuration on top. → See also: LattiQ, VQE, QAOA.

FEAST Eigensolver [QB]: A contour-integration-based eigensolver used in quantum_basis for computing eigenvalues within a specified energy window. FEAST is suited to mid-spectrum queries where Lanczos iteration is less efficient. → See also: Eigensolver, Lanczos Method.

Fermionic Degree of Freedom [QB]: A quantum degree of freedom obeying Fermi–Dirac statistics, characterized by anti-commutation relations. quantum_basis supports arbitrary combinations of fermionic and bosonic degrees of freedom in constructing the many-body basis. → See also: Bosonic Degree of Freedom, Many-Body Basis.

Formal Postulate [QB] [META]: An axiomatic constraint, defined in quantum_basis, that determines what counts as a valid quantum-theoretic object within TriadicFrameworks. Formal postulates are the canon's ontological commitments. No module in this cluster may define quantum objects that violate these postulates. → See also: Canonical Unit Type, Quantum Ontology.

Foundation Layer [QB] [META]: The TriadicFrameworks designation for quantum_basis in its role as the rigorous, fully quantum-mechanical ground truth of the cluster. The Foundation Layer provides exact spectra against which all phenomenological and variational approximations (Layers II and III) are validated. → See also: Dependency Terminus, Primitive (Layer I), quantum_basis.

Generalized Lin Table [QB]: A data structure used in quantum_basis for encoding the many-body basis with or without translational symmetry. The Lin Table provides an efficient bijection between basis states and their integer indices, enabling fast matrix-vector products in the Lanczos loop. → See also: Many-Body Basis, Translational Symmetry.

Governance [ALL] [META]: The set of policies governing modification, addition, and deprecation of modules within docs/theories/repos/. Governance is enforced at the cluster level: changes to any module are evaluated for their impact on the full triadic dependency chain, not in isolation. → See also: Canonical Review, Triadic Slot.

Green's Function [QL]: A many-body propagator computed by quantum-lattice, characterizing how a quantum system responds to perturbations as a function of energy and momentum. Green's functions are used to compute spectral functions and local density of states, and are fundamental inputs to recursive boundary methods for semi-infinite systems. → See also: Local Density of States, Spectral Function.

Hamiltonian [QB] [QL] [LQ]: The operator encoding the total energy of a quantum system, central to all three modules. In quantum_basis, it is represented as a sparse matrix in the many-body basis and passed to eigensolvers. In quantum-lattice, it is constructed phenomenologically from tight-binding and mean-field parameters. In LattiQ, it appears as an Ising-mapped form encoding the lattice optimization problem. → See also: Exact Diagonalization, Ising Spin Hamiltonian, Tight-Binding Model.

Hilbert Space [QB] [META]: The complete complex vector space of quantum states for a given system, as defined and constructed by quantum_basis. The Hilbert space is the mathematical container within which all quantum-theoretic operations in this cluster are performed. Its dimension grows exponentially with system size, motivating the approximation strategies of Layers II and III. → See also: Many-Body Basis, State-Space Primitive.

Interface (Layer III) [LQ] [META]: The third and outermost layer in any triadic cluster; the module that makes the cluster's internal structure legible and composable to external consumers. In this cluster, LattiQ occupies Layer III. The Layer III interface is the only sanctioned point of contact between this cluster and the broader TriadicFrameworks canon. → See also: Abstraction Boundary, Extension Hook, LattiQ.

Integration Seam [ALL] [META]: The defined boundary at which two modules or two clusters connect. Within the cluster, integration seams exist between Layer I and Layer II (at the point where quantum_basis types are consumed by quantum-lattice) and between Layer II and Layer III (where quantum-lattice structural constructs are translated by LattiQ). Canonical unit types enforce coherence at these seams. Across clusters, integration seams are always Layer-III–to–Layer-III. → See also: Canonical Unit Type, Compositional Direction, Type Coherence.

Intel MKL [QB]: Intel Math Kernel Library, used by quantum_basis as a high-performance backend for sparse linear algebra operations underlying Hamiltonian construction and eigensolver routines. → See also: Eigensolver, Hamiltonian.

Ising Spin Hamiltonian [LQ]: The binary optimization formulation used in LattiQ, in which the target problem (SVP) is encoded as an energy function over spin-½ variables (±1 or 0/1). The Ising encoding is the bridge between classical lattice geometry and quantum circuit execution. → See also: Hamiltonian, Shortest Vector Problem, VQE, QAOA.

Join / Meet Structure [QL] [META]: The lattice-theoretic operations — least upper bound (join, ∨) and greatest lower bound (meet, ∧) — through which quantum-lattice encodes the partial-order relationships between quantum states. The join/meet structure is the algebraic backbone of the lattice topology. → See also: Lattice Topology, Partial Order.

Kagome Lattice [QB]: One of the bundled example lattice geometries in quantum_basis, used to demonstrate exact diagonalization on geometrically frustrated systems. → See also: Lattice Geometry, Exact Diagonalization.

Kernel Polynomial Method (KPM) [QB]: A spectral expansion technique used in quantum_basis to compute spectral functions and the density of states without full diagonalization. KPM approximates spectral quantities via Chebyshev polynomial expansion of the Hamiltonian, enabling access to large system sizes. → See also: Spectral Function, Eigensolver.

Kondo Lattice Model [QB]: A bundled example model in quantum_basis representing itinerant electrons coupled to localized spins. Used to validate the many-body basis construction and ED pipeline for models with competing energy scales. → See also: Exact Diagonalization, Many-Body Basis.

Landau Level [QL]: A quantized energy level arising in a two-dimensional electron system under a perpendicular magnetic field, computable by quantum-lattice. Landau levels underpin the integer quantum Hall effect; their structure in lattice models is accessible through the module's Hamiltonian machinery. → See also: Quantum Hall Edge State, Band Structure.

Lanczos Method [QB]: An iterative Krylov-subspace algorithm used by quantum_basis to compute low-energy eigenvalues and eigenvectors of large sparse Hamiltonians without full diagonalization. The Lanczos method is the primary eigensolver for ground-state and low-spectrum queries. → See also: Eigensolver, FEAST Eigensolver.

LattiQ [LQ]: The Layer III — Interface module of this triadic cluster. A C++ framework, built atop FastVQA, for solving lattice optimization problems on quantum hardware or simulators. Its primary target is the Shortest Vector Problem, encoded as an Ising spin Hamiltonian and solved via VQE or QAOA. As Layer III, LattiQ is the public face of the quantum-theoretic cluster: the sole sanctioned entry point for external canon modules and higher-order theories. Author: Miloš Prokop. License: MIT. → See also: FastVQA, Interface (Layer III), Ising Spin Hamiltonian, Shortest Vector Problem.

Lattice Geometry [QB] [QL] [LQ]: The spatial arrangement and connectivity of sites in a lattice model — including square, honeycomb, Kagome, and arbitrary geometries. Lattice geometry is specified in quantum-lattice, consumed by both quantum_basis (for ED benchmarking) and LattiQ (for Ising encoding). → See also: Hamiltonian, Tight-Binding Model.

Lattice Topology [QL] [META]: The partial-order and join/meet structure through which quantum-lattice encodes how quantum states relate to one another — which states are reachable from which, under what operations. Lattice topology is the structural core of Layer II; it is what quantum-lattice contributes to the cluster that neither Layer I nor Layer III provide. → See also: Join / Meet Structure, Partial Order, Structure (Layer II).

Layer [ALL] [META]: One of three positions in a triadic cluster, designated I (Primitive), II (Structure), or III (Interface). Each module in docs/theories/repos/ occupies exactly one layer. Layers are not ranks of importance but roles in a dependency and composability chain. → See also: Primitive (Layer I), Structure (Layer II), Interface (Layer III).

Lin Table → See Generalized Lin Table.

Living Document [META]: A canonical document designation indicating that the document is maintained and updated as the canon evolves, but does not grow by arbitrary revision — only by justified canonical instantiation. Both ABOUT.md and this GLOSSARY.md carry Living Document status. → See also: Governance.

Load-Bearing Node [ALL] [META]: The designation for each of the three modules within the triadic cluster, conveying that each module is architecturally necessary — the cluster's coherence depends on the presence and integrity of all three. The removal or failure of any single node dissolves the cluster's functional arc. → See also: Cluster, Triadic Unit.

Local Density of States (LDOS) [QL]: A spatially-resolved spectral quantity computed by quantum-lattice, characterizing the distribution of electronic states as a function of energy at a specific lattice site. LDOS is computed via Green's function methods and recursive techniques for semi-infinite systems. → See also: Green's Function, Spectral Function.

Many-Body Basis [QB]: The complete set of basis vectors spanning the many-body Hilbert space for a given lattice model, as constructed by quantum_basis. The many-body basis is the fundamental data structure of Layer I: it is the object over which all subsequent operations — diagonalization, spectral computation, variational optimization — are ultimately defined. → See also: Basis Vector, Hilbert Space, Generalized Lin Table.

Mean-Field Theory [QL]: A classical approximation scheme implemented in quantum-lattice that replaces quantum many-body interactions with an effective single-particle field. Mean-field calculations produce phenomenological Hamiltonians that can be passed upward to LattiQ for variational quantum optimization. → See also: Hamiltonian, Tight-Binding Model.

Module [ALL] [META]: Within this directory, a module is a curated ontological record pairing an upstream open-source repository with its TriadicFrameworks canonical role, layer assignment, and integration contract. A module is not merely a library; it is a load-bearing node in the triadic architecture. → See also: Load-Bearing Node, Triadic Slot.

Module JSON [ALL] [META]: The structured metadata file (*_module.json) that constitutes the TriadicFrameworks canonical record for each upstream repository. Module JSON files are metadata documents, not derivative works of the upstream source code; they are therefore not subject to upstream licenses (GPL-3.0 or MIT). → See also: Module, Governance.

Nambu Basis [QL]: A spinor representation used in quantum-lattice for BdG Hamiltonians, pairing particle and hole degrees of freedom to encode superconducting order parameters and particle-hole symmetry explicitly. → See also: Bogoliubov–de Gennes (BdG) Formalism, Superconductivity (Unconventional).

Non-Collinear Magnetism [QL]: A magnetic configuration in which spin orientations are neither fully parallel nor fully anti-parallel, supported by quantum-lattice through its full treatment of spin-orbit coupling and general spin texture. → See also: Spin-Orbit Coupling.

Ontological Record [ALL] [META]: The authoritative canonical description of a module's conceptual territory, triadic role, and integration contract within TriadicFrameworks. Each module JSON and associated documentation in this directory functions as an ontological record. The directory as a whole is the canonical ontological registry for the quantum-theoretic cluster. → See also: Module, Module JSON.

Operator [QB] [QL]: A matrix-valued object acting on the Hilbert space or single-particle state space. In quantum_basis, operators are defined by the user in terms of matrix representations of elementary degrees of freedom and used to construct the Hamiltonian. In quantum-lattice, operators appear as Hamiltonian terms encoding hopping, coupling, and symmetry-breaking fields. → See also: Hamiltonian, State-Space Primitive.

Partial Order [QL] [META]: A binary relation ≤ on the set of quantum states that is reflexive, antisymmetric, and transitive, forming the structural backbone of the lattice as implemented in quantum-lattice. The partial order, together with join/meet operations, defines the lattice topology that LattiQ translates into canonical interface types. → See also: Join / Meet Structure, Lattice Topology.

Post-Quantum Cryptography [LQ]: The domain of cryptographic algorithms designed to resist attacks by quantum computers. LattiQ's primary application target — the Shortest Vector Problem — is a cornerstone hardness assumption in post-quantum cryptographic schemes. The module thus situates the quantum-theoretic cluster at the intersection of quantum simulation and cryptographic security research. → See also: Shortest Vector Problem.

Primitive (Layer I) [QB] [META]: The first and foundational layer of a triadic cluster, responsible for defining the mathematical substrate and state-space vocabulary from which all higher layers are constructed. quantum_basis occupies Layer I in this cluster. Primitives without structure are unordered — mathematically present but operationally inert. → See also: Dependency Terminus, Foundation Layer, quantum_basis.

Quantum-Theoretic Primitive Cluster [ALL] [META]: The formal TriadicFrameworks designation for the collective {quantum_basis, quantum-lattice, LattiQ} as a closed functional arc. As a primitive cluster, it provides the quantum-theoretic grounding for the broader canon; no other cluster within TriadicFrameworks is authorized to define independent quantum ontology without grounding through this cluster's LattiQ interface. → See also: Cluster, Downward Grounding, Upward Composition.

Quantum Hall Edge State [QL]: A conducting state localized at the boundary of a topological insulator or quantum Hall system, arising from bulk-boundary correspondence. quantum-lattice computes edge states as part of its topological invariant and band structure machinery. → See also: Chern Number, Landau Level, Topological Invariant.

Quantum Ontology [QB] [META]: The set of ontological commitments — what kinds of objects exist, what their properties are, and what operations on them are valid — established by the formal postulates of quantum_basis. Quantum ontology is the canon's source of truth for all quantum-theoretic reasoning within TriadicFrameworks. → See also: Formal Postulate, Canonical Unit Type.

quantum_basis [QB]: The Layer I — Primitive module of this triadic cluster. A C++ library for constructing the many-body Hilbert-space basis for arbitrary condensed-matter lattice models. Supports any combination of bosonic and fermionic degrees of freedom; eigensolvers include Lanczos and FEAST; spectral access via KPM; sparse algebra backed by Intel MKL and ARPACK-NG. Upstream: wztzjhn/quantum_basis. Author: Zhentao Wang. License: GPL-3.0. → See also: Exact Diagonalization, Foundation Layer, Primitive (Layer I).

quantum-lattice [QL]: The Layer II — Structure module of this triadic cluster. A Python (+ Fortran kernel) toolkit for designing and solving single-particle and mean-field Hamiltonians on arbitrary lattice geometries. Capabilities include non-collinear magnetism, spin-orbit coupling, unconventional superconductivity, topological invariants, Green's functions, Landau levels, and an interactive PyQt GUI. Upstream: joselado/quantum-lattice. Author: Jose Lado. License: GPL-3.0. → See also: Hamiltonian, Simulation Layer, Structure (Layer II).

Rank [LQ]: A configurable hyperparameter in LattiQ controlling the truncation or approximation rank of the SVP encoding, affecting circuit depth and solution accuracy on near-term quantum hardware. → See also: Shortest Vector Problem, LattiQ.

Recursive Method [QL]: A numerical technique used in quantum-lattice for computing Green's functions and local density of states in semi-infinite systems, enabling the simulation of surface and interface physics without modeling a full finite slab. → See also: Green's Function, Local Density of States.

Simulation Layer [QL] [META]: The TriadicFrameworks designation for quantum-lattice in its role as the primary Hamiltonian factory of the cluster. The Simulation Layer generates phenomenological models that feed both the ED benchmarking engine (quantum_basis) and the variational quantum optimizer (LattiQ). → See also: Hamiltonian, Structure (Layer II), quantum-lattice.

Shortest Vector Problem (SVP) [LQ]: The computational problem of finding the shortest non-zero vector in a given lattice. SVP is the primary application target of LattiQ, which encodes it as an Ising spin Hamiltonian and solves it via VQE or QAOA. SVP hardness underlies many post-quantum cryptographic security proofs. → See also: Ising Spin Hamiltonian, Post-Quantum Cryptography, VQE, QAOA.

Simulation Layer → See quantum-lattice.

Sparse Matrix Algebra [QB]: The set of numerical operations on matrices with predominantly zero entries, used by quantum_basis to represent and manipulate large Hamiltonians efficiently. Backed by Intel MKL and ARPACK-NG. → See also: Hamiltonian, Intel MKL, ARPACK-NG.

Spectral Function [QB] [QL]: A frequency-domain quantity characterizing how a quantum system responds to perturbations at each energy. In quantum_basis, the spectral function is computed via KPM. In quantum-lattice, it is accessible through momentum-resolved Green's functions. Spectral functions from the ED layer serve as high-fidelity benchmarks for the variational layer. → See also: Kernel Polynomial Method, Green's Function.

Spin-Orbit Coupling [QL]: An interaction between a particle's spin and its orbital motion, implemented in quantum-lattice as a Hamiltonian term that mixes spin-up and spin-down channels. Spin-orbit coupling is essential for modeling topological insulators, anomalous Hall effects, and non-collinear magnetic textures. → See also: Non-Collinear Magnetism, Topological Invariant.

State-Space Primitive [QB] [META]: The foundational objects — basis vectors, Hilbert space representations, and the algebraic rules governing their composition — established by quantum_basis. State-space primitives are the raw material from which all structural and interface constructs in Layers II and III are built. → See also: Basis Vector, Canonical Unit Type, Hilbert Space.

Structure (Layer II) [QL] [META]: The second layer of a triadic cluster, responsible for arranging the primitives of Layer I into topologically coherent relational structures. quantum-lattice occupies Layer II in this cluster. Structure without primitives is form without content — valid topology over empty space. Layer II is the module most likely to evolve as theoretical refinement occurs, because structure is where theoretical maturation lives. → See also: Lattice Topology, quantum-lattice, Simulation Layer.

Superconductivity (Unconventional) [QL]: Superconducting phenomena not explained by conventional BCS theory — including d-wave, p-wave, and topological superconductors — modeled in quantum-lattice via BdG Hamiltonians with general pairing symmetries. → See also: Bogoliubov–de Gennes (BdG) Formalism, Nambu Basis.

Superposition Arrangement [QL] [META]: The structural encoding of quantum superposition relationships within quantum-lattice as explicit first-class objects. Superposition arrangements are not computed on demand; they are declared as structural elements of the lattice, making them accessible to LattiQ for interface translation. → See also: Entanglement Geometry, Lattice Topology.

Synthesis Module [LQ] [META]: An alternate designation for the Layer III module (LattiQ), emphasizing its role as the operational synthesis of the cluster's theoretical content into a composable interface. → See also: Interface (Layer III), LattiQ.

t-J Model [QB]: A bundled example model in quantum_basis describing strongly correlated electrons with nearest-neighbor hopping (t) and antiferromagnetic exchange (J), relevant to high-temperature superconductivity. → See also: Exact Diagonalization, Kondo Lattice Model.

Tight-Binding Model [QL]: A single-particle lattice model in which electrons hop between sites with specified amplitudes, implemented in quantum-lattice. Tight-binding models are the primary phenomenological tool of Layer II, producing band structures and Hamiltonians that feed into both the ED solver and the VQA optimizer. → See also: Band Structure, Hamiltonian, Mean-Field Theory.

Topological Invariant [QL]: A quantity characterizing the global topology of the quantum state space of a band structure that cannot change under smooth deformations of the Hamiltonian. quantum-lattice computes Chern numbers and Z₂ invariants. → See also: Chern Number, Z₂ Invariant, Quantum Hall Edge State.

Traversal Rule [QL] [META]: One of the composition rules defined by quantum-lattice that specify how the lattice topology may be navigated — which transitions between states are permitted and under what conditions. Traversal rules constitute the grammar that LattiQ translates into canonical interface operations. → See also: Lattice Topology, Partial Order.

Triadic Dependency Chain [ALL] [META]: The directed dependency sequence quantum_basis → quantum-lattice → LattiQ that governs the flow of types, constructs, and interface contracts within the cluster. Changes anywhere in the chain must be evaluated for propagation effects toward the chain's output end (LattiQ). → See also: Dependency Terminus, Layer, Cluster.

Triadic Slot [ALL] [META]: One of the three designated positions (Layer I, II, or III) that a module may occupy within a triadic cluster. A triadic slot is a role, not merely a position: it carries specific responsibilities (primitive definition, structural arrangement, interface synthesis) and governance implications. New modules must occupy a defined triadic slot; no module may augment an existing slot. → See also: Layer, Governance, Canonical Review.

Triadic Unit [ALL] [META]: A complete, closed cluster of exactly three modules — one per layer — that together constitute a single coherent functional arc from raw formalism to applied interface. The {quantum_basis, quantum-lattice, LattiQ} set is a triadic unit. A triadic unit is the minimal composable element in the TriadicFrameworks canon. → See also: Cluster, Quantum-Theoretic Primitive Cluster.

Twisted Bilayer Graphene [QL]: A lattice system composed of two graphene layers with a small relative rotation angle, supported by quantum-lattice as a superlattice model. At the magic angle (~1.1°), the system exhibits flat bands and strongly correlated phenomena accessible via tight-binding Hamiltonians. → See also: Tight-Binding Model, Band Structure.

Type Coherence [ALL] [META]: The property of a triadic cluster in which all modules share a common type vocabulary — the canonical unit types — without implicit conversion or re-definition at integration seams. Type coherence is enforced in this cluster at the substrate level by quantum_basis. → See also: Canonical Unit Type, Integration Seam.

Upward Composition [ALL] [META]: One of two integration directions of the quantum-theoretic cluster within the broader TriadicFrameworks canon. Upward composition means the cluster, via LattiQ's extension hooks, can be composed with other triadic clusters. Cross-cluster dependencies are always Layer-III–to–Layer-III bindings. → See also: Downward Grounding, Extension Hook, Compositional Direction.

Variational Quantum Algorithm (VQA) [LQ]: A class of hybrid quantum-classical algorithms in which a parameterized quantum circuit is optimized by a classical optimizer to minimize an energy or cost function. LattiQ implements VQA via VQE and QAOA as interchangeable solvers for the SVP encoding. → See also: VQE, QAOA, FastVQA.

Variational Quantum Eigensolver (VQE) [LQ]: A VQA that estimates the ground-state energy of a Hamiltonian by optimizing a parameterized quantum circuit's output expectation value. In LattiQ, VQE is applied to the Ising Hamiltonian encoding of SVP. → See also: Quantum Approximate Optimization Algorithm (QAOA), Ising Spin Hamiltonian, CVaR.

Quantum Approximate Optimization Algorithm (QAOA) [LQ]: A VQA designed for combinatorial optimization, alternating between problem and mixer Hamiltonians through p layers of parameterized unitaries. In LattiQ, QAOA is an interchangeable solver with VQE for the Ising-encoded SVP. → See also: VQE, Ising Spin Hamiltonian, Shortest Vector Problem.

Z₂ Invariant [QL]: A topological invariant taking values in {0, 1}, classifying systems with time-reversal symmetry into topologically trivial (0) or non-trivial (1) phases. The Z₂ invariant is computed by quantum-lattice and is the relevant topological index for time-reversal-invariant topological insulators. → See also: Chern Number, Topological Invariant.

Module Reference Summary#

Cross-Reference Index#

This document is maintained as part of the TriadicFrameworks canonical record. It describes vocabulary and conceptual architecture, not implementation. For implementation details, consult each module's own documentation.

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