DualityForge: Algorithmic Duality Frameworks

• DualityForge is a framework that systematically constructs and analyzes dualities across mathematical, physical, and machine learning domains by encoding transformations and algebraic structures.
• It enables the automated generation of dual Hamiltonian families, field theory duals, and gauge theory duality cascades, enhancing both theoretical investigation and practical implementation.
• The framework facilitates counterfactual data synthesis in deep learning, providing a robust pipeline for contrastive learning and anomaly detection.

DualityForge is a term denoting a class of systematic, algorithmic, and software-oriented frameworks for constructing, analyzing, and leveraging dualities across a wide array of mathematical, physical, and machine learning domains. The precise construction and usage of “DualityForge” depend on context, spanning quantum and classical dualities in lattice models, gauge theories, field theory, and, more recently, controllable data synthesis and contrastive learning in deep learning architectures. Across these domains, the defining feature is the explicit, stepwise encoding of duality transformations, their algebraic structures, and operational recipes for automating the search for dual pairs, duality cascades, and contrastive datasets.

1. Local Algebraic and Operator-Theoretic Duality Construction

DualityForge was first instantiated in the context of quantum and classical statistical mechanics as a local, algorithmic framework for generating and verifying dualities via the notion of bond algebras. A Hamiltonian $H=\sum_i \lambda_i b_i$ is decomposed into local or quasi-local “bonds” $b_i$, generating a bond algebra $\mathcal{A}[\{b_i\}]$—a von Neumann algebra of bounded operators closed under all algebraic operations. Duality is then defined as the existence of an algebra isomorphism $\Phi:\mathcal{A}[\{b_i\}]\to\mathcal{A}[\{b'_i\}]$ mapping $b_i$ to $b'_i$ and preserving all products and adjoint operations. Such dualities are necessarily unitarily implementable: any $*$-isomorphism between von Neumann algebras on Hilbert spaces $\mathcal{H}_1,\mathcal{H}_2$ is realized by a unitary operator $U : \mathcal{H}_1 \to \mathcal{H}_2$, i.e., $\Phi(O) = U O U^\dagger$ for all $O \in \mathcal{A}$.

The software implementation of DualityForge in this context systematically parses the set of bonds, computes their algebraic relations, proposes candidate dual operators on possibly different Hilbert spaces, and verifies full relation-preservation. If successful, it outputs the dual Hamiltonian, the duality map, and the corresponding unitary or partial isometry. This methodology readily accommodates both local and non-local dualities (e.g., Jordan–Wigner mappings), dimensional reduction, and connects quantum and classical dualities through transfer-matrix formalism. It scales to arbitrary system size, dimension, and gauge group, providing a “forge” for discovering new dualities in complex lattice systems (Cobanera et al., 2011).

2. Systematic Generation of Dual Hamiltonian Families

DualityForge methodologies have also been formalized for the systematic generation of parameterized Hamiltonian families possessing a prescribed duality. Suppose a duality operator $D$ acts on a finite-dimensional Hilbert space $\mathcal{H}$, and there is an induced map $f$ on parameter space $\mathcal{P}$. The defining relation is

$D H(g) D^{-1} = H(f(g))$

where $H(g)$ is a member of the Hamiltonian family, $g \in \mathcal{P}$, and $D^n=I$, $f^n = \text{Id}$. To guarantee this property, one decomposes the operator space $\mathrm{End}(\mathcal{H})$ into irreducible representations of $C_n$, constructing basis operators $H_m^\alpha$ and associated parameter functions $a_m^\alpha(g)$ transforming covariantly under $f$.

The Hamiltonian ansatz becomes $H(g) = \sum_{m,\alpha} a_m^\alpha(g) H_m^\alpha$. Further constraints such as Hermiticity or locality are imposed via optimization: the residual $\mathcal{F}(g,g') = D H(g') D^{-1} - H(g)$ is minimized, with physical constraints incorporated as penalties. Newton or trust-region methods are used for root-finding and numerical continuation to trace out self-dual manifolds in parameter space.

This approach is applicable to tight-binding models, coupled-oscillator networks, and general metamaterials. Importantly, the explicit construction of the dual-covariant basis and minimization of the residual enables on-demand generation of systems with guaranteed duality, including the description of universal behavior near self-dual points (Fruchart et al., 2021).

3. DualityForge in Field Theory: Duality Families and Solution Transfer

In scalar field theory, DualityForge formalizes the classification and construction of dual field pairs (and families) via explicit dualization formulas. Two scalar fields $\phi(x)$ and $\psi(y)$ with Lagrangians $$L[\phi] = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2 - V(\phi)$$ and $$L[\psi] = \frac{1}{2} \partial_\mu \psi \partial^\mu \psi - \frac{1}{2} m^2 \psi^2 - U(\psi)$$, are dual if there exists an exponent map $a \leftrightarrow A=2a/(a-2)$, a monomial field map $\phi(x) = \psi(y)^{2-a}$, coordinate rescaling $x^\mu = \sqrt{(2-A)/(2-a)}\, y^\mu$, and coupling identification, such that the equations of motion map into each other.

A field with a monomial potential has a unique dual; a polynomial $n$-term potential yields $n+1$ duals (one per monomial term), and a nonpolynomial function (e.g., sine-Gordon) generates infinitely many duals. The practical algorithm involves expanding the potential, selecting a seed monomial, constructing the dual potential, and transferring exact solutions between duals using the stated field-coordinate transformation (Li et al., 2019).

4. Algorithmic Construction of Duality Cascades in Gauge Theory

DualityForge was adapted to encode chains of Seiberg-like dualities, particularly in $\mathcal{N}=1$ supersymmetric gauge theories with two-index tensor matter. Each step in a duality cascade is represented as a template mapping “electric” theories (specified by gauge group, matter representations, superpotential) into their dual “magnetic” counterparts, with precise rules for matter content, meson/baryon identification, and superpotential deformation. Three archetypal cascade classes are encoded: vector-like adjoint, chiral SU$\times$SO, and free-field-based SU$\times$SU$^4$. Each cascade tracks the evolution of gauge groups, matter, F-flatness, 't Hooft anomaly consistency, and IR phase (confinement, CFT, IR free, meta-stable SUSY breaking).

Standard criteria, such as the rank condition for supersymmetry breaking and the analysis of the Coulomb branch for dynamical restoration, are encoded stepwise. Worked-out examples are given in LaTeX-ready form, supporting both theoretical analysis and automated software implementation (Hook, 2014).

Application Area	Principal Objects	DualityForge Role
Lattice models/Hamiltonians	Bonds, operator algebras	Detect, implement dualities
Tight-binding/coupled oscillators	Parameterized Hamiltonians	Systematic construction
Field theories (scalar)	Potentials, equations of motion	Dual family classification
Supersymmetric gauge theories	Gauge group/matter/W	Cascade generation, IR analysis

5. Data Synthesis and Contrastive Learning in Deep Models

In multimodal deep learning, DualityForge refers to a framework for counterfactual data synthesis and contrastive training aimed at correcting over-reliance on language priors. The pipeline leverages controllable, diffusion-based video editing to generate counterfactual scenarios from real-world video clips under structured contexts $\mathcal{C}$ (“erase the ball”, “reverse event order”, etc.). Three specialized editing sub-pipelines (visual, semantic, commonsense) produce context-guided altered videos, automatically subjected to quality checks by an MLLM ensemble.

QA generation leverages the same structured context, producing dense anomaly-annotated captions and generating contrastive question-answer (QA) pairs such that the same question $Q$ admits different correct answers for the original and edited videos. This paired real/edit construction directly targets hallucination reduction by training models to ground their answers in visual evidence rather than linguistic priors.

The resulting DualityVidQA dataset comprises 144,000 QA pairs from 81,000 unique clips, spanning four counterfactual anomaly classes, and supports large-scale supervised and RL fine-tuning. Duality-Normalized Advantage Training (DNA-Train) is introduced: a two-stage SFT+RL pipeline where RL is performed using pairwise $\ell_1$ advantage normalization to balance real/counterfactual gradients, improving convergence and performance, especially in counterfactual tasks. Empirically, this approach achieves significant absolute improvements (e.g., +24% pairwise accuracy over baseline on DualityVidQA-Test) and generalizes across tasks (Huang et al., 30 Dec 2025).

6. Limitations, Domain-Specific Considerations, and Future Directions

While DualityForge frameworks have demonstrated broad applicability, each instantiation is limited by the fidelity of underlying algorithms—e.g., diffusion-based video editors for counterfactual generation, or the completeness of bond algebra representations in quantum models. In deep learning, some classes of longer-range or multi-object counterfactuals and extreme anomalies remain underrepresented. In gauge theory, not all possible dualities or IR phases can be exhaustively mapped without further data and computational power.

Future advances include stronger diffusion models for richer video edits, greater integration of human-in-the-loop curation, the extension of normalization schemes beyond dual pairs, and the routine transfer of duality-forged pipelines to new modalities and physical systems (multimodal tasks, novel materials, higher-dimensional field theories). The systematic, local, and algorithmic nature of these frameworks positions DualityForge as a critical toolkit for both theoretical investigation and machine learning practice across disciplines.