Formal Distributions Framework

• Formal Distributions Framework is a comprehensive structure that unifies classical probability distributions, generalized (Schwartz) distributions, and stochastic process representations using formal mathematical methods.
• It employs rigorous techniques on formal manifolds, oscillatory integrals, and probabilistic logics to enable detailed analysis in model checking and deformation quantization.
• The framework supports diverse applications—from geometric analysis and quantization theory to machine learning and random structure generation—by leveraging duality, sheaf theory, and continuity.

The formal distributions framework encompasses a spectrum of mathematical and computational structures that abstract, generalize, and unify key aspects of classical probability distributions, generalized (Schwartz) distributions, and stochastic process representations. It plays a central role in several research domains, including functional analysis, formal geometry, stochastic modeling, probabilistic logics, and learning theory, providing the apparatus to rigorously handle distributions in settings ranging from smooth manifolds to logic programming and stochastic systems. Below, the foundational structures, methodologies, and applications of the formal distributions framework are systematically presented, with particular emphasis on the technical details underlying these diverse lines of research.

1. Formal Distributions on Formal Manifolds

Formal manifolds generalize the notion of smooth manifolds by equipping a topological space with a sheaf of formal functions—power series in formal variables with smooth coefficients. On such a manifold $(M, \widehat{\mathcal{O}}_M)$, formal distributions extend classical distributions and generalized functions to the formal category. Four core functorial spaces are defined (Chen et al., 10 Jul 2024, Chen et al., 8 Aug 2024):

• Formal Functions: $$\widehat{\mathcal{O}}(U) = C^\infty(U)[[y_1,\ldots,y_k]]$$ on an open set $U$, the topology given by completion in the formal variables.
• Compactly Supported Formal Densities: A cosheaf $\mathcal{D}_o(\mathcal{F})$, locally modeled as $$\mathcal{D}_c(\mathbb{R}^n) \widehat{\otimes} \mathbb{C}[y_1,\ldots,y_k]$$.
• Formal Generalized Functions: Sheaf $\mathcal{C}^{-\infty}(\mathcal{F}; E)$ of continuous linear functionals on the cosheaf of formal densities, equipped with strong topology.
• Formal Distributions: Sheaf $\mathcal{D}^{-\infty}(\mathcal{F}; E)$ of continuous linear functionals on compactly supported sections, also admitting cosheaf versions for densities with compact support.

The duality and tensor product identifications mirror the classical Schwartz theory: if $\mathcal{F}$ is locally free of rank $r$ and $E$ a complete locally convex space, then

$$\mathcal{C}^{-\infty}(M; \mathcal{F}; E) \cong \mathcal{C}^{-\infty}(M; \mathcal{F}) \widehat{\otimes}_\pi E.$$

Distributions supported at a point are identified with polynomial algebras in local coordinates and formal variables, yielding stalks isomorphic to $\mathbb{C}[x_1, \ldots, x_n][y_1, \ldots, y_k]$ (Chen et al., 10 Jul 2024).

De Rham Complexes and Poincaré Lemma (Formal Setting): The formal de Rham sheaf $\widehat{\Omega}_M^r(E)$ is constructed from alternating $r$-multilinear forms on the sheaf of derivations. The global de Rham complex

$$0 \rightarrow \widehat{\mathcal{O}}(M) \widehat{\otimes} E \rightarrow \widehat{\Omega}_M^1(M;E) \rightarrow \cdots \rightarrow \widehat{\Omega}_M^{n+k}(M;E) \rightarrow 0$$

admits a strong Poincaré lemma: if $M$ is contractible (or isomorphic to $(\mathbb{R}^n)^{(k)}$), all cochain complexes constructed using any of formal functions, generalized functions, or distributions are split exact, admitting continuous linear homotopies (Chen et al., 8 Aug 2024).

2. Oscillatory Formal Distributions and Quantization Theory

Oscillatory formal distributions provide an algebraic framework to capture formal analogues of oscillatory integrals in microlocal analysis and deformation quantization (Karabegov, 2020). Let $M$ be a smooth manifold and $x_0\in M$. A $v$-formal distribution supported at $x_0$ is a formal series

$$A = \sum_{r\geq0} v^r A_r,\qquad A_r \in D_{x_0}(M)$$

where $D_{x_0}(M)$ denotes Schwartz distributions supported at $x_0$. Such a distribution is called oscillatory if it has the form

$A = \delta_{x_0} \circ \exp(v^{-1} X)$

with $X = \sum_{r\geq2} v^r X_r$ a differential operator whose leading symbol defines a nondegenerate symmetric bilinear form on $T_{x_0}M$.

Characterization Theorem: A formal distribution $A$ is a formal oscillatory integral (FOI) if and only if it is oscillatory with nondegenerate leading quadratic term. There is a jet-recovery algorithm which reconstructs the infinite jet of the phase and amplitude from the knowledge of all $A_r$ (Karabegov, 2020).

Natural Star Products: In deformation quantization, a star product $\star$ is natural if and only if its kernel distributions $(f,g) \mapsto (f\star g)(x)$ are oscillatory for all $x$. The construction yields an explicit criterion for the naturality of star products via oscillatory formal bidistributions.

3. Formal Distributions in Probabilistic Logics and Model Checking

3.1. Distributional Probabilistic Model Checking

In stochastic model checking, the formal distributions framework enables direct computation and optimization of full distributional properties—not just expected values—for both discrete-time Markov chains (DTMCs) and Markov decision processes (MDPs) (ElSayed-Aly et al., 2023). One systematically computes the pmf $\mu$ of cumulative rewards until a target event, allowing queries on expectation, variance, VaR, and CVaR.

• Algorithmic Core:
  • For DTMCs: Graph-based fixed-point or forward algorithms yield the full distribution $\mu(i) = \mathbb{P}[X = i]$ over nonnegative integer reward accumulations, with controlled truncation error $\epsilon$.
  • For MDPs: Distributional value iteration carries per-state full reward distributions (using, e.g., categorical or quantile projections), supporting risk-neutral (expectation) and risk-sensitive (CVaR) policy optimization.
  • Convergence and soundness theorems establish that the computed pmfs over-approximate the true distribution up to $\epsilon$, and the computed policy is CVaR-optimal up to discretization error.

3.2. Probabilistic Team Semantics

Probabilistic team semantics formalizes the paper of logical dependencies in probabilistic databases and random structures. Here, "teams" are generalized from sets of assignments to probability distributions over assignments, and formal distribution identities (e.g., marginal identity, distribution equivalence, probabilistic independence) can be expressed and manipulated in the logic (Hannula et al., 2018). The resulting logics admit strict hierarchies of expressive power and have tight connections to two-sorted real arithmetic.

• Marginal identity atoms ($\vec{x} \approx \vec{y}$), marginal-distribution equivalence ($\approx^*$), and conditional independence atoms ($\vec{y}\perp\!\!\!\perp_{\vec{x}}\vec{z}$) provide fully formal syntactic and semantic interfaces for reasoning about probabilistic dependencies.
• The expressive hierarchy FO($\approx$) < FO($\approx,$dep) = FO($\approx^*$) ≤ FO($\perp\!\!\!\perp$) is established.

3.3. Generalized Distribution Semantics

In probabilistic logic programming and its generalizations, a formal separation is maintained between a tuple-independent "free" random component and a deterministic expansion via logic programs (Weitkämper, 2022). For finite relational worlds, a generalized probabilistic logic program is specified by:

• A tuple-independent base measure $P^w$ on the extensional vocabulary,
• A functorial deterministic expansion $\Pi$ (an acyclic logic program or lifted query), thus producing, via pushforward, a projective family of formal distributions across all finite domains. Only projective families satisfying the strong independence property (SIP) and lacking essential asymmetry are representable by such semantics.

4. Formal Distributions in Inverse Problems and Machine Learning

4.1. Data Consistent (DC) Inversion and LUQ

The DC framework treats parameter and observable spaces as measure spaces linked by a measurable QoI map $Q$ (Roper et al., 4 Mar 2024). The formal distributions associated to these spaces—via pullback and pushforward—enable exact inversion for measures: $$Q_{\rightarrow}\#\mu_{\text{upd}} = P \implies \mu_{\text{upd}}(\lambda) \propto \mu_{\text{init}}(\lambda) \frac{\pi_{\text{obs}}(Q(\lambda))}{\pi_{\text{pred}}(Q(\lambda))}$$ Machine-learned QoI maps are obtained by filtering, clustering, and kernel-PCA, producing features robust to noisy data (epistemic uncertainty) and ensuring the quantified distributions on QoIs match observed ones (aleatoric uncertainty). A suite of diagnostics (e.g., predictability, sufficiency tests via RKHS projections) underpin rigorous, iterative updates of the parameter measure.

4.2. Generative Modeling and Error Decomposition

A unifying mathematical framework expresses all major generative modeling paradigms as combinations of formal distribution representations (potential/vertical, pushforward/horizontal, optimal transport) and loss functionals (density-based, IPM, regression) (Yang, 2022). The formal machinery supports:

• Quantitative decomposition of generalization/approximation/training errors, with dimension-independent rates achieved via early stopping and (implicit or explicit) regularization.
• Abstract characterization of critical points in the geometry of the loss landscape, including conditions for and mechanisms of phenomena such as GAN mode collapse.

5. Stochastic Formal Distributions and Random Structure Generation

5.1. Formal Context Generation

The classical random "coin-toss" method for generating formal contexts (incidence matrices) is extended by introducing Dirichlet-distributed categorical laws on row object attribute counts (Felde et al., 2018). This allows arbitrary discrete distributions over row-sums, encompassing and vastly generalizing the binomial model:

• For each object $g$, a Dirichlet sample $p$ yields $\theta_g \sim\text{Categorical}(p)$, and the resulting contexts span a strictly larger variety of incidence patterns than coin-toss models.
• Empirically, Dirichlet-based models produce significantly more diverse context statistics (I–PI coordinates), supporting benchmark and null-model studies in formal concept analysis.

6. Summary Table: Domains of Formal Distributions Frameworks

Domain/Framework	Structural Emphasis	Technical Core
Formal manifolds (geometry)	Sheaves and cosheaves of formal densities, distributions, and de Rham complexes	Nuclear LCS, dualities, strong exactness, support at a point
Microlocal analysis, quantization	Oscillatory formal distributions, star products	Natural differential/algebraic structures, jet-recovery
Probabilistic logic/model checking	Projective semantical families, logical atoms	Pushforward semantics, projectivity, risk/loss measures
Inverse problems and ML	Pushforward/pullback measures, learning representations	Data consistent inversion, feature extraction, measure-theoretic update
Random context/structure generation	Dirichlet-based categorical laws	Categorical/Dirichlet parametrization, null-model construction

This multifaceted formal distributions framework establishes a rigorous yet flexible apparatus for extending distributional machinery to new domains, balancing analytic control (topology, duality, cohomology) with algorithmic tractability (online updates, explicit representations, invariance properties). These foundational structures underpin ongoing advances in geometric analysis, stochastic modeling, symbolic learning, and computable logic.