Hypothesis h: Cross-domain Mathematical Insights

• Hypothesis h is a formal proposition that encompasses model constraints, testable assertions, and deep structural conjectures across various mathematical fields.
• It is methodologically applied in areas such as Markov process potential theory, analytic formulations for Lévy processes, and multivariate statistical matrix testing.
• The hypothesis informs exact inference techniques, underpins key number theory conjectures, and advances understanding in nonextensive statistical mechanics.

A hypothesis $h$ refers, in mathematical statistics, probability theory, statistical mechanics, potential theory, and analytic number theory, to a formally stated proposition, constraint set, or structural assertion regarding parameters, functional forms, processes, or algebraic objects. The foundational role and technical formulation of a hypothesis $h$ vary sharply by context—ranging from model constraints and testable parametric assertions to deep structural conjectures in analysis and probability. Below, salient incarnations and frameworks for hypothesis $h$ are detailed across representative domains.

1. Hypothesis $h$ in Markov Process Potential Theory: Hunt’s Hypothesis (H)

Hunt’s Hypothesis (H) is a central technical condition in the fine potential theory of right Markov processes on a locally compact separable metric space $(E,\mathcal{B})$. For a standard Markov process $X=(X_t)$, let $T_D = \inf\{ t>0 : X_t \in D \}$ denote the hitting time of $D\subset E$. A set $D$ is thin if $\exists C\in \mathcal{B}^*$ with $D\subset C$ and $P^x(T_C = 0)=0$ for all $x$, semipolar if contained in a countable union of thin sets, and polar if $\exists C\in\mathcal{B}^*$ with $D\subset C$ and $P^x(T_C < \infty)=0$ for all $x$.

Hunt’s hypothesis (H):

$\text{Every semipolar set is polar},$

formally,

$$(H)\qquad \forall\,\text{semipolar }D\subset E,\ \forall\,x\in E,\quad P^x(T_D < \infty) = 0.$$

This encapsulates a subtle structural property of the process’s exceptional sets and is equivalent (assuming duality and lower semicontinuity of excessive functions) to several potential-theoretic maxims: the bounded maximum principle, the dichotomy of fine and cofine topologies modulo polar sets, and positivity principles for potentials. The hypothesis is pivotal, e.g., in Getoor’s conjecture, which posits (H) for all Lévy processes except degenerate and pathological cases (Hu et al., 2019, Hansen et al., 2014, Hu et al., 2011, Hu et al., 2012, Hu et al., 2014, Hu et al., 2019, Hu et al., 2017).

2. Analytic and Probabilistic Formulations: Lévy and Hunt Processes

For a Lévy process $X$ on $\mathbb{R}^n$ with characteristic exponent $\psi(z)$,

$$\mathbf{E}[e^{i\langle z, X_t \rangle}] = e^{-t\psi(z)}$$

and

$$\psi(z) = i\langle a, z \rangle + \tfrac{1}{2}\langle z, Qz \rangle + \int_{\mathbb{R}^n \setminus\{0\}} [1 - e^{i\langle z,x\rangle} + i\langle z, x\rangle 1_{|x|<1}]\,\mu(dx).$$

Hunt's hypothesis (H) is known to hold in the following settings:

• If the Gaussian part $A$ is non-degenerate (i.e., $\det A>0$), by the Kanda–Forst theorem, (H) holds as $$|\operatorname{Im}\psi(z)| \leq M(1+\operatorname{Re}\psi(z))$$ and transition densities are bounded and continuous (Hu et al., 2011).
• In degenerate Gaussian cases, (H) holds if $$\mu(\mathbb{R}^n \setminus \sqrt{A}\mathbb{R}^n) < \infty$$ and a specific drift-cancellation equation admits a solution, reflecting an intricate coupling between the jump and diffusive components.
• For pure-jump subordinators, (H) can only hold when the drift $d=0$ (Hu et al., 2011, Hu et al., 2012, Hu et al., 2014).

Key analytic equivalences involve:

• Slicing Fourier space via the conditions on $A(z) = 1 + \operatorname{Re} \psi(z)$ and $B(z) = |1+\psi(z)|$, yielding necessary and sufficient integrability criteria for the resolvent operators and potentials (Hu et al., 2012, Hu et al., 2014).
• Stability of (H) under operations such as convolution (sum of independent Lévy processes), subordination by suitable time-changes, and absolutely continuous changes of measure (Hu et al., 2019, Hu et al., 2017).

3. Hypothesis Matrix $H$ in Wald-Type Statistics (Statistical Inference)

In multivariate hypothesis testing, $H$ refers to the hypothesis matrix specifying linear constraints in a general null hypothesis for a $d$-dimensional parameter: $$\mathcal{H}_0: H\theta = y,\qquad H\in\mathbb{R}^{m\times d},\quad y\in\mathbb{R}^m,\ m\leq d,$$ where $\theta$ may represent group means, regression coefficients, quantiles, or covariance entries. The Wald-type statistic (WTS) for testing $H_0$ takes the quadratic form

$$\mathrm{WTS}(H, y) = N (H\hat{\theta} - y)^\top (H\widehat{\Sigma}H^\top)^{-1} (H\hat{\theta} - y),$$

where $\widehat{\Sigma}$ estimates $\mathrm{Cov}(\hat{\theta})$.

Crucially, the choice of $H$ (up to row operations and scaling) does not affect the solution set nor the WTS value: $\mathrm{WTS}(H_1, y_1) = \mathrm{WTS}(H_2, y_2)$ if $(H_1, y_1)$ and $(H_2, y_2)$ specify the same hypothesis. Computational efficiency is, however, highly sensitive to the dimension and structure of $H$ (Sattler et al., 2023).

4. h-Function Hypothesis Testing: Unified Exact Inference

The $h$-function in statistical hypothesis testing unifies exact level-$\alpha$ tests and confidence intervals: $$h(x,\theta) = \sup_{\psi\in\Theta_0}P_\psi(T(X,\psi) \leq T(x,\theta)),$$ with $T$ a test statistic and $\Theta_0$ the null parameter set. For simple nulls $\Theta_0 = \{\theta_0\}$, $h(x,\theta_0)$ reduces to the traditional $p$-value.

The method provides both:

• Exact size-$\alpha$ tests by rejecting when $h(X,\theta)\leq \alpha$ [Theorem 2.1].
• Level $1-\alpha$ confidence intervals as the set $\{\theta: h(x,\theta) > \alpha\}$ (Wang, 2021).

Iterative refinement using the $h$-function can modify approximate intervals into exact ones, or further shorten exact intervals.

5. Hypothesis (H) in Number Theory: Schinzel's Hypothesis H and Rudnick–Sarnak's Hypothesis H

• Schinzel's Hypothesis H (number theory): States that for irreducible, primitive polynomials $P_1, ..., P_n\in\mathbb{Z}[t]$ with positive leading coefficients and local conditions, there exist infinitely many integers $t$ such that all $P_i(t)$ are simultaneously prime. Recent advances show that for $100\%$ of polynomials, there exists at least one such integer (the “100% version”) (Skorobogatov et al., 2020).
• Rudnick–Sarnak's Hypothesis H (automorphic forms): For a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n$ over a number field, and each fixed $k \geq 2$,

$$\sum_{\mathfrak{p}} \frac{(\log N\mathfrak{p})^2 |a_\pi(\mathfrak{p}^k)|^2}{N\mathfrak{p}^k} < \infty,$$

where $a_\pi(\mathfrak{p}^k)$ are sums of Satake parameters. This hypothesis underpins central conjectures in the analytic theory of automorphic $L$-functions, including GUE statistics for zeros, strong multiplicity one, and Selberg orthogonality (Jiang, 28 Jul 2025), and has recently been proven in full generality for all $\mathrm{GL}_n$ over any number field.

6. Generalized Chaos Hypothesis in Nonextensive Statistical Mechanics

In kinetic theory, the molecular chaos hypothesis (Stosszahlansatz) postulates that the two-particle distribution factorizes, $f^{(2)}(\mathbf{r},\mathbf{v},\mathbf{v}_1,t)$ $=$ $$f(\mathbf{r},\mathbf{v},t)f(\mathbf{r},\mathbf{v}_1,t)$$, giving rise to Boltzmann’s H-theorem. In nonextensive statistical mechanics, this is generalized via a $q$-deformed ast product: $x\ast_q y = [x^{q-1} + y^{q-1} - 1]_+^{1/(q-1)}$ and the associated $q$-distribution for two-body correlations. This leads to a generalized H-theorem for Tsallis entropy, but kinetic theory uniquely singles out the normal (linear) average as consistent with the generalized chaos hypothesis; the widely-used $q$-average is formally inconsistent and fails to yield a valid H-theorem in this setting (0903.2441).

7. Table: Hypothesis $h$ in Distinct Mathematical Contexts

Domain	Hypothesis $h$ Formulation	Key Technical Role
Markov process potential theory	Every semipolar set is polar (Hunt's H)	Determines structure of exceptional sets
Lévy processes	Fourier-analytic/integral/energy criteria (H)	Classifies polarity, connects with exponents
Multivariate statistics	Linear system $H\theta = y$ (matrix form)	Specification of test constraints
Nonparametric statistics	$h$-function: tail probability (test and CI construction)	Unifies exact inference
Analytic number theory	Schinzel’s H/RS Hypothesis H: primality/coefficient bounds	Controls prime value distribution/zero stats
Kinetic theory/statistical mechanics	Molecular chaos/Stosszahlansatz ($q$-deformation)	Structural closure of kinetic equations

8. Connections, Invariance, and Open Problems

• Hunt’s (H) is preserved under locally absolutely continuous change of measure (Hu et al., 2019), under convolution with independent compound Poisson processes, and under (in suitable cases) subordination by independent processes with positive drift.
• For generic classes of Markov, Lévy, and Hunt processes (including isotropic unimodal processes with convolution-type Green kernels) (H) can be resolved using translation-invariance and triangle estimates on Green functions (Hansen et al., 2014).
• Fundamental conjectures for (H), such as Getoor’s, remain open in the most delicate regimes (pure-jump, driftless subordinators in higher dimensions) (Hu et al., 2012, Hu et al., 2019).
• In number theory, the resolution of Hypothesis H directly unlocks a range of equidistribution and zero statistics for automorphic $L$-functions (Jiang, 28 Jul 2025).

9. Summary

Hypothesis $h$ is domain-dependent, ranging from high-level qualitative assertions about the structure of exceptional sets in potential theory, to explicit computational matrix formulations in multivariate statistics, to deep analytic conjectures about primes and automorphic forms in number theory. Across these settings, the specification, invariance, and consequences of hypothesis $h$ are central for the formulation and solution of major structural problems.