Yury N. Berdinsky | Searcharxiv

arXiv2026

Classification of Conformally Covariant 2-Tensors from the Kulkarni--Nomizu Product in Dimension 4

We classify all natural, conformally covariant, symmetric (0,2)-tensors of conformal weight -2 and differential order less than or equal to 4 in dimension 4, built from the metric, the Schouten tensor, covariant derivatives, and the Kulkarni-Nomizu product. We prove that the space of such tensors is exactly 2-dimensional, spanned by the Bach tensor (order 2) and the Eastwood-Singer tensor (order 4). The algebraic core is formally verified using Lean 4 with Mathlib. The 8x11 constraint matrix and divergence-free condition are verified computationally with exact rational arithmetic. Both English and Russian versions provided; Russian version available as ancillary.

math-ph↗

arXiv2026

Henstock--Kurzweil Path Integral in Financial Mathematics: A Machine-Verified Pricing of European and Barrier Options

We apply the Henstock--Kurzweil (HK) gauge integral to the Black--Scholes model of option pricing and obtain the European call price directly from a Gaussian cylindrical kernel, without stochastic calculus. Under the risk- neutral measure, the log-price is a Brownian motion with drift nu = r - sigma^2/2. Its transition density is the Gaussian kernel G_t(x,y) = (2 pi sigma^2 t)^{-1/2} exp( - (y - x - nu t)^2 / (2 sigma^2 t) ). We give a machine-checked formalization in Lean 4 / Mathlib of the following: the Chapman--Kolmogorov (semigroup) property, the fact that G_t is a probability density, strong continuity of the pricing operator, the closed-form price C = S_0 N(d_1) - K e^{-rT} N(d_2) with the standard normal CDF N, and the exactness of the drift--diffusion Chernoff splitting at every level. The entire proof is "sorry"-free and depends only on propext, Classical.choice, and Quot.sound. Digital and barrier options are treated as further examples, illustrating the universality of the method, and we show that the construction is compatible with the classical Ito calculus in the continuum limit.

q-fin.PR↗