{
  "abstract": "We develop a real-time functional-integral framework for stochastic neural fields based on the Henstock–Kurzweil gauge integral and the Kuelbs–Steadman (KS2) spaces. Unlike classical approaches that rely on a Wick rotation to convert an ill-defined oscillatory path integral into a Wiener integral, the gauge integral works directly in real time: a point-dependent gauge replaces the translation-invariant measure, and conditionally convergent sums become genuine gauge limits. This is essential for biological systems whose dynamics are intrinsically real-time and punctuated by discontinuous spikes. As a concrete test we treat the subjective-centered model of Titov, in which conscious cognitive activity arises from the interaction of a noisy «intentiating» neural field with an «intential» repertoire of stable patterns (exitoms) coupled through the connectome. Discretising the cortex into a finite grid and encoding the linearised Wilson–Cowan dynamics by a symmetric positive- definite matrix A = αI – βW, we obtain a closed-form partition function Z = (2π)^(n/2) (det A)^(-1/2) and the two-point correlation ⟨φ_k φ_l⟩ = Z (A⁻¹)_kl. The cognitive transition probability into an exitom reduces to a one-dimensional Gaussian tail, yielding the formula P = N((m–θ)/√a) in terms of the standard normal CDF. The Wilson–Cowan evolution operator is split via the Chernoff/Lie–Trotter product formula, and we prove that time-sliced approximants converge to the exact propagator exp(t(D+C)), which satisfies the associated Fokker–Planck equation. Every statement is formalised in the Lean 4 / Mathlib file HkNeuralField.lean; all proofs are sorry-free and depend only on propext, Classical.choice, and Quot.sound. The paper also discusses parametric sensitivity of cognitive transitions, the extension to spike-driven activity on the Skorokhod space, an algorithmic bridge from connectome to real-time brain–computer interfaces, and a philosophical interpretation of the gauge integral as a mathematical model of subjectivity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "name": "К В Титов"
    },
    {
      "name": "Y. N. Berdinsky"
    },
    {
      "name": "A. S. Ushakov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b58jq1zbm1jq0.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.21726881",
  "fulltext_url": null,
  "id": "b58jq1zbm1jq0",
  "issns": [],
  "landing_url": "https://hal.science/hal-05711261v3",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "0.phys"
    ],
    [
      "hal",
      "0.scco"
    ],
    [
      "hal",
      "0.stat"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05711261/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b58jq1zbm1jq0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "hal-05711261",
      "source": "hal",
      "url": "https://hal.science/hal-05711261v3"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.stat",
      "subfield": "statistics"
    },
    {
      "field": "physics",
      "origin": "hal:0.phys",
      "subfield": "general"
    }
  ],
  "title": "A real-time path-integral framework for stochastic neural fields via the Henstock–Kurzweil integral and Külbs–Steadman spaces",
  "title_text": "A real-time path-integral framework for stochastic neural fields via the Henstock–Kurzweil integral and Külbs–Steadman spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/b58jq1zbm1jq0",
  "venue": "HAL",
  "versions": []
}