Bismuth algebra, field and integration

This work is a direct extension of Zenodo DOI: 10.5281/zenodo.18462228.

: a real-analytic algebra and field on \mathbb{C}^2 generated by globally bounded, non-constant functions, together with a natural integration theory and a precise analysis of its spectral size.

The construction is based on a fundamental bounded generator

B(\bar z, \sigma) = \tanh(\mathrm{Re}\,\bar z) + i\,\tanh(\mathrm{Im}\,\sigma),

where \bar z and \sigma are treated as independent complex variables. This choice bypasses Liouville-type collapse and forces a rich asymptotic structure at infinity.

The paper develops three core components:

1. The Bismuth Algebra and Spectrum
   A uniformly closed algebra of bounded real-analytic functions on \mathbb{C}^2 is constructed. Using Gelfand theory, its maximal spectrum is shown to be necessarily huge, with cardinality 2^{\mathfrak c}, reflecting genuine asymptotic separation at infinity rather than a single compactification point.
2. Bismuth Integration and Vanishing Theorem
   A canonical two-form and torus integration are defined. A Bismuth–Stokes vanishing theorem is proved: integrals of globally Bismuth-analytic (bounded) functions over any Bismuth torus vanish identically. Non-zero integrals arise precisely from meromorphic obstructions, providing a residue-like mechanism intrinsic to the framework.
3. The Global Bismuth Field and Cardinality
   By assembling local fields of meromorphic germs over the maximal spectrum, a global Bismuth field is defined. Its cardinality is shown to be 2^{2^{\mathfrak c}}, not by design but as an unavoidable consequence of boundedness, asymptotic separation, and Stone–Čech–type phenomena.

The work demonstrates that once globally bounded non-constant real-analytic functions are admitted, a radically different notion of infinity, spectrum, and residue theory emerges. The resulting field is structurally distinct from classical holomorphic or meromorphic frameworks and opens a new analytic setting where infinity is inherently non-collapsed.

To the author’s knowledge, no larger globally defined analytic field admitting a coherent infinitesimal, differential, and integral calculus has previously been constructed