Euclidean-Lorentzian Dichotomy and Algebraic Causality in Finite Ring Continuum

Area of Science:

• Number Theory
• Abstract Algebra
• Theoretical Physics

Background:

• The Finite Ring Continuum (FRC) program explores algebraic structures.
• Prime shells Fp, where p=4t+1, are investigated for their properties.
• Understanding the interplay between Euclidean and Lorentzian structures is crucial.

Purpose of the Study:

• To present an extension of the FRC program.
• To demonstrate a Euclidean-Lorentzian dichotomy in symmetry-complete prime shells Fp.
• To establish an intrinsic relativistic algebra within finite-field arithmetic.

Main Methods:

• Analysis of symmetry-complete prime shells Fp with p=4t+1.
• Derivation of a finite-field Lorentz transformation.
• Generation of a finite orthogonal group O(Qν,Fp2) of split type.

Main Results:

• Prime shells Fp exhibit a fundamental Euclidean-Lorentzian dichotomy.
• A genuine Lorentzian quadratic form cannot be realized within a single space-like prime shell Fp.
• Finite-field Lorentz transformations preserve the Minkowski form.
• The invariant interval and Lorentz symmetry emerge naturally within finite-field arithmetic.
• Causality originates algebraically from the dichotomy.

Conclusions:

• The study establishes an intrinsic relativistic algebra within FRC.
• Euclidean invariants are confined to space-like shells Fp.
• Lorentzian structure and causal separation arise in the quadratic spacetime extension Fp2.