A Unified Finitist Indefinite-Hermitian Framework for 3+1D Spacetime: Clifford Simplex Dynamics, Quadratic Character Identities, and a Conjectural Finitist-to-Continuum Program

Abstract

This research paper establishes a limit-free, finite-dimensional algebraic framework for discrete spacetime geometry and quantum kinematics in (3+1) dimensions. Operating over the finite Galois field F_p (with prime characteristic p > 5 and p = 3 mod 4) and its quadratic extension F_p[delta] (isomorphic to the field of order p^2), the paper bridges Universal Hyperbolic Geometry (UHG) in projective 3-space P^3(F_p) with finite-state indefinite-Hermitian dynamics. The framework serves as the foundational master paper for an integrated finitist theory of spacetime and fundamental interactions, supplying the algebraic substrate for companion models of U(1) electromagnetism, SU(2) x SU(3) non-abelian gauge fields, and emergent gravitational dynamics. Key Mathematical Results and Contributions Witt-Index Incompatibility and 4x4 Clifford Engine:The paper proves via finite-field quadratic form classification that the (3+1)D Lorentzian form on 4D vector space has elliptic (minus) Witt type over F_p when p = 3 mod 4. This establishes a structural obstruction precluding linear quadratic-form isometries into 2x2 real matrix determinant spaces (which are split/hyperbolic). The paper resolves this obstruction by constructing an explicit minimal 4x4 Clifford algebra over F_p. Bivector edge anchors satisfy an exact quadratic polynomial identity, generating rational Cayley spinor transports whose induced adjoint action on vector space is proven algebraically to be special orthogonal (SO(3,1) transformations). The Parity-Quadrume Character Theorem:For every non-degenerate projective tetrahedron in P^3(F_p), the paper proves the exact finite-field theorem:chi_p(V) = -epsilon,where V is the Projective Quadrume (the normalized 4x4 Gram determinant), chi_p is the Legendre quadratic character, and epsilon is a Z_2 parity grading computed from the product of the vertex quadratic norms. This identity algebraically locks the geometric orientation and volume opening of the tetrahedron to the metric discriminant of the Lorentzian signature (-1). Finite-State Indefinite-Hermitian (Krein-Type) Dynamics:The state space is spanned by Cyclic Oriented Polygonal Splines (COPS) of non-degenerate tetrahedra modulo dihedral D_4 loop symmetries. Using a parity-preserving mutation graph, the paper defines an effective Hamiltonian with a reference-configuration potential that measures deviations from standard orthonormal cells. The resulting rational Cayley evolution operator satisfies exact Galois-Hermitian unitarity (U_dag U = I), Krein-type isometry (U_dag G U = G), and exact parity-sector decoupling. Exact Finite Path Expansion:The paper derives an exact finite matrix-product expansion for transition amplitudes, expressing discrete time evolution as an algebraic sum over mutation paths without invoking continuous path integrals. Conjectural Finitist-to-Continuum Program:The paper rigorously separates proved finite-field theorems from open continuum questions, laying out a structured six-step research program to investigate how discrete quadrume dynamics, additive character partition sums, and simplicial refinement limits may connect to higher-derivative Regge calculus and classical General Relativity (G_uv + Lambda g_uv = 0). Computational Verification and Reproducibility All algebraic identities, point-count distributions, Clifford matrix relations, Cayley unitarities, and path expansions are fully verified symbolically and numerically in Python (SymPy) over F_7 and F_7[delta] (field order 49). Complete automated verification scripts, state-space generators, and raw dataset tables are included and openly accessible in the accompanying software-and-data archive (Zenodo DOI: 10.5281/zenodo.21998225).