Central Idempotent Clifford Signature Sectors: Matching-Relative Lorentzian–Euclidean Symbol Couplings and Certified Finite-Dimensional Reduction

This paper develops a finite-dimensional algebraic framework in which fixed Clifford
signatures are organized by central sign observables rather than chosen as scalar background
data at the outset. The construction begins with the one-generator relation J2 = S, S2 = I,
and extends to Clifford-type systems satisfying
EiEj + Ej Ei = 2δij Si,
where the Si are central commuting involutions at the kinematic level. A central point of
the paper is that this central-sign layer is a structured branch decomposition of ordinary
fixed-signature Clifford sectors, not by itself a noncentral dynamical theory of fluctuating
metrics. The algebraic object produced by the signs is a central algebra-valued Clifford-
signature form in a fixed finite-dimensional frame, not a coordinate-covariant metric tensor
on a smooth manifold.
The claim architecture is therefore stratified: the central layer supplies only branch
projectors and fixed-signature sector restrictions; branch transition enters only through
declared noncentral data; Schur complements are matching-relative matrix-symbol reductions;
and PDE, Fredholm, scattering, or global spectral conclusions require an independent
operator-realization certificate.
A controlled four-dimensional model is then introduced using a single signature operator
Λ, whose eigensectors satisfy Lorentzian (3, 1) and Euclidean (4, 0) real-signature Clifford
relations whose complexifications are represented on a common complex spinor carrier. This
is deliberately a one-sign Lorentzian–Euclidean transition slice, not the full family of diagonal
four-sign Clifford-signature patterns. Noncentral branch transitions are introduced only in
the extended algebra, through a declared branch-coupling datum (χ, BLE , BEL); its same-
coordinate identity-coupled specialization has minimal common-carrier gauge T = σx ⊗ I4
with [T, Λ]̸ = 0. At the constant-background Fourier-symbol level, the Lorentzian branch
is normalized to recover the standard massive matrix-symbol shell, while the Euclidean
branch supplies only a controlled resolvent channel or an explicitly certified reduced channel
through the matched symbol AE (χ(kL)). The branch mass splitting is defined as the
branch restriction of the declared branch-odd mass operator MΛ(Φ0) = mI8 + gΦ0Λ. Schur
complement reduction yields an O(Ω2) finite-dimensional symbol-level Lorentzian self-energy
correction, whose trace part is treated only as an algebraic identity component unless a
certified identity-only or mass-shift window is declared; the remaining part is a non-identity
matrix-symbol correction rather than a pure mass shift. At operator level this correction is
generally pseudodifferential and nonlocal. The identity off-diagonal coupling is treated as a
minimal common-carrier gauge and is embedded in a more general branch-map formulation.
Reduced Hermitian and non-Hermitian two-mode normal forms are used only after declaring
an exact two-mode Feshbach datum and, when the displayed normal form is approximate,
a controlled remainder datum. Non-Hermitian splittings are treated as complex algebraic
splittings unless a real-gap certificate is supplied, and exceptional points are separated
from ordinary scalar degeneracies and from uncertified displayed candidates by explicit
defectiveness and robustness conditions. No claim is made here about a closed PDE operator,
self-adjoint domain theory, Fredholmness, scattering theory, a local Dirac-type effective PDE
after Schur elimination, or a global operator-level gap.