Split-Complex Scalars
The Split-Complex Scalars (I.D20) are the elements of ℝ[j]/(j²−1) treated as the canonical scalar algebra of the τ-framework. Every τ-categorical scalar — including ι_τ, the rank-(n, k) algebraic invariants, and the spectral trichotomy values — is a split-complex scalar. The algebra is forced by the Split-Complex Forced theorem (T07) and is unique up to canonical isomorphism.
τ-Definition
The Split-Complex Scalars (I.D20) are the elements of ℝ[j]/(j²−1) treated as the canonical scalar algebra of the τ-framework. Every τ-categorical scalar — including ι_τ, the rank-(n, k) algebraic invariants, and the spectral trichotomy values — is a split-complex scalar. The algebra is forced by the Split-Complex Forced theorem (T07) and is unique up to canonical isomorphism.
Categorical invariant. ℝ[j]/(j²−1) viewed as the underlying scalar algebra of the τ-categorical kernel.
Primary registry anchor:
I.D20
Supporting items:
I.T10
τ-Derivation Chain
Mathematical content
The Split-Complex Scalars are the algebra ℝ[j]/(j²−1) — i.e., elements a + bj where a, b ∈ ℝ and j² = 1. The algebra is commutative with idempotents e_+ = (1+j)/2 and e_- = (1-j)/2 satisfying e_+ + e_- = 1 and e_+ · e_- = 0.
Consequences:
- Master constant ι_τ = 2/(π+e) lives in ℝ ⊂ ℝ[j].
- Central theorem rank-(3, 15) invariant lives in calibrated ℝ[j] (D09).
- Spectral trichotomy values {B, I, S} read off split-complex polarity.
Key features:
- Two canonical idempotents e_+, e_- (corresponding to {+, −} polarity).
- Zero divisors (e.g., (1+j)(1-j) = 0) — distinguishes split-complex from complex.
- Hyperbolic-trigonometric structure (cosh, sinh) instead of Euclidean (cos, sin).
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Boundary.SplitComplex
Lean kind: def
Lean symbol: Tau.BookI.Boundary.SplitComplex
Cross-domain bridges
This glossary term sits on the boundary between domains. The τ-framework's cross-domain pivots are the structural junctions where physics, life, and metaphysics readouts meet.
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PG-C18-kappa-tauGravity-sector coupling κ_τ →MathG-D10-split-complex-scalarsSplit-Complex Scalars -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-D10-split-complex-scalarsSplit-Complex Scalars -
PG-L13-tau-mercury-precessionτ-Mercury Perihelion Precession →MathG-D10-split-complex-scalarsSplit-Complex Scalars -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-D10-split-complex-scalarsSplit-Complex Scalars -
PG-Q24-velocityVelocity →MathG-D10-split-complex-scalarsSplit-Complex Scalars