Calibrated Split-Complex Codomain
The Calibrated Split-Complex Codomain (II.D35) is the split-complex algebra ℝ[j]/(j²−1) equipped with the τ-categorical calibration — a graded structure on real and j-imaginary components reflecting the K1 strict order. With 25 incoming edges, it is the canonical codomain for τ-internal Hartogs extensions (T12) and Yoneda-as-theorem hom-objects (D04).
τ-Definition
The Calibrated Split-Complex Codomain (II.D35) is the split-complex algebra ℝ[j]/(j²−1) equipped with the τ-categorical calibration — a graded structure on real and j-imaginary components reflecting the K1 strict order. With 25 incoming edges, it is the canonical codomain for τ-internal Hartogs extensions (T12) and Yoneda-as-theorem hom-objects (D04).
Categorical invariant. ℝ[j]/(j²−1) equipped with a calibration grading c : ℝ[j] → ℕ₀ × ℕ₀ compatible with K1 strict order.
Primary registry anchor:
II.D35
τ-Derivation Chain
Mathematical content
The Calibrated Split-Complex Codomain is the algebra ℝ[j]/(j²−1) equipped with a calibration grading c : ℝ[j] → ℕ₀ × ℕ₀ that maps a + bj to (a-degree, b-degree) where the degrees are determined by K1 strict-order placement of a and b.
Consequences:
- Hartogs extensions (T12) target this calibrated codomain — the calibration carries through the extension uniqueness.
- Yoneda hom-objects (D04 / II.D50) are calibrated split-complex objects — the calibration provides the τ-internal grading needed for the Yoneda enrichment ladder (T05).
- Central theorem (T04) — the rank-(3, 15) algebraic invariant lives in calibrated split-complex.
Lean Coverage
Status: Formalized
Module: TauLib.BookII.Hartogs.CalibratedSplitComplex
Lean kind: structure
Lean symbol: Tau.BookII.Hartogs.CalibratedSplitComplex
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-D09-calibrated-split-complexCalibrated Split-Complex Codomain -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-D09-calibrated-split-complexCalibrated Split-Complex Codomain -
PG-L13-tau-mercury-precessionτ-Mercury Perihelion Precession →MathG-D09-calibrated-split-complexCalibrated Split-Complex Codomain -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-D09-calibrated-split-complexCalibrated Split-Complex Codomain -
PG-Q24-velocityVelocity →MathG-D09-calibrated-split-complexCalibrated Split-Complex Codomain -
MathG-D09-calibrated-split-complexCalibrated Split-Complex Codomain →PG-C02-iota-tauMaster constant ι_τ