Algebraic Lemniscate
The Algebraic Lemniscate (I.D18) is the canonical 1-dimensional boundary structure on the τ-categorical kernel — a real-algebraic curve with two crossing branches (the lemniscate of Bernoulli, x² − y² = (x²+y²)²). Within τ, the lemniscate is the unique boundary geometry compatible with the K2 labelled-boundary axiom and the Split-Complex Forced theorem. With 28 incoming edges, it is the geometric substrate for Book I's holomorphy tower (S02) and Book III's spectral correspondence (A02).
τ-Definition
The Algebraic Lemniscate (I.D18) is the canonical 1-dimensional boundary structure on the τ-categorical kernel — a real-algebraic curve with two crossing branches (the lemniscate of Bernoulli, x² − y² = (x²+y²)²). Within τ, the lemniscate is the unique boundary geometry compatible with the K2 labelled-boundary axiom and the Split-Complex Forced theorem. With 28 incoming edges, it is the geometric substrate for Book I's holomorphy tower (S02) and Book III's spectral correspondence (A02).
Categorical invariant. A real-algebraic curve Lem ⊂ τ-boundary equipped with the {+1, −1} polarity grading and the j² = +1 split-complex action.
Primary registry anchor:
I.D18
τ-Derivation Chain
Mathematical content
The τ-categorical boundary admits a unique 1-dimensional real-algebraic curve, the Algebraic Lemniscate, defined by the equation x² − y² = (x² + y²)² in split-complex coordinates and equipped with the polarity grading from Prime Polarity (T06). The curve has two crossing branches at the origin (the polarity transition).
Proof sketch (expand)
K2 forces the boundary to be a labelled topological space; T10 (Split-Complex Forced) forces the labels to live in ℝ[j]/(j²−1); the unique 1-dimensional smooth curve in this algebra compatible with K1 strict order is the Bernoulli lemniscate. Uniqueness follows from a degree-counting argument on the K1-graded boundary.
Consequences:
- Boundary geometry of Books I–III spectral analysis lives on Lem.
- Holomorphy tower (S02) is built over Lem.
- Spectral correspondence (A02) operates on degree-3 polynomial sub-structures of Lem.
Lean Coverage
Status: Formalized
Module: TauLib.BookI.Polarity.Lemniscate
Lean kind: theorem
Lean symbol: Tau.BookI.Polarity.algebraicLemniscate
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-T09-algebraic-lemniscateAlgebraic Lemniscate -
PG-L05-tau-newton-gravityτ-Newton's Law of Gravity →MathG-T09-algebraic-lemniscateAlgebraic Lemniscate -
PG-L13-tau-mercury-precessionτ-Mercury Perihelion Precession →MathG-T09-algebraic-lemniscateAlgebraic Lemniscate -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T09-algebraic-lemniscateAlgebraic Lemniscate -
PG-Q24-velocityVelocity →MathG-T09-algebraic-lemniscateAlgebraic Lemniscate