Central Theorem: O(τ³) ≅ A_spec(L)

Overview

The Central Theorem (II.T40) establishes that O(τ³) ≅ A_spec(L): the algebra of τ-holomorphic functions on the fibered product τ³ = τ¹ ×_f T² is isomorphic to the spectral algebra of the lemniscate boundary L = S¹ ∨ S¹. The isomorphism is functorial, bipolar-compatible, tower-graded, and ι_τ-calibrated. It means that the full three-dimensional interior is completely determined by one-dimensional boundary data — a holographic principle internal to the mathematical framework.

Detail

The Central Theorem is the climax of Book II and the most-cited result in the series. τ³ = τ¹ ×_f T² is the fibered product of the base (macroscopic/cosmological sector, τ¹) and fiber (microscopic/quantum sector, T²). The lemniscate L = S¹ ∨ S¹ is the boundary of τ³ — two circles joined at a crossing point, each lobe corresponding to one polarity class of primes (Prime Polarity, I.T05). A_spec(L) is the spectral algebra generated by boundary characters on L. II.T40 proves that the restriction map O(τ³) → A_spec(L) is an algebra isomorphism. Functoriality means the isomorphism commutes with categorical maps between τ³ and L. Bipolar compatibility means γ-even and η-odd components are separately respected. Tower grading means the isomorphism respects the enrichment layers E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. ι_τ-calibration means all dimension constants in the isomorphism are rational functions of ι_τ = 2/(π+e). This result is the structural basis for all holographic reasoning in Books IV–V: every physical prediction from the boundary (observable universe) to the interior (quantum structure) uses II.T40.

Result Statement

The algebra of holomorphic functions on τ³ is isomorphic to the spectral algebra of the lemniscate boundary: O(τ³) ≅ A_spec(L) (II.T40). This isomorphism is functorial, bipolar-compatible, tower-graded, and ι_τ-calibrated. Interior is fully determined by boundary data.