TauLib · API Book III

TauLib.BookIII.Doors.SplitComplexZeta

TauLib.BookIII.Doors.SplitComplexZeta

Split-Complex Zeta Function, Functional Equation Involution, and Bipolar Euler Product.

Registry Cross-References

  • [III.D26] Split-Complex Zeta ζ_τ – split_zeta_b, split_zeta_c, split_zeta_check

  • [III.D27] Functional Equation Involution J – fe_involution, fe_involution_check

  • [III.T16] Bipolar Euler Product – bipolar_euler_check

Mathematical Content

III.D26 (Split-Complex Zeta): ζ_τ(s) = e₊·ζ_B(s) + e₋·ζ_C(s), encoding the Riemann zeta in split-complex coordinates. B-lobe = B-type primes, C-lobe = C-type primes.

III.D27 (Functional Equation Involution): J exchanges B-lobe and C-lobe components: J(b, c, x) = (c, b, x). J² = id.

III.T16 (Bipolar Euler Product): ζ_τ(s) = ∏_p (1 - Label(p)·p^{-s})^{-1}. CRT decomposition at each primorial level recovers the partial products.

Tau.BookIII.Doors.split_zeta_b

source def Tau.BookIII.Doors.split_zeta_b (k : Denotation.TauIdx) :Denotation.TauIdx

[III.D26] B-lobe partial zeta: product of B-type primes up to depth k. Delegates to the label product from Trichotomy. Equations

  • Tau.BookIII.Doors.split_zeta_b k = Tau.BookIII.Spectral.compute_label_product Tau.BookIII.Spectral.PrimeLabel.B k Instances For

Tau.BookIII.Doors.split_zeta_c

source def Tau.BookIII.Doors.split_zeta_c (k : Denotation.TauIdx) :Denotation.TauIdx

[III.D26] C-lobe partial zeta: product of C-type primes up to depth k. Equations

  • Tau.BookIII.Doors.split_zeta_c k = Tau.BookIII.Spectral.compute_label_product Tau.BookIII.Spectral.PrimeLabel.C k Instances For

Tau.BookIII.Doors.split_zeta_x

source def Tau.BookIII.Doors.split_zeta_x (k : Denotation.TauIdx) :Denotation.TauIdx

[III.D26] X-type partial product (crossing prime contribution). Equations

  • Tau.BookIII.Doors.split_zeta_x k = Tau.BookIII.Spectral.compute_label_product Tau.BookIII.Spectral.PrimeLabel.X k Instances For

Tau.BookIII.Doors.split_zeta_check

source def Tau.BookIII.Doors.split_zeta_check (db : Denotation.TauIdx) :Bool

[III.D26] Split-complex zeta check: B · C · X = Prim(k). The three label products account for all primes in the primorial. Equations

  • Tau.BookIII.Doors.split_zeta_check db = Tau.BookIII.Doors.split_zeta_check.go db 1 (db + 1) Instances For

Tau.BookIII.Doors.split_zeta_check.go

source@[irreducible]

**def Tau.BookIII.Doors.split_zeta_check.go (db : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.fe_involution

source def Tau.BookIII.Doors.fe_involution (nf : Spectral.BoundaryNF) :Spectral.BoundaryNF

[III.D27] Functional equation involution: exchanges B-lobe and C-lobe components of a boundary normal form. J(b, c, x) = (c, b, x). Equations

  • Tau.BookIII.Doors.fe_involution nf = { b_part := nf.c_part, c_part := nf.b_part, x_part := nf.x_part, depth := nf.depth } Instances For

Tau.BookIII.Doors.fe_involution_check

source def Tau.BookIII.Doors.fe_involution_check (bound db : Denotation.TauIdx) :Bool

[III.D27] Involution check: J² = identity for all BNFs in range. Equations

  • Tau.BookIII.Doors.fe_involution_check bound db = Tau.BookIII.Doors.fe_involution_check.go bound db 0 1 ((bound + 1) * (db + 1)) Instances For

Tau.BookIII.Doors.fe_involution_check.go

source@[irreducible]

**def Tau.BookIII.Doors.fe_involution_check.go (bound db : Denotation.TauIdx)

(x k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.fe_swap_check

source def Tau.BookIII.Doors.fe_swap_check (bound db : Denotation.TauIdx) :Bool

[III.D27] J swaps B and C but fixes X. Equations

  • Tau.BookIII.Doors.fe_swap_check bound db = Tau.BookIII.Doors.fe_swap_check.go bound db 0 1 ((bound + 1) * (db + 1)) Instances For

Tau.BookIII.Doors.fe_swap_check.go

source@[irreducible]

**def Tau.BookIII.Doors.fe_swap_check.go (bound db : Denotation.TauIdx)

(x k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.bipolar_euler_check

source def Tau.BookIII.Doors.bipolar_euler_check (db : Denotation.TauIdx) :Bool

[III.T16] Bipolar Euler product: B · C · X = Prim(k) and B, C coprime. Equations

  • Tau.BookIII.Doors.bipolar_euler_check db = Tau.BookIII.Doors.bipolar_euler_check.go db 1 (db + 1) Instances For

Tau.BookIII.Doors.bipolar_euler_check.go

source@[irreducible]

**def Tau.BookIII.Doors.bipolar_euler_check.go (db : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.euler_tower_check

source def Tau.BookIII.Doors.euler_tower_check (db : Denotation.TauIdx) :Bool

[III.T16] Tower coherence: products at depth k+1 extend depth k. Equations

  • Tau.BookIII.Doors.euler_tower_check db = Tau.BookIII.Doors.euler_tower_check.go db 1 (db + 1) Instances For

Tau.BookIII.Doors.euler_tower_check.go

source@[irreducible]

**def Tau.BookIII.Doors.euler_tower_check.go (db : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.split_zeta_5

source theorem Tau.BookIII.Doors.split_zeta_5 :split_zeta_check 5 = true

Tau.BookIII.Doors.fe_involution_15_4

source theorem Tau.BookIII.Doors.fe_involution_15_4 :fe_involution_check 15 4 = true

Tau.BookIII.Doors.fe_swap_15_4

source theorem Tau.BookIII.Doors.fe_swap_15_4 :fe_swap_check 15 4 = true

Tau.BookIII.Doors.bipolar_euler_5

source theorem Tau.BookIII.Doors.bipolar_euler_5 :bipolar_euler_check 5 = true

Tau.BookIII.Doors.euler_tower_4

source theorem Tau.BookIII.Doors.euler_tower_4 :euler_tower_check 4 = true

Tau.BookIII.Doors.b_zeta_3

source theorem Tau.BookIII.Doors.b_zeta_3 :split_zeta_b 3 = 5

[III.D26] Structural: B-zeta at depth 3 = 5 (only B-type prime ≤ p₃).

Tau.BookIII.Doors.c_zeta_3

source theorem Tau.BookIII.Doors.c_zeta_3 :split_zeta_c 3 = 3

[III.D26] Structural: C-zeta at depth 3 = 3 (only C-type prime ≤ p₃).

Tau.BookIII.Doors.fe_involution_involutive

source theorem Tau.BookIII.Doors.fe_involution_involutive (nf : Spectral.BoundaryNF) :fe_involution (fe_involution nf) = nf

[III.D27] Structural: involution is own inverse.

Tau.BookIII.Doors.euler_product_3

source theorem Tau.BookIII.Doors.euler_product_3 :split_zeta_b 3 * split_zeta_c 3 * split_zeta_x 3 = Polarity.primorial 3

[III.T16] Structural: B · C · X = Prim(3) at depth 3.