TauLib · API Book III

TauLib.BookIII.Doors.GrandGRH

TauLib.BookIII.Doors.GrandGRH

Grand GRH (Generalized Riemann Hypothesis), Prime Polarity Scaling, and L-Functions as Spectral Determinants.

Registry Cross-References

  • [III.D31] Grand GRH (τ-effective) – grand_grh_finite_check [AXIOM at adelic level]

  • [III.T20] Prime Polarity Scaling Theorem – prime_polarity_scaling_check

  • [III.D32] L-Function as Spectral Determinant – l_function_spectral_check

Mathematical Content

III.D31 (Grand GRH): At adelic level: for all boundary characters on 𝔸_τ, the corresponding L-function has all non-trivial zeros on Re(s) = ½. CONJECTURAL at the adelic extension beyond finite primorial cutoff.

III.T20 (Prime Polarity Scaling): The GRH at each primorial depth decomposes into three independent statements via Label_n: purity of B-sector zeros, purity of C-sector zeros, and balance of X-sector zeros.

III.D32 (L-Function as Spectral Determinant): All L-functions as spectral determinants of the universal operator at finite cutoff.

Tau.BookIII.Doors.grand_grh_finite_check

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

[III.D31] Grand GRH at finite primorial level k: the finite Euler product decomposes correctly via B/C/X labels, and each sector has the correct zero structure at this level. Equations

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

Tau.BookIII.Doors.grand_grh_finite_check.go

source@[irreducible]

**def Tau.BookIII.Doors.grand_grh_finite_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.grand_grh_adelic

source axiom Tau.BookIII.Doors.grand_grh_adelic (k : ℕ) :grand_grh_finite_check k = true

[III.D31] Grand GRH Axiom: the adelic extension of GRH beyond finite primorial cutoff. CONJECTURAL SCOPE. All finite checks pass; the axiom asserts the infinite/adelic limit.

Tau.BookIII.Doors.prime_polarity_scaling_check

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

[III.T20] Prime polarity scaling: the GRH decomposition into B/C/X sectors is compatible across primorial levels. Scaling from level k to k+1 preserves polarity type of each prime. Equations

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

Tau.BookIII.Doors.prime_polarity_scaling_check.go

source@[irreducible]

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

(i k fuel : ℕ) :Bool**

Equations

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

Tau.BookIII.Doors.sector_growth_check

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

[III.T20] Sector growth rates: B-product and C-product grow at different rates (B = multiplicative/exponent, C = additive/tetration). Equations

  • Tau.BookIII.Doors.sector_growth_check db = Tau.BookIII.Doors.sector_growth_check.go db 2 (db + 1) Instances For

Tau.BookIII.Doors.sector_growth_check.go

source@[irreducible]

**def Tau.BookIII.Doors.sector_growth_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.l_function_spectral_check

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

[III.D32] L-function as spectral determinant at finite level: L_{≤k}(s) = ∏_{p ≤ p_k} (1 - p^{-s})^{-1}. At τ-level: the finite Euler product equals the primorial when all factors are included, and decomposes via the B/C/X labels. Equations

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

Tau.BookIII.Doors.l_function_spectral_check.go

source@[irreducible]

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

(i k fuel : ℕ) :Bool**

Equations

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

Tau.BookIII.Doors.grand_grh_finite_5

source theorem Tau.BookIII.Doors.grand_grh_finite_5 :grand_grh_finite_check 5 = true

Tau.BookIII.Doors.prime_polarity_scaling_5

source theorem Tau.BookIII.Doors.prime_polarity_scaling_5 :prime_polarity_scaling_check 5 = true

Tau.BookIII.Doors.sector_growth_5

source theorem Tau.BookIII.Doors.sector_growth_5 :sector_growth_check 5 = true

Tau.BookIII.Doors.l_function_spectral_5

source theorem Tau.BookIII.Doors.l_function_spectral_5 :l_function_spectral_check 5 = true

Tau.BookIII.Doors.grand_grh_3

source theorem Tau.BookIII.Doors.grand_grh_3 :grand_grh_finite_check 3 = true

[III.D31] Structural: Grand GRH finite check at depth 3.

Tau.BookIII.Doors.label_5_is_B

source theorem Tau.BookIII.Doors.label_5_is_B :Spectral.label_direct 5 = Spectral.PrimeLabel.B

[III.T20] Structural: label classification of prime 5 (5 ≡ 1 mod 4 → B).

Tau.BookIII.Doors.label_3_is_C

source theorem Tau.BookIII.Doors.label_3_is_C :Spectral.label_direct 3 = Spectral.PrimeLabel.C

[III.T20] Structural: label classification of prime 3 (3 ≡ 3 mod 4 → C).

Tau.BookIII.Doors.l_function_3

source theorem Tau.BookIII.Doors.l_function_3 :split_zeta_b 3 * split_zeta_c 3 * split_zeta_x 3 = Polarity.primorial 3

[III.D32] Structural: L-function at depth 3 = Prim(3).

Tau.BookIII.Doors.grand_grh_from_axiom

source theorem Tau.BookIII.Doors.grand_grh_from_axiom (k : ℕ) :grand_grh_finite_check k = true

[III.D31] Structural: Grand GRH from axiom.