TauLib · API Book III

TauLib.BookIII.Sectors.LanglandsReflection

TauLib.BookIII.Sectors.LanglandsReflection

Langlands₁ Reflection Bridge, Template Invariance, Universal Operator, and Spectral Polarity.

Registry Cross-References

  • [III.D15] Langlands₁ Reflection Bridge – langlands_reflection_check

  • [III.T06] Template Invariance Under Reflection – template_invariance_check

  • [III.D16] Universal Operator – universal_operator

  • [III.D17] Spectral Polarity – spectral_polarity

  • [III.P04] Balanced Sector Uniqueness – balanced_uniqueness_check

Mathematical Content

III.D15 (Langlands₁): The enrichment functor F_E restricted to boundary characters produces a correspondence between E₀-level sectors and E₁-level sectors. Template invariant; carrier changes.

III.T06 (Template Invariance): The layer template (Carrier, Predicate, Decoder, Invariant) is preserved under the Langlands₁ reflection.

III.D16 (Universal Operator): H_∞ on L²(Char(L)) unifies all L-functions as spectral determinants.

III.D17 (Spectral Polarity): For each sector S_g, the spectral polarity pol(S_g) = Σ|m| / Σ|n| over characters in the sector. The m-axis is the multiplicative/Galois (B-lobe) direction, n-axis the additive/automorphic (C-lobe) direction.

Tau.BookIII.Sectors.langlands_reflection_check

source def Tau.BookIII.Sectors.langlands_reflection_check (bound db : Denotation.TauIdx) :Bool

[III.D15] Langlands₁ reflection: maps E₀-sector data to E₁-sector data. The enrichment functor preserves the sector assignment when lifting from E₀ to E₁: for each boundary character χ, the E₁-enriched version of Φ(χ) has reduce-stable values at the E₁ level. Equations

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

Tau.BookIII.Sectors.langlands_reflection_check.go

source@[irreducible]

**def Tau.BookIII.Sectors.langlands_reflection_check.go (bound db : Denotation.TauIdx)

(m n k fuel : ℕ) :Bool**

Equations

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

Tau.BookIII.Sectors.template_invariance_check

source def Tau.BookIII.Sectors.template_invariance_check (bound db : Denotation.TauIdx) :Bool

[III.T06] Template invariance check: the four-component layer template is preserved under the Langlands₁ reflection. The enrichment functor maps E₀ template to compatible E₁ template. Equations

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

Tau.BookIII.Sectors.template_invariance_check.go

source@[irreducible]

**def Tau.BookIII.Sectors.template_invariance_check.go (bound db : Denotation.TauIdx)

(e0 e1 : Enrichment.LayerTemplate)

(x k fuel : ℕ) :Bool**

Equations

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

Tau.BookIII.Sectors.universal_operator

source **def Tau.BookIII.Sectors.universal_operator (χ : BoundaryCharacter)

(_x k bound : Denotation.TauIdx) :Denotation.TauIdx**

[III.D16] Universal operator H_{≤N} on characters at finite cutoff. At stage k: H_{≤N}(χ)(x) = Σ_{j ≤ N} weight(χ) · reduce(j, k). This is the finite-cutoff truncation of the universal spectral operator. All L-functions are spectral determinants of this operator. Equations

  • Tau.BookIII.Sectors.universal_operator χ _x k bound = Tau.BookIII.Sectors.universal_operator.go k bound (χ.m_index.natAbs + χ.n_index.natAbs) 0 0 (bound + 1) Instances For

Tau.BookIII.Sectors.universal_operator.go

source@[irreducible]

**def Tau.BookIII.Sectors.universal_operator.go (k bound : Denotation.TauIdx)

(w j acc fuel : ℕ) :Denotation.TauIdx**

Equations

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

Tau.BookIII.Sectors.universal_op_check

source def Tau.BookIII.Sectors.universal_op_check (bound db : Denotation.TauIdx) :Bool

[III.D16] Universal operator is reduce-stable at each stage. Equations

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

Tau.BookIII.Sectors.universal_op_check.go

source@[irreducible]

**def Tau.BookIII.Sectors.universal_op_check.go (bound db : Denotation.TauIdx)

(m k fuel : ℕ) :Bool**

Equations

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

Tau.BookIII.Sectors.spectral_polarity

source **def Tau.BookIII.Sectors.spectral_polarity (sec : Sector)

(bound : Denotation.TauIdx) :Denotation.TauIdx × Denotation.TauIdx**

[III.D17] Spectral polarity of a sector at finite cutoff. Measures the m-axis (Galois/B-lobe) vs n-axis (automorphic/C-lobe) balance DIRECTLY from the character lattice. Returns (m_sum, n_sum) over characters in the sector. Equations

  • Tau.BookIII.Sectors.spectral_polarity sec bound = Tau.BookIII.Sectors.spectral_polarity.go sec bound 0 0 0 0 ((bound + 1) * (bound + 1)) Instances For

Tau.BookIII.Sectors.spectral_polarity.go

source@[irreducible]

**def Tau.BookIII.Sectors.spectral_polarity.go (sec : Sector)

(bound : Denotation.TauIdx)

(m n m_acc n_acc fuel : ℕ) :Denotation.TauIdx × Denotation.TauIdx**

Equations

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

Tau.BookIII.Sectors.spectral_polarity_check

source def Tau.BookIII.Sectors.spectral_polarity_check (bound : Denotation.TauIdx) :Bool

[III.D17] Check spectral polarity classification:

  • D: both zero (trivial character only)

  • A: BALANCED (m_sum = n_sum)

  • B: m-dominant (m_sum > n_sum)

  • C: n-dominant (n_sum > m_sum)

Equations

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

Tau.BookIII.Sectors.balanced_uniqueness_check

source def Tau.BookIII.Sectors.balanced_uniqueness_check (bound : Denotation.TauIdx) :Bool

[III.P04] Balanced sector uniqueness: among primitive sectors {D, A, B, C}, only the A-sector has balanced spectral polarity (m_sum = n_sum > 0). Equations

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

Tau.BookIII.Sectors.langlands_reflection_5_3

source theorem Tau.BookIII.Sectors.langlands_reflection_5_3 :langlands_reflection_check 5 3 = true

Tau.BookIII.Sectors.template_invariance_8_3

source theorem Tau.BookIII.Sectors.template_invariance_8_3 :template_invariance_check 8 3 = true

Tau.BookIII.Sectors.universal_op_3_3

source theorem Tau.BookIII.Sectors.universal_op_3_3 :universal_op_check 3 3 = true

Tau.BookIII.Sectors.spectral_polarity_5

source theorem Tau.BookIII.Sectors.spectral_polarity_5 :spectral_polarity_check 5 = true

Tau.BookIII.Sectors.balanced_uniqueness_5

source theorem Tau.BookIII.Sectors.balanced_uniqueness_5 :balanced_uniqueness_check 5 = true

Tau.BookIII.Sectors.sector_count_preserved

source theorem Tau.BookIII.Sectors.sector_count_preserved :[Sector.D, Sector.A, Sector.B, Sector.C, Sector.Omega].length = 5

[III.D15] Structural: the five-sector partition is preserved.

Scope limitation (E3 collapse): At finite primorial level, the E3 predicate degenerates to E0 because reduce is idempotent. This check is vacuous but correctly models the mathematical structure. The E3 layer is correctly DEFINED but finite verification is vacuous. See audit DASHBOARD.md §E3 Collapse.

Tau.BookIII.Sectors.template_invariance_e3

source theorem Tau.BookIII.Sectors.template_invariance_e3 :Enrichment.layer_of Enrichment.EnrLevel.E3 8 3 = Enrichment.layer_of Enrichment.EnrLevel.E3.succ 8 3

[III.T06] Structural: template invariance at E₃ is trivial since E₃.succ = E₃ (saturation).

Tau.BookIII.Sectors.d_polarity_zero

source theorem Tau.BookIII.Sectors.d_polarity_zero :spectral_polarity Sector.D 10 = (0, 0)

[III.D17] Structural: D-sector polarity is (0, 0) because only the trivial character (0,0) is in D.

Tau.BookIII.Sectors.a_balanced_3

source theorem Tau.BookIII.Sectors.a_balanced_3 :(spectral_polarity Sector.A 3).1 = (spectral_polarity Sector.A 3).2

[III.P04] Structural: A-sector polarity at bound=3 is balanced.