TauLib · API Book III

TauLib.BookIII.Doors.MasterSchema

TauLib.BookIII.Doors.MasterSchema

Master Schema Theorem: all eight Millennium Problems as instances of Mutual Determination at varying enrichment levels.

Registry Cross-References

  • [III.T23] Master Schema Theorem – MasterSchemaEntry, master_schema_check

Mathematical Content

III.T23 (Master Schema): All eight Millennium Problems are instances of Mutual Determination at varying enrichment levels:

  • E₀: RH (Part IV), Poincaré (Part IV)

  • E₁: NS (Part V), YM (Part V), Hodge (Part V)

  • E₁→E₂: BSD (Part VI), Langlands (Part VI)

  • E₂: P vs NP (Part VII)

The spectral algebra provides the common language, the primorial ladder the common tower, the CRT the common local-global bridge.

Tau.BookIII.Doors.MillenniumProblem

source inductive Tau.BookIII.Doors.MillenniumProblem :Type

The eight Millennium Problems (including Langlands as the 8th force).

  • RH : MillenniumProblem
  • Poincare : MillenniumProblem
  • NS : MillenniumProblem
  • YM : MillenniumProblem
  • Hodge : MillenniumProblem
  • BSD : MillenniumProblem
  • Langlands : MillenniumProblem
  • PvsNP : MillenniumProblem Instances For

Tau.BookIII.Doors.instReprMillenniumProblem

source instance Tau.BookIII.Doors.instReprMillenniumProblem :Repr MillenniumProblem

Equations

  • Tau.BookIII.Doors.instReprMillenniumProblem = { reprPrec := Tau.BookIII.Doors.instReprMillenniumProblem.repr }

Tau.BookIII.Doors.instReprMillenniumProblem.repr

source def Tau.BookIII.Doors.instReprMillenniumProblem.repr :MillenniumProblem → ℕ → Std.Format

Equations

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

Tau.BookIII.Doors.instDecidableEqMillenniumProblem

source instance Tau.BookIII.Doors.instDecidableEqMillenniumProblem :DecidableEq MillenniumProblem

Equations

  • Tau.BookIII.Doors.instDecidableEqMillenniumProblem x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

Tau.BookIII.Doors.instBEqMillenniumProblem

source instance Tau.BookIII.Doors.instBEqMillenniumProblem :BEq MillenniumProblem

Equations

  • Tau.BookIII.Doors.instBEqMillenniumProblem = { beq := Tau.BookIII.Doors.instBEqMillenniumProblem.beq }

Tau.BookIII.Doors.instBEqMillenniumProblem.beq

source def Tau.BookIII.Doors.instBEqMillenniumProblem.beq :MillenniumProblem → MillenniumProblem → Bool

Equations

  • Tau.BookIII.Doors.instBEqMillenniumProblem.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx) Instances For

Tau.BookIII.Doors.problem_level

source def Tau.BookIII.Doors.problem_level (p : MillenniumProblem) :Enrichment.EnrLevel

[III.T23] Enrichment level assignment for each problem. Equations

  • Tau.BookIII.Doors.problem_level Tau.BookIII.Doors.MillenniumProblem.RH = Tau.BookIII.Enrichment.EnrLevel.E0
  • Tau.BookIII.Doors.problem_level Tau.BookIII.Doors.MillenniumProblem.Poincare = Tau.BookIII.Enrichment.EnrLevel.E0
  • Tau.BookIII.Doors.problem_level Tau.BookIII.Doors.MillenniumProblem.NS = Tau.BookIII.Enrichment.EnrLevel.E1
  • Tau.BookIII.Doors.problem_level Tau.BookIII.Doors.MillenniumProblem.YM = Tau.BookIII.Enrichment.EnrLevel.E1
  • Tau.BookIII.Doors.problem_level Tau.BookIII.Doors.MillenniumProblem.Hodge = Tau.BookIII.Enrichment.EnrLevel.E1
  • Tau.BookIII.Doors.problem_level Tau.BookIII.Doors.MillenniumProblem.BSD = Tau.BookIII.Enrichment.EnrLevel.E2
  • Tau.BookIII.Doors.problem_level Tau.BookIII.Doors.MillenniumProblem.Langlands = Tau.BookIII.Enrichment.EnrLevel.E2
  • Tau.BookIII.Doors.problem_level Tau.BookIII.Doors.MillenniumProblem.PvsNP = Tau.BookIII.Enrichment.EnrLevel.E2 Instances For

Tau.BookIII.Doors.problem_part

source def Tau.BookIII.Doors.problem_part (p : MillenniumProblem) :Denotation.TauIdx

[III.T23] Part assignment (which Book III part treats each problem). Equations

  • Tau.BookIII.Doors.problem_part Tau.BookIII.Doors.MillenniumProblem.RH = 4
  • Tau.BookIII.Doors.problem_part Tau.BookIII.Doors.MillenniumProblem.Poincare = 4
  • Tau.BookIII.Doors.problem_part Tau.BookIII.Doors.MillenniumProblem.NS = 5
  • Tau.BookIII.Doors.problem_part Tau.BookIII.Doors.MillenniumProblem.YM = 5
  • Tau.BookIII.Doors.problem_part Tau.BookIII.Doors.MillenniumProblem.Hodge = 5
  • Tau.BookIII.Doors.problem_part Tau.BookIII.Doors.MillenniumProblem.BSD = 6
  • Tau.BookIII.Doors.problem_part Tau.BookIII.Doors.MillenniumProblem.Langlands = 6
  • Tau.BookIII.Doors.problem_part Tau.BookIII.Doors.MillenniumProblem.PvsNP = 7 Instances For

Tau.BookIII.Doors.MasterSchemaEntry

source structure Tau.BookIII.Doors.MasterSchemaEntry :Type

[III.T23] Master Schema entry: each problem has a Mutual Determination instance (B ↔ I ↔ S) at its enrichment level. The finite check verifies that the MD infrastructure is available at the required level.

  • problem : MillenniumProblem
  • level : Enrichment.EnrLevel
  • part : Denotation.TauIdx
  • md_check : Denotation.TauIdx → Denotation.TauIdx → Bool Instances For

Tau.BookIII.Doors.rh_schema_check

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

[III.T23] RH schema: boundary = Euler product, interior = zeta values, spectral = H_L eigenvalues. Equations

  • Tau.BookIII.Doors.rh_schema_check _bound db = (Tau.BookIII.Doors.bipolar_euler_check db && Tau.BookIII.Doors.critical_line_multi_check db && Tau.BookIII.Doors.tau_effective_rh_check db) Instances For

Tau.BookIII.Doors.poincare_schema_check

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

[III.T23] Poincaré schema: boundary = local patches, interior = Hartogs bulk, spectral = fundamental group. Equations

  • Tau.BookIII.Doors.poincare_schema_check bound db = (Tau.BookIII.Doors.simply_connected_check bound db && Tau.BookIII.Doors.gluing_guarantee_check bound db) Instances For

Tau.BookIII.Doors.generic_schema_check

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

[III.T23] Generic E₁/E₂ schema: uses the full MD infrastructure at the appropriate enrichment level. The specific content for NS, YM, Hodge, BSD, Langlands, P vs NP is developed in Parts V-VII. At this level we verify the template is available. Equations

  • Tau.BookIII.Doors.generic_schema_check bound db = Tau.BookIII.Doors.mutual_det_check bound db Instances For

Tau.BookIII.Doors.master_schema_check

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

[III.T23] Master Schema: assemble all eight problem schemas. Equations

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

Tau.BookIII.Doors.level_coverage_check

source def Tau.BookIII.Doors.level_coverage_check :Bool

[III.T23] All eight problems have distinct enrichment levels covering the full E₀-E₂ range. Equations

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

Tau.BookIII.Doors.part_assignment_check

source def Tau.BookIII.Doors.part_assignment_check :Bool

[III.T23] Each problem maps to a unique part (4, 5, 6, or 7). Equations

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

Tau.BookIII.Doors.rh_schema_10_4

source theorem Tau.BookIII.Doors.rh_schema_10_4 :rh_schema_check 10 4 = true

Tau.BookIII.Doors.poincare_schema_10_3

source theorem Tau.BookIII.Doors.poincare_schema_10_3 :poincare_schema_check 10 3 = true

Tau.BookIII.Doors.master_schema_10_3

source theorem Tau.BookIII.Doors.master_schema_10_3 :master_schema_check 10 3 = true

Tau.BookIII.Doors.level_coverage

source theorem Tau.BookIII.Doors.level_coverage :level_coverage_check = true

Tau.BookIII.Doors.part_assignment

source theorem Tau.BookIII.Doors.part_assignment :part_assignment_check = true

Tau.BookIII.Doors.rh_level

source theorem Tau.BookIII.Doors.rh_level :problem_level MillenniumProblem.RH = Enrichment.EnrLevel.E0

[III.T23] Structural: RH is at E₀.

Tau.BookIII.Doors.pvsnp_level

source theorem Tau.BookIII.Doors.pvsnp_level :problem_level MillenniumProblem.PvsNP = Enrichment.EnrLevel.E2

[III.T23] Structural: P vs NP is at E₂.

Tau.BookIII.Doors.ns_level

source theorem Tau.BookIII.Doors.ns_level :problem_level MillenniumProblem.NS = Enrichment.EnrLevel.E1

[III.T23] Structural: NS is at E₁.

Tau.BookIII.Doors.rh_part

source theorem Tau.BookIII.Doors.rh_part :problem_part MillenniumProblem.RH = 4

[III.T23] Structural: all 8 problems have parts in {4,5,6,7}.

Tau.BookIII.Doors.ns_part

source theorem Tau.BookIII.Doors.ns_part :problem_part MillenniumProblem.NS = 5

Tau.BookIII.Doors.bsd_part

source theorem Tau.BookIII.Doors.bsd_part :problem_part MillenniumProblem.BSD = 6

Tau.BookIII.Doors.pvsnp_part

source theorem Tau.BookIII.Doors.pvsnp_part :problem_part MillenniumProblem.PvsNP = 7

Tau.BookIII.Doors.e0_before_e1

source theorem Tau.BookIII.Doors.e0_before_e1 :(problem_level MillenniumProblem.RH).toNat < (problem_level MillenniumProblem.NS).toNat

[III.T23] Structural: enrichment levels are ordered.

Tau.BookIII.Doors.e1_before_e2

source theorem Tau.BookIII.Doors.e1_before_e2 :(problem_level MillenniumProblem.NS).toNat < (problem_level MillenniumProblem.PvsNP).toNat