TauLib · API Book III

TauLib.BookIII.Physics.PhysicsAssembly

TauLib.BookIII.Physics.PhysicsAssembly

Physics Layer Assembly: NS + YM + Hodge = E₁ Complete.

Registry Cross-References

  • [III.T29] Physics Assembly – physics_assembly_check

Mathematical Content

III.T29 (Physics Assembly): The three E₁ Millennium Problems form a complete description of the physics layer:

  • Navier-Stokes (existence): Hartogs flow stabilizes → positive regularity

  • Yang-Mills (mass gap): B/C coprimality → spectral gap → mass existence

  • Hodge (addressability): BNF uniqueness + sector decomposition → NF-addressability

Together, these exhaust the E₁ content: every E₁ object has admissible fluid data (NS), carries a mass gap (YM), and is NF-addressable (Hodge).

The assembly theorem verifies that all three components are simultaneously satisfied at each primorial level, confirming E₁ completeness.

Tau.BookIII.Physics.ns_component_check

source def Tau.BookIII.Physics.ns_component_check (bound db : Denotation.TauIdx) :Bool

[III.T29] NS component: fluid data admissibility + flow stabilization + positive regularity. Equations

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

Tau.BookIII.Physics.ym_component_check

source def Tau.BookIII.Physics.ym_component_check (db : Denotation.TauIdx) :Bool

[III.T29] YM component: strong sector + mass gap + coupling well-defined. Equations

  • Tau.BookIII.Physics.ym_component_check db = (Tau.BookIII.Physics.strong_sector_check db && Tau.BookIII.Physics.yang_mills_gap_check db && Tau.BookIII.Physics.ym_coupling_check db) Instances For

Tau.BookIII.Physics.hodge_component_check

source def Tau.BookIII.Physics.hodge_component_check (bound db : Denotation.TauIdx) :Bool

[III.T29] Hodge component: NF-addressability + Hodge filtration + spectral compatibility. Equations

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

Tau.BookIII.Physics.physics_assembly_check

source def Tau.BookIII.Physics.physics_assembly_check (bound db : Denotation.TauIdx) :Bool

[III.T29] Physics assembly: all three E₁ components are satisfied simultaneously at each primorial level. Equations

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

Tau.BookIII.Physics.e1_completeness_check

source def Tau.BookIII.Physics.e1_completeness_check (db : Denotation.TauIdx) :Bool

[III.T29] E₁ completeness: the three problems cover distinct aspects. NS = dynamics (flow), YM = spectrum (gap), Hodge = addressing (filtration). Verify non-redundancy: each component checks something the others don’t. Equations

  • Tau.BookIII.Physics.e1_completeness_check db = Tau.BookIII.Physics.e1_completeness_check.go db 3 (db + 1) Instances For

Tau.BookIII.Physics.e1_completeness_check.go

source@[irreducible]

**def Tau.BookIII.Physics.e1_completeness_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.Physics.physics_is_e1

source def Tau.BookIII.Physics.physics_is_e1 :Bool

[III.T29] Enrichment level assignment: all three problems are at E₁. Equations

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

Tau.BookIII.Physics.physics_in_part5

source def Tau.BookIII.Physics.physics_in_part5 :Bool

[III.T29] Part assignment: all three problems are in Part V. Equations

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

Tau.BookIII.Physics.ns_component_15_4

source theorem Tau.BookIII.Physics.ns_component_15_4 :ns_component_check 15 4 = true

Tau.BookIII.Physics.ym_component_5

source theorem Tau.BookIII.Physics.ym_component_5 :ym_component_check 5 = true

Tau.BookIII.Physics.hodge_component_15_3

source theorem Tau.BookIII.Physics.hodge_component_15_3 :hodge_component_check 15 3 = true

Tau.BookIII.Physics.physics_assembly_15_3

source theorem Tau.BookIII.Physics.physics_assembly_15_3 :physics_assembly_check 15 3 = true

Tau.BookIII.Physics.e1_completeness_5

source theorem Tau.BookIII.Physics.e1_completeness_5 :e1_completeness_check 5 = true

Tau.BookIII.Physics.physics_is_e1_thm

source theorem Tau.BookIII.Physics.physics_is_e1_thm :physics_is_e1 = true

Tau.BookIII.Physics.physics_in_part5_thm

source theorem Tau.BookIII.Physics.physics_in_part5_thm :physics_in_part5 = true

Tau.BookIII.Physics.ns_at_e1

source theorem Tau.BookIII.Physics.ns_at_e1 :Doors.problem_level Doors.MillenniumProblem.NS = Enrichment.EnrLevel.E1

[III.T29] Structural: NS is at E₁.

Tau.BookIII.Physics.ym_at_e1

source theorem Tau.BookIII.Physics.ym_at_e1 :Doors.problem_level Doors.MillenniumProblem.YM = Enrichment.EnrLevel.E1

[III.T29] Structural: YM is at E₁.

Tau.BookIII.Physics.hodge_at_e1

source theorem Tau.BookIII.Physics.hodge_at_e1 :Doors.problem_level Doors.MillenniumProblem.Hodge = Enrichment.EnrLevel.E1

[III.T29] Structural: Hodge is at E₁.

Tau.BookIII.Physics.ns_part5

source theorem Tau.BookIII.Physics.ns_part5 :Doors.problem_part Doors.MillenniumProblem.NS = 5

[III.T29] Structural: all three problems in Part V.

Tau.BookIII.Physics.ym_part5

source theorem Tau.BookIII.Physics.ym_part5 :Doors.problem_part Doors.MillenniumProblem.YM = 5

Tau.BookIII.Physics.hodge_part5

source theorem Tau.BookIII.Physics.hodge_part5 :Doors.problem_part Doors.MillenniumProblem.Hodge = 5

Tau.BookIII.Physics.three_e1_problems

source theorem Tau.BookIII.Physics.three_e1_problems :[Doors.problem_level Doors.MillenniumProblem.NS, Doors.problem_level Doors.MillenniumProblem.YM, Doors.problem_level Doors.MillenniumProblem.Hodge] = [Enrichment.EnrLevel.E1, Enrichment.EnrLevel.E1, Enrichment.EnrLevel.E1]

[III.T29] Structural: the three E₁ problems exhaust the E₁ level.