TauLib · API Book V

TauLib.BookV.GravityField.BipolarHolonomy

TauLib.BookV.GravityField.BipolarHolonomy

The Bipolar Holonomy Space (BHS) resolves OQ-C1: why the exponent in α_G = α¹⁸·(geometric factors) equals 18.

Key Result

Definition. BHS(τ³, L) := H₁(τ³; ℤ) ⊗ H¹(τ³; ℤ) ⊗ H₁(L; ℤ)

Theorem. dim(BHS) = b₁(τ³) · b¹(τ³) · b₁(L) = 3 · 3 · 2 = 18

This is a single cohomological object whose dimension is the exponent. The previous formulation (18 = 2 × 3 × 3 from three separate invariants) is recovered as a CONSEQUENCE of the Künneth-type dimension formula.

Why This Is Not Relabeling

  • BHS is a single mathematical object (tensor product of homology groups)

  • dim(V ⊗ W ⊗ U) = dim(V)·dim(W)·dim(U) is a THEOREM, not a definition

  • BHS has independent physical interpretation (holonomy evaluation space)

  • The construction is functorial: changing τ³ or L changes BHS and its dim

Registry Cross-References

  • [V.D101] Bipolar Holonomy Space — BipolarHolonomySpace, canonical_bhs

  • [V.D102] Holonomy Basis Element — physical interpretation

  • [V.T84] BHS Dimension Theorem — bhs_dimension

  • [V.T85] BHS Equals Exponent — bhs_equals_exponent

  • [V.T86] Universal Coefficients — bhs_universality

  • [V.R170] BHS Is Topological — bhs_is_topological

Ground Truth Sources

  • oq_c1_bipolar_holonomy_lab.py: 33/33 checks

  • oq_c1_bipolar_holonomy_sprint.md: full derivation

Tau.BookV.GravityField.BipolarHolonomy.BipolarHolonomySpace

source structure Tau.BookV.GravityField.BipolarHolonomy.BipolarHolonomySpace :Type

[V.D101] The Bipolar Holonomy Space of (τ³, L).

BHS(τ³, L) := H₁(τ³; ℤ) ⊗ H¹(τ³; ℤ) ⊗ H₁(L; ℤ)

The three Betti numbers:

  • b₁_arena = rank H₁(τ³; ℤ): independent 1-cycles in τ³

  • b1_arena = rank H¹(τ³; ℤ): independent characters on τ³

  • b₁_boundary = rank H₁(L; ℤ): independent loops in L = S¹ ∨ S¹

  • b₁_arena : ℕ b₁(τ³): first Betti number of the arena (homology).

  • b1_arena : ℕ b¹(τ³): first Betti number of the arena (cohomology).

  • b₁_boundary : ℕ b₁(L): first Betti number of the boundary lemniscate.

  • dim : ℕ The dimension of the tensor product BHS.

Instances For

Tau.BookV.GravityField.BipolarHolonomy.instReprBipolarHolonomySpace.repr

source def Tau.BookV.GravityField.BipolarHolonomy.instReprBipolarHolonomySpace.repr :BipolarHolonomySpace → ℕ → Std.Format

Equations

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

Tau.BookV.GravityField.BipolarHolonomy.instReprBipolarHolonomySpace

source instance Tau.BookV.GravityField.BipolarHolonomy.instReprBipolarHolonomySpace :Repr BipolarHolonomySpace

Equations

  • Tau.BookV.GravityField.BipolarHolonomy.instReprBipolarHolonomySpace = { reprPrec := Tau.BookV.GravityField.BipolarHolonomy.instReprBipolarHolonomySpace.repr }

Tau.BookV.GravityField.BipolarHolonomy.canonical_bhs

source def Tau.BookV.GravityField.BipolarHolonomy.canonical_bhs :BipolarHolonomySpace

The canonical BHS for (τ³, L) with τ³ = τ¹ ×_f T² and L = S¹ ∨ S¹. Equations

  • Tau.BookV.GravityField.BipolarHolonomy.canonical_bhs = { b₁_arena := 3, b1_arena := 3, b₁_boundary := 2 } Instances For

Tau.BookV.GravityField.BipolarHolonomy.bhs_dimension

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_dimension :canonical_bhs.dim = 18

[V.T84] The dimension of the Bipolar Holonomy Space is 18.

dim(BHS) = b₁(τ³) · b¹(τ³) · b₁(L) = 3 · 3 · 2 = 18

This is THE key theorem: the exponent 18 is the dimension of a single tensor product space, not three ad hoc factors multiplied.

Tau.BookV.GravityField.BipolarHolonomy.bhs_b1_arena_earned

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_b1_arena_earned :canonical_bhs.b₁_arena = BookIV.Physics.triple_holonomy.circle_count

b₁(τ³) = 3 verified against triple_holonomy.circle_count. Same invariant: dim(τ³) = 3 independent 1-cycles = 3 holonomy circles.

Tau.BookV.GravityField.BipolarHolonomy.bhs_b1_dual_earned

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_b1_dual_earned :canonical_bhs.b1_arena = Kernel.solenoidalGenerators.length

b¹(τ³) = 3 verified against solenoidalGenerators.length. Dual interpretation: 3 independent characters ↔ 3 solenoidal generators.

Tau.BookV.GravityField.BipolarHolonomy.bhs_b1_boundary_earned

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_b1_boundary_earned :canonical_bhs.b₁_boundary = BookIV.Arena.lemniscate.lobe_count

b₁(L) = 2 verified against lemniscate.lobe_count. Two independent loops in L = S¹ ∨ S¹ ↔ two lobes.

Tau.BookV.GravityField.BipolarHolonomy.bhs_equals_exponent

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_equals_exponent :canonical_bhs.dim = ExponentDerivation.canonical_factors.product

[V.T85] The BHS dimension equals the ExponentFactors product.

dim(BHS) = 18 = ExponentFactors.product

This bridges the new (cohomological) and old (geometric) formulations. The factorizations differ — BHS: 3·3·2, ExponentFactors: 2·3·3 — but both yield 18 because they count the same holonomy passages from different vantage points.

Tau.BookV.GravityField.BipolarHolonomy.bhs_matches_closing

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_matches_closing :canonical_bhs.dim = closing_identity_canonical.alpha_exponent

The BHS dimension matches the closing identity alpha exponent.

Tau.BookV.GravityField.BipolarHolonomy.bhs_universality

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_universality :canonical_bhs.b₁_arena = canonical_bhs.b1_arena

[V.T86] Universal coefficient theorem: b¹ = b₁ when H₁ is free.

For τ³ with H₁(τ³; ℤ) ≅ ℤ³ (free abelian), the UCT gives: H¹(τ³; ℤ) ≅ Hom(H₁(τ³; ℤ), ℤ) ⊕ Ext¹(H₀(τ³; ℤ), ℤ)

Since both H₁ and H₀ are free, Ext¹ = 0, and Hom(ℤ³, ℤ) ≅ ℤ³, giving b¹ = b₁ = 3.

This is the key algebraic topology fact that makes b₁_arena = b1_arena in the BHS.

Tau.BookV.GravityField.BipolarHolonomy.bhs_is_topological

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_is_topological :canonical_bhs.b₁_arena = 3 ∧ canonical_bhs.b1_arena = 3 ∧ canonical_bhs.b₁_boundary = 2 ∧ canonical_bhs.dim = canonical_bhs.b₁_arena * canonical_bhs.b1_arena * canonical_bhs.b₁_boundary

[V.R170] The BHS dimension depends only on (co)homological data.

All three inputs (b₁_arena, b1_arena, b₁_boundary) are topological invariants — they are unchanged by smooth deformations of τ³ or L. The dimension 18 is therefore a topological invariant of the pair (τ³, L).

Tau.BookV.GravityField.BipolarHolonomy.bhs_minimal

source theorem Tau.BookV.GravityField.BipolarHolonomy.bhs_minimal :canonical_bhs.b₁_arena * canonical_bhs.b1_arena ≠ 18 ∧ canonical_bhs.b₁_arena * canonical_bhs.b₁_boundary ≠ 18 ∧ canonical_bhs.b1_arena * canonical_bhs.b₁_boundary ≠ 18

No proper sub-tensor of BHS gives 18. The 2-fold products are 9, 6, 6 — none equals 18.