TauLib · API Book V

TauLib.BookV.Gravity.CoRotorCoupling

TauLib.BookV.Gravity.CoRotorCoupling

The co-rotor coupling distance κ_n on T² at the lemniscate crossing.

Registry Cross-References

  • [V.D10] Co-Rotor Coupling — CoRotorCoupling

  • [V.D11] Gravitational Closing Identity — canonical_coupling

  • [V.T04] κ_n Required Value — kn_required_range

  • [V.T05] c₁ = 3/π Identification — c1_matches_three_over_pi

  • [V.P02] Tree-Level Range — kn_tree_in_range

  • [V.R03] R Formula Independence — deficit_positive

Mathematical Content

The Co-Rotor Coupling Distance

On T² with shape ratio r/R = ι_τ, two co-rotating loops (toroidal and poloidal circles) interact at the lemniscate crossing L = S¹ ∨ S¹. The spectral coupling distance combines:

  • √3 from the three-fold sector structure: |1 - ω| = √3 where ω = e^{2πi/3} (Eisenstein cube root, see IV.T11)

  • Factor 2 from two independent rotation axes (b₁(T²) = 2)

Tree-level result: κ_n^{tree} = 2√3 = 3.4641…

The α-Order Correction

The physical coupling distance receives a first-order EM correction:

χ·κ_n/2 = √3 · (1 - c₁·α)

Sprint 4 numerical laboratory (224 candidates tested) identified:

c₁ = 3/π = 0.95493...

matching the required value 0.95453 to 0.04% (G deviation: 0.0003%).

Physical interpretation: three lemniscate sectors each contribute 1/π holonomy units to the correction, giving c₁ = 3 × (1/π) = 3/π.

Closing Identity

The gravitational fine-structure constant satisfies:

α_G = α¹⁸ · (χ·κ_n/2) = α¹⁸ · √3 · (1 - (3/π)·α)

This is the closing identity from Sprint 3 (SS1-SS6) with the correction identified in Sprint 4 (SS17).

Scope

  • The tree-level κ_n = 2√3 is conjectural

  • The correction c₁ = 3/π is conjectural (0.04% match)

  • The closing identity structure is tau-effective

  • The required value κ_n = 3.44 is established (CODATA arithmetic)

Ground Truth Sources

  • kappa_n_geometric_derivation_sprint.md (Sprint 4)

  • kappa_n_closing_identity_sprint.md (Sprint 3)

  • electron_mass_first_principles.md §28

Tau.BookV.Gravity.CoRotorCoupling

source structure Tau.BookV.Gravity.CoRotorCoupling :Type

[V.D10] Co-rotor coupling structure on T².

Encodes the tree-level coupling distance κ_n^{tree} = 2√3 and the α-order correction coefficient c₁.

The physical κ_n = κ_n^{tree} × (1 - c₁·α):

  • tree_factor = 2 (from b₁(T²) = 2, two rotation axes)

  • spectral_distance_sq = 3 (from 1-ω ² = 3, three-fold lemniscate)
  • correction_c1_numer/denom ≈ 3/π ≈ 0.95493

With these values and CODATA α: κ_n ≈ 2√3 × (1 - (3/π)·α) ≈ 3.4400 (0.0003% from CODATA G).

  • tree_factor : ℕ Tree-level factor (number of rotation axes on T²).

  • spectral_distance_sq : ℕ Spectral distance squared |1-ω|² at the crossing.

  • correction_c1_numer : ℕ c₁ numerator (rational approximation of 3/π).

  • correction_c1_denom : ℕ c₁ denominator.

  • denom_pos : self.correction_c1_denom > 0 Denominator positive.

  • scope : String Scope label.

Instances For

Tau.BookV.Gravity.instReprCoRotorCoupling.repr

source def Tau.BookV.Gravity.instReprCoRotorCoupling.repr :CoRotorCoupling → ℕ → Std.Format

Equations

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

Tau.BookV.Gravity.instReprCoRotorCoupling

source instance Tau.BookV.Gravity.instReprCoRotorCoupling :Repr CoRotorCoupling

Equations

  • Tau.BookV.Gravity.instReprCoRotorCoupling = { reprPrec := Tau.BookV.Gravity.instReprCoRotorCoupling.repr }

Tau.BookV.Gravity.c1_three_over_pi_numer

source def Tau.BookV.Gravity.c1_three_over_pi_numer :ℕ

3/π rational approximation (7 significant digits). 3/π = 0.9549296585… ≈ 9549297/10000000

Quality: 9549297 × π_denom ≈ 3 × 10000000 × π_denom (verified by range bounds below). Equations

  • Tau.BookV.Gravity.c1_three_over_pi_numer = 9549297 Instances For

Tau.BookV.Gravity.c1_three_over_pi_denom

source def Tau.BookV.Gravity.c1_three_over_pi_denom :ℕ

Equations

  • Tau.BookV.Gravity.c1_three_over_pi_denom = 10000000 Instances For

Tau.BookV.Gravity.canonical_coupling

source def Tau.BookV.Gravity.canonical_coupling :CoRotorCoupling

The canonical co-rotor coupling with c₁ = 3/π. Equations

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

Tau.BookV.Gravity.kn_required_numer

source def Tau.BookV.Gravity.kn_required_numer :ℕ

κ_n required value from the closing identity (rational approximation).

κ_n = 2 · α_G / α¹⁸ = 3.4399723239…

Using 8-digit approximation: 34399723/10000000. Equations

  • Tau.BookV.Gravity.kn_required_numer = 34399723 Instances For

Tau.BookV.Gravity.kn_required_denom

source def Tau.BookV.Gravity.kn_required_denom :ℕ

Equations

  • Tau.BookV.Gravity.kn_required_denom = 10000000 Instances For

Tau.BookV.Gravity.kn_required_range

source theorem Tau.BookV.Gravity.kn_required_range :3439 * kn_required_denom < 1000 * kn_required_numer ∧ 1000 * kn_required_numer < 3441 * kn_required_denom

[V.T04] κ_n is in the range (3.439, 3.441).

This range is established by CODATA arithmetic: κ_n = 2 · G · m_n² / (α¹⁸ · ℏc) is fixed by measured constants.

34399 × 10000 < 10000 × 34399723 < 34410 × 10000.

Tau.BookV.Gravity.kn_tree_numer

source def Tau.BookV.Gravity.kn_tree_numer :ℕ

κ_n^{tree} = 2√3 rational approximation. 2√3 = 3.4641016… ≈ 2 × 17320508/10000000. Equations

  • Tau.BookV.Gravity.kn_tree_numer = 2 * Tau.BookIV.Physics.sqrt3_numer Instances For

Tau.BookV.Gravity.kn_tree_denom

source def Tau.BookV.Gravity.kn_tree_denom :ℕ

Equations

  • Tau.BookV.Gravity.kn_tree_denom = Tau.BookIV.Physics.sqrt3_denom Instances For

Tau.BookV.Gravity.kn_tree_in_range

source theorem Tau.BookV.Gravity.kn_tree_in_range :3464 * kn_tree_denom < 1000 * kn_tree_numer ∧ 1000 * kn_tree_numer < 3465 * kn_tree_denom

[V.P02] The tree-level κ_n = 2√3 is in range (3.464, 3.465).

Tau.BookV.Gravity.tree_exceeds_required

source theorem Tau.BookV.Gravity.tree_exceeds_required :kn_tree_numer * kn_required_denom > kn_required_numer * kn_tree_denom

Tree level exceeds required value. 2√3 > κ_n(required), confirming the correction is negative.

Tau.BookV.Gravity.c1_target_numer

source def Tau.BookV.Gravity.c1_target_numer :ℕ

The c₁ target from the deficit analysis (rational approximation). c₁ = (√3 - χκ_n/2) / (α·√3) = 0.9545278697… ≈ 9545279/10000000. Equations

  • Tau.BookV.Gravity.c1_target_numer = 9545279 Instances For

Tau.BookV.Gravity.c1_target_denom

source def Tau.BookV.Gravity.c1_target_denom :ℕ

Equations

  • Tau.BookV.Gravity.c1_target_denom = 10000000 Instances For

Tau.BookV.Gravity.c1_matches_three_over_pi

source theorem Tau.BookV.Gravity.c1_matches_three_over_pi :c1_three_over_pi_numer < c1_target_numer + 5000 ∧ c1_target_numer < c1_three_over_pi_numer + 5000

[V.T05] c₁ = 3/π matches the target to better than 0.05%.

c₁(3/π) - c₁(target) < 5000/10000000 = 0.0005

Verified: |9549297 - 9545279| = 4018 < 5000. Relative error: 4018/9545279 ≈ 0.042% < 0.05%.

Tau.BookV.Gravity.c1_in_range

source theorem Tau.BookV.Gravity.c1_in_range :954 * c1_three_over_pi_denom < 1000 * c1_three_over_pi_numer ∧ 1000 * c1_three_over_pi_numer < 956 * c1_three_over_pi_denom

c₁ = 3/π is in range (0.954, 0.956).

Tau.BookV.Gravity.canonical_spectral_distance

source theorem Tau.BookV.Gravity.canonical_spectral_distance :canonical_coupling.spectral_distance_sq = 3

The tree-level coupling has spectral distance² = 3 (from 1-ω ² = 3).

Tau.BookV.Gravity.canonical_tree_factor

source theorem Tau.BookV.Gravity.canonical_tree_factor :canonical_coupling.tree_factor = 2

The tree-level coupling has factor 2 (from b₁(T²) = 2).

Tau.BookV.Gravity.canonical_denom_pos

source theorem Tau.BookV.Gravity.canonical_denom_pos :canonical_coupling.correction_c1_denom > 0

The correction denominator is positive.

Tau.BookV.Gravity.deficit_positive

source theorem Tau.BookV.Gravity.deficit_positive :kn_tree_numer * kn_required_denom > kn_required_numer * kn_tree_denom

The deficit is positive: 2√3 > κ_n(required), so the correction reduces the tree-level value.