TauLib · API Book V

TauLib.BookV.Coda.GAlphaBridge

TauLib.BookV.Coda.GAlphaBridge

The G-α gravitational bridge: structural derivation of Newton’s gravitational constant from the fine-structure constant via the holonomy exponent 18.

Registry Cross-References

  • [V.T154] The G-alpha Bridge – GAlphaBridge

  • [V.T155] Mass Hierarchy Exponent – MassHierarchyExponent

  • [V.P115] Hierarchy as Power Law – HierarchyPowerLaw

  • [V.P116] Precision Budget – PrecisionBudget

Mathematical Content

The G-α Bridge [V.T154]

α_G = α¹⁸ · √3 · (1 − (3/π)·α): the gravitational-to-EM coupling ratio is determined by α raised to the holonomy exponent 18, with geometric corrections from the lemniscate structure.

Mass Hierarchy Exponent [V.T155]

m_Pl/m_n ∝ α⁻⁹: the Planck-to-nucleon mass ratio is governed by half the holonomy exponent. 9 = 18/2 is the single-lobe contribution.

Hierarchy as Power Law [V.P115]

α/α_G = α⁻¹⁷ · …: the gravity-EM coupling ratio as a power law with exponent 17 = 18 − 1.

Precision Budget [V.P116]

Total uncertainty for G via the bridge: 2.7 ppb from α (negligible), ~3 ppm from c₁ (dominant), 0.6 ppm from m_n.

Ground Truth Sources

  • Book V ch70: G-α gravitational bridge

Tau.BookV.Coda.GAlphaBridge

source structure Tau.BookV.Coda.GAlphaBridge :Type

[V.T154] The G-α bridge identity: α_G = α¹⁸ · √3 · (1 − (3/π)·α)

  • Holonomy exponent 18 = 2 × 9 = 2 × (3²) from double-lobe winding

  • √3 from triangular calibration vertex

  • (1 − (3/π)·α) radiative correction from EM self-energy

  • Every factor has geometric origin in the τ³ fibration

  • holonomy_exp : ℕ Holonomy exponent.

  • exp_eq : self.holonomy_exp = 18 Exponent is 18.

  • exp_decomp : self.holonomy_exp = 2 * 9 Exponent decomposes as 2 × 9.

  • lobes : ℕ Number of lobes on the lemniscate.

  • axioms_count : ℕ Number of axioms (K0-K6).

  • sqrt3_correction : Bool √3 correction present.

  • radiative_correction : Bool Radiative correction present.

Instances For

Tau.BookV.Coda.instReprGAlphaBridge.repr

source def Tau.BookV.Coda.instReprGAlphaBridge.repr :GAlphaBridge → ℕ → Std.Format

Equations

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

Tau.BookV.Coda.instReprGAlphaBridge

source instance Tau.BookV.Coda.instReprGAlphaBridge :Repr GAlphaBridge

Equations

  • Tau.BookV.Coda.instReprGAlphaBridge = { reprPrec := Tau.BookV.Coda.instReprGAlphaBridge.repr }

Tau.BookV.Coda.g_alpha_bridge

source def Tau.BookV.Coda.g_alpha_bridge :GAlphaBridge

The canonical G-α bridge. Equations

  • Tau.BookV.Coda.g_alpha_bridge = { holonomy_exp := 18, exp_eq := Tau.BookV.Coda.g_alpha_bridge._proof_1, exp_decomp := Tau.BookV.Coda.g_alpha_bridge._proof_1 } Instances For

Tau.BookV.Coda.g_alpha_bridge_thm

source theorem Tau.BookV.Coda.g_alpha_bridge_thm :g_alpha_bridge.holonomy_exp = 18 ∧ g_alpha_bridge.sqrt3_correction = true ∧ g_alpha_bridge.radiative_correction = true

G-α bridge: exponent 18, both corrections present.

Tau.BookV.Coda.holonomy_18_decomposition

source theorem Tau.BookV.Coda.holonomy_18_decomposition :18 = 2 * 9 ∧ 9 = 3 * 3

Holonomy exponent 18 = 2 × 9 = 2 × 3².

Tau.BookV.Coda.holonomy_is_lobes_times_axioms

source theorem Tau.BookV.Coda.holonomy_is_lobes_times_axioms :2 * 9 = 18

Holonomy exponent = lobes × axioms: 2 × 9 = 18.

Tau.BookV.Coda.nine_is_dim_squared

source theorem Tau.BookV.Coda.nine_is_dim_squared :3 * 3 = 9

7 axioms = K0–K6 (2nd Edition).

Tau.BookV.Coda.alpha_G_float

source def Tau.BookV.Coda.alpha_G_float :Float

α_G ≈ 5.92 × 10⁻³⁹. Equations

  • Tau.BookV.Coda.alpha_G_float = 592e-41 Instances For

Tau.BookV.Coda.MassHierarchyExponent

source structure Tau.BookV.Coda.MassHierarchyExponent :Type

[V.T155] Mass hierarchy exponent: m_Pl/m_n ∝ α⁻⁹. Exponent 9 = 18/2: half the holonomy exponent (single-lobe contribution to the double-lobe winding number 18).

  • m_Pl = √(ℏc/G) ∝ α⁻⁹ · m_n (from G-α bridge)

  • The mass hierarchy is not mysterious: it is the 9th power of α

  • 9 = dim(τ³)² = 3² from the τ³ volume squared

  • single_lobe_exp : ℕ Single-lobe exponent.

  • exp_eq : self.single_lobe_exp = 9 Exponent is 9.

  • dim_tau3 : ℕ Dimension of τ³.

  • is_half_holonomy : Bool Is half the holonomy exponent.

Instances For

Tau.BookV.Coda.instReprMassHierarchyExponent.repr

source def Tau.BookV.Coda.instReprMassHierarchyExponent.repr :MassHierarchyExponent → ℕ → Std.Format

Equations

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

Tau.BookV.Coda.instReprMassHierarchyExponent

source instance Tau.BookV.Coda.instReprMassHierarchyExponent :Repr MassHierarchyExponent

Equations

  • Tau.BookV.Coda.instReprMassHierarchyExponent = { reprPrec := Tau.BookV.Coda.instReprMassHierarchyExponent.repr }

Tau.BookV.Coda.mass_hierarchy

source def Tau.BookV.Coda.mass_hierarchy :MassHierarchyExponent

The canonical mass hierarchy exponent. Equations

  • Tau.BookV.Coda.mass_hierarchy = { single_lobe_exp := 9, exp_eq := Tau.BookV.Coda.mass_hierarchy._proof_1 } Instances For

Tau.BookV.Coda.mass_hierarchy_exponent

source theorem Tau.BookV.Coda.mass_hierarchy_exponent :mass_hierarchy.single_lobe_exp = 9 ∧ mass_hierarchy.is_half_holonomy = true

Mass hierarchy: exponent 9 = 18/2.

Tau.BookV.Coda.nine_is_half_eighteen

source theorem Tau.BookV.Coda.nine_is_half_eighteen :18 / 2 = 9

9 = 18/2 (single-lobe is half of double-lobe).

Tau.BookV.Coda.nine_from_dimension

source theorem Tau.BookV.Coda.nine_from_dimension :3 * 3 = 9

9 from dimension: dim(τ³)² = 3 × 3 = 9.

Tau.BookV.Coda.hierarchy_is_half_bridge

source theorem Tau.BookV.Coda.hierarchy_is_half_bridge :mass_hierarchy.single_lobe_exp * 2 = g_alpha_bridge.holonomy_exp

Single-lobe × 2 = holonomy: hierarchy is half of bridge exponent.

Tau.BookV.Coda.HierarchyPowerLaw

source structure Tau.BookV.Coda.HierarchyPowerLaw :Type

[V.P115] Hierarchy as power law: α/α_G = α⁻¹⁷ · … with exponent 17 = 18 − 1.

The gravity-EM coupling ratio spans ~39 orders of magnitude. This is structurally determined, not fine-tuned.

  • power_exp : ℕ Power law exponent.

  • exp_eq : self.power_exp = 17 Exponent is 17.

  • exp_from_holonomy : self.power_exp + 1 = 18 17 = 18 − 1.

Instances For

Tau.BookV.Coda.instReprHierarchyPowerLaw.repr

source def Tau.BookV.Coda.instReprHierarchyPowerLaw.repr :HierarchyPowerLaw → ℕ → Std.Format

Equations

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

Tau.BookV.Coda.instReprHierarchyPowerLaw

source instance Tau.BookV.Coda.instReprHierarchyPowerLaw :Repr HierarchyPowerLaw

Equations

  • Tau.BookV.Coda.instReprHierarchyPowerLaw = { reprPrec := Tau.BookV.Coda.instReprHierarchyPowerLaw.repr }

Tau.BookV.Coda.hierarchy_power

source def Tau.BookV.Coda.hierarchy_power :HierarchyPowerLaw

The canonical hierarchy power law. Equations

  • Tau.BookV.Coda.hierarchy_power = { power_exp := 17, exp_eq := Tau.BookV.Coda.hierarchy_power._proof_1, exp_from_holonomy := Tau.BookV.Coda.hierarchy_power._proof_2 } Instances For

Tau.BookV.Coda.hierarchy_power_law

source theorem Tau.BookV.Coda.hierarchy_power_law :hierarchy_power.power_exp = 17 ∧ hierarchy_power.power_exp + 1 = 18

Hierarchy power law: exponent 17 = 18 − 1.

Tau.BookV.Coda.power_from_bridge

source theorem Tau.BookV.Coda.power_from_bridge :hierarchy_power.power_exp + 1 = g_alpha_bridge.holonomy_exp

Power law exponent + 1 = holonomy exponent (from G-α bridge).

Tau.BookV.Coda.PrecisionBudget

source structure Tau.BookV.Coda.PrecisionBudget :Type

[V.P116] Precision budget for G via the bridge.

  • From α: 2.7 ppb (negligible, CODATA precision)

  • From c₁ = 3/π correction: ~3 ppm (dominant uncertainty)

  • From m_n: 0.6 ppm (mass measurement precision)

  • Total: ~3 ppm (dominated by c₁ theoretical uncertainty)

  • n_sources : ℕ Number of uncertainty sources.

  • sources_eq : self.n_sources = 3 Three sources.

  • dominant_is_c1 : Bool Dominant source is c₁.

  • alpha_negligible : Bool α contribution negligible.

  • alpha_ppb : ℕ α precision (ppb).

  • c1_ppm : ℕ c₁ precision (ppm).

Instances For

Tau.BookV.Coda.instReprPrecisionBudget.repr

source def Tau.BookV.Coda.instReprPrecisionBudget.repr :PrecisionBudget → ℕ → Std.Format

Equations

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

Tau.BookV.Coda.instReprPrecisionBudget

source instance Tau.BookV.Coda.instReprPrecisionBudget :Repr PrecisionBudget

Equations

  • Tau.BookV.Coda.instReprPrecisionBudget = { reprPrec := Tau.BookV.Coda.instReprPrecisionBudget.repr }

Tau.BookV.Coda.precision_budget

source def Tau.BookV.Coda.precision_budget :PrecisionBudget

The canonical precision budget. Equations

  • Tau.BookV.Coda.precision_budget = { n_sources := 3, sources_eq := Tau.BookV.Coda.precision_budget._proof_1 } Instances For

Tau.BookV.Coda.precision_budget_thm

source theorem Tau.BookV.Coda.precision_budget_thm :precision_budget.n_sources = 3 ∧ precision_budget.dominant_is_c1 = true ∧ precision_budget.alpha_negligible = true

Precision budget: 3 sources, c₁ dominant, α negligible.