Gravitational Coupling from Fζ Hierarchy

This module derives the gravitational coupling constant from Fζ hierarchy.

Main Results

The electron-to-Planck mass ratio: m_e / m_Pl = φ^(-107 - 5/63)

Where:

• 107 = C(5,2) × (C(5,2)+1) - C(3,2) = 10 × 11 - 3
• 5/63 = activeDLevels / (C(3,2) × C(7,2))

Physical Interpretation

Gravity emerges from the complete Fζ hierarchy through:

• Lepton mass structure (107)
• Electromagnetic structure (5 = active D-levels)
• Weak structure (3 = C(3,2))
• Full N₆ hierarchy (21 = C(7,2))

Pair Counts C(m,2)

theorem FUST.GravitationalCoupling.C32_eq : Nat.choose 3 2 = 3

theorem FUST.GravitationalCoupling.C52_eq : Nat.choose 5 2 = 10

theorem FUST.GravitationalCoupling.C62_eq : Nat.choose 6 2 = 15

theorem FUST.GravitationalCoupling.C72_eq : Nat.choose 7 2 = 21

Lepton Mass Exponent: 107

107 = p₃ + e₃ + d = Σ L(k)³ + Π L(k) + kerDim(N₆) = 92 + 12 + 3.

theorem FUST.GravitationalCoupling.leptonMassExponent_eq : Nat.choose 5 2 * (Nat.choose 5 2 + 1) - Nat.choose 3 2 = 107

107 = C(5,2) × (C(5,2)+1) - C(3,2) = 10 × 11 - 3

Fractional Correction: 5/63

theorem FUST.GravitationalCoupling.gravityCorrectionDenom_eq : Nat.choose 3 2 * Nat.choose 7 2 = 63

Denominator 63 = C(3,2) × T(6) = 3 × 21

theorem FUST.GravitationalCoupling.alpha_exponent_eq : 6 - 2 + 1 = 5

Numerator 5 = active D-levels = 6 - 2 + 1

@[reducible, inline]

abbrev FUST.GravitationalCoupling.leptonExponent : ℕ

Lepton exponent: C(5,2) × (C(5,2)+1) - C(3,2) = 10 × 11 - 3 = 107

Equations

• FUST.GravitationalCoupling.leptonExponent = Nat.choose 5 2 * (Nat.choose 5 2 + 1) - Nat.choose 3 2

@[reducible, inline]

abbrev FUST.GravitationalCoupling.activeDLevels : ℕ

Active D-levels = 6 - 2 + 1 = 5

Equations

• FUST.GravitationalCoupling.activeDLevels = 6 - 2 + 1

@[reducible, inline]

abbrev FUST.GravitationalCoupling.correctionDenom : ℕ

Correction denominator = C(3,2) × C(7,2) = 3 × 21 = 63

Equations

• FUST.GravitationalCoupling.correctionDenom = Nat.choose 3 2 * Nat.choose 7 2

@[reducible, inline]

noncomputable abbrev FUST.GravitationalCoupling.gravitationalExponent : ℝ

Full gravitational exponent: -(leptonExponent + activeDLevels/correctionDenom)

Equations

• FUST.GravitationalCoupling.gravitationalExponent = -(↑FUST.GravitationalCoupling.leptonExponent + ↑FUST.GravitationalCoupling.activeDLevels / ↑FUST.GravitationalCoupling.correctionDenom)

theorem FUST.GravitationalCoupling.gravitationalExponent_derivation : leptonExponent = 107 ∧ activeDLevels = 5 ∧ correctionDenom = 63

Gravitational exponent derivation

@[reducible, inline]

noncomputable abbrev FUST.GravitationalCoupling.electronPlanckRatio : ℝ

The electron-to-Planck mass ratio formula

Equations

• FUST.GravitationalCoupling.electronPlanckRatio = FUST.φ ^ FUST.GravitationalCoupling.gravitationalExponent

theorem FUST.GravitationalCoupling.electronPlanckRatio_eq : electronPlanckRatio = φ ^ (-(107 + 5 / 63))

Gravitational Coupling Constant

@[reducible, inline]

noncomputable abbrev FUST.GravitationalCoupling.gravitationalCoupling : ℝ

Gravitational coupling α_G = (m_e/m_Pl)²

Equations

• FUST.GravitationalCoupling.gravitationalCoupling = FUST.GravitationalCoupling.electronPlanckRatio ^ 2

theorem FUST.GravitationalCoupling.gravitationalCoupling_exponent : gravitationalCoupling = φ ^ (-(214 + 10 / 63))

D-Hierarchy Pair Counts

theorem FUST.GravitationalCoupling.totalDHierarchyPairs_eq : Nat.choose 2 2 + Nat.choose 3 2 + Nat.choose 4 2 + Nat.choose 5 2 + Nat.choose 6 2 = 35

theorem FUST.GravitationalCoupling.D_structure_pairs : Nat.choose 2 2 = 1 ∧ Nat.choose 3 2 = 3 ∧ Nat.choose 4 2 = 6 ∧ Nat.choose 5 2 = 10 ∧ Nat.choose 6 2 = 15

CMB Temperature: 152

T_CMB/T_Pl = φ^(-152). Decomposition: 152 = 107 + 45. φ^(-107) ≈ m_e/m_Pl (mass scale), φ^(-45) = T_CMB/m_e (thermal factor). Both terms are dimensionless exponents. 152 = 2 × L(9).

@[reducible, inline]

abbrev FUST.GravitationalCoupling.cmbDecouplingFactor : ℕ

CMB decoupling factor = C(3,2) × C(6,2) = 3 × 15 = 45

Equations

• FUST.GravitationalCoupling.cmbDecouplingFactor = Nat.choose 3 2 * Nat.choose 6 2

theorem FUST.GravitationalCoupling.cmbDecouplingFactor_eq : Nat.choose 3 2 * Nat.choose 6 2 = 45

CMB decoupling: C(3,2)×C(6,2) = 45

@[reducible, inline]

abbrev FUST.GravitationalCoupling.cmbTemperatureExponent : ℕ

CMB temperature exponent = leptonExponent + cmbDecouplingFactor

Equations

• FUST.GravitationalCoupling.cmbTemperatureExponent = FUST.GravitationalCoupling.leptonExponent + FUST.GravitationalCoupling.cmbDecouplingFactor

theorem FUST.GravitationalCoupling.cmbTemperatureExponent_eq : leptonExponent + cmbDecouplingFactor = 152

CMB exponent: 107 + 45 = 152

theorem FUST.GravitationalCoupling.cmbTemperatureExponent_value : cmbTemperatureExponent = 152

@[reducible, inline]

noncomputable abbrev FUST.GravitationalCoupling.cmbTemperatureRatio : ℝ

Equations

• FUST.GravitationalCoupling.cmbTemperatureRatio = FUST.φ ^ (-↑FUST.GravitationalCoupling.cmbTemperatureExponent)

Cosmological Constant: 582

ρ_Λ/ρ_Pl = φ^(-582). Stefan-Boltzmann ρ ∝ T⁴: φ^(-582) = (T_CMB/T_Pl)⁴ × φ^26 582 = 4 × 152 - 26 where 26 = Σ L(k)² (sector trace squares).

@[reducible, inline]

abbrev FUST.GravitationalCoupling.sectorTraceSq : ℕ

Sector trace square sum (ℕ version)

Equations

• FUST.GravitationalCoupling.sectorTraceSq = 1 ^ 2 + 3 ^ 2 + 4 ^ 2

theorem FUST.GravitationalCoupling.sectorTraceSq_eq : sectorTraceSq = 26

@[reducible, inline]

abbrev FUST.GravitationalCoupling.cosmologicalExponent : ℕ

Cosmological exponent: 4 × 152 - 26 = 582

Equations

• FUST.GravitationalCoupling.cosmologicalExponent = 4 * FUST.GravitationalCoupling.cmbTemperatureExponent - FUST.GravitationalCoupling.sectorTraceSq

theorem FUST.GravitationalCoupling.cosmologicalExponent_value : cosmologicalExponent = 582

@[reducible, inline]

noncomputable abbrev FUST.GravitationalCoupling.cosmologicalDensityRatio : ℝ

Equations

• FUST.GravitationalCoupling.cosmologicalDensityRatio = FUST.φ ^ (-↑FUST.GravitationalCoupling.cosmologicalExponent)

Summary Theorem

theorem FUST.GravitationalCoupling.gravitational_coupling_summary : Nat.choose 5 2 * (Nat.choose 5 2 + 1) - Nat.choose 3 2 = 107 ∧ 6 - 2 + 1 = 5 ∧ Nat.choose 3 2 * Nat.choose 7 2 = 63 ∧ Nat.choose 2 2 + Nat.choose 3 2 + Nat.choose 4 2 + Nat.choose 5 2 + Nat.choose 6 2 = 35

theorem FUST.GravitationalCoupling.cosmological_summary : cmbTemperatureExponent = 152 ∧ cosmologicalExponent = 582 ∧ cmbDecouplingFactor = 45 ∧ sectorTraceSq = 26

N₆ Coefficient Structure

The N₆ coefficients [1, -3, 1, -1, 3, -1] satisfy:

• Signed sum = 0 (kills constants)
• Absolute sum = 10 = C(5,2)
• Positive/negative part sums = 5 = activeLevels
• Evaluation point sum φ³+φ²+φ+ψ+ψ²+ψ³ = 8 = L(1)+L(2)+L(3)

theorem FUST.GravitationalCoupling.Diff6_coeff_sum : 1 + -3 + 1 + -1 + 3 + -1 = 0

theorem FUST.GravitationalCoupling.Diff6_coeff_abs_sum : 1 + 3 + 1 + 1 + 3 + 1 = Nat.choose 5 2

theorem FUST.GravitationalCoupling.Diff6_coeff_positive_sum : 1 + 1 + 3 = 6 - 2 + 1

theorem FUST.GravitationalCoupling.Diff6_coeff_negative_sum : 3 + 1 + 1 = 6 - 2 + 1

theorem FUST.GravitationalCoupling.Diff6_eval_multiplier_sum : φ ^ 3 + φ ^ 2 + φ + ψ + ψ ^ 2 + ψ ^ 3 = 8

Sum of N₆ evaluation multipliers: φ³+φ²+φ+ψ+ψ²+ψ³ = 8

N₆ Spectral Invariants

The N₆ evaluation multipliers {φ³,φ²,φ,ψ,ψ²,ψ³} have elementary symmetric polynomials:

• e₁ = 8 (trace, proven above as Diff6_eval_multiplier_sum)
• e₂ = 18 (pairwise products)
• e₃ = 6 (triple products)
• e₄ = -12 (4-tuple products)
• e₅ = -2 (5-tuple products)
• e₆ = 1 (determinant = (φψ)⁶ = 1)

The characteristic polynomial p(x) = x⁶ - 8x⁵ + 18x⁴ - 6x³ - 12x² + 2x + 1 determines the 6th-order recurrence for dissipation coefficients.

noncomputable def FUST.GravitationalCoupling.Diff6_charPoly (x : ℝ) : ℝ

N₆ characteristic polynomial: x⁶ - 8x⁵ + 18x⁴ - 6x³ - 12x² + 2x + 1

Equations

• FUST.GravitationalCoupling.Diff6_charPoly x = x ^ 6 - 8 * x ^ 5 + 18 * x ^ 4 - 6 * x ^ 3 - 12 * x ^ 2 + 2 * x + 1

theorem FUST.GravitationalCoupling.Diff6_eval_multiplier_product : φ ^ 3 * φ ^ 2 * φ * ψ * ψ ^ 2 * ψ ^ 3 = 1

Product of all N₆ evaluation multipliers: e₆ = (φψ)⁶ = 1

theorem FUST.GravitationalCoupling.Diff6_eval_pairwise_sum : φ ^ 3 * φ ^ 2 + φ ^ 3 * φ + φ ^ 3 * ψ + φ ^ 3 * ψ ^ 2 + φ ^ 3 * ψ ^ 3 + φ ^ 2 * φ + φ ^ 2 * ψ + φ ^ 2 * ψ ^ 2 + φ ^ 2 * ψ ^ 3 + φ * ψ + φ * ψ ^ 2 + φ * ψ ^ 3 + ψ * ψ ^ 2 + ψ * ψ ^ 3 + ψ ^ 2 * ψ ^ 3 = 18

e₂ = 18: sum of pairwise products of N₆ evaluation multipliers

theorem FUST.GravitationalCoupling.Diff6_eval_triple_sum : ψ ^ 6 + φ * ψ ^ 3 + φ * ψ ^ 4 + φ * ψ ^ 5 + φ ^ 2 * ψ ^ 3 + φ ^ 2 * ψ ^ 4 + φ ^ 2 * ψ ^ 5 + φ ^ 3 * ψ + φ ^ 3 * ψ ^ 2 + 2 * (φ ^ 3 * ψ ^ 3) + φ ^ 3 * ψ ^ 4 + φ ^ 3 * ψ ^ 5 + φ ^ 4 * ψ + φ ^ 4 * ψ ^ 2 + φ ^ 4 * ψ ^ 3 + φ ^ 5 * ψ + φ ^ 5 * ψ ^ 2 + φ ^ 5 * ψ ^ 3 + φ ^ 6 = 6

e₃ = 6: sum of triple products (φ^a·ψ^b form)

theorem FUST.GravitationalCoupling.Diff6_eval_4tuple_sum : φ * ψ ^ 6 + φ ^ 2 * ψ ^ 6 + φ ^ 3 * ψ ^ 5 + φ ^ 3 * ψ ^ 4 + φ ^ 3 * ψ ^ 3 + φ ^ 3 * ψ ^ 6 + φ ^ 4 * ψ ^ 5 + φ ^ 4 * ψ ^ 4 + φ ^ 4 * ψ ^ 3 + φ ^ 5 * ψ ^ 5 + φ ^ 5 * ψ ^ 4 + φ ^ 5 * ψ ^ 3 + φ ^ 6 * ψ ^ 3 + φ ^ 6 * ψ ^ 2 + φ ^ 6 * ψ = -12

e₄ = -12: sum of 4-tuple products

theorem FUST.GravitationalCoupling.Diff6_eval_5tuple_sum : φ ^ 3 * ψ ^ 6 + φ ^ 4 * ψ ^ 6 + φ ^ 5 * ψ ^ 6 + φ ^ 6 * ψ ^ 5 + φ ^ 6 * ψ ^ 4 + φ ^ 6 * ψ ^ 3 = -2

e₅ = -2: sum of 5-tuple products

theorem FUST.GravitationalCoupling.Diff6_charPoly_roots : Diff6_charPoly (φ ^ 3) = 0 ∧ Diff6_charPoly (φ ^ 2) = 0 ∧ Diff6_charPoly φ = 0 ∧ Diff6_charPoly ψ = 0 ∧ Diff6_charPoly (ψ ^ 2) = 0 ∧ Diff6_charPoly (ψ ^ 3) = 0

Each N₆ evaluation multiplier is a root of the characteristic polynomial

theorem FUST.GravitationalCoupling.Diff6_spectral_invariants : φ ^ 3 + φ ^ 2 + φ + ψ + ψ ^ 2 + ψ ^ 3 = 8 ∧ φ ^ 3 * φ ^ 2 * φ * ψ * ψ ^ 2 * ψ ^ 3 = 1 ∧ Diff6_charPoly (φ ^ 3) = 0 ∧ Diff6_charPoly (φ ^ 2) = 0 ∧ Diff6_charPoly φ = 0 ∧ Diff6_charPoly ψ = 0 ∧ Diff6_charPoly (ψ ^ 2) = 0 ∧ Diff6_charPoly (ψ ^ 3) = 0

N₆ spectral invariants summary

N₆ Sector Factorization

The characteristic polynomial factors into three quadratic sectors: p(x) = (x²-x-1)(x²-3x+1)(x²-4x-1) corresponding to matter (φ,ψ), gauge (φ²,ψ²), gravity (φ³,ψ³).

theorem FUST.GravitationalCoupling.Diff6_charPoly_factorization (x : ℝ) : Diff6_charPoly x = (x ^ 2 - x - 1) * (x ^ 2 - 3 * x + 1) * (x ^ 2 - 4 * x - 1)

N₆ charPoly factors as product of three sector polynomials

theorem FUST.GravitationalCoupling.sector_traces : φ + ψ = 1 ∧ φ ^ 2 + ψ ^ 2 = 3 ∧ φ ^ 3 + ψ ^ 3 = 4

Sector traces: φᵏ+ψᵏ for k=1,2,3

theorem FUST.GravitationalCoupling.gravity_trace_eq_four : φ ^ 3 + ψ ^ 3 = 4

Gravity sector trace: φ³+ψ³ = 4

theorem FUST.GravitationalCoupling.gravity_sector_det : (φ * ψ) ^ 3 = -1

Gravity sector determinant = -1: (φψ)³ = (-1)³ = -1

theorem FUST.GravitationalCoupling.gravity_sector_discriminant : 4 ^ 2 + 4 * 1 = Nat.choose 6 3

Gravity sector discriminant = C(6,3) = spacetimeDim × activeDLevels

theorem FUST.GravitationalCoupling.gravity_disc_eq : Nat.choose 6 3 = 20

theorem FUST.GravitationalCoupling.matter_gauge_discriminant : 1 ^ 2 + 4 * 1 = 5 ∧ 3 ^ 2 - 4 * 1 = 5

Matter and gauge sectors have equal discriminant = 5 = activeDLevels

theorem FUST.GravitationalCoupling.sector_discriminants : 1 ^ 2 + 4 * 1 = 5 ∧ 3 ^ 2 - 4 * 1 = 5 ∧ 4 ^ 2 + 4 * 1 = Nat.choose 6 3

Complete sector discriminant structure

theorem FUST.GravitationalCoupling.sector_trace_sq_sum : 1 ^ 2 + 3 ^ 2 + 4 ^ 2 = 26

Sector trace squares: 1²+3²+4² = 26

Sector Spectral Invariants

The N₆ characteristic polynomial factors as p(x) = (x²-x-1)(x²-3x+1)(x²-4x-1) with sector traces tₖ = L(k): t₁=1 (matter), t₂=3 (gauge), t₃=4 (gravity).

Spectral invariants: p₃ = Σ tₖ³ = 92 (sector self-interaction) e₃ = Π tₖ = 12 (cross-sector coupling) σ = Σ tₖ² = 26 (sector trace square sum)

theorem FUST.GravitationalCoupling.sector_trace_cube_sum : 1 ^ 3 + 3 ^ 3 + 4 ^ 3 = 92

1³+3³+4³ = 92: third power sum of sector traces

theorem FUST.GravitationalCoupling.sector_trace_product : 1 * 3 * 4 = 12

1×3×4 = 12: product of all sector traces

Inverse Square Law from N₆ Algebra

Newton's inverse square law F ∝ 1/r² is derived purely from the N₆ operator structure:

1. φ⁻¹ = -ψ (golden conjugate inversion)
2. N₆(t⁻²)(z) = 0 (inverse-square monomial is in the numerator kernel)
3. N₆(t⁻¹)(z) = 6·(φ-ψ)/z (force is inverse-square)
4. □(t⁻¹) = 0 (1/r potential is harmonic under iterated N₆-normalized operator)

theorem FUST.GravitationalCoupling.Diff6_inv_sq_zero (z : ℂ) (hz : z ≠ 0) : DζOperator.Diff6 (fun (t : ℂ) => t⁻¹ ^ 2) z = 0

N₆ annihilates t⁻²: the inverse-square monomial is in the numerator kernel.

theorem FUST.GravitationalCoupling.Diff6_inv_one (z : ℂ) : DζOperator.Diff6 (fun (t : ℂ) => t⁻¹) z = 6 * (↑φ - ↑ψ) / z

N₆(t⁻¹)(z) = 6·(φ-ψ)/z: the gravitational force is inverse-square.

theorem FUST.GravitationalCoupling.Diff6_congr_nonzero (f g : ℂ → ℂ) (z : ℂ) (hz : z ≠ 0) (hfg : ∀ (y : ℂ), y ≠ 0 → f y = g y) : DζOperator.Diff6 f z = DζOperator.Diff6 g z

N₆ preserves pointwise equality at evaluation points

theorem FUST.GravitationalCoupling.Diff6_homogeneous (c : ℂ) (f : ℂ → ℂ) (z : ℂ) : DζOperator.Diff6 (fun (t : ℂ) => c * f t) z = c * DζOperator.Diff6 f z

noncomputable def FUST.GravitationalCoupling.FUSTDAlembertian (f : ℂ → ℂ) : ℂ → ℂ

FUST d'Alembertian: iterated N₆-normalized operator □ = Diff6norm ∘ Diff6norm

Equations

• FUST.GravitationalCoupling.FUSTDAlembertian f = FUST.GravitationalCoupling.Diff6norm✝ (FUST.GravitationalCoupling.Diff6norm✝¹ f)

theorem FUST.GravitationalCoupling.dAlembertian_inv_zero (z : ℂ) (hz : z ≠ 0) : FUSTDAlembertian (fun (t : ℂ) => t⁻¹) z = 0

The 1/r potential is harmonic under the FUST d'Alembertian: □(t⁻¹) = Diff6norm(Diff6norm(t⁻¹)) = 0.

theorem FUST.GravitationalCoupling.inverse_square_law_derivation : (∀ (z : ℂ), z ≠ 0 → DζOperator.Diff6 (fun (t : ℂ) => t⁻¹ ^ 2) z = 0) ∧ (∀ (z : ℂ), DζOperator.Diff6 (fun (t : ℂ) => t⁻¹) z = 6 * (↑φ - ↑ψ) / z) ∧ ∀ (z : ℂ), z ≠ 0 → FUSTDAlembertian (fun (t : ℂ) => t⁻¹) z = 0

Inverse square law derivation from N₆ structure

Dimensional Derivation Structure

N₆tⁿ = C(n)·xⁿ where C(n) is the dissipation coefficient. The monomial eigenvalue C(n) vanishes for n ∈ {0,1,2}: Δ=0 (constants), Δ=1 (mass), Δ=2 (kinetic energy). Since N₆ annihilates Δ=1, mass ratios m_e/m_Pl are boundary data, not eigenvalue data. Physical exponents thus form a two-layer structure: Layer 1: N₆ eigenvalues → σ=26, F∝1/r² Layer 2: Physical assembly with dimensional intermediates → 152, 582

theorem FUST.GravitationalCoupling.Diff6_force_dimension_active (z : ℂ) (hz : z ≠ 0) : DζOperator.Diff6 (fun (t : ℂ) => t⁻¹) z ≠ 0

N₆ does NOT annihilate Δ=-1: the force operator is outside the kernel

theorem FUST.GravitationalCoupling.derivation_layer1_eigenvalues : sectorTraceSq = 26 ∧ ∀ (z : ℂ), DζOperator.Diff6 (fun (t : ℂ) => t⁻¹) z = 6 * (↑φ - ↑ψ) / z

Layer 1: N₆ eigenvalue structure determines physical framework

theorem FUST.GravitationalCoupling.derivation_layer3_assembly : cmbTemperatureExponent = leptonExponent + cmbDecouplingFactor ∧ cosmologicalExponent = 4 * cmbTemperatureExponent - sectorTraceSq ∧ cmbTemperatureExponent = 152 ∧ cosmologicalExponent = 582

Layer 2: physical assembly with dimensional intermediates.

Graviton Structural Prediction

The graviton is predicted (not postulated) by the N₆ operator structure:

1. Existence: N₆ charPoly = (matter)(gauge)(gravity) has a gravity sector (x²-4x-1)
2. Massless: □(t⁻¹) = 0 (graviton propagator has no mass term)
3. Inverse square: N₆(t⁻¹) ∝ 1/z (force law from operator algebra)
4. Coupling: m_e/m_Pl = φ^(-107-5/63) from Fζ hierarchy combinatorics

theorem FUST.GravitationalCoupling.graviton_massless (z : ℂ) : z ≠ 0 → FUSTDAlembertian (fun (t : ℂ) => t⁻¹) z = 0

Graviton masslessness: □(t⁻¹) = 0

theorem FUST.GravitationalCoupling.graviton_prediction : (∀ (x : ℝ), Diff6_charPoly x = (x ^ 2 - x - 1) * (x ^ 2 - 3 * x + 1) * (x ^ 2 - 4 * x - 1)) ∧ φ ^ 3 + ψ ^ 3 = 4 ∧ (φ * ψ) ^ 3 = -1 ∧ 4 ^ 2 + 4 * 1 = Nat.choose 6 3 ∧ (∀ (z : ℂ), z ≠ 0 → FUSTDAlembertian (fun (t : ℂ) => t⁻¹) z = 0) ∧ ∀ (z : ℂ), DζOperator.Diff6 (fun (t : ℂ) => t⁻¹) z = 6 * (↑φ - ↑ψ) / z

Complete graviton structural prediction