Gravitational Collapse from Fζ Structure

All results derived from Fζ = 5z·Dζ operator algebra and φ-ψ duality only. No continuous theory (GR, QFT, Bekenstein-Hawking) is assumed.

Discrete Scale Lattice

Physical scales form a discrete lattice φⁿ (n ∈ ℤ). No continuous limit exists.

@[reducible, inline]

noncomputable abbrev FUST.BlackHole.discreteScale (n : ℤ) : ℝ

Discrete scale: φⁿ

Equations

• FUST.BlackHole.discreteScale n = FUST.φ ^ n

theorem FUST.BlackHole.discreteScale_pos (n : ℤ) : discreteScale n > 0

theorem FUST.BlackHole.no_continuous_limit (n : ℤ) : discreteScale n ≠ 0 ∧ discreteScale (n + 1) / discreteScale n = φ

theorem FUST.BlackHole.discrete_levels_distinct (n m : ℤ) : n ≠ m → discreteScale n ≠ discreteScale m

Discrete levels are distinct (φ > 1 implies injectivity)

φ-ψ Duality (Algebraic Reversibility)

φ·|ψ| = 1 from the characteristic equation x² = x + 1. This is a purely algebraic identity, not a physical assumption.

theorem FUST.BlackHole.phi_psi_duality : φ * |ψ| = 1

theorem FUST.BlackHole.frame_duality : φ > 1 ∧ |ψ| < 1 ∧ φ * |ψ| = 1

theorem FUST.BlackHole.algebraic_reversibility (n : ℕ) : φ ^ n * |ψ| ^ n = 1

φⁿ · |ψ|ⁿ = 1: algebraic reversibility at every scale

theorem FUST.BlackHole.unitarity_from_duality (n : ℤ) : discreteScale n * discreteScale (-n) = 1

Scale inversion: φⁿ · φ⁻ⁿ = 1

Fζ Dissipation Rate

Fζ output grows as φ per time step via Dζ gauge covariance. This exponential growth from Fζ algebra replaces the continuous Hawking formula.

The dissipation ratio |ψ| = φ⁻¹ between successive scales is a purely algebraic consequence of φ·|ψ| = 1, not imported from continuous thermal physics.

@[reducible, inline]

noncomputable abbrev FUST.BlackHole.dissipationRate (n : ℕ) : ℝ

Fζ dissipation ratio at scale level n: |ψ|ⁿ

Equations

• FUST.BlackHole.dissipationRate n = |FUST.ψ| ^ n

theorem FUST.BlackHole.dissipationRate_pos (n : ℕ) : dissipationRate n > 0

theorem FUST.BlackHole.dissipation_geometric (n : ℕ) : dissipationRate (n + 1) = dissipationRate n * |ψ|

Dissipation decreases geometrically by |ψ| per step

theorem FUST.BlackHole.dissipation_ratio (n : ℕ) : dissipationRate (n + 1) / dissipationRate n = |ψ|

Dissipation ratio between successive levels is |ψ|

Scale Separation (Discrete Resolution)

The minimum scale separation is φ^{-k} for k steps. This is the Fζ resolution limit, not a "horizon width" from GR.

@[reducible, inline]

noncomputable abbrev FUST.BlackHole.scaleResolution (k : ℕ) : ℝ

Scale resolution at k steps: φ^{-k}

Equations

• FUST.BlackHole.scaleResolution k = FUST.φ ^ (-↑k)

theorem FUST.BlackHole.scaleResolution_pos (k : ℕ) : scaleResolution k > 0

theorem FUST.BlackHole.scaleResolution_decreasing (k : ℕ) : scaleResolution (k + 1) < scaleResolution k

State Space Dimension from Fζ

The number of independent degrees of freedom at scale level k is determined by the Diff6 kernel dimension (= 3, via Φ_A = Diff6+Diff2-Diff4 in Fζ) and the number of scale steps. This replaces Bekenstein-Hawking S ∝ A without importing continuous GR.

@[reducible, inline]

abbrev FUST.BlackHole.degreesOfFreedom (k : ℕ) : ℕ

Degrees of freedom at k scale levels: Diff6 kernel dim × k levels = 3k (This counts spatial modes per scale step from Fζ AF channel structure)

Equations

• FUST.BlackHole.degreesOfFreedom k = 3 * k

theorem FUST.BlackHole.dof_monotone (k : ℕ) : degreesOfFreedom k ≤ degreesOfFreedom (k + 1)

ψ-Contraction (Time-Reversed Evolution)

|ψ|ⁿ is the contraction dual of φⁿ expansion. This is purely algebraic (φ·|ψ| = 1), not "white holes" from GR.

@[reducible, inline]

noncomputable abbrev FUST.BlackHole.contractionScale (n : ℕ) : ℝ

Contraction scale: |ψ|ⁿ

Equations

• FUST.BlackHole.contractionScale n = |FUST.ψ| ^ n

theorem FUST.BlackHole.contractionScale_pos (n : ℕ) : contractionScale n > 0

theorem FUST.BlackHole.expansion_contraction_identity (n : ℕ) : discreteScale ↑n * contractionScale n = 1

Expansion × contraction = 1 (algebraic identity)

theorem FUST.BlackHole.contraction_rate (n : ℕ) : contractionScale (n + 1) = contractionScale n * |ψ|

Scale Unboundedness

φⁿ grows without bound (φ > 1). Purely algebraic.

theorem FUST.BlackHole.scale_unbounded (M : ℝ) : ∃ (k : ℕ), discreteScale ↑k > M

Fζ Critical Scale

For mass m = Δ·x₀², the conserved energy E_cons = φ^(4n)·m² reaches 1 at a critical scale n_crit. The corresponding coordinate distance is r_crit = x₀·|ψ|, derived purely from Fζ algebra and φ-ψ duality.

Key identity: m·t_FUST = x₀² (since Δ = 1/t_FUST).

theorem FUST.BlackHole.abs_psi_eq_inv_phi : |ψ| = φ⁻¹

|ψ| = φ⁻¹: algebraic consequence of φ·|ψ| = 1

noncomputable def FUST.BlackHole.criticalScale (x₀ : ℝ) : ℝ

Critical scale: x₀·|ψ| (proper time stalls beyond this coordinate)

Equations

• FUST.BlackHole.criticalScale x₀ = x₀ * |FUST.ψ|

theorem FUST.BlackHole.criticalScale_pos (x₀ : ℝ) (hx₀ : x₀ > 0) : criticalScale x₀ > 0

theorem FUST.BlackHole.criticalScale_lt (x₀ : ℝ) (hx₀ : x₀ > 0) : criticalScale x₀ < x₀

theorem FUST.BlackHole.criticalScale_at_lattice (k : ℕ) : criticalScale (φ ^ (-↑k)) = φ ^ (-(↑k + 1))

For x₀ = φ^(-k), criticalScale = φ^(-(k+1)) = scaleResolution(k+1)

theorem FUST.BlackHole.criticalScale_eq_scaleResolution (k : ℕ) : criticalScale (scaleResolution k) = scaleResolution (k + 1)

criticalScale is exactly one lattice step below x₀

theorem FUST.BlackHole.criticalScale_monotone (x₁ x₂ : ℝ) (hle : x₁ ≤ x₂) : criticalScale x₁ ≤ criticalScale x₂

Larger x₀ → larger critical scale