FUST Thermodynamics: Three Laws from Fζ Structure

The three laws of thermodynamics derived from the unified operator Fζ = 5z·Dζ.

Physical correspondence:

• Entropy S(f)(z) := |Fζ f z|² measures departure from ker(Fζ)
• Time evolution f ↦ f(φ·) is identified with temporal direction via Poincaré I4 decomposition (Gravity.lean: Φ_A_coeff built from φ,ψ with η(inr 0, inr 0) = -1)
• ker(Fζ) = thermal equilibrium = light cone (no proper time)
• ker(Fζ)⊥ = active states = massive particles (proper time > 0)

Laws:

• Zeroth: ker(Fζ) states share zero entropy (thermal equilibrium)
• First: FζLagrangian is φ-equivariant (energy conservation via Noether)
• Second: φ > 1 amplifies ker(Fζ)⊥ components (entropy increase)
• Third: f ∉ ker(Fζ) ⟹ ∃z, S(f)(z) > 0 (absolute zero unreachable)

Zeroth Law: Thermal Equilibrium as ker(Fζ)

Two systems in ker(Fζ) have identical entropy: S ≡ 0. Same-degree scalar multiples have gauge-invariant entropy ratio.

theorem FUST.Thermodynamics.zeroth_law_kernel_equilibrium (f g : ℂ → ℂ) (hf : TimeStructure.IsInKerFζ f) (hg : TimeStructure.IsInKerFζ g) (z : ℂ) : TimeStructure.entropyAtFζ f z = TimeStructure.entropyAtFζ g z

ker(Fζ) states share zero entropy everywhere

theorem FUST.Thermodynamics.zeroth_law_same_degree_thermal (a b : ℂ) (f : ℂ → ℂ) (z : ℂ) (_hb : b ≠ 0) (hfz : FζOperator.Fζ f z ≠ 0) : TimeStructure.entropyAtFζ (fun (t : ℂ) => a * f t) z / TimeStructure.entropyAtFζ (fun (t : ℂ) => b * f t) z = Complex.normSq a / Complex.normSq b

Entropy ratio for same-degree states is gauge-invariant

First Law: Energy Conservation

FζLagrangian is φ-equivariant: Lf(φ·) = Lf. This is the FUST Noether current: φ-scale symmetry ⟹ energy conservation.

theorem FUST.Thermodynamics.first_law_equivariance (f : ℂ → ℂ) (z : ℂ) : TimeStructure.FζLagrangian (TimeStructure.timeEvolution f) z = TimeStructure.FζLagrangian f (↑φ * z)

φ-equivariance of Lagrangian = energy conservation

theorem FUST.Thermodynamics.first_law_vacuum_energy (f : ℂ → ℂ) (hf : TimeStructure.IsInKerFζ f) (n : ℤ) : actionFζ f n = 0

Vacuum (ker) energy is zero

theorem FUST.Thermodynamics.first_law_action_monotone (f : ℂ → ℂ) (N M : ℕ) (h : N ≤ M) : partialActionFζ f N ≤ partialActionFζ f M

Partial action is monotone: enlarging the scale window cannot decrease energy

Second Law: Entropy Increase

φ-scaling (time evolution) amplifies ker(Fζ)⊥ components by φⁿ > 1. Reversible processes remain in ker(Fζ); irreversible stay outside.

theorem FUST.Thermodynamics.second_law_entropy_shift (f : ℂ → ℂ) (z : ℂ) : TimeStructure.entropyAtFζ (TimeStructure.timeEvolution f) z = TimeStructure.entropyAtFζ f (↑φ * z)

Time evolution shifts entropy evaluation point by φ

theorem FUST.Thermodynamics.second_law_monomial_amplification (n : ℕ) (t : ℂ) : TimeStructure.timeEvolution (fun (s : ℂ) => s ^ n) t = ↑φ ^ n * t ^ n

Monomial amplification: timeEvolution(tⁿ) = φⁿ·tⁿ

theorem FUST.Thermodynamics.second_law_reversible (f : ℂ → ℂ) (hf : TimeStructure.IsInKerFζ f) : TimeStructure.IsInKerFζ (TimeStructure.timeEvolution f)

Reversible: ker(Fζ) is invariant under time evolution

theorem FUST.Thermodynamics.second_law_irreversible (f : ℂ → ℂ) (hf : ¬TimeStructure.IsInKerFζ f) : ¬TimeStructure.IsInKerFζ (TimeStructure.timeEvolution f)

Irreversible: non-ker states remain non-ker

theorem FUST.Thermodynamics.second_law_amplification (n : ℕ) (hn : n ≥ 1) : φ ^ n > 1

φⁿ > 1 for n ≥ 1: amplification factor exceeds unity

Third Law: Absolute Zero Unreachable

Active states (f ∉ ker(Fζ)) cannot reach absolute zero (S ≡ 0). Finite iterations of time evolution preserve non-ker status.

theorem FUST.Thermodynamics.third_law_positive_entropy (f : ℂ → ℂ) (hf : ¬TimeStructure.IsInKerFζ f) : ∃ (z : ℂ), TimeStructure.entropyAtFζ f z > 0

Active states have positive entropy somewhere

theorem FUST.Thermodynamics.third_law_absolute_zero_iff (f : ℂ → ℂ) : (∀ (z : ℂ), TimeStructure.entropyAtFζ f z = 0) ↔ TimeStructure.IsInKerFζ f

S ≡ 0 ⟺ f ∈ ker(Fζ)

theorem FUST.Thermodynamics.third_law_finite_unreachable (f : ℂ → ℂ) (hf : ¬TimeStructure.IsInKerFζ f) (n : ℕ) : ¬TimeStructure.IsInKerFζ (TimeStructure.timeEvolution^[n] f)

Finite time evolution cannot reach ker(Fζ) from outside

theorem FUST.Thermodynamics.third_law_mass_gap : massGapSq > 0

Mass gap: minimum energy of active states is positive

theorem FUST.Thermodynamics.third_law_lightlike_zero (f : ℂ → ℂ) (hf : TimeStructure.IsInKerFζ f) (z : ℂ) : TimeStructure.entropyAtFζ f z = 0

ker states can reach zero entropy everywhere

Structural Properties of Entropy

theorem FUST.Thermodynamics.entropy_pos_iff_active (f : ℂ → ℂ) (z : ℂ) : TimeStructure.entropyAtFζ f z > 0 ↔ FζOperator.Fζ f z ≠ 0

Positive entropy ⟺ Fζ is active

theorem FUST.Thermodynamics.entropy_scalar_scaling (c : ℂ) (f : ℂ → ℂ) (z : ℂ) : TimeStructure.entropyAtFζ (fun (t : ℂ) => c * f t) z = Complex.normSq c * TimeStructure.entropyAtFζ f z

Entropy scales quadratically under scalar multiplication