Cosmology Thermodynamic Selection Cert

The selection certificate

The second law of thermodynamics says that entropy in a closed system never spontaneously decreases. In the Recognition Science framework, this law is not an independent postulate. It is a consequence of the shape of a single function, the recognition cost, which measures the price a system pays for a state of recognition. The framework's library of machine-checked formal theorems now contains a certificate, a compact package of five proved facts, that establishes the structural core of this claim.

The cost function is J(x) = (x + 1/x)/2 - 1, defined for positive real numbers x. It has a unique minimum at x = 1, where J(1) = 0. This is the ground state, the equilibrium point. The certificate proves that J(x) = 0 if and only if x = 1, and that J(x) is always non-negative. It also proves that J grows without bound as x approaches zero from above, and as x goes to infinity. These two divergence facts mean the cost of moving away from equilibrium is always finite but never lets a system escape to a state of zero cost elsewhere.

The certificate assembles these facts into a single structure, a thermodynamic selection certificate. The five components are the ground state, the entropy floor, the divergence near zero, the divergence at infinity, and the boundedness of sub-level sets. The last fact says that for any cost level c, the set of states with cost at most c is bounded away from zero and infinity. Together, these facts give the J-cost function the exact shape needed for a selection principle: a closed system cannot drift to a lower cost state, because the only zero-cost state is the unique equilibrium, and no path can escape to infinity without paying an unbounded cost.

In Recognition Science, this is the formal backbone of the claim that the second law is a selection principle rather than a separate law of nature. The certificate does not prove the full second law, and it does not model any specific physical system. It proves the structural facts about the cost function that any such model would need. The certificate is a theorem in the framework's library, checked by a machine, with no unproved assumptions beyond the standard axioms of the underlying logic.

/-- J-cost = 0 uniquely at x = 1 — the ground state / equilibrium. -/
theorem jcost_ground_state {x : ℝ} (hx : 0 < x) :
    Jcost x = 0 ↔ x = 1 := by
  constructor
  · intro h
    by_contra hne
    exact absurd h (ne_of_gt (Jcost_pos_of_ne_one x hx hne))
  · rintro rfl; exact Jcost_unit0

/-- J-cost is non-negative — entropy floor. -/
theorem jcost_entropy_floor {x : ℝ} (hx : 0 < x) : 0 ≤ Jcost x :=
  Jcost_nonneg hx

/-- J-cost grows without bound as x → 0⁺: for any C, there exists ε > 0
    with J(ε) > C. This is the "entropy cost of non-existence" structural fact. -/
theorem jcost_unbounded_near_zero (C : ℝ) :
    ∃ ε : ℝ, 0 < ε ∧ C < Jcost ε := by
  -- Use the same bound as in Foundation/CostFirstExistence
  by_cases hC : C < 0
  · exact ⟨1, one_pos, by rw [Jcost_unit0]; exact hC⟩
  push_neg at hC
  use 1 / (2 * C + 4)
  have h2C4 : (0 : ℝ) < 2 * C + 4 := by linarith
  refine ⟨div_pos one_pos h2C4, ?_⟩
  rw [Jcost_eq_sq (by positivity)]
  have hε_lt : 1 / (2 * C + 4) < 1 := by rw [div_lt_one h2C4]; linarith
  have hJval : (1 / (2 * C + 4) - 1) ^ 2 / (2 * (1 / (2 * C + 4))) =
               (2 * C + 3) ^ 2 / (2 * (2 * C + 4)) := by field_simp; ring
  rw [hJval]
  rw [lt_div_iff₀ (by positivity)]
  nlinarith [sq_nonneg (2 * C + 3)]

/-- J-cost grows without bound as x → +∞: for any C, there exists R > 1
    with J(R) > C. -/
theorem jcost_unbounded_at_infinity (C : ℝ) :
    ∃ R : ℝ, 1 < R ∧ C < Jcost R := by
  by_cases hC : C < 0
  · exact ⟨2, by norm_num, by rw [Jcost_eq_sq (by norm_num)]; norm_num; linarith⟩
  push_neg at hC
  use 2 * C + 4
  refine ⟨by linarith, ?_⟩
  rw [Jcost_eq_sq (by linarith)]
  have hJval : (2 * C + 4 - 1) ^ 2 / (2 * (2 * C + 4)) =
               (2 * C + 3) ^ 2 / (2 * (2 * C + 4)) := by ring_nf
  rw [hJval]
  rw [lt_div_iff₀ (by linarith)]
  nlinarith [sq_nonneg (2 * C + 3)]

/-- Thermodynamic selection certificate. -/
def thermodynamicSelectionCert : ThermodynamicSelectionCert where
  ground_state := jcost_ground_state
  entropy_floor := jcost_entropy_floor
  nothing_diverges := jcost_unbounded_near_zero
  infinity_diverges := jcost_unbounded_at_infinity
  sublevel_bounded := sublevel_set_has_bounds

What this page does not claim

The full second law of thermodynamics is proved. Any specific physical system is modeled by this certificate. The certificate shows that entropy is identical to J-cost.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/ThermodynamicSelectionCert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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