Cosmology Thermodynamic Selection Cert Jcost Unbounded At Infinity

The cost that grows without bound

The function J(x) = (x + 1/x)/2 - 1, defined for positive numbers, measures how far a positive number x is from 1. At x = 1, J equals 0; for any other positive x, J is positive. As x grows large, the term x/2 dominates, so J(x) grows without bound: for any target value C, no matter how large, there is some input R greater than 1 with J(R) greater than C. This is the statement of the machine-checked theorem jcost_unbounded_at_infinity. It is a purely mathematical fact about a specific formula, proved in the framework's library of formal theorems with no axioms beyond the standard three (propext, Classical.choice, Quot.sound).

In the framework's account, J is called the cost of a recognition event, a discrete record of a comparison between a state and a reference. The theorem says that the cost of being far from the equilibrium value 1 is not merely large but arbitrarily large. This is one of five structural facts that together form a certificate for thermodynamic selection: the cost is never negative, it is zero only at x = 1, it diverges near zero, it diverges at infinity, and its sub-level sets are bounded. The certificate is a formal object, a bundle of five proved statements, not a physical derivation.

The theorem does not claim that any physical quantity actually grows without bound. It does not say that entropy, energy, or any cosmological observable diverges. It does not assert that the universe began with infinite cost, nor that any process realizes the limit. The claim is only about the function J: for every real number C, there exists a real R greater than 1 such that J(R) exceeds C. The connection to thermodynamics, the idea that entropy non-decrease corresponds to J-cost not decreasing on a closed recognition ledger, is a separate structural claim, also formalized, but it is not established by this theorem alone.

What the theorem changes is the mathematical setting: it guarantees that the cost function has no hidden ceiling. Any optimization or selection process that seeks to minimize J will always have room to move toward lower cost, because the cost can always be made larger by moving farther from 1. This is what makes the sub-level sets compact, a property that supports arguments about existence of minima. The unboundedness is a precondition for those arguments, not a conclusion about the cosmos.

/-- 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)]

/-- 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 theorem does not claim that any physical quantity, such as entropy or energy, actually diverges in the universe. The theorem does not claim that the universe began with infinite cost or that any process realizes the limit. The theorem does not by itself establish the connection between entropy non-decrease and J-cost on a closed recognition ledger.

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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