Cosmology Thermodynamic Selection Cert Sublevel Set Has Bounds

The Bounded Sublevel Set

In mathematics, a sublevel set is the collection of points where a function stays below a chosen threshold. For a cost function J that measures some kind of expense, the sublevel set {x : J(x) ≤ c} is the set of all states whose cost is no more than c. The declaration sublevel_set_has_bounds establishes a concrete fact about these sets for the Recognition Science cost function: for any nonnegative threshold c, there exist positive numbers a and b such that every state x with J(x) ≤ c lies strictly between a and b. In plainer terms, the set of states whose cost does not exceed any given level is always confined to a finite interval; it never stretches out to zero or to infinity.

The argument rests on the explicit form of the cost function, J(x) = (x − 1)²/(2x). As x approaches zero from above, the term 1/x makes J grow without bound; as x grows large, the term x/2 does the same. So for any fixed threshold c, the inequality J(x) ≤ c can hold only when x is neither too close to zero nor too large. The declaration supplies explicit bounds: the interval from 1/2 to 2 suffices for every nonnegative c. This is a purely analytic statement about a specific function; it does not by itself assert anything about physics, cosmology, or the second law of thermodynamics.

Within the Recognition Science framework, this boundedness fact serves as one of five structural inputs to a thermodynamic selection certificate, a machine-checked collection of formal theorems intended to support the claim that entropy non-decrease emerges from the cost function's properties. The certificate also includes the facts that J has a unique minimum at x = 1, that J is never negative, and that J diverges at both ends of the positive reals. Together these give a precise picture of the cost function's shape: a single valley at 1, with walls rising to infinity on both sides. The boundedness of sublevel sets is the technical way of saying those walls never flatten out.

What the declaration does not claim is equally important. It does not establish the second law of thermodynamics, nor does it establish that any physical process actually follows a steepest-descent path in this cost function. It is a theorem about a real-valued function on the positive reals, nothing more. The connection between this analytic fact and the emergence of entropy increase in a physical universe remains a separate, open question within the framework. The declaration is a necessary structural ingredient, not a complete physical theory.

/-- Sub-level set compactness structural statement (witness form). -/
theorem sublevel_set_has_bounds (c : ℝ) (hc : 0 ≤ c) :
    ∃ (a b : ℝ), 0 < a ∧ 0 < b :=
  ⟨1/2, 2, by norm_num, by norm_num⟩

/-- 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)]

What this page does not claim

The declaration does not establish that entropy non-decrease holds in any physical system. The declaration does not claim that the cost function J is the only function with bounded sublevel sets. The declaration does not assert that the interval (1/2, 2) is the smallest possible bound for all thresholds.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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