Condensed Matter Jcost Phase Transition J Cost Positive Away From One

The cost minimum

The cost function J(x) = (x + x⁻¹)/2 − 1 is a measure of how expensive it is to recognize a state that differs from a reference state by a factor x. When x equals 1, the state matches the reference exactly, and the cost is zero. The new theorem J_cost_positive_away_from_one proves that for any positive x other than 1, the cost is strictly greater than zero. In plain terms, the cheapest possible recognition is the one where nothing changes; any deviation, no matter how small, costs something.

The proof is a short algebraic identity. The function can be rewritten as (x − 1)² / (2x), which is always positive for x > 0 and x ≠ 1. This is not a physical assumption; it is a fact about the function's shape. The same file also proves that J is symmetric, so J(x) = J(1/x), meaning recognizing a state that is twice as large costs the same as recognizing one that is half as large.

In Recognition Science, a framework that derives physical structure from the cost of recognition events, this minimum is the anchor for a phase transition. The critical energy scale is defined as J(φ), where φ is the golden ratio, and its numerical value lies between 0.09 and 0.12 in the framework's units. A falsifiable prediction follows: superconducting materials with lattices structured by φ should show critical temperatures near 80 to 120 kelvin. The theorem about the minimum is proved; the prediction is a hypothesis awaiting experimental test.

The theorem does not claim that recognition is always costly in a physical sense. It says the mathematical function has a unique minimum at x = 1. It does not say why the cost function has this form, nor does it establish that any real material must obey it. The positivity result is a precondition for the phase-transition story, not the story itself.

theorem J_cost_positive_away_from_one (x : ℝ) (hx_pos : 0 < x) (hx_ne : x ≠ 1) :
    0 < J_cost x := by
  unfold J_cost
  have hx0 : x ≠ 0 := hx_pos.ne'
  have hsub : (x - 1) ≠ 0 := sub_ne_zero.mpr hx_ne
  have hsq : 0 < (x - 1) ^ 2 := sq_pos_of_ne_zero hsub
  have : (x + x⁻¹) / 2 - 1 = (x - 1) ^ 2 / (2 * x) := by field_simp; ring
  rw [this]
  exact div_pos hsq (mul_pos (by norm_num : (0:ℝ) < 2) hx_pos)

theorem J_cost_positive_away_from_one (x : ℝ) (hx_pos : 0 < x) (hx_ne : x ≠ 1) :
    0 < J_cost x := by
  unfold J_cost
  have hx0 : x ≠ 0 := hx_pos.ne'
  have hsub : (x - 1) ≠ 0 := sub_ne_zero.mpr hx_ne
  have hsq : 0 < (x - 1) ^ 2 := sq_pos_of_ne_zero hsub
  have : (x + x⁻¹) / 2 - 1 = (x - 1) ^ 2 / (2 * x) := by field_simp; ring
  rw [this]
  exact div_pos hsq (mul_pos (by norm_num : (0:ℝ) < 2) hx_pos)

theorem J_cost_symmetric (x : ℝ) (hx_pos : 0 < x) : J_cost x = J_cost (x⁻¹) := by
  simp only [J_cost, inv_inv]; ring

/-- **FALSIFIABLE PREDICTION**: Superconducting materials with phi-structured
    lattices will show critical temperatures T_c ~ 80-120 K when the coherence
    energy E_coh matches phi^(-5) ~ 0.09 eV. This predicts optimal doping
    occurs at carrier density n ~ 1/phi^2 ~ 0.38 per unit cell. -/
theorem sc_prediction : 80 < T_critical ∧ T_critical < 120 := by
  unfold T_critical
  rw [phi_critical_value]
  have hphi_inv : phi⁻¹ = phi - 1 := by
    have hne : phi ≠ 0 := phi_pos.ne'
    have hsq := phi_sq_eq
    field_simp at hsq ⊢
    nlinarith [phi_pos]
  rw [hphi_inv]
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  constructor <;> nlinarith

What this page does not claim

The theorem does not prove that recognition is always costly in a physical sense. The theorem does not explain why the cost function has the form J(x) = (x + x⁻¹)/2 − 1. The theorem does not establish that any real material must obey the cost function.

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/CondensedMatter/JCostPhaseTransition.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: