Condensed Matter Jcost Phase Transition J Cost Symmetric

The symmetry of the cost function

The declaration J_cost_symmetric proves a basic algebraic fact about the canonical J-cost function, which the framework defines as J(x) = (x + x⁻¹)/2 - 1. The theorem states that for any positive real number x, J(x) equals J(1/x). In plain words: the cost of a ratio and the cost of its reciprocal are the same. This is a symmetry property, and the proof in the machine-checked library of formal theorems is a direct calculation, not an empirical claim.

The J-cost function itself has a simple shape. It has a minimum at x = 1, where its value is 0, and it is positive for every other positive x. The symmetry means the function is unchanged when you replace a number by its reciprocal. For example, J(2) equals J(1/2). This is a property of the formula, and it follows from the definition alone. The theorem J_cost_symmetric is a statement about real numbers, not about any physical material.

In the framework's condensed matter section, this cost function is used to define scales. The constant phi, the golden ratio, appears as a critical point: the energy scale phi_critical_energy is J(phi), which the framework proves lies between 0.09 and 0.12. A temperature scale T_critical is defined as that energy times 1000, and a theorem sc_prediction proves that this T_critical lies between 80 and 120. These are definitions and derived bounds, not measurements.

What the symmetry theorem does not claim is important. It does not say that any real material has a phase transition at that temperature. It does not say that the J-cost function is the only possible cost function. The theorem is a formal identity about a specific definition. The framework's falsifiable prediction, stated in the same file, is that superconducting materials with phi-structured lattices will show critical temperatures near 80 to 120 K when a coherence energy matches a certain scale. That prediction is a hypothesis, not a consequence of the symmetry theorem.

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

theorem J_cost_minimum_at_one : J_cost 1 = 0 := by
  unfold J_cost
  norm_num

theorem phi_critical_numeric : 0.09 < phi_critical_energy ∧ phi_critical_energy < 0.12 := by
  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 <;> linarith

/-- **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

No real material is proven to exhibit a phase transition at the derived T_critical scale. The symmetry theorem does not establish that J-cost is the unique or only possible cost function. The sc_prediction theorem is a derived bound on a defined scale, not a measurement of any physical sample.

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

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