Condensed Matter Jcost Phase Transition J Cost Minimum At One

The cost minimum

The function at the center of this page is J(x) = (x + x⁻¹)/2 − 1. For any positive number x, this expression measures a kind of balance between x and its reciprocal. When x equals 1, the two terms are equal, and the value is exactly 0. The theorem J_cost_minimum_at_one proves this in a machine-checked library of formal theorems: J(1) = 0. A second theorem, J_cost_positive_away_from_one, shows that for every positive x other than 1, the value is strictly greater than 0. Together they establish that 1 is the unique minimum of this cost function on the positive real numbers.

This minimum is not an isolated curiosity. The same function, under the name recognition cost (a measure of how far a system's state is from a balanced reference), appears in the framework's account of phase transitions. The framework defines a critical energy scale as J(φ), where φ is the golden ratio, and proves numerically that this value lies between 0.09 and 0.12. From that scale it derives a critical temperature T_critical = J(φ) × 1000, and a theorem proves that this temperature falls strictly between 80 and 120 kelvin. The minimum at 1 is the anchor: it fixes the zero of the cost scale, so that all other values are measured relative to this balanced point.

What the declaration does not claim is equally important. The theorem J_cost_minimum_at_one is a statement about the function's value at a single point; it does not by itself establish any physical law. The connection to superconductivity is a separate, falsifiable prediction, not a consequence of the minimum. The prediction states that superconducting materials with phi-structured lattices will show critical temperatures near 80 to 120 kelvin when the coherence energy matches φ⁻⁵, about 0.09 eV. That prediction names its own falsifier: a phi-structured superconductor with the stated coherence energy whose critical temperature falls outside that window would count against it.

In plain terms, the minimum at one is the zero point of a scale. It tells you where the cost of imbalance vanishes, and it gives every other value a reference. The framework's further claims about superconductors build on this zero point, but they stand or fall on their own evidence, not on the simple fact that J(1) = 0.

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

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 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

The theorem does not prove that any real material superconducts. The minimum at one does not by itself force the critical temperature prediction. The framework does not claim that all superconductors follow this phi-structured pattern.

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: