Foundation Jcost Cosh Identity Jcost Exp Symm

The cost symmetry

The exponential function maps addition to multiplication: e^y times e^−y equals 1. The cost function J, which measures the price of a recognition event, treats these two reciprocal factors as equals. The theorem jcost_exp_symm states that for every real number y, J(e^y) = J(e^−y). This is a symmetry under inversion: swapping a quantity with its reciprocal leaves the cost unchanged.

In the Recognition Science framework, the cost function J is not chosen freely. It is forced by five plain conditions, and its unique form is J(x) = (x + 1/x)/2 − 1. The symmetry theorem is a direct consequence of that form. When x is written as e^y, the cost becomes (e^y + e^−y)/2 − 1, which is the hyperbolic cosine minus one. Since cosh is an even function, the symmetry is immediate. The theorem is proved in the framework's machine-checked library of formal theorems, with no unproved assumptions.

The symmetry has a concrete meaning. If a recognition event costs J(x), then the reciprocal event costs exactly the same. Doubling a scale and halving it are equally expensive. This is not an arbitrary choice; it follows from the forced form of J. The same library also proves that J(e⁰) = 0, that J is always non-negative, and that J is strictly positive whenever y is not zero. These properties together describe a cost that vanishes only at the identity and grows symmetrically as y moves away from zero in either direction.

What the theorem does not claim is broader. It does not say that all physical costs are symmetric under inversion; it applies only to the specific cost function J. It does not assert that the exponential parametrization is the only useful one, nor does it derive the value of any physical constant. The symmetry is a local property of J, not a statement about the whole forcing chain that produces J from its five conditions.

/-- J(eʸ) = J(e⁻ʸ). -/
theorem jcost_exp_symm (y : ℝ) :
    Jcost (Real.exp y) = Jcost (Real.exp (-y)) := by
  rw [jcost_exp_cosh_form, jcost_exp_cosh_form]
  rw [neg_neg]; ring

/-- J(eʸ) = (eʸ + e⁻ʸ)/2 - 1. -/
theorem jcost_exp_cosh_form (y : ℝ) :
    Jcost (Real.exp y) = (Real.exp y + Real.exp (-y)) / 2 - 1 := by
  rw [Jcost_eq_sq (Real.exp_ne_zero y)]
  rw [Real.exp_neg]
  field_simp [Real.exp_ne_zero y]
  ring

/-- J(e⁰) = 0. -/
theorem jcost_exp_zero : Jcost (Real.exp 0) = 0 := by
  rw [jcost_exp_cosh_form]; simp

/-- J(eʸ) ≥ 0. -/
theorem jcost_exp_nonneg (y : ℝ) : 0 ≤ Jcost (Real.exp y) := by
  rw [jcost_exp_cosh_form]
  have := Real.add_one_le_exp y
  have := Real.add_one_le_exp (-y)
  nlinarith [Real.exp_pos y, Real.exp_pos (-y)]

/-- J(eʸ) > 0 for y ≠ 0. -/
theorem jcost_exp_pos {y : ℝ} (hy : y ≠ 0) : 0 < Jcost (Real.exp y) := by
  have hexp_ne_one : Real.exp y ≠ 1 := by
    intro h; exact hy (by rwa [Real.exp_eq_one_iff] at h)
  exact Jcost_pos_of_ne_one _ (Real.exp_pos y) hexp_ne_one

What this page does not claim

The theorem does not claim that all physical costs are symmetric under inversion. The theorem does not derive the value of any physical constant. The theorem does not assert that the exponential parametrization is the only useful one for J.

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/Foundation/JCostCoshIdentity.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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