Foundation Jcost Cosh Identity

The cosh form of cost

The hyperbolic cosine, usually written cosh, is the average of a number and its reciprocal: cosh(y) = (e^y + e^−y)/2. It is the curve a hanging chain makes, and it appears whenever a quantity is symmetric under flipping a sign. In Recognition Science, the same shape describes the cost of a recognition event when the event is measured on a logarithmic scale.

Recognition Science starts from a ledger: a discrete record of events, where each event has a cost that is forced, not chosen. The central cost function, written J(x), is proved from five plain conditions to equal (x + 1/x)/2 − 1. On a logarithmic scale, writing x = e^y, that identity becomes J(e^y) = (e^y + e^−y)/2 − 1, which is exactly cosh(y) − 1. The framework's library, a machine-checked collection of formal theorems, proves this equivalence as a theorem named jcost_exp_cosh_form, with no gaps and no extra axioms.

The module JCostCoshIdentity establishes four properties of this cosh form. First, the cost is zero exactly when y = 0, meaning the event is a fixed point, no cost is paid. Second, the cost is symmetric: J(e^y) = J(e^−y), so flipping the logarithm leaves the price unchanged. Third, the cost is never negative, and fourth, it is strictly positive whenever y is not zero. The library packs these four facts into a single certificate structure, a formal bundle that any later proof can cite as one unit.

In plain language, the module says the cost function, when viewed on the right scale, is a simple, symmetric, non-negative bowl that touches zero at exactly one point. This matters because the cosh form is the nonlinear version that appears in the strong-field Regge action, a context where the linear approximation fails. The identity gives a clean handle on that regime: the same J that governs weak fields, written in exponential coordinates, becomes a familiar hyperbolic function with all its classical properties intact.

What the module changes is how later work can talk about cost. Instead of carrying the algebraic definition everywhere, a proof can invoke the cosh form, the symmetry, or the positivity as a single certified step. The certificate bundles the four properties so that a downstream theorem about strong fields, or about the shape of the cost near its minimum, can proceed without re-deriving the basics.

/-- 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(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) = 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) ≥ 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 module does not prove that J is the only cost function satisfying the five conditions. The module does not derive the value of any physical constant such as the fine-structure constant. The cosh identity does not by itself establish the golden ratio or the eight-tick cycle.

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)

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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