Foundation Jcost Cosh Identity Jcost Exp Pos

The positivity theorem

The hyperbolic cosine, written cosh, is a standard function that describes the shape of a hanging chain or a catenary. Its average of an exponential and its reciprocal, (e^y + e^-y)/2, is always at least 1, and equals 1 only when y is 0. The theorem jcost_exp_pos concerns a closely related expression: it subtracts 1 from that average, so the result is 0 at y = 0 and strictly positive for every other real number y.

In the framework of Recognition Science, this expression is the cost, a measure of the price reality pays to recognize an event, where a discrete record of events is kept. The theorem states that this cost, when evaluated at an exponential of y, is zero exactly when y is zero, and is greater than zero otherwise. In plain terms, recognition costs nothing only in the trivial case of no change; any actual departure from that point carries a positive price. The proof is machine-checked, meaning a computer verified every step of the reasoning, with no gaps and no unproved assumptions.

The result is not a claim about physics directly. It does not say that any physical system must pay this cost, nor that the cost function is the only possible one. It is a mathematical statement about a specific function, derived from the framework's definition of cost. The theorem's role is foundational: it confirms a basic property that the cost function must have, serving as a building block for later results in the framework, such as the derivation of the golden ratio and the structure of recognition cycles.

This positivity property is intuitively necessary for a meaningful cost: if recognition could have negative cost, the framework would allow strange situations where events are rewarded for happening. The theorem rules that out, ensuring the cost is always non-negative and strictly positive except at the fixed point. It is a small but essential piece, a guarantee of good behavior that the rest of the framework can rely on.

/-- 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

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

What this page does not claim

This theorem does not claim that any physical system must pay this cost. It does not establish that this cost function is the only possible one. It does not say anything about the empirical values of physical constants.

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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