Foundation Universal Forcing Reciprocal Generator Recip Fixed Iff Cost Zero

The fixed point and the zero

The reciprocal of a positive number is 1 divided by that number. The reciprocal of 2 is 1/2, the reciprocal of 5 is 1/5, and the reciprocal of 1 is 1 itself. This last case is the only one where a positive number equals its own reciprocal. That trivial observation, applied within a specific framework, becomes a structural statement about cost.

In Recognition Science, recognition is the act of registering a quantity, and cost is the price of doing so, a number that measures how much work the registration requires. The framework's cost function J assigns a cost to every positive number. The declaration recip_fixed_iff_cost_zero proves, for any positive x, that x equals its reciprocal if and only if the cost J(x) is zero. Both conditions hold exactly when x is 1. The reciprocal operation's only fixed point is also the unique zero of the cost function.

The theorem does not claim that the cost function is derived from the reciprocal operation, nor that the reciprocal operation is derived from the cost. It establishes an equivalence between two properties of the same point. The result is a theorem in the machine-checked library of formal theorems, meaning its proof is verified by a computer. The statement is precise: for all positive real numbers, the equation 1/x = x holds exactly when J(x) = 0, and both are equivalent to x = 1.

What the declaration does not claim is broader. It says nothing about why the cost function has this property, nothing about the physical meaning of zero cost, and nothing about any other point where cost might vanish. It is a single, sharp equivalence, not a theory of economics or a statement about the universe. The result is a lemma within a larger framework, not a standalone law of nature.

/-- **The involution's fixed point is the cost's zero.** For positive `x`,
`ι x = x ↔ J x = 0` — both hold iff `x = 1`. So `ι`'s own fixed point is not
arbitrary: it is the unit, the unique point of zero recognition cost. This is
the sharp form of the cost side — not merely that `J` is `ι`-symmetric, but that
the symmetry axis of `ι` coincides with the null set of `J`. -/
theorem recip_fixed_iff_cost_zero {x : ℝ} (hx : 0 < x) :
    recip x = x ↔ Cost.Jcost x = 0 := by
  rw [recip_fixed_iff hx, Cost.Jcost_eq_zero_iff x hx]

/-- `ι` has a unique fixed point among positive reals, namely the unit `1`. -/
theorem recip_fixed_iff {x : ℝ} (hx : 0 < x) : recip x = x ↔ x = 1 := by
  unfold recip
  have hxne : x ≠ 0 := ne_of_gt hx
  constructor
  · intro h
    have hsq : x * x = 1 := by
      have hmul : x⁻¹ * x = x * x := by rw [h]
      rw [inv_mul_cancel₀ hxne] at hmul
      exact hmul.symm
    nlinarith [hsq, hx]
  · intro h; subst h; simp

What this page does not claim

The cost function is not derived from the reciprocal operation. The reciprocal operation is not derived from the cost function. The theorem says nothing about the physical meaning of zero cost.

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

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