Algebra Cost Algebra Continuous Bijective Preserves J Eq Id Or Inv

The automorphism rigidity

The positive real numbers, the set of all numbers greater than zero, carry a natural notion of cost: a measure of how far a number is from 1, the neutral element of multiplication. In the Recognition Science framework, this cost is not chosen freely but forced by a few plain conditions to be J(x) = (x + 1/x)/2 − 1, a function that assigns zero cost to 1 and grows as a number moves away from 1 in either direction, whether toward zero or toward infinity.

A central structural question is: which transformations of the positive reals leave this cost unchanged? The answer, established in the framework's machine-checked library of formal theorems, is strikingly small. Consider a function f from the positive reals to itself that is continuous, bijective (one-to-one and onto), and preserves the cost, meaning J(f(x)) = J(x) for every positive x. The theorem eq_id_or_reciprocal establishes that such a function must be either the identity map f(x) = x or the reciprocal map f(x) = 1/x. There are no other possibilities.

The result is a rigidity statement: the cost function is so sensitive to the multiplicative structure that it admits only two symmetries. The reciprocal map is a genuine symmetry because J(1/x) = J(x), a fact established as J_reciprocal. The identity map trivially preserves cost. The theorem rules out any exotic continuous bijection that might, for example, permute numbers while leaving all costs intact. This classification is not merely an observation; it is a established theorem in the library, with the two automorphisms shown to be distinct and exhaustive.

In Recognition Science, this rigidity anchors the algebraic layer. The cost algebra, built from the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y), is the foundational object from which the framework derives further structure, including the golden ratio and the eight-tick cycle. The automorphism classification tells us that the cost function's symmetries are exactly the two trivial ones, leaving no hidden freedom in the algebra. This is what the declaration continuous_bijective_preserves_J_eq_id_or_inv establishes in plain terms.

What the result does not claim is equally important. It does not assert that every continuous bijection preserving cost is a symmetry of the full cost algebra, only of the cost function itself. It does not claim that the identity and reciprocal are the only automorphisms of the positive reals as a multiplicative group; that group has many automorphisms. The theorem is specifically about maps that preserve the cost function J, and it relies on continuity and bijectivity as hypotheses. Without continuity, pathological non-continuous bijections could exist that preserve cost, though the theorem does not address them. The classification is exact and complete under the stated conditions.

/-- Exact classification: every `J`-automorphism is either identity or reciprocal. -/
theorem eq_id_or_reciprocal (f : JAut) : f = id ∨ f = reciprocal := by
  rcases eq_self_or_inv f posTwo with h2 | h2
  · left
    exact eq_id_of_map_two_eq_two f h2
  · right
    have hcomp : comp reciprocal f = id := by
      apply eq_id_of_map_two_eq_two
      calc
        comp reciprocal f posTwo = reciprocal (f posTwo) := rfl
        _ = reciprocal (posInv posTwo) := by rw [h2]
        _ = posTwo := by simp [reciprocal]
    apply JAut.ext
    intro x
    have hx : comp reciprocal f x = id x := congrArg (fun g : JAut => g x) hcomp
    have hx' : posInv (posInv (f x)) = posInv x := congrArg posInv hx
    simpa [comp, reciprocal, id] using hx'

/-- **Reciprocal symmetry**: Cost is invariant under inversion.
    This is the algebraic encoding of "double-entry": every ratio x
    and its reciprocal 1/x carry the same cost. -/
theorem J_reciprocal (x : ℝ) (hx : 0 < x) : J x = J x⁻¹ :=
  Jcost_symm hx

/-- The reciprocal automorphism is genuinely nontrivial. -/
theorem reciprocal_ne_id : reciprocal ≠ id := by
  intro h
  have htwo : reciprocal posTwo = id posTwo := congrArg (fun f : JAut => f posTwo) h
  have hval : ((reciprocal posTwo : PosReal) : ℝ) = ((id posTwo : PosReal) : ℝ) :=
    congrArg Subtype.val htwo
  norm_num [reciprocal, id, posInv, posTwo] at hval

What this page does not claim

The theorem does not classify automorphisms of the multiplicative group of positive reals, only those preserving the cost function. The theorem does not claim that continuity and bijectivity are necessary for a map to preserve cost; it only states they are sufficient to force the dichotomy. The result does not extend to maps that preserve the cost function only approximately or on a restricted domain.

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/Algebra/CostAlgebra.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: