costCompose_comm
Cost composition induced by the Recognition Composition Law is commutative for real numbers. Algebraists working within the Recognition Science framework cite this commutativity when reordering cost terms in derivations. The proof proceeds by unfolding the explicit definition of the operation and invoking the ring tactic for algebraic verification.
claimFor all real numbers $a$ and $b$, $a ★ b = b ★ a$, where the cost composition is given by $a ★ b = 2ab + 2a + 2b$.
background
The module CostAlgebra defines the cost composition operation ★ induced by the Recognition Composition Law. Given costs a and b corresponding to J-values, the operation is a ★ b = 2 a b + 2 a + 2 b, which can be rewritten as 2(a + 1)(b + 1) - 2. This encodes the combination rule from the RCL: J(xy) + J(x/y) = 2 J(x) J(y) + 2 J(x) + 2 J(y). The defect functional coincides with J on positive reals. Upstream results include the costCompose definition itself and supporting theorems ensuring the algebraic setting, such as collision-free properties and self-reference structures from the Law of Existence and related modules.
proof idea
This is a one-line term proof. It unfolds the costCompose definition to obtain the bilinear expression and applies the ring tactic to confirm equality via the commutative ring axioms on the reals.
why it matters in Recognition Science
The commutativity result is used directly in the proof of costCompose_right_cancel to derive right cancellation under non-negative conditions. It underpins the algebraic manipulations required for Recognition Cost Systems and window aggregation. This symmetry is consistent with the RCL and contributes to the overall forcing chain from T0 to the eight-tick octave and three-dimensional structure.
scope and limits
- Does not establish associativity of the ★ operation.
- Does not apply when costs take negative values.
- Does not reference the phi-ladder or Berry threshold.
- Does not discharge any scaffolding or hypothesis interfaces.
Lean usage
rw [costCompose_comm a₁ b, costCompose_comm a₂ b] at hformal statement (Lean)
121theorem costCompose_comm (a b : ℝ) : a ★ b = b ★ a := by
proof body
Term-mode proof.
122 unfold costCompose; ring
123
124/-- **THEOREM: Associator defect for raw RCL composition.**
125 The unnormalized RCL form is not strictly associative; the defect is `2*(a-c)`. -/