costCompose_flexible
The theorem establishes flexibility of the raw ★ composition operation from the Recognition Composition Law, proving (a ★ b) ★ a equals a ★ (b ★ a) for all real a and b. Algebraists checking non-associative structures within the Recognition Science framework cite this when building power-associativity. The proof is a one-line wrapper instantiating the associator defect theorem at c = a, which cancels the defect term.
claimFor all real numbers $a, b$, $(a ⋆ b) ⋆ a = a ⋆ (b ⋆ a)$, where ⋆ denotes the binary operation on $ℝ$ satisfying the Recognition Composition Law.
background
The CostAlgebra module derives algebraic identities for the raw ★ operation, which is the unnormalized composition obeying the Recognition Composition Law (RCL): J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). This law originates from the J-cost function in the imported Cost and FunctionalEquation modules. The local setting focuses on basic closure properties such as commutativity and limited associativity before higher-power results.
proof idea
The proof is a one-line wrapper that applies costCompose_assoc_defect with arguments a, b, a. The defect equation states (a ★ b) ★ c = a ★ (b ★ c) + 2*(a - c); substituting c = a makes the defect term vanish, yielding the equality directly.
why it matters in Recognition Science
This result feeds the downstream theorem costCompose_power_assoc, which establishes unambiguous third-power associativity: (a ★ a) ★ a = a ★ (a ★ a). It fills an algebraic step in the Recognition framework by clarifying the defect structure of the RCL operation, prior to full associativity questions or links to the forcing chain (T0-T8).
scope and limits
- Does not prove full associativity of ★ for arbitrary triples a, b, c.
- Does not extend the equality beyond real numbers.
- Does not derive an explicit closed form for the ★ operation.
- Does not connect to phi-ladder, mass formulas, or physical constants.
Lean usage
theorem costCompose_power_assoc (a : ℝ) : (a ★ a) ★ a = a ★ (a ★ a) := by simpa using (costCompose_flexible a a)formal statement (Lean)
132theorem costCompose_flexible (a b : ℝ) : (a ★ b) ★ a = a ★ (b ★ a) := by
proof body
Term-mode proof.
133 simpa using (costCompose_assoc_defect a b a)
134
135/-- **THEOREM: Left-zero evaluation for raw RCL composition.** -/