costCompose_power_assoc
The cost composition operation satisfies third-power associativity, so that for any real a the expressions (a ★ a) ★ a and a ★ (a ★ a) coincide. Algebraists verifying flexible binary operations and Recognition Science modelers building phi-ladder mass formulas would cite the result when confirming that triple compositions remain unambiguous. The proof is a one-line specialization of the flexible law to equal arguments.
claimFor any real number $a$, $(a ★ a) ★ a = a ★ (a ★ a)$, where $★$ denotes the cost composition operation induced by the J-cost function.
background
The ★ operation is the binary cost composition on reals induced by the J function, J(x) = (x + x^{-1})/2 - 1, that satisfies the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The upstream theorem costCompose_flexible establishes that this operation is flexible: (a ★ b) ★ a = a ★ (b ★ a). The local setting is the algebra of costs inside IndisputableMonolith.Algebra.CostAlgebra, which imports the functional equation and Aczel variants to ground the composition in the Recognition Science forcing chain.
proof idea
One-line wrapper that applies costCompose_flexible to the pair (a, a).
why it matters in Recognition Science
The declaration confirms that triple powers are unambiguous under cost composition, a prerequisite for consistent use of the phi-ladder and eight-tick octave constructions. It directly supports the RCL holding and sits beneath any later checks on higher associativity or spectral emergence structures. No downstream uses are recorded, leaving open whether the same flexibility extends to the full power-associativity failure noted for four copies.
scope and limits
- Does not establish full associativity for four or more factors.
- Does not prove commutativity of the ★ operation.
- Does not extend the result beyond real numbers.
- Does not supply numerical estimates or bounds on the defect.
formal statement (Lean)
175theorem costCompose_power_assoc (a : ℝ) : (a ★ a) ★ a = a ★ (a ★ a) := by
proof body
Term-mode proof.
176 simpa using (costCompose_flexible a a)
177
178/-- Four copies already witness failure of full power-associativity. -/