costCompose_zero_left
plain-language theorem explainer
The cost composition operation satisfies 0 ★ a = 2a for every real a. Researchers deriving algebraic properties of the J-cost structure in Recognition Science cite this identity when proving the ★-magma has no identity element. The proof is a direct unfolding of the binary operation definition followed by ring normalization.
Claim. For all real numbers $a$, $0 ★ a = 2a$, where the operation is defined by $a ★ b := 2ab + 2a + 2b$.
background
In the CostAlgebra module the binary operation ★ is introduced by the definition costCompose(a, b) := 2ab + 2a + 2b. This formula is induced by the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y), which encodes how J-costs combine under multiplication of ratios. The upstream declaration costCompose supplies the explicit algebraic expression and the infix notation used throughout the module.
proof idea
The proof is a one-line wrapper that unfolds the definition of costCompose and applies ring_nf to simplify the resulting polynomial expression.
why it matters
This identity is invoked in the downstream theorem costCompose_no_identity to establish that the nonnegative ★-magma possesses no identity element. It supplies a basic algebraic fact required for the cost algebra that sits inside the Recognition framework and ultimately feeds the forcing chain from the RCL through the phi-ladder to the derivation of three spatial dimensions.
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