rcl_is_inevitable
plain-language theorem explainer
Any combiner P satisfying the multiplicative consistency relation with the J-cost on positive reals must equal the bilinear form 2uv + 2u + 2v on the non-negative quadrant. Recognition Science researchers cite this to establish that the Recognition Composition Law is forced by the definitions of symmetric comparison and normalized cost rather than chosen. The proof is a one-line wrapper applying the unconditional inevitability theorem.
Claim. Suppose a function $P : ℝ → ℝ → ℝ$ satisfies $J(xy) + J(x/y) = P(J(x), J(y))$ for all $x > 0$, $y > 0$, where $J$ is the J-cost. Then $P(u, v) = 2uv + 2u + 2v$ for all $u ≥ 0$, $v ≥ 0$.
background
The J-cost is the unique function satisfying symmetry $J(x) = J(1/x)$, normalization $J(1) = 0$, and calibration $J''(1) = 1$, given explicitly by $J(x) = (x + x^{-1})/2 - 1$. The Recognition Composition Law is the identity $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$. This theorem sits in the Ultimate Inevitability module, which reduces the RCL to three primitive requirements: symmetry as the definition of comparison, normalization as the definition of cost, and consistency as the definition of multiplicative structure.
proof idea
The proof is a one-line wrapper that applies the unconditional inevitability theorem rcl_unconditional from the prior module.
why it matters
This result shows the RCL is inevitable once symmetric normalized comparison is accepted, completing the reduction from five assumptions to three primitives. It aligns with the forcing chain landmarks T5 J-uniqueness and the Recognition Composition Law identity. The module doc-comment states there is no alternative to the RCL, in contrast to Euclidean geometry, making Recognition Science inevitable at the level of comparison itself.
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