aestheticCost_zero_at_optimum
plain-language theorem explainer
The theorem shows that aesthetic cost on the familiarity ratio reaches exactly zero at unit ratio. Researchers modeling Berlyne-style preference curves or fluency-affect accounts would cite it to fix the peak-appeal point in the J-cost landscape. The proof is a one-line term that substitutes the definition of aestheticCost into the unit lemma for Jcost.
Claim. Let $J(r)$ denote the recognition cost on the familiarity ratio $r$. Then $J(1) = 0$.
background
In the Recognition Science treatment of aesthetics, appeal is quantified by the J-cost incurred on the familiarity ratio $r :=$ observed similarity over target familiarity. The function aestheticCost is defined directly as the J-cost: aestheticCost $r :=$ Jcost $r$. The module sets the maximum appeal at $r = 1$, where parseability is perfect without excess novelty or boredom. This rests on the Jcost unit lemma, which states Jcost 1 = 0 by direct simplification of the squared-ratio expression $J(x) = (x-1)^2/(2x)$ from the Cost module.
proof idea
The proof is a one-line term that applies the Jcost_unit0 lemma. Because aestheticCost is defined by direct substitution as Cost.Jcost, the equality at argument 1 follows immediately from the corresponding property of Jcost.
why it matters
This theorem supplies the optimum_zero field inside the CulturalAestheticCert structure, which bundles the core properties needed to certify the RS account of cultural preference. It anchors the zero-cost point that corresponds to the peak of the Berlyne inverted-U and the reciprocal-symmetry signature of J-cost. The result closes the basic case for the familiarity-ratio model before the module extends to off-optimum penalties and symmetry.
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