no_hidden_state_comparison_forces_rcl
plain-language theorem explainer
An operative positive-ratio comparison obeying no-hidden-state composition forces its derived cost function to lie in the RCL family. Researchers deriving the phi-ladder and eight-tick octave from functional equations cite this when showing that scale-invariant comparisons without hidden memory imply the specific bilinear form of the Recognition Composition Law. The proof is a one-line wrapper that reduces the no-hidden-state hypothesis to counted-once composition and invokes the corresponding forcing theorem.
Claim. Let $C$ be a comparison operator on positive reals. If $C$ satisfies the operative positive-ratio conditions (identity, non-contradiction, continuity, scale invariance, non-triviality) and admits a no-hidden-state composition (a symmetric counted-once resource expression whose evaluation on the derived costs recovers the composite cost), then the derived cost $F(r) = C(r,1)$ belongs to the RCL family: there exist a combiner $P$ and constant $c$ such that $F(xy) + F(x/y) = P(F(x),F(y))$ with $P(u,v) = 2u + 2v + c uv$ for all positive $x,y$.
background
A comparison operator assigns a real-valued cost to any pair of positive reals. The derived cost fixes the second argument at the multiplicative identity, yielding a function on positive ratios. No-hidden-state composition requires a counted-once resource expression that is symmetric and whose evaluation on the two derived costs exactly reproduces the composite derived cost for products and quotients. Operative positive-ratio comparison encodes the four Aristotelian constraints plus continuity and non-triviality. The upstream theorem counted_once_combiner_forces_rcl already shows that counted-once composition plus the operative conditions forces the RCL family.
proof idea
The proof is a one-line wrapper. It applies counted_once_combiner_forces_rcl to the operative hypothesis together with the reduction that no-hidden-state composition implies counted-once composition (via the sibling lemma no_hidden_state_implies_counted_once).
why it matters
This result shows that no-hidden-state operative comparisons force the RCL family and feeds directly into the main theorem no_hidden_state_logic_forces_rcl. In the Recognition Science framework it supplies the composition law step that generates the phi-ladder, the eight-tick octave, and the constants $c=1$, $hbar=phi^{-5}$, $G=phi^5/pi$. It touches the open question whether every such comparison must arise from the J-cost functional equation.
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