morphism_preserves_order
plain-language theorem explainer
Octave morphisms preserve the weaklyBetter relation induced by strain functions. Researchers transferring equilibrium orderings across scales in Recognition Science cite this when establishing pattern invariance between octaves. The proof is a direct one-line application of the morphism's internal preservation field.
Claim. Let $f$ be an octave morphism from octave $O_1$ to octave $O_2$. For states $x,y$ in the state space of $O_1$, if $x$ is weakly better than $y$ under the strain ordering of $O_1$, then the image $f(x)$ is weakly better than $f(y)$ under the strain ordering of $O_2$.
background
In the RRF setting an octave carries a state space together with a strain function that induces the weaklyBetter partial order. An OctaveMorphism is a map between state spaces equipped with a proof that the map respects this ordering. The module develops transfer results for such structures, showing how orderings and equilibria move when octaves are related by morphisms or equivalences.
proof idea
The proof is a one-line wrapper that applies the preserves_order field of the given OctaveMorphism structure to the supplied states and hypothesis.
why it matters
The result lets ordering properties transfer between octaves, supporting the module's claim that equivalent octaves share the same equilibrium structures and thereby realize the same pattern at different scales. It sits inside the octave-transfer block that builds on the eight-tick octave from the forcing chain and on the Recognition Composition Law for consistent J-cost functions. It directly enables sibling statements such as equiv_equilibria_iff.
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