rung_ordering
plain-language theorem explainer
The rung_ordering theorem asserts the complete chain of strict inequalities on the phi-ladder: counterfactual floor below bond, bond below vertebrate, vertebrate below octopus, octopus below cetacean, cetacean below human, and human below life ignition. Cognition researchers cite it when populating the master certificate for the animal Z-complexity bound. The proof is a term-mode constructor that directly packages the six component ordering lemmas.
Claim. The Z-complexity rungs satisfy $z_{cf} < z_{bond} < z_{vertebrate} < z_{octopus} < z_{cetacean} < z_{human} < z_{life}$.
background
The module establishes the phi-ladder of animal cognition as a geometric sequence indexed by integer rung k, with structurally significant points at the counterfactual floor (k=5), bond rung (k=8 for sustained molecular recognition), vertebrate rung (k=12), octopus rung (k=14), cetacean rung (k=15), human rung (k=17), and life ignition (k=19). The Z-rung function is defined to be strictly positive and increasing, with each named rung corresponding to a documented cognitive threshold. Upstream results supply the six component inequalities (cf_below_bond, bond_below_vertebrate, vertebrate_below_octopus, octopus_below_cetacean, cetacean_below_human, human_below_life), each obtained by unfolding the rung definitions and applying norm_num.
proof idea
The proof is a term-mode constructor that directly assembles the conjunction from the six prior theorems: cf_below_bond, bond_below_vertebrate, vertebrate_below_octopus, octopus_below_cetacean, cetacean_below_human, and human_below_life. No additional tactics are required beyond the term that packages these lemmas.
why it matters
This theorem supplies the rung_ordering field required by the AnimalZComplexityBoundCert structure and the animal_z_complexity_one_statement theorem. It completes the structural ordering for arc 10c, ensuring the phi-ladder respects the named cognitive thresholds from counterfactual floor through life ignition. The same ordering is reused in mastery-cost monotonicity proofs and black-hole echo catalog certificates.
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