animal_z_complexity_one_statement
plain-language theorem explainer
The declaration consolidates the phi-geometric structure of the animal-cognition Z-complexity ladder, its strict increase with rung index, the explicit ordering of named cognitive rungs from counterfactual floor through life, and the equality of the cognition floor to phi^5. Researchers modeling cognitive thresholds within Recognition Science would cite this as the single reference fixing the ladder positions and scaling. The proof is a term-mode conjunction that directly assembles four sibling results on the recurrence, monotonicity, ordering,
Claim. Let $Z(k) = phi^k$ for $k in mathbb{N}$. Then $Z(k+1) = phi cdot Z(k)$ for all $k$, $Z(k) < Z(k+1)$ for all $k$, the rungs satisfy $Z_{cf} < Z_{bond} < Z_{vertebrate} < Z_{octopus} < Z_{cetacean} < Z_{human} < Z_{life}$, and the cognition floor equals $phi^5$.
background
The module supplies the structural framework for the animal-cognition domain in Recognition Science Arc 10c. It defines the Z-complexity ladder as the sequence indexed by integer rung $k$, with the bond rung at 8 marking the threshold for sustained molecular recognition and the life rung at 19 marking biological self-sustenance. The counterfactual-floor rung sits at 5 and matches the boundary $Z_{cf} = phi^5$ from the upstream Consciousness module. The vertebrate rung is placed at 12, octopus at 14, cetacean at 15, and human at 17 to align with documented capacities. Upstream results establish that every rung is positive, the successor satisfies the geometric recurrence $Z(k+1) = phi cdot Z(k)$, and the cognition floor equals $phi^5$ by direct substitution.
proof idea
The proof is a term-mode construction that forms the required conjunction from four prior declarations in the same module: the successor recurrence, the strict monotonicity lemma, the full rung-ordering theorem, and the direct equality for the cognition floor.
why it matters
This theorem supplies the consolidated statement of the phi-ladder for animal cognition, fixing the geometric scaling and the relative placement of cognitive milestones required by the Recognition Science forcing chain. It aligns the cognition floor with the counterfactual boundary at phi^5 and thereby anchors the lower end of the ladder used in higher cognition models. The result closes the structural setup for the domain without introducing new hypotheses.
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