z_rung_zero
plain-language theorem explainer
z_rung_zero establishes that the base rung of the Z-complexity ladder equals unity. Cognition researchers in Recognition Science cite it to anchor the geometric sequence of animal thresholds. The proof reduces directly by unfolding the definition of z_rung and simplifying the power.
Claim. $Z_0 = 1$, where the Z-complexity at integer rung $k$ is given by $Z_k := phi^k$.
background
The Animal Z-Complexity Bound module introduces the φ-ladder for animal cognition in Recognition Science. The function z_rung maps each natural number rung k to Z_k = φ^k, with φ the self-similar fixed point from the J-cost equation in PhiForcingDerived.of. Upstream results include the definition of z_rung itself and supporting structures from NucleosynthesisTiers.of for physical tiers and DAlembert.LedgerFactorization.of for the underlying (R_+, ×) calibration. The module establishes that the ladder is positive, strictly increasing, and orders key thresholds such as the bond rung at k=8 and the life rung at k=19.
proof idea
The proof is a one-line wrapper that unfolds the definition of z_rung and applies the simp tactic to evaluate φ^0 as 1.
why it matters
This result supplies the base case for the master certificate animalZComplexityBoundCert, which aggregates rung properties for the cognition domain. It aligns with the Recognition Science forcing chain at T6, where φ emerges as the fixed point, and supports the placement of cognitive milestones on the ladder without claiming empirical derivations.
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