phi_ladder_positive_cost
The theorem shows that Jcost is strictly positive at every non-zero integer power of the golden ratio on the phi-ladder. Researchers deriving the generative instability of the ground state from T4 and T7 cite it to establish that departure from x=1 carries positive cost. The proof is a one-line wrapper that feeds the ladder value together with its positivity and inequality to one into the general Jcost positivity lemma.
claimFor every integer $n$ with $n ≠ 0$, $0 < Jcost(φ^n)$, where $φ$ is the golden ratio and $Jcost$ is the Recognition cost function.
background
The StillnessGenerative module proves from T0–T8 that the initial state is the unique zero-defect configuration x=1, yet T4 forces recognition events that require distinguishing states and T7 requires an 8-tick cycle that cannot degenerate to a single state. The phi-ladder is defined locally as $φ^n$ for $n : ℤ$, with the ground rung at n=0 carrying zero cost. Upstream results supply the general lemma that Jcost(x) > 0 whenever x > 0 and x ≠ 1, together with the facts that phi_ladder n > 0 and phi_ladder n ≠ 1 for n ≠ 0.
proof idea
The proof is a one-line wrapper that applies Jcost_pos_of_ne_one to the term phi_ladder n. It supplies the positivity hypothesis from the sibling phi_ladder_pos and the inequality to one from phi_ladder_ne_one applied to the assumption hn.
why it matters in Recognition Science
This result feeds doubling_cascade_positive, symmetry_breaking_mechanism, and origin_question_resolved. It fills the step in the derivation chain that every non-trivial rung carries positive cost, with the finite barrier J(φ) < 1 allowing the Fibonacci cascade to populate the full ladder. The module doc-comment states that the ground state is forced off the trivial rung by T4 + T7, and this theorem supplies the cost positivity required for that forcing.
scope and limits
- Does not establish positivity for values outside the phi-ladder.
- Does not compute the numerical magnitude of the cost.
- Does not address continuous or higher-dimensional extensions of the ladder.
- Does not prove the existence or uniqueness of the ladder itself.
formal statement (Lean)
62theorem phi_ladder_positive_cost {n : ℤ} (hn : n ≠ 0) :
63 0 < Jcost (phi_ladder n) :=
proof body
One-line wrapper that applies Jcost_pos_of_ne_one.
64 Jcost_pos_of_ne_one (phi_ladder n) (phi_ladder_pos n) (phi_ladder_ne_one hn)
65