IsStablePosition
IsStablePosition declares a real number x stable precisely when it is positive and lies in the phi-ladder set. Researchers modeling J-cost minimization under self-similarity cite this predicate to mark quantized equilibrium states. The definition is a direct conjunction of positivity with membership in the set of all phi powers.
claimA real number $x$ is a stable position when $x > 0$ and $x = phi^n$ for some integer $n$.
background
The module develops the golden ratio as the fixed point that minimizes J-cost for self-similar patterns, where stability requires the ratio to satisfy $r^2 = r + 1$. PhiLadder is the set of all $phi^n$ for $n$ in the integers; upstream definitions in both PhiEmergence and YangMillsMassGap characterize these elements as the positive candidates that survive J-cost minimization. The local setting is therefore the emergence of discrete positions from continuous self-similar cost reduction.
proof idea
The definition is a direct conjunction: positivity of $x$ together with membership in the set PhiLadder. It applies the upstream PhiLadder construction without further reduction.
why it matters in Recognition Science
This predicate supplies the left-hand side of the hypothesis H_StableIffPhiLadder, the core claim that J-cost minimization forces stable states exactly onto the phi-ladder. It is invoked by the theorem phi_ladder_stable, which shows every ladder element satisfies the predicate. In the Recognition framework the definition closes the step from the self-similar fixed point (T6) to the discrete ladder that underlies mass quantization and the eight-tick octave.
scope and limits
- Does not compute the explicit J-cost value at a given x.
- Does not prove that every stable position lies on the ladder.
- Does not treat zero or negative reals.
- Does not specify the local-minimum condition beyond the conjunction.
Lean usage
example (n : ℤ) : IsStablePosition (phi ^ n) := phi_ladder_stable nformal statement (Lean)
198def IsStablePosition (x : ℝ) : Prop :=
proof body
Definition body.
199 x > 0 ∧ x ∈ PhiLadder
200
201/-- All φ-ladder positions are stable -/