phiLadderPosition
The definition assigns to each integer n the real number phi raised to the power n, fixing the coordinate system for rungs on the phi-ladder. Partition theorists and Recognition Science researchers cite it when deriving stability constraints from J-cost. It is introduced as a direct abbreviation of the exponential map on the golden ratio.
claimThe position on the φ-ladder at integer index $n$ is given by $φ^n$.
background
In the φ-Ladder Stability module, positions are defined as powers of the golden ratio φ. This coordinate system interprets the Rogers-Ramanujan identities as J-cost admissibility constraints, where the golden recurrence φ² = φ + 1 forces adjacent rungs (gap 1) to collapse into a single higher rung. The J-cost for ratio φ is positive, marking such occupations as unstable minima. The module links this to Zeckendorf's theorem, in which non-consecutive Fibonacci sums correspond exactly to gap ≥ 2 occupations on the ladder.
proof idea
Direct definition using real exponentiation: phiLadderPosition n is set equal to phi ^ n.
why it matters in Recognition Science
This definition supplies the basic rung coordinates for the Rogers-Ramanujan stability theorem, which derives the classical 'differ by ≥ 2' rule from positive J-cost and absorption on the phi-ladder. It is referenced by adjacent_collapses, adjacent_absorptive, adjacent_phi_ratio, phiLadderPosition_pos, and rogers_ramanujan_stability. In the framework it realizes the self-similar fixed point phi as the generator of the ladder structure.
scope and limits
- Does not prove positivity or monotonicity of the positions.
- Does not compute J-cost or gapCost values.
- Does not derive the full Rogers-Ramanujan identities.
- Does not address physical constants or forcing chain steps beyond the ladder.
formal statement (Lean)
125def phiLadderPosition (n : ℤ) : ℝ := phi ^ n
proof body
Definition body.
126
127/-- φ-ladder positions are always positive. -/