rogers_ramanujan_stability
plain-language theorem explainer
The Rogers-Ramanujan stability theorem states that gap-1 occupations on the phi-ladder carry strictly positive J-cost and collapse under the golden recurrence while gap-0 is costless, thereby enforcing the differ-by-at-least-two rule for stable partitions. Partition theorists and number theorists would cite it when grounding the classical identities in a cost-minimization principle on the phi-ladder. The proof is a one-line wrapper that assembles three prior lemmas on cost positivity, absorption, and triviality.
Claim. On the golden-ratio ladder with positions at powers of the golden ratio, the J-cost of a unit gap satisfies $J > 0$, every pair of adjacent rungs sums to a single higher rung via the relation $x + x r = x r^2$ where $r = (1 + 2)/2$, and the zero-gap configuration has cost exactly zero.
background
The phi-ladder consists of discrete positions at powers of the golden ratio for integer indices, with J-cost defined as $J(x) = (x + x^{-1})/2 - 1$ for positive ratios x. Gap cost measures the interaction penalty of occupying two rungs separated by a given integer distance, while phiLadderPosition maps an index to its corresponding power. The module derives the Rogers-Ramanujan differ-by-at-least-two rule directly from J-cost admissibility on this ladder. Upstream results supply the cost function via the multiplicative recognizer comparator and the observer-forcing cost definition, both of which return non-negative real values derived from the J functional.
proof idea
The proof is a one-line wrapper that applies the three lemmas gap1_cost_positive, adjacent_absorptive, and gap0_cost_zero. It constructs the conjunction by pairing the strict positivity of unit-gap cost, the existence of a collapse target for every adjacent pair, and the zero cost of the trivial gap.
why it matters
This declaration supplies the J-cost justification for the classical Rogers-Ramanujan partition rule inside the Recognition Science framework. It closes the link between the phi-ladder geometry (T5 J-uniqueness and T6 self-similar fixed point) and the stability of non-adjacent occupations, consistent with Zeckendorf representations. No downstream theorems are recorded yet, leaving open the question of whether the same cost constraint generates the mod-5 congruence side of the identities.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.