RNAState
plain-language theorem explainer
Discrete RNA conformational states are indexed by a natural-number φ-rung, with J-cost fixed to φ^rung − 1 relative to the unfolded reference. Chemists and biophysicists modeling RNA-binding small molecules (Risdiplam-class splicing modulators) cite this carrier when placing stem-loop costs on the Recognition Science ladder. It is a pure structure with an equality witness, not a proved theorem.
Claim. An RNA conformational state is a pair $(n, c)$ with rung index $n \in \mathbb{N}$ and real cost $c$ satisfying the identity $c = \varphi^n - 1$, where $\varphi$ is the golden ratio and $c$ is the J-cost measured from the unfolded reference conformation.
background
The module treats Element 85 (Domain C): small molecules that bind RNA secondary or tertiary structure and thereby modulate splicing, translation, or stability. Recognition Science predicts that the binding J-cost is quantized on the φ-ladder of stem-loop conformations.
The J-cost is the standard Recognition cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced unique by the T5 step of the unified forcing chain. Here the cost field stores that cost relative to an unfolded reference, specialized to the closed form $\varphi^n - 1$ on rung $n$. Upstream cost definitions (observer forcing, multiplicative recognizers, PRC bridge) all identify event cost with this same $J$.
The golden ratio $\varphi$ itself is the self-similar fixed point forced at T6; the φ-ladder is the discrete mass/cost grading used throughout the monolith (yardstick $\cdot \varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$ in the particle sector).
proof idea
No proof obligations beyond the structure itself. Three fields are declared: a natural rung, a real cost, and a propositional equality pinning cost to $\varphi^{\mathrm{rung}} - 1$. Inhabitants are built by supplying a rung and discharging the equality (typically by rfl after setting cost to that closed form), as the downstream constructor rnaStateAt does in one line.
why it matters
This is the data carrier for the RNA φ-ladder inside Element 85. The sole direct consumer is rnaStateAt, which builds the state at each natural rung and unlocks the monotone-cost and zero-minimum lemmas that follow in the same file. Those lemmas underwrite the claim that RNA-targeted compound binding costs sit on a discrete, strictly increasing ladder rather than a continuum of arbitrary free energies.
In the broader framework the construction is the chemistry-side instance of the same φ-grading that appears in the mass formula and in the eight-tick / D = 3 forcing chain (T6–T8). It does not yet close a paper proposition by itself; it supplies the typed state space on which the certified cost inequalities of RNATargetedCompoundsCert are stated.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.