rna_state_zero_minimum
plain-language theorem explainer
Among the φ-ladder RNA conformational states, the rung-0 reference state has globally minimal recognition cost: its cost is at most the cost of the state at any natural rung n. Chemists and RS modelers of RNA-targeted binders (Risdiplam-class) cite this as the baseline minimum on the ladder. The proof reduces to zero cost at rung 0, then shows φ^n − 1 ≥ 0 by induction from φ ≥ 1.
Claim. For every natural number $n$, the recognition cost of the rung-$0$ RNA state is at most the cost of the rung-$n$ state: $C_0 \le C_n$, where $C_k = \varphi^k - 1$.
background
This module treats Element 85 (Domain C): small molecules that bind RNA secondary or tertiary structure and modulate splicing, translation, or stability. Recognition Science predicts that binding J-cost is quantized on a φ-ladder of stem-loop conformations.
Each ladder state is an RNAState with rung index $n$ and cost $\varphi^n - 1$. The golden ratio satisfies $\varphi \ge 1$ (in fact $\varphi > 1$). The companion fact that the reference state has zero cost, $C_0 = 0$, is already proved by unfolding the ladder definition at $n = 0$.
The same ladder is monotone in rung (cost nondecreasing), so the zero rung is the natural candidate for a global minimum among discrete ladder states.
proof idea
Rewrite the left-hand side with the zero-cost lemma, so the goal is $0 \le (\mathrm{rnaStateAt}, n).\mathrm{cost}$. Unfold the ladder definition to $0 \le \varphi^n - 1$. From $\varphi \ge 1$, prove $1 \le \varphi^n$ by induction on $n$: the zero case is trivial; the successor step uses $\mathrm{pow_succ}$ and multiplicative monotonicity of nonnegative factors. Close with linarith.
why it matters
This is the module's master minimum theorem for the RNA φ-ladder. It is packaged into the master certificate rnaTargetedCompoundsCert as the reference_minimum field, alongside monotonicity and zero reference cost. That certificate is the Lean-side closure for Domain C claims about RNA-targeted compounds: binding costs sit on a ladder whose floor is the rung-0 conformation.
In the broader RS picture the cost is the standard J-derived ladder excess $\varphi^n - 1$, consistent with φ forced as the self-similar scale (T6) and with recognition cost structure used across chemistry modules. No sorry remains in the module.
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