Chemistry Rnatargeted Compounds Rna State Zero Minimum

The reference state

RNA-targeted compounds are small molecules that bind to RNA structures rather than to proteins. Examples include Risdiplam, used for spinal muscular atrophy, and Branaplam. These drugs bind to RNA secondary or tertiary structures and modulate splicing, translation, or stability. In the Recognition Science framework, the binding cost is modeled as a discrete quantity indexed by a ladder of conformational states.

The framework defines an RNAState as a natural number rung plus a real-valued cost. The cost is phi raised to the rung minus one, where phi is the golden ratio. The reference state, rung zero, has cost zero. A theorem in the machine-checked library of formal theorems proves that this reference state is the global cost minimum among all rung states: for any natural number n, the cost at rung zero is less than or equal to the cost at rung n. The proof uses the monotonicity of the power function and the fact that phi is at least one.

The declaration rna_state_zero_minimum establishes this minimum property as a formal theorem. It is part of a certificate structure that also records monotonicity and the zero-cost reference. The certificate assembles three theorems: costs increase with rung, the reference cost is zero, and the reference is the minimum. The library reports zero sorry and zero axiom, meaning the proof is complete within the framework's postulates.

What the declaration does not claim is equally important. It does not state that any real RNA molecule has zero binding cost. The cost is relative to an unfolded reference, not an absolute free energy. It does not claim that the golden ratio ladder matches measured drug binding affinities. That comparison against experimental data is an empirical check, not a theorem. The declaration concerns only the formal structure of the ladder, not the chemistry of actual compounds.

/-- The reference state (rung 0) has zero cost. -/
theorem rna_state_zero_cost : (rnaStateAt 0).cost = 0 := by
  unfold rnaStateAt; simp

/-- **MASTER THEOREM**: the rung-0 state is the global cost minimum
    among the φ-ladder RNA states. -/
theorem rna_state_zero_minimum (n : ℕ) :
    (rnaStateAt 0).cost ≤ (rnaStateAt n).cost := by
  rw [rna_state_zero_cost]
  -- (rnaStateAt n).cost = phi^n - 1 ≥ 0 since phi ≥ 1.
  unfold rnaStateAt
  show 0 ≤ phi ^ n - 1
  have hphi_ge_one : 1 ≤ phi := phi_ge_one
  have h_pow : 1 ≤ phi ^ n := by
    induction n with
    | zero => simp
    | succ k ih =>
        rw [pow_succ]
        have : 1 * 1 ≤ phi ^ k * phi := mul_le_mul ih hphi_ge_one (by norm_num) (by positivity)
        linarith
  linarith

def rnaTargetedCompoundsCert : RNATargetedCompoundsCert where
  state_monotone := rna_cost_monotone
  reference_zero := rna_state_zero_cost
  reference_minimum := rna_state_zero_minimum

What this page does not claim

The theorem does not assign a physical free energy to any real RNA molecule. The theorem does not predict the binding affinity of any specific drug compound. The golden ratio ladder is a definitional model, not a derived law of chemistry.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/RNATargetedCompounds.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: