Chemistry Rnatargeted Compounds Rna Cost Monotone

The cost ladder

RNA-targeted compounds are small molecules that bind to RNA structures, not to proteins. Examples include Risdiplam and Branaplam, which are being developed for spinal muscular atrophy. These compounds can change how RNA is spliced, translated, or stabilized. In the Recognition Science (RS) framework, the binding cost of such a compound is not a free parameter. It is a number that comes from a discrete ladder of possible RNA conformations.

The framework models each RNA conformational state as a point on a ladder indexed by a natural number, its rung. The cost of a state is defined as phi raised to the rung number, minus one, where phi is the golden ratio, about 1.618. The unfolded reference state, at rung zero, therefore has a cost of zero. The formal theorem rna_cost_monotone proves that this cost increases monotonically with the rung: if one rung is less than or equal to another, then its cost is also less than or equal. In plain terms, moving up the ladder never decreases the binding cost.

This theorem is a proved statement in the framework's machine-checked library of formal theorems. It is not an empirical claim about real molecules. The theorem does not say that any particular RNA structure actually has a cost equal to phi to some power. It only establishes a formal ordering property of a defined cost function. The framework's library also proves that the rung-zero state is the global minimum among these ladder states, which follows directly from the monotonicity result.

What the theorem does not claim is just as important. It does not claim that Risdiplam or Branaplam actually bind with these exact costs in a laboratory. It does not claim that the golden ratio ladder is the only possible way to assign costs to RNA states. The monotonicity result is a structural fact about a chosen model, not a measurement of nature. The framework's prediction is that real binding costs, if they follow the model, should respect this ordering. That prediction remains to be tested against experimental data.

/-- The φ-ladder of RNA states. -/
def rnaStateAt (n : ℕ) : RNAState where
  rung := n
  cost := phi ^ n - 1
  cost_eq := rfl

/-- The cost increases monotonically with rung. -/
theorem rna_cost_monotone (m n : ℕ) (h : m ≤ n) :
    (rnaStateAt m).cost ≤ (rnaStateAt n).cost := by
  unfold rnaStateAt
  show phi ^ m - 1 ≤ phi ^ n - 1
  have hphi_ge_one : 1 ≤ phi := phi_ge_one
  have h_pow : phi ^ m ≤ phi ^ n := pow_le_pow_right₀ hphi_ge_one h
  linarith

/-- **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

What this page does not claim

This theorem does not claim that any real RNA molecule has a measured binding cost equal to phi to some power. This theorem does not claim that the golden ratio ladder is the only possible model for RNA binding costs. This theorem does not claim that Risdiplam or Branaplam actually follow this cost ordering in experiments.

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.

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