Chemistry Rnatargeted Compounds

RNA-targeted compounds

RNA-targeted compounds are small molecules that bind to RNA secondary or tertiary structures rather than to proteins, modulating processes like splicing, translation, or stability. Examples include Risdiplam for spinal muscular atrophy and Branaplam. This approach expands the druggable space beyond proteins and has already produced approved therapies.

In Recognition Science, the framework models each discrete RNA conformational state as a rung on a ladder. The ledger, a discrete record of recognition events, assigns each state a cost relative to the unfolded reference. The framework's central forcing theorem, which derives the cost function J(x) = (x + 1/x)/2 - 1 from five plain conditions, determines that these costs are quantized by the golden ratio φ. The definition is explicit: a state at rung n has cost φ^n - 1.

The framework establishes three facts. First, cost increases monotonically with rung: higher rungs, representing more folded or complex conformations, always carry greater cost. Second, the reference state at rung zero has exactly zero cost. Third, that reference state is the global minimum among all ladder states. These are proved theorems in the machine-checked library of formal theorems, with no unproved assumptions.

The practical picture is a discrete energy profile for RNA conformations, where binding a small molecule corresponds to moving between rungs. The framework predicts that the binding J-cost is quantized by the φ-ladder, meaning only certain conformational changes are recognized at low cost. This is a hypothesis with a named falsifier: if a measured binding energy for an RNA-targeted compound deviates from the predicted φ-ladder spacing by more than experimental error, the prediction fails.

What this changes is how one thinks about drug discovery: instead of screening blindly, the framework offers a principled rule for which RNA conformations are cheap to reach. The monotonicity theorem guarantees that the unfolded reference is always the cheapest state, so any binding event must pay a positive cost that grows with structural complexity.

/-- A discrete RNA conformational state, indexed by the φ-rung. -/
structure RNAState where
  rung : ℕ
  /-- The state's J-cost relative to the unfolded reference. -/
  cost : ℝ
  cost_eq : cost = phi ^ rung - 1

/-- 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

/-- 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

/-- **MASTER CERTIFICATE.** -/
structure RNATargetedCompoundsCert where
  state_monotone :
    ∀ m n : ℕ, m ≤ n → (rnaStateAt m).cost ≤ (rnaStateAt n).cost
  reference_zero : (rnaStateAt 0).cost = 0
  reference_minimum : ∀ n : ℕ, (rnaStateAt 0).cost ≤ (rnaStateAt n).cost

What this page does not claim

No measured binding energies are cited; the quantization is a prediction, not a measurement. The framework does not derive the existence of RNA-targeted compounds; it models their states after the fact. No claim is made that all RNA conformations fall on the ladder, only that the framework models them so.

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)

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