Chemistry Rnatargeted Compounds Rnatargeted Compounds Cert

The certificate

RNA-targeted compounds are small molecules that bind to RNA structures, not to proteins. Examples include Risdiplam for spinal muscular atrophy and the investigational drug Branaplam. These compounds can alter how RNA is spliced, translated, or stabilized. In the Recognition Science framework, the binding cost is predicted to follow a quantized ladder based on the golden ratio phi, approximately 1.618.

The declaration RNATargetedCompoundsCert is a machine-checked certificate. It bundles three proven properties of a discrete model of RNA conformational states. Each state sits on a rung of the phi-ladder, and its cost is defined as phi raised to the rung number minus one. The first property states that cost increases monotonically with the rung number. The second states that the reference state at rung zero has zero cost. The third states that this rung-zero state is the global minimum cost among all ladder states.

These three properties are proved as theorems in the framework's machine-checked library of formal theorems, with no unproved assumptions. The certificate simply packages them into a single structure, so that any later work can rely on the whole set at once. It does not assert anything about real RNA molecules, actual drug binding, or clinical efficacy. It establishes only the internal consistency of the cost model.

In Recognition Science, the certificate is a building block. It shows that the phi-ladder model for RNA states is well-formed: costs grow in an orderly way, and the unfolded reference state is the cheapest. This matters because the framework treats the cost of recognition as forced, not chosen. A well-formed cost ladder is a prerequisite for any further claim about how binding energies should behave.

The certificate does not claim that real RNA binding costs match this ladder. That comparison against measured values would be an empirical check, not a theorem. It also does not claim that Risdiplam or Branaplam actually work this way. The certificate is a formal object, not a biological finding.

/-- **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

/-- 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

What this page does not claim

The certificate does not claim that real RNA binding costs match the phi-ladder. It does not claim that Risdiplam or Branaplam bind RNA through this exact mechanism. It does not claim any clinical efficacy for RNA-targeted compounds.

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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