Chemistry Rnatargeted Compounds Rna State At

The RNA state ladder

RNA-targeted compounds are small molecules that bind to RNA structures, not to proteins. Drugs such as Risdiplam for spinal muscular atrophy and the investigational Branaplam work this way, modulating how RNA is spliced, translated, or stabilized. The Recognition Science framework models each possible RNA conformation as a discrete state on a ladder indexed by a natural number, called its rung. The framework's declaration rnaStateAt constructs the state at any given rung and assigns it a cost, defined as phi raised to the rung number, minus one.

Here phi is the golden ratio, approximately 1.618, the same number that appears in pentagons and Fibonacci sequences. The cost formula means the states are not evenly spaced: each step up the ladder multiplies the cost by phi. The rung-0 state, the unfolded reference, has cost exactly zero. The framework proves three properties of this ladder: cost increases monotonically as the rung number rises, the reference state has zero cost, and the reference state is the global minimum among all ladder states. These are formal theorems in the framework's machine-checked library of formal theorems, with no unproved axioms.

The declaration itself is a definition, not a measurement or a physical law. It establishes what a state is within the framework and how its cost is computed, nothing more. The framework predicts that the binding cost for real RNA-targeted compounds is quantized along this phi-ladder, but that prediction is a hypothesis, not a proved theorem. The declaration does not name any specific drug, any particular RNA sequence, or any measured binding affinity. It does not claim that every RNA structure in a cell actually sits on this ladder, only that the framework models them this way.

What the declaration does establish is a clean, testable structure. A researcher can take a measured binding cost for a compound, compare it to the nearest rung value, and see whether the phi-ladder pattern holds. The monotonicity theorem guarantees that higher rungs always cost more, which gives the ladder a definite order. The zero-cost reference gives a natural baseline. These properties make the model useful for organizing data, even before any experimental confirmation arrives.

/-- 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

What this page does not claim

The declaration does not claim any specific drug binds to any specific RNA sequence. The declaration does not claim the phi-ladder is a measured property of real RNA molecules. The declaration does not claim every RNA conformation in a cell lies on this discrete ladder.

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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