Chemistry Rnatargeted Compounds Rnastate

The RNA state ladder

RNAState is a formal structure in the Recognition Science framework's machine-checked library. It defines a discrete RNA conformational state as a pair: a natural number rung (an index on a ladder) and a real number cost (a measure of energetic penalty relative to an unfolded reference). The defining equation is cost = phi ^ rung - 1, where phi is the golden ratio (1.618...). This is a definition, not a theorem; it chooses a model of how RNA states are priced.

The library proves three properties of this definition. First, the cost increases monotonically with the rung: if m is at most n, then the cost at rung m is at most the cost at rung n. Second, the reference state at rung 0 has cost exactly 0. Third, that rung-0 state is the global cost minimum among all phi-ladder RNA states. These are proved theorems, checked by the library's kernel with no axioms beyond the standard three. The first two are direct consequences of the definition; the third follows from the first two.

In plain terms, the model says: an RNA molecule can occupy a discrete set of states, each labeled by a natural number; the cost of a state grows exponentially with its label, and the lowest state costs nothing. The framework's broader claim is that this phi-ladder of costs predicts the binding J-cost of RNA-targeted compounds such as Risdiplam and Branaplam. That prediction is a hypothesis, not a theorem; the library does not prove that real small molecules actually bind according to this ladder.

The declaration does not claim that all RNA conformational changes follow this ladder, nor that the model matches any measured binding data. It also does not claim that the golden ratio is the only possible scaling; the definition simply chooses it. The library's contribution is the formal structure and the monotonicity and minimum theorems, not an empirical validation.

/-- 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

The declaration does not prove that real small molecules bind according to the phi-ladder. The declaration does not claim that the golden ratio is the only possible scaling for RNA costs. The declaration does not provide any measured binding energies or empirical validation.

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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