Chemistry Rnatargeted Compounds Rna State Zero Cost

The reference state

RNA-targeted compounds are small molecules that bind to RNA structures rather than proteins, and they include approved drugs such as Risdiplam for spinal muscular atrophy. In the Recognition Science framework, this binding is modeled as a recognition event with a forced cost. The framework defines a discrete RNA conformational state, indexed by a natural number rung, and assigns each state a cost relative to an unfolded reference. The cost formula is cost = phi ^ rung - 1, where phi is the golden ratio.

The declaration rna_state_zero_cost is a machine-checked theorem stating that the reference state, at rung 0, has cost exactly zero. This is not an assumption or a definition; it is proved by unfolding the definition of the state and simplifying. The same library also proves that cost increases monotonically with rung and that the rung-0 state is the global cost minimum among all ladder states. These three results form a master certificate for the RNA-targeted compounds domain.

In Recognition Science, the framework's library is a machine-checked collection of formal theorems. The theorem establishes a structural fact about the model: the unfolded reference is the cheapest possible state. This is a definitional consequence of the cost formula, not an empirical measurement. The framework predicts that the binding cost is quantized by the phi-ladder of RNA stem-loop conformations, but the theorem itself only concerns the formal model.

The practical consequence is that the framework's model has a well-defined baseline: the unfolded RNA is the zero-cost state, and every folded state costs more. This makes the model internally consistent for comparing relative binding costs. The theorem does not claim that real RNA molecules have zero energy when unfolded, nor does it claim that the golden ratio ladder has been experimentally verified. It establishes a property of the formal construction.

/-- The reference state (rung 0) has zero cost. -/
theorem rna_state_zero_cost : (rnaStateAt 0).cost = 0 := by
  unfold rnaStateAt; simp

/-- 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

The theorem does not claim real unfolded RNA molecules have zero physical energy. The theorem does not claim the golden ratio ladder has been experimentally verified for any drug. The theorem does not derive the fine-structure constant or any coupling constant.

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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