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theorem proved term proof

stability_bound_at_anchor

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formal statement (Lean)

 172theorem stability_bound_at_anchor (f_residue : Fermion → ℝ → ℝ)
 173    (h_bounded : ∀ f, |deriv (deriv (fun t => f_residue f (Real.exp t))) (Real.log muStar)| < 1) :
 174    ∀ (A : AnchorSpec), A.equalWeight →
 175      (A.muStar = muStar) →  -- Anchor alignment
 176      ∃ (ε : ℝ), 0 < ε ∧ ∀ (f : Fermion),
 177        |deriv (deriv (fun t => f_residue f (Real.exp t))) (Real.log A.muStar)| < ε := by

proof body

Term-mode proof.

 178  -- Follows from the smoothness of RG flow in the vicinity of the anchor.
 179  -- REDUCTION: Perturbative unitarity ensures derivatives of anomalous dimensions are bounded.
 180  intro A _hA hA_eq
 181  -- The second derivative is bounded by 1 at muStar by hypothesis.
 182  -- Since A.muStar = muStar, the bound transfers directly.
 183  use 1
 184  constructor
 185  · norm_num
 186  · intro f
 187    rw [hA_eq]
 188    exact h_bounded f
 189
 190/-- **DISPLAY IDENTITY THEOREM**
 191
 192    At μ⋆, the RG residue equals F(Z) = gap(Z).
 193
 194    **Proof Structure**:
 195    1. The Display Function $F(Z) = \log_\phi (1 + Z/\phi)$ represents the geometric cost of a ledger state.
 196    2. The Single Anchor Policy posits a one-to-one mapping between geometric charges $Z$ and residues $f$.
 197    3. Specifically, the integrated anomalous dimension from $\mu_{\star}$ to the physical mass scale
 198       is forced by RS to match the geometric gap $F(Z)$.
 199    4. This matches the `RGTransport.anchorClaimHolds` predicate.
 200
 201    **STATUS**: HYPOTHESIS (RS-SM bridge) -/

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