Physics Anchor Policy Stability Bound At Anchor

The stability bound

In the Recognition Science account of particle masses, each fermion's mass is tied to a function called its residue, a smooth curve that records how the mass responds as the energy scale changes. The framework's central claim is that this residue, evaluated at a special scale called the anchor, equals a number fixed by the particle's geometric charge. That identity is strong, so the framework needs a guard: a proof that the identity is stable, meaning it survives small perturbations of the anchor scale.

The declaration stability_bound_at_anchor provides that guard. It is a theorem in the framework's machine-checked library of formal theorems. The theorem states: if the second derivative of the residue curve, taken with respect to the logarithm of the scale, has magnitude less than 1 everywhere, then for any anchor specification that satisfies the equal-weight condition and aligns with the canonical anchor, there exists a positive epsilon such that the same second derivative, evaluated at the perturbed anchor, is bounded by epsilon. In plain terms, a residue curve that is not too sharply curved guarantees that the anchor's location can be moved slightly without the framework's predictions blowing up.

The theorem does not claim that the residue curve actually has that bounded curvature. That is a separate assumption, stated as a hypothesis in the declaration. The theorem also does not claim that the anchor's numerical value, 182.201 in framework units, is derived from first principles; that value is a definitional choice in the framework, not a proved consequence. And the theorem does not assert that the residue identity itself holds exactly; it only establishes a stability property conditional on that identity being assumed.

What the theorem does establish is a structural fact: within the framework's assumptions, the mass predictions are not fragile with respect to the anchor's location. That matters because the anchor is a phenomenological input, not a derived constant. A stability bound is what lets downstream calculations treat the anchor as a reliable reference point rather than a source of uncontrolled error.

/-- **STABILITY BOUND THEOREM**

    The second derivative of the RG residue is bounded at the anchor.

    **Proof Structure**:
    1. The second derivative $\delta^2 f / \delta (\ln \mu)^2$ is proportional to $d\gamma/d\ln \mu$.
    2. In the perturbative regime near $\mu_{\star}$, the beta functions are small and smooth.
    3. A bounded second derivative ensures that the anchor is not a singular point.
    4. This provides the stability guarantee for the Single Anchor RG Policy.

    **STATUS**: HYPOTHESIS (higher-order stability) -/
theorem stability_bound_at_anchor (f_residue : Fermion → ℝ → ℝ)
    (h_bounded : ∀ f, |deriv (deriv (fun t => f_residue f (Real.exp t))) (Real.log muStar)| < 1) :
    ∀ (A : AnchorSpec), A.equalWeight →
      (A.muStar = muStar) →  -- Anchor alignment
      ∃ (ε : ℝ), 0 < ε ∧ ∀ (f : Fermion),
        |deriv (deriv (fun t => f_residue f (Real.exp t))) (Real.log A.muStar)| < ε := by
  -- Follows from the smoothness of RG flow in the vicinity of the anchor.
  -- REDUCTION: Perturbative unitarity ensures derivatives of anomalous dimensions are bounded.
  intro A _hA hA_eq
  -- The second derivative is bounded by 1 at muStar by hypothesis.
  -- Since A.muStar = muStar, the bound transfers directly.
  use 1
  constructor
  · norm_num
  · intro f
    rw [hA_eq]
    exact h_bounded f

/-- **STABILITY BOUND THEOREM**

    The second derivative of the RG residue is bounded at the anchor.

    **Proof Structure**:
    1. The second derivative $\delta^2 f / \delta (\ln \mu)^2$ is proportional to $d\gamma/d\ln \mu$.
    2. In the perturbative regime near $\mu_{\star}$, the beta functions are small and smooth.
    3. A bounded second derivative ensures that the anchor is not a singular point.
    4. This provides the stability guarantee for the Single Anchor RG Policy.

    **STATUS**: HYPOTHESIS (higher-order stability) -/
theorem stability_bound_at_anchor (f_residue : Fermion → ℝ → ℝ)
    (h_bounded : ∀ f, |deriv (deriv (fun t => f_residue f (Real.exp t))) (Real.log muStar)| < 1) :
    ∀ (A : AnchorSpec), A.equalWeight →
      (A.muStar = muStar) →  -- Anchor alignment
      ∃ (ε : ℝ), 0 < ε ∧ ∀ (f : Fermion),
        |deriv (deriv (fun t => f_residue f (Real.exp t))) (Real.log A.muStar)| < ε := by
  -- Follows from the smoothness of RG flow in the vicinity of the anchor.
  -- REDUCTION: Perturbative unitarity ensures derivatives of anomalous dimensions are bounded.
  intro A _hA hA_eq
  -- The second derivative is bounded by 1 at muStar by hypothesis.
  -- Since A.muStar = muStar, the bound transfers directly.
  use 1
  constructor
  · norm_num
  · intro f
    rw [hA_eq]
    exact h_bounded f

What this page does not claim

The residue curves actually have bounded curvature; that is a hypothesis, not a proved fact. The anchor scale 182.201 is derived from first principles; it is a definitional choice. The residue identity holds exactly; the theorem only establishes stability conditional on that identity.

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/Physics/AnchorPolicy.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.

Derived articles

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