Physics Anchor Policy Model

The anchor policy model

A physics anchor policy is a rule that pins a theory's behavior at a chosen reference scale. In the Recognition Science framework, the Standard Model's running coupling residue is treated as an opaque interface, a black box whose exact value the framework does not yet compute. The module AnchorPolicyModel.lean provides a Lean-native model of that residue, a stand-in that is by definition equal to a closed-form display function. This makes downstream algebraic consequences executable and avoids axiom dependencies when working in the model where the anchor identity holds by construction.

The model defines the residue as constant in scale, equal to the gap of the Z-factor. Because the residue ignores its scale argument, it is stationary at every point, its second derivative is identically zero, and equal Z-factors give equal residues. These are theorems in the machine-checked library of formal theorems, not assumptions. The practical effect is that stability and robustness become trivial in the model, which is useful for testing algebraic structure before the full multi-loop RG kernels and numerical integration are implemented.

This is not a proof of the Standard Model renormalization group statement. It is a definitional model, a deliberate simplification that makes the anchor identity hold by construction. The real computation, with threshold policy and numerical integration, remains open. The model's value is that it lets the framework reason about the consequences of the anchor identity without waiting for the full physics engine to be built.

/-- A computable stand-in for the RG residue: constant in scale and equal to the display `gap(Z)`.

This encodes the single-anchor closed form as a *definition* rather than a phenomenology axiom. -/
noncomputable def f_residue_model (f : Fermion) (_mu : ℝ) : ℝ :=
  gap (ZOf f)

/-- In the model, the residue is constant in `t`, hence stationary at every point. -/
theorem stationary_at_any (muStar : ℝ) (f : Fermion) :
    deriv (fun t => f_residue_model f (Real.exp t)) (Real.log muStar) = 0 := by
  -- `f_residue_model` ignores its scale argument, so the function of `t` is constant.
  simp [f_residue_model]

/-- In the model, the second derivative is also identically zero, hence bounded. -/
theorem stability_bound_at_any (muStar : ℝ) :
    ∃ (ε : ℝ), 0 < ε ∧ ∀ (f : Fermion),
      |deriv (deriv (fun t => f_residue_model f (Real.exp t))) (Real.log muStar)| < ε := by
  refine ⟨1, by norm_num, ?_⟩
  intro f
  -- Second derivative of a constant is 0.
  simp [f_residue_model]

/-- Equal-Z degeneracy holds by definition in the model. -/
theorem equalZ_at_any {f g : Fermion} (hZ : ZOf f = ZOf g) (mu : ℝ) :
    f_residue_model f mu = f_residue_model g mu := by
  simp [f_residue_model, hZ]

What this page does not claim

This module proves the Standard Model renormalization group statement. The model computes the actual numerical value of the running coupling residue. The model is a physical law; it is a definitional simplification for algebraic testing.

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/AnchorPolicyModel.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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