Physics Anchor Policy Model F Residue Model

The model residue

In the Recognition Science framework's machine-checked library of formal theorems, recognition (the framework's term for a discrete record of events) often appears alongside conventional physics. For the Standard Model's renormalization group, the library does not yet implement the full multi-loop kernels and numerical integration inside its proof system. Instead, it defines a computable stand-in called f_residue_model.

The definition is simple: for any fermion and any scale, the model residue equals a quantity written as gap(ZOf f). The function ignores its scale argument entirely, so it is constant in scale. That makes two consequences immediate: the residue is stationary at every point, and its second derivative is identically zero, hence bounded. If two fermions have the same ZOf value, the model gives them the same residue by definition.

These are algebraic conveniences, not physical discoveries. The docstring states plainly that this is not a proof of the Standard Model renormalization group statement. The model exists to make downstream algebraic consequences executable and to avoid axiom dependencies when working in a setting where the anchor identity holds by construction. The full physics, with real scale dependence and threshold policy, remains a target.

/-- 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]

/-- 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 does not prove the Standard Model renormalization group statement. This does not show that the real residue is constant in scale. This does not derive the value of the residue from first principles.

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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