Physics Anchor Policy Display Identity At Anchor

The anchor identity

In particle physics, a fermion's mass is not a single number but a function of the energy scale at which you probe it, a phenomenon called the running mass. The Recognition Science framework (RS) organizes this running behavior around a special scale, the anchor scale, where the framework's geometric charges and physical masses are meant to line up. The declaration display_identity_at_anchor states the core assumption of this organization: at the anchor scale, the integrated mass correction for any fermion equals a fixed function F of that fermion's geometric charge Z. In plain terms, the correction is completely determined by the charge, with no extra dependence on the fermion's identity or the exact scale.

The declaration is a theorem in the machine-checked library, but its content is a hypothesis about physics, not a derived law. The theorem proves a conditional statement: if the exact identity holds for all scales (that is, if the residue function always equals F of the charge), then it holds at the anchor scale. The library also proves a stability bound: if the residue's curvature at the anchor is small, then small perturbations keep the identity approximately true. These are formal statements about a mathematical model, not measurements of nature.

What the declaration does not claim is the important part. It does not assert that the exact identity actually holds in the real world; that is left as an open hypothesis, with the actual numerical transport of quantum chromodynamics and quantum electrodynamics performed by external tools, not by the proof assistant. It does not claim that the anchor scale has a specific measured value; the canonical anchor at 182.201 is a definitional choice, a placeholder for numerical checks. And it does not claim that the mass spectrum is derived from first principles; the ratio of muon to electron mass, for instance, follows only under the assumed identity and the specific charge assignments, which are themselves definitions.

In Recognition Science, the value of this declaration is architectural. It gives downstream modules a clean, auditable interface: a single hypothesis about the residue function, from which the framework can derive statements about mass ratios and stability. It isolates the empirical residue as an assumption, so that any failure of the framework's predictions can be traced to that assumption rather than to an error in the formal derivation. The declaration is a scaffold, not a discovery.

/-- **DISPLAY IDENTITY THEOREM**

    At μ⋆, the RG residue equals F(Z) = gap(Z).

    **Proof Structure**:
    1. The Display Function $F(Z) = \log_\phi (1 + Z/\phi)$ represents the geometric cost of a ledger state.
    2. The Single Anchor Policy posits a one-to-one mapping between geometric charges $Z$ and residues $f$.
    3. Specifically, the integrated anomalous dimension from $\mu_{\star}$ to the physical mass scale
       is forced by RS to match the geometric gap $F(Z)$.
    4. This matches the `RGTransport.anchorClaimHolds` predicate.

    **STATUS**: HYPOTHESIS (RS-SM bridge) -/
theorem display_identity_at_anchor (_γ : AnomalousDimension) (f_residue : Fermion → ℝ → ℝ)
    (h_exact : ∀ f μ, f_residue f μ = F (ZOf f)) :  -- Exact RS identity
    ∀ (A : AnchorSpec), A.equalWeight →
      ∀ (f : Fermion), f_residue f A.muStar = F (ZOf f) := by
  -- The fundamental mapping from RS geometric charges Z to physical mass residues.
  -- With the exact RS identity as hypothesis, this is immediate.
  intro A _hA f
  exact h_exact f A.muStar

/-- **DISPLAY IDENTITY THEOREM**

    At μ⋆, the RG residue equals F(Z) = gap(Z).

    **Proof Structure**:
    1. The Display Function $F(Z) = \log_\phi (1 + Z/\phi)$ represents the geometric cost of a ledger state.
    2. The Single Anchor Policy posits a one-to-one mapping between geometric charges $Z$ and residues $f$.
    3. Specifically, the integrated anomalous dimension from $\mu_{\star}$ to the physical mass scale
       is forced by RS to match the geometric gap $F(Z)$.
    4. This matches the `RGTransport.anchorClaimHolds` predicate.

    **STATUS**: HYPOTHESIS (RS-SM bridge) -/
theorem display_identity_at_anchor (_γ : AnomalousDimension) (f_residue : Fermion → ℝ → ℝ)
    (h_exact : ∀ f μ, f_residue f μ = F (ZOf f)) :  -- Exact RS identity
    ∀ (A : AnchorSpec), A.equalWeight →
      ∀ (f : Fermion), f_residue f A.muStar = F (ZOf f) := by
  -- The fundamental mapping from RS geometric charges Z to physical mass residues.
  -- With the exact RS identity as hypothesis, this is immediate.
  intro A _hA f
  exact h_exact f A.muStar

/-- **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 exact display identity is not asserted to hold in nature; it is a hypothesis. The anchor scale 182.201 is not a measured value but a definitional placeholder. The mass spectrum is not derived from first principles; it depends on assumed charge assignments and the identity hypothesis.

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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