Foundation Pair Kernel Source Normalization Scaled Source Equation Scale

The scaling freedom

The declaration scaledSourceEquation_scale establishes a scaling freedom in a class of source equations. A source equation here relates a field to a source profile through a weighted graph Laplacian, a discrete analogue of the Laplacian operator. The theorem states that if a field e solves the equation with source profile rho and normalization scale scale, then for any real number c, the rescaled field c times e solves the same equation with normalization scale c times scale. In symbols: if the Laplacian of e equals scale times rho, then the Laplacian of c times e equals c times scale times rho.

This is a homogeneity statement: the equation is linear in the field and linear in the scale. The proof is a direct calculation using the fact that the Laplacian is linear. A companion theorem, every_source_scale_admitted_of_unit_solution, draws out the consequence: if you have one solution with scale 1, you can rescale it to obtain a solution for any real source scale. The docstring for this premise package calls this a mathematical no-go. It means the current framework, at this layer, cannot distinguish between different absolute sizes of the source scale. The scale is a free parameter, not a derived quantity.

In Recognition Science, this is not a claim that the physical source scale is absent. The framework explicitly states that a physical scale-selection law must break the rescaling family proved here. The theorem is a statement about the formal premises, not about the world. It says: under these definitions, the scale is not forced. The framework's broader forcing chain, which derives constants like the golden ratio and particle masses, operates at a different layer and is not affected by this local freedom.

The declaration also supports an energy identity: under symmetric weights, the action equals the scale times the source-field pairing. This identity is generic and holds for any scaled source equation. It shows that the scaling freedom is consistent with the energy structure, not in tension with it. The action scales quadratically with the field, while the pairing scales linearly, and the identity balances the two.

What the declaration does not claim is equally important. It does not claim that the source scale is physically undetermined. It does not claim that all scales are equivalent in the full framework. It does not claim that the source equation itself is the final law. The theorem is a precise, limited statement about a specific formal object. It is a no-go for this premise package, not a no-go for the framework.

The practical consequence is that any attempt to derive a physical source scale from these equations alone will fail. The framework must add a new principle, a scale-selection law, to break the freedom. This is an open target, not a contradiction. The theorem clarifies what the framework needs, which is exactly what a good no-go result should do.

/-- Scaling a solution field scales the source-equation normalization. -/
theorem scaledSourceEquation_scale
    (w : ι → ι → ℝ) (rho : ι → ℝ) (scale c : ℝ) (e : ι → ℝ)
    (hsource : ScaledSourceEquation w rho scale e) :
    ScaledSourceEquation w rho (c * scale) (fun i => c * e i) := by
  intro i
  rw [laplacian_scale, hsource i]
  ring

/-- From an explicit unit-normalized solution, every source scale is admitted
by rescaling the field. -/
theorem every_source_scale_admitted_of_unit_solution
    (w : ι → ι → ℝ) (rho : ι → ℝ) (sourceScale : ℝ) (e : ι → ℝ)
    (hunit : ScaledSourceEquation w rho 1 e) :
    ScaledSourceEquation w rho sourceScale (fun i => sourceScale * e i) := by
  simpa using
    scaledSourceEquation_scale
      (w := w) (rho := rho) (scale := (1 : ℝ)) (c := sourceScale)
      (e := e) hunit

/-- Generic energy identity for any scaled source equation under symmetric
weights. -/
theorem action_eq_scale_sourcePairing_of_scaledSourceEquation
    (w : ι → ι → ℝ) (rho : ι → ℝ) (scale : ℝ) (e : ι → ℝ)
    (hsymm : ∀ i j, w i j = w j i)
    (hsource : ScaledSourceEquation w rho scale e) :
    action w e = scale * sourcePairing rho e := by
  rw [action_eq_sum_mul_laplacian w e hsymm]
  unfold sourcePairing
  calc
    (∑ i, e i * laplacian w e i)
        = ∑ i, e i * (scale * rho i) := by
          apply Finset.sum_congr rfl
          intro i _
          rw [hsource i]
    _ = ∑ i, scale * (e i * rho i) := by
          apply Finset.sum_congr rfl
          intro i _
          ring
    _ = scale * ∑ i, e i * rho i := by
          rw [Finset.mul_sum]

/-- Scaling a solution field scales the source-equation normalization. -/
theorem scaledSourceEquation_scale
    (w : ι → ι → ℝ) (rho : ι → ℝ) (scale c : ℝ) (e : ι → ℝ)
    (hsource : ScaledSourceEquation w rho scale e) :
    ScaledSourceEquation w rho (c * scale) (fun i => c * e i) := by
  intro i
  rw [laplacian_scale, hsource i]
  ring

What this page does not claim

The theorem does not claim that the physical source scale is absent or undetermined in the world. The theorem does not claim that all source scales are equivalent in the full Recognition Science framework. The theorem does not claim that the source equation itself is the final law of source dynamics.

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/Foundation/PairKernelSourceNormalization.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: