Encyclopedia Foundation Foundation Pair Kernel Source Normalization Action Eq Scale Source Pairing Of Sc

ARTICLE 6 claims 6 theorems

Foundation Pair Kernel Source Normalization Action Eq Scale Source Pairing Of Sc

A machine-checked theorem ties a field's energy to its source strength, but only under a symmetry condition and only for a defined equation.

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The energy identity

The declaration action_eq_scale_sourcePairing_of_scaledSourceEquation proves an energy identity for a discrete field theory. It states that for a symmetric pairwise interaction kernel and any field solving a scaled source equation, the action equals the scale times the source pairing. In plain terms: if the interaction between two sites is the same in both directions, and the field satisfies a linear equation with a source profile and a scale, then the total energy of the field is exactly the scale multiplied by the sum of field times source at each site. This is a theorem in the machine-checked library of formal theorems, with no unproved axioms beyond the standard logical ones.

The identity is a direct consequence of two homogeneity facts. The Laplacian is linear in the field, so scaling the field by a constant scales the Laplacian by that constant. The action is quadratic, so scaling the field by a constant scales the action by the square of that constant. Combining these with the source equation yields the identity. The proof is short and fully formalized.

What the theorem does not claim is equally important. It does not assert that any particular source scale is physically preferred. The same module proves that from a single unit-normalized solution, one can rescale the field to admit any source scale whatsoever. This is a mathematical no-go for the current premise package: the source equation alone cannot select a scale. The theorem also requires the kernel to be symmetric; without that condition the identity does not follow. Finally, the identity is about the discrete finite graph, not about any continuous or physical limit.

In Recognition Science, this result is part of a broader investigation into what the framework's axioms force. Here it shows a limitation: the source normalization freedom is real and must be broken by an additional physical law, not by the equation itself. The theorem is a precise statement of that freedom, and a reminder that a formal proof of an identity does not by itself determine a physical scale.

THEOREM action_eq_scale_sourcePairing_of_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean
action_eq_scale_sourcePairing_of_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean:95 
/-- 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]
THEOREM action_eq_scale_sourcePairing_of_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean
action_eq_scale_sourcePairing_of_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean:95 
/-- 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]
THEOREM laplacian_scale · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean
laplacian_scale · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean:34 
/-- The weighted graph Laplacian is homogeneous in the field. -/
theorem laplacian_scale
    (w : ι → ι → ℝ) (e : ι → ℝ) (c : ℝ) (i : ι) :
    laplacian w (fun i => c * e i) i = c * laplacian w e i := by
  unfold laplacian
  rw [Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro j _
  ring
THEOREM action_scale · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean
action_scale · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean:44 
/-- The quadratic pair action is homogeneous of degree two in the field. -/
theorem action_scale
    (w : ι → ι → ℝ) (e : ι → ℝ) (c : ℝ) :
    action w (fun i => c * e i) = c ^ 2 * action w e := by
  unfold action
  calc
    (1 / 2 : ℝ) *
        ∑ i, ∑ j, w i j * ((c * e i) - (c * e j)) ^ 2
        = (1 / 2 : ℝ) *
            ∑ i, ∑ j, c ^ 2 * (w i j * (e i - e j) ^ 2) := by
          congr 1
          apply Finset.sum_congr rfl
          intro i _
          apply Finset.sum_congr rfl
          intro j _
          ring
    _ = (1 / 2 : ℝ) *
          (c ^ 2 * ∑ i, ∑ j, w i j * (e i - e j) ^ 2) := by
          congr 1
          rw [Finset.mul_sum]
          apply Finset.sum_congr rfl
          intro i _
          rw [Finset.mul_sum]
    _ = c ^ 2 *
          ((1 / 2 : ℝ) * ∑ i, ∑ j, w i j * (e i - e j) ^ 2) := by
          ring
THEOREM every_source_scale_admitted_of_unit_solution · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean
every_source_scale_admitted_of_unit_solution · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean:80 
/-- 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
THEOREM action_eq_scale_sourcePairing_of_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean
action_eq_scale_sourcePairing_of_scaledSourceEquation · IndisputableMonolith/Foundation/PairKernelSourceNormalization.lean:95 
/-- 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]

What this page does not claim

The theorem does not claim that any particular source scale is physically preferred or derived. The theorem does not claim the identity holds for non-symmetric kernels. The theorem does not claim anything about continuous or infinite systems.

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

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