Foundation Pair Kernel Source Normalization Every Source Scale Admitted Of Unit

Source scaling freedom

In the Recognition Science framework, the basic object is a ledger, a discrete record of events on a finite set of points, with a field value at each point and a weight between each pair of points. The framework's source equation asks when a field e, given weights w, produces a source profile rho with a chosen strength scale: it writes the weighted graph Laplacian of e equal to scale times rho. The declaration every_source_scale_admitted_of_unit_solution proves a striking freedom: if you have one field that solves this equation with scale 1, then for any real number sourceScale you can multiply the field by sourceScale and get a solution with that new scale. In symbols, from Laplacian w e = 1 * rho, the theorem derives Laplacian w (sourceScale * e) = sourceScale * rho.

The proof is short and mechanical. The framework's library of machine-checked formal theorems first shows the Laplacian is homogeneous in the field: multiplying the field by a constant multiplies the Laplacian by the same constant. It also shows the quadratic pair action, a sum over pairs of weighted field products, scales as the square of the field. From these two facts, the theorem scales a unit solution to any desired source scale by direct substitution. The library also records a companion result: the same unit solution yields both scale 1 and scale 2 equations on the same carrier and source profile, a concrete demonstration that the scaling freedom is real and not an artifact of a single choice.

What the declaration does not claim is the interesting part. The docstring is explicit: this is a mathematical no-go for the premise package, not a claim that the physical source scale is absent. The theorem shows that within this particular framework setup, the source scale cannot be pinned down by the equation alone. A physical theory would need a separate scale-selection law to break the rescaling family. The framework's own energy identity, which relates the action to scale times a source-field pairing, holds for any scale, so it provides no selection either. The declaration is a precise statement of what the framework's current source law can and cannot determine, and it leaves the question of the true physical source scale open.

/-- 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

/-- 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

/-- Explicit normalization decoy: one unit solution gives both scale `1`
and scale `2` equations on the same carrier and source profile. -/
theorem decoy_scale_one_and_two_of_unit_solution
    (w : ι → ι → ℝ) (rho : ι → ℝ) (e : ι → ℝ)
    (hunit : ScaledSourceEquation w rho 1 e) :
    ScaledSourceEquation w rho 1 e ∧
      ScaledSourceEquation w rho 2 (fun i => (2 : ℝ) * e i) ∧
      (1 : ℝ) ≠ 2 := by
  refine ⟨hunit, ?_, by norm_num⟩
  simpa using
    every_source_scale_admitted_of_unit_solution
      (w := w) (rho := rho) (sourceScale := (2 : ℝ)) (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]

What this page does not claim

The declaration does not claim that the physical source scale is absent or meaningless. The declaration does not claim that the scaling freedom survives for nonlinear source equations. The declaration does not identify any particular source scale as physically correct.

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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