Foundation Pair Kernel Source Normalization Scaled Source Equation

The source equation and its scaling freedom

In graph theory and discrete mathematics, a Laplacian is a matrix that measures how a value at a vertex differs from the average of its neighbors; the equation L e = rho, where L is the Laplacian, e is a field on the vertices, and rho is a source profile, is a discrete analogue of Poisson's equation from electrostatics. The declaration ScaledSourceEquation in the Recognition Science framework records exactly this equation with one extra parameter: it states that the Laplacian of the field e equals the source profile rho multiplied by a real number called scale. In plain terms, the declaration names the condition that a field solves the source equation with a chosen normalization scale; it is a definition, not an assertion that any such field exists.

The framework's library then proves a consequence that looks like a limitation but is actually a precise mathematical fact. Because the Laplacian is linear, multiplying a solution field by any constant c multiplies the left side by c, so the same source profile admits the same equation with scale multiplied by c and field multiplied by c. The theorem every_source_scale_admitted_of_unit_solution states that if a field solves the equation with scale 1, then for any real number s, the field multiplied by s solves the equation with scale s. A companion result, decoy_scale_one_and_two_of_unit_solution, makes the point concrete: one unit solution yields both a scale-1 and a scale-2 equation on the same carrier and source profile. The framework calls this a normalization decoy: without additional structure, the overall scale of the source is not fixed by the equation alone.

In Recognition Science, this result is a no-go for the current premise package, not a claim about the physical world. The framework's own documentation says it plainly: the theorem introduces no source scale by existence, and a physical scale-selection law must break the rescaling family proved here. The declaration and its theorems are proved in the machine-checked library of formal theorems with no sorry and no new axiom, so the mathematical content is secure. But the framework does not claim that the physical source scale is absent; it claims only that this particular equation, with this premise package, cannot determine it.

What this means for a reader is that the declaration is a definition plus a theorem about that definition, not a physical law. It establishes a scaling freedom that any complete theory must break, and it provides the formal vocabulary for naming that freedom. The consequence is a clear target: a future theory must add a condition that selects one scale, and this result shows exactly where that condition must act.

/-- A field `e` solves the source equation with source profile `rho` and
normalization scale `scale`. -/
def ScaledSourceEquation
    (w : ι → ι → ℝ) (rho : ι → ℝ) (scale : ℝ) (e : ι → ℝ) : Prop :=
  ∀ i, laplacian w e i = scale * rho i

/-- 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

/-- 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

/-- 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

What this page does not claim

The declaration does not assert that any solution field exists for a given source profile. The scaling freedom does not imply that the physical source scale is absent in the framework's full theory. The theorem does not prove that scale is unmeasurable, only that this equation alone cannot determine it.

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: