Foundation Pair Kernel Source Normalization Decoy Scale One And Two Of Unit Solu

The normalization decoy

In the Recognition Science framework, a ledger is a discrete record of events on a network, and a recognition is the act of accounting for one event in that record. The framework models physical fields as solutions to a source equation on such a ledger, where a field e responds to a source profile rho through a weighted graph Laplacian. The equation is written L(e) = scale · rho, where the scale is a real number multiplying the source.

The declaration decoy_scale_one_and_two_of_unit_solution proves a precise statement about that equation. If a field e solves the equation with scale 1, then the same field e also solves it with scale 1, and the doubled field 2e solves it with scale 2, on the same ledger and with the same source profile. The proof is a direct consequence of the Laplacian's linearity: multiplying the field by a constant multiplies the Laplacian by the same constant, so the equation stays balanced when both field and scale are doubled. The declaration also records that 1 and 2 are distinct real numbers, which makes the two solutions genuinely different equations.

This is a mathematical no-go result for this specific premise package, not a physical claim. The theorem shows that the source equation, as defined here, cannot select a preferred normalization scale: from one explicit unit-normalized solution, every real scale is admitted by rescaling the field. The framework's own library states this plainly: a physical scale-selection law must break the rescaling family proved here. The declaration does not claim that the physical source scale is absent from nature, only that this particular equation does not determine it.

The result matters because it draws a boundary around what the framework's current axioms can force. A reader who expects the framework to derive the absolute size of a source from its ledger equation will not find it here. What the framework does provide is a clean statement of the freedom: the homogeneous rescaling family is proved, and any future law that fixes the scale must add structure beyond the source equation itself.

/-- 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

/-- 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

What this page does not claim

The physical source scale is absent from nature. The source equation alone determines the absolute normalization of any field. The rescaling freedom applies to equations with nonlinear source terms.

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.

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