Foundation Pair Kernel Gap2a Common Type Bridge Pullback Along Unit Drop Eq Scal

The bridge theorem

In the Recognition Science framework, physical quantities are represented by linear maps between vector spaces. Two such maps are at issue. One, the Noether momentum map, lives on a one-dimensional scalar line. The other, the Gauss source, lives on a higher-dimensional space of field variations. The theorem pullback_along_unit_drop_eq_scale_smul_id builds a bridge between these two spaces and proves a precise equality condition.

The bridge works by pulling the Gauss source back to the scalar line along a special embedding. This embedding sends a scalar t to a field variation whose potential drop between two fixed sites equals t. The theorem states that for any such embedding, the pulled-back Gauss source equals the scalar sourceScale times the identity map. In plain terms: after pulling back, the Gauss source becomes a simple scaling of the scalar coordinate.

The theorem's real content is the if and only if condition. The pulled-back Gauss source equals the Noether momentum map exactly when the source scale equals a specific constant, the unique cotangent coordinate. This is a formal theorem in the framework's machine-checked library. The framework proves the equivalence; it does not prove that the physical source actually has that scale.

What the theorem does not claim is the physical identification itself. The equality between the two descriptions remains an unforced premise, a typed residual. The framework isolates this as a separate statement, RemainingPhysicalEquality, and explicitly proves that the premises do not force it. A decoy unit scale is shown to fail the equality. The bridge certifies the mathematics; the physics stays open.

/-- Along any unit-drop embedding, pullback of the scaled Gauss covector is
multiplication by `sourceScale`. -/
theorem pullback_along_unit_drop_eq_scale_smul_id {n : ℕ}
    (sourceScale : ℝ) (a b : Fin n)
    (ι : ℝ →ₗ[ℝ] (Fin n → ℝ))
    (hι : IsUnitPotentialDropEmbedding ι a b) :
    pullbackToPostingDual
        (pairKernelPhysicalSourceCovector sourceScale a b) ι =
      sourceScale • (LinearMap.id : ℝ →ₗ[ℝ] ℝ) := by
  apply LinearMap.ext
  intro t
  have hdrop := hι t
  simp [pullbackToPostingDual, pairKernelPhysicalSourceCovector_apply, hdrop]

theorem pulledBackGauss_eq_noether_iff_sourceScale_eq_uniqueCotangent
    {n : ℕ} [DecidableEq (Fin n)]
    (sourceScale : ℝ) (a b : Fin n) (hab : a ≠ b) :
    PulledBackGaussEqualsNoetherMomentumMap sourceScale a b ↔
      sourceScale = uniqueCotangentCoordinate := by
  have hpull := pulledBackGaussSource_eq_scale_smul_id sourceScale a b hab
  have hnoeth := noetherMomentumMapCovectorLinear_eq_uniqueCotangent_smul_id
  constructor
  · intro heq
    have hfun :
        sourceScale • (LinearMap.id : ℝ →ₗ[ℝ] ℝ) =
          uniqueCotangentCoordinate • (LinearMap.id : ℝ →ₗ[ℝ] ℝ) := by
      calc
        sourceScale • (LinearMap.id : ℝ →ₗ[ℝ] ℝ) =
            pulledBackGaussSource sourceScale a b := hpull.symm
        _ = noetherMomentumMapCovectorLinear := heq
        _ = uniqueCotangentCoordinate • (LinearMap.id : ℝ →ₗ[ℝ] ℝ) := hnoeth
    have h1 :=
      congrArg (fun φ : PostingDual => φ (1 : ℝ)) hfun
    simpa using h1
  · intro hs
    -- Rewrite both sides to the same `s • id` form.
    calc
      pulledBackGaussSource sourceScale a b =
          sourceScale • (LinearMap.id : ℝ →ₗ[ℝ] ℝ) := hpull
      _ = uniqueCotangentCoordinate • (LinearMap.id : ℝ →ₗ[ℝ] ℝ) := by
            rw [hs]
      _ = noetherMomentumMapCovectorLinear := hnoeth.symm

theorem remainingPhysicalEquality_unforced :
    ¬ (∀ sourceScale : ℝ,
      CurrentPremisesWithUniqueRealCotangent sourceScale →
        RemainingPhysicalEquality sourceScale) :=
  uniqueRealCotangent_does_not_force_identification

What this page does not claim

The theorem does not prove that the physical source has the unique cotangent scale. The theorem does not derive the value of any physical constant such as hbar or G. The theorem does not identify the physical source with the unique cotangent.

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/PairKernelGap2aCommonTypeBridge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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