Foundation Pair Kernel Gap2a Common Type Bridge

The common-type bridge

In Recognition Science, two central objects live on different mathematical stages. One, the ledger (a discrete record of events), has a momentum map that acts on a single real number. The other, a physical source covector, acts on field variations, which are functions from a finite set of points to real numbers. These types are not directly equal, so the framework's library builds a bridge between them.

The bridge works by pulling the physical source back to the scalar line. A physical source at unit scale evaluates a field variation by taking the difference in value between two points, a potential drop. The library constructs a specific linear map that sends a real number to a field variation with that exact potential drop. Composing the source with this map yields a covector on the scalar line, the same type as the momentum map. This construction is a definitional choice, a model, not a forced result.

The library then proves the strongest honest comparison. The pulled-back source equals the momentum map if and only if a single real parameter, the source scale, equals a specific value called the unique cotangent coordinate. This equivalence is a theorem. The library also proves that the physical equality itself, the identification of the source with the cotangent, is not forced by the current premises. It remains a typed residual, an unforced premise the framework isolates rather than hides.

This matters because it draws a clean line. The bridge itself is fully established, with no gaps in its logic. But it does not manufacture a physical identification. A decoy unit source scale is shown to fail the equality, and a candidate source magnitude is rejected. The framework's library certifies the bridge and explicitly records that it does not derive the physical equality. This is a deliberate act of honesty: the structure is complete, and the remaining physical question is clearly marked as open.

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

theorem decoyUnitSourceScale_rejected {n : ℕ} [DecidableEq (Fin n)]
    (a b : Fin n) (hab : a ≠ b) :
    ¬ PulledBackGaussEqualsNoetherMomentumMap decoyUnitSourceScale a b := by
  intro heq
  exact decoyUnitSourceScale_ne_uniqueCotangent
    ((pulledBackGauss_eq_noether_iff_sourceScale_eq_uniqueCotangent
        decoyUnitSourceScale a b hab).1 heq)

What this page does not claim

The module does not prove that the physical source is identical to the unique cotangent. The bridge construction is a definitional model, not a theorem about the physical world. No claim is made about the value of the source scale or the unique cotangent coordinate.

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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Derived articles

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