Foundation Pair Kernel Gap2a Common Type Bridge Noether Momentum Map Covector Li

The bridge and its limit

In linear algebra, a covector is a linear map that takes a vector and returns a number. The declaration noetherMomentumMapCovectorLinear_eq_uniqueCotangent_smul_id concerns one specific covector, called the Noether momentum map covector, which lives on the scalar line: it takes a real number and returns a real number. The theorem states that this covector is exactly the unique cotangent coordinate multiplied by the identity map. In plainer terms, the Noether momentum map covector is just scalar multiplication by a single fixed number, and that number is the unique cotangent coordinate.

The proof is a direct computation. It expands the definition of the Noether momentum map covector, uses the fact that the atomic tick action generator equals the native action quantum, and then applies the definition of the unique cotangent coordinate. The result is an equality of linear maps, verified by checking what each map does to an arbitrary input. This is a theorem in the machine-checked library of formal theorems, meaning the equality is proved from the definitions with no additional assumptions.

The theorem does not claim that the unique cotangent coordinate is physically meaningful. A separate theorem in the same module, remainingPhysicalEquality_unforced, proves that the current premises do not force the identification of the physical source with the unique cotangent. In other words, the equality of linear maps is a mathematical fact, but whether the physical source scale equals the unique cotangent coordinate remains an open premise, a typed residual that the framework does not derive.

This distinction matters for reading the framework correctly. The covector equality is a clean algebraic result, useful for translating between two descriptions of the same linear structure. The physical identification is a separate question, explicitly marked as unforced. The framework is honest about the gap: it proves what it can prove and isolates what it cannot.

/-- Core typed residual: pullback equals Noether carrier iff the free Gauss
scale equals the unique action-lattice cotangent coordinate. -/
theorem noetherMomentumMapCovectorLinear_eq_uniqueCotangent_smul_id :
    noetherMomentumMapCovectorLinear =
      uniqueCotangentCoordinate • (LinearMap.id : ℝ →ₗ[ℝ] ℝ) := by
  apply LinearMap.ext
  intro x
  simp [noetherMomentumMapCovectorLinear_apply,
    atomicTickActionGenerator_eq_nativeActionQuantum,
    uniqueCotangentCoordinate]

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

What this page does not claim

The theorem does not claim that the physical source is identified with the unique cotangent. The theorem does not derive the value of the unique cotangent coordinate from first principles. The theorem does not involve Planck's constant, pi, or the gravitational constant G.

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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