Foundation Pair Kernel Gap2a Noether Momentum Map Carrier Noether Momentum Map C

The unique covector

In classical mechanics, a conserved quantity often comes from a symmetry: Noether's theorem pairs a continuous symmetry with a quantity that does not change along the motion. The Recognition Science framework builds a discrete analogue. Its basic unit is a ledger, a discrete record of posting events, and each legal atomic tick is one unit L1 posting step. The declaration noetherMomentumMapCovectorLinear_pairs_generator establishes that a specific linear functional, the momentum-map covector, is the unique real-linear functional that pairs the action generator to the primitive character 1. In plainer terms, it is the one linear map that assigns the value 1 to the fundamental action displacement, and by doing so it recovers the primitive positive character on every orbit displacement.

The construction starts with the action generator, a positive real number representing the native action quantum of a primitive posting. The continuous action orbit is the translation one-parameter group on the action line with that generator displacement. The momentum-map covector is then defined as the inverse of the generator times the identity linear map; applying it to the generator yields 1. The theorem proves this covector extends the primitive character along the action, and the companion theorem noetherMomentumMapCovector_unique shows it is the unique such covector. A second theorem, noetherMomentumMapCovector_eq_canonical, identifies it with the canonical real character extension, and noetherMomentumMapCoordinate_eq_uniqueCotangent shows its coordinate coincides with the already-derived unique action-lattice cotangent coordinate. The covector is also independent of the specific tick, as noetherMomentumMapCovector_independent_of_tick states.

The theorem does not claim that this covector is the physical pair-kernel source. The docstring is explicit: the construction builds the carrier but does not identify it with the physical source, and does not assume source · action = unit, NativeActionDualSourceLaw, Green-sum continuum, or endpoint data. The equality to an independently constructed pair-kernel physical source covector, recorded as MomentumMapLedgerPostingAdjacency.LegalAtomicTick_eq_PairKernelSourceCovector, remains open. The theorem also does not claim that the unpriced primitive covector, the identity map, works; in fact unpricedPrimitiveCovector_rejected proves it fails the pairing condition because the action generator is less than 1, not equal to 1.

What the theorem changes is the status of the carrier: it is no longer a free choice but a forced consequence of the Noether/symplectic structure already certified. The framework's library, a machine-checked collection of formal theorems, proves that any covector satisfying the Noether momentum-map condition must equal this one. The carrier is unique, operationally defined, and independent of the particular tick. The remaining gap is the physical identification, which is a separate, unproved target.

/-- Uniqueness: any operational momentum-map covector equals the constructed
carrier. -/
theorem noetherMomentumMapCovector_unique
    (μ : PostingDual) (hμ : IsNoetherMomentumMapCovector μ) :
    μ = noetherMomentumMapCovectorLinear := by
  have hExt : ExtendsPrimitiveCharacterAlongAction μ := hμ.2
  have hCanon : μ = canonicalRealCharacterExtension :=
    uniqueRealCharacterExtension μ hExt
  exact hCanon.trans noetherMomentumMapCovectorLinear_eq_canonical.symm

/-- All realized LedgerPostingAdjacency.LegalAtomicTicks share the same momentum-map covector. -/
theorem noetherMomentumMapCovector_independent_of_tick
    {d₁ d₂ : Nat}
    {L₁ L₁' : LedgerPostingAdjacency.LedgerState d₁} {L₂ L₂' : LedgerPostingAdjacency.LedgerState d₂}
    (tick₁ : LedgerPostingAdjacency.LegalAtomicTick (d := d₁) L₁ L₁')
    (tick₂ : LedgerPostingAdjacency.LegalAtomicTick (d := d₂) L₂ L₂') :
    noetherMomentumMapCovector tick₁ = noetherMomentumMapCovector tick₂ :=
  rfl

theorem unpricedPrimitiveCovector_rejected
    {d : Nat} {L L' : LedgerPostingAdjacency.LedgerState d}
    (_tick : LedgerPostingAdjacency.LegalAtomicTick (d := d) L L') :
    ¬ IsNoetherMomentumMapCovector unpricedPrimitiveCovector := by
  intro h
  have hExt : ExtendsPrimitiveCharacterAlongAction unpricedPrimitiveCovector := h.2
  exact unpricedPrimitiveCovector_fails_pairs_generator
    (extends_implies_pairs_generator _ hExt)

What this page does not claim

The covector is not identified with the physical pair-kernel source, whose equality remains open. No assumption is made that source · action = unit, NativeActionDualSourceLaw, Green-sum continuum, or endpoint data hold. The theorem does not prove that the unpriced primitive covector (the identity map) is a valid momentum-map covector; it proves the opposite.

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/PairKernelGap2aNoetherMomentumMapCarrier.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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