Foundation Pair Kernel Gap2a Noether Momentum Map Carrier

The carrier construction

A ledger (a discrete record of events) in the Recognition Science framework advances through primitive posting steps, each one a legal atomic tick. This construction yields a real-valued covector, a linear functional that assigns a number to any displacement, for each such tick. The construction is motivated by the framework's existing Noether and symplectic structure, where a conserved charge accompanies a continuous symmetry. Here, the symmetry is the one-parameter translation group along the action line, and the charge is the momentum-map covector.

The central result is a uniqueness theorem: for any legal atomic tick, there exists exactly one covector that extends the primitive character along the action orbit, meaning it pairs the action generator to the value 1. This unique covector is shown to coincide with the already-derived action-lattice cotangent coordinate. The construction is deliberately independent of any particular tick, so the same covector serves every atomic posting step. A naive alternative, the unpriced primitive covector that simply reads off the identity, is explicitly rejected because it fails to pair the generator correctly.

In Recognition Science, this construction builds the missing real covector carrier for the momentum-map / Noether charge of a tick's normalized action orbit. It does not identify this carrier with the physical pair-kernel source, and it does not assume several later laws, such as the source-action unit law or the Green-sum continuum. The equality of this carrier to an independently constructed pair-kernel physical source covector remains open, recorded as a target for a future equality lane. The construction is axiom-clean, with no new axioms and no sorry, and it introduces no Planck constant, no pi, and no gravitational constant.

What this establishes in plain language is a firm, machine-checked bridge between the ledger's discrete posting steps and the continuous action-orbit picture. The framework now has a unique, well-defined price for each atomic step, derived from symmetry principles rather than chosen by hand. That price is not yet tied to any physical source, but the carrier exists, is unique, and is conserved along the orbit. This gives the framework a concrete handle for later connecting its internal accounting to physical pair-kernel sources, a step that remains explicitly open.

theorem legalAtomicTickMomentumMap_exists_unique
    {d : Nat} {L L' : LedgerPostingAdjacency.LedgerState d}
    (tick : LedgerPostingAdjacency.LegalAtomicTick (d := d) L L') :
    ∃! μ : PostingDual, IsNoetherMomentumMapCovector μ := by
  refine ⟨noetherMomentumMapCovector tick,
    noetherMomentumMapCovector_isOperational tick, ?_⟩
  intro μ hμ
  exact (legalAtomicTickMomentumMap_unique tick μ).1 hμ

theorem noetherMomentumMapCoordinate_eq_uniqueCotangent :
    noetherMomentumMapCoordinate = uniqueCotangentCoordinate := by
  simp [noetherMomentumMapCoordinate, uniqueCotangentCoordinate,
    atomicTickActionGenerator_eq_nativeActionQuantum]

/-- 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)

theorem noetherMomentumMapCarrier_cert :
    NoetherMomentumMapCarrierCert where
  ambient_noether_symplectic := existingNoetherSymplecticCertificate
  generator_priced_by_native_action :=
    atomicTickActionGenerator_eq_nativeActionQuantum
  pairs_generator := noetherMomentumMapCovectorLinear_pairs_generator
  operational := fun tick => noetherMomentumMapCovector_isOperational tick
  unique := noetherMomentumMapCovector_unique
  decoy_rejected := unpricedPrimitiveCovector_fails_pairs_generator
  coincides_with_unique_cotangent :=
    noetherMomentumMapCoordinate_eq_uniqueCotangent
  orbit_charge_conserved := atomicTickOrbitNoetherCharge_conserved
  physical_source_equality_open := carrierDoesNotIdentifyPhysicalSource

What this page does not claim

The carrier is not identified with any physical pair-kernel source. The construction does not assume the source-action unit law, the Green-sum continuum, or endpoint data. The equality to a physical source covector is not proved here.

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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