Foundation Pair Kernel Gap2a Noether Momentum Map Carrier Atomic Tick Action Gen

The action generator

In the Recognition Science framework, a ledger is a discrete record of events, and a primitive posting is the smallest possible single entry in that record. The declaration atomicTickActionGenerator_eq_nativeActionQuantum establishes that the action generator for such a primitive posting equals the framework's native action quantum. In plain terms, it proves that the cost or action associated with one indivisible posting step is a fixed, positive number, not something that varies from tick to tick. The theorem atomicTickActionGenerator_pos confirms this value is strictly greater than zero, so a primitive posting always carries a real, nonzero amount of action.

The construction builds this generator as the action displacement of one primitive posting, embedded into a real cotangent residual. The theorem itself is a direct consequence of the action-lattice embedding already in place. What matters for the reader is the consequence: every legal atomic tick, the framework's term for one valid posting step between ledger states, moves along an action orbit with this same generator. The covector that pairs with this generator, the momentum-map covector, is then uniquely determined: it is the unique real-linear functional that recovers the primitive positive character on every orbit displacement, effectively assigning the value 1 to the generator itself. The theorem noetherMomentumMapCoordinate_eq_uniqueCotangent shows this coordinate coincides with the already-derived unique action-lattice cotangent coordinate.

What this declaration does not claim is equally important. It does not identify the carrier with the physical pair-kernel source. The framework explicitly keeps separate the momentum-map covector constructed here from any independently constructed physical source covector. The equality between them, recorded as MomentumMapLedgerPostingAdjacency.LegalAtomicTick_eq_PairKernelSourceCovector, is an open target, not a proved theorem. The declaration also does not assume source · action = unit, nor does it rely on endpoint data or any Green-sum continuum. It builds the carrier purely from the generator and character pairing inside the Noether/symplectic package, with no new axioms and no sorry.

The practical upshot is a clean, machine-checked uniqueness result: for any legal atomic tick, there exists exactly one momentum-map covector satisfying the defining property, and it is the same for every tick. This gives the framework a stable, tick-independent notion of action cost for a primitive posting, a necessary ingredient for any further physical interpretation. The open question of whether this mathematical carrier matches a physical pair-kernel source remains exactly that: open.

theorem atomicTickActionGenerator_eq_nativeActionQuantum :
    atomicTickActionGenerator = nativeActionQuantum :=
  actionLatticeEmbed_primitive

theorem atomicTickActionGenerator_pos : 0 < atomicTickActionGenerator := by
  rw [atomicTickActionGenerator_eq_nativeActionQuantum]
  exact nativeActionQuantum_pos

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

/-- 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 noetherMomentumMapCoordinate_eq_uniqueCotangent :
    noetherMomentumMapCoordinate = uniqueCotangentCoordinate := by
  simp [noetherMomentumMapCoordinate, uniqueCotangentCoordinate,
    atomicTickActionGenerator_eq_nativeActionQuantum]

What this page does not claim

The momentum-map covector is identified with the physical pair-kernel source. The declaration assumes source · action = unit or relies on endpoint data. The equality to a pair-kernel source covector is proved.

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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Derived articles

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