Foundation Pair Kernel Gap2a Noether Momentum Map Carrier Unpriced Primitive Cov

The unpriced covector

A covector is a linear machine that consumes a vector and returns a number. In the Recognition Science framework, a ledger (a discrete record of events) assigns to each legal atomic tick, a single unit posting step, a covector that acts as the tick's momentum-map or Noether charge. The framework's library, a machine-checked collection of formal theorems, builds this covector from the tick's action generator, the displacement along its continuous action orbit. The natural candidate is the identity map, which sends every number to itself. The declaration unpricedPrimitiveCovector_fails_pairs_generator proves that this identity map is not the right covector: it fails to pair the action generator with the primitive character value 1.

The proof is short and exact. The identity map sends the action generator to itself, not to 1. The framework has already proved that the action generator is a positive number strictly less than 1, a fact recorded as nativeActionQuantum_lt_one. Since the generator is not 1, the identity map cannot satisfy the pairing condition. The theorem unpricedPrimitiveCovector_rejected then follows: the identity map is not a Noether momentum-map covector at all. The correct covector, built by scaling the identity map by the inverse of the generator, is unique and is proved to exist for every legal atomic tick.

This negative result is a positive step. It rules out the cheapest possible construction and forces the framework to use the scaled covector, which is independently derived from the Noether structure. The uniqueness theorems noetherMomentumMapCovector_unique and legalAtomicTickMomentumMap_exists_unique guarantee that there is exactly one such covector, and that it is the same for every tick. The framework does not claim that this covector is the physical pair-kernel source; that equality remains an open target, recorded but not proved here.

theorem unpricedPrimitiveCovector_fails_pairs_generator :
    ¬ PairsGeneratorToPrimitiveCharacter unpricedPrimitiveCovector := by
  intro h
  have hval : atomicTickActionGenerator = 1 := by
    simpa [PairsGeneratorToPrimitiveCharacter, unpricedPrimitiveCovector] using h
  have hlt : atomicTickActionGenerator < 1 := by
    rw [atomicTickActionGenerator_eq_nativeActionQuantum]
    exact nativeActionQuantum_lt_one
  exact (ne_of_lt hlt) hval

theorem nativeActionQuantum_lt_one : nativeActionQuantum < 1 := by
  have hone : (1 : ℝ) < nativeActionQuantumInv := one_lt_nativeActionQuantumInv
  have hmul : nativeActionQuantum * nativeActionQuantumInv = 1 :=
    nativeActionQuantumInv_mul
  have haeq : nativeActionQuantum = (nativeActionQuantumInv)⁻¹ := by
    -- from a * b = 1 derive a = b⁻¹
    exact eq_inv_of_mul_eq_one_left hmul
  rw [haeq]
  exact inv_lt_one_of_one_lt₀ hone

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 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

What this page does not claim

The identity map is the correct momentum-map covector. The momentum-map covector is identical to the physical pair-kernel source. The action generator equals 1.

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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