Foundation Ledger Canonicality Neutral Sector

The neutral sector

The ledger, a discrete record of events, assigns each state a scalar quantity called charge. The neutral sector is simply the collection of states whose charge is exactly zero. This is a definition, not a discovery: the declaration neutralSector in the framework's machine-checked library of formal theorems packages the set-builder expression { s | charge s = 0 } into a named object.

Why isolate the zero-charge states? The framework's later theorems about hierarchy, factorization, and neutral dynamics all consume this single interface. A ledger with at least one neutral state is the generic case for a zero-parameter ledger, and the library records this as a class assumption, HasNeutralStates, rather than as a derived consequence. The distinction matters: the nonemptiness of the neutral sector is an input condition for downstream results, not something the definition itself guarantees.

The declaration does not claim conservation. A charge function alone, without a transition map describing how states evolve, cannot assert that charge is conserved over time. The library is explicit on this point: a former field that tried to encode conservation by requiring equal charges to imply the trivial proposition True was deleted because it excluded nothing. Conservation requires a separate structure, a closed framework with a transition map and a proof that charge is invariant under it.

In Recognition Science, the neutral sector is where the action is not: it is the reference point against which nonzero charges are measured. The definition is deliberately minimal, carrying no external knobs and no claims about dynamics. What it provides is a stable vocabulary for stating the framework's stronger results, which is exactly what a primitive definition should do.

/-- The neutral sector of a ledger is the set of states with zero charge. -/
def neutralSector (L : ZeroParameterComparisonLedger) : Set L.Carrier :=
  { s | L.charge.charge s = 0 }

/-- A ledger scalar quantity on a type `α` with values in `ℝ`.

Conservation under dynamics is not a property of a bare charge function: it
requires a transition map (see `ClosedFramework.ClosedObservableFramework`,
field `charge_conserved : ∀ s, charge (T s) = charge s`). Former field
`charge_conserved : ∀ s₁ s₂, charge s₁ = charge s₂ → True` deleted (Move A):
equal charges implying `True` excluded nothing. -/
structure ConservedCharge (α : Type) where
  charge : α → ℝ

What this page does not claim

The neutral sector is not asserted to be nonempty in general; that is a separate class assumption. The declaration does not claim that charge is conserved under any dynamics. The neutral sector is not a physical object; it is a mathematical definition within the framework.

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/LedgerCanonicality.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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