Foundation Ledger Canonicality Conserved Charge

A charge that is only a label

A conserved charge is a quantity that stays the same while a system changes. In physics, electric charge is the familiar example: the total charge in an isolated region does not change as particles move and interact. The Recognition Science framework builds its own version of this idea from a ledger, a discrete record of events. The framework's declaration ConservedCharge establishes the first, minimal piece: a charge is simply a function that assigns a real number to every state of a system. That is all the declaration itself proves.

The framework is explicit that conservation is not a property of a bare charge function. A charge only becomes conserved when it is paired with a transition map, a rule saying how states evolve. The declaration's documentation records a deliberate correction: an earlier version claimed that equal charges implied a trivial statement, which excluded nothing. The corrected definition separates the label from the dynamics. A charge is a label; conservation is a separate requirement that the label survives a transition.

In the framework's fuller structure, the ZeroParameterComparisonLedger packages this charge together with other primitive pieces: countable states, a symmetric comparison cost, and closure under composition. The neutral sector, the set of states with zero charge, is defined from the charge function and is nonempty in the generic case. These pieces feed downstream theorems about hierarchy and dynamics, but the ConservedCharge declaration itself does not force any of that. It establishes only the labeling, and the framework says so plainly.

/-- 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

/-- 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

/-- 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 declaration does not prove that any particular charge is conserved under any dynamics. The declaration does not derive the existence of a transition map or any specific evolution rule. The declaration does not establish that the neutral sector is nonempty for all ledgers, only for those with zero-charge states.

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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