Ethics Moral State Globally Admissible Append

The append rule

A moral state, in the Recognition Science framework, is a projection of a universal ledger (a discrete record of recognition events) onto a single agent's domain. Each state carries a reciprocity skew σ, a real number tracking the log-multiplier imbalance between what the agent gives and receives. A list of moral states is globally admissible when the sum of all its agents' skews is exactly zero, the conservation law that admissible worldlines satisfy.

The theorem globally_admissible_append states a closure property: if two lists of moral states are each globally admissible, then their concatenation is also globally admissible. The proof is a direct fold over the list structure, adding the two zero sums. This is a formal theorem in the framework's machine-checked library of formal theorems, not a definitional choice or a hypothesis.

The theorem does not claim that any particular list is admissible, only that admissibility is preserved under the append operation. It does not address what happens when two individually admissible lists are interleaved, reordered, or transformed in ways that change individual skews. It also does not assert that global admissibility of a concatenation implies admissibility of each part; that converse direction is not part of the theorem.

/-- A collection of moral states is globally admissible if total skew is zero -/
def globally_admissible (states : List MoralState) : Prop :=
  total_skew states = 0

/-- Global admissibility is preserved under list concatenation if both parts are admissible -/
theorem globally_admissible_append {xs ys : List MoralState}
  (hx : MoralState.globally_admissible xs)
  (hy : MoralState.globally_admissible ys) :
  MoralState.globally_admissible (xs ++ ys) := by
  unfold MoralState.globally_admissible MoralState.total_skew at *
  simp [List.foldl_append]
  -- Let f be the accumulator for total_skew
  let f : ℝ → MoralState → ℝ := fun acc s => acc + s.skew
  -- From hypotheses, both partial totals are zero
  have hx0 : List.foldl f 0 xs = 0 := by
    simpa [MoralState.total_skew, MoralState.globally_admissible] using hx
  have hy0 : List.foldl f 0 ys = 0 := by
    simpa [MoralState.total_skew, MoralState.globally_admissible] using hy
  -- Then the concatenated fold is also zero
  have : List.foldl f (List.foldl f 0 xs) ys = 0 := by
    simpa [hx0]
      using hy0
  simpa [MoralState.total_skew, MoralState.globally_admissible, List.foldl_append, f]
    using this

What this page does not claim

The theorem does not claim that admissibility of a concatenation implies admissibility of each part. It does not claim that any particular list of moral states is admissible. It does not address interleaving or reordering of states, only appending one list after another.

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/Ethics/MoralState.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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