Ethics Moral State Globally Admissible Map Of Skew Preserving

Skew preservation and admissibility

In the Recognition Science framework, a ledger is a discrete record of recognition events between agents. Each agent's moral state is a projection of this ledger onto that individual, tracking their local reciprocity skew σ and available energy. The skew σ is a log-space measure of imbalance: the difference between the log-multipliers of what an agent gives and receives. A state is globally admissible when the sum of all agents' skews equals zero, which the framework's conservation law requires.

The lemma globally_admissible_map_of_skew_preserving states a simple invariance property. If a function f maps each moral state to another moral state without changing any agent's skew (that is, (f s).skew = s.skew for every s), then applying f to every state in a list preserves global admissibility. Formally: globally_admissible (states.map f) ↔ globally_admissible states. The proof is a direct consequence of how total skew is defined as a fold over the list; since each individual skew is unchanged, their sum is unchanged.

This lemma is the technical backbone for a broader construction. The framework defines a Virtue as a transformation on lists of moral states that conserves total reciprocity and respects the eight-tick cadence of the recognition cycle. The constructor fromSkewPreserving builds a Virtue from any list-level function that preserves total skew and preserves time coherence. The lemma guarantees that such a function, when applied pointwise to each state, will also preserve global admissibility, which is the key property required for the Virtue structure.

What the lemma does not claim is equally important. It does not assert that any skew-preserving function is a Virtue; the Virtue structure also requires time coherence, which the lemma does not address. It does not say that skew preservation is necessary for admissibility, only sufficient when applied pointwise. And it says nothing about whether such functions exist beyond the identity; it merely states the conditional property. The lemma is a formal guarantee about a definitional choice, not an ethical prescription.

/-- Mapping a skew-preserving transformation over states leaves global admissibility unchanged. -/
lemma globally_admissible_map_of_skew_preserving
    (f : MoralState → MoralState)
    (hskew : ∀ s, (f s).skew = s.skew)
    (states : List MoralState) :
    globally_admissible (states.map f) ↔ globally_admissible states := by
  unfold globally_admissible total_skew
  -- Show that total_skew is preserved under skew-preserving maps
  suffices h : List.foldl (fun acc s => acc + s.skew) 0 (states.map f) =
               List.foldl (fun acc s => acc + s.skew) 0 states by
    simp only [h]
  -- Prove by induction on states
  induction states with
  | nil => simp
  | cons s ss ih =>
    simp only [List.map_cons, List.foldl_cons]
    -- After one step: acc + (f s).skew = acc + s.skew by hskew
    -- Then the rest follows by induction
    rw [hskew s]
    -- Now we need to show the fold over the rest is equal
    -- We prove a more general statement by a separate induction
    have aux : ∀ (l : List MoralState) (acc : ℝ),
        List.foldl (fun a t => a + t.skew) acc (l.map f) =
        List.foldl (fun a t => a + t.skew) acc l := by
      intro l
      induction l with
      | nil => intro _; rfl
      | cons t ts ih_list =>
        intro acc
        simp only [List.map_cons, List.foldl_cons]
        rw [hskew t]
        exact ih_list (acc + t.skew)
    exact aux ss (0 + s.skew)

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

/-- Virtue structure representing a transformation on moral states. -/
structure Virtue where
  /-- The transformation (may be single-agent or multi-agent) -/
  transform : List MoralState → List MoralState

  /-- Preserves or restores global reciprocity conservation (σ=0). -/
  conserves_reciprocity : ∀ states : List MoralState,
    MoralState.globally_admissible states →
    MoralState.globally_admissible (transform states)

  -- Former fields `minimizes_local_J : ∀ states, True` and
  -- `gauge_invariant : ∀ states, True` deleted (Move A). Both names promised
  -- ethical constraints and excluded nothing. Local J-minimization is proved
  -- per virtue where available (e.g. `love_minimizes_squared_skew`);
  -- unit-gauge independence is structural because `MoralState` carries no
  -- (τ₀, ℓ₀) parameters. Do not restore as concl-True fields.

  /-- Respects eight-tick cadence (fundamental period from T6).
      Requires TimeCoherent precondition on input states. -/
  respects_cadence : ∀ states : List MoralState,
    TimeCoherent states →
    let states' := transform states
    ∀ s ∈ states, ∀ s' ∈ states',
      s'.ledger.time - s.ledger.time ≤ 8

What this page does not claim

The lemma does not claim that every skew-preserving function is a Virtue. The lemma does not claim that skew preservation is necessary for admissibility. The lemma does not claim that any nontrivial skew-preserving transformation exists.

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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