Explanation of sequential_preserves_conservation

(1) What the declaration says in plain English

If a state satisfies a conservation property and every single event posting from a finite history H preserves that property, then sequentially applying all events according to any valid one-per-tick schedule for H yields a final state that still satisfies conservation. The proof proceeds by induction over the schedule list using list recursion.

(2) Why it matters in Recognition Science

Recognition Science requires finite recognition histories to be serialized into atomic (one-event-per-tick) schedules while preserving invariants such as ledger balance or cost conservation. This generic lemma guarantees that any property preserved under single postings remains intact after full serialization, supporting the constructive atomicity result for T2 (discreteness) without introducing axioms.

(3) How to read the formal statement

The theorem is declared as:

theorem sequential_preserves_conservation
    {S : Type _} (Conservation : S → Prop) (post : S → E → S)
    (prec : Precedence E)
    [DecidableRel prec] [DecidableEq E]
    (H : Finset E) (σ : Schedule E)
    (hcover : σ.order.toFinset = H)
    (s0 : S)
    (h0 : Conservation s0)
    (preserve_single : ∀ s e, e ∈ H → Conservation s → Conservation (post s e)) :
    Conservation (σ.order.foldl post s0)

It takes a state type S, a conservation predicate, a posting function, a precedence relation, a finite event set H, a schedule σ whose order exactly covers H, an initial conserving state s0, and the single-step preservation assumption; it concludes that the folded result conserves. The proof uses classical logic and list recursion on σ.order.

(4) Visible dependencies or certificates in the supplied source

The theorem is defined inside the Atomicity namespace and relies on the Schedule structure and the covering condition hcover. It is adjacent to exists_sequential_schedule (which produces qualifying schedules) and atomic_tick (the atomic serialization theorem). The module docstring explicitly states the lemma's purpose as a generic preservation result for finite histories under well-founded precedence. No sorry appears in the declaration.

(5) What this declaration does not prove

It assumes single-step preservation rather than proving any concrete conservation law. It applies only to finite histories (the countable serialization section appears later in the same module). It does not address cost functions, φ-forcing, or the full T5/T6 chain, consistent with the module's note that it remains independent of cost/convexity. It also does not prove existence of schedules (handled by separate theorems) or physical interpretations of Conservation.