eventCompose_left_id
The left identity law for narrative event composition states that prepending the zero event state to any event state a leaves a unchanged. Researchers confirming the algebraic content of strict domain realizations in the Universal Forcing framework cite this result when checking that narrative costs obey the required monoid laws. The proof reduces the claim by unfolding the componentwise addition and applying case analysis plus simplification on the natural-number fields.
claimFor every event state $a$, the left composition of the zero narrative state with $a$ equals $a$, where an event state is a pair of natural numbers (action count, beat count), the zero state is the pair of zeros, and composition adds the corresponding components.
background
EventState is the structure carrying two natural-number fields act and beat. narrativeZero is the pair with both fields set to zero. eventCompose adds the act fields of its arguments and adds the beat fields separately, yielding the additive monoid structure on pairs of naturals. The present theorem sits inside the Rich Domain Cost Theorems module, whose purpose is to replace the placeholder composition_law := True in each of the five domain modules with concrete, axiom-free statements.
proof idea
The proof is a one-line wrapper that unfolds the definitions of eventCompose and narrativeZero, performs case analysis on the structure of a, and invokes the simplifier to discharge the resulting componentwise equalities on natural numbers.
why it matters in Recognition Science
This theorem supplies the left-identity half of the composition laws required for the narrative domain's StrictLogicRealization instance. It therefore contributes directly to the module's claim that each of the five domains (Music, Biology, Narrative, Ethics, Metaphysical) carries non-trivial, law-bearing content rather than mere placeholders. In the larger Recognition Science development it ensures the forcing chain can incorporate narrative costs while preserving the composition requirements of the universal forcing.
scope and limits
- Does not prove the corresponding right-identity law.
- Does not establish associativity of composition.
- Does not supply numerical cost values or triangle inequalities.
- Does not address the other four domains treated in the same module.
formal statement (Lean)
124theorem eventCompose_left_id (a : EventState) :
125 eventCompose narrativeZero a = a := by
proof body
Term-mode proof.
126 unfold eventCompose narrativeZero
127 cases a; simp
128
129end NarrativeRich
130
131/-! ## Ethics -/
132
133namespace EthicsRich
134
135open Ethics
136