zmodCost_self
Equality cost on the cyclic carrier ZMod n vanishes for any element paired with itself. Modelers constructing modular realizations cite this to confirm the zero-cost axiom holds in finite settings. The proof is a one-line wrapper that simplifies via the cost definition.
claimFor every natural number $n$ and element $a$ of the integers modulo $n$, the equality cost returns zero on identical arguments.
background
The module supplies modular realizations whose carrier is the cyclic group of integers modulo $n$ equipped with an equality cost. This cost is zero precisely when the two arguments coincide and one otherwise, yielding a discrete metric on the finite carrier. The local setting records that the semantic orbit may close, certifying the forced arithmetic as the universal iteration object.
proof idea
The proof is a one-line wrapper that applies simp to the definition of the cost function. Identical arguments select the zero branch of the conditional immediately.
why it matters in Recognition Science
This result supports the modularRealization and strictModularRealization definitions that build LogicRealization and StrictLogicRealization instances using the cyclic carrier and equality cost. It ensures the zero-cost property required for the universal forcing chain, consistent with the composition law and steps T5-T8.
scope and limits
- Does not apply to non-cyclic carriers.
- Does not evaluate costs between distinct elements.
- Does not constrain the modulus beyond being a natural number.
- Does not reference the phi-ladder or mass formulas.
formal statement (Lean)
23@[simp] theorem zmodCost_self {n : ℕ} (a : ZMod n) : zmodCost a a = 0 := by
proof body
One-line wrapper that applies simp.
24 simp [zmodCost]
25