triangular_5
The lemma states that the fifth triangular number equals fifteen. Researchers deriving the forty-five tick synchronization number from the eight-tick cycle closure would cite this intermediate value when accumulating phase contributions. The proof reduces immediately by reflexivity to the triangular number definition.
claim$T(5) = 15$, where the triangular number function satisfies $T(n) = n(n+1)/2$.
background
The triangular number function is defined by $T(n) = n(n+1)/2$ for natural numbers $n$, with $T$ serving as the abbreviation for fundamental periods. In the Gap45 physical motivation module, the number 45 arises as the ninth triangular number $T(9)$ because an eight-tick cycle requires a closure step to return to the initial phase, yielding nine steps total and cumulative phase $T(9) = 45$. Upstream results supply the definition of triangular and the identification of $T$ with the naturals.
proof idea
One-line term proof that applies reflexivity directly to the triangular number definition evaluated at five.
why it matters in Recognition Science
This lemma supplies an explicit intermediate value used in the module's derivation of the forty-five tick synchronization from the eight-tick octave and closure principle. It supports the cumulative phase accumulation step that addresses the physical motivation gap for the 45-tick argument. No downstream theorems currently depend on it.
scope and limits
- Does not prove the general closed-form formula for triangular numbers.
- Does not supply the physical interpretation of phase accumulation.
- Does not connect the value to synchronization constraints or ledger neutrality.
- Does not address other triangular numbers such as T(9) or T(8).
formal statement (Lean)
102@[simp] lemma triangular_5 : triangular 5 = 15 := rfl
proof body
Term-mode proof.
103
104/-- T(8) = 36. -/