triangular_0
The base case states that the zeroth triangular number vanishes. Researchers deriving the 45-tick synchronization from cumulative phase accumulation in closed 8-tick cycles cite this initialization step. The proof reduces immediately by reflexivity to the definition of the triangular number function.
claim$T(0) = 0$ where $T(n) = n(n+1)/2$ denotes the $n$th triangular number.
background
Triangular numbers are defined by $T(n) = n(n+1)/2$, the sum of the first $n$ positive integers. The abbreviation T stands for fundamental periods in the Breath1024 foundation. The Gap45.PhysicalMotivation module supplies a physically grounded derivation of the number 45 as the ninth triangular number, addressing the paper's gap on the unmotivated 45-tick synchronization in the 8-tick framework.
proof idea
One-line wrapper that applies reflexivity to the definition of triangular.
why it matters in Recognition Science
This base case anchors the triangular sequence that motivates 45 as cumulative phase over the closed 9-step cycle required by the 8-tick octave (T7) and D=3 (T8). It supports sibling results such as triangular_9_is_45 that close the synchronization argument in the dimension-forcing chain.
scope and limits
- Does not establish triangular values for positive integers.
- Does not derive the 45-tick synchronization constraint.
- Does not incorporate phase accumulation or closure principle.
formal statement (Lean)
87@[simp] lemma triangular_0 : triangular 0 = 0 := rfl
proof body
Term-mode proof.
88
89/-- T(1) = 1. -/