triangular_0
plain-language theorem explainer
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
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.
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