triangular_2
plain-language theorem explainer
The second triangular number equals 3. Workers deriving the 45-tick synchronization in the gap-45 module cite this base case when accumulating phase sums over the closed 8-tick cycle. The proof is a direct reflexivity reduction once the triangular definition is applied.
Claim. The second triangular number satisfies $T(2) = 3$.
background
Triangular numbers are defined by the formula $T(n) = n(n+1)/2$, which sums the first $n$ positive integers and models cumulative phase accumulation in the ledger. The module treats the 8-tick cycle (arising from $2^D$ with $D=3$) as requiring a ninth closure step to return to the initial phase state, so the cumulative phase over nine steps is the ninth triangular number 45. This supplies the physical motivation for the synchronization constraint that closes the identified gap in the dimension-forcing argument.
proof idea
The proof is a one-line reflexivity application that matches the triangular definition at input 2 directly to the integer 3.
why it matters
This base case anchors the sequence of triangular numbers used to obtain $T(9) = 45$ and thereby motivates the 45-tick synchronization requirement. It feeds the parent derivation of the closed-cycle phase sum in the same module and supports the eight-tick octave together with the closure principle for three spatial dimensions. The result addresses the paper's open question on physical motivation for the 45-tick count.
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