triangular_3
plain-language theorem explainer
The third triangular number equals 6. Workers on the gap45 synchronization argument cite it when counting cumulative phase steps over the closed cycle. The proof reduces immediately by reflexivity on the triangular definition.
Claim. The third triangular number equals 6: $T(3) = 6$, where $T(n) := n(n+1)/2$.
background
Triangular numbers accumulate phase contributions in the ledger model. The definition states $T(n) = 1 + 2 + ... + n = n(n+1)/2$. The module derives the 45-tick count as $T(9)$ by extending the 8-tick cycle with one closure step, analogous to fence posts, so that cumulative phase returns to the initial state after nine steps.
proof idea
One-line wrapper that applies reflexivity to the triangular definition.
why it matters
This lemma supports the derivation of $T(9) = 45$ that closes the physical motivation gap for the 45-tick synchronization. It grounds the count in cumulative phase over the nine-step cycle required by the eight-tick octave closure principle. The result sits inside the gap45 sequence that addresses the unmotivated 45 in the dimension-forcing argument.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.