triangular_rec_at_8
The declaration confirms that the ninth triangular number equals the eighth plus nine. Researchers modeling cumulative phase accumulation over the closed 8-tick cycle cite this identity when deriving the 45-tick synchronization. The proof is a one-line reflexivity check that follows immediately from the explicit formula for triangular numbers.
claim$T(9) = T(8) + 9$, where $T(n) = n(n+1)/2$ denotes the $n$th triangular number.
background
Triangular numbers are defined explicitly by $T(n) = n(n+1)/2$, the sum of the first $n$ positive integers. The Gap45.PhysicalMotivation module uses this to supply a physically grounded derivation of the number 45, interpreting it as the cumulative phase over a closed cycle of nine steps (the 8-tick octave from $D=3$ plus one closure step). Upstream, Breath1024 defines $T$ as the fundamental period in natural numbers, while the sibling triangular definition supplies the closed-form expression applied here.
proof idea
The proof is a one-line term proof that applies reflexivity (rfl) directly to the definition of triangular, which satisfies the recurrence by algebraic construction.
why it matters in Recognition Science
This identity supplies the concrete verification step needed to reach $T(9)=45$ in the 45-tick synchronization argument. It fills the physical-motivation gap for the eight-tick framework by confirming the cumulative-phase formula at the closure point. The result supports the synchronization requirement between the 8-tick ledger neutrality constraint and the phase-accumulation constraint, though no downstream theorems currently depend on it.
scope and limits
- Does not prove the recurrence for arbitrary $n$.
- Does not derive the physical constants or the full synchronization constraint.
- Does not address the Berry creation threshold or the phi-ladder mass formula.
formal statement (Lean)
84theorem triangular_rec_at_8 : triangular 9 = triangular 8 + 9 := by rfl
proof body
Term-mode proof.
85
86/-- T(0) = 0. -/