triangular_formula
The triangular number formula equates the nth triangular number to n(n+1)/2. Researchers deriving the 45-tick synchronization from the 8-tick cycle in Recognition Science cite this to compute cumulative phase accumulation over a closed cycle. The proof is a one-line reflexivity that matches the definition directly to the closed form.
claimThe nth triangular number satisfies $T(n) = n(n+1)/2$.
background
The Gap45.PhysicalMotivation module supplies a physically grounded derivation of the number 45 in the dimension-forcing argument. It identifies 45 as the 9th triangular number arising from cumulative phase in the 8-tick ledger cycle: each tick k contributes phase proportional to k, so the total over a closed cycle of 9 steps is T(9). The sibling definition states that the nth triangular number is T(n) = 1 + 2 + ... + n. Upstream structures from LedgerFactorization and PhiForcingDerived calibrate the J-cost and ledger neutrality that make the phase accumulation constraint meaningful.
proof idea
This is a one-line wrapper that applies reflexivity to equate the definition of triangular to its closed-form expression.
why it matters in Recognition Science
This declaration confirms the arithmetic identity required for the cumulative phase argument in the 45-tick synchronization. It directly supports the closure principle (8 ticks plus one step yields 9) that produces T(9) = 45, linking the eight-tick octave (T7) and D = 3 spatial dimensions from the forcing chain. The result fills the physical motivation gap noted in the module for the Recognition Science ledger model.
scope and limits
- Does not prove the formula for non-natural numbers.
- Does not derive the physical interpretation of the 45-tick cycle.
- Does not connect to phi-ladder mass formulas or alpha-band constants.
- Does not address ledger neutrality constraints beyond the phase sum.
formal statement (Lean)
80theorem triangular_formula (n : ℕ) : triangular n = n * (n + 1) / 2 := rfl
proof body
Term-mode proof.
81
82/-- Recursive definition of triangular numbers.
83 We verify this by direct computation for specific values. -/