derivations_equivalent
plain-language theorem explainer
The equivalence shows that the ninth triangular number equals nine times five, identifying the cumulative phase sum over a closed eight-tick cycle with the closure-fibonacci factorization. Researchers filling the physical-motivation gap in the 45-tick synchronization step of the dimension-forcing argument would cite this result. The proof is a one-line reflexivity check that both sides compute definitionally to 45.
Claim. Let $T(n) = n(n+1)/2$ be the nth triangular number. Then $T(9) = 9 × 5$, where 9 is the closure step count for the eight-tick cycle and 5 is the Fibonacci factor.
background
The Gap45.PhysicalMotivation module supplies a physically grounded derivation of the synchronization number 45 that appears in the dimension-forcing chain. The eight-tick cycle follows from $2^D$ with $D=3$, but the cycle is not closed; the closure principle adds one step, producing nine total steps. Cumulative phase over these steps is the triangular number $T(9) = 45$. Upstream results include the phase definition $kπ/4$ for $k=0..7$ in EightTick and the abbreviation $T := ℕ$ used for fundamental periods in Breath1024.
proof idea
The proof is a one-line reflexivity check. Both sides of the equality are definitionally identical once triangular is expanded via its formula and closure_number and fibonacci_factor are substituted with their concrete values 9 and 5.
why it matters
This theorem supplies the missing physical motivation for the 45-tick synchronization that the paper flags as unmotivated. It connects the triangular-number reading directly to the closure-fibonacci product, thereby supporting the eight-tick octave (T7) and the forcing of $D=3$ (T8) in the UnifiedForcingChain. No downstream uses are recorded yet; the result stands as a self-contained algebraic bridge inside the Gap45 development.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.