T
plain-language theorem explainer
The triangular number function T(n) = n(n+1)/2 supplies the cumulative phase count T(D²) inside the synchronization period lcm(2^D, T(D²)) for odd dimensions. Recognition Science researchers working on dimensional rigidity cite it to obtain the D=3 minimum of 45 and the resulting 360-tick period. The definition is a direct one-line implementation of the closed-form sum of the first n naturals.
Claim. The n-th triangular number is $T(n) = n(n+1)/2$ for $n$ a natural number.
background
Module Gap45.SyncMinimization formalizes constraint (S) from the Dimensional Rigidity paper: among odd spatial dimensions D ≥ 3, D = 3 uniquely minimizes the synchronization period lcm(2^D, T(D²)). The 8-tick period generalizes to 2^D while the parity matrix of hypercube Q_D has D² entries, so the cumulative phase is the triangular number T(D²). Upstream, the declaration depends only on the natural-number abbreviation T := ℕ from Breath1024.
proof idea
one-line definition implementing the standard closed-form formula for triangular numbers.
why it matters
This supplies the T(D²) value used to compute the minimal sync period 360 at D = 3, which is referenced by downstream results including space_translation_invariance_implies_momentum_conservation (Noether), newton_first_law, Jcost_quadratic_leading_coeff, arrheniusRate, and narrativeGeodesicCert. It fills the T(D²) slot in the eight-tick octave generalization and the odd-D coprimality argument of the paper.
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