fragility
plain-language theorem explainer
fragility defines a dimensionless proxy for glass fragility at the k-th multiple of the eight-beat relaxation period. Researchers modeling supercooled liquids and vitrification in the Recognition Science framework cite it to encode universal decay of viscosity departure from Arrhenius form. The definition is a direct algebraic power expression using the fixed eight-beat period and the successor of k.
Claim. Let $k$ be a natural number and let $φ$ be the golden ratio. Then fragility$(k) = (1/φ)^{8(k+1)}$.
background
The module treats glass transition as the point where a supercooled liquid vitrifies into an amorphous solid, with all properties derived from 8-tick relaxation dynamics. Fragility quantifies the rapidity of viscosity increase near Tg: strong glasses such as SiO₂ exhibit low values while fragile glasses such as polymers exhibit high values. The eight-beat period is the constant 8 that sets the fundamental relaxation timescale, and Constants.phi supplies the scaling factor; upstream results include the definition of eight_beat_period and the successor operation on the index.
proof idea
This is a one-line definition that substitutes eight_beat_period for the constant 8, applies the successor to the index k, and raises the reciprocal of Constants.phi to that power.
why it matters
The definition supplies the fragility expression invoked by downstream results including glass_univ (positivity for every k) and fragility_one_lt_zero (strict decay with k). It implements the RS prediction that fragility scales with φ^k and ties directly to the eight-tick octave (T7) of the forcing chain. The module documentation states that fragility correlates with molecular structure complexity and supports the universal Tg/Tm ratio near 2/3.
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