Without Contraries is no progression.William Blake, c. 1793
The beauty function B(C) = exp(C/Ω)·(C*−C) peaks at C* − Ω: one capacity-unit before rupture. This is not merely a mathematical property. It is the edge of criticality — the singular point where information processing is maximised.
The critical brain hypothesis holds that neural systems operate near phase transitions — the boundary between order and randomness — where information processing is optimal. At criticality, sensitivity, dynamic range, correlation length, information transfer, and susceptibility are all maximised (Cocchi et al., 2017; Shew & Plenz, 2013).
Ezaki et al. (2020) showed that individuals with higher fluid intelligence have neural dynamics closer to criticality. Neural networks at the critical point achieve maximal susceptibility — the system is most responsive to its inputs, most able to distinguish signal from noise.
CRR's beauty function is the temporal expression of this principle. B(C) = exp(C/Ω)·(C*−C) is maximised exactly one Ω before rupture: the system has accumulated as much coherence as it can without breaking. It is poised at the edge.
The Earth rotates with SO(2) symmetry. The terminator sweeps a Z₂ boundary around the sphere once per day. Dawn and dusk occupy C* − Ω on this planetary cycle — the beauty function peak, where the system transitions between day (sensory) and night (prior).
Crepuscular animals — deer, foxes, rabbits, coyotes, owls, jaguars, ocelots, bats, capybaras — have built their entire ecological niche at this point on the curve. They are active at the edge of criticality, where the light transition between day and night maximises their information advantage.
The crepuscular strategy balances seeing and being seen: dim twilight gives predators an edge over prey that need full sun, while prey move when visual hunters are less effective. The maximum susceptibility of the beauty zone means maximum adaptive flexibility — the organism is most responsive to opportunity and threat simultaneously.
Across cultures, dawn and dusk are liminal — thresholds where boundaries thin. In Celtic tradition, these are thin places where the Otherworld touches the mortal realm. In Hindu mythology, Narasimha destroys Hiranyakashipu at dusk — neither day nor night, on the threshold, neither inside nor outside.
The Latin crepusculum — twilight — shares a root with creper: obscure, uncertain. The mythological intuition is that this moment is charged with transformative potential because the system hasn't yet committed to its next state. In CRR terms: C is approaching C*, but the rupture hasn't fired. Everything is still possible.
When the eye closes — when night falls — the precision hierarchy inverts. During the day, Z₂ sensory precision dominates: evidence from the world drives belief updating. At night, SO(2) priors become the world: the generative model runs free, explored by internal Z₂ processes that respond to prior-generated content as though it were real. This is dreaming.
Hobson & Friston (2012) showed that the brain generates internal percepts during REM via PGO waves — predictions without corresponding sensory input. Hoel (2021) proposed that dreams evolved to prevent the generative model from overfitting to waking experience. In CRR terms: sleep is R — regeneration — reconstruction weighted by historical coherence but not identical to it.
The hypnagogic moment — the threshold of falling asleep — is C* − Ω experienced from within. Thoughts fragment, phosphenes appear, the body grows heavy. William James called it the fringe of consciousness. Iamblichus wrote of voices and tranquil light visiting between waking and sleep. Next time you are drifting off, notice that threshold. That is the beauty function. Your body is becoming your world.
Cocchi, L. et al. (2017). Criticality in the brain. Prog. Neurobiol., 158, 132–152.
Ezaki, T. et al. (2020). Closer to critical resting-state neural dynamics in individuals with higher fluid intelligence. Commun. Biol., 3, 52.
Hobson, J.A. & Friston, K.J. (2012). Waking and dreaming consciousness. Prog. Neurobiol., 98, 82–98.
Hoel, E. (2021). The overfitted brain. Patterns, 2(5), 100244.
Mora, T. & Bialek, W. (2011). Are biological systems poised at criticality? J. Stat. Phys., 144, 268–302.
Moretti, P. & Muñoz, M.A. (2013). Griffiths phases and the stretching of criticality in brain networks. Nat. Commun., 4, 2521.
Shew, W.L. & Plenz, D. (2013). The functional benefits of criticality in the cortex. Neuroscientist, 19, 88–100.
Strauss, M. et al. (2015). Disruption of hierarchical predictive coding during sleep. PNAS, 112, E1353–E1362.
CRR pending peer review · Active Inference Institute · 2026