√2 as the Informational Stability Attractor: Why Persistent Structures Converge to the Simplest Irrational

Why does √2 appear in protein geometry, mitochondrial architecture, photosynthetic complexes, and cosmological structure? Coincidence — or a universal selection principle? We propose that √2, as the smallest irrational algebraic number, occupies a unique position as the informational stability attractor for persistent structures. The argument is rooted in a fundamental tension: any structure that persists in time must simultaneously avoid two failure modes. Rigidification — when internal frequency ratios are rationally related, modes lock into exact resonance, energy accumulates destructively, and the structure becomes fragile. Dissipation — when ratios are highly irrational (transcendental), modes cannot maintain coherent structure and energy disperses. √2 sits at the critical point between these regimes: irrational enough to avoid mode-locking, yet the simplest possible irrational (continued fraction 1; 2, 2, 2, ...), allowing coherent structure to persist. Convergent empirical evidence from four independent domains supports the hypothesis. In mitochondrial bioenergetics, the mammalian ATP synthase c₈-ring produces a 45° proton step (cos 45° = √2/2 exactly), and cristae dimensions follow √2 scaling with 1% deviation — a system persistent for ~200 million years. In photosynthesis, LHCII inter-pigment distances follow a √2 geometric series (mean error 4.8%), outperforming 98.6% of random scaling factors — persistent for ~2.5 billion years. In protein geometry, the three fundamental β-sheet dimensions form a geometric sequence with ratio √2 (1% deviation) — persistent across the entire tree of life (>3 billion years). In cosmology, the galaxy correlation function slope γ ≈ 1.59 matches 3 − √2 ≈ 1.586 — persistent for ~13 billion years. The paper makes three categories of predictions. The rational regime: biological systems deviating toward rational ratios should exhibit pathological rigidity (epileptic synchrony, low heart rate variability, amyloid crystallization). The golden regime: φ should appear in systems optimized for exploration over efficiency (phyllotaxis, extreme-environment organisms). The √2 regime: long-lived structures should statistically cluster around √2 ± 3%. Falsification criteria are explicit: no peak at √2 in a PDB analysis of >10,000 protein structures, no deviation correlation with disease states in cristae morphometry, no log-periodic modulation in galaxy surveys, or a different constant fitting the cross-domain data better. The paper directly addresses the numerology concern and distinguishes the hypothesis through mechanism (Diophantine theory), specificity (persistent structures only), and falsifiability.