A single algebraic diagnostic detects structurally identical phase transitions from quarks to the cosmic horizon

The ratio of geometric to arithmetic coherence (IC/F), computed from a single algebraic kernel applied to normalised physical measurements, exhibits structurally identical phase-transition signatures across systems spanning 61 orders of magnitude — from quark confinement (10−1510 −15 m) to dark-energy-dominated cosmology (102610 26 m). At the quark–hadron boundary, IC/F collapses 23× as colour confinement drives a single measurement channel to zero; atomic structure subsequently restores coherence 35× through emergent degrees of freedom; dark energy reproduces the identical collapse mechanism at 102610 26 m scales. The heterogeneity gap 𝛥=𝐹−𝐼𝐶Δ=F−IC spikes at every phase boundary tested, providing a single-number diagnostic for structural fragility independent of the underlying dynamics. These patterns emerge from a six-output kernel function 𝐾:[0,1]𝑛×𝛥𝑛→(𝐹,𝜔,𝑆,𝐶,𝜅,𝐼𝐶)K:[0,1] n ×Δ n →(F,ω,S,C,κ,IC), derived from a single admissibility axiom, that maps any bounded measurement vector to fidelity F (weighted arithmetic mean), log-integrity κ (weighted log-mean), and composite integrity IC = exp(κ) (weighted geometric mean), among other quantities. Three algebraic identities — 𝐹+𝜔≡1F+ω≡1 (duality), 𝐼𝐶≤𝐹IC≤F (integrity bound), and 𝐼𝐶=exp⁡(𝜅)IC=exp(κ) (log-integrity relation) — guarantee the diagnostic's structural reliability across all domains. The integrity bound IC ≤ F instantiates the classical weighted inequality of arithmetic and geometric means; the contribution is not the bound itself but its systematic application as a scale-independent phase-transition diagnostic across 20 scientific domains. Exhaustive computational verification — 11,342 automated proofs across 406 physical objects — confirms all three identities to machine precision (<10−15<10 −15 ) with zero violations.