Stronger at the Broken Places: A Free-Energy Theory of Golden Repair in Fractured Solids and Social Networks

A repaired object is usually assumed to be, at best, as good as it was before it broke. We report that this is false. We define the $\text{Kintsugi coefficient}$ $\kappa$ — the ratio of a specimen’s fracture toughness (the energy it takes to advance a crack) after a visible golden repair to its toughness while pristine — and show, across ceramics, engineering teams, and simulation, that $\kappa >1$: a body broken and mended with a conductive golden seam is $\text{stronger}$ than one that never broke. The result follows from a Landau free energy (the quantity a system settles into minimizing at equilibrium) whose global minimum is the repaired state and whose pristine state is merely metastable, together with a crack-shielding law $$\kappa (\varphi )=\frac{1}{(1-\eta \varphi {)}^2}>1.$$ In three-point bending of $N=180$ soda-lime specimens we measure $\kappa =1.61\pm 0.07$ for golden repair, against $0.71$ for an epoxy sham, with the Weibull modulus rising from $5.4$ to $9.2$. Rescaling the seam length by the Griffith length collapses ceramics, a $312$-team retrospective cohort, and a spring-network simulation onto one master curve, with exponent $\beta =1/\phi \approx 0.618$ and saturation at ${\kappa }_{\infty }=\phi \approx 1.618$ — the golden ratio, a coincidence we were unable to remove. We conclude that pristineness is a liability, that repaired-and-public systems occupy a deeper free-energy well than untested ones, and that the thermodynamically optimal manufacturing step is to fracture the product on purpose before shipping it.