Three-body unitary coupled-channel analysis on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>η</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1405</mml:mn><mml:mo>/</mml:mo><mml:mn>1475</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>

The recent BESIII data on $J/\psi\to\gamma(K_SK_S\pi^0)$, which is significantly more precise than earlier $\eta(1405/1475)$-related data, enables quantitative discussions $\eta(1405/1475)$ at the previously unreachable level. We conduct a three-body unitary coupled-channel analysis of experimental Monte-Carlo outputs for radiative $J/\psi$ decays via $\eta(1405/1475)$: $K_SK_S\pi^0$ Dalitz plot distributions from BESIII, and branching ratios $\gamma(\eta\pi^+\pi^-)$ $\gamma(\gamma\pi^+\pi^-)$ final states relative to that $\gamma(K\bar{K}\pi)$. Our model systematically considers (multi-)loop diagrams an associated triangle singularity, critical in making excellent predictions $\eta(1405/1475)\to \pi\pi\pi$ lineshapes ratios. pole locations are revealed first time. Two poles $\eta(1405)$ found different Riemann sheets $K^*\bar{K}$ channel, while one $\eta(1475)$. described with two bare dressed by continuum states. lower state would be excited $\eta^\prime$, higher could $\eta^{(\prime)}$, hybrid, glueball, or their mixture. This work presents first-ever determination based manifestly framework applied (Dalitz plots).