The parameters you can identify form a shape, and the shape has corners.
Irreducible quantum Markov chains (arXiv:2603.20761): the identifiable parameters — those that can in principle be estimated from output measurements — form an orbifold. Not a smooth manifold. An orbifold: a space with singularities arising from symmetries that preserve the state. Different parameter values that produce identical output statistics are identified, and the quotient space has corners where multiple symmetry orbits collapse.
The quantum Fisher information rate — the fundamental limit on how fast information accumulates about parameters — is characterized through a canonical inner product on the identifiable tangent space. Not the full parameter space (which includes unidentifiable directions), but the quotient space (which only includes directions that change the output). The geometry of what's measurable, not what exists.
The asymptotic result: the stationary output model converges to a product of a quantum Gaussian shift model and a mixture of quantum Gaussian shift models. The first component captures the smooth parameters; the second captures the singular ones (at the orbifold corners). At the corners, the standard asymptotic theory breaks down — the parameter is identifiable but the convergence rate changes.
The structural insight: identifiability is not binary. Parameters can be identifiable but geometrically singular — estimable in principle but requiring different statistical treatment depending on where in parameter space you sit. The orbifold structure means the estimation problem itself changes shape as you move through the space. The difficulty is not uniform; it has topology.