Examples

Every certificate below ships with the package (twindex example NAME) and is replayed in CI on Python 3.10–3.13.

name	kind	verdict	what it shows
regression_pair	pair	VALID	analytic case on [0,4]⁴: p = x(1−x)/2 certifies θ = 0 and θ = (4,4,4,4), gap 4
theta500_pair	pair	VALID	data from θ̄ = 500: twins at θ₀ = 0 and θ₀ = 1000, so the certified diameter is the full prior width
witness_theta500	witness	VALID	the generating parameter is data-compatible
box_inside	box	VALID	the whole box [0,2]⁴ is data-compatible
box_outside	box	VALID	the whole box [0,4]⁴ misses the θ̄ = 500 data
paving_regression	paving	VALID	certified 4 ≤ D ≤ 4 on [0,4]⁴
paving_theta500	paving	VALID	certified 2 ≤ D ≤ 2 on the local prior [499,501]⁴
invalid_gap	pair	INVALID	the claimed gap exceeds the exact gap
invalid_bc	pair	INVALID	a candidate violates p(0) = 0
inconclusive	witness	INCONCLUSIVE	a crude candidate cannot decide compatibility

The headline result

With five sensors at x = 1/6, …, 5/6, noise ε = 10⁻³, data generated at θ̄ = (500, 500, 500, 500) and prior [0, 1000]⁴, the certified twins are

θ  = (0,    999.999999, 24.179257, 827.878564)
θ' = (1000,   0.000001, 975.820743, 172.121436)

These data do not reduce the worst-case uncertainty in θ₀ below the prior bound: every estimator has worst-case error at least 500. This is a prior-limited result. It does not claim that every intermediate θ₀ is compatible, and it says nothing about the other coordinates.