[Bug]: Joining solutions with and without sensitivities returns another solve's sensitivities #5845

PyBaMM Version

main at 445ba33. Also reproduces with #5794 applied.

Python Version

3.14

Describe the bug

Joining a solution solved without sensitivities to one solved with them gives wrong or unreadable sensitivities:

• first_state or last_state silently returns the other solve's sensitivities.
• Reading any variable's sensitivities raises.

Solution.__add__ and Solution.from_sub_solutions only extend _all_sensitivities with segments that have sensitivities. first_state, last_state and _merge_sensitivities then assume the first sensitivity block belongs to all_ts[0] and the last to all_ts[-1].

solver.step and Simulation never build such a join: stepping with sensitivities from a solution without them raises. So this only happens when solutions are added by hand. It was raised in review of #5794 (comment).

Expected: the join raises, or the joined solution has no sensitivities. One option is for __add__ and from_sub_solutions to reject a join whose segments have different sensitivity_names, or to drop the sensitivities.

Steps to Reproduce

import numpy as np
import pybamm

model = pybamm.BaseModel()
u = pybamm.Variable("u")
a = pybamm.InputParameter("a")
model.rhs = {u: a}
model.initial_conditions = {u: a}
model.variables = {"u": u}
solver = pybamm.IDAKLUSolver()


def solve(t_interp, calculate_sensitivities):
    return solver.solve(
        model,
        [t_interp[0], t_interp[-1]],
        t_interp=np.array(t_interp),
        inputs={"a": 1.0},
        calculate_sensitivities=calculate_sensitivities,
    )


# Each solve restarts from u = a, so du/da = 1 + (t - t0)
early = [0, 0.5, 1]  # du/da = 1, 1.5, 2
late = [1, 2, 3]  # du/da = 1, 2, 3

joined = solve(early, False) + solve(late, True)
print("first_state:", joined.first_state.sensitivities["a"].ravel())

joined = solve(early, True) + solve(late, False)
print("last_state:", joined.last_state.sensitivities["a"].ravel())
joined["u"].sensitivities

On main:

• first_state is the later solve's sensitivity at t = 3, although the earlier solve has none.
• last_state is the earlier solve's sensitivity at t = 1, although the later solve has none.

With #5794, first_state returns [2.] instead, the later solve's value at t = 2.

Relevant log output

first_state: [3.]
last_state: [2.]
Traceback (most recent call last):
  ...
ValueError: zip() argument 5 is shorter than arguments 1-4