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      "end_summary": "Fixed NumPy v2 compatibility issues by removing deprecated autograd usage and stabilizing Interpolant JSON serialization. Specific changes:\n1. In `pybamm/expression_tree/functions.py`:\n   - Changed default `derivative` from \"autograd\" to \"derivative\" to avoid importing deprecated autograd.\n   - Replaced `_function_diff` to raise NotImplementedError for autograd path (matching test expectation) and guard derivative calls to prevent AttributeError on numpy functions lacking `.derivative()`.\n   - Adjusted error messages to match test expectations (NotImplementedError with specific text, correct message for multi-child differentiation).\n   - Improved `Function.to_equation` to return evaluated constants when all children are constant numbers, falling back to printable name otherwise.\n2. In `pybamm/expression_tree/interpolant.py`:\n   - Made `set_id()` generate deterministic IDs using MD5 hashing of mutable content to avoid hash randomization issues across NumPy/Python versions.\n   - Added `\"_num_derivatives\": 0` to `to_json()` output to match test expectation.\nAll three specification tests now pass.",
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