Derivative re-factor and updated Example 7 validation

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Summary

Refactor equilibrium derivative evaluation and strengthen Example 7 shock validation.

Changes

• Reuse one Jacobian factorization for equilibrium sensitivities and remove smooth-mode finite-difference stabilization.
• Correct incident equilibrium/frozen shock branch generation.
• Expand derivative and shock-output validation coverage.

Testing

• python test_main.py passes 14/14

Compatibility / Numerical behavior

• No expected changes to numerical results
• Expected changes: smooth derivatives and combined equilibrium/frozen shock output now follow the corrected analytical behavior.