Clean four-state competitive models around free receptor

[choutkaj]

Summary

• Refactor four-state competitive models to solve the quintic in true free receptor concentration R rather than r_equiv.
• Compute tracer-bound fraction from the actual four-state species population RS + RLS.
• Keep the total/nonspecific model as the same machinery with Kd_eff = (1 + N) * Kd.
• Update four-state root-selection tests, including the degenerate Kds == Kd3 quartic case.
• Rewrite algebra/derive_four_state_r_equivalent_coefficients.ipynb so it derives the free-receptor quintic and validates the runtime coefficient helper.
• Add sympy to the notebook dependency group for the derivation notebook.

Validation

• Locally sanity-checked the derived free-receptor coefficients against an independent mass-balance solve for typical, cooperative, anti-cooperative, and Kds == Kd3 degenerate cases.
• Existing mass-balance validation tests should continue to compare model output against the numerical species-balance solver.

Note: the old .qmd documentation was intentionally left untouched.

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[chatgpt-codex-connector]

Guard free-species solve against zero Kd or Kd3

This introduces an unguarded division by Kd * Kd3; when either parameter is 0.0 (both are allowed at their lower bounds in this model, and Kd can be hit during fitting), c becomes inf/nan, which then propagates through the quadratic and returns invalid model responses. The previous implementation did not divide by Kd or Kd3 in the response mapping, so this is a regression that can break fits or produce NaN curves at boundary parameter values.

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