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Paper mapping

Mapping the paper to the codebase

This guide cross-references “A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode” with symbols and modules in this repository.

  • Geometric polynomial equation and the series solution S[t2,t3,...]:
  • Paper: Equation 0 = 1 − α + t2 α^2 + t3 α^3 + ..., with α = S[...].

  • Code: geodepoly/hyper_catalan.py implements:

    • hyper_catalan_coefficient(m_counts) for the array coefficients.
    • evaluate_hyper_catalan(t_values, max_weight) to numerically sum a truncated S.
    • evaluate_quadratic_slice(t2, ...) and catalan_number(n) for the Catalan slice.

  • Lagrange inversion / series reversion for a shifted polynomial:

  • Paper: Sections on Lagrange inversion and series bootstrap.

  • Code: geodepoly/series_solve.py:

    • shift_expand, inverseseries_g_coeffs, series_step, series_one_root.

  • Finishing methods and polishing:

  • Paper: Practical computation beyond the formal series.

  • Code: durand_kerner, halley_refine, newton_refine, composed in series_solve_all.

  • Resummation and acceleration:

  • Paper: discusses summation/acceleration themes.

  • Code: geodepoly/resummation.py supports Padé and Borel(-Padé) options.

  • The Geode array and combinatorial structure:

  • Paper: factorization and conjectures about the array.

  • Code: hyper_catalan_coefficient and evaluate_hyper_catalan expose the array numerics; future work can add factorization utilities once conjectures are finalized.

  • Bridges and examples:

  • Paper: Worked examples (e.g., Wallis cubic) and CAS bridges.
  • Code: bridges/geodepoly_cli.py, examples/quickstart.py, and tests in tests/.