PCF Spectral Classes — Supplementary Data and Code

Supplementary material for:

Spectral Classes of Polynomial Continued Fractions: A Hypergeometric Framework and Arithmetic Obstructions
Submitted to Experimental Mathematics, April 2026.

Contents

File	Description
spectral_class_phase1.py	Computes the Phase 1 taxonomy: spectral class tuple (d, Λ, Δ, ρ, τ) for all 37 families
spectral_class_phase2_congruence.py	Computes z0-orbit analysis, PGL2(Q) orbit assignments, and obstruction diagnostics
pcf_spectral_phase1_taxonomy.csv	Complete 37-family taxonomy with numerical limits to 50+ digits
pcf_spectral_phase2_congruence.csv	Hypergeometric z0 data and 2F1 fit results for all identified d=1 families

Requirements

Python 3.10 or later.

Install dependencies:

pip install mpmath sympy

Tested with mpmath 1.3, sympy 1.12.

Reproducing the results

Reproduce Table 1 (37-family taxonomy):

python spectral_class_phase1.py

Expected runtime: approximately 5 minutes.
Output: pcf_spectral_phase1_taxonomy.csv

Reproduce z0 analysis and obstruction diagnostics:

python spectral_class_phase2_congruence.py

Expected runtime: approximately 10 minutes.
Output: pcf_spectral_phase2_congruence.csv

Key results reproduced by these scripts

• Poincaré characteristic lemma: verified to 65-digit precision for all d=1 families (residual < 1e-64 in all cases)
• Catalan logarithmic obstruction: 3/(2n) Frobenius residual, PSLQ exclusion at 220-digit precision
• Hypergeometric z0 framework: 15/15 identified d=1 families admit a 2F1 representation
• Three-tier obstruction hierarchy: Catalan (Delta=0), V_quad (d=2, confluent-Heun), Apéry-zeta(3) (d=3, Picard-Fuchs)

Citation

If you use this code or data, please cite:
[Citation will be updated upon acceptance]