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High-precision measurements of transition frequencies to the $P$-states of helium [1] for principal quantum number $n$ as high as 102 have confirmed a 9$\sigma$ disagreement between theory and experiment for the ionization energy of the $1s2s\;^3S_1$ state. However, traditional theoretical methods of calculation fail in this range of high $n$ for comparison. This paper presents high precision variational calculations in Hylleraas coordinates for all singlet and triplet $P$-states of helium up to $n = 35$ with a uniform accuracy of 1 part in $10^{22}$ for the nonrelativistic energy [2]. Mass polarization, relativistic and quantum electrodynamic effects are included to achieve a final accuracy of $\pm$1 kHz or better for the ionization energy of the Rydberg states of $^4$He in the range $24\le n \le 35$. The results are combined with 11 transition frequency measurements of Clausen et al. [1] to obtain complementary measurements of the ionization energy of the $1s2s\;^3S_1$ state that do not depend on quantum defect extrapolations to the series limit. The result from the triplet spectrum yields an ionization energy of 1152\,842\,742.728(6) MHz, which agrees with but is larger than the experimental value by 14 $\pm$17 kHz. However, it confirms a much larger 9$\sigma$ discrepancy of $0.468\pm0.055$ MHz with the theoretical ionization energy of Patk\'o\v{s} et al. [3]. The results provide a test of the quantum defect extrapolation method at the level of $\pm$17 kHz, and insights into the validity of the quantum defect method. A combined $1/n$ expansion and quantum defect analysis yields theoretical values that cover the full range of measurements up to $n=102$.\newline
[1] G. Clausen et al.\ Phys.\ Rev. A {\bf 111}, 012817 (2025).\newline
[2] G. W. F. Drake et al. Phys.\ Rev.\ A {\bf 113}, 012810 (2026).\newline
[3] Patk\'o\v{s} et al.\ Phys.\ Rev.\ A {\bf 103}, 042809 (2021).