Exact Bipartite M2 Magic Saturation Laws over GF(d^2): Golden Ratio Trace, Quantized Schmidt Spectra, and the Euler-Pi Limit - RJW

Abstract Through exact numerical construction and 1,000-digit precision verification across all 28 prime dimensions from $d=5$ to $d=113$, we present closed-form mathematical laws governing the maximization of bipartite stabilizer Rényi entropy (magic) $M_2^*(d) = \ln(d^4/(2d^2-1))$. We introduce the zero-parameter Golden-Lucas trace generator extending over $\mathrm{GF}(d^2)$ for primes $d \equiv \pm 2 \pmod 5$, prove an exact quantized half-rank tripartite Schmidt spectrum law yielding $S_{\mathrm{ent}} = \ln(d) - \frac{d-1}{d}\ln(2)$, and demonstrate that the infinite product over our exact dimension-bounds recovers $\pi$ strictly through the Euler-Basel identity over dimension limits. Because these bounds are verified through exact Weyl-Heisenberg operators, the conclusions rest entirely on reproducible arbitrary-precision number theory and linear algebra. The complete numerical dataset, including the 1,000-digit precision tables and explicit complex state amplitudes for all 28 primes, is available in the supplemental Magic_All.Txt archive attached to this deposit.Authors Note: Greetings dear reader. I will keep it simple tonight. Math does not lie and in my opinion, it is God's gift to see reality in the universe or prove its wrong. May your path to elegance be quick and sure. Thanks for reading. RJW