u/Delicious_Sort5580

...working through the Rice–Mele Zak phase at golden tuning v/w = 1/φ, the elliptic reduction gives Legendre parameter n = 4φ⁻³

Pluging in: since φ⁻³ satisfies x² + 4x − 1 = 0, you get 1 − n + n² = 17x², and the Legendre j-invariant collapses to

j(4φ⁻³) = 16·17³ = 78608.

and this matches the rational elliptic curve y² = x³ + x² − 28x + 48 (LMFDB 200.b1, conductor 200), 2-isogenous to a curve with j = 2048.

Generalizing to metallic ratios ρ_p = (p + √(p²+4))/2:

j(n_p) = 16(p² + 16)³ / p⁴

Integral iff p ∈ {1, 2, 4, 8, 16}. Proof is short — any odd prime in p doesn't divide p² + 16, so p must be a power of 2; for p = 2^a, integrality forces a ≤ 4.

quick question pls : is this collapse documented somewhere? feels like it should overlap with Zagier-style dilogarithm work or known CM j-values, but I haven't found a direct reference. Any help appreciated 👍

Hello everyone. Can somebody give me an honest review about the work i've made? please do not analize the "language" and/or the claims, nor the "grammar"; i was not paying attention to the academics, just focused on the essence/ideea; i think is bigger than it may appear; thank you;

Hello everyone. Can somebody give me an honest review about the work i've made? please do not analize the "language" and/or the claims, nor the "grammar"; i was not paying attention to the academics, just focused on the essence/ideea; i think is bigger than it may appear; thank you;