function-field-density

• Digit-restricted coprimality over F_3[t]: among the 2^n polynomials of degree below n with every coefficient in {0, 1}, the ordered coprime density over all 4^n pairs including the zero polynomial, by Moebius inversion over the exact factorisation of every polynomial, cross-checked against Euclid on every pair at n = 1..4.
• The marginal divisibility pi(p) at n = 10, the Euler product on those marginals through degree 5, and the exact chance of sharing no prime of degree at most 5 that the product misses.
• Mixed radix: five digit schedules at 12 positions, 4096 points each, coprime counts among unordered distinct pairs by Moebius inversion, the Euler product through 13 against the exact small-prime joint, and the per-prime local factors.
• Domain as stated on the page: n up to 14 for the function field, 12 digit positions for the schedules.

RUN

• uv run python research/lab/py/function-field-density/density.py
• uv run python research/lab/py/function-field-density/radix.py

WITNESSES

• coprime.md:194 0.564176, 0.563471, 0.562833 at n = 10, 12, 14, gap 0.000333 from 9/16
• coprime.md:195 9/16 = 0.562500 and pi(t) = 1/2
• coprime.md:196 0.333984, 0.333008, mean deviations 0.003798 and 0.001881 at degrees 2 and 3
• coprime.md:197 0.560193 through degree 5, crossing 9/16 between degrees 3 and 4, exact 592189/1048576 = 0.564755, difference -0.004563
• coprime.md:200 gamma = 1.261860, gamma/2 = 0.630930
• coprime.md:205 4096 points, 8386560 pairs, pure base 2 0.607874
• coprime.md:209 4286664, 0.511135; :210 5640929, 0.672615; :211 5097972; :212 4316493, 0.514692; :213 0
• coprime.md:215 alternating gap 0.161480
• coprime.md:216 residues 1024, 1024, 1024, 1024, 0, 0, halves 1/2 and 1/4, log advantage 0.223144
• coprime.md:217 prime 5 at -0.014253, equal factors at 2, 7, 11, 13
• coprime.md:218 0.520112 and 0.640940, +0.000012, underprediction 0.043340
• research/claims/:318 the F_3[t] line, every number above
• research/claims/:467 1.261860 and 0.630930