sides-holding-a-number

• The numbers of Object E on Two bases: the even-digit design E_N at odd side N, its integers Z_N = 2 K_N, and which sides hold one number.
• Exact arithmetic throughout: integers, Fraction, and numpy integer arrays; the only floats are the printed ratios, rounded outward where they are bounds.
• Membership of p/q is the orbit rule N^j p mod 2q in {0, ..., q}; its control is an independent walk on the digits of p/q by Fraction.

VERBS

• period Q: for every p/q in (0, 1) with q <= Q, the residues R(p/q) of the odd sides holding it mod 2q; checks the orbit rule against the digit walk at every p/q in [0, 1] with q <= Q, over odd sides 3..6q+1 at q <= 40 and one period 3..2q+1 above, then the symmetry p -> q - p, the least period 2q, the bounds 2/q and (q+1)/(2q) or 1/2, the rule mod q at odd q, the cyclic-subgroup count of the unit residues, the prime formula, and the invariance r -> q - r at even q; prints the largest shares and the odd primes with share(1/q) > 2/q. Runtime 19.6 s at Q = 200.
• eisenstein Q: at every odd prime q <= Q and every p, the parity of the number of odd sides 3 <= N < q whose first digit of p/q is odd, against the Legendre symbol of the odd one of p, q - p by Euler's criterion; also the exact sum of floor(pN/q) over the odd N < 2q against (2p - 1)(q - 1)/2 + p, and how often its parity matches the symbol. Runtime 0.03 s at Q = 200.
• count K: c(k) = card{odd N : 3 <= N <= 2k, 2k in Z_N} at every k <= K, the small sides by enumerating K_N and the sides above sqrt(2K) by the parity of floor(2k/N) on a difference array; checks every k <= 3000 by the direct digit test; prints the extremes of (c(k) - (1 - log 2) k)/sqrt(k) and the largest error against the bound sqrt(2k) + 1. Runtime 2.1 s at K = 10^6.
• second E: c(k) by an O(sqrt k) block count, checked against the direct test at every k <= 3000, at 60 random k in each [10^e, 2 10^e), e = 6..E, seed 376; prints the mean and spread of the error over sqrt(k) and of the small-side part against kappa and beta. Runtime 7.6 s at E = 11.
• family: the integers below 3000 held by every odd side; the rejection of every p/q, q <= 60, at side 2q - 1; E_N inside E_(N^e) on integers and on fractions, and its strictness at e >= 2 by N + 1 and (N + 1)/N^e; Kummer's reading of K_p at odd primes p <= 23 by math.comb; the carry reading of K_(p^a); and the least witnesses at sides 9 and 15. Runtime 1.2 s.
• inter X [FILE]: the integers below X held by each of nine side sets, by a walk over K_3 from the top digit that cuts a branch when the least member of another K_N at or above the branch's least value lies past its largest; checks the walk against the direct digit test below k = 10^6 at four side sets, and, when FILE is a copy of the A030979 record from https://raw.githubusercontent.com/oeis/oeisdata/main/seq/A030/A030979.seq, the sides {3, 5, 7} against its terms; without FILE that comparison is skipped and the URL printed. Runtime 1.2 s at X = 10^12.
• deep D: the same walk for the sides {3, 5, 7, 11} below 10^D. Runtime 14.6 s at D = 1000.
• weyl M L: for five irrationals from 90-digit truncations, the share of the odd sides 3 <= N <= 2M + 1 holding each to level L, times 2^L. Runtime 0.2 s at M = 10^5, L = 6.

RUN

uv run python mrlyprod/research/lab/py/sides-holding-a-number/sides.py period 200
uv run python mrlyprod/research/lab/py/sides-holding-a-number/sides.py eisenstein 200
uv run python mrlyprod/research/lab/py/sides-holding-a-number/sides.py count 1000000
uv run python mrlyprod/research/lab/py/sides-holding-a-number/sides.py second 11
uv run python mrlyprod/research/lab/py/sides-holding-a-number/sides.py family
uv run python mrlyprod/research/lab/py/sides-holding-a-number/sides.py inter 1e12 A030979.seq
uv run python mrlyprod/research/lab/py/sides-holding-a-number/sides.py deep 1000
uv run python mrlyprod/research/lab/py/sides-holding-a-number/sides.py weyl 100000 6

WITNESSES

• cobham.md, Object E: the orbit rule against the digit walk, 1661996 checks, the least period 2q, the bounds and the maximum 2/3, the rule mod q, the coset count and the prime formula at all 12231 fractions with q <= 200, and the shares printed there (period 200).
• The half-period parity law and the full-period identity at all 4180 pairs with q an odd prime <= 200 (eisenstein 200).
• The readings 0.9821 to 1.0496 of the share to level L times 2^L (weyl 100000 6).
• The error extremes 0.547191, 0.357533, [-0.348228, -0.082525] and c(10^6) = 306665 (count 1000000).
• The means -0.220703 to -0.215000 against kappa = 0.213864 and the small-side readings against beta = 0.139689 (second 11).
• The family checks, the strictness at e >= 2, and the witnesses 20, 6, 10, 2, 3, 15 (family).
• The counts 10072 and 50, the 17 integers equal to twice the A030979 terms, and the lists at sides adding 9, 11, 13, 15 (inter 1e12); 0, 2, 6320 alone below 10^1000 (deep 1000).