Burnol Residue

• The residue of K(s) = sum n^(-s) over the positive integers whose base-3 digits lie in {0,1} at the off-real poles s_(0,k) = log_3 2 + 2 pi i k/log 3, k = 0..10, enclosed in a complex rectangle by interval arithmetic (mpmath.iv, 128 bits) with every truncation bounded by a proved tail; the same engine on base 3 {0,2} (the residues scale by 2^(-s_(0,k))) and on the full digit set {0,1,2}, where K is the Riemann zeta function and the off-real residues are exactly zero.
• The engine is the level recursion Lev_(j+1)(w) = base^(-w) sum_(l >= 0) (-1)^l ((w)_l/l!) base^(-l) gamma_l Lev_j(w+l) on the vector of shifts w = s_(0,k) + i, i <= I, run to level; the l-truncation is bounded by a hypergeometric tail and the levels past level by a geometric one; R_k = lim_j Lev_j(s_(0,k)) = lambda_(0,k) log base by Burnol's Proposition 5.1.
• A second certified enclosure by the functional-equation route: R_k = 1 + sum_(m >= 1) (-1)^m ((s)_m/m!) 2^(-1) 3^(-m) K(s+m) with each K(s+m) summed directly in intervals to a second level plus its tail, the m-series cut at M plus its tail; wider, and it must meet the first.
• Two controls in floats: the k-th Fourier coefficient of the log-periodic profile Phi(u) = lim_j A(3^(u+j))/2^(u+j) from exact counts A(x) on a grid, through Res_(s_(0,k)) K = s_(0,k) c_k, and Burnol's level-sum limit by direct enumeration.
• The column: lambda_(m,k) for m = 1..3 by Burnol's Proposition 7.1 recurrence, checked at m = 1, 2 against the closed forms (s-1)/4 and (s-1)(s-2)/64 times lambda_(0,k).
• Safe rounding: every endpoint is converted to an exact fraction before printing, lower endpoints floored, upper endpoints ceiled.

RUN

• uv run python research/lab/py/burnol-residue/burnol_residue.py
• uv run python research/lab/py/burnol-residue/burnol_residue.py --band adds k = 2..10.
• Prints only, writes nothing; the default run takes about 8 seconds, the band about 55.

WITNESSES

• dimensions.md the arithmetic pole, certified, "The string above lives in [0,1]": Burnol's Proposition 5.1 limit formula and the statement that the paper does not study the off-real residues further.
• dimensions.md the arithmetic pole, certified, "Write G(u) = A(3^u)/2^u": the profile lemma: Phi(u) = 2^(1-u) on [1 - s_0, 1], Phi(1 - s_0) = 2^(s_0) = 1.548562.
• dimensions.md the arithmetic pole, certified, "For Re s > s_0": the dictionary Res_(s_(0,k)) K = s_(0,k) c_k.
• dimensions.md the arithmetic pole, certified, "Let Lev_j(w) = sum n^(-w)": lambda_(0,1) in [0.231891517689918, 0.231891517689919] + i [-0.501067414481069, -0.501067414481068], width 3.4e-18, level tail 2.4e-19, distance from zero at least 0.552125193, I = 66, level 40.
• dimensions.md the arithmetic pole, certified, "The same certificate at k = 0..10" and its table: the band k = 0..10, I from 52 to 164, widths from 2e-18 to 4e-13.
• dimensions.md the arithmetic pole, certified, "Two independent floating-point computations": the level-sum control 0.231891518 - 0.501067414 i at distance 1.9e-10 against its bound 7.5e-10, the Fourier control c_1 = -0.082138842 - 0.049607499 i at distance 7.2e-08, the second enclosure [0.23189098, 0.23189280] + i [-0.50106814, -0.50106632], the {0,2} enclosure [0.135292875345483, 0.135292875345484] + i [0.329874091244466, 0.329874091244467], the zeta boxes around 1 and 0.
• dimensions.md the arithmetic pole, certified, "Burnol's Proposition 7.1 recurrence carried down": the column lambda_(1,1), lambda_(2,1), lambda_(3,1) and the closed forms with [t^1] = -1/4, [t^2] = 1/64.