periodic-determinant

• Rigorous zeros of the periodic-point determinant D(s) = det(I - L_s) of a continued fraction transfer operator, after Jenkinson and Pollicott 2018: the coefficients delta_n(s) of det(I - z L_s) computed exactly from the closed walks of length n <= P, the tail sum_(n > P) delta_n bounded by their Euler bound on approximation numbers, all arithmetic in mpmath.iv interval arithmetic.
• The operator is (L_s F)_u(z) = sum_(u -a-> v) (z + a)^(-2s) F_v(1/(z + a)) over a finite deterministic labelled graph, on H^2(D) per state, D the disc of rational centre c and radius r; one state is L_(A,s). The contraction h is max_a max(abs(T_a(c - r) - c), abs(T_a(c + r) - c))/r, exact in rationals.
• The weights are conjugated by g(z) = (z + beta)^(-2s): w_a(z) = ((z + beta)/(beta (z + a) + 1))^(2s), whose modulus on the circle is bounded by interval arithmetic on arcs; the determinant is unchanged and the constant drops, which is what makes complex s reachable. The disc and beta are chosen by a float grid minimising the tail; the certificate never uses the grid's numbers.
• A trace class is a (period, trace) pair: a closed walk with labels a_1 ... a_n contributes mu^(-2s)/(1 - (-1)^n mu^(-2)), mu the larger root of x^2 - t x + (-1)^n, t the trace of [[0,1],[1,a_1]] ... [[0,1],[1,a_n]], enumerated in numpy integers; delta_n follow by Newton's identities.
• zero certifies the zero s_1 of D at A = {1,2} by a Krawczyk test on the printed box itself, rounded outward: the Euler tail at the centre, a Cauchy tail on the derivative at radius 0.001, the image of the box strictly inside it; every arc asserts the real parts of z + a, z + beta and beta (z + a) + 1 positive. It then prints interval upper bounds on the constant K of Jenkinson and Pollicott's (47) on their own disc at dim E_2 and at s_1, unconjugated and conjugated.
• control certifies dim E_A for A = {1,2}, {1,3}, {2,3}, {1,2,3} and checks each bracket against Jenkinson and Pollicott 2018 Theorem 1 and Pollicott and Vytnova 2022 Table 3, read at source.
• orphans takes every width-4 orphan with rho > 1 and the named code 54, builds the graph of lab/py/question-mark verb graph, keeps the strongly connected components with a branch, minimises each by follower language, and certifies each component's pressure zero; rows with a cofinite edge on a cycle are listed and skipped. The collocation reading of the same sibling, truncated at twelve digits, is printed beside the bracket, and the generator asserts that it is the bracket truncated at twelve digits or lies inside the bracket, printing the count of each: 11 and 1.
• A real zero is certified at s- < s+ by two facts at each end: the winding number of the truncated det(I - z L_s) around abs(z) = 2, the tail at radius 2 excluded on every arc, equals the period p of the graph (the count sums float angle steps between arc endpoints, sound because each arc's enclosure lies in an open half-plane that the float endpoints are asserted to share, so each step is the true one in (-pi, pi)), and D(s-) + tail < 0 < D(s+) - tail; together they give lambda_0(s-) > 1 > lambda_0(s+) for the leading eigenvalue.

RUN

• uv run python mrlyprod/research/lab/py/periodic-determinant/determinant.py zero, 15 seconds.
• uv run python mrlyprod/research/lab/py/periodic-determinant/determinant.py control, 31 seconds, {1,2,3} at periods to 13.
• uv run python mrlyprod/research/lab/py/periodic-determinant/determinant.py orphans, 45 seconds, periods to 22 capped by 4 * 10^6 walks.
• uv run python mrlyprod/research/lab/py/periodic-determinant/determinant.py all, 83 seconds.
• --period 18 (the zero and control truncation, orphans takes it plus 4), --arcs 2000 and --modes 40 are the dials.
• Prints only, writes nothing, reads one sibling, lab/py/question-mark/question.py, for the orphan graphs and their collocation readings.

WITNESSES

• The certificate paragraph of apollonian, ## The horocycle.
• The finite orphans paragraph and table of beneath, ### Every rule is a graph-directed continued fraction set, the table printed by orphans, every h and tail rounded up.
• zero: 16189 trace classes to period 18, disc 7/10, 4/5, beta = 9/5, h = 29/56, K <= 5.73393 at s_1, tail at most 4.95e-40, derivative tail at most 5.08e-37, printed box half-widths 3.5e-39 and 4.0e-39; exactly one zero of D with Re s in [0.457015235230560657361880338944633917910, 0.457015235230560657361880338944633917917] and Im s in [6.958882679527224185470967589885884480867, 6.958882679527224185470967589885884480875]; the constant on the Jenkinson-Pollicott disc at most 4.098461 unconjugated and 2.406247 at beta = 33/20 at dim E_2, 5123.603 and 6.847291 at s_1 at beta = 9/5, every bound rounded up.
• control: dim E_2 in [0.5312805062772051416244686473684717854930591089, 0.5312805062772051416244686473684717854930591092], containing the Jenkinson-Pollicott interval; {1,3}, {2,3}, {1,2,3} within 1e-20 of Pollicott-Vytnova, {2,3} bracketed to 3e-60.
• orphans: 12 rows certified, the 8 cofinite rows 54699, 55275, 59799, 60375, 60855, 61431, 63903, 64479 listed; every minimal graph has 1 or 2 states; code 31710 reaches only [0.57451176, 0.57451180] at period 14, its graph growing at 2.73 per letter.