Picard-Fuchs operators of the four non-polylogarithmic period geometries through 5PM
(normalization of Klemm, Nega, Sauer and Plefka, arXiv:2401.07899; theta = z d/dz).
This file accompanies the value files bound_frobenius/values_*.txt and Table 6 of the paper.

CONVENTIONS
  theta form: L = sum_s z^s R_s(theta); each R_s is listed by its coefficients in ascending powers of theta.
  d/dz form:  L = sum_i p_i(z) (d/dz)^i, obtained from the theta form with theta^j = sum_i S2(j,i) z^i (d/dz)^i
              (Stirling numbers of the second kind); exact integer coefficients.
  Frobenius basis at the MUM point z = 0 (all four operators, after the z^(1/2) conjugation of L_K3 noted below):
     varpi_k(z) = sum_{j=0..k} (ln z)^j / j! * h_{k-j}(z),  k = 0..r-1,  h_0(0) = 1,  h_j(0) = 0 for j >= 1,
  so varpi_0 is the power-series solution, varpi_1 = varpi_0 ln z + h_1, etc.  The value files list the Wronskian
  matrix W[i,k] = (d/dz)^(i-1) varpi_(k-1)(z), i,k = 1..r, continued along the stated path (ln z continued along the same path).
  In this frame the unipotent MUM monodromy acts by ln z -> ln z + 2 pi i, i.e. varpi_k -> sum_j (2 pi i)^(k-j)/(k-j)! varpi_j.

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L_K3:  (2theta-1)^3 + z^2 (2theta+1)^3 - 4 z theta (4theta^2+1)     variable z = x^2
  4PM K3 (symmetric square of the Legendre operator).  The value files use the equivalent operator (1/8) z^(-1/2) L_K3 z^(1/2) = theta^3 - z (2theta^3 + 3theta^2 + 2theta + 1/2) + z^2 (theta+1)^3, which is MUM at z = 0 with all local exponents 0 and holomorphic solution varpi_0 = 2F1(1/2,1/2;1;z)^2 = (2/pi)^2 K(z)^2.
  theta rows (paper form):
    z^0: [-1, 6, -12, 8]   i.e. z^0 * ((-1)*theta^0 + (6)*theta^1 + (-12)*theta^2 + (8)*theta^3)
    z^1: [0, -4, 0, -16]   i.e. z^1 * ((-4)*theta^1 + (-16)*theta^3)
    z^2: [1, 6, 12, 8]   i.e. z^2 * ((1)*theta^0 + (6)*theta^1 + (12)*theta^2 + (8)*theta^3)
  theta rows (form used for the value files, (1/8) z^(-1/2) L_K3 z^(1/2)):
    z^0: [0, 0, 0, 1]   i.e. z^0 * ((1)*theta^3)
    z^1: [-1/2, -2, -3, -2]   i.e. z^1 * ((-1/2)*theta^0 + (-2)*theta^1 + (-3)*theta^2 + (-2)*theta^3)
    z^2: [1, 3, 3, 1]   i.e. z^2 * ((1)*theta^0 + (3)*theta^1 + (3)*theta^2 + (1)*theta^3)
  d/dz form (of the form used for the value files):
    p_0(z) = z*(2*z - 1)/2
    p_1(z) = z*(7*z**2 - 7*z + 1)
    p_2(z) = 3*z**2*(z - 1)*(2*z - 1)
    p_3(z) = z**3*(z - 1)**2
  check: operator applied to the holomorphic series (engine form, varpi_0 = (2/pi)^2 K(z)^2 generated from 2F1(1/2,1/2;1;z)^2) vanishes identically through z^30 in both forms: yes
  first coefficients of varpi_0: 1, 1/2, 11/32, 17/64, 1787/8192, 3047/16384, ...
  leading coefficient p_3(z) = z**3*(z - 1)**2
  local exponents (roots of the indicial equation):
    z = 0 (~0): {0, 0, 0}
    z = 1 (~1.0000000): {0, 0, 0}
    z = infinity (in w = 1/z): {1, 1, 1}

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L_A:  theta^3 - z (2theta+1)(17theta^2+17theta+5) + z^2 (theta+1)^3     variable z = x^2
  5PM-2SF topology 40, Apery / Beukers-Peters K3 surface K3'.  varpi_0 = sum_n A_n z^n, A_n = sum_k C(n,k)^2 C(n+k,k)^2 = 1, 5, 73, 1445, ...
  theta rows (paper form):
    z^0: [0, 0, 0, 1]   i.e. z^0 * ((1)*theta^3)
    z^1: [-5, -27, -51, -34]   i.e. z^1 * ((-5)*theta^0 + (-27)*theta^1 + (-51)*theta^2 + (-34)*theta^3)
    z^2: [1, 3, 3, 1]   i.e. z^2 * ((1)*theta^0 + (3)*theta^1 + (3)*theta^2 + (1)*theta^3)
  d/dz form:
    p_0(z) = z*(z - 5)
    p_1(z) = z*(7*z**2 - 112*z + 1)
    p_2(z) = 3*z**2*(2*z**2 - 51*z + 1)
    p_3(z) = z**3*(z**2 - 34*z + 1)
  check: operator applied to the holomorphic series (varpi_0 generated from the Apery numbers) vanishes identically through z^30 in both forms: yes
  first coefficients of varpi_0: 1, 5, 73, 1445, 33001, 819005, ...
  leading coefficient p_3(z) = z**3*(z**2 - 34*z + 1)
  local exponents (roots of the indicial equation):
    z = 0 (~0): {0, 0, 0}
    z = 17 - 12*sqrt(2) (~0.029437252): {0, 1/2, 1}
    z = 12*sqrt(2) + 17 (~33.970563): {0, 1/2, 1}
    z = infinity (in w = 1/z): {1, 1, 1}

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L_CY3:  theta^4 - 2^8 z (theta+1/2)^4     variable z = 2^-8 x^4
  5PM-1SF topology 3, Calabi-Yau threefold CY3.  varpi_0 = 4F3(1/2,1/2,1/2,1/2;1,1,1;2^8 z).
  theta rows (paper form):
    z^0: [0, 0, 0, 0, 1]   i.e. z^0 * ((1)*theta^4)
    z^1: [-16, -128, -384, -512, -256]   i.e. z^1 * ((-16)*theta^0 + (-128)*theta^1 + (-384)*theta^2 + (-512)*theta^3 + (-256)*theta^4)
  d/dz form:
    p_0(z) = -16*z
    p_1(z) = -z*(1280*z - 1)
    p_2(z) = -z**2*(3712*z - 7)
    p_3(z) = -2*z**3*(1024*z - 3)
    p_4(z) = -z**4*(256*z - 1)
  check: operator applied to the holomorphic series (varpi_0 generated from the 4F3 coefficients ((1/2)_n/n!)^4 2^(8n)) vanishes identically through z^30 in both forms: yes
  first coefficients of varpi_0: 1, 16, 1296, 160000, 24010000, 4032758016, ...
  leading coefficient p_4(z) = -z**4*(256*z - 1)
  local exponents (roots of the indicial equation):
    z = 0 (~0): {0, 0, 0, 0}
    z = 1/256 (~0.0039062500): {0, 1, 1, 2}
    z = infinity (in w = 1/z): {1/2, 1/2, 1/2, 1/2}

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L_CY3':  theta^4 - 2^4 z (192theta^4+128theta^3+112theta^2+48theta+7) + 2^14 z^2 (192theta^4+256theta^3+208theta^2+64theta+7) - 2^30 z^3 (theta+1/2)^4     variable z = (1 - gamma^2)/2^10
  5PM-2SF topology 37, Calabi-Yau threefold CY3' (Hadamard type, chi = 80).  varpi_0 = 1 + 112 z + 47376 z^2 + ...
  theta rows (paper form):
    z^0: [0, 0, 0, 0, 1]   i.e. z^0 * ((1)*theta^4)
    z^1: [-112, -768, -1792, -2048, -3072]   i.e. z^1 * ((-112)*theta^0 + (-768)*theta^1 + (-1792)*theta^2 + (-2048)*theta^3 + (-3072)*theta^4)
    z^2: [114688, 1048576, 3407872, 4194304, 3145728]   i.e. z^2 * ((114688)*theta^0 + (1048576)*theta^1 + (3407872)*theta^2 + (4194304)*theta^3 + (3145728)*theta^4)
    z^3: [-67108864, -536870912, -1610612736, -2147483648, -1073741824]   i.e. z^3 * ((-67108864)*theta^0 + (-536870912)*theta^1 + (-1610612736)*theta^2 + (-2147483648)*theta^3 + (-1073741824)*theta^4)
  d/dz form:
    p_0(z) = -16*z*(4194304*z**2 - 7168*z + 7)
    p_1(z) = -z*(5368709120*z**3 - 11796480*z**2 + 7680*z - 1)
    p_2(z) = -z**2*(1024*z - 1)*(15204352*z**2 - 22272*z + 7)
    p_3(z) = -2*z**3*(1024*z - 1)**2*(4096*z - 3)
    p_4(z) = -z**4*(1024*z - 1)**3
  check: operator applied to the holomorphic series (varpi_0 generated by the MUM recurrence (first terms 1, 112, 47376 as in the paper)) vanishes identically through z^30 in both forms: yes
  first coefficients of varpi_0: 1, 112, 47376, 27846400, 19020643600, 14161273120512, ...
  leading coefficient p_4(z) = -z**4*(1024*z - 1)**3
  local exponents (roots of the indicial equation):
    z = 0 (~0): {0, 0, 0, 0}
    z = 1/1024 (~0.00097656250): {0, 1/2, 3/2, 2}
    z = infinity (in w = 1/z): {1/2, 1/2, 1/2, 1/2}

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CONTINUATION PATHS used for bound_frobenius/values_*.txt (gamma = cos(theta) in (0,1), x = exp(-i theta) on the physical branch):
  K3 and K3':  z = x^2 on |z| = 1, lower arc.  Two homotopic polygonal chains in the region Im z < 0, starting at z = 1/200:
     path A: 1/200 -> 3/100 - 3i/100 -> 0.55 z_t -> z_t ;   path B: 1/200 -> 8/1000 - 6i/100 -> 0.8 exp(-i/5) z_t -> z_t .
     All finite singular points of both operators lie on the positive real axis, so the region Im z < 0 contains none of them
     and any two paths in it are homotopic; for K3' the physical path from the scattering segment (z_+, 1) through z - i0 encloses no singular point.
  CY3 (4F3):   z = x^4/2^8 lies ON the singular circle |z| = 2^-8 at arg z = -4 theta in (-2 pi, 0).  Physical class: from z = 2^-8/50,
     clockwise along a spiral inside the disk (radius 0.70 * 2^-8, angular steps 1/2, path A; radius 0.85 * 2^-8, steps 0.35, path B),
     monotone in angle and never crossing arg z = 0, ending at z_t.  This is the class of z - i0 continued past z = 2^-8 (gamma = 1).
     The short counterclockwise route inside the disk to the same endpoint differs from the physical class by one turn around
     z = 0 alone, so the two Frobenius bases differ by the unipotent MUM monodromy (the 4F3 path check of the paper at gamma = 1/2).
  CY3' (Hadamard):  z = (1-gamma^2)/2^10 is real in (0, 2^-10); path A is the real segment from z = 2^-13, path B a detour through
     Im z < 0 (homotopic; no singular point in between).  All entries are real.
  Working precision 600 bits; the two paths agree and the rigorous ball radii are as listed per point in README_bound.md.