# Bound-arc Frobenius matrices of the four period geometries that occur through 5PM (the 4PM K3 and the three 5PM geometries; Table 6 of the paper)

Files: values_K30_1_12.txt (K3, 3x3 at 12 points), values_K3P_1_12.txt (K3', 3x3 at 12 points),
values_F43_1_6.txt and values_F43_7_12.txt (CY3, 4x4 at 6 + 6 points), values_HAD_1_12.txt (CY3', 4x4 at 12 points):
48 point blocks, 1200 real numbers (real and imaginary parts of 600 matrix entries).
Operators, variables, the Frobenius normalization and the path classes are specified in ../operators.txt.

## Format
Each file starts with '#' comment lines.  A block

    POINT <family> <tag> gamma=<exact value>
      z_re=<Re z>
      z_im=<Im z>
      W[i,k] = <Re W_ik>  <Im W_ik>        (i, k = 1..r; r = 3 for K3, K3'; r = 4 for CY3, CY3')

gives the Wronskian matrix W[i,k] = (d/dz)^(i-1) varpi_(k-1)(z) of the MUM Frobenius basis (varpi_0, ..., varpi_(r-1)),
continued from z = 0 to the bound-arc point z(gamma) along the physical path.  Row 1 is the Frobenius basis itself;
W[1,1] = varpi_0 is the entry printed (rounded to ten decimals) in Table 6.  The tags g995 ... g500 label the twelve
energies gamma = 199/200, 99/100, 49/50, 97/100, sqrt(8/9), 23/25, 9/10, sqrt(3)/2, 4/5, sqrt(2)/2, 3/5, 1/2 (gisco = sqrt(8/9),
the specific energy of the Schwarzschild innermost stable circular orbit).

## Precision and enclosures
All numbers are midpoints of complex balls computed in ball arithmetic at 600-bit working precision (about 180 decimal
digits are printed).  Each matrix was transported along two homotopically equivalent paths (A and B, see ../operators.txt).
The table lists, per point, the number of decimal digits to which the two transports agree (floor of -log10 of the
largest entrywise difference divided by the largest entry) and the number of digits guaranteed by the rigorous ball radius of path A
(floor of -log10 of the largest entrywise radius divided by the largest entry).  Every printed midpoint is therefore
correct to at least the number of digits in the last column, relative to the largest entry of its matrix; digits beyond
that are not significant.

| geometry | tag | gamma | z (6 decimals) | two-path agreement (digits) | enclosure radius (digits) |
|---|---|---|---|---|---|
| K3 (Legendre) | g995 | 199/200 | 0.980050-0.198751 i | 176 | 171 |
| K3 (Legendre) | g990 | 99/100 | 0.960200-0.279313 i | 176 | 171 |
| K3 (Legendre) | g980 | 49/50 | 0.920800-0.390035 i | 176 | 171 |
| K3 (Legendre) | g970 | 97/100 | 0.881800-0.471624 i | 176 | 171 |
| K3 (Legendre) | gisco | sqrt(8/9) | 0.777778-0.628539 i | 176 | 171 |
| K3 (Legendre) | g920 | 23/25 | 0.692800-0.721130 i | 176 | 172 |
| K3 (Legendre) | g900 | 9/10 | 0.620000-0.784602 i | 176 | 172 |
| K3 (Legendre) | g866 | sqrt(3)/2 | 0.500000-0.866025 i | 176 | 172 |
| K3 (Legendre) | g800 | 4/5 | 0.280000-0.960000 i | 177 | 172 |
| K3 (Legendre) | g707 | sqrt(2)/2 | -0.000000-1.000000 i | 177 | 173 |
| K3 (Legendre) | g600 | 3/5 | -0.280000-0.960000 i | 177 | 172 |
| K3 (Legendre) | g500 | 1/2 | -0.500000-0.866025 i | 177 | 172 |
| K3' (Apery) | g995 | 199/200 | 0.980050-0.198751 i | 177 | 167 |
| K3' (Apery) | g990 | 99/100 | 0.960200-0.279313 i | 177 | 167 |
| K3' (Apery) | g980 | 49/50 | 0.920800-0.390035 i | 177 | 167 |
| K3' (Apery) | g970 | 97/100 | 0.881800-0.471624 i | 177 | 167 |
| K3' (Apery) | gisco | sqrt(8/9) | 0.777778-0.628539 i | 177 | 167 |
| K3' (Apery) | g920 | 23/25 | 0.692800-0.721130 i | 177 | 167 |
| K3' (Apery) | g900 | 9/10 | 0.620000-0.784602 i | 177 | 167 |
| K3' (Apery) | g866 | sqrt(3)/2 | 0.500000-0.866025 i | 177 | 167 |
| K3' (Apery) | g800 | 4/5 | 0.280000-0.960000 i | 177 | 168 |
| K3' (Apery) | g707 | sqrt(2)/2 | -0.000000-1.000000 i | 177 | 169 |
| K3' (Apery) | g600 | 3/5 | -0.280000-0.960000 i | 177 | 168 |
| K3' (Apery) | g500 | 1/2 | -0.500000-0.866025 i | 178 | 167 |
| CY3 (4F3) | g995 | 199/200 | 0.003598-0.001522 i | 177 | 172 |
| CY3 (4F3) | g990 | 99/100 | 0.003297-0.002095 i | 177 | 171 |
| CY3 (4F3) | g980 | 49/50 | 0.002718-0.002806 i | 177 | 171 |
| CY3 (4F3) | g970 | 97/100 | 0.002169-0.003249 i | 177 | 171 |
| CY3 (4F3) | gisco | sqrt(8/9) | 0.000820-0.003819 i | 177 | 170 |
| CY3 (4F3) | g920 | 23/25 | -0.000156-0.003903 i | 177 | 170 |
| CY3 (4F3) | g900 | 9/10 | -0.000903-0.003800 i | 177 | 170 |
| CY3 (4F3) | g866 | sqrt(3)/2 | -0.001953-0.003383 i | 177 | 169 |
| CY3 (4F3) | g800 | 4/5 | -0.003294-0.002100 i | 177 | 169 |
| CY3 (4F3) | g707 | sqrt(2)/2 | -0.003906+0.000000 i | 177 | 168 |
| CY3 (4F3) | g600 | 3/5 | -0.003294+0.002100 i | 177 | 168 |
| CY3 (4F3) | g500 | 1/2 | -0.001953+0.003383 i | 177 | 167 |
| CY3' (Hadamard) | g995 | 199/200 | 0.000010+0.000000 i | 177 | 176 |
| CY3' (Hadamard) | g990 | 99/100 | 0.000019+0.000000 i | 177 | 176 |
| CY3' (Hadamard) | g980 | 49/50 | 0.000039+0.000000 i | 177 | 176 |
| CY3' (Hadamard) | g970 | 97/100 | 0.000058+0.000000 i | 177 | 176 |
| CY3' (Hadamard) | gisco | sqrt(8/9) | 0.000109+0.000000 i | 177 | 177 |
| CY3' (Hadamard) | g920 | 23/25 | 0.000150+0.000000 i | 177 | 176 |
| CY3' (Hadamard) | g900 | 9/10 | 0.000186+0.000000 i | 177 | 176 |
| CY3' (Hadamard) | g866 | sqrt(3)/2 | 0.000244+0.000000 i | 177 | 176 |
| CY3' (Hadamard) | g800 | 4/5 | 0.000352+0.000000 i | 177 | 176 |
| CY3' (Hadamard) | g707 | sqrt(2)/2 | 0.000488+0.000000 i | 178 | 175 |
| CY3' (Hadamard) | g600 | 3/5 | 0.000625+0.000000 i | 178 | 175 |
| CY3' (Hadamard) | g500 | 1/2 | 0.000732+0.000000 i | 178 | 175 |

Smallest enclosure-radius entry in the table: 167 digits.  The K3 column of Table 6 additionally agrees with
(2/pi)^2 K(z)^2 on the principal arithmetic-geometric-mean branch, and the CY3' column with the direct series and with
evaluation through the Hadamard factorization of the operator, as stated in the paper.

Run logs read for this table (sha256, first 16 hex digits): grid_K30.log 0348530b908491cd, grid_K3P.log 048dd8ef9013131b, grid_F43_1_6.log 2eb2c273274e71c5, grid_F43_7_12.log 3fde690a0a5790bf, grid_HAD.log 7b86022482556f4e.