Paper IV: figure and atlas descriptions
These reading aids describe the 6 chapter figures, including their arrows, axes, scales and matrix entries. They retain the assumptions and example scope of the manuscript.
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Figure 1: Exact residue lanes for order three
The horizontal axis is exponent \(m\), from 0 through 17, and the vertical lanes are \(j=0,1,2\). Lane 0 contains 0, 3, 6, 9, 12, 15; lane 1 contains 1, 4, 7, 10, 13, 16; lane 2 contains 2, 5, 8, 11, 14, 17. Each labeled dot is an exponent address. Equal spacing and lane height carry no information about Taylor coefficient size.
Figure 2: The signed shift of order four
Read the cycle in order:
\[
e_0\to e_1\to e_2\to e_3\to e_0.
\]
The first three arrows have weight \(+1\) and the closing \(e_3\to e_0\) arrow has weight \(-1\). The four-by-four matrix uses target rows and source columns numbered 0 through 3. The table gives every entry. Four applications traverse the negative edge once, giving \(J_4^4=-I\).
| Target / source | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | -1 |
| 1 | 1 | 0 | 0 | 0 |
| 2 | 0 | 1 | 0 | 0 |
| 3 | 0 | 0 | 1 | 0 |
Figure 3: Two normalizations of secant coefficients
The columns use \(n=0\) through 4. The upper row is the block exponential coefficient \(P_{2,n}(1)\), and the lower row is the ordinary coefficient of \(z^{2n}\) in \(\sec z\). Dividing each upper value by \((2n)!\) gives the lower value. Both rows encode the same five terms of the secant function.
| Coefficient / index | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Block exponential coefficient | 1 | 1 | 5 | 61 | 1385 |
| Ordinary Taylor coefficient | 1 | 1/2 | 5/24 | 61/720 | 1385/40320 |
Figure 4: Reciprocal-order parameter polynomials
The horizontal axis is reciprocal order \(r\) on \([0,2]\), and the vertical axis is block coefficient. The straight curve evaluates \(P_{2,1}(r)=r\); the curved one evaluates \(P_{2,2}(r)=r(3r+2)\). Both begin at zero. At \(r=1\) the marked points are 1 and 5; at \(r=2\) the formulas give 2 and 16. The curves evaluate exact formulas; they are not fitted parameter estimates.
Figure 5: Four zeros become one reduced zero
The left panel has horizontal axis \(\operatorname{Re}z\) and vertical axis \(\operatorname{Im}z\). For \(m=0,1,2\), put
\[
\tau_m=\sqrt{2}\,\pi\left(m+\frac12\right).
\]
The twelve dots are \(\pm\tau_m\) and \(\pm i\tau_m\). The right panel sends each group of four to
\[
\lambda_m=\tau_m^4=4\pi^4\left(m+\frac12\right)^4
\]
on a positive horizontal axis with logarithmic scale. Its three labeled positions correspond to \(m=0,1,2\). This exact order-four model uses
\[
D_4(z)=\cos\left(\frac{z}{\sqrt{2}}\right)\cosh\left(\frac{z}{\sqrt{2}}\right).
\]
It does not classify finite exceptional zeros at other orders.
Figure 6: Finite partial sums of zero moments
Each horizontal axis counts \(M\) retained reduced zeros, from 1 through 40, where
\[
\lambda_k=\pi^2\left(k+\frac12\right)^2.
\]
The left vertical axis is the sum of reciprocal zeros,
\[
\sum_{k=0}^{M-1}\lambda_k^{-1},
\]
and the right is the sum of their squared reciprocals,
\[
\sum_{k=0}^{M-1}\lambda_k^{-2}.
\]
Both vertical scales are linear. The solid curves are finite numerical sums, while the dashed levels are the exact infinite sums \(1/2\) and \(1/6\). The second sum approaches its limit much faster because the larger reciprocal powers give less weight to distant zeros. The exact levels come from the product and coefficient identities.