Finite prime families admit explicit separating knot signature bumps

Let $\mathrm{\Theta }$ be the finite set $\{0,\pi \}\cup \{2\pi r\mathrm{/}{p}_j:1\le j\le m,1\le r\le (p_j-1)\mathrm{/}2\}$. All displayed interior angles are distinct except when both indices and residues agree: equality $r\mathrm{/}{p}_j=s\mathrm{/}{p}_{\mathrm{\ell }}$ for distinct primes would force $p_j$ to divide $r$ or $p_{\mathrm{\ell }}$, impossible in the stated residue ranges. The target ${\theta }_i=2\pi \mathrm{/}{p}_i$ is one of these interior angles. Its distance from every other member of $\mathrm{\Theta }$ is positive; let ${\delta }_i$ be one third of the finite minimum of these distances. Thus $I_i=({\theta }_i-{\delta }_i,{\theta }_i+{\delta }_i)\subset (0,\pi )$ contains none of the other angles in $\mathrm{\Theta }$. Choose the least positive integer $N_i$ with $2N_i>C_i$, which exists by the Archimedean property of the reals. All these are finite operations on the supplied list.

Use [F1] with ${\theta }_i,I_i$ to obtain the exact matrix $B_i$ there, and let ${\tilde{V}}_i$ be the block sum of $N_i$ copies of $B_i$. Its signature is $N_i$ times that of $B_i$ by finite inertia additivity. In each positive four-dimensional skew block reorder $(e_1,e_2,f_1,f_2)$ as $(e_1,f_1,e_2,f_2)$; in each negative skew block use $(e_1,-f_1,e_2,-f_2)$. Their block sum is an explicit signed permutation $P_i$ for which $V_i=P_i^T{\tilde{V}}_i{P}_i$ has standard symplectic skew part. Congruence preserves every Hermitian signature by [F1]. Apply [F2] with $g=4N_i$ to realize exactly $V_i$ on an actual smooth one-boundary surface, whose oriented boundary is $K_i$. Thus the constructed knot and chosen matrix have the required nonnegative, conjugate-symmetric signature function, equal to $2N_i$ at ${\omega }_i$. No mirroring or change of knot orientation is needed to make its sign positive.

At any nonidentity $p_j$-th root, complex conjugation puts its angle in $\mathrm{\Theta }\setminus \{0,\pi \}$. If $j\mathrm{\ne }i$, that angle is outside $I_i$, so its signature is zero by [F1]. If $j=i$, only the residues $r\equiv \pm 1$ give the angle ${\theta }_i$; every other nonidentity residue gives another angle of $\mathrm{\Theta }$ outside $I_i$ and has signature zero. At the identity all Hermitian matrices in the definition are zero, so their signature is zero as well. The two residues $\pm 1$ are distinct because $p_i$ is odd. Their values are both $2N_i$, so summing the exact finite list gives the displayed averages, with no integration or analytic identification. These constructions and the matrix-realization choices are finite and explicit; the statement concerns the actual supplied Seifert surfaces. This proves the full finite separation and height assertions.