Lemma 6: entire regularization and interpolation bounds

For every complex period lattice $\Lambda$ there exists a normalized entire sigma function $\sigma$ and real constants $R_0>0$, $c_{13}>1$, and $c_{14}>1$, chosen before all polynomial, radius, and derivative data, such that both estimates below hold. Normalization means $\sigma(0)=0$, $\sigma'(0)=1$, and $\sigma'(z)=\zeta(z)\sigma(z)$ off the lattice. The same sigma function and threshold $R_0$ serve both estimates.

For a complex period lattice $\Lambda$, let $\wp,\zeta,\wp'$ denote its canonical Weierstrass functions. For positive integers $d,l$ and a complex coefficient array $p=(p_{ijk})$, set

The exponents are bounded separately by $(d,l,l)$; they need not attain these bounds. For a complex number $v$ put

The maximum is finite; its constant monomial contributes $2$.

For every positive $d,l,M$, every array with $|p_{ijk}|\le M$, and every $u\in\mathbb C$, the expression

has an entire extension. Choose this extension before the radii and derivative data. If $2<r<R$, $R>R_0$, $|u|<R$, $|v|<r-2$, and $F$ has at least $N\ge0$ zeros counted with multiplicity in $|z|<r$, then for every integer $t\ge0$,

For every positive $d,l,M$, every array with $|p_{ijk}|\le M$, and every $v\notin\Lambda$, the expression

has an entire extension. Choose this extension before the radii and derivative data. If $2<r<R$, $R>R_0$, $|v|<R$, $|u|<r-2$, and $G$ has at least $N\ge0$ zeros counted with multiplicity in $|z|<r$, then for every integer $t\ge0$,

These are both analytic estimates of Lemma 6. They provide the entire functions and derivative bounds used for the auxiliary-polynomial argument, with no grid, arithmetic-model, or geometric zero-estimate assumption.

Formalization note. A lower bound on the number of zeros is expressed by an arbitrary finite set $s\subset\{|z|<r\}$ and arbitrary multiplicities $m(a)\ge0$ with $N\le\sum_{a\in s}m(a)$, together with vanishing of every derivative of order below $m(a)$ at each $a\in s$. The estimate holds for every such witness. Empty sets, $N=0$, $t=0$, and the zero coefficient array are included. The product identities specify the extensions only where all meromorphic factors are regular: $z+u\notin\Lambda$ in the first case, and $z,z+v\notin\Lambda$ in the second. Zeros and derivative evaluation points may lie at excluded points of those product formulas, because they refer to the entire extensions. No condition $2r<R$ is imposed. The constants raised to $R^2l$ use real powers; all other displayed exponents are integers.