A lattice-theoretic approach to the abc conjecture: Weighted arithmetic derivatives, a master inequality, and the structure of the residual family
We develop a self-contained lattice-theoretic framework for the abc conjecture. For coprime positive integers $a+b=c$ with $k=\omega(ab)$, the framework rests on a weighted arithmetic derivative $d^\psi$ whose additivity equation $d^\psi(a+b)=d^\psi(a)+d^\psi(b)$ is solvable modulo the modulus $g_c=\gcd_{p\parallel c}\bigl(\frac{c}{p}v_p(c)\bigr)$, and on a Wronskian identity $W=a\,d^\psi(b)-b\,d^\psi(a)=M_{ab}\,H(u)$, where $M_{ab}=ab/\operatorname{rad}(ab)$ and $H$ is an explicit linear form. A divisibility lemma $M_aM_bM_c\mid W$ and an elementary upper bound yield the master inequality…
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