A δ-Integer and Wave-Integer Structural Decomposition of the ABC Conjecture(Letter)

We present a structural reformulation of the classical ABC conjecture using two related representations of arithmetic data: **δ-integers**, which encode additive and multiplicative composition by convolution, and **wave integers**, which encode the same structures through Fourier, Mellin, and prime-branch representations. For coprime \(abc\)-triples \[a+b=c,(a,b)=1,\] we introduce the normalized rational coordinate \[t=a/c,1-t=b/c,\] and define \[m_p(t)=max\{ |v_p(t)|,\, |v_p(1-t)|\}.\] Coprimality gives the exact identity \[m_p(t)=v_p(abc),\] so that the complete finite-prime multiplicity structure of the \(abc\)-triple is encoded by \(t\) and \(1-t\). The principal structural result of the note is the exact decomposition \[{3h(t)=R(t)+E(t)+A_∞(t)}\] with \[h(t)=log c,\] where \[R(t)=(abc)\] measures distinct prime support, \[E(t)=/rad(abc)=∑p abc(v_p(abc)-1)log p\] measures excess prime multiplicity, and \[A_∞(t)= -log(t(1-t))\] is an Archimedean imbalance term associated with the additive decomposition \(t+(1-t)=1\). This decomposition separates the height exactly into three arithmetic components: radical support, repeated prime multiplicity, and Archimedean additive imbalance. The classical ABC conjecture is then equivalently reformulated as \[E(t)+A_∞(t)≤(2+ε)R(t)+C_ε.\] The note also connects the isolated multiplicity term \(E(t)\) with existing results on linear forms in \(p\)-adic logarithms. Individual valuation bounds arising from Yu's theory, together with simultaneous valuation estimates discussed by Pasten, provide unconditional information on the quantities \(v_p(abc)\). This leads to a second, distinct research problem: whether these local and simultaneous valuation estimates can be aggregated in a way that directly exploits the logarithmically weighted structure \[E(t)=∑p abc(v_p(abc)-1)log p\] and yields bounds substantially sharper than those obtained indirectly from estimates for the total height or the maximal valuation. No proof of the ABC conjecture is claimed. The contribution is structural: it provides a δ-integer and wave-integer representation of the additive and multiplicative arithmetic channels, derives the exact decomposition \(3h=R+E+A_∞\), and isolates a concrete unconditional direction for studying the finite-prime multiplicity term \(E(t)\) through existing \(p\)-adic valuation theory.