The Linear Slicing Method for Equal Sums of Like Powers: Modular and Geometric Constraints

We study the Diophantine equation $a^k+b^k=c^k+d^k$ with integer variables and exponent $k>1$, under the linear constraint $(c+d)-(a+b)=h$. We analyze the geometry and arithmetic of these linear slices. On the central slice $h=0$, we prove strictly convex uniqueness: distinct unordered pairs with the same sum yield distinct power sums. For shifted slices $h\mathrm{\ne }0$, we establish a Modular Divisibility Obstruction (MDO): any solution requires $h$ to be divisible by a specific squarefree modulus $M_k={\prod }_{p-1\mid k-1}p$. This condition creates a strong divisibility filter; for example, if $k=13$, the obstruction eliminates $99.96\mathrm{\%}$ of all possible shifts. We combine this arithmetic constraint with a geometric exclusion zone principle and a global overlap bound, showing that the slice size must satisfy $\min \{S,S+h\}\gg \mathrm{\mid }h\mathrm{\mid }$. Finally, we prove an asymptotic dominance bound $k\le \max \{S,S+h\}\log 2$, implying that for any fixed slice, solutions cannot exist for sufficiently large $k$.