We study the set of weights occurring in the high-order binary Reed-Muller codes $RM(m-7,m)$. For every $m\geq 14$, we prove that the weight spectrum contains all even weights in $[336,2^m-336]$, together with an explicit finite set of low weights and their complements. The proof first establishes the corresponding base-code inclusion in $RM(7,14)$. This base step uses four-block concatenation, explicit degree-six monomial witnesses, and known spectrum inputs for $RM(5,10)$ and $RM(6,12)$. The base inclusion is then lifted to all $m\geq 14$ by a concatenation-based induction. We also analyze the remaining base-code target weights $322,326,330,334$ in several restricted families, including four-block branches, affine-flat and structured inner corrections in a fixed Carlet-type Maiorana-McFarland outer shell, and sparse sums of degree-7 coordinate monomials. The residual analysis delineates several construction routes that cannot yield the four unresolved base-code weights, thereby sharpening the remaining $c=7$ spectrum problem.

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