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The graph of the accelerated Collatz map \( T \) carries structure that is irrelevant to the Collatz conjecture: one third of its vertices are multiples of three, which no orbit can revisit, and the negative integers cannot be used at all because \( T \) admits non-trivial cycles on them. We construct an explicit conjugacy \( J \) between the conjecture-relevant vertices \( {[}{1}{]}_{3}\cup{[}{2}{]}_{3} \) and the nonzero integers \( \mathbb{Z}{*} \), yielding a map \( {K}{\colon}\ \mathbb{Z}{*}\to \mathbb{Z}{*} \) whose graph is isomorphic to the pruned Collatz graph. The Collatz conjecture becomes the statement that every \( K \)-orbit reaches the two-cycle \( \{1,-1\} \); in these coordinates the sign of an iterate records its residue class modulo \( 3 \), so every nonzero integer indexes a conjecture-relevant vertex and no non-trivial cycles are introduced. We analyze the edge structure of the new graph, identify a spanning acyclic subgraph that is bipartite under the sign \( 2 \)-coloring, and introduce an accelerated map \( \hat{K} \) whose graph is bipartite outright and whose in-degree sequence is governed by the \( 3 \)-adic valuation \( {\nu} 3(2k-1) \), recovering OEIS A254046 with mean in-degree \( 3/2 \). Finally, we observe that \( K \)-orbits, plotted as signed time series, behave like damped seismic signals: they oscillate across zero with limiting sign-change frequency \( 2/3 \) while their envelope decays at the heuristic rate \( \lambda=\tfrac12\ln\tfrac43 \) per step, suggesting dynamical and spectral tools as instruments for studying Collatz dynamics. No proof of the conjecture is claimed; the aim is a coordinate system in which its dynamics are easier to see.