Write \(\tau(x)\) for the first time at which \(D^\tau(x)=0\). Exact enumeration gives \[ \max_{x\in\{0,\ldots,100\}^4}\tau(x)=13. \] One witness is \((0,7,20,44)\). Its trajectory is \[ \begin{aligned} &(0,7,20,44),(7,13,24,44),(6,11,20,37),(5,9,17,31),\\ &(4,8,14,26),(4,6,12,22),(2,6,10,18),(4,4,8,16),\\ &(0,4,8,12),(4,4,4,12),(0,0,8,8),(0,8,0,8),\\ &(8,8,8,8),(0,0,0,0). \end{aligned} \] The candidate witness \((57,81,37,44)\) belongs to the same class: subtracting 37 gives \((20,44,0,7)\), a rotation of the displayed witness.

The number of ordered starting tuples at each stopping time \(0,1,\ldots,13\) is \[ (1,100,20100,1353400,53030200,17977176,22886192,5951312,2093072,540976,165040,31136,8880,2816). \] These counts sum to \(101^4=104060401\).

The displayed trajectory from (0,7,20,44) takes thirteen Ducci steps, which proves the lower bound. For the upper bound, normalize each nonconstant quadruple by subtracting its smallest coordinate, dividing by the positive greatest common divisor of its coordinates, and taking the least rotation or reflection. Translation, positive scaling, rotation, and reflection preserve the stopping time, so this assigns every nonconstant quadruple to one primitive canonical representative. If that representative has maximum coordinate m and dihedral orbit size o, the number of tuples in the box that it represents is o times the sum of 101-gm over positive integers g with gm at most 100. The companion C17 certificate enumerates all 425,479 primitive representatives, computes this weight for each one, and handles the 101 constant tuples separately. The weights sum to 101^4, so the classes exhaust the box without omission. The last nonzero stopping-time class is thirteen. Together with the displayed witness, this proves that the exact maximum is thirteen.

The primitive representatives, chosen lexicographically within their dihedral orbits, are \[ \begin{gathered} (0,7,20,44),(0,10,29,64),(0,12,34,75),(0,13,37,81),\\ (0,13,38,84),(0,15,43,95),(0,24,37,44),(0,35,54,64),\\ (0,41,63,75),(0,44,68,81),(0,46,71,84),(0,52,80,95). \end{gathered} \] Every ordered maximizer has the unique form \[ c\mathbf 1+g\sigma(p), \] where \(p\) is on this list, \(\sigma(p)\) is one of its eight distinct rotations and reflections, \(1\leq g\leq\lfloor100/\max(p)\rfloor\), and \(0\leq c\leq100-g\max(p)\). Conversely, every tuple of this form has stopping time 13.

For a representative with maximum coordinate \(m\), the number contributed is \[ 8\sum_{g=1}^{\lfloor100/m\rfloor}(101-gm). \] The representative maxima are \(44,64,75,81,84,95\), each occurring twice. Their contributions sum to 2,816.

OEIS A065677 defines \(a(n)\) as the maximal Ducci length among nonnegative quadruples whose entries are at most \(n\). This is the present problem with \(n=100\). Its inverse, A065678, records the least bound at which a given length occurs. The two relevant values are \[ A065678(13)=44,\qquad A065678(14)=105. \] It follows that \(A065677(100)=13\). The independently computed values through bound 20 in the candidate record agree term for term with A065677.

The literature also explains the quotient used in the certificate. Behn, Kribs-Zaleta, and Ponomarenko treat translation, positive scaling, rotation, and reflection as equivalences and classify four-number longevity in a canonical two-dimensional region. The exhaustive computation here supplies the finite-box multiplicities and complete list of maximizers.

For each nonconstant tuple \(x\), subtract \(c=\min_i x_i\), divide by the positive coordinate gcd \(g\), and choose the lexicographically least rotation or reflection \(p\). Translation, positive scaling, rotation, and reflection preserve the stopping time. This gives a disjoint classification of every nonconstant box tuple.

If \(m=\max(p)\) and the dihedral orbit of \(p\) has size \(o\), its full-box weight is \[ o\sum_{g=1}^{\lfloor100/m\rfloor}(101-gm). \] The program enumerates 425,479 primitive canonical representatives. It handles the 101 constant tuples separately. The weighted histogram sums to \(101^4\), the largest stopping time is 13, and exactly twelve canonical representatives attain it.

The FNV-1a trace serializes each accepted representative in lexicographic loop order as the little-endian eight-byte word \[ a+(b\ll7)+(c\ll14)+(d\ll21)+(\tau\ll28). \] The trace is `05d8bd77b00e039b`. The stable three-line output has SHA-256 digest `651e1796aa233be72ada58e1affa3d8884162e266e3b4c82065d66c3e9e05ad6`.