How is the sum_to_zero_vector implemented under the hood?

Modeling
4

Ok try this

data {
  int<lower=0> N;
}
parameters {
  vector<lower=0>[N - 1] raw;
 // real<lower=0> scale; // optional, for flexibility
}
transformed parameters {
  vector[N] diff;
  diff[1] = 0;  // Anchor point
  diff[2:N] = cumulative_sum(raw);
 // vector[N] z = scale * (diff - mean(diff)); // if you want to scale
 vector[N] z = (diff - mean(diff));
}
model {
  raw ~ exponential(sqrt(N - 1) - 1);     // or any distribution with mass on the positive reals
 // scale ~ normal(0, 1);  // or any appropriate prior

}
generated quantities {
  real z_sum = sum(z);
}

setting N = 10 results in

   variable   mean median    sd   mad      q5     q95  rhat ess_bulk ess_tail
   <chr>     <dbl>  <dbl> <dbl> <dbl>   <dbl>   <dbl> <dbl>    <dbl>    <dbl>
 1 z[1]     -2.23  -2.13  0.830 0.805 -3.74   -1.06   1.00     9589.    6458.
 2 z[2]     -1.74  -1.66  0.695 0.689 -2.99   -0.774  1.00     9136.    6455.
 3 z[3]     -1.24  -1.16  0.596 0.568 -2.34   -0.426  1.000    8817.    6267.
 4 z[4]     -0.754 -0.692 0.516 0.473 -1.69   -0.0278 1.000    8956.    6929.
 5 z[5]     -0.253 -0.228 0.450 0.406 -1.03    0.430  1.00     9963.    7553.
 6 z[6]      0.249  0.225 0.452 0.405 -0.445   1.02   1.000    9626.    7165.
 7 z[7]      0.736  0.675 0.504 0.471  0.0286  1.65   1.00    10229.    5943.
 8 z[8]      1.24   1.16  0.600 0.561  0.430   2.34   1.000    9507.    6623.
 9 z[9]      1.75   1.65  0.720 0.690  0.761   3.08   1.00     8376.    5692.
10 z[10]     2.24   2.14  0.852 0.826  1.06    3.80   1.00     7868.    5606.