Editing Forum reference mechanisms (section)

== Completeness ==
A set of reference primitives is complete if any finite WDAG can be changed into any other finite WDAG through a finite series of new posts. (Equality of WDAGs here is measured by weights. A WDAG which includes a post with 0 REP value is equal to a WDAG that deletes that post.)

The single mechanism of simultaneous donation and leaching is complete when combined with the incinerator mechanism, as we now demonstrate.

'''Proof of completeness:'''

Given <code>post1</code> with weight <code>W<sub>1</sub></code> we can reweight it to any other weight <code>W</code> by creating <code>post2</code> which donates to <code>post1</code>  if <code>W<sub>1</sub></code> < <code>W</code> or leaches from <code>post1</code> if <code>W<sub>1</sub></code> > <code>W</code>. If leaching is required, <code>post2</code> simultaneously donates to the incinerator with strength 1. Thus <code>post2</code> ends with 0 REP value.

This is not the end of the proof, since other posts in the sub-DAG of referenced ancestors are now changed, as well as the set of descendants affected by leaching from <code>post1</code>. We need to balance all the changes so that the leaching and donating processes affect the entire DAG correctly. We will need to solve a system of linear equations in order to find the right reweighting.  

To organize the process, first note that any DAG can be given a total order. Since our Forum is a finite WDAG, we can define a Layer 0 consisting of all the roots of the DAG, i.e., those posts with no ancestors, i.e., those posts which have no references going out, but only references possibly coming in. Layer 1 is the set of all posts which reference out only to Layer 0 posts. Layer 2 posts are those which have references out to Layer 1 and possibly Layer 0. Inductively, Layer <math display="inline">i+1</math> posts are those which reference posts in Layer <math display="inline">i</math> and possibly Layer <math display="inline">k</math> for any <math display="inline">k<i</math>.  

We now work inductively on layers. The posts in Layer 0 may be reweighted without difficulty by leaching or donating. The posts in Layer 1 can be reweighted, but any change in such a post must also reweight any post from Layer 0 that is referenced. To do this we solve the following linear equation:  

??mathy matheta MATH-a math-math?? 

'''End proof.'''