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Regarding your edited version of this question: the degrees of the vertices in the path don't really matter. If $\alpha, \beta$ are both in $I_n$ then $\alpha \succeq \beta$ if there is an alternating path from $\alpha$ to $\beta$ of either the type $(n,n-1,\ldots,n)$ or the type $(n,n+1,\ldots,n)$: in either case, you have $\alpha \succeq \beta$. In the first case, you will have $\alpha, \beta \in M^- \cup M^0$ and in the second case you will have $\alpha, \beta \in M^+ \cup M^0$. In general, this partial order might relate $\alpha$ and $\beta$ lying across different $I_n$'s too.

There is a typo at the end of your calculation: one has $\pi\tilde{d} = d\pi$ instead of $d\phi$. If you already accept that $\rho$ and $\pi$ are interested in algebraic discrete Morse theoryinverses, then there is really no trouble:$$\tilde{d}\rho = (\rho d \pi) \rho = \rho d(\pi \rho) = \rho d$$Maybe you should also examine Jollenbeck and Welker's similar and concurrently developed monograph mean to ask some other question here: even the proof that has been cited $\pi$ is a chain map (which you worked out in your question) follows easily by Skoldberga similar argument if you accept that $\rho$ is its inverse.Chapter 4

You are right, this is not true in general: the image of my dissertation contains $\tilde{d}$ must be critical (that is, in the span of $M^0$) whereas there is no such requirement on the image of $d$ restricted to $\pi(K)$.

The result follows from recursive application of Lemma 5. Instead the calculation that you have performed so far, look at $(d - d\phi d)x$ as $(1 - d \phi) (dx)$ and write $dx$ out as a filtered version linear combination of $(n-1)$-dimensional cells. By Lemma 3, only the cells from $M^-$ will contribute. Now apply Lemma 5 to each piece of this theory with applications towards persistent homology computationcombination...

It's always nice to see people working on discrete Morse theory.

It is an "if and only if". Meaning: the partial order $\prec$ is defined by $\alpha \prec \gamma$ if and only if $\gamma$ precedes $\alpha$ in a path of the matching. The idea goes back to Forman's "Morse theory for cell complexes" where it is not explicitly stated as a partial order. I think the origins of the partial order are in Chari's reformulation of discrete Morse functions as acyclic matchings, see his page 7 of "On discrete Morse functions and combinatorial decompositions"decompositions".

As you noted, $\phi(dx_\alpha) = \sum_{\beta \prec \alpha}y_\beta$ and your question asks why this quantity should be trivial. In fact, it is not trivial in general, but its image under $\rho$ is trivial (which is all you need): by definition the image of $\phi$ is never critical, so $\rho\circ\phi$ is trivial.

Yes, $\sigma$ ranges over all critical cells of dimension $(n-1)$. This is a common problem with notation: I suspect that the author intended to specify the multiplicity $m(\gamma:\sigma\to\alpha)$ when writing $m(\gamma)$ where $\gamma$ is a path of the matching. In this case, one should define $m(\gamma) = 0$ whenever the source cell $\sigma$ is non-critical, which allows you to sum over arbitrary cells.

If you are interested in algebraic discrete Morse theory, you should also examine Jollenbeck and Welker's similar and concurrently developed monograph here that has been cited by Skoldberg. Chapter 4 of my dissertation contains a filtered version of this theory with applications towards persistent homology computation.

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It's always nice to see people working on discrete Morse theory.

It is an "if and only if". Meaning: the partial order $\prec$ is defined by $\alpha \prec \gamma$ if and only if $\gamma$ precedes $\alpha$ in a path of the matching. The idea goes back to Forman's "Morse theory for cell complexes" where it is not explicitly stated as a partial order. I think the origins of the partial order are in Chari's reformulation of discrete Morse functions as acyclic matchings, see his "On discrete Morse functions and combinatorial decompositions".
As you noted, $\phi(dx_\alpha) = \sum_{\beta \prec \alpha}y_\beta$ and your question asks why this quantity should be trivial. In fact, it is not trivial in general, but its image under $\rho$ is trivial (which is all you need): by definition the image of $\phi$ is never critical, so $\rho\circ\phi$ is trivial.
Yes, $\sigma$ ranges over all critical cells of dimension $(n-1)$. This is a common problem with notation: I suspect that the author intended to specify the multiplicity $m(\gamma:\sigma\to\alpha)$ when writing $m(\gamma)$ where $\gamma$ is a path of the matching. In this case, one should define $m(\gamma) = 0$ whenever the source cell $\sigma$ is non-critical, which allows you to sum over arbitrary cells.