We thank the Referee for a thoughtful report. Below we provide replies to the questions and concerns raised in the report, and describe the changes that have been made to the manuscript to address those concerns.

In the "Weaknesses" section, the Referee writes:

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2- The manuscript mainly focuses on the generic definition of currents with respect to the permutation of electric fields. It is difficult (to me) to foresee practical consequences of the present manuscript in real experiments.

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As a minor clarification, we note that the permutations we consider are not just with respect to the electric fields (represented by the last n indices of the n-th order conductivity tensor), but with respect to all indices, including the 0-th index representing the current: see Eqs .(19,20).

As for the practical consequences of our analysis, they are discussed below.

In the third paragraph of the Report, the Referee writes:

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But it is not clear from the manuscript if different prescriptions for nonlinear conductivities have any experimental impact, beyond precise values of parameters. If I understand it well, the authors discuss that different prescriptions lead to different overall numerical prefactors, that can be accounted for in unknown parameters, like lifetimes or similar.

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In the specific example discussed around Eqs.(6-13), different prescriptions that do not pass our sanity criteria give different overall numerical prefactors for the Hall current. More generally, incorrect Hall vs Ohmic partitions may also lead to Hall currents point along different directions. Moreover, Eq.(17) shows that even when the Hall current only changes by a numerical prefactor, the Ohmic current -- which in general is not parallel to the applied field -- may change direction. Finally, the numerical prefactors are important by themselves to allow the experimental benchmarking of parameter-free theories. In particular, the lifetimes may either be evaluated from first principles, or extracted from the measured linear Ohmic resistivity.

In the fourth paragraph of the Report, the Referee writes:

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Taking the second order conductivity as an example, it is well known that inversion symmetry must be broken to get a non-zero response. According to the authors, this information is irrelevant for the partitioning they describe, but is of paramount importance in experiments, so I find this shift between what is important in the present manuscript, a little bit confusing.

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As the Referee points out, if inversion symmetry is present the entire quadratic conductivity tensor vanishes; this is now mentioned explicitly below Eq. 4. In that case, our prescription gives the trivially correct result that the Hall and Ohmic parts of the quadratic current response vanish separately. Our analysis becomes nontrivial when the quadratic current is symmetry allowed, i.e., in acentric crystals, as elaborated in our answer to the comment below.

In the fifth and final paragraph of the Report and in the Requested Changes, the Referee writes:

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To frame a little bit my concern, the question I would find quite useful to be answered in the present manuscript is How the present discussion about partitioning might be relevant to understand experimental results? Discussing it for the second nonlinear conductivity would be enough (for me).

1- I strongly encourage the authors to enlarge the discussion section with the role of inversion symmetry and experimental consequences of their results in the case of second nonlinear conductivity, paying attention to any available physical example (the way shift , photogalvanic currents modify when performing the partioning procedure, etc).

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To address this concern and following the subsequent recommendation, we have made substantial revisions to the manuscript, namely:

• We added a new Sec. 5 where we carry out a systematic symmetry analysis of the quadratic conductivity, broken down into four fundamental contributions: Hall vs Ohmic, and time-reversal even vs time-reversal odd. The results, summarized in Table 2, could guide the search for materials displaying only one type of quadratic response. We call attention to the time-even Ohmic component, which is purely disorder mediated. We correct a result from the recent literature, and point out a possible oversight in recent papers on disorder-mediated quadratic responses, where the Ohmic part of those response may have been overlooked. These changes are reflected in part in the expanded abstract.

• We revised the Discussion section, which now contains two paragraphs only. The second paragraph is new. In it, we comment on the relevance of our analysis for interpreting recent measurements of (i) a higher-order (cubic) nonlinear current, and (ii) spontaneous unidirectional magnetoresistance. In particular, we provide a sharp phenomenological definition of the latter effect, without invoking specific microscopic mechanisms.

We thank the Referee and the Editor for maintaining the discussion during this holiday period. We would like to start by clarifying that our goal is not to "get something out of this effort". The goal of publishing a manuscript is not to increase one's publication list, but to share ideas, and to get feedback from a community which is broader than the one we could reach via email (as suggested by the Referee in his initial report).

One of the possible outcomes of our submission was that someone would point us to exactly which wheel we are reinventing. That was the reason why we chose SciPost Physics - a journal with a public peer-review process. However, we have not received so far any reference to a previous work that preempts ours. And naturally, we are unable to comment on the relation of our work to theorems that the Referee does not quite remember.

It is evident that a tensor may be separated into "something and the rest", and there are multiple ways to do so. Fully symmetric combination, fully antisymmetric one, and "the rest" are just particular cases of linear combinations of permutations of indices as defined by our Eq. 19, all of which transform as tensors. So, what is the reason to exclude all other possible combinations? What makes the symmetric combination unique, and in what sense is it "unique"? We have all been taught that to solve a mathematical problem means to find *all* the solutions and to prove that no other solutions may exist, and this is what we have achieved. Conversely, statements such as "it appears" or "it seems" are not accepted as valid solutions.

The Referee writes

"I maintain that asking the question about "current partitioning" is just not a good idea. The fact that other groups made associated mistakes basically attests to this, they shouldn't have tried."

We beg to differ:
the fact that we were able to formulate that question with mathematical rigor, and to obtain a rigorous answer for it, demonstrates that it is in fact a valid question to ask. If anything, it shows that other groups should have tried harder, rather than not trying at all. Moreover, the question we posed is a physics question about currents, not some linear algebra problem that the Referee claims is well known even though he/she cannot quite pin down or provide a reference for.

In closing, we wish to make the following comment. We are puzzled by the Referee's insistence that there is only "one reasonable way to approach the problem". Approaching a problem from different angles can only enrich our understanding of it. Our approach to the problem at hand offers a general solution which contains as special cases the one favored by the Referee, as well as a corrected version of an alternative solution discussed in the recent literature. This suggests that our approach has some merit to it, and that it constitutes a timely addition to the literature. Efforts to bring clarity to topics of current interest should be encouraged, not dismissed.

Dear Referee, we thank you for the thoughtful remarks on our work, and
apologize for the late reply. We will do our best to maintain a more
lively conversation in the future.

We think that the source of our disagreement lies within the following
lines of your reply:

"One can ask two questions related to the form of this tensor:"
"[...] this is the sortest path to my understanding."
"We seem to disagree on whether anything needs to be added to it."
"Now my question is what the present work adds to this discussion?"
"[...] in what way you think [this point of view] is incomplete."

We do not think that Q1 and Q2 are the only valid and interesting
questions that one can ask regarding the form of conductivity
tensors. Apart from stating the obvious answer, our work adds nothing
to the discussion of Q1; it also does not add anything new to the
discussion of Q2. What our work does instead is to address a different
question which we believe is interesting in its own right, and whose
answer provides a fresh perspective on the Hall vs Ohmic partition of
the current.

The question we chose to address in our manuscript is the following:

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What are all the possible ways of partitioning the nonlinear current
into physically meaningful parts, without taking into account the
symmetries of the system?
*****************************************************

The answer we found is that the only nontrivial generic partition
possible is precisely the one that you discuss. (In itself, this
answer is not terribly surprising or exciting. But having been able to
rigoroulsy exclude any other possibilities was new to us, and rather
satisfying.) While the partial nonlinear currents are unique, the
partial nonlinear conductivities are - unlike their linear
counterparts - strongly nonunique. This is fully accounted for by our
answer to the question above, which takes the form of a set of
conditions that any meaningful partial conductivities must fulfill.

As the you noted, those conditions are satisfied by the "fully
symmetric" and "remainder" partial conductivities:

"separating the conductivity tensor into its symmetric part, and `the
rest' is perfectly meaningful"

One reason why we find this point of view incomplete is that those
conditions are also satisfied by other equally valid choices of Ohmic
and Hall nonlinear conductivities (which, of course, yield the same
Ohmic and Hall currents).

This brings us to a very practical reason why our work constitutes a
useful addition to the literature: the form of the quadratic Hall
conductivity tensor that has been frequently used in recent works
(e.g., Refs. 15 and 20) is different from the one you find to be the
most meaningful. This goes to show that one person's shortest path to
understanding may be different from that of another person, especially
when the question being asked affords multiple correct answers, as is
the case here. By taking into account their inherent nonuniqueness,
our work clarifies the most general definitions of nonlinear Hall and
Ohmic conductivities.

Finally, we feel that the "games that we are playing" around Eq. 17 do
serve a purpose: they demonstrate that without imposing the two
necessary conditions that we postulate, one can get literally any
value for the Hall current (this is, in fact, what happened in
Refs. 15 and 20). You are absolutely right that

"separating the conductivity tensor into its symmetric part, and 'the
rest' is perfectly meaningful,"

But what makes this procedure "unique", except for the fact that it is
formulated in an elegant way? Nature likes beauty, but aesthetics is
not a solid criteria for physics. After reshuffling via Eq. 17, both
the Hall and Ohmic parts remain "tensors from the point of view of
their components transformation" (because they are linear combination
of tensors). Our work strictly proves that the elegant partition (of
the current) that was obvious to you from the beginning is indeed the
only valid one. In addition, it provides a general recipe for all
possible valid separations between Hall and Ohmic conductivities.

We hope you find these comments helpful to understand our point of
view, and look forward to your further questions and comments.