Is Witten's Proof of the Positive Mass Theorem Rigorous?
$begingroup$
I noticed that the only official reason given for awarding Edward Witten the Fields Medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully rigorous.
However, I came across this paper https://projecteuclid.org/download/pdf_1/euclid.cmp/1103921154 by Taubes and Parker which claims to make Witten's proof 'mathematically rigorous' and to justify assumptions which Witten made about Dirac operators. Does this mean that the Witten proof is not rigorous, or is it just the case that there were some unjustified lemmas to clear up which do not affect the validity or rigour of the argument (similar to the case of Perelman's proof of the Poincare conjecture, where some lemmas and slight gaps had to be filled in)?
I am just curious as I have never really heard of the Taubes-Parker paper so I was assuming that the Witten paper was fully rigorous.
mp.mathematical-physics general-relativity dirac-operator spinor
asked 2 days ago
TomTom
410311
$endgroup$
add a comment |
$begingroup$
I noticed that the only official reason given for awarding Edward Witten the Fields Medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully rigorous.
However, I came across this paper https://projecteuclid.org/download/pdf_1/euclid.cmp/1103921154 by Taubes and Parker which claims to make Witten's proof 'mathematically rigorous' and to justify assumptions which Witten made about Dirac operators. Does this mean that the Witten proof is not rigorous, or is it just the case that there were some unjustified lemmas to clear up which do not affect the validity or rigour of the argument (similar to the case of Perelman's proof of the Poincare conjecture, where some lemmas and slight gaps had to be filled in)?
I am just curious as I have never really heard of the Taubes-Parker paper so I was assuming that the Witten paper was fully rigorous.
mp.mathematical-physics general-relativity dirac-operator spinor
asked 2 days ago
TomTom
410311
$endgroup$
2
$begingroup$
I think some clarification on what is meant by 'fully rigorous' would help, and possibly avoid some close votes -- questions like 'is xyz's proof of conjecture abc valid' are too opinion based, but if you could point to specific arguments/lemmas that appear nebulous for specific mathematical reasons it seems a good question. For example we could interpret 'fully rigorous' as 'being proved directly from the axioms of some set theory/logic' in which case the answer is obviously no, but this is not generally the bar for rigor in mathematics outside logic and set theory.
$endgroup$
– Alec Rhea
2 days ago
4
$begingroup$
Following @AlecRhea’s good points, I’d suggest the most productive way to frame this question isn’t “Is Witten’s proof rigorous?” so much as “How rigorous was Witten’s proof, and what is its relationship to later more rigorous elaborations like Taubes–Parker?” There is a spectrum different levels of rigour short of a full proof: a full sketch with some details missing; a sketch with most details omitted; a detailed outline; a heuristic argument which turns out to guide an eventual full proof; a heuristic argument that motivates the result but doesn’t form the basis of any proof…
$endgroup$
– Peter LeFanu Lumsdaine
2 days ago
$begingroup$
@AlecRhea I think questions about the validity of the proof of conjecture abc are referring to Mochizuki, not xyz...
$endgroup$
– user1728
2 days ago
add a comment |
$begingroup$
I noticed that the only official reason given for awarding Edward Witten the Fields Medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully rigorous.
However, I came across this paper https://projecteuclid.org/download/pdf_1/euclid.cmp/1103921154 by Taubes and Parker which claims to make Witten's proof 'mathematically rigorous' and to justify assumptions which Witten made about Dirac operators. Does this mean that the Witten proof is not rigorous, or is it just the case that there were some unjustified lemmas to clear up which do not affect the validity or rigour of the argument (similar to the case of Perelman's proof of the Poincare conjecture, where some lemmas and slight gaps had to be filled in)?
I am just curious as I have never really heard of the Taubes-Parker paper so I was assuming that the Witten paper was fully rigorous.
mp.mathematical-physics general-relativity dirac-operator spinor
asked 2 days ago
TomTom
410311
$endgroup$
I noticed that the only official reason given for awarding Edward Witten the Fields Medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully rigorous.
However, I came across this paper https://projecteuclid.org/download/pdf_1/euclid.cmp/1103921154 by Taubes and Parker which claims to make Witten's proof 'mathematically rigorous' and to justify assumptions which Witten made about Dirac operators. Does this mean that the Witten proof is not rigorous, or is it just the case that there were some unjustified lemmas to clear up which do not affect the validity or rigour of the argument (similar to the case of Perelman's proof of the Poincare conjecture, where some lemmas and slight gaps had to be filled in)?
I am just curious as I have never really heard of the Taubes-Parker paper so I was assuming that the Witten paper was fully rigorous.
mp.mathematical-physics general-relativity dirac-operator spinor
mp.mathematical-physics general-relativity dirac-operator spinor
asked 2 days ago
TomTom
410311
asked 2 days ago
TomTom
410311
asked 2 days ago
TomTom
410311
asked 2 days ago
TomTom
410311
asked 2 days ago
TomTom
410311
410311
2
$begingroup$
I think some clarification on what is meant by 'fully rigorous' would help, and possibly avoid some close votes -- questions like 'is xyz's proof of conjecture abc valid' are too opinion based, but if you could point to specific arguments/lemmas that appear nebulous for specific mathematical reasons it seems a good question. For example we could interpret 'fully rigorous' as 'being proved directly from the axioms of some set theory/logic' in which case the answer is obviously no, but this is not generally the bar for rigor in mathematics outside logic and set theory.
$endgroup$
– Alec Rhea
2 days ago
4
$begingroup$
Following @AlecRhea’s good points, I’d suggest the most productive way to frame this question isn’t “Is Witten’s proof rigorous?” so much as “How rigorous was Witten’s proof, and what is its relationship to later more rigorous elaborations like Taubes–Parker?” There is a spectrum different levels of rigour short of a full proof: a full sketch with some details missing; a sketch with most details omitted; a detailed outline; a heuristic argument which turns out to guide an eventual full proof; a heuristic argument that motivates the result but doesn’t form the basis of any proof…
$endgroup$
– Peter LeFanu Lumsdaine
2 days ago
$begingroup$
@AlecRhea I think questions about the validity of the proof of conjecture abc are referring to Mochizuki, not xyz...
$endgroup$
– user1728
2 days ago
add a comment |
2
$begingroup$
I think some clarification on what is meant by 'fully rigorous' would help, and possibly avoid some close votes -- questions like 'is xyz's proof of conjecture abc valid' are too opinion based, but if you could point to specific arguments/lemmas that appear nebulous for specific mathematical reasons it seems a good question. For example we could interpret 'fully rigorous' as 'being proved directly from the axioms of some set theory/logic' in which case the answer is obviously no, but this is not generally the bar for rigor in mathematics outside logic and set theory.
$endgroup$
– Alec Rhea
2 days ago
4
$begingroup$
Following @AlecRhea’s good points, I’d suggest the most productive way to frame this question isn’t “Is Witten’s proof rigorous?” so much as “How rigorous was Witten’s proof, and what is its relationship to later more rigorous elaborations like Taubes–Parker?” There is a spectrum different levels of rigour short of a full proof: a full sketch with some details missing; a sketch with most details omitted; a detailed outline; a heuristic argument which turns out to guide an eventual full proof; a heuristic argument that motivates the result but doesn’t form the basis of any proof…
$endgroup$
– Peter LeFanu Lumsdaine
2 days ago
$begingroup$
@AlecRhea I think questions about the validity of the proof of conjecture abc are referring to Mochizuki, not xyz...
$endgroup$
– user1728
2 days ago
2
2
$begingroup$
I think some clarification on what is meant by 'fully rigorous' would help, and possibly avoid some close votes -- questions like 'is xyz's proof of conjecture abc valid' are too opinion based, but if you could point to specific arguments/lemmas that appear nebulous for specific mathematical reasons it seems a good question. For example we could interpret 'fully rigorous' as 'being proved directly from the axioms of some set theory/logic' in which case the answer is obviously no, but this is not generally the bar for rigor in mathematics outside logic and set theory.
$endgroup$
– Alec Rhea
2 days ago
$begingroup$
I think some clarification on what is meant by 'fully rigorous' would help, and possibly avoid some close votes -- questions like 'is xyz's proof of conjecture abc valid' are too opinion based, but if you could point to specific arguments/lemmas that appear nebulous for specific mathematical reasons it seems a good question. For example we could interpret 'fully rigorous' as 'being proved directly from the axioms of some set theory/logic' in which case the answer is obviously no, but this is not generally the bar for rigor in mathematics outside logic and set theory.
$endgroup$
– Alec Rhea
2 days ago
4
4
$begingroup$
Following @AlecRhea’s good points, I’d suggest the most productive way to frame this question isn’t “Is Witten’s proof rigorous?” so much as “How rigorous was Witten’s proof, and what is its relationship to later more rigorous elaborations like Taubes–Parker?” There is a spectrum different levels of rigour short of a full proof: a full sketch with some details missing; a sketch with most details omitted; a detailed outline; a heuristic argument which turns out to guide an eventual full proof; a heuristic argument that motivates the result but doesn’t form the basis of any proof…
$endgroup$
– Peter LeFanu Lumsdaine
2 days ago
$begingroup$
Following @AlecRhea’s good points, I’d suggest the most productive way to frame this question isn’t “Is Witten’s proof rigorous?” so much as “How rigorous was Witten’s proof, and what is its relationship to later more rigorous elaborations like Taubes–Parker?” There is a spectrum different levels of rigour short of a full proof: a full sketch with some details missing; a sketch with most details omitted; a detailed outline; a heuristic argument which turns out to guide an eventual full proof; a heuristic argument that motivates the result but doesn’t form the basis of any proof…
$endgroup$
– Peter LeFanu Lumsdaine
2 days ago
$begingroup$
@AlecRhea I think questions about the validity of the proof of conjecture abc are referring to Mochizuki, not xyz...
$endgroup$
– user1728
2 days ago
$begingroup$
@AlecRhea I think questions about the validity of the proof of conjecture abc are referring to Mochizuki, not xyz...
$endgroup$
– user1728
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3).
6. Conclusion
From this very brief summary of Witten's achievements it should be clear that
he has made a profound impact on contemporary mathematics. In his hands
physics is once again providing a rich source of inspiration and insight in
mathematics. Of course physical insight does not always lead to immediately
rigorous mathematical proofs but it frequently leads one in the right direction,
and technically correct proofs can then hopefully be found. This is the case with
Witten's work. So far his insight has never let him down and rigorous proofs, of
the standard we mathematicians rightly expect, have always been forthcoming.
There is therefore no doubt that contributions to mathematics of this order are
fully worthy of a Fields Medal.
answered 2 days ago
Francesco PolizziFrancesco Polizzi
48.3k3129211
$endgroup$
4
$begingroup$
I have read it and do not doubt Witten's contributions to physics or to mathematics: I was asking specifically about the proof of the positive mass theorem which was given as the official reason on the citation.
$endgroup$
– Tom
2 days ago
1
$begingroup$
The proof of positive mass theorem is treated in Part 3 of the laudatio. It is called a "outline", so actually some technical detail was missing, but the idea was completely correct. That said, I do not know why only that result has been given as official reason for the prize, since the laudatio contains much more.
$endgroup$
– Francesco Polizzi
2 days ago
$begingroup$
OK, maybe that was my confusion, as that is literally the only reason given and that has always seemed very strange to me. I'm not saying that his proof of the positive mass theorem was not a very significant achievement.
$endgroup$
– Tom
2 days ago
$begingroup$
@Tom : What do you mean by "literally the only reason given"? I don't think that the IMU or the Fields Medal Committee produces a piece of text for each medalist and declares that text, and only that text, to be the "official reason" for awarding the medal. No such "official citations" appear on the IMU website, for example. I suspect that if you ask any former committee member, they will say that the reasons for the choices are complicated, and that any public statement or press release is not intended to be a comprehensive statement of those reasons.
$endgroup$
– Timothy Chow
yesterday
add a comment |
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$begingroup$
You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3).
6. Conclusion
From this very brief summary of Witten's achievements it should be clear that
he has made a profound impact on contemporary mathematics. In his hands
physics is once again providing a rich source of inspiration and insight in
mathematics. Of course physical insight does not always lead to immediately
rigorous mathematical proofs but it frequently leads one in the right direction,
and technically correct proofs can then hopefully be found. This is the case with
Witten's work. So far his insight has never let him down and rigorous proofs, of
the standard we mathematicians rightly expect, have always been forthcoming.
There is therefore no doubt that contributions to mathematics of this order are
fully worthy of a Fields Medal.
answered 2 days ago
Francesco PolizziFrancesco Polizzi
48.3k3129211
$endgroup$
4
$begingroup$
I have read it and do not doubt Witten's contributions to physics or to mathematics: I was asking specifically about the proof of the positive mass theorem which was given as the official reason on the citation.
$endgroup$
– Tom
2 days ago
1
$begingroup$
The proof of positive mass theorem is treated in Part 3 of the laudatio. It is called a "outline", so actually some technical detail was missing, but the idea was completely correct. That said, I do not know why only that result has been given as official reason for the prize, since the laudatio contains much more.
$endgroup$
– Francesco Polizzi
2 days ago
$begingroup$
OK, maybe that was my confusion, as that is literally the only reason given and that has always seemed very strange to me. I'm not saying that his proof of the positive mass theorem was not a very significant achievement.
$endgroup$
– Tom
2 days ago
$begingroup$
@Tom : What do you mean by "literally the only reason given"? I don't think that the IMU or the Fields Medal Committee produces a piece of text for each medalist and declares that text, and only that text, to be the "official reason" for awarding the medal. No such "official citations" appear on the IMU website, for example. I suspect that if you ask any former committee member, they will say that the reasons for the choices are complicated, and that any public statement or press release is not intended to be a comprehensive statement of those reasons.
$endgroup$
– Timothy Chow
yesterday
add a comment |
$begingroup$
You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3).
6. Conclusion
From this very brief summary of Witten's achievements it should be clear that
he has made a profound impact on contemporary mathematics. In his hands
physics is once again providing a rich source of inspiration and insight in
mathematics. Of course physical insight does not always lead to immediately
rigorous mathematical proofs but it frequently leads one in the right direction,
and technically correct proofs can then hopefully be found. This is the case with
Witten's work. So far his insight has never let him down and rigorous proofs, of
the standard we mathematicians rightly expect, have always been forthcoming.
There is therefore no doubt that contributions to mathematics of this order are
fully worthy of a Fields Medal.
answered 2 days ago
Francesco PolizziFrancesco Polizzi
48.3k3129211
$endgroup$
4
$begingroup$
I have read it and do not doubt Witten's contributions to physics or to mathematics: I was asking specifically about the proof of the positive mass theorem which was given as the official reason on the citation.
$endgroup$
– Tom
2 days ago
1
$begingroup$
The proof of positive mass theorem is treated in Part 3 of the laudatio. It is called a "outline", so actually some technical detail was missing, but the idea was completely correct. That said, I do not know why only that result has been given as official reason for the prize, since the laudatio contains much more.
$endgroup$
– Francesco Polizzi
2 days ago
$begingroup$
OK, maybe that was my confusion, as that is literally the only reason given and that has always seemed very strange to me. I'm not saying that his proof of the positive mass theorem was not a very significant achievement.
$endgroup$
– Tom
2 days ago
$begingroup$
@Tom : What do you mean by "literally the only reason given"? I don't think that the IMU or the Fields Medal Committee produces a piece of text for each medalist and declares that text, and only that text, to be the "official reason" for awarding the medal. No such "official citations" appear on the IMU website, for example. I suspect that if you ask any former committee member, they will say that the reasons for the choices are complicated, and that any public statement or press release is not intended to be a comprehensive statement of those reasons.
$endgroup$
– Timothy Chow
yesterday
add a comment |
$begingroup$
You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3).
6. Conclusion
From this very brief summary of Witten's achievements it should be clear that
he has made a profound impact on contemporary mathematics. In his hands
physics is once again providing a rich source of inspiration and insight in
mathematics. Of course physical insight does not always lead to immediately
rigorous mathematical proofs but it frequently leads one in the right direction,
and technically correct proofs can then hopefully be found. This is the case with
Witten's work. So far his insight has never let him down and rigorous proofs, of
the standard we mathematicians rightly expect, have always been forthcoming.
There is therefore no doubt that contributions to mathematics of this order are
fully worthy of a Fields Medal.
answered 2 days ago
Francesco PolizziFrancesco Polizzi
48.3k3129211
$endgroup$
You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3).
6. Conclusion
From this very brief summary of Witten's achievements it should be clear that
he has made a profound impact on contemporary mathematics. In his hands
physics is once again providing a rich source of inspiration and insight in
mathematics. Of course physical insight does not always lead to immediately
rigorous mathematical proofs but it frequently leads one in the right direction,
and technically correct proofs can then hopefully be found. This is the case with
Witten's work. So far his insight has never let him down and rigorous proofs, of
the standard we mathematicians rightly expect, have always been forthcoming.
There is therefore no doubt that contributions to mathematics of this order are
fully worthy of a Fields Medal.
answered 2 days ago
Francesco PolizziFrancesco Polizzi
48.3k3129211
edited 2 days ago
answered 2 days ago
Francesco PolizziFrancesco Polizzi
48.3k3129211
answered 2 days ago
Francesco PolizziFrancesco Polizzi
48.3k3129211
answered 2 days ago
Francesco PolizziFrancesco Polizzi
48.3k3129211
48.3k3129211
4
$begingroup$
I have read it and do not doubt Witten's contributions to physics or to mathematics: I was asking specifically about the proof of the positive mass theorem which was given as the official reason on the citation.
$endgroup$
– Tom
2 days ago
1
$begingroup$
The proof of positive mass theorem is treated in Part 3 of the laudatio. It is called a "outline", so actually some technical detail was missing, but the idea was completely correct. That said, I do not know why only that result has been given as official reason for the prize, since the laudatio contains much more.
$endgroup$
– Francesco Polizzi
2 days ago
$begingroup$
OK, maybe that was my confusion, as that is literally the only reason given and that has always seemed very strange to me. I'm not saying that his proof of the positive mass theorem was not a very significant achievement.
$endgroup$
– Tom
2 days ago
$begingroup$
@Tom : What do you mean by "literally the only reason given"? I don't think that the IMU or the Fields Medal Committee produces a piece of text for each medalist and declares that text, and only that text, to be the "official reason" for awarding the medal. No such "official citations" appear on the IMU website, for example. I suspect that if you ask any former committee member, they will say that the reasons for the choices are complicated, and that any public statement or press release is not intended to be a comprehensive statement of those reasons.
$endgroup$
– Timothy Chow
yesterday
add a comment |
4
$begingroup$
I have read it and do not doubt Witten's contributions to physics or to mathematics: I was asking specifically about the proof of the positive mass theorem which was given as the official reason on the citation.
$endgroup$
– Tom
2 days ago
1
$begingroup$
The proof of positive mass theorem is treated in Part 3 of the laudatio. It is called a "outline", so actually some technical detail was missing, but the idea was completely correct. That said, I do not know why only that result has been given as official reason for the prize, since the laudatio contains much more.
$endgroup$
– Francesco Polizzi
2 days ago
$begingroup$
OK, maybe that was my confusion, as that is literally the only reason given and that has always seemed very strange to me. I'm not saying that his proof of the positive mass theorem was not a very significant achievement.
$endgroup$
– Tom
2 days ago
$begingroup$
@Tom : What do you mean by "literally the only reason given"? I don't think that the IMU or the Fields Medal Committee produces a piece of text for each medalist and declares that text, and only that text, to be the "official reason" for awarding the medal. No such "official citations" appear on the IMU website, for example. I suspect that if you ask any former committee member, they will say that the reasons for the choices are complicated, and that any public statement or press release is not intended to be a comprehensive statement of those reasons.
$endgroup$
– Timothy Chow
yesterday
4
4
$begingroup$
I have read it and do not doubt Witten's contributions to physics or to mathematics: I was asking specifically about the proof of the positive mass theorem which was given as the official reason on the citation.
$endgroup$
– Tom
2 days ago
$begingroup$
I have read it and do not doubt Witten's contributions to physics or to mathematics: I was asking specifically about the proof of the positive mass theorem which was given as the official reason on the citation.
$endgroup$
– Tom
2 days ago
1
1
$begingroup$
The proof of positive mass theorem is treated in Part 3 of the laudatio. It is called a "outline", so actually some technical detail was missing, but the idea was completely correct. That said, I do not know why only that result has been given as official reason for the prize, since the laudatio contains much more.
$endgroup$
– Francesco Polizzi
2 days ago
$begingroup$
The proof of positive mass theorem is treated in Part 3 of the laudatio. It is called a "outline", so actually some technical detail was missing, but the idea was completely correct. That said, I do not know why only that result has been given as official reason for the prize, since the laudatio contains much more.
$endgroup$
– Francesco Polizzi
2 days ago
$begingroup$
OK, maybe that was my confusion, as that is literally the only reason given and that has always seemed very strange to me. I'm not saying that his proof of the positive mass theorem was not a very significant achievement.
$endgroup$
– Tom
2 days ago
$begingroup$
OK, maybe that was my confusion, as that is literally the only reason given and that has always seemed very strange to me. I'm not saying that his proof of the positive mass theorem was not a very significant achievement.
$endgroup$
– Tom
2 days ago
$begingroup$
@Tom : What do you mean by "literally the only reason given"? I don't think that the IMU or the Fields Medal Committee produces a piece of text for each medalist and declares that text, and only that text, to be the "official reason" for awarding the medal. No such "official citations" appear on the IMU website, for example. I suspect that if you ask any former committee member, they will say that the reasons for the choices are complicated, and that any public statement or press release is not intended to be a comprehensive statement of those reasons.
$endgroup$
– Timothy Chow
yesterday
$begingroup$
@Tom : What do you mean by "literally the only reason given"? I don't think that the IMU or the Fields Medal Committee produces a piece of text for each medalist and declares that text, and only that text, to be the "official reason" for awarding the medal. No such "official citations" appear on the IMU website, for example. I suspect that if you ask any former committee member, they will say that the reasons for the choices are complicated, and that any public statement or press release is not intended to be a comprehensive statement of those reasons.
$endgroup$
– Timothy Chow
yesterday
add a comment |
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I think some clarification on what is meant by 'fully rigorous' would help, and possibly avoid some close votes -- questions like 'is xyz's proof of conjecture abc valid' are too opinion based, but if you could point to specific arguments/lemmas that appear nebulous for specific mathematical reasons it seems a good question. For example we could interpret 'fully rigorous' as 'being proved directly from the axioms of some set theory/logic' in which case the answer is obviously no, but this is not generally the bar for rigor in mathematics outside logic and set theory.
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– Alec Rhea
2 days ago
4
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Following @AlecRhea’s good points, I’d suggest the most productive way to frame this question isn’t “Is Witten’s proof rigorous?” so much as “How rigorous was Witten’s proof, and what is its relationship to later more rigorous elaborations like Taubes–Parker?” There is a spectrum different levels of rigour short of a full proof: a full sketch with some details missing; a sketch with most details omitted; a detailed outline; a heuristic argument which turns out to guide an eventual full proof; a heuristic argument that motivates the result but doesn’t form the basis of any proof…
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– Peter LeFanu Lumsdaine
2 days ago
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@AlecRhea I think questions about the validity of the proof of conjecture abc are referring to Mochizuki, not xyz...
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– user1728
2 days ago