The status of the Scholze-Stix Report and an analysisof the Mochizuki-Scholze-Stix Controversy
Kirti Joshi
June 28, 2024
§ 1.1 The need for this document After Mochizuki’s [Mochizuki, 2021a,b,c,d] preprints werereleased in 2012, there was tremendous excitement in the math community, and a serious effort
was made to understand his work. Unfortunately, neither Mochizuki, nor any of his ‘experts’
could provide satisfactory answers or examples to explain his theory. So the Scholze-Stix Report([Scholze and Stix, 2018]) which talked about the flaws in Mochizuki’s work ([Mochizuki,
2021a,b,c,d]) was widely accepted as a valid conclusion.
However, I have always believed that if Mochizuki’s proof has to be rejected, then it must
be based on a complete understanding of its ideas and present correct mathematical arguments,
precisely identifying the issue(s) with its claimed strategy.
Despite having discussed the flaws in [Scholze and Stix, 2018], [Scholze, 2021] in my papers
and the clarity and precision that my work has brought to Mochizuki’s work, [Scholze and Stix,
2018], [Scholze, 2021] are still considered by many as definitive conclusions on [Mochizuki,
2021a,b,c,d]. Hence the need for this document.
This document provides quantitative evidence (§ 1.2, § 1.3) that due to the lack of clarity in
Mochizuki’s work, [Scholze and Stix, 2018] understanding of Mochizuki’s work is flawed at afundamental level, leading to an inaccurate analysis of [Mochizuki, 2021a,b,c,d]. My works
on this subject provide far more accurate critiques of [Mochizuki, 2021a,b,c,d] and how thehighlighted issues can be circumvented.
I have tried my best to be fair in my critiques of [Mochizuki, 2021a,b,c,d] and [Scholze andStix, 2018]. However, both have been reticent in publicly acknowledging my work. Taking on
the claims of two powerful mathematicians, I completely understand that the professional fallout
for me is substantial and academically debilitating (this has already played out as may be evident
from Mochizuki’s colorful language and analogies in rejecting my work while many arithmetic
geometers have simply distanced themselves from the conversation). I believe that more voices
should participate in the discussion of the abc-conjecture because the public comments byboth Mochizuki and Scholze (and some others who have echoed Scholze) make it clear thatthey simply do not wish to be second-guessed on this matter. This has emboldened somemathematicians to publicly dismiss my work even though it brings many new ideas to the table.
The conclusion of this document is independent of the proof of the abc-conjecture.

1§ 1.2 Debunking some popular myths surrounding Mochizuki’s Theory I want to debunksome of the popular myths about Mochizuki’s Theory.
Let me set out some notation for this discussion. Let E be a p-adic field and let X/E be ageometrically connected, smooth, hyperbolic curve over E and assume that X is definable oversome number field. Mochizuki’s theory works with tempered fundamental groups [André, 2003],[Lepage, 2010]. This requires some additional notation. Let K ⊃ E be an algebraically closed,
complete (rank one) valued field containing E isometrically [such a field is an algebraicallyclosed, perfectoid field [Scholze, 2012]]. Let XEan be the Berkovich analytic space of X/E
and let ∗ : M (K) → XEan be a K-valued point of the analytic space XEan . Such a point is ageometric base-point and allows one to compute the tempered fundamental group
temp
Πtemp
(XEan , ∗K : M (K) → XEan )X/E;K := π1using the K-geometric base-point ∗K : M (K) → XEan . The isomorphism class of this group is
temp
independent of the chosen geometric base-point, so often one writes Πtemp
X/E instead of ΠX/E;K .
§ 1.2.1 Myth Mochizuki’s [Mochizuki, 2021a,b,c,d] is about étale fundamental group of X/E.
Sources For This Myth May date back to Mochizuki’s preprints in 2012, but also see [Fesenko,
2015], [Scholze and Stix, 2018], the discussion on Peter Woit’s Blog [Scholze et al., April 2020].
This Myth is False Mochizuki’s IUTT is about tempered fundamental groups and Mochizuki’s
Key Principle of Inter-Universality [Mochizuki, 2021a, § I3, Pages 25–26] requires us to remember the geometric base-points for tempered fundamental groups i.e. one works with Πtemp
X/E;K andtemp
not ΠX/E . [Note that Mochizuki does not use perfectoid fields and that is why I have objected tohis papers. Once this information is thrown in, I show that there is a valid theory satisfying all
the claimed properties.] [The table given below is taken from [Joshi, 2024b].]Comparison of geometric base-points for étale and tempered fundamental groupsétale fund. grouptempered fund. group

Input datum(X/Qp , ∗K : Spec(K) → X)

(XQanp , ∗K : M (K) → XQanp )where X/Qp is a scheme, K ⊃ Qp is an alg. where XQanp is an analytic space over Qp , K ⊃closed field, and ∗K : Spec(K) → X is a Qp an alg. closed, perfectoid field, and ∗K :K-geom. base point of X
M (K) → XQanp is a K-geom. base point of
XQanp .
isom. class of alg. closed fields K ⊃ Qp is isom. class of alg. closed, perfectoid fields
determined by the cardinality of K
K ⊃ Qp is not determined by cardinality ofK.
Caution For important reasons, one works with an alg. closed perfectoid field K, together with tilting data i.e.
one works with untilts rather than K. So readers should not walk away with the impression that working withperfectoid fields is adequate. See [Joshi, 2022, Definition 5.1] for the precise datum required.

2Let me also remark that Mochizuki’s Anabelian Reconstruction Theory methods of [Mochizuki,
2012, 2013, 2015], cannot provide reconstruction of the perfectoid field in the geometric basepoint datum. For example, the algebraically closed perfectoid field Cp , cannot be reconstructedfrom the topological absolute Galois group GQp as was shown in [Mochizuki, 1997]. Hence
dealing with the geometric base-point datum (as required by Mochizuki’s Key Principle of InterUniversality) would necessarily require taking the approach I have followed in my theory (andby doing so I obtain a canonical theory in which all of Mochizuki’s ideas have intrinsic meaning).
The failure of [Mochizuki, 2021a,b,c,d] to treat its own principle clearly and accurately forms
the core of my mathematical objections to [Mochizuki, 2021a,b,c,d].
§ 1.2.2 Myth Validity of the Absolute Grothendieck Conjecture for hyperbolic curves implies
Mochizuki’s proof [Mochizuki, 2021a,b,c,d] is impossible.
Sources For This Myth Advertisement by Mochizuki (and others) about anabelian aspects of[Mochizuki, 2021a,b,c,d] (and use of [Mochizuki, 2015]), are a possible source of this myth(and in my opinion, Mochizuki failed to bring clarity on this point). But it may have taken strongroots because of [Scholze and Stix, 2018] Footnote 8.
This Myth is False Scholze-Stix conflated Teichmuller Theory and Moduli theory because ofthis belief. Existence of Teichmuller Theory of the sort Mochizuki claims (and established in
my work) is independent of the validity of the Absolute Grothendieck Conjecture over number
fields and p-adic fields. From my point of view the non-validity of the Absolute Grothendieck
Conjecture over algebraically closed, perfectoid fields (or over C) is a better way of establishing
Mochizuki’s Teichmuller Theory claim. Also see [Joshi, 2021b], [Joshi, 2023c].
§ 1.2.3 Myth Mochizuki’s anabelian reconstruction theorem [Mochizuki, 2015, Theorem 1.9],
Πtemp
X/E;K determines the isomorphism class of X/E. Hence [Mochizuki, 2021a,b,c,d] is aproblematic theory.
Sources For This Myth [Scholze and Stix, 2018], Discussions on Peter Woit’s Blog [Scholze
et al., April 2020].
This Myth is False Using Πtemp
X/E;K , Mochizuki’s reconstruction theorem [Mochizuki, 2015,
Theorem 1.9] determines the isomorphism class of X/E but it does not determine K. [Remember
Mochizuki’s Key Principle of Inter-Universality requires us to remember geometric base-points.]§ 1.2.4 Myth [Scholze and Stix, 2018, Remark 9] is correct and hence there is no valid theory
in [Mochizuki, 2021a,b,c,d].
Sources For This Myth [Scholze and Stix, 2018], Scholze’s comments to my work on MathOverFlow; where Will Sawin backs this assertion by Scholze.
This Myth is False While [Scholze and Stix, 2018, Remark 9] is the central thesis of the ScholzeStix Report, this remark is completely false. [Mochizuki’s discussion of this remark appearsin [Mochizuki, 2018, (C5)] (Mochizuki refers to Remark 9 as Remark 8) and my discussion
here should be read with his comments and with § 1.2.4.1.] Unfortunately, despite having
pointed this out explicitly in my response (on David Robert’s blog), this incorrect argument
has been repeated by Scholze and few others in defending [Scholze and Stix, 2018, Remark 9].
3

The said remark is based on an incorrect understanding of what [Mochizuki, 2015, Theorem
1.9] provides and what is actually needed in [Mochizuki, 2021a,b,c,d]. Let me explain thispoint. The important difference is this: using the group Πtemp
X/E;K , the said theorem ([Mochizuki,
2015, Theorem 1.9]) provides the isomorphism class of the curve X/E as asserted but it does
not provide the geometric base-point information (given by the alg. closed perfectoid fieldK). However, Mochizuki’s Key Principle of Inter-Universality requires one to remember thegeometric base-point information. One clear way of saying this, via the main theorem of [Joshi,
2021a], [Joshi, 2022], is this: [Mochizuki, 2015, Theorem 1.9] does not determine the arithmetic
holomorphic structure at all; however such structures are central to [Mochizuki, 2021a,b,c,d].
This is the sense in which Mochizuki uses the cited theorem in the context of [Mochizuki,
2021a,b,c,d]. This is an important point and Mochizuki should have alerted Scholze-Stix andother readers (myself included) to this point with complete clarity and adequate emphasis (thiswould require dealing with geometric base-points the way my work does). At any rate, theconclusion of the said theorem cannot be used to arrive at the conclusion [Scholze and Stix,
2018, Remark 9] in the context of [Mochizuki, 2021a,b,c,d]. [Also see § 1.2.1.]
Note § 1.2.4.1 : This document was sent to Scholze for an early read and his comments. Afterdetailed conversations with Scholze (June 2024), I can say that [Scholze and Stix, 2018, Remark
9] arose because of Mochizuki’s emphasis, and advertisement by him and others, of the primacyof group theory (anabelian) as the raison d’etre of [Mochizuki, 2021a,b,c,d]. On the other hand,
Scholze and I are now in agreement (June 2024) that Mochizuki’s emphasis should have been on
the primacy of geometric and arithmetic objects of the theory (with an adequate demonstrationof the existence of such objects). As an algebraic geometer, well-versed in classical Teichmuller
Theory, I have always recognized that Teichmuller Theory (of any sort: classical, p-adic,
arithmetic) is about geometric objects and in my work, I have emphasized the primacy ofgeometric and arithmetic objects from the very beginning, with relevant groups arising as
symmetries of such objects, and my reading and criticism of both [Mochizuki, 2021a,b,c,d] and[Scholze and Stix, 2018, Scholze, 2021] has been based on this viewpoint.
Note § 1.2.4.2 : This document has also been sent to Mochizuki for an early read and hiscomments. I am currently awaiting his response.
§ 1.2.5 Myth All Hodge-Theaters in [Mochizuki, 2021a,b,c,d] are isomorphic, hence there isno theory in [Mochizuki, 2021a,b,c,d].
Sources For This Myth [Scholze, 2021], Comments by Scholze and others on Peter Woit’s
blog [Scholze et al., April 2020].
This Myth is False Collection of distinct Hodge-Theaters (with a fixed decoration) is a proxy
for a variation of (p-adic) Hodge Structures (at all primes p simultaneously) [Mochizuki, 2021a,
Page 25] and in particular Hodge-Theaters arise from distinct geometric and arithmetic data.
It is certainly true that all Hodge-Theaters (with a fixed decoration) are all isomorphic, butsuch an isomorphism comes at the expense of forgetting the arithmetic and geometric data the
Hodge-Theaters arise from and doing this is analogous to asserting that all Riemann surfacesof a fixed genus are homeomorphic–a statement which is true but not the goal in the studyof Riemann surfaces. So the identification of Hodge-Theaters asserted in [Scholze, 2021]
4

serves no useful mathematical purpose as we are averaging over contributions of distinct
arithmetic geometric data in [Mochizuki, 2021a,b,c,d]. [Mochizuki does not provide an adequate
justification for the existence of distinct Hodge-Theaters, but my work provides the most natural
way of understanding how p-adic Hodge Theory enters the picture and demonstrates clearly how
distinct Hodge-Theaters arise from distinct geometric data (see [Joshi, 2024b]).]§ 1.2.6 Myth There is no purpose to Mochizuki’s weird definitions and formalism in [Mochizuki,
2021a,b,c,d].
Sources For This Myth [Scholze and Stix, 2018], Discussion on Peter Woit’s blog [Scholze
et al., April 2020].
This Myth is False In hindsight gained from my work, Mochizuki’s formalism is an approximate
proxy for modern p-adic Hodge Theory (at all primes simultaneously). This is an important
point many p-adic Hodge Theorists have missed. Notably, Mochizuki’s use of perfect Frobenioids (together with their realifications) in [Mochizuki, 2021a,b,c,d] is to placate the absenceof machinery of p-adic Hodge theory such as tilting which was not available to Mochizuki.
[Mochizuki’s group theoretic approach does make this aspect extremely difficult to see but mywork makes this completely transparent (see [Joshi, 2021a, 2022, 2023a]) and [Joshi, 2024b]
contains a comparison of these perfect Frobenioids and perfectoids in one of the appendices.]§ 1.2.7 Myth The theory of [Joshi, 2021a, 2022, 2023a] adds on information which serves no
purpose in this discussion related to [Mochizuki, 2021a,b,c,d].
Sources For This Myth This myth may have its origins in conversations (2023) about my workon MathOverFlow and on Peter Woit’s Blog.
This Myth is False As pointed out in [Joshi, 2021a, 2022, Remark 4.3], the local Arithmetic
Teichmuller Space constructed in [Joshi, 2022, Definition 5.1] bears a formal similarity to thediamond XL♢v associated to the analytic adic space XLadv given in [Scholze, 2017, Definition
15.5]. I had already pointed out (as far back as 2022) this similarity on a number of social media
sites. [Note For precise mathematical assertion see [Joshi, 2022, Remark 4.3 and Definition
5.1]. [Strictly speaking, [Joshi, 2022, Def. 5.1] deals with a slightly more general object than
XL♢v –but this technical point may be ignored for simplicity.]
To put it clearly: in dimension one and genus one, the theory developed in [Joshi, 2021a,
2022, 2023a] so closely intertwines the theories of [Scholze, 2012, 2017], [Fargues and Fontaine,
2018], [Mochizuki, 2021a,b,c,d] that any attempt at separation based on this myth has no factual
mathematical foundation.

5§ 1.3 Mochizuki, Scholze-Stix claims compared The following table provides a comparisonof hypotheses of Mochizuki and Joshi and the omissions of these hypothesis by Scholze-Stix
leading to incorrect mathematical conclusions.
Comparison of claims of Mochizuki, Scholze-Stix and Joshi
objectgeom.
point

how is
used?

base

this

Valuation data

Why needed?

Mochizuki
Central role of arbitrary geometric base-points as a proxyfor deformations of arithmetic[Mochizuki, 2021a, § I3, Page
25]
domain and codomain of all
key operations refer to distinct
geom. base-points [Mochizuki,
2021a, § I3, Page 25]
encoded in realified Frobenioids[Mochizuki, 2021a]

for computing local and global
arith. degrees
log-Links
log-Links aka Mochizuki’s
proxy for Frobenius (at eachprime) [Mochizuki, 2021c]
Theta-Links
Central role of Theta-Links
Claimed via theory of Frobenioids [Mochizuki, 2021a]
distinct Arith. Asserted in [Mochizuki, 2021a]
Holomorphic
but existence is not clearly esStructures
tablished

Scholze-Stix
ignored(see § 1.2.1, § 1.2.3)

Joshi
included (in the data of an arithmeticoid) as arbitrary alg. closed
perfectoid fields [Joshi, 2021a,
2022]

incorrectly identify thedomains and codomains
leading to incorrect conclusions
no mention of distinct
valuation data

naturally show Mochizuki’s requirements [Joshi, 2024b]

ignoredignored (see § 1.2.5)

argue that such objects
cannot exist (see § 1.2.5)
declare that these cannot
exist (see § 1.2.1, § 1.2.4)

works with valuation data insteadof Frobenioids [Joshi, 2021a,
2022, 2023a]
as in Mochizuki but without using
Frobenioids [Joshi, 2023a, 2022]
works with the Frobenius morphism instead of a proxy [Joshi,
2023b,a].
demonstrates the existence bothat local and global level. [Joshi,
2023b,a, 2024b]
Demonstrates the existence anddeformation property via arbitrary alg. closed perfectoid fields
or equivalently by using arbitrarygeom. base-points [Joshi, 2021a,
2022, 2023a]

NOTE(1) Despite the fact that Mochizuki asserts his Key Principle of Inter-Universality, there is no mention of therequired input data (see the table accompanying § 1.2.1) in [Mochizuki, 2021a,b,c,d], and on two separate
occasions, Mochizuki denied the relevance of alg. closed perfectoid fields to [Mochizuki, 2021a,b,c,d].
(2) One cannot build a valid theory, as [Mochizuki, 2021a,b,c,d] claims to do, by requiring arbitrary geometricbase-points for tempered fundamental groups but work solely over Q̄p . This is the reason for my mathematical objections to [Mochizuki, 2021a,b,c,d]. My work fixes this central issue and builds a robust (and evenmore general) theory with all properties claimed by Mochizuki for his IUTT.

6§ 1.4 Conclusion(1) The conclusion of this report is independent of the proof of the abc-conjecture.
(2) As § 1.2, § 1.3 shows, the reports [Scholze and Stix, 2018], [Scholze, 2021] omit most
key mathematical assumptions and aspects of Mochizuki’s Theory (see Note § 1.2.4.1).
Mochizuki’s report on [Scholze and Stix, 2018] is here. My work and this document showsthat arguments and claims of [Scholze and Stix, 2018], [Scholze, 2021] and [Mochizuki,
2018], [Mochizuki, 2022] have no mathematical importance except as a matter of historical
record.
(3) I think that Mochizuki has mounted an incredibly weak defense of his work, but it would
be a travesty of mathematics to not acknowledge that the assessment of Mochizuki’s
claims was based on a poor and unsound understanding, by many, of the mathematicalprinciples which have guided his claims. [This is independent of whether or not the
abc-conjecture has been proven in his works.](4) However, let me make this absolutely clear: Mochizuki’s unwillingness to have mathematical conversations or to recognize my work, which bring mathematical clarity to his
work, continues to be a vexing issue (anabelian geometers around Mochizuki have also
followed his suit on this). Moreover, I strongly protest the unprofessional behavior (e.g.
name-calling and personal insults) that he has resorted to in this matter.
(5) Notably, there exists a valid Teichmuller Theory of Number Fields as Mochizuki has
claimed (this is fully demonstrated in [Joshi, 2023a]).
(6) I demonstrate that Mochizuki’s formalism in [Mochizuki, 2021a,b,c,d] is a proxy formodern p-adic Hodge Theory (at all primes simultaneously) and my work provides amathematically precise way of arriving at his claims by circumventing his formalism.
(7) The ideas espoused by Mochizuki in [Mochizuki, 2021a,b,c,d] are highly non-trivial eventhough, he and all his ‘experts,’ have been less than lucid about what is it that his theory
claims. [While Mochizuki may be unaware, I have heard that some of the ‘experts’ havestudied my work to better understand his claims.](8) My position on whether or not Mochizuki has proved the abc-conjecture is still open
(as my preprint [Joshi, 2024a] still remains under consideration). In other words, I’mcurrently neutral on the matter of the abc-conjecture. However, I continue to work on[Joshi, 2024b,a] to tie up all the loose ends. Updates of [Joshi, 2023a, 2019, 2024b]
posted.

7§ 1.5 Time-line of events Here is a brief time-line of events as I have witnessed them:March 2012 Mochizuki’s papers are released online and published as RIMS (Kyoto) Preprints.
Oct 2012 Vesselin Dimitrov posts his counter examples to Mochizuki’s stated estimate in the
March 2012 (and RIMS Preprint) version of [Mochizuki, 2021d]. Mochizuki points out
corrections in this response and Dimitrov withdraws his objections in his MO post (andMochizuki’s corrections appear in [Mochizuki, 2021d]).
2015 [Fesenko, 2015] is published (see § 1.2.1).
2015 First IUTT Workshop.
2015 Brian Conrad’s Report on the above conference.
2016 Second IUTT Workshop.
2016 [Mochizuki, 2016] is published and it provides insights into his proof by means of proofsof [Amorós et al., 2000, Zhang, 2001] of geometric case of Szpiro inequality. [On theother hand [Scholze and Stix, 2018], perhaps unaware of this, ignore the geometric caseand miss the most important of all clues to the claims of [Mochizuki, 2021a,b,c,d].]2016 In the context of [Mochizuki, 2016], Taylor Dupuy posts a two part video explaining theBogomolov Zhang Proofs. [I was not aware of Taylor’s videos, however I learned of thegeometric case from Mochizuki, see the next point.]
Spring 2018 I visit Mochizuki for my sabbatical. Scholze and Stix visit Mochizuki in April.
Mochizuki lectures to me on the Bogomolov Zhang Proofs (discussed in entry 2016 above)
based on [Mochizuki, 2016] and I came to further understand the geometric case fromthis point of view through Mochizuki’s lectures (and my independent reading of [Amorós
et al., 2000, Zhang, 2001]). Because of my prior understanding of Mochizuki’s work on
Teichmuller Theory ([Mochizuki, 1996], [Mochizuki, 1999]), I recognized that he may beonto something highly non-trivial.
May 2018 [Scholze and Stix, 2018] is put into circulation; Mochizuki posts his report.
2019 [Joshi, 2019] (first version is posted on the arxiv).
April 2020 [Joshi, 2020a] is posted on the arxiv. [The March 2020 version had some errors
in one section which were fixed in this release.] Scholze asserts that anabelomorphy(introduced in [Joshi, 2020a]) has nothing to do with Mochizuki’s work; I invite Mochizuki
to explain the relationship and he does by contributing to [Joshi, 2020a, 1.7], and it clearly
contradicts Scholze’s assertion.
April 2020 Discussion on Peter Woit’s blog. Taylor Dupuy argues that Scholze-Stix havemissed some key points of [Mochizuki, 2021a,b,c,d]. [Also see § 1.2.1.]

8

July 2020 Early versions of [Joshi, October 2020] are sent to Mochizuki and Scholze. [Scholze
kindly verifies the proof, but denies its role in Mochizuki’s work; perfectoid fields don’t
sit well with Anabelian minded Mochizuki and Hoshi and they dismiss it, contradicting
Mochizuki’s Key Principle of Inter-Universality [Mochizuki, 2021a, § I3, Pages 25–26]
even though my results provide the clearest evidence that there is an arithmetic Teichmuller
Theory of the sort Mochizuki claims. Such a theory is detailed by me in [Joshi, 2021a].]July 2020 Early version of a ‘Rosetta Stone’ fragment arising in my work and establishing
the connection between Mochizuki’s log-Links and Fargues-Fontaine Theory was sent
to Mochizuki. Mochizuki denies its relevance. [The relevant mathematics is detailed in[Joshi, 2023b] and [Joshi, 2023a].]August 2020 I posted [Joshi, 2020b] on the arxiv.
Oct 2020 Early version of the first of my arithmetic Teichmuller Theory papers [Joshi, 2021a]
is posted on the arxiv.
March 2021 Cancellation (by the organizers) of my invited talk in this June 2021 IUTT meeting.
March 2021 Mochizuki’s papers [Mochizuki, 2021a,b,c,d] are published. On the eve of the
publication Mochizuki posts [Mochizuki, 2022] arguing ‘change of logic.’ [Mochizuki
does not recognize my approach ([Joshi, October 2020], [Joshi, 2022]), which provides abetter way of demonstrating that his theory has many distinct objects (for natural reasons).]
Summer 2021 I give several Zoom lectures to Taylor Dupuy on my approach to Mochizuki’swork.
June 2021 During this meeting, Mochizuki and Hoshi (on Zoom/Slack) deny the role ofalgebraically closed perfectoid fields (i.e of geometric base-point datum) contradicting
Mochizuki’s Key Principle of Inter-Universality.
July 2021 [Scholze, 2021] is published.
Nov 2021 [Joshi, 2021b] is posted on the arxiv.
Nov 2021 David Roberts’ blogpost. [I agree with this article: Mochizuki has repeatedly citedthe analogy with P1 in the context [Mochizuki, 2021a,b,c,d]. But using this example iscompletely misleading (as David also notes) and does not even remotely represent the ΘLink in [Mochizuki, 2021a,b,c,d]. See [Joshi, 2023b, 2024b] for the precise construction.]
2022 [Saı̈di, 2022] is published (contains no discussion of [Scholze and Stix, 2018, Scholze,2021]).
Spring 2022 I give several lectures on Zoom to Minhyong Kim and Dinesh Thakur. [First draftof [Joshi, 2024b] is readied.]
Oct 2022 I post a new version of [Joshi, 2022] and discuss relationship between my results,[Mochizuki, 2021a,b,c,d], and [Scholze and Stix, 2018].
9

Nov 2022 Correspondence with Will Sawin (Nov 11–Nov 18) explaining my work. [It was
clear that he had a limited understanding of the relevant mathematics.] Scholze (Nov22–) argues on MathOverFlow that [Scholze and Stix, 2018, Remark 9] is still valid (see§ 1.2.4). Subsequently, Will Sawin backs Scholze’s assertion and dismisses my work.
Nov 2022 I respond with this mathematical explanation on David Roberts’ blog and in it I
clearly point out the issue of geometric base-points (which is missing in [Scholze and Stix,
2018]). No response is received from Scholze or Sawin.
March 2023 [Joshi, 2023b] is posted on the arxiv.
April 2023 In response to the [Scholze and Stix, 2018] claims about Mochizuki’s Corollary
3.12, I post this essay based on [Joshi, 2023b] outlining my ideas.
May 2023 [Joshi, 2023a] is posted on the arxiv. This describes an arithmetic Teichmuller
Theory of Number Fields (whose existence is asserted by Mochizuki) and I also establish
the geometric case of Mochizuki’s Corollary 3.12.
June 2023 I post this response to “Grouchy Expert’s” comments on David Roberts’ blogpost
regarding my work.
July 2023 Peter Woit declares victory for Scholze-Stix in this blogpost. I post a number ofcomments to his blogpost to remind Woit’s readers of the flaws in the Scholze-Stix report.
Concerning Will Sawin’s posted comments see § 1.2.4.
Jan 2024 [Joshi, 2024b] is posted on the arxiv.
March 2024 [Joshi, 2024a] is posted on the arxiv. Mochizuki posts his comments on my work.
Neither Mochizuki nor Scholze publicly recognize any of my work done so far but bothargue that [Joshi, 2024a] must be incorrect.
[Work in Progress: My work (in dimension one and genus one) so closely mirrors
Mochizuki’s Theory that if there are any counter examples, to [Joshi, 2024a], as Mochizuki
claims in his comments, then they will also apply to [Mochizuki, 2021d]. Since Mochizuki
is asserting these counter examples, it is clear to me that either he has a poor understandingof my work [Joshi, 2021a, 2023b,a, 2024b,a] or the main constructions of [Joshi, 2024b]
needs some minor corrections. Unfortunately, so far, Mochizuki has not shown willingness
to engage in any mathematical conversations with me, and hence it will take me sometime to tie up any remaining issues.]
May-June 2024 Work in Progress: Updated several preprints ([Joshi, 2019], [Joshi, 2020a],[Joshi, 2023a], [Joshi, 2024b]) on the arxiv. In updates to [Joshi, 2023a] and [Joshi, 2024b]
I provide additional clarifications about how I deal with local and global phenomena.
Meanwhile, Scholze and I are having a respectful and professional conversation (on going)as I work to clarify his questions; while I continue to wait for Mochizuki’s response to myemails.

10References
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M ATH . DEPARTMENT, U NIVERSITY OF A RIZONA , 617 N S ANTA R ITA , T UCSON 85721-0089, USA.
Email: kirti@math.arizona.edu

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