REPORT ON THE RECENT SERIESOF PREPRINTS BY K. JOSHI

Shinichi Mochizuki
March 2024

Contents:§1. General Overview§2. Local tilts and global arithmetic inequalities§3. Further discussion
Section 1: General Overview
Over the past few years, K. Joshi has released a series of preprints, culminatingin the following two recently released preprints:[CnstIII] K. Joshi, Construction of Arithmetic Teichmuller Spaces III: A ‘Rosetta Stone’
and a proof of Mochizuki’s Corollary 3.12 (January 25, 2024 version), preprint.
[CnstIV] K. Joshi, Construction of Arithmetic Teichmuller Spaces IV: Proof of the abcconjecture (March 10, 2024 version), preprint.
Since this series of preprints contains numerous assertions concerning inter-universal
Teichmüller theory, a theory exposed in the surveys [Alien], [EssLgc] [cf. also theoriginal papers [IUTchI-IV], [ExpEst]], I have been requested by various people to
make public my understanding concerning the content of this series of preprints.
In a word, my short answer to such requests may be summarized as follows:(ShtAns) although Joshi, in this series of preprints, makes references to and often
uses certain portions of the terminology and notation of inter-universal
Teichmüller theory, it is conspicuously obvious to any reader of thesepreprints who is equipped with a solid, rigorous understanding of the actual mathematical content of inter-universal Teichmüller theory that theauthor of this series of preprints is profoundly ignorant of the actual
mathematical content of inter-universal Teichmüller theory, and, in particular, that this series of preprints does not contain, at least from thepoint of view of the mathematics surrounding inter-universal Teichmüller
theory, any meaningful mathematical content whatsoever.
To my knowledge,(NotPb) none of these preprints has been published in an internationally recognizedmathematical journal.
Typeset by AMS-TEX

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SHINICHI MOCHIZUKI

Nevertheless, in accordance with the spirit of [i.e., as opposed to the strictly interpreted ethical duties implied by] the passage [namely, Article (6.)] of the subsection entitled “Responsibilities of authors” of the Code of Practice of the European
Mathematical Society [cf. [EMSCOP]] that I often quote — i.e.,
Mathematicians should not make public claims of potential new theorems
or the resolution of particular mathematical problems unless they are able
to provide full details in a timely manner.
— it should be recalled that(WrEx) it is of fundamental importance to the sound development of the ﬁeldof mathematics, especially from a historical point of view, to produce detailed, explicit, mathematically substantive, and readily accessible written expositions of the logical structure underlying publicly
articulated mathematical assertions,
i.e., as opposed to resorting to justiﬁcations of such mathematical assertions via
invocations of such mathematically meaningless — and indeed largely psychologically/socially/politically based — notions as the notion of “common sense” [cf. thediscussion of [EssLgc], §1.3, §1.5, §1.8]. The present report may be understood asa written exposition in the spirit of (WrEx).
In this context, it should be recalled [cf., e.g., [Rpt2023]] that the original
papers on inter-universal Teichmüller theory, i.e., [IUTchI-IV], as well as a sequel[ExpEst] to these original papers, were published a number of years ago in leading international journals. Moreover, there is by now a quite substantial body ofmathematicians who are not only thoroughly familiar with the original theory [i.e.,of [IUTchI-IV]], but have also been quite actively engaged for a number of years
now in research on further enhanced versions of inter-universal Teichmüller theory.
Over the past few years, and especially over the past few months, I consulted withmany of these researchers concerning the issue of what would be the optimal way
to respond to Joshi’s series of preprints. The present report may be understood,
to a substantial extent, as the product of these consultations. The reactions that I
met with during these consultations may be summarized as follows:(Reac1) First of all, there was an entirely unanimous consensus that Joshi’s
series of preprints was obviously mathematically meaningless, andthat it was obvious that he did not have any idea what he was talking
about.
(Reac2) In light of the blatant obviousness underlying (Reac1), many consultees
were not interested in investing the time and eﬀort necessary to discuss
Joshi’s series of preprints in detail and indeed strongly encouraged me tosimply ignore them as well.
(Reac3) On the other hand, in response to (Reac2), I emphasized [cf. the above
discussion surrounding (WrEx)] the importance of producing detailed, explicit, mathematically substantive, and readily accessible written expositions of the mathematical content underlying the unanimous reactionof researchers of the inter-universal Teichmüller theory community [cf.
(Reac1)].

REPORT ON JOSHI PREPRINT SERIES

3(Reac4) In the context of (Reac3), I would also emphasize another important aspect of (Reac3), namely, that the “exercise” of making the mathematicalcontent mentioned in (Reac3) explicit, in accordance with (WrEx), should
be understood as a valuable pedagogical tool, both for seasoned researchers and novices in inter-universal Teichmüller theory, for deepening
one’s understanding of the mathematics involved [cf. the discussion of[EssLgc], §1.6, §1.7, §1.8, §1.10, §3.1].
Finally, in the context of the theme of “computers and inter-universal Teichmüller theory” discussed in [EssLgc], §1.12, I should mention that I could not
help but notice the following stimulating interconnections between this theme of“computers and inter-universal Teichmüller theory” and Joshi’s series of preprints:(AI1) When browsing through Joshi’s series of preprints, i.e., whose content
consists of a sort of rough concatenation of various “fragments” of interuniversal Teichmüller theory that is nonetheless devoid of any substantive
mathematical understanding [cf. (ShtAns)], I could not help but be reminded of the so-called “hallucinations” produced by artiﬁcial intelligence algorithms, such as ChatGPT, i.e., which are synthesized precisely
by means of various mechanically searched contextual concatenations thatare entirely devoid of any genuine “human” understanding of the actual
content of the text involved.
(AI2) The observation of (AI1) suggests that in the future, it is quite possiblethat the production, or indeed mass production (!), of similar “pseudomathematical texts” by articial intelligence algorithms may become
more widespread and, in particular, pose a substantial threat to the sounddevelopment of the ﬁeld of mathematics by sewing the seeds of entirely unnecessary confusion in the worldwide mathematical community concerning
established mathematical theories.
(AI3) On the other hand, it is also quite possible that in the future, the “exercise” of producing suitable written expositions in response [cf. (Reac4)] —which can often require a quite substantial investment of time and eﬀort ofthe sort that many researchers in the ﬁeld, who are quite busy with theirown research projects, are simply not willing to make [cf. (Reac2)] —could be performed, as least partially, via artiﬁcial intelligence algorithms
for the automated parsing of texts, so that researchers would not needto invest the time and eﬀort necessary to do this themselves. Indeed, one
important aspect of the valuable pedagogical nature of the “exercise”
of producing suitable written expositions in response to manuscripts suchas [CnstIII], [CnstIV] [cf. (Reac4)] lies precisely in the fact this sort of“exercise” may be thought of as a sort of initial, preparatory step, relative
to the ultimate goal of designing such automated text parsing algorithms.
(AI4) In the context of the issue of substantial investments of time and effort [cf. (Reac2), (AI3)], I should also mention that I was saddened, asI browsed through Joshi’s series of preprints, to contemplate the quiteconsiderable investment of time and eﬀort that Joshi must have put, presumably without any signiﬁcant use of artiﬁcial intelligence algorithms,
into writing this series of preprints.

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In the context of (AI4), it is also interesting to note that during the past few years
during which Joshi wrote this series of preprints,(TmEﬀ) quite a number of mathematicians were able to study and achieve agenuine mathematical understanding of inter-universal Teichmüller theory
in the usual, conventional way [often with the help of [Alien], [EssLgc]],
while expending surely no more than a tiny fraction of the time and eﬀort
that Joshi must have put into writing this series of preprints during thesame time period.
Section 2: Local tilts and global arithmetic inequalities
Joshi’s central assertion concerning the mathematical validity of inter-universal
Teichmüller theory [cf. [CnstIII], §1] consists of reaﬃrming the assertions of the
RCS, or “redundant copies school”, concerning the essential logical structureof inter-universal Teichmüller theory. In particular, Joshi denies the mathematical validity of inter-universal Teichmüller theory on the basis of the assertionsof the RCS. The confusion surrounding these assertions of the RCS is explained
in detail throughout [EssLgc], especially in [EssLgc], §3. Joshi does not, in [CnstIII], [CnstIV], add any new essentially new content to the assertions of the RCS.
As discussed throughout [EssLgc], these assertions of the RCS amount to a very
elementary and meaningless misunderstanding that corresponds, in a very
precise fashion, to any one of the following well-known elementary examples:(RC-Ex1) the “contradiction T = T −1 ” concerning the standard coordinate on
the projective line that arises if one arbitrarily identiﬁes the two copies of
the aﬃne line that appear in the usual gluing construction of the projective line, or, alternatively, the northern and southern hemispheres on the
Riemann sphere [cf. the discussion of [EssLgc], Example 2.4.7, for moredetails];(RC-Ex2) the “contradiction” to the eﬀect that the multiplicative map given byraising to the N -th power, for N an integer ≥ 2, on an integral domainof characteristic zero, is a ring homomorphism if one arbitrarily identiﬁes
the domain and codomain of this multiplicative map [cf. the discussion of[EssLgc], Example 2.4.8, for more details];(RC-Ex3) the “contradiction” — i.e., to the eﬀect that a non-holomorphic Teichmüller map is holomorphic — that arises from arbitrarily identifying
the two copies of the complex plane [regarded, say, as a Riemann surface]obtained by considering the distinct holomorphic structures on a single
copy of the Euclidean plane that arise from nontrivial Teichmüller deformations — i.e., of the form
Λ : C  x + iy → λ · x + iy ∈ C,
for some real number λ > 1 — as in [EssLgc], Example 3.3.1 [cf. also
[EssLgc], Example 3.5.2, (iii)].
In this context, it is perhaps somewhat ironic to observe that the very deﬁnition of the notion of a tilt, which plays a fundamental role in [CnstIII], [CnstIV],REPORT ON JOSHI PREPRINT SERIES

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becomes invalid/self-contradictory if one applies the RCS approach of arbitrarily identifying isomorphic objects in some nontrivial system to the distinctly
labeled copies of “Ok ” that appear in the inverse limit
Ok 

=

lim
←− Ok

used to deﬁne the tilt. Indeed, this approach which would imply that the above
inverse limit is in fact given by the set{x ∈ Ok | xp = x}
— i.e., a set that is manifestly completely diﬀerent from the set “Ok ” obtained bytaking the inverse limit as in the deﬁnition of the tilt [in the conventional way!]with the distinct labels intact [cf. the discussion of [EssLgc], Example 3.5.3, (v), formore details, e.g., concerning the notation]. When viewed relative to the analogy
constituted by the present discussion of applying the RCS approach to the deﬁnitionof a tilt, Joshi’s series of preprints may be understood as(EpEx) a sort of epic exercise in mental gymnastics devoted to showing that atheory based on the “RCS approach-motivated deﬁnition” of a tilt
by some sort of inverse limit of the formlim←− Okn— where the {kn }n∈N is a collection of p-adic ﬁelds indexed by the nonnegative integers such that, for distinct nonnegative integers n, m, thetopological ﬁelds kn and km are non-isomorphic — in fact satisﬁes essentially the same properties as the properties asserted in the theory oftilts relative to the conventional deﬁnition of “Ok ”.
Of course, it will be immediately obvious to any mathematician who has a genuine
mathematical understanding of the conventional theory of tilts that such an epic
exercise is nothing more than a completely meaningless exercise in futility,
and that it would be much more meaningful and productive [cf. (TmEﬀ)] for theauthor of such a series of preprints to devote his time and energy to reexamining why
he arrived at the [patently erroneous!] conclusion that the RCS approach — i.e., tothe eﬀect that arbitrarily identifying isomorphic objects in some nontrivial
system has no substantive mathematical/logical eﬀect on the mathematics
involved — is valid in the ﬁrst place.
Of the examples (RC-Ex1), (RC-Ex2), (RC-Ex3) cited above, (RC-Ex3) is
especially closely related to the point of view taken in [CnstIII], §1. Indeed, in [CnstIII], §1, refers to the assertions of the RCS as the issue of showing the “pluralityof arithmetic holomorphic structures” in inter-universal Teichmüller theory.
Relative to this terminology, the central assertions of [CnstIII] may be summarized
as follows:(Js1) The “plurality of arithmetic holomorphic structures” in inter-universal
Teichmüller theory is not proven in [IUTchI-IV].
(Js2) As a consequence of (Js1), the proofs of the main results of [IUTchI-IV],such as [IUTchIII], Corollary 3.12, are incorrect.

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SHINICHI MOCHIZUKI(Js3) A correct proof of [IUTchIII], Corollary 3.12, is given in [CnstIII] by
applying the p-adic theory of tilts/untilts/Fontaine-Fargues curves.
In a word, all three of these main assertions (Js1), (Js2), (Js3) are mathematically false, as we explain in substantial detail in the discussion below.
First of all, with regard to (Js1), there is a confusion in the use of the phrase“plurality of arithmetic holomorphic structures”, especially, in the context of theanalogy with Teichmüller deformations of Riemann surfaces, relative to the two
related [but by no means equivalent!] issues of(Js1-1) whether or not the domain and codomain Riemann surfaces in a Teichmüller deformation are required to be non-isomorphic;(Js1-2) whether or not, when considering holomorphic isomorphisms between
the domain and codomain Riemann surfaces as in (Js1-1), one imposes acertain compatibility condition, namely, that one requires such a holomorphic isomorphism to induce the same map on underlying topologicalsurfaces as the Teichmüller deformation under consideration.
That is to say, even in the case of the most classical and fundamental type of
Teichmüller deformation discussed in (RC-Ex3), the non-isomorphicity conditionof (Js1-1) is not satisﬁed. Nevertheless, because in this context, one typically only
considers holomorphic isomorphisms that are subject to the compatibility conditionof (Js1-2), one conventionally regards, in classical complex Teichmüller theory, thevarious holomorphic structures arising from Teichmüller deformations as in (RCEx3) as distinct, hence, in particular, as constituting a “plurality” of holomorphic
structures.
As explained in detail in [EssLgc], Example 3.5.3, (vi) [cf., especially, thediscussion of (RCRS1), (RCRS2), (RCRS3), (RCΘ1), (RCΘ2), (RCΘ3)], there isa very precise correspondence between this situation for Riemann surfaces and thesituation surrounding the domain and codomain [Θ±ell NF-]Hodge theaters of the Θlink in inter-universal Teichmüller theory. That is to say, relative to the analogous
compatibility condition [cf. (Js1-2)], the “plurality” of arithmetic holomorphic
structures is valid in inter-universal Teichmüller theory and indeed is an immediate
consequence of the deﬁnition of a ring [i.e., in essence, the property observed ineﬀect in (RC-Ex2) that the multiplicative map given by raising to the N -th power,
for N an integer ≥ 2, on an integral domain of characteristic zero, is not a ringhomomorphism] — cf. the discussion surrounding [EssLgc], Example 3.5.3, (vi),(RCRS1), (RCRS2), (RCΘ1), (RCΘ2).
Moreover, by considering tilts, one can indeed construct a situation [cf. [EssLgc],
Example 3.5.3, (vi), (RCRS3), (RCΘ3)] that is roughly reminiscent of the situation
surrounding the domain and codomain of the Θ-link in inter-universal Teichmüller
theory [cf. [EssLgc], Example 3.5.3, (iv), (TltSim)], and which, moreover, satisﬁes the non-isomorphicity condition of (Js1-1), but which is, however, completely
useless from the point of constructing a theory that is structurally similar tointer-universal Teichmüller theory, on account of the numerous and quite fundamental structural diﬀerences between this tilt-based construction and thecorresponding constructions in inter-universal Teichmüller theory [cf. [EssLgc], Example 3.5.3, (iv), (vii), (TltDf1), (TltDf2), (TltDf3), (TltDf4), (TltDf5), (TltDf6),(TltDf7)].

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With regard to (Js2), we observe [cf. [EssLgc], Example 3.5.3, (vi), (RCΘlg)]
that the “plurality” of arithmetic holomorphic structures discussed above [i.e., relative to a compatibility condition analogous to the condition of (Js1-2)], which is valid
in inter-universal Teichmüller theory and indeed is an immediate consequence of thedeﬁnition of a ring, is in fact never logically applied in the development or proofsof the main results [such as, for instance, [IUTchIII], Theorem 3.11, or [IUTchIII],
Corollary 3.12] of inter-universal Teichmüller theory. That is to say, even if one
takes the position that one does not know whether or not the “plurality” of arithmetic holomorphic structures in this sense holds, there is no eﬀect whatsoever on
the essential logical structure of the development or proofs of the main results [suchas, for instance, [IUTchIII], Theorem 3.11, or [IUTchIII], Corollary 3.12] of interuniversal Teichmüller theory. Put another way, the only eﬀect of taking such a
position is that it implies that there is a possibility that the theory involves a sortof “overkill”, i.e., that one is possibly doing more than is in fact necessary in orderto prove the desired results.
Next, we consider (Js3). Tracing back through the proofs of the main results of[CnstIII], [CnstIV], starting with the ﬁnal conclusion concerning the ABC/Szpiro
inequality in [CnstIV], Theorem 7.2.1, one ﬁnds the following chain of main implications:[CnstIV], Theorem 7.2.1⇐=[CnstIV], Theorem 7.1.1⇐=[CnstIV], Theorem 6.1.1⇐=[CnstIV], Theorem 6.10.1⇐=[CnstIII], Corollary 9.11.1.1⇐=[CnstIII], Theorem 9.11.1[where we note that it is not clear whether or not the number “9.11...” assigned by
the author to these key results in [CnstIII] was purely coincidental or a consequenceof some sort of sense of rhetoric or humor that lies beyond my understanding].
In a word, the central and quite fundamental problem with the proof of [CnstIII], Theorem 9.11.1, which in fact underlies all of the main results of [CnstIII],[CnstIV] lies in the fact that
(LcGlIq) the argument given to prove [CnstIII], Theorem 9.11.1 [which is in fact
entirely similar to the argument used in the ﬁnal portion of the proof of[CnstIII], Theorem 7.3.1, or the proof of [CnstIII], Theorem 9.9.1] reduces
the proof of a global inequality to the veriﬁcation of local inequalities[i.e., at each prime of a number ﬁeld], which are then summed over inorder to obtain the global inequality.
Here, we recall that it is well-known — and indeed can be easily veriﬁed by considering abc triples of the form (1, pn , 1+pn ), for p a prime number and n an arbitrarily
large positive integer — that
(EssGlIq) any essentially global inequality — i.e., such as the ABC/Szpiro
inequalities or [IUTchIII], Corollary 3.12 — can never be obtained in thisway, i.e., as a result of summing up local inequalities at each prime of anumber ﬁeld.

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Indeed, it follows immediately from the computations in the proofs of [IUTchIV],
Theorem 1.10 [which are summarized in Step (viii) of this proof]; [IUTchIV], Corollary 2.2, (ii), as well as the corresponding portions of the proofs of [ExpEst], Theorem 5.1; [ExpEst], Corollary 5.2, that if· one collects the various terms in the log-volumes that appear in the global
inequality of [IUTchIII], Corollary 3.12, and then multiplies by a suitable
positive real number, and, moreover,· one restricts one’s attention, for simplicity, to abc triples/elliptic curvesdeﬁned over a number ﬁeld which is a quadratic imaginary ﬁeld,
then the local portion of this global log-volume inequality at a nonarchimedean
valuation v of residue characteristic pv of the number ﬁeld under consideration atwhich the elliptic curve under consideration has bad multiplicative reduction is of
the form
α · log(qv ) ≤ β— where we write qv for the q-parameter of the elliptic curve under consideration atv and log(qv ) for the unique positive rational number λ such that pλv /qv is a pv -adic
unit; α ∈ A for some compact subset A of the set of positive real numbers that is
independent of the moduli of the elliptic curve under consideration; β ∈ B for somecompact subset B of the set of real numbers that is independent of the moduli ofthe elliptic curve under consideration. In particular, if one considers abc triples ofthe form (1, pn , 1 + pn ) at a nonarchimedean valuation v such that pv = p, then
α · log(qv ) is bounded below by a positive real multiple γ · n of n [where the positive
real number γ is independent of n], hence → +∞ as n → +∞, thus contradicting
the inequality of the above display.
More generally, at least as far as I could see, the only nontrivial “global” objects
that appear in the theory developed in [CnstIII], [CnstIV] consist of collections oflocal objects, i.e., one object at each prime of a number ﬁeld. Put another way,
(EssGl) no essentially global phenomena appear anywhere in the theory of[CnstIII], [CnstIV].
In this context, we recall that, by contrast, the essentially global nature of
the arithmetic line bundles treated in [IUTchI-IV] may be seen in the subtle correspondence between additive and multiplicative representations of arithmetic line
bundles, which involves, in an essential way, the global ring structure of a number ﬁeld and, in particular, the way in which the relations arising from this global
ring structure necessarily involve all of the primes of the number ﬁeld [cf. [AbsTopIII], Remark 5.10.2, (iv); [IUTchII], Remark 4.11.2, (iii); [IUTchIII], Remark
3.6.2, (i)]. This correspondence between additive and multiplicative structures isclosely related, in inter-universal Teichmüller theory, to the delicate interplay between Frobenius-like and étale-like structures, which is in turn closely related
to the various coricity, symmetry, and commutativity properties surrounding
the log-theta-lattice, all of which play a fundamental role in the essential logicalstructure of inter-universal Teichmüller theory [cf. the discussion of [EssLgc], §3.1,§3.2, §3.3, §3.4; [EssLgc], Example 3.2.2].
Here, we note that although the terms “Frobenius-like” and “étale-like” appear
in [CnstIII], §8.4, the discussion involving these terms in [CnstIII], §8.4, neverREPORT ON JOSHI PREPRINT SERIES

9

mentions these fundamental coricity, symmetry, and commutativity properties and,in particular, bespeaks the apparently quite profound ignorance, on the part of Joshi,of the fundamental role that such properties play in the essential logical structure ofinter-universal Teichmüller theory, especially in the proof of [IUTchIII], Corollary
3.12 [cf. the discussion of [EssLgc], §3.10, §3.11]. Indeed, these properties are
entirely irrelevant to the argument given in the proof of [CnstIII], Theorem 9.11.1.
Finally, in passing, we note yet another fundamental problem with the statement [i.e., the chain of inequalities in the ﬁrst display of the statement] and proofof [CnstIII], Theorem 9.11.1:(HllVl) the inequality concerning the volume of the region obtained after passing to the hull is obtained as a consequence of a stronger inequality
concerning the volume of the region that arises before passing to the hull.
Here, the issue of ﬁnding a direct contradiction to such a stronger inequality issomewhat subtle and, at the time of writing, at least to my knowledge, strictly
speaking, inconclusive. On the other hand, from the point of view of the essential
logical structure of the argument in [IUTchIII] that is actually used to prove theinequality of [IUTchIII], Corollary 3.12, we recall that this inequality cannot be
concluded until one completes the closed loop that is obtained by, in particular,passing to the hull [cf. the discussion of [EssLgc], §3.10, (Stp4), (Stp5), (Stp6),(Stp7), (Stp8)]. Moreover, if one is willing to ignore certain technicalities at thearchimedean primes and the nonarchimedean primes that divide the rational prime
2, then a stronger inequality as in (HllVl) contradicts the existence of certain wellknown types of sequences of abc triples due to Masser and others [cf. the discussionof [IUTchIV], Remark 1.10.5, (ii)].
That is to say, a stronger inequality as in (HllVl) would imply that, in Step(v) of the proof of [IUTchIV], Theorem 1.10, one could apply the ﬁrst displayed inequality, of [IUTchIV], Proposition 1.4, (iii) — i.e., rather than the second displayed
inequality, of [IUTchIV], Proposition 1.4, (iii), as is actually done — which would
mean that one could replace the terms that are linear in l in the inequalities of the
ﬁnal display in the statement of [IUTchIV], Theorem 1.10, by terms that are linear
in log(l). Combining this strengthening of the inequalities of [IUTchIV], Theorem
1.10, with a suitably modiﬁed version of the estimates of [IUTchIV], Corollary 2.2,(ii), would yield a contradiction to certain well-known types of sequences of abc
triples due to Masser and others, that is to say, if one is willing to ignore certain
technicalities at the archimedean primes and the nonarchimedean primes that divide the rational prime 2. Here, we note in passing that these technicalities at
the nonarchimedean primes that divide the rational prime 2 were in fact resolved
in [ExpEst], and that it is strongly expected that the remaining technicalities at
the archimedean primes will be resolved in an enhanced version of inter-universal
Teichmüller theory that is currently under development.
Thus, in summary, although, strictly speaking, the existence of such technicalities means that, at a completely rigorous level, the stronger inequality of (HllVl)does not in fact yield an immediate contradiction, it is nevertheless strongly expected,
especially in light of work in progress on various enhanced versions of inter-universal
Teichmüller theory, that such technicalities are inessential and can be overcome,i.e., which would imply that the stronger inequality of (HllVl) does indeed imply anumerical contradiction in an entirely rigorous sense.

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Section 3: Further discussion
The discussion of §2 may be understood as an exposition of my understanding,at the time of writing, of the mathematical content underlying the general overview
given in §1. On the other hand, since my understanding of the p-adic theory oftilts/untilts/Fontaine-Fargues curves [which lies beyond my range of specialty] isquite limited and superﬁcial, I would certainly be open to further discussions concerning this theory, in the context of the mathematical content exposed in §2, withexperts in the ﬁeld. More generally, I would like to take this opportunity to state
explicitly that I would be open to — and indeed, in many [though not necessarily
all!] cases, welcome — the possibility of conducting further discussions concerning the content of the present report with mathematicians who belong to at least
one of the following categories:(MthCat1) mathematicians who are experts in inter-universal Teichmüller theory or related anabelian geometry, in the sense that they have published paper(s) proving nontrivial result(s) in these ﬁelds in internationallyrecognized mathematical journal(s);(MthCat2) mathematicians who are experts in the p-adic theory surrounding tilts/
untilts/Fontaine-Fargues curves, again in the sense that they havepublished paper(s) proving nontrivial result(s) in these ﬁelds in internationally recognized mathematical journal(s);(MthCat3) senior mathematicians in arithmetic geometry, with the rank of full
professor at a world-class center of research, who may not belong to(MthCat1) or (MthCat2), but who are nonetheless widely recognized asleading researchers in their ﬁeld of specialty.
Here, it should be noted that Joshi does not belong to any of these categories.
Indeed, the fact that he does not belong to (MthCat1) or (MthCat2) is closely
related to the numerous issues discussed in the present report. Moreover, since
the technical core of Joshi’s series of preprints revolves around the p-adic theoryof tilts/untilts/Fontaine-Fargues curves [cf. (MthCat2), i.e., a class of mathematicians to which neither Joshi nor I belong!], it seems essentially self-evident that thenext natural step, with regard to the possibility of pursuing further discussions concerning the mathematical content of Joshi’s series of preprints, must involve some
sort of interaction between (MthCat2) mathematicians and myself [or perhaps other(MthCat1) mathematicians]. As a result, over the past few months, I have madequite substantial eﬀorts to contact (MthCat2) mathematicians who were aware of
Joshi’s series of preprints to obtain information from them concerning their understanding of the mathematical content of this series of preprints. The only responses,
however, that I was able to extract from such (MthCat2) mathematicians were to
the eﬀect that
(MC2Rsp) it was obvious from their point of view, i.e., from the point of view of
the theory surrounding tilts/untilts/Fontaine-Fargues curves, that Joshi’s
series of preprints did not contain any meaningful mathematicalcontent, and that, since they were busy with their own research projects,
they were simply not interested in spending any more time or eﬀort examining/discussing Joshi’s series of preprints.

REPORT ON JOSHI PREPRINT SERIES

11Interestingly enough, the mathematical content that was pointed out by (MthCat2)
mathematicians as justiﬁcation for the evaluation of (MC2Rsp) was essentially identical to the mathematical content discussed above in (EssGlIq), (EssGl) [i.e., whicharose independently from my own investigations, as well as, again independently,
from the investigations of other (MthCat1) mathematicians], namely, the observation that
(MC2EssGl) the theory developed in Joshi’s series of preprints essentially only concerns “global data” that consists solely of collections of local data, where
the “local data” was not particularly novel, but rather based on essentially known mathematical constructions, i.e., in short that there was no
meaningful essentially global content to the theory.
Finally, it should be noted that previous [and indeed quite substantial] attempts,
both on my part and on the part of other (MthCat1) mathematicians, to engage
in discussions with Joshi for the purpose of explaining aspects of inter-universal
Teichmüller theory concerning which he appeared to be confused failed to result ina productive dialogue.

Bibliography[EMSCOP] European Mathematical Society, Code of Practice, approved by the EMS Executive Committee (October 2012), available at the following URL:
https://euromathsoc.org/code-of-practice[AbsTopIII] S. Mochizuki, Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms, J. Math. Sci. Univ. Tokyo 22 (2015), pp. 939-1156.
[IUTchI] S. Mochizuki, Inter-universal Teichmüller Theory I: Construction of Hodge
Theaters, Publ. Res. Inst. Math. Sci. 57 (2021), pp. 3-207.
[IUTchII] S. Mochizuki, Inter-universal Teichmüller Theory II: Hodge-Arakelov-theoretic
Evaluation, Publ. Res. Inst. Math. Sci. 57 (2021), pp. 209-401.
[IUTchIII] S. Mochizuki, Inter-universal Teichmüller Theory III: Canonical Splittings of
the Log-theta-lattice, Publ. Res. Inst. Math. Sci. 57 (2021), pp. 403-626.
[IUTchIV] S. Mochizuki, Inter-universal Teichmüller Theory IV: Log-volume Computations and Set-theoretic Foundations, Publ. Res. Inst. Math. Sci. 57 (2021), pp.
627-723.
[ExpEst] S. Mochizuki, I. Fesenko, Y. Hoshi, A. Minamide, W. Porowski, Explicit Estimates in Inter-universal Teichmüller Theory, Kodai Math. J. 45 (2022), pp.
175-236.
[Rpt2023] S. Mochizuki, Brief Report on the Current Situation Surrounding Inter-universal
Teichml̈ler Theory (IUT), available at the following URL:
http://www.kurims.kyoto-u.ac.jp/~motizuki/2023-08%20Brief%20report%20on%20the%20current%20situation%20surrounding%20inter-universal%20Teichmuller%20theory%20(IUT).pdf[Alien] S. Mochizuki, The Mathematics of Mutually Alien Copies: from Gaussian
Integrals to Inter-universal Teichmüller Theory, Inter-universal Teichmuller

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SHINICHI MOCHIZUKI

Theory Summit 2016, RIMS Kōkyūroku Bessatsu B84, Res. Inst. Math. Sci.
(RIMS), Kyoto (2021), pp. 23-192.
[EssLgc] S. Mochizuki, On the essential logical structure of inter-universal Teichmüller
theory in terms of logical AND “∧”/logical OR “∨” relations: Report on the occasion of the publication of the four main papers on inter-universal Teichmüller
theory, RIMS Preprint 1968 (November 2022); updated version available at thefollowing URL:
http://www.kurims.kyoto-u.ac.jp/~motizuki/Essential%20Logical%20Structure%20of%20Inter-universal%20Teichmuller%20Theory.pdf