9b. Object identity: the real-line diagram and the hull

The question in plain language. The Scholze--Stix (SS) objection compares numbers in several distinct real-line copies. IUT III also forms a containing set of possible outputs by taking a hull. A change of numerical coordinates and an inclusion of sets are different operations. The fact that the hull is drawn in IUT III but not in SS's hexagon does not tell us whether the hexagon is required elsewhere in the proof, or whether the specific native \(q\)-value is bounded by the hull's output.

The logic without IUT vocabulary. If \(A\subseteq B\) and every number in \(B\) is at most \(-T\), a separately supplied number \(q_0\) is bounded only after a further premise, such as \(q_0\in B\). For example, \(A=(-\infty,-2]\), \(B=(-\infty,-1]\), \(T=1\), and \(q_0=-1/2\) satisfy the strict inclusion and output bound, but \(q_0\notin B\). This is a countermodel to the unlicensed inference from set inclusion alone, not a counterexample to IUT: the latter may supply other hypotheses. The critical question is which source-stated operation connects its native \(q\)-value to the bounded output.

This is a source-audited working ledger, not a formal model of either party's entire argument. It refines 09's provisional M1--M10 ledger. Citations refer to the hosted versions in the source register, not an unavailable 2018 edition of IUT III. The precise Step-(xi) dispute remains DISPUTED; the checked drawings, elementary calculations, and reported conjectures have different statuses.

Graduate-level walkthrough (intuition only)

To read SS's diagram, assign names to six real-line nodes before trying to commute its paths. If two proposed routes \(f,g:A\to B\) are supposed to agree but one acts like \(f(x)=j^2x\) and the other like \(g(x)=x\), then at \(j=2\), \(x=1\) they give \(4\) and \(1\). This toy calculation shows the force of a compatibility condition; it does not establish that both routes in the real IUT argument have these types or must agree. SS discuss the \(j^2\) factor in their text; it is not a labeled arrow in their figure.

Now read IUT III's hull on a different diagram. Inclusion \(U\subseteq\overline U\) compares regions. With a compatible monotone log-volume, an upper bound on the hull can bound eligible outputs from \(U\); inclusion alone cannot place the separately fixed input value \(r=-|\log q|\) under that bound. Remark 3.9.5 (Ob8)/(Ob9) gives a vertical bijection of hull log-volumes, so the real question is where a choice-compatible native-value comparison enters, not whether the paper has no vertical map.

Optional background, not a model of the IUT hull: isomorphisms and ordinary convex hulls on Wikipedia. An IUT holomorphic hull is not an ordinary convex hull; for source-located arrows consult the table below and the paired dispute.

One key for reading every arrow

Mark or kind	What it can mean	What it does not establish alone
=	Literal equality in a specified context, or an equality proposed by a diagram; those are different claims	Identity of earlier structures just because their labels look alike
\(\cong\)	A specified isomorphism of specified structures	Equality of elements in two numerical copies without that map and its normalization
Poly-isomorphism / correspondence	A family or relation of allowed transports between structured data	A unique real-valued map, or preservation of every operation not named by the source
\(U\subseteq\overline U\)	Inclusion in a hull, generally non-invertible	That a particular external value belongs to the bounded output set
\(x\in A\)	A membership proposition needing a proof	An arrow that is justified by drawing a path near \(A\)

The IDs below have source prefixes. An SS- real line, an I- paper object, and an R- report model are not definitionally the same object. UNKNOWN means this audit has no checked source-level map with the claimed type, not that such a map cannot exist.

SS's six nodes: their diagram, not IUT III's hull

SS, section 2.2, PDF pp. 9--10 introduces three kinds of ordered real line in two columns. Their diagram groups the indexed concrete \(\Theta\) lines into one displayed family node, not a single unindexed line. Subscripts in the following table were checked against the rendered PDF image; \(\ell^\star\) is our notation for its raised star-like mark.

ID	Notation in the SS figure	Role in SS's description	Attached to
SS-A	\(\mathbb R_{\odot,\Theta}\)	Abstract \(\Theta\)-pilot line	\(\Theta\) side, \(HT_1\)
SS-B	\(\mathbb R_{\odot,q}\)	Abstract \(q\)-pilot line	\(q\) side, \(HT_2\)
SS-C_j	\((\mathbb R_{\odot_c,\Theta_j})_{j=1,\ldots,\ell^\star}\)	Family of concrete \(\Theta\)-component lines	\(\Theta\) side, indexed by \(j\)
SS-D	\(\mathbb R_{\odot_c,q}\)	Concrete \(q\)-pilot line	\(q\) side
SS-E	\(\mathbb R_\Theta\)	Arithmetic-degree line	\(\Theta\) side
SS-F	\(\mathbb R_q\)	Arithmetic-degree line	\(q\) side

Here is a topological redrawing in our IDs, with the arrowheads of SS's unnumbered display on PDF p. 10. The four diagonals are unlabelled in the source; downward arrowheads point along the slanted edges. This is not a facsimile of the PDF.

                SS-A --[Theta-link, isomorphism]--> SS-B
              /                                       \
             v                                         v
          SS-C_j                                     SS-D
             \                                         /
              v                                       v
                SS-E ----------[=]---------------> SS-F

Edge	What the figure actually prints	What an agent must check next
SS-A -> SS-B	Top \(\Theta\)-link with isomorphism sign	Whether this numerical real-line isomorphism is supplied, with the required normalization, by IUT's prime-strip poly-isomorphism
SS-A -> SS-C_j, SS-B -> SS-D	Two directed, unlabelled upper diagonals	The exact abstract-to-concrete maps and their effect on degrees for each \(j\)
SS-C_j -> SS-E, SS-D -> SS-F	Two directed, unlabelled lower diagonals	How the indexed family is assembled/normalized into one degree line
SS-E -> SS-F	Bottom arrow labelled =	Whether the asserted identification of the two previously distinguished copies is licensed by the IUT construction

No arrow is labelled \(j^2\) or "hull" in the printed hexagon. SS's prose, PDF pp. 9--10, says that encoding the \(j\)-th concrete \(\Theta\) degree requires inserting a \(j^2\) rescaling somewhere on the left; equation (1.5), PDF p. 4 uses this factor. The text argues that enforcing all the proposed identifications would then make the loop inconsistent or lose the useful inequality. The figure does not specify which left comparison is rescaled. Do not put \(j^2\) on a guessed edge. Comparing paths for a particular \(j\) requires selecting that component and the aggregation/normalization maps; calling the entire family node one ordered real line would conceal this choice.

IUT III's distinct, source-named operations

These rows describe what the hosted IUT III, PDF pp. 181--184 states in Step (xi-a)--(xi-f). "Checked" means the specific notation/statement was read; it does not certify the disputed inference or the paper's cited earlier lemmas.

Edge and distinct objects	Kind; paper locator	Source-stated content	Separate obligation
I-1: \(\Theta\)-pilot at \((0,0)\) to \(q\)-pilot at \((1,0)\)	Prime-strip poly-isomorphism; Def. 3.8(i--ii), pp. 112--113; (xi-a), p. 181	Pilot objects correspond across the link	A log-volume identity across theaters is not stated by this pilot correspondence alone
I-2: multiradial representations at \((0,\circ)\) and \((1,\circ)\)	Different permutation-symmetry poly-isomorphism; Thm. 3.11(i), pp. 153--155; (xi-b), pp. 181--182	Transports collections of possible representation outputs	Which admissible choices and numerical evaluation survive this transport?
I-3: \(q\)-pilot's representation among \({}^{1,\circ}\mathcal U^{\mathbb Q}\) possibilities to \({}^{1,\circ}\mathcal U\)	Linked via prime-strip isomorphisms; (xi-c), p. 182	Related possibilities in the \((1,\circ)\) column	No particular valued-region map or log-volume law was extracted from this sentence
I-4: \({}^{1,\circ}\mathcal U\) to \({}^{1,\circ}\overline{\mathcal U}\)	Hull inclusion; Rem. 3.9.5(i--ii), p. 127; (xi-c), p. 182	The paper visibly writes \({}^{1,\circ}\overline{\mathcal U}\supseteq{}^{1,\circ}\mathcal U\); the overline disappears from plain PDF text extraction	Inclusion is not a bijective real-line map and by itself says nothing about the native \(q\)-value
I-5: the overlined hull to its normalized determinant and an output half-line	Determinant/log-volume; Rem. 3.9.5(vii), (Ob1)--(Ob5), pp. 131--134; (xi-d), p. 183	A rank-one output is formed and a half-line $\mathbb R_{\le-	\log(\Theta)
I-5v: hull log-volumes on the two sides of a vertical log-link, after a shift	Log-Kummer comparison and a natural bijection of hull log-volumes via realified semi-simplification; Rem. 3.9.5(vii), (Ob8), PDF pp. 137--138, and (Ob9), pp. 138--139; cited in (xi-d), p. 183	These statements give a meaningful vertical comparison even with log-link iterates; (Ob9-1) treats determinant and iterate indeterminacies as compatibility conditions	This bijection does not itself select the native \((1,0)\) \(q\) value as a log-volume of a particular \((1,\circ)\) output hull, or give an (Ind1)--(Ind3)-uniform bound for it
I-6: the \((1,0)\) \(q\)-pilot to its native number $-	\log(q)	$	\(q\)-pilot log-volume; Cor. 3.12, p. 174; (xi-d), p. 183
I-7: native $-	\log(q)	$ to the output half-line	(xi-e)--(xi-f), p. 184: values described as linked via prime-strip isomorphisms, then membership asserted

Theorem 3.11(ii), PDF pp. 155--156, states compatibility for particular vertical Kummer packets; its (Ind3) includes only upper semi-compatibility as a vertical label varies. (Ob8)/(Ob9) are not absent: I-5v records their vertical comparability and bijection. The bounded source check did not find in them the additional selection and numerical law carrying I-6 through I-5v to I-7 for the allowed choices. Claiming that no passage anywhere supplies this link would go beyond this audit.

Crosswalk: which alleged equalities are still questions?

Tempting match	What has actually been checked	What would make the match usable
SS-A -> SS-B and I-1	The labels describe the corresponding pilots; SS draws a numeric isomorphism, IUT specifies a prime-strip poly-isomorphism	Type a chosen map on the particular numerical copies, including its preservation or scaling law
SS-C_j -> SS-E and I-2/I-5	The \(j^2\) rescaling appears in SS's prose; IUT additionally forms an output hull and determinant	Match the actual \(j\)-indexed normalization and verify whether the paper's comparison must factor through SS's six-node loop
SS-E = SS-F and I-5v/I-6	Equality is printed in SS's drawing; IUT's (Ob9) instead states a vertical bijection of hull log-volumes, while Step (xi-d) separately produces the native input	Identify an authorized common real copy, select the native value among admissible outputs, and check (xi-e)'s relationship before (xi-f)'s membership
I-4 and an SS diagram edge	The overlined hull inclusion is visible in IUT III; SS draws no corresponding inclusion	Establish which SS comparison, if any, the hull replaces or bypasses, without assuming the desired bound
R-1: Project LANA's \(\eta_q,\eta^{\mathrm{anab}}_S:R_{\mathrm{val}}\to R_{\mathrm{ss}}\) and I-7	The interim report, §9.2, equation (9-1), PDF p. 46 proposes equality of these suitably identified maps for admissible \(S\); §10.5, p. 49 says no proof is known to its team	Prove the map compatibility and its passage through reconstructed membership, output bound, and shared normalization; the report's maps are a later model, not named objects of Step (xi)

Mochizuki's 2018 comments, (C12)--(C14), PDF pp. 3--4 call the all-scalar-identification assumption (Lin) and reject it for indeterminacy-subject log-volumes. (Lin) is his label for SS's alleged assumption, not a term printed by SS. His separate planar-region analogy, (LVEx), PDF pp. 24--26 illustrates a non-invertible set enlargement alongside a linear gluing map; it does not supply a map from I-6 to I-5 in the IUT paper.

Two small schemas, not rival formalizations of IUT

All-isomorphism schema. Fix a particular \(j>1\) and a nonzero ordered real-line input \(v\). If two paths with the same typed source and target must agree, but one is \(j^2\) times the other, their output on \(v\) cannot both agree: for an isomorphism \(f\), \(f(v)=j^2f(v)\) would force \(j^2=1\). This is the elementary monodromy test under the specified all-isomorphism and commutativity hypotheses. It does not prove that IUT requires this loop or dictate which SS diagonal gets \(j^2\).

Hull schema. A set operation \(H(A)=A\cup\sigma(A)\) need not factor through a number \(\log\mu(A)\); 09's finite four-point example gives equal-sized inputs with different hull sizes. The exact \((a,b,\lambda)\) experiment in 05 shows the same effect in Mochizuki's illustrative region family and corrects this guide's earlier numerical example, not the source. A hull can therefore coexist with a separate linear gluing without having to be an arrow in SS's real-line diagram. This does not prove the IUT operation has the needed compatibility: I-7 remains the live obligation. The typed conditional bridge in 09 states what source-level premises would yield the bound, including a weaker one-sided numerical candidate whose quantifiers also remain unproved.

The first independently checkable question for a mathematician is thus not "which agent wins?" It is: Does the published Step (xi-a)--(xi-f), with all cited hypotheses, provide a typed comparison that takes the particular native I-6 value into the bounded I-5 output for the paper's allowed choices? A proof of that arrow would answer this local audit; a source-level counterexample must satisfy the paper's actual hypotheses. Until then, the two diagrams give distinct conditional tests, not a proof, a refutation, or agreement about abc. The repeatable experiment protocol is in 09a; the XI-002 source trace tests I-6 -> I-7 with a separate normalization mini-audit.