RESEARCH DEVELOPMENT REPORT # From Stable Families to Exceptional Fibrations A record of the collaboration between Zhuchao Ji and ChatGPT 13–15 October 2026 This report reconstructs the available discussions leading to the manuscript Stable algebraic families of polarized endomorphisms and exceptional fibrations. It follows the questions, partial results, identified gaps, changes of formulation, proof-development requests, and final revisions, rather than presenting the endpoint as a result obtained in a single step. The endpoint is the 151-page PDF supplied on 15 October 2026, under the filename Stable algebraic famillies.pdf. Prepared at the author’s request, 15 October 2026. All dates and times in the narrative are expressed in China Standard Time (Asia/Shanghai, UTC+8). ## Scope and documentary basis The requested period begins on 13 October 2026 and ends with the manuscript supplied on the afternoon of 15 October. The report combines the recovered conversation records, the visible excerpts of earlier exchanges, the recorded sequence of files, and a direct reading of the supplied final PDF. The earliest recoverable project-specific exchange in this period is Ji’s request at 21:43 on 13 October to explain an already existing manuscript. Its wording shows that the project had begun before that exchange. The prompt that originally produced that manuscript, and a complete verbatim export of every intervening turn, were not recoverable. Accordingly, this is a detailed reconstruction of the available interaction history, not a certified complete transcript. That gap is left visible; no initial question or missing answer has been invented. User prompts printed in quotation blocks are English translations of recovered wording. Other exchanges are paraphrased. Exact times are used only where a turn-level timestamp was recovered; a thread date by itself is not assigned to all of its later turns. Earlier draft page counts record what the assistant reported at the time. Only the final PDF’s page count was checked directly. This is a history of mathematical development. Statements that an earlier response had completed a proof are recorded as such, alongside the occasions on which a gap was acknowledged. The final mathematical account describes the supplied manuscript; the report does not constitute a new independent verification of its proofs or priority claims. The distinction is especially relevant where the conversation moved from an incomplete argument to a much stronger completion claim. Source labels. [C1]–[C8] identify groups of recovered conversations in Appendix C. [M] identifies the supplied final manuscript; its section, proposition, and printed page numbers are used throughout. ## Contents | Section | Printed PDF page | | --- | --- | | Scope and documentary basis | i | | 1 The problem and the overall development | 1 | | 2 13 October: the existing surface manuscript and the one-dimensional test | 1 | | 3 14 October: what a locally product dynamical fibration should mean | 3 | | 4 14 October: equilibrium webs as a parallel foundation | 3 | | 5 14 October: the surface classification and the first higher-dimensional attempt | 5 | | 6 15 October, morning: changing the geometric conclusion | 7 | | 7 15 October, midday: reviews, varying fibers, and the parameter curve | 8 | | 8 15 October, afternoon: final mathematical and expository corrections | 9 | | 9 The result stated in the supplied final manuscript | 11 | | 10 How the final proof architecture answers the earlier obstacles | 12 | | 11 What the interaction contributed | 13 | | A Interaction register | 14 | | B Draft and file chronology | 17 | | C Source register and manuscript references | 18 | ## 1 The problem and the overall development The central question was how to characterize algebraic families of polarized endomorphisms that are dynamically stable. The one-dimensional reference point was McMullen’s characterization of stable algebraic families of rational maps: the non-isotrivial possibility is the flexible Lattès phenomenon. The collaboration sought a higher-dimensional formulation that could accommodate exceptional dynamics in some directions and constant dynamics in the remaining directions. The project changed along three axes. The phase space expanded from projective surfaces to arbitrary-dimensional normal projective varieties, and ultimately to families in which the normal projective variety itself varies. The geometric conclusion changed from a surface trichotomy expressed using semiconjugacies and products to a rational exceptional factor with a precisely specified system of fiber arrows. The stability hypothesis and conclusion were refined to distinguish stability on the whole parameter space from stability on a nonempty algebraic open subset. | Stage | Main question or change | | --- | --- | | 13 October | Explain an existing surface argument; identify its original idea; extract a simpler proof on $\mathbf P^{1}$. | | 14 October | Define dynamics on nontrivial fibrations; develop the equilibrium-web criterion; formulate the surface classification and attempt induction. | | Night of 14–15 | Record explicit gaps for normal X; work on reducible symbols, integrability, algebraic factors, varying covers, and local product realization. | | 15 October, morning | Replace a literal abelian target by an abelian-type exceptional factor of the original system; require an if-and-only-if theorem and a projective-space specialization. | | 15 October, midday | Review proof omissions; allow varying normal fibers; return to a curve parameter space; retain generic stability where removability is not established. | | 15 October, afternoon | Correct three technical statements; add the Lyapunov criterion; sharpen the account of new bifurcation results; compile the supplied manuscript. | The user’s interventions supplied both the expanding research objective and many of the constraints that determined the eventual theorem. The assistant supplied candidate arguments, partial results, counterexample tests, explanatory summaries, and successive manuscripts. A substantial part of the work consisted of changing the statement so that the geometric conclusion expressed exactly what the proposed proof was intended to construct. ## 2 13 October: the existing surface manuscript and the one-dimensional test ### 2.1 The first recoverable question At 21:43 on 13 October, Ji asked: > Summarize the main results and proof strategy of this article. What is its most original idea? The assistant’s response at 21:47 described the subject as the classification of stable algebraic families on $\mathbf P^{2}$. It presented three model outcomes: isotrivial families, homogeneous lifts associated with one-dimensional flexible Lattès families, and two-dimensional Lattès families. This is the earliest recovered account of the proposed surface classification, rather than evidence of the exact wording of the original problem. [C1] The response emphasized a passage from periodic multipliers to traces of matrix words, and from those traces to an “observation rank” that should distinguish rigidity from the existence of geometric structure. Equilibrium webs, monodromy, and non-Archimedean annuli also appeared in that explanation. In the historical development, this was the first recovered articulation of the idea that periodic spectral data should reconstruct enough local geometry to force a global classification. ### 2.2 Asking where non-Archimedean dynamics entered At 21:58 Ji asked: > Does this article use non-Archimedean dynamics? The assistant answered that non-Archimedean dynamics was used essentially in the then-current Chapter 4: valuations, Tate or Berkovich geometry, inverse-image annuli, and coefficient estimates were described as the bridge from stability to homogeneity. Chapters 2 and 3 were characterized as mainly complex-dynamical. [C1] That answer concerns the earlier manuscript. It should not be read as a section map of the final 151-page document, whose organization changed substantially. The question nevertheless shows that Ji was already separating the complex local rigidity mechanism from the arithmetic or degeneration input used to globalize it. ### 2.3 Using the method to revisit McMullen’s theorem At approximately 21:59 Ji redirected the method toward dimension one: > Can the method of this article give a new proof of the characterization of stable algebraic families of rational maps on $\mathbf P^{1}$? This made the classical one-dimensional theorem a test of the proposed approach. Instead of adding dimensions immediately, the discussion asked whether the local return mechanism could recover a known rigidity theorem in a simpler setting. The assistant developed an argument involving inverse returns in linearizing coordinates, long-return multipliers, finite scalar monodromy, and an exceptional-map alternative. Ji then asked, at approximately 22:06: > How is $h_s$ defined? The recovered answer at 22:07 specified $$ h_s=(\sigma_s|_U)^{-1}\circ\beta_s\circ\sigma_s,\qquad F_s^r\circ\sigma_s\circ h_s=\sigma_s,\qquad h_s(0)=b_s, $$ where $\beta_s$ is a local inverse branch of $F_s^r$ and $\sigma_s$ is the local coordinate map used in the construction. This clarification supplied the actual map behind the return notation. [C1] Ji requested a simplified English proof in the supplied template, a compiled PDF, and references wherever existing results could be cited. A further instruction explicitly required direct, non-defensive mathematical writing. At 22:40 the assistant reported an eight-page proof, with a PDF and LaTeX source archive. Its summary cited long-return multiplier limits, finite scalar monodromy, Poincaré–Ritt-type exceptionality, and Baker’s result. The recorded deliverables were p1-stability.pdf and a source archive. [C1] This stage produced a separate one-dimensional document and a vocabulary that persisted into the higher-dimensional work. In the final abstract, Ji explicitly required the relationship with Ji–Xie’s proof of McMullen’s theorem to be stated, including homoclinic orbits and the multipliers of adjoint periodic points. ## 3 14 October: what a locally product dynamical fibration should mean In a thread recorded around 18:00 on 14 October, Ji considered finite self-maps of curves V and W and a fibration with base V whose fibers are isomorphic to W. He asked how the two maps could define dynamics on this fibration. The original wording referred to a self-map on W; the ensuing exchange concerned compatibility of fiber dynamics with the fibration. [C2] The assistant pointed out that the fiber map must be compatible with the gluing maps. Ji followed up: > A product map has this property. Such maps are more general than product maps. Have they been studied, and do they have a name? The recovered record preserves the question and the compatibility observation, but not a sufficiently complete answer to attribute a particular proposed name or reference to that exchange. Its mathematical importance for the later project is clear. Having mutually isomorphic fibers does not by itself specify a dynamical system with constant fiber dynamics. In local trivializations, the same fiber map must transform correctly under changes of coordinates. Moreover, an endomorphism of the total space maps the fiber over z to the fiber over g(z); it usually does not define a self-map of a single fiber. The final manuscript resolves this issue through finite forward fiber paths. It asks for simultaneous polarized identifications of successive fibers with a fixed system $$ W\xrightarrow{h}W\xrightarrow{h}\cdots\xrightarrow{h}W, $$ and requires compatibility through a finite-length frame torsor. This is a retrospective description of the endpoint, not a claim that Definition 1.5 had already been formulated in the initial fibration discussion. [M, pp. 3–4] ## 4 14 October: equilibrium webs as a parallel foundation ### 4.1 The proposed extension beyond surfaces At 19:32 on 14 October, Ji opened a related proof-development thread: > Berteloot–Bianchi–Dupont proved, for holomorphic families of endomorphisms of $\mathbf P^{2}$, that the existence of an equilibrium web is equivalent to J-stability. Prove that the equivalence also holds on $\mathbf P^{N}$. He subsequently asked for a rigorous English proof for polarized endomorphisms of a normal projective variety, written in the supplied template and using existing references where possible. At 20:04 the assistant reported an eleven-page normal projective version, equilibrium_webs_normal_projective.pdf, with its LaTeX source. [C3] At 20:06 Ji objected that the proof was too complicated and requested a simpler version on $\mathbf P^{N}$. At 20:15 the assistant reported a four-page replacement, equilibrium_webs_PN.pdf. The proposed mechanism had three parts: uniqueness of web sections, convergence or lifting of inverse histories, and the use of local unstable manifolds. [C3] ### 4.2 The user asks what was already known Ji then asked whether the result had already been proved and what Berteloot–Buff established about equilibrium webs. This forced a distinction between several statements that could otherwise have been conflated: - existence of an arbitrary equilibrium web; - existence of an acritical ergodic web; - existence of an equilibrium lamination; - holomorphic motion of all repelling Julia cycles. In the recovered response at 21:05, the assistant distinguished the known lamination-based statements from the proposed implication starting with an arbitrary web. It presented almost-everywhere uniqueness of a web section at a fixed parameter as the missing input. This discussion later became a specific novelty paragraph in Section 2 of the final manuscript. [C3] ### 4.3 Uniqueness of the leaf through a typical point At 21:14 Ji asked: > Assume there is an equilibrium web. At a fixed parameter, is there only one leaf of the web through almost every point for the maximal entropy measure? Did Berteloot–Buff, or someone else, prove this? The assistant’s answer distinguished uniqueness or nonintersection under acriticality hypotheses from the intended assertion for arbitrary webs. It also offered a short argument using compact forward-invariant support and positive Lyapunov exponents. [C3] The eventual manuscript expresses the idea through first differing jets. Two sections through the same point have a first nonzero difference in their Taylor expansions. Iteration transports that coefficient by the derivative along the orbit. Expansion forces it to grow, while compactness of the web and Cauchy estimates bound it. The manuscript uses this to obtain uniqueness at almost every point for each fixed parameter, then constructs inverse histories. [M, Lemmas 2.6–2.7, pp. 10–11] The qualifier “at each fixed parameter” matters. The historical question was not merely whether one could choose some lamination subordinate to a web. It asked for uniqueness inside the given web itself at a prescribed parameter. ### 4.4 Simple connectivity and global markings At 21:20 Ji showed the statement of the BBD theorem and asked why the parameter space was assumed simply connected and where that assumption entered the proof. He then asked whether the proposed argument allowed the parameter space to be any simply connected complex manifold. [C3] The recovered answer explained simple connectivity in terms of trivializing periodic-point coverings and choosing global periodic markings, together with decomposition of inverse-image graphs. The archive does not preserve enough of the ensuing answer to reconstruct a complete theorem for arbitrary simply connected manifolds. The final document takes a more specific route: its main parameter space is an algebraic curve, and the local analytic criterion is stated over coordinate disks. Thus the broad parameter-manifold question should be recorded as a question raised during the development, rather than silently identified with the final theorem. ## 5 14 October: the surface classification and the first higher-dimensional attempt ### 5.1 A requested trichotomy on normal projective surfaces In the surface-proof thread beginning at 19:47, and in the formulation recovered at 20:16, Ji asked for a rigorous classification with three alternatives: isotriviality; a locally product fibration over a one-dimensional exceptional system with isotrivial fiber dynamics, related through a finite semiconjugacy; or a two-dimensional exceptional system. The proof was to use the chosen definition of fibered dynamics and follow the supplied writing template. [C4] At 20:20 the assistant explicitly acknowledged that the general normal-surface trichotomy had not been established. The available argument covered product or finite-quotient constructions, but the implication from non-isotrivial stable dynamics to the required exceptional or fibered structure remained a substantive step. The response also exposed the importance of distinguishing a morphism to a curve from a rational fibration: projective space imposes restrictions on the former that do not apply to the latter. At 21:04 the assistant subsequently reported an eighteen-page proof on a nonempty algebraic open subset after finite parameter change, with a finite semiconjugacy starting from the original surface. The reported files were stable_surface_families.pdf, its LaTeX source, and a source archive. A later recovered summary records a fifty-page rewrite that removed three article references; the precise intervening prompt and the identity of those three references were not recovered. The report therefore records the rewrite without assigning it an unsupported mathematical change. [C4] ### 5.2 The instruction to use induction Ji next asked to generalize the two-dimensional proof to arbitrary dimension: > Extend the present two-dimensional proof to arbitrary dimension, using induction. Work with a one-parameter algebraic family of polarized endomorphisms of a normal projective variety X, with dim X = N, and assume the family is J-stable. The project was no longer just a classification on $\mathbf P^{2}$. It required an induction that could survive passage to covers, factors, and return maps on periodic fibers. [C5] An early test concerned how much global product structure could reasonably be required. At 21:47 the assistant proposed a family on $\mathbf P^{3}$ combining a varying Lattès component with fixed polynomial dynamics. It described this as an obstruction to an overly strong global finite or quasi-finite product semiconjugacy, while allowing a product description on dense open sets. This is recorded as the assistant’s test of the formulation, not as a separately verified counterexample supplied by this report. Ji then selected the wording: > Use the formulation “mapping onto a dense open subset of the target.” At approximately 22:01 this became an explicit requirement, followed by a request to continue under the updated hypotheses and compile the result. The working convention was a finite surjection between suitable dense phase opens, with semiconjugacy understood on the common domain. [C5] ### 5.3 A genuine partial result, with the remaining problem stated At 22:29 the assistant reported a twenty-five-page manuscript. It described complete arguments for special branches, including product phase spaces of the form $(\mathbf P^1)^N$ and abelian-torsor or trivial-base cases, while explicitly stating that the arbitrary-normal-X trichotomy remained unproved. Uniform algebraization and the higher-dimensional geometric classification were still missing. [C5] Ji did not accept the special cases as the endpoint: > Continue the proof for normal X. Think for two hours. This request is part of the historical record. The requested duration is not treated as evidence of measured reasoning time. The subsequent response, timestamped 00:38 on 15 October, reported a 135-page manuscript. One stated result, then numbered Theorem 13.3, concerned a flat torsion-free connection regular at smooth repelling periodic points and a full-dimensional finite abelian–torus presentation on a dense open set. The response nevertheless retained the statement that the complete trichotomy had not been proved. It identified three specific obstacles. [C5] 1. Treat general reducible symbols, establish integrability of the resulting distribution, and construct an algebraic dynamical factor. 2. Prove the varying-cover statement needed for relative induction. 3. Realize product coordinates on dense open sets as a global model satisfying the chosen locally product definition. Ji responded: > Continue thinking for two hours and complete these three steps. At 02:52 the assistant reported that the first two steps had been closed in an expanded draft: it cited then-numbered Theorems 12.14 and 19.27 for reducible symbols and integrable foliations, and Theorem 20.24 with subsequent corollaries for bounded varying covers and factor constructions. The third step remained unresolved. The recovered record also mentions an obstruction involving ramified component descent on a normal sixfold, but does not preserve enough detail to reproduce that example here. [C5] This sequence is a central part of the collaboration. The argument did not move directly from “induct on dimension” to a full normal-projective theorem. It passed through an explicit inventory of local, relative, and globalization problems, and the user repeatedly directed work at those problems. ## 6 15 October, morning: changing the geometric conclusion ### 6.1 From products to fiber arrows At 09:05 on 15 October, Ji again asked for the arbitrary-dimensional result for normal projective X. The recovered discussion contrasts an initially desired product description with a more flexible formulation over an exceptional base. At 09:17 the assistant reiterated that invariant distributions require integrability and algebraic leaves, and that a product statement is stronger than a finite semiconjugacy. Ji then requested completion and a non-product fiber-product formulation. [C6] At 09:46 the assistant reported a 225-page completion. Its geometric formulation used a rational factor $$ \pi:X_{K'}\dashrightarrow Z,\qquad \pi f=g\pi,\qquad 1\le r:=\dim Z\le N, $$ after finite parameter extension, together with a map to a dynamical fiber product. On appropriate opens the map was described by $$ x\longmapsto\bigl(\pi(x),f(x)\bigr),\qquad D\longrightarrow Z\times_{g,Z,\pi}U, $$ with degree $q^{N-r}$. The point of this formulation was that the fiber over z maps to the fiber over g(z), so the relevant datum is a fiber arrow. The reported page count refers to that intermediate draft, not the supplied final document. [C6] ### 6.2 An abelian target is not the same as an abelian-type factor The discussion then confronted a basic directional issue. If Z itself is required to be an abelian variety, one may need a finite dynamical cover $\widetilde X\to X$ before constructing a map $\widetilde X\dashrightarrow Z$. A varying Lattès family on $\mathbf P^{1}$ makes the distinction visible: $\mathbf P^{1}$ does not have a positive-dimensional abelian rational factor, although its dynamics can be presented using a finite map from an elliptic curve. [C6] At 10:06 Ji specified the desired resolution: do not replace the source of the main factor by a finite phase cover; retain $\pi:X\dashrightarrow Z$ and require the map on Z to be exceptional of abelian variety type. He also required one fixed complex polarized fiber system, simultaneous local product coordinates, the removal of irrelevant material, and the necessary definitions and references. At 10:20 the assistant reported a revision retaining the original-source factor. [C6] This correction affected the theorem itself. A finite extension of the parameter field, a finite cover of the phase variety, and a rational dynamical factor are different operations. The final statement permits the first and has the third as its conclusion. Finite phase covers appear inside the proof and must be descended; they do not replace the source in Theorem 1.6. ### 6.3 Requiring an equivalence and a projective-space version At 10:44 Ji instructed: > State the main theorem as “J-stable if and only if . . . ” and prove that the latter two cases imply J-stability. Give the corresponding version for X = $\mathbf P^{N}$, where the statement can be simplified. He further asked that, if only abelian-variety-type exceptional maps were used, the article refer to Ji–Song–Xie without introducing an unused general definition. [C6] The word “characterization” now required both directions. The implication from stability to structure had occupied most of the previous development. The converse required a separate construction showing that the proposed exceptional-base and fixed-fiber geometry actually produces stable dynamics. At 10:49 Ji also asked to remove the one-parameter-family assumption, reasoning that general quasi-projective parameter spaces should be reducible to curves. At 11:21 the assistant reported a 139-page manuscript with an equivalence theorem, a $\mathbf P^{N}$ corollary, and sections on curve reduction and the converse. The files were recorded as stable_families_exceptional_fibrations.pdf and its LaTeX source. [C6] The final document no longer has exactly this scope or numbering. A later user instruction deliberately restored a curve base, and the main theorem eventually became Theorem 1.6 with a generic-stability qualifier in the varying-normal-fiber setting. ## 7 15 October, midday: reviews, varying fibers, and the parameter curve ### 7.1 The first recorded error review At 11:58 Ji asked the assistant to check the 139-page manuscript for mathematical errors. At 12:05 the assistant identified two omissions: a normalized-pullback compatibility requirement in then-numbered Lemma 12.5, and the need for an existing normal projective polarized model in Proposition 13.11. The assistant reported that it had not found a fatal counterexample. This describes the scope of that response; it is not a general proof-validity conclusion of the present report. [C7] Ji immediately requested repairs and recompilation in the existing template. At 12:13 the assistant reported a revised 140-page manuscript, stable_families_exceptional_fibrations_revised.pdf, with source. The two omissions are significant because functorial compatibility and the existence of a suitable compactified model cannot be supplied merely by local coordinates or a statement about generic function fields. ### 7.2 Allowing the underlying normal variety to vary At 12:22 Ji asked whether the assumption that all phase fibers were isomorphic could be removed. The ensuing discussion first considered varying smooth projective varieties, and Ji then requested treatment of normal families as well. The assistant reported a normal-family revision at 12:56. [C7] The final setup is consequently a flat projective morphism $$ p:\mathcal X\longrightarrow S $$ with geometrically integral normal fibers, an S-endomorphism f, and a relatively ample line bundle $\mathcal L$ satisfying $f^*\mathcal L\simeq\mathcal L^{\otimes q}$. It is not restricted to the product family S × X. The recovered revision requests also call for the main theorem to retain only J-stability on a nonempty Zariski open subset in this generality, to remove an earlier full-parameter condition and Question 19.9, and to place the definition of J-stability in a definition environment. These changes made the unresolved extension issue visible in the statement itself. [C7] ### 7.3 Returning from general parameter spaces to a curve At 13:36 Ji deliberately narrowed the presentation: > The present article proves the result for a general parameter space S. Change the proof to the case where S is a curve, and simplify the statements and proofs accordingly. He simultaneously requested the title Stable algebraic families of polarized endomorphisms and exceptional fibrations. The abstract was to say that the work extends the one-dimensional theorem of McMullen and borrows from Ji–Xie’s homoclinic-orbit and periodic multiplier approach. [C7] This was a later choice, not the unchanged scope of the whole project. Earlier that morning the user had requested a general parameter-space theorem; later he preferred the curve-base formulation for a simpler article. The final manuscript works over a smooth connected complex algebraic curve and its function field K = C(S). ### 7.4 Can “generically” be removed? At 13:47 Ji asked: > Now that S is a curve, can “generically” be removed from “generically J-stable”? At 13:54 the assistant answered that the then-current proof did not remove it for arbitrary varying normal fibers. The issue was whether a smooth repelling periodic section could meet a singular locus at an omitted parameter. A local degeneration model $$ X_t=\{xy-z^2=tw\} $$ was used to illustrate the possible obstruction. The response explicitly did not present this local model as a global projective polarized counterexample. [C7] The final document preserves the resulting distinction. Theorem 1.6 is an equivalence for generic J-stability of varying normal fibers. On a fixed normal phase variety, Lemma 2.20 and Corollary 2.21 provide removability of omitted parameters. In particular, Corollary 1.7 for fixed $\mathbf P^{N}$ uses J-stability on all of S. The earlier response also discussed smooth varying fibers, but the final projective-space statement rests on the fixed-phase-space result. [M, pp. 25–29, 147–148] ## 8 15 October, afternoon: final mathematical and expository corrections ### 8.1 The abstract, terminology, and attribution At approximately 14:05 Ji asked that Section 2 identify which bifurcation results were new, including statements previously unavailable for general $\mathbf P^{N}$ families with N ≥3. He also shortened the abstract’s description of the exceptional-factor alternative to a positive-dimensional exceptional factor whose generic fiber has simultaneous local product coordinates with one isotrivial polarized fiber system. The technical abelian-variety-type definition was to remain in the body. [C7] Ji requested acknowledgment that the general algebraic notion of exceptional maps in arbitrary dimension was recently introduced by Ji–Song–Xie. A separate global terminology instruction replaced “multipliers of associated periodic points” by “multipliers of adjoint periodic points.” These were distinct attributions: McMullen supplied the theorem being generalized, Ji–Xie supplied the motivating proof method, and Ji–Song–Xie supplied the broader exceptional-map framework. There is one useful distinction between the request and the final contents. The main theorem defines only exceptional maps of abelian variety type, but Section 11 still introduces general abelian–torus presentations for use in the induction. The final manuscript therefore did not remove every general exceptional-system definition. [M, Definition 11.1, p. 72] ### 8.2 The second recorded error review and three repairs At 14:22 Ji requested another mathematical-error check. At 14:32 he asked the assistant to make the three identified corrections directly in the manuscript. The final file contains the following forms of those repairs. [C8] An effective torus action. In Proposition 13.16 the torus is divided by the full geometric-generic kernel of its action, including finite components. This makes the induced action effective. The descended transport is an isogeny, and the dense orbit can then be treated as a torsor with trivial stabilizer. The distinction matters because removing only the connected kernel would not exclude a residual finite stabilizer. [M, pp. 107–109] The domain of deck germs. Proposition 8.6 makes explicit how a local deck germ is related to an actual inverse return on the fixed coordinate ball. Suitable normal-form prefixes give $$ h=N^{-m}gN^{-\ell},\qquad g=N^mhN^\ell. $$ The first map must genuinely be defined as an inverse return on the chosen domain before invariance for return maps can be transferred to the deck germ. This repairs a domain issue in the descent argument. [M, pp. 59–61] Cover classes and the stack of covers. Lemma 12.4 separates the finite étale coarse algebraic space representing isomorphism classes from the locally constant Deligne–Mumford stack of covers. The latter retains automorphisms and has finite geometric fiber groupoids. This avoids replacing a moduli problem with automorphisms by an unjustified fine moduli scheme. [M, pp. 82–83] ### 8.3 Adding Lyapunov stability as an equivalent definition In the same 14:32 instruction, Ji requested another equivalent definition of stability in Section 2: > Introduce the equivalent definition $dd^cL$ = 0, where L is the Lyapunov exponent, and prove that it is equivalent to the other definitions. In higher dimension the manuscript makes the necessary clarification that $$ L(s)=\sum_{j=1}^{N}\chi_j(s)=\int\log|\operatorname{Jac}f_s|\,d\mu_s. $$ Thus L is the sum of Lyapunov exponents, counted with multiplicity. On a parameter curve, $dd^cL$ = 0 means that L is harmonic; it is not the assertion that each individual exponent is harmonic, or that L must be constant. [M, Definition 2.2 and Remark 2.18] At 14:37 Ji again requested an explicit statement of novelty in the bifurcation section. The assistant’s response at 14:55 reported the repairs, the new definition and equivalence proof, and the revised account of prior work, with PDF and LaTeX deliverables. The supplied PDF was compiled at 15:02:33 according to its metadata and recorded as uploaded at about 15:04. It is the 151-page endpoint used in this report. [C8; M] ## 9 The result stated in the supplied final manuscript ### 9.1 The general theorem Let S be a smooth connected complex algebraic curve, K = C(S), and let $(p:\mathcal X\to S,f,\mathcal L)$ be a flat projective algebraic family of q-polarized endomorphisms, q > 1, with geometrically integral normal fibers of dimension N ≥1. Theorem 1.6 states that the family is generically J-stable if and only if one of the following holds: 1. The family is isotrivial. 2. After a finite extension K′/K, there are a normal projective K′-variety Z, a q-polarized endomorphism g of Z exceptional of abelian variety type, and a rational fibration $$ \pi:X_{\eta,K'}\dashrightarrow Z,\qquad \pi f_{\eta,K'}=g\pi,\qquad 1\le\dim Z\le N, $$ whose generic fiber arrows have simultaneous local product coordinates with one fixed complex polarized fiber system $(W,L_W,h)$ of dimension N −dim Z. Here a rational fibration has geometrically integral generic fiber. Isotriviality means that after a finite extension of the parameter field the generic dynamical system is isomorphic to one defined over C. Generic J-stability means J-stability after removing finitely many points of S. [M, Definitions 1.1–1.5 and Theorem 1.6, pp. 2–5] The exceptional condition has an explicit abelian presentation: for some e ≥1, there are an abelian variety A, a finite surjective map η : A →Z, and a $q^e$-polarized isogeny α such that $$ \eta\circ\alpha=g^e\circ\eta. $$ This makes precise the morning correction: Z need not itself be an abelian variety, and the factor map starts at the original generic phase variety after parameter extension. ### 9.2 What the fixed fiber system means The fiber condition is stronger than pairwise isomorphism of the fibers. On an equivariant projective graph model and over the function field of Z, it concerns every finite path of successive fiber arrows. Each path must be geometrically polarized isomorphic to a constant chain built from one h : W →W. A frame consists of compatible polarized isomorphisms at all vertices. Definition 1.5 also asks that frames of a sufficiently long finite path determine frames of every longer forward path: restriction to that initial finite path is an isomorphism of frame schemes. This is the simultaneous condition that makes the local descriptions compatible across time. After spreading and passing to suitable étale charts, these frames provide holomorphic product coordinates on dense phase opens. [M, pp. 3–4] This definition records the outcome of several earlier questions: what a nontrivial dynamical bundle is, why individual fiber isotriviality is not enough, why a return self-map differs from a fiber arrow, and how to express the required compatibility without asserting a global product decomposition. ### 9.3 The specialization to projective space For a degree-d family on fixed $\mathbf P^{N}$, Corollary 1.7 replaces generic stability by stability on all of S. The isotrivial case is written as conjugacy after finite parameter extension: $$ f_{K'}=M^{-1}f_0M,\qquad M\in\operatorname{PGL}_{N+1}(K'),\qquad f_0\in\operatorname{End}_d(\mathbf P^N)(\mathbb C). $$ In the other case the factor Z is geometrically unirational, its dynamics is exceptional of abelian variety type, and it has the same simultaneous fiber-path condition. When dim Z = 1, the factor is a Lattès map on a rational curve; when dim Z = N, the original map is itself Lattès. [M, p. 5] ### 9.4 The stability equivalences and their stated novelty For a holomorphic polarized family on a fixed normal projective variety over a parameter disk, Theorem 2.3 equates existence of an arbitrary equilibrium web, holomorphic motion of all repelling Julia cycles, holomorphic motion of all smooth repelling Julia cycles, and $dd^cL$ = 0. Its treatment of singular repellers uses the action on the Zariski tangent space. The relative normal-family statement is made after restricting to a nonempty parameter open. [M, pp. 7, 27–29] Section 2 expressly credits the previously established projective-space Lyapunov, lamination, and repelling-cycle equivalences to the cited work of Berteloot–Bianchi–Dupont and Berteloot–Buff. Its claimed addition is the implication beginning with an arbitrary equilibrium web, without acriticity, ergodicity, or a pre-existing lamination, together with fixed-parameter leaf uniqueness and the normal-variety extensions. The manuscript describes this as new even for arbitrary one-parameter families on $\mathbf P^{N}$ with N ≥3. These are the manuscript’s carefully differentiated novelty claims; the Lyapunov criterion itself is not presented as new in projective space. [M, pp. 5–6] ## 10 How the final proof architecture answers the earlier obstacles The final manuscript does not merely append an induction to the surface proof. It introduces intermediate structures that match the specific obstacles exposed in the discussion. The following is a retrospective reading of that architecture. ### 10.1 From stability to a hypothesis preserved by induction Section 3 obtains constant characteristic polynomials along marked repelling periodic branches. The stated argument uses algebraic periodic covers and bounded reciprocals of repelling eigenvalues on algebraic curves. Section 5 then introduces a pointed reference spectral class that is preserved under the covers, factors, and periodic-fiber returns used later. [M, Proposition 3.1; Definition 5.1 and Proposition 5.2] This responds to a problem hidden in a naive induction: a subsystem created by the proof cannot simply be declared globally J-stable. The intermediate hypothesis carries the periodic spectral information actually needed at a geometrically generic reference parameter. ### 10.2 From return data to geometry without symbol irreducibility Sections 6–9 develop good return centers, normal forms, finite jet data, continuation, and local alternatives. Theorem 9.9 gives three kinds of output: local conjugacy, a transitive finite-dimensional polynomial symmetry algebra, or a finite web of proper integrable distributions. Corollary 9.10 globalizes these structures. [M, pp. 40–70] This is where the final manuscript locates the answer to the first named gap: a general reducible symbol cannot be handled merely by finding an invariant linear subspace. The distribution needs an integrability argument, and the finite collection of branches must be treated without silently choosing a globally defined one. ### 10.3 From foliations and periodic fibers to algebraic factors Sections 10–15 connect the local alternatives to global algebraic geometry. Under its Poincaré-map hypotheses, Theorem 14.11 turns an invariant foliation into a proper polarized rational factor or exceptionality. Section 12 treats the bounded irregular covers of periodic fibers, with Theorem 12.9 performing the gluing into an independent exceptional factor. Section 15 closes the phase-dimension induction within the pointed spectral class. [M, pp. 71–139] In particular, Theorem 15.12 concludes that a nonisotrivial family in that class has a positive-dimensional trace-zero abelian-quotient factor. Uniform bounds on the relevant covers are the response to the second explicitly identified gap. Separate control of cover classes, automorphisms, and descent explains why the stack correction in Lemma 12.4 mattered to the overall proof. ### 10.4 From a maximal factor to one fixed fiber dynamics Section 16 takes a maximal connected trace-zero exceptional factor. If residual variation remained in sufficiently many periodic fiber returns, the preceding construction would produce another independent factor and contradict maximality. Periodic Betti observations make the finite polarized path classes constant. The stabilization of the frame groups and their cosets then produces one stationary complex system $(W,L_W,h)$. [M, Lemma 16.1, Proposition 16.2, and Corollary 16.3, pp. 139–142] This addresses the third gap in the form adopted by the final theorem: the conclusion uses simultaneous local product coordinates for compatible paths, rather than asserting without qualification that the whole phase space is a global product. The definition and the proof were adjusted together. ### 10.5 The converse required by the user Sections 18–19 supply the reverse implication. The manuscript extends compatible frames over an abelian phase base after finite étale cover and iteration, constructs asymptotically full families of moving repelling periodic points, and obtains an equilibrium web. The relative criterion then yields stability on a nonempty parameter open. The fixed-projective-space removability statement upgrades this to all parameters for Corollary 1.7. [M, Lemma 18.4, Proposition 18.5, and Section 19, pp. 144–148] This part closes the logical requirement introduced at 10:44 on 15 October: the exceptional-factor alternatives must imply stability, not merely be consequences of it. ## 11 What the interaction contributed The historical record supports a concrete division of roles. Ji set the classification problem and repeatedly chose its scope: first the test in dimension one, then surfaces and arbitrary normal X, then varying normal fibers. He insisted on completing the identified gaps, required converse implications, rejected a finite phase cover as a substitute for the desired original-source factor, and clarified the fiber dynamics. He also directed the final curve-base presentation, the two error-review rounds, the Lyapunov definition, and the account of prior work. The assistant supplied the candidate proof mechanisms and their written realizations: the spectral-to-return reconstruction, section-uniqueness arguments, intermediate special cases, local and relative constructions, counterexample tests, and a succession of compiled drafts. The record includes both explicit admissions of incompleteness and subsequent reports of completion. The latter should not erase the former when describing how the manuscript was developed. The final formulation retains several visible consequences of this interaction: an exceptional quotient rather than a literal abelian target; a rational factor from the original generic phase variety; finite compatible fiber paths rather than unqualified product structure; generic stability for varying normal fibers; full stability for fixed projective space; and a distinction between previously known Lyapunov criteria and the claimed arbitrary-web strengthening. ## A Interaction register This register groups the recoverable requests and responses by conversation branch, with chronological order within each branch. Entries without a precise time are placed within their documented thread or phase; they are not reconstructed timestamps. The substantive discussion appears in the body of the report. ### 01. Explain the existing article 13 Oct, 21:43–21:47 Ji asks for its main results, proof strategy, and most original idea. The assistant describes a surface classification and the passage from multipliers to matrix-word traces and geometric reconstruction. [C1] ### 02. Identify the non-Archimedean input 13 Oct, 21:58 Ji asks whether non-Archimedean dynamics is used. The assistant assigns a central role to valuations, Tate/Berkovich geometry, inverse annuli, and coefficient estimates in the earlier Chapter 4. [C1] ### 03. Recover the one-dimensional theorem 13 Oct, about 21:59 Ji asks whether the method gives a new proof of the stable-family characterization on $\mathbf P^{1}$. The assistant proposes the local inverse-return route. [C1] ### 04. Define the return map 13 Oct, 22:06–22:07 Ji asks how $h_s$ is defined. The assistant writes it as the conjugate of a local inverse branch using $\sigma_s$, with the return identity recorded in Section 2. [C1] ### 05. Write and compile the simplified proof 13 Oct, later thread Ji requests English, the supplied template, references, and a compiled file. A separate correction says not to use defensive writing. At 22:40 the assistant reports an eight-page $\mathbf P^{1}$ proof and source archive. [C1] ### 06. Define dynamics on a fibration 14 Oct, around 18:00 Ji asks how maps of the base and model fiber define dynamics on a fibration with isomorphic fibers. The assistant identifies compatibility with gluing. [C2] ### 07. Ask for terminology beyond products 14 Oct, same thread Ji asks whether the broader class has been studied or named. The available record does not preserve the complete answer. [C2] ### 08. Extend the equilibrium-web equivalence 14 Oct, 19:32 Ji asks for the $\mathbf P^{N}$ extension of the surface web/J-stability equivalence. [C3] ### 09. Treat normal projective phase space 14 Oct, 19:43–20:04 Ji requests a rigorous general polarized version in the template. The assistant reports an eleven-page proof and LaTeX source. [C3] ### 10. Simplify the web argument 14 Oct, 20:06–20:15 Ji finds the proof too complicated and asks for a simpler $\mathbf P^{N}$ proof. The assistant reports a four-page version. [C3] ### 11. Check earlier literature 14 Oct, response at 21:05 Ji asks whether the result was already known and what Berteloot–Buff proves. The assistant distinguishes arbitrary webs, acritical webs, and laminations. [C3] ### 12. Uniqueness through a typical point 14 Oct, 21:14–21:17 Ji asks for almost-everywhere uniqueness at a fixed parameter and its prior status. The assistant discusses existing acritical results and proposes an arbitrary-web argument. [C3] ### 13. Explain simple connectivity 14 Oct, 21:20–21:26 Ji asks why BBD assumes a simply connected parameter space and whether the new approach handles arbitrary simply connected complex manifolds. The recovered answer discusses global markings and inverse-image graphs. [C3] ### 14. Prove the surface trichotomy 14 Oct, 19:47–20:20 thread Ji requests the surface classification with the agreed fibered-map definition. The assistant initially acknowledges that the general normal-surface case is incomplete. [C4] ### 15. Produce the surface manuscript 14 Oct, response at 21:04 The assistant reports an eighteen-page proof after finite parameter change on dense opens. A later summary records a fifty-page rewrite; its full intervening exchange is unavailable. [C4] ### 16. Generalize by induction 14 Oct, higher-dimensional thread Ji requests the theorem for N-dimensional normal projective X. The assistant examines the difference between finite product semiconjugacy and dense-open structures. [C5] ### 17. Test an overly global formulation 14 Oct, response at 21:47 The assistant proposes a $\mathbf P^{3}$ family combining varying Lattès and fixed polynomial dynamics as an obstruction to the strongest product formulation. [C5] ### 18. Choose the dense-open target wording 14 Oct, around 22:01 Ji selects “mapping onto a dense open subset of the target,” then requests continuation and compilation under the revised assumptions. [C5] ### 19. Report a partial higher-dimensional result 14 Oct, 22:29 The assistant reports a twenty-five-page draft with complete special cases, but explicitly leaves the arbitrary-normal-X trichotomy unproved. [C5] ### 20. Continue for normal X 14–15 Oct, same thread Ji asks to continue for two hours rather than stop with the special cases. [C5] ### 21. Identify three remaining gaps 15 Oct, 00:38 The assistant reports a 135-page draft and a connection/abelian–torus result, while listing reducible-symbol algebraization, relative varying covers, and global local-product realization as unresolved. [C5] ### 22. Explain the originality again 15 Oct, response at 01:02 In the surface-manuscript discussion, the assistant describes the chain from constant spectra to long inverse-return word traces, local geometry, majority continuation, and exceptional factors. This was a further explanation of the proof’s proposed core mechanism. [C4] ### 23. Work on all three gaps 15 Oct, response at 02:52 After Ji’s further two-hour request, the assistant reports progress closing the first two gaps and retains the third, including an obstruction to arbitrary ramified-component descent. [C5] ### 24. Resume the full-dimensional proof 15 Oct, 09:05–09:30 Ji again asks for completion. The assistant reiterates the integrability and algebraicity issue; the user requires a non-product fiber-product formulation. [C6] ### 25. Report a long completion draft 15 Oct, 09:46 The assistant reports a 225-page draft using an exceptional rational factor and a finite map to the dynamical fiber product, of fiber degree $q^{N-r}$. [C6] ### 26. Resolve the abelian-target issue 15 Oct, morning The assistant explains why an abelian target may require a finite dynamical source cover, using the Lattès example on $\mathbf P^{1}$. [C6] ### 27. Retain a factor of the original source 15 Oct, 10:06–10:20 Ji rejects using a phase cover in place of the desired source, requires an abelian-type exceptional quotient and fixed compatible fiber system, and asks for definitions and references. The assistant reports the revision. [C6] ### 28. Make the theorem an equivalence 15 Oct, 10:44 Ji requests “if and only if,” the reverse implications, and a corresponding $\mathbf P^{N}$ version. [C6] ### 29. Limit unused exceptional terminology 15 Oct, same request Ji asks to cite JSX and define only the abelian-variety-type class if that is all the argument uses. The final main theorem follows this restriction, while the proof still uses AT-presentations. [C6; M] ### 30. Enlarge the parameter space 15 Oct, 10:49–11:21 Ji requests removal of the one-parameter hypothesis by curve reduction. The assistant reports a 139-page equivalence manuscript with general parameter spaces. [C6] ### 31. Perform a mathematical-error check 15 Oct, 11:58–12:05 Ji requests a review. The assistant identifies normalized-pullback compatibility and an existing polarized-model hypothesis as omissions. [C7] ### 32. Repair and recompile 15 Oct, 12:05–12:13 Ji requests the fixes in the established template. The assistant reports a revised 140-page PDF and source. [C7] ### 33. Allow nonisomorphic phase fibers 15 Oct, 12:22 onward Ji asks to remove the assumption that all fibers are isomorphic; after a smooth-family discussion, he requests the normal-family version. The assistant reports an extension at 12:56. [C7] ### 34. Keep the appropriate generic statement 15 Oct, same revision thread The requests retain stability on a nonempty parameter open, remove an earlier full-parameter condition and Question 19.9, and place J-stability in a definition environment. [C7] ### 35. Return to a curve and change the title 15 Oct, 13:36 Ji narrows the manuscript to a smooth connected curve S and requests the final title, with McMullen and Ji–Xie in the abstract. [C7] ### 36. Ask whether “generic” is removable 15 Oct, 13:47–13:54 Ji asks to remove the qualifier. The assistant retains it for varying normal fibers, explaining the possible singular specialization of a smooth repelling section. [C7] ### 37. Shorten the exceptional-factor description 15 Oct, around 14:05 Ji replaces the longer abstract phrase by a shorter formulation involving one isotrivial polarized fiber system, while retaining the technical definition in the body. [C7] ### 38. Identify new bifurcation statements 15 Oct, same phase Ji asks what is new, including for general $\mathbf P^{N}$ families with N ≥3, and requests mention of JSX’s arbitrary-dimensional framework. [C7] ### 39. Standardize “adjoint periodic points” 15 Oct, same phase Ji requests a global replacement of “associated” by “adjoint” in the multiplier terminology. [C7] ### 40. Conduct a second mathematical-error review 15 Oct, 14:22 onward The review identifies the effective-torus, deck-germ domain, and stack-formulation issues. [C8] ### 41. Apply the three repairs 15 Oct, 14:32 Ji requests direct corrections in the manuscript. Their final forms are visible in Proposition 13.16, Proposition 8.6, and Lemma 12.4. [C8; M] ### 42. Add the Lyapunov criterion 15 Oct, 14:32 Ji requests $dd^cL$ = 0 as an equivalent stability definition, with proof. The manuscript defines L as the sum of exponents and develops the normal-space criterion. [C8; M] ### 43. Refine novelty once more 15 Oct, 14:37–14:55 Ji repeats the request to identify new higher-dimensional bifurcation results. The assistant reports the final revised section and compiled PDF/TeX. [C8] ### 44. Supply the endpoint and request this report 15 Oct, 15:02–15:06 The 151-page final PDF is compiled and supplied. Ji requests an English PDF documenting the interaction from the first question through partial results, discussion, and the final result. [M; current request] ## B Draft and file chronology Page counts below are historical delivery claims unless identified as checked. Version suffixes show the circulation of files, not independently established mathematical differences between their contents. | Stage | Document or recorded filename | Historical role | | --- | --- | --- | | 13 Oct | p1-stability.pdf | Eight-page one-dimensional proof reported at 22:40. | | 14 Oct | stable-algebraic-families-P2 series | Surface drafts circulated alongside successive $\mathbf P^{1}$ files. | | 14 Oct | equilibrium_webs_normal_projective.pdf | Eleven-page general web criterion reported. | | 14 Oct | equilibrium_webs_PN.pdf | Four-page simplified projective-space version reported. | | 14 Oct | stable_surface_families.pdf | Eighteen-page surface proof reported; later expanded rewrites. | | 14–15 Oct | normal_projective_isospectral_families series | Related spectral-rigidity working manuscript; not identified with the final theorem solely by filename. | | 14–15 Oct | stable_families_higher_dimension.pdf | Twenty-five-page and later 135-page stages reported; explicit unresolved normal-case steps. | | 15 Oct, morning | Higher-dimensional completion draft | A 225-page completion was reported; subsequently reformulated and shortened. | | 15 Oct, 11:21 | stable_families_exceptional_fibrations.pdf | A 139-page equivalence theorem, initially with general parameter spaces. | | 15 Oct, 12:13 | Revised exceptional-fibrations manuscript | A 140-page repair of two identified omissions. | | 15 Oct, afternoon | stable_families_normal_extension.pdf/.tex | Varying normal fibers, curve base, generic stability, later technical repairs and Lyapunov addition. | | 15 Oct, 15:04 | Stable algebraic famillies.pdf | Supplied endpoint: 151 pages, checked directly; title is the final exceptional-fibrations title. | The file trail also includes an 08:58 question on 15 October asking why a document from a shared conversation had not been delivered. It records a delivery issue; the recovered material does not establish a mathematical change arising from it. Two contemporaneous research questions asked whether related methods could treat isospectral Hénon families and positive-entropy K3 automorphism families. They are adjacent applications, not steps in the final polarized-endomorphism theorem. The recovered K3 answer retained an unresolved non-Kummer isotriviality step. This report does not absorb those separate problems into the stable-family proof. ## C Source register and manuscript references [C1] Existing manuscript and the $\mathbf P^{1}$ proof. 13 October evening, conversation on explaining the classification proof and its original idea; subsequent questions on non-Archimedean dynamics, McMullen’s theorem, $h_s$, and the compiled one-dimensional write-up. [C2] Dynamics on a fibration. 14 October, discussion of maps on base and fiber, compatibility with transition functions, and terminology for maps broader than products. [C3] Equilibrium webs and J-stability. 14 October evening, normal-projective and $\mathbf P^{N}$ proofs; literature comparison; fixed-parameter leaf uniqueness; simple connectivity. [C4] Surface classification and explanation of the proof. 14 October evening and continuation into 15 October, requested surface trichotomy, initial incompleteness, compiled surface drafts, and explanation of the spectral/return reconstruction. [C5] Higher-dimensional extension and explicit gaps. 14 October evening through the early hours of 15 October, dense-open formulation, special cases, repeated requests to continue for normal X, and the three named obstacles with subsequent progress reports. [C6] Exceptional factor and equivalence theorem. 15 October morning, fiber-product formulation, original-source requirement, abelian-type quotient, simultaneous product coordinates, converse implications, $\mathbf P^{N}$ specialization, and temporary generalization of the parameter space. [C7] First error review and normal-family revision. 15 October midday and early afternoon, normalized-pullback/model fixes, varying normal fibers, curve-base simplification, generic stability, title, abstract, and adjoint-periodic-point terminology. [C8] Final error review and bifurcation revision. 15 October afternoon, effective torus, deck-germ domains, cover-stack wording, Lyapunov stability, and revised novelty statements. [M] Stable algebraic families of polarized endomorphisms and exceptional fibrations. Supplied file: Stable algebraic famillies.pdf; 151 pages; PDF creation metadata: 15 October 2026, 15:02:33 CST. The report uses this file for the final theorem, definitions, section map, correction details, and stated relationship to prior literature. The principal mathematical sources are identified here as they function in the manuscript and the discussions. This is not an additional literature survey: - McMullen [McM87]. The one-dimensional rigidity theorem being generalized; cited in the manuscript as Theorem 2.2. - Ji–Xie [JX23]. Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics, Forum of Mathematics, Pi 11 (2023), e11. The manuscript cites Sections 2.2–2.3 and 6 for the motivating approach. - Berteloot–Bianchi–Dupont [BBD18]. Equilibrium webs and stability; the final discussion refers in particular to Theorems 1.1 and 4.1. - Berteloot–Buff [BB26]. The final manuscript uses Proposition 1.2 and Theorems 2.2 and 4.1 in separating known lamination/Lyapunov criteria from its claimed arbitrary-web implication. - Ji–Song–Xie [JSX26]. Exceptional maps and abelian points in backward orbits; the manuscript cites Definitions 1.2–1.4 for the arbitrary-dimensional exceptional-map framework.