\documentclass[11pt]{amsart} % DRAFT v0.11, October 2026 (CURRENT). Answers outside referee report 008 (ACCEPT WITH MINOR EDITORIAL CORRECTIONS; suitable for a scoped preprint): R8-01 the certificate tag in the proof of prop:H now cites certificate/blind-weights/; R8-02 57 representative rows covering 46 distinct orbits, and the five provisional mismatches of yun-cmp2.log explained; R8-03 the referee's independent recomputation of the three rank-one c = 6 rows cited. % v0.10: A blind re-implementation of the Section 6 weight and stabiliser computations (R7-05; certificate/blind-weights/) reproduces every earlier row and Table K, with no numerical disagreement; Table 6.2's caption and the certificate section now cite it as the full list, since the old sweepQ.log lacks C_6 and the three rank-one c = 6 orbits of B_1, B_3, B_4; prop:orbits now says the nilpotent stabiliser part is spanned by ad h-eigenvectors (one is a combination of two root vectors). % v0.9: answers outside referee report 007 (ACCEPT WITH REQUIRED MINOR REVISIONS; census repair confirmed by an independent implementation): R7-06 the certificate description now names the combined evidence (the legacy run_all.sh and reference-logs/sweepQ.log, which predate the repair and omit C_6, together with certificate/census-repair/); R7-01 the sharper search box |mu_1| <= 5, |mu_2| <= 8 added to the proof of lem:types; R7-04 the retraction in the proof of thm:U (a) displayed. Changed blocks carry the comment [referee report 007, R7-xx]; responses in review/RESPONSES.md, round 8. % v0.8: corrections from the programme's first blind recheck (CARD VERIFY, 2026-10-19; register r1360, r1364): Lemma 6.1 had omitted the seventh bundle type C_6 = t^{(-2,-2)} s_1 (j = 6); C_6 added to lem:types, prop:evimage, prop:orbits and tab:weights (it satisfies (G1), (G2) with lambda = (1,1)); the census certificate replaced; the domain of P_clean (odd block) and the positional labelling of the Conventions paragraph stated. % v0.7: answers outside referee report 006 (ACCEPT WITH MINOR REVISIONS); responses in review/RESPONSES.md, round 6. % v0.7: R6-01 (this header), R6-02 (a single congruence subgroup for the whole finite diagram in lem:stackdescent), % R6-03 and R6-04 (provenance and locator wording); changed blocks carry the comment [referee report 006, R6-xx]. % --- Revision history (earlier versions) --- % v0.6: R5-01 (lem:stackdescent, release-test remark, audit hodge-YAUD10), R5-02 (auxiliary prime in lem:rc), % R5-03 (canonical Matsuki certificate), R5-04 (Andre 1996, 2.5 a) Scolie, p. 17), lem:NS and cor:stalks corrections. % v0.5: R4-01 (naturality diagram, audit hodge-YAUD7), R4-02 (global weighted resolution and defect, audit hodge-YAUD8), % R4-03 (Betti comparison, audits hodge-YAUD7 and hodge-YAUD9), R4-05 (certificate rerun and the Sugiura reference), % R4-07 (date and certificate provenance); changed blocks carry the comment [referee report 004, R4-xx]. % (was v0.4:) v0.3 plus the round-4 audits hodge-YAUD2R2 (R3-02), YAUD3-YAUD6 (R3-03..R3-06) and MATSUKI-C0 (R3-01). \usepackage{amsmath,amssymb,amsthm} \usepackage{url} \usepackage{booktabs} \usepackage{mathrsfs} % DRAFT v0.3, 18 October 2026. Not for circulation. Answers outside referee report 002; responses in review/RESPONSES.md, round 2. % v0.3 answers outside referee report 002 (ChatGPT_Review_017): R2-01 and R2-02 (rational % quaternionic coefficients; cycle-level localisation; counter-2026-10-18-R2COEF.md with the % repairs of theory-2026-10-18-R2COEFX.md), R2-03, R2-04 and R2-05 (theory-2026-10-18-R2LINK.md), % R2-09 and R2-10 (orchestrator). Section 7 and Proposition 6.4 are unchanged pending separate runs. % Changed blocks carry the comment [referee report 002, R2-xx]. % (was v0.2:) % DRAFT v0.2, 18 October 2026. Not for circulation. % v0.2 answers outside referee report 001 (hodge-fleet notes/yun-g2-algebraic-review/review/ % Yun_G2_Referee_Report_001.md); item-by-item responses in review/RESPONSES.md, round 1. % Changed: Section 3 naturality square (A-CHAIN); Section 4 defect vanishing (U-DEFECT, with % the internal hostile-check repairs D1-D5); Section 5 typed lemmas (G-DESCENT); Section 6 % characteristic-zero distinctness and completeness, certificate bundle (C-CERT); Section 7 % Lemma 7.1 and Proposition 7.2 (T-CHOW, EHKV 2.18/2.20); scope sentence after Theorem B. % (was v0.1:) % DRAFT v0.1, 17 October 2026. Not for circulation. % Scope frozen at register entry F-1650 as updated in r1309 (outside referee hodge-BREF4: % CONFIRMED-WITH-REPAIRS). Every repair in the cumulative ledger of ref-BREF4-report.md % section 7 is meant to be incorporated below; the correspondence is recorded in the % comments marked [ledger n]. Before outside review: pin all theorem numbers marked [pin], % attach the computational certificates (testbed-lab exp/enum6, exp/korb, exp/hcheck), % and run the outside red-team round. \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem*{theoremA}{Theorem A} \newtheorem*{theoremB}{Theorem B} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{hypothesis}[theorem]{Hypothesis} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \setlength{\emergencystretch}{2em} \newcommand{\Q}{\mathbb Q} \newcommand{\C}{\mathbb C} \newcommand{\Z}{\mathbb Z} \newcommand{\PP}{\mathbb P} \newcommand{\A}{\mathbb A} \newcommand{\G}{\mathbb G} \newcommand{\F}{\mathbb F} \newcommand{\cO}{\mathcal O} \newcommand{\cL}{\mathcal L} \newcommand{\cG}{\mathcal G} \newcommand{\cF}{\mathcal F} \newcommand{\Bun}{\mathrm{Bun}} \newcommand{\Gr}{\mathrm{Gr}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\End}{\operatorname{End}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Stab}{\operatorname{Stab}} \newcommand{\MT}{\operatorname{MT}} \newcommand{\Proj}{\operatorname{Proj}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\ev}{\operatorname{ev}} \newcommand{\cl}{\operatorname{cl}} \newcommand{\tA}{\widetilde{A}} \newcommand{\tK}{\widetilde{K}} \newcommand{\Gtwo}{\mathrm{G}_2} \newcommand{\IC}{\mathrm{IC}} \newcommand{\GR}{\mathrm{GR}} \newcommand{\trd}{\operatorname{trd}} \newcommand{\bL}{\mathbb L} \newcommand{\bF}{\mathbb F} \newcommand{\odd}{\mathrm{odd}} \newcommand{\tg}[1]{\textsf{[#1]}} \newcommand{\PD}{\operatorname{PD}} \newcommand{\clH}{\cl_{H}} \newcommand{\Qi}{\Q(i)} \newcommand{\R}{\mathbb R} \title[Algebraicity of the $\mathrm{G}_2$-invariant tensors on Yun's motives]{Algebraicity of the $\mathrm{G}_2$-invariant tensors on Yun's exceptional motives} \author{Ben Severn} \email{benzsevern@gmail.com} \subjclass[2020]{Primary 14C30; Secondary 14D24, 22E57, 14C25} \keywords{Hodge conjecture, motivated cycles, exceptional motives, geometric Satake, Hecke eigensheaves, weighted blow-ups} \date{Draft v0.11, October 2026} \begin{document} \begin{abstract} Zhiwei Yun constructed, from a rigid Hecke eigensheaf for $\Gtwo$ on $\PP^1$ minus three points, a family of rank-seven motives whose motivic Galois groups are dense in $\Gtwo$. We show that on this family the invariant cubic form is the class of an algebraic cycle: at every Hodge-generic parameter, every rational Hodge class on every tensor construction of the rank-seven piece is algebraic, and at every parameter the transported $\Gtwo$-invariant tensors are algebraic. The cycles are top components of Satake convolution fibres transported by Hecke correspondences. The projector that cuts out the rank-seven piece is made algebraic by a canonical weighted resolution of Yun's boundary divisor, built on the moduli stack of bundles from the stabiliser tori of its points, together with a characteristic-zero census of those points and a cycle-class comparison on varieties with quotient singularities. Yun announced that the cubic is motivated in Andr\'e's sense; the step from motivated to algebraic is new. \end{abstract} \maketitle \section{Introduction} Let $\Gtwo$ be the split exceptional group of that type over $\Q$, and let $S=\PP^1\setminus\{0,1,\infty\}$. In \cite{Yun} Yun constructs, for each complex point $x\in S$, a pure rank-seven rational Hodge structure % [referee report 002, R2-01] \[ M_x \;=\; \varepsilon\,H^6_c(T_x,\Q), \] of weight six, whose Tate twist $M_x(3)$ is the Betti realisation of Yun's motive $M_{\chi,x}\in\mathrm{Mot}(\Q)$, with $M_x(3)\otimes\C=E(V_7)_x$, occurring in the cohomology of an explicit smooth variety attached to a Hecke modification at $x$ of a principal $\Gtwo$-bundle on $\PP^1$ with parahoric level structure at $0$ and $\infty$ (Section~\ref{sec:setup}). The variation $x\mapsto M_x$ has monodromy Zariski dense in $\Gtwo$ \cite[Thm.~1.2.1, \S\S4.5.1--4.5.2]{Yun} (for the Betti local system, Lemma~\ref{lem:dense}), and $M_x$ carries a nondegenerate invariant quadratic form $b$ and an invariant alternating cubic form $t$, the images under Yun's tensor functor of the corresponding $\Gtwo$-invariants of the seven-dimensional representation $V_7$. Write $N_x=M_x(3)$, of weight zero. In \cite[\S1.2.3]{Yun} Yun notes that the argument of Dettweiler and Reiter \cite{DR10}, based on Andr\'e's theorem \cite{Andre96}, shows that these motives have the expected motivic Galois group in the sense of Andr\'e, and omits the details. Our first result is a checked construction of the invariant cubic as a motivated class, by a route that does not use the normaliser argument of \cite{DR10}. \begin{theoremA} For every $x\in S(\C)$ the invariant cubic $t\in (M_x^{\otimes 3})(9)$ is a motivated class in the sense of Andr\'e \cite{Andre96}, given explicitly as a motivated projector applied to the class of an algebraic cycle obtained from the unique top-dimensional component of a Satake convolution fibre. If $x\in S(\Q)$ it is motivated over $\Q$. \end{theoremA} Our main result upgrades this to algebraicity. \begin{theoremB} Let $\overline X_x$ be the smooth projective variety of Section~\ref{sec:proofB}, whose cohomology contains $M_x$ as the image of an algebraic correspondence acting on $H^*(\overline X_x)$ as an idempotent of rank seven (idempotent in cohomology, not claimed in Chow). % [referee report 002, R2-01; report 003, R3-07] \begin{enumerate} \item For every $x\in S(\C)$ and every $n$, the image of every $\Gtwo$-invariant tensor of $V_7^{\otimes n}$ under Yun's construction is the class of an algebraic cycle on $\overline X_x^{\,n}$. \item If $x$ is Hodge-generic for the variation $M$, the Mumford--Tate group of $N_x$ is $\Gtwo$, and every rational Hodge class in every tensor construction $T^{a,b}(N_x)$ (mixed tensor powers, their symmetric and exterior summands, and duals) is the class of an algebraic cycle. \end{enumerate} \end{theoremB} So the Hodge conjecture holds for all tensor powers of the Hodge structure $M_x$ at Hodge-generic $x$, in the following projector-relative sense: $M_x$ is the image on $H^*(\overline X)$ of the algebraic correspondence $P_X$ of Proposition~\ref{prop:transfer}, whose action on cohomology is idempotent of rank seven (we do not claim $P_X\circ P_X=P_X$ in $\mathrm{CH}(\overline X\times\overline X)_\Q$, so no Chow motive is constructed; Remark~\ref{rem:chowidem}), and every Hodge class in $(M_x^{\otimes a}\otimes M_x^{\vee\otimes b})(c)$, viewed in $H^*(\overline X^{\,a+b})(c')$ through $P_X^{\otimes(a+b)}$, is the class of an algebraic cycle on $\overline X^{\,a+b}$. This says nothing about the rest of $H^*(\overline X)$. [referee report 001, B2] The Mumford--Tate group $\Gtwo$ is not of abelian type, so the standard tools for the Hodge conjecture (divisors and the Lefschetz $(1,1)$ theorem, abelian varieties, Kuga--Satake constructions) do not reach these classes. We make no claim about Hodge classes that may appear at special parameters where the Mumford--Tate group drops (Remark~\ref{rem:special}), about the family of Dettweiler and Reiter \cite{DR10} (Remark~\ref{rem:DR}), or about the Hodge conjecture in general. \subsection*{Outline of the proof} The cycle in Theorem~B is explicit. Satake theory identifies $\Gtwo$-invariants of $V_7^{\otimes n}$ with top Borel--Moore homology of the convolution fibre $m_n^{-1}(e)$; because the quasi-minuscule Schubert variety of $\Gtwo$ is rationally smooth, these classes are spanned by top-dimensional irreducible components, which are algebraic cycles (Section~\ref{sec:A}). Hecke correspondences, made finite by framing the endpoints, transport them to cycles on products of the Hecke fibre over $x$. The difficulty is the projector. The rank-seven piece lives in the compactly supported cohomology of an open variety $T_x$, a finite \'etale cover of the complement of a divisor $D_x$ (Yun's divisor) in the six-dimensional Schubert cell $Y=\Gr_{x,\alpha^\vee}$. Yun's cleanness theorem \cite[Thm.~2.4.2]{Yun} says that compactly supported and ordinary cohomology agree on the relevant isotypic block. On a smooth compactification this gives a splitting which is \emph{motivated} (Theorem~A) but a priori not algebraic, because it needs inverse Lefschetz operators on boundary components. We make it algebraic by choosing the compactification well: if $(Y,D_x)$ admits a model whose normalised cover has only quotient singularities and whose boundary is a labelled normal crossing divisor up to finite quotients, then the cohomology of the normalised cover in the odd block \emph{is} the clean piece, cut out by a finite-group idempotent (Theorem~\ref{thm:U}). Such a model comes from a canonical sequence of weighted blow-ups, built not on $Y$ but on the moduli stack $\Bun$ of bundles, along its finitely many residual gerbes; the weights come from contracting cocharacters of the stabiliser tori (Theorem~\ref{thm:G}). The hypotheses of that construction are checked on a complete characteristic-zero census of the points of $\Bun$ of codimension at most six, which reduces to orbit classifications of explicit subgroups of $\Gtwo$ on the flag variety (Section~\ref{sec:census}). Finally the projector is transferred to a smooth model by a cycle-class comparison on varieties with quotient singularities (Section~\ref{sec:transfer}). \subsection*{Relation to earlier work} Dettweiler and Reiter \cite{DR10} constructed the first motives for motivated cycles with motivic Galois group $\Gtwo$, from a rigid local system obtained by middle convolution; their statements for Grothendieck motives are conditional on the standard conjectures. Yun \cite{Yun} constructed the family studied here, and later stated that his $\Gtwo$-local system is the same as that of Dettweiler and Reiter \cite[Thm.~6.1.5 and the paragraph after Thm.~6.2.3]{YunCDM}; equality of the local systems does not by itself give an algebraic correspondence between the two families of motives. F{\ae}rgeman \cite{Faergeman} gives motivic realisations in Katz's sense. As far as we know, no algebraicity statement for the invariant cubic of a $\Gtwo$-motive was previously in print. \subsection*{Conventions} Cohomology has rational coefficients unless stated otherwise. Varieties are over $\C$. $\Gtwo$ has simple roots $\alpha_1$ (short) and $\alpha_2$ (long), and we write roots and coweights as $a\alpha_1+b\alpha_2$ and $m_1\alpha_1^\vee+m_2\alpha_2^\vee$. ``Hodge-generic'' is relative to the variation $M$ over $S$. % [referee report 002, R2-01] \emph{Coefficients for $\tA$.} All sheaves and cohomology are $\Q$-linear. Since $\tA=Q_8$ has no two-dimensional rational representation, we never use one: the odd block of $\Q[\tA]$ is the quaternion division algebra $D=e_-\Q[\tA]=(-1,-1)_\Q$, $e_-=(1-z)/2$. For a $\Q[\tA\times\tA]$-module $H$ put $H_{\odd}=e_-^{(1)}e_-^{(2)}H$, a module over $D\otimes D^{\mathrm{op}}\cong M_4(\Q)$, where $(a_1,a_2)$ acts on $D$ by $x\mapsto a_1xa_2^{-1}$ (Yun's convention: left and inverse-right translation \cite[\S3.2.5]{Yun}). Here the superscripts are positional: the first factor is the \emph{target} and the second the \emph{source} (deck) group, so that in the notation $\tA(1)$ (source), $\tA(2)$ (target) used from Section~2 on, $\tA(2)$ acts by $x\mapsto gx$ and $\tA(1)$ by $x\mapsto xg^{-1}$. % [blind recheck 2026-10-19, VYUN1B / YUNADJ: labelling clarified, no change of convention] On every space used below the odd part for the deck group alone, $e_-^{(2)}H$, equals $H_{\odd}$: for $H^6_c(T_x)$ by \cite[(3.16)]{Yun}, where the two-sided $\chi$-block is $V_\chi\boxtimes V_\chi^*$, and for the pieces over $\star$ because the fibre of $T_x$ there is an $\tA\times\tA$-bitorsor. The two-dimensional $V_\chi$ occurs only after extension of scalars to $\Qi$, where $D\otimes\Qi\cong\End_{\Qi}(V_\chi)$, and only in comparisons with \cite{Yun}. \section{Yun's construction}\label{sec:setup} We recall what we use from \cite{Yun}, with Yun's notation. \subsection{The moduli stack} Let $G=\Gtwo$, let $P_0\subset L_0G$ be the parahoric subgroup attached by Gross--Reeder--Yu to $-1\in W$, and let $P_\infty$ be the parahoric of the same type containing the opposite Iwahori subgroup at $\infty$ \cite[\S\S2.1.3--2.1.4, 2.2.1]{Yun}. Their reductive quotients are both the symmetric subgroup $K\subset G$, of type $A_1\times A_1$. Yun works with a double-cover variant $\Bun=\Bun_G(\widetilde P_0,P_\infty)$ \cite[\S2.2]{Yun}. It contains an open point $U\cong B\tA$, the classifying stack of a finite group $\tA$; for $\Gtwo$ the group $\tA$ is the quaternion group $Q_8$, and its centre $\langle z\rangle$ has order two \cite[\S\S2.2.5--2.3]{Yun}. Let $\chi$ be the odd character of $\tA$. % [referee report 002, R2-01] Let $D=e_-\Q[\tA]$ be the odd block. The odd sheaf $\bF$ on $U=B\tA$ is the rank-four rational local system attached to the left $\tA$-module $D$; right multiplication makes $D$ act on $\bF$ by endomorphisms. Over $\Qi$, $\bF\otimes\Qi\cong F_\chi\otimes_{\Qi}V_\chi^{*}$, where $V_\chi$ is the two-dimensional irreducible representation on which $z$ acts by $-1$ and $F_\chi$ is Yun's odd sheaf. \begin{theorem}[Cleanness, {\cite[Thm.~2.4.2]{Yun}}]\label{thm:clean} For $j:U\hookrightarrow\Bun$, $j_!\bF\cong Rj_*\bF$. \end{theorem} Yun proves $j_!F_\chi\cong Rj_*F_\chi$; the cone of $j_!\bF\to Rj_*\bF$ vanishes after $\otimes_\Q\Qi$, hence vanishes. This is an $\ell$-adic statement; its Betti form is used only on finite-type smooth charts and on $Y$, through Lemma~\ref{lem:ratclean}. % [referee report 004, R4-03; audits hodge-YAUD7, hodge-YAUD9] \subsection{The Birkhoff decomposition} By \cite[\S2.2.2]{Yun}, the points of the stack $\Bun_G(P_0,P_\infty)$ are the bundles $\mathcal P_{\tilde w}$, indexed by the double cosets \[ \tilde w\in W_K\backslash\widetilde W/W_K , \] with \[ \Aut(\mathcal P_{\tilde w}) \;=\; \Gamma_0\cap \operatorname{Ad}(\tilde w)P_0 . \] For $\tilde w=t^\mu w$ the Lie algebra of $\Aut(\mathcal P_{\tilde w})$ is $\mathfrak t\oplus\bigoplus_\alpha\bigoplus_{n=\ell_\alpha}^{h_\alpha} t^nE_\alpha$ with \begin{equation}\label{eq:interval} \ell_\alpha=\lceil -\operatorname{ht}(\alpha)/2\rceil,\qquad h_\alpha=\langle\alpha,\mu\rangle-\lceil \operatorname{ht}(w^{-1}\alpha)/2\rceil , \end{equation} and we write $N_\alpha=\max(h_\alpha-\ell_\alpha+1,0)$ for the number of loop root vectors in direction $\alpha$. The constant torus $T$ is a maximal torus of $\Aut(\mathcal P_{\tilde w})$. A point of $\Bun$ over $\mathcal P_{\tilde w}$ also records a Borel subgroup $b\in G/B$ (the level structure at $1$), and $\Aut(\mathcal P_{\tilde w})$ acts on $G/B$ through evaluation at $1$, \[ \ev_1:\Aut(\mathcal P_{\tilde w})\to G . \] The Iwahori level at $1$ is \cite[\S2.2.5]{Yun}, the variant parahoric $\tilde{\mathbf P}_0$ is \cite[\S2.2.4]{Yun}, its canonical double cover is \cite[\S2.1.6]{Yun}, and the base point $u_0$ is \cite[\S2.2.6]{Yun}. \subsection{The Hecke fibre and Yun's divisor} Let $\lambda=\alpha^\vee$ be the quasi-minuscule coweight, so that $V_\lambda=V_7$. For $x\in S$, the Schubert variety $\Gr_{x,\le\lambda}$ has dimension $\langle 2\rho,\lambda\rangle=2h^\vee-2=6$ and consists of the base point $\star$ and the open orbit $Y=\Gr_{x,\lambda}$ \cite[\S3.4]{Yun}, \cite[Lemma~5.22]{HNY}. Modifying the bundle $u_0\in U$ at $x$ by a point of $Y$ defines \[ \omega:Y\longrightarrow\Bun , \] and Yun's divisor is $D_x=\omega^{-1}(\Bun\setminus U)$, so that $\Gr^U_{x,\le\lambda}=\Gr_{x,\le\lambda}\setminus D_x$ \cite[\S3.4]{Yun}. Here $u_0$ is the fixed \emph{target} of the modification and $\omega$ is the \emph{source} map, as in \cite[(3.3), \S3.2.3]{Yun}. Over $Y\setminus D_x$ the framing of the source endpoint $\omega(y)\in U$ gives a finite \'etale $\tA$-torsor $T_x\to Y\setminus D_x$, whose deck group is Yun's source action $\tA(1)$ and on which $\tA(2)=\Aut(u_0)$, the target action, acts compatibly \cite[(3.5)]{Yun}; this is Yun's $\widetilde Y_x$ \cite[\S\S3.3.2--3.3.3]{Yun}, and we also write $\widetilde Y_x=T_x$. % [referee report 002, R2-01] % [referee report 003, R3-02; audits YAUD2, YAUD2R, YAUD2R2] Let $\bL=e_-p_*\Q_{T_x}$, a rank-four local system with fibre $D$ (monodromy by left multiplication; $D$ acts on the right through the deck group $\tA(1)$). Since $D=\Q[\tA]e_-$ is a cyclic projective $\Q[\tA]$-module, $\bL=\Hom_{\tA}(D,p_*\Q_{T_x})$. \begin{lemma}\label{lem:omega} The map $\omega$ is smooth, and $j'_!\bL\cong Rj'_*\bL$ for $j':Y\setminus D_x\hookrightarrow Y$. \end{lemma} \begin{proof} On the relative-position-$\lambda$ orbit either Hecke projection is smooth with fibre $\Gr_\lambda$; pulling one endpoint back to $u_0$ is the base change along the \'etale map $\mathrm{pt}\to B\tA\to\Bun$, so the other endpoint map $\omega$ is smooth. Smooth base change transports Theorem~\ref{thm:clean} to $Y$ for Yun's $\ell$-adic sheaves; the Betti statement is Lemma~\ref{lem:ratclean}. % [referee report 004, R4-03; audits hodge-YAUD7, hodge-YAUD9] \end{proof} \begin{remark} In the chart $C_0$ of $Y\cong\A^6$ used in our computations, $D_x$ is cut out by a polynomial of degree $30$ that factors as $F_6G_8H_{16}$ (one component through the open stratum of each of the three relevant boundary types). Nothing below depends on these equations: the whole argument is carried out on $\Bun$. \end{remark} \subsection{The rank-seven motive} Over $\Qi$, Yun's eigenvalue is $E_\chi(V_7)_x=(V_\chi^*\otimes H^*_c(T_x,\Qi))^{\tA}$ in the relevant degree \cite[(3.12), (3.18)]{Yun}. The base point $\star$ lies in the open set $\Gr^U_{x,\le\lambda}$ \cite[(3.13)]{Yun}: it is not a boundary point. % [referee report 002, R2-01] The group $\tA(2)=\Aut(u_0)$ acts on $Y$ and on $T_x$, commuting with the deck group $\tA(1)$; under $V_\chi^*\boxtimes V_\chi$ of \cite[(3.9)]{Yun}, $\tA(1)$ acts on $V_\chi^*$ and $\tA(2)$ on $V_\chi$. % [referee report 003, R3-02; audits YAUD2, YAUD2R, YAUD2R2] Put \[ \varepsilon=\frac1{32}\sum_{a_1,a_2\in\tA}\phi_\chi(a_1a_2^{-1})\,a_1\otimes a_2\in\Q[\tA\times\tA], \qquad M_x=\varepsilon\,H^6_c(T_x,\Q), \] with $\phi_\chi$ the ($\Q$-valued) character of $V_\chi$. Up to a scalar this is Yun's idempotent \cite[\S3.3.2]{Yun} ($\Q'=\Q$ for $\Gtwo$). The normalisation is forced: with $\Phi=\sum_b\phi_\chi(b)(b,1)$ and $\Delta=\sum_a(a,a)$, which commute, $\Delta^2=8\Delta$ and $\Phi^2=4\Phi$, so $(c\,\Phi\Delta)^2=32c\cdot c\,\Phi\Delta$, and $c\Phi\Delta$ is idempotent exactly for $c=1/32=\dim V_\chi/|\tA|^2$. On $D$, $\varepsilon$ acts by $x\mapsto\frac12\trd(x)$, the rank-one projection onto the identity line $\Q\cdot1$, and it kills the even block; $\varepsilon\otimes\Qi$ is the projection $a\mapsto\frac12\operatorname{tr}(a)\,\mathrm{id}$ on $\End(V_\chi)$. As a correspondence, $\varepsilon$ has coefficient $\frac1{16}$ on the graphs of the pairs with $a_1=a_2$, $-\frac1{16}$ on those with $a_1=za_2$, and $0$ otherwise. By \cite[(3.16)]{Yun}, $H^6_c(T_x)_{\odd}\otimes\Qi\cong\End(V_\chi)\otimes_{\Qi}E_\chi(V_7)_x(-3)$ has dimension $28$. As $\varepsilon$ is a primitive idempotent of $D\otimes D^{\mathrm{op}}\cong M_4(\Q)$, Morita theory gives $H^6_c(T_x)_{\odd}\cong D\otimes_\Q M_x$, so $\dim_\Q M_x=7$ and $M_x(3)\otimes\Qi=E_\chi(V_7)_x$. The odd part is pure of weight six by Lemma~\ref{lem:omega} (compactly supported cohomology has weights $\le6$, ordinary cohomology $\ge6$), so $M_x(3)=\varepsilon\,\mathrm{Gr}^W_6H^6_c(\widetilde Y_x)(3)$ (Yun's $e_\chi$ applied to the pure part) is the Betti realisation of Yun's $M_{\chi,x}$ \cite[Prop.~3.3.3]{Yun}; as $x$ varies, $N_x=M_x(3)$ is the fibre of the $\Q$-local system $N_S$ with $N_S\otimes\Q_\ell\cong E_\chi(V_7)$ as local systems (Lemma~\ref{lem:NS}). % [referee report 004, R4-03; audits hodge-YAUD7, hodge-YAUD9] A conjugate primitive idempotent gives an image isomorphic to $M_x$ through an algebraic correspondence. [ledger 10; referee report 001, A4] \begin{lemma}[Rational smoothness]\label{lem:ratsmooth} $\Gr_{x,\le\lambda}$ is rationally smooth: its intersection complex is $\Q[6]$. \end{lemma} \begin{proof} In characteristic zero the opposite-cell chart of $\Gr_{\le\lambda}$ completed by $\star$ is the minimal nilpotent cone, and blowing up the vertex gives the negative line bundle over the adjoint variety $E=\Gtwo/P$ \cite[(5.21), Lemma~5.22]{HNY}. The variety $E$ has one Schubert class in each even degree $0,\dots,10$, and $h^5\ne 0$ for the ample class $h$, so all the maps $H^{2i}(E)\to H^{2i+2}(E)$ given by $h$ are isomorphisms between one-dimensional spaces. The Gysin sequence of the punctured cone then shows that the link of $\star$ is a rational homology sphere. \end{proof} \section{The motivated cubic}\label{sec:A} \subsection{Top components of convolution fibres} Let $C_n$ be the $n$-fold convolution of $\Gr_{\le\lambda}$ and $m_n:C_n\to\Gr$ the multiplication. By Lemma~\ref{lem:ratsmooth} the iterated twisted products are rationally smooth, $IC(C_n)=\Q[6n]$, and $\dim m_n^{-1}(e)\le 3n$ by stratified semismallness \cite[\S4]{MV}. Hence \[ \Hom(\mathbf 1,V_7^{\otimes n})\;=\;H^{BM}_{6n}(m_n^{-1}(e),\Q), \] spanned by the classes of the $3n$-dimensional irreducible components. For $n=3$ the space is one-dimensional ($V_7\otimes V_7=\mathbf1\oplus V_7\oplus V_{14}\oplus V_{27}$), so there is exactly one nine-dimensional component $Z_t$, and the Satake image of $t$ is a nonzero multiple of $[Z_t]$. \subsection{Finite framing correspondences} An $n$-step chain of modifications at $x$ with fixed target $u_0$, variable source in $U$ and all intermediate bundles in $U$ becomes, after choosing frames for its $n-1$ intermediate bundles and for its source, a point of $X^n$, where $X$ is the one-step space; the forgetful maps form the tower $P=X^n\to Q\to\Omega$ of Section~\ref{ssec:achain}. Pulling a top component back to the chain space, restricting to the locus where the intermediate bundles lie in $U$, and passing to the framing covers gives an algebraic correspondence into $X^n$; we keep the coefficient $D$ at the source, the endpoint labels and the identity-section condition on the composite. [ledger 10] % [referee report 002, R2-01] % [referee report 003, R3-02; audits YAUD2, YAUD2R, YAUD2R2] The odd block is $D$; Yun's evaluation maps contract adjacent $D$-factors by multiplication, the transported tensor is the cyclic tensor $J_n$, the adjoint of $n$-fold multiplication for the invariant form $\trd(x\bar y)$ (Lemma~\ref{lem:kunneth}), and the endpoint character contributes a factor $4$; applying $\varepsilon$ in every factor gives the nonzero coefficient $2^{1-n}/4$ on Yun's tensor unit (Proposition~\ref{prop:value}). This is why the endpoint coefficient system must be retained: the unweighted image of a component need not give the desired class. Cleanness removes chains with an intermediate bundle outside $U$, and a top component missing the all-intermediate-$U$ locus contributes zero. Denote by $W_u$ the resulting rational cycle on $\widetilde Y_x^{\,n}$ for an invariant $u\in(V_7^{\otimes n})^{\Gtwo}$ (made precise in Section~\ref{ssec:localize}), and by $\overline W_u$ its closure in a smooth projective compactification. % [referee report 001, A1-A3: release test A-CHAIN; text from counter-2026-10-17-ACHAIN.md] \subsection{The naturality square}\label{ssec:achain} Throughout, coefficients are $\Q$ and sheaves are constructible sheaves for the analytic topology on complex points. % [referee report 004, R4-03; audit hodge-YAUD7, section 6.3] (the former unrestricted assertion that every cited statement of \cite{Yun} holds verbatim in this setting is deleted) For each bounded fixed-parameter diagram used in the component comparison, we compare the specified geometric complexes and maps, starting from constant sheaves and finite-monodromy integral lattices, first with finite coefficients and then in their adic realizations. Their rational Betti forms are constructed by the same geometric operations. We do not assert an equivalence with all algebraically constructible analytic $\Q_\ell$-complexes. % [referee report 004, R4-03; audits hodge-YAUD7, hodge-YAUD9] The comparison statements used, and the downstream statements that rest on them, are collected in Section~\ref{ssec:betti}: the relative comparison over $S$ (Lemmas~\ref{lem:rc}--\ref{lem:relcomp}), the clean extension (Lemma~\ref{lem:ratclean}), the transported tensors (Lemmas~\ref{lem:flat}--\ref{lem:alt}), and arithmetic fixedness (Lemma~\ref{lem:arith}). The symmetric tensor functor of \cite[(3.19)]{Yun} is not compared; Lemma~\ref{lem:alt} does not use it. Fix $x\in S(\C)$, $\lambda=\alpha^\vee$, $d=\langle 2\rho,\lambda\rangle=6$, and an integer $n\ge1$. \paragraph{Chain spaces.} Let $C_n=\Gr_{x,\le\lambda}\tilde\times\cdots\tilde\times\Gr_{x,\le\lambda}$ ($n$ factors) be the space of chains $b_n\to b_{n-1}\to\cdots\to b_1\to b_0=u_0$ of modifications at $x$, each of relative position $\le\lambda$, with fixed \emph{target} $u_0$ (Yun's convention \cite[(3.3), \S3.2.3]{Yun}: the coefficient sits on the source and is pushed to the target). Step $i$ runs from the source $b_i$ to the target $b_{i-1}$; the $i$-th convolution coordinate is the position of $b_i$ relative to $b_{i-1}$, so coordinate $1$ is adjacent to the fixed target, and $b_n$ is the variable source endpoint. In Yun's numbering of the two endpoint actions the source comes first ($\tA(1)$) and the target second ($\tA(2)$); in multiplication matrices and in convolution slots the target comes first. % [referee report 003, R3-02; audits YAUD2, YAUD2R, YAUD2R2] Write $\omega_i:C_n\to\Bun$ for the bundle $b_i$ and $m_n:C_n\to\Gr_{x,\le n\lambda}$ for the composite relative position, so that $m_n^{-1}(e)$ is the set of chains whose composite is the identity modification of $u_0$. Put \[ C_n^{(U)}=\omega_n^{-1}(U),\qquad C_n^{U^n}=\bigcap_{i=1}^{n}\omega_i^{-1}(U)\subset C_n^{(U)} \quad(\text{open}). \] Since $\omega_n(m_n^{-1}(e))=u_0\in U$, the compact set $m_n^{-1}(e)$ lies in $C_n^{(U)}$. \begin{lemma}\label{lem:top} \textup{(a)} $\dim m_n^{-1}(e)\le 3n$. \textup{(b)} $C_n$ is rationally smooth of dimension $nd$, and $m_{n*}\Q_{C_n}[nd](nd/2)\cong \IC_\lambda^{*n}$. \textup{(c)} There is a canonical isomorphism \[ \Hom_{\mathrm{Sat}}(\IC_0,\IC_\lambda^{*n})\;\cong\;H^{BM}_{2\cdot 3n}\bigl(m_n^{-1}(e),\Q\bigr)(-3n), \] and the right side has the basis $\{[Z]\}$, $Z$ running over the $3n$-dimensional irreducible components of $m_n^{-1}(e)$ with their reduced structure and complex orientation. \end{lemma} \begin{proof} (a) is stratified semismallness of convolution \cite[Lemma~4.4]{MV}. (b): $C_n$ is an iterated locally trivial fibration with fibre the rationally smooth $\Gr_{\le\lambda}$ (Lemma~\ref{lem:ratsmooth}), so $\IC(C_n)=\Q[nd]$; the convolution $\IC_\lambda^{*n}$ is by definition $m_{n*}$ of the twisted external product of the $\IC_\lambda=\Q[d](d/2)$. (c): $m_n$ is proper, so with $i:m_n^{-1}(e)\hookrightarrow C_n$ and base change, $\Hom(\delta_e,m_{n*}\Q[nd](nd/2))=H^0(m_n^{-1}(e),i^!\Q_{C_n}[nd](nd/2))$. Rational smoothness gives $\Q_{C_n}[2nd](nd)\cong\omega_{C_n}$, hence $i^!\Q_{C_n}[nd](nd/2)=\omega_{m_n^{-1}(e)}[-nd](-nd/2)$ and the $\Hom$ equals $H^{BM}_{nd}(m_n^{-1}(e))(-nd/2)$, with $nd=6n=2\cdot 3n$. By (a) this is the top Borel--Moore group, freely spanned by the fundamental classes of the $3n$-dimensional components. Both sides of (c) are $\Hom$s in the abelian category of perverse sheaves because $\IC_\lambda^{*n}$ is perverse semisimple: it is perverse by \cite[Prop.~4.2]{MV}, and semisimple by the decomposition theorem \cite{BBD}, since $m_n$ is proper and $\Q_{C_n}[nd]$ is pure. \end{proof} For $u\in(V_7^{\otimes n})^{\Gtwo}$ let $s_u:\IC_0\to\IC_\lambda^{*n}$ be the morphism attached to $u$ by the Satake equivalence \cite[Thm.~7.3]{MV}, and let $Z_u=\sum_Z c_Z(u)\,Z$ be the rational cycle with $[Z_u]=s_u$ under Lemma~\ref{lem:top}(c). For $n=3$, $u=t$: the space is one-dimensional, so $Z_t=c\,Z_{9}$ for the unique nine-dimensional component and some $c\in\Q^\times$. \paragraph{The framing torsors and the coefficients.} % [referee report 003, R3-02; audits YAUD2, YAUD2R, YAUD2R2] Over $U=B\tA$ the base point $u_0:\mathrm{pt}\to B\tA$ is an $\tA$-torsor. Let $X=\widetilde{\GR}{}^U_x$ be Yun's one-step space, with its source action $\tA(1)$ and target action $\tA(2)$ \cite[(3.5)--(3.6)]{Yun}. Factor $i$ of $P=X^n$ is step $i$ of the chain, so factor $1$ is adjacent to the fixed target $u_0$. In matrices we write the target first: by \cite[(3.9)]{Yun} and $f\otimes w\mapsto wf$, \[ H^*_c(X)_{\odd}\otimes\Qi\;\cong\;V_\chi^*\boxtimes V_\chi\otimes E_\chi\;\cong\;\Hom(V_{\rm source},V_{\rm target})\otimes E_\chi , \] row index $=$ target, column index $=$ source; rationally $\tA(2)$ acts on $D$ by $x\mapsto gx$ and $\tA(1)$ by $x\mapsto xg^{-1}$. The endpoint bookkeeping is: \begin{center} \small \begin{tabular}{p{1.9cm}p{3.2cm}p{2.9cm}p{3.1cm}} endpoint & Yun's action & action on coefficients & slot\\\hline $b_0=u_0$ & $\tA(2)$ of step $1$ & $gx_1$ & external target, row of slot $1$\\ $b_i$, $00$ on $W$), $\mu_x=\ker(T\to\Aut W)$ (finite), $D_W=f^{-1}(\mathcal B\setminus\mathcal U)$, $W^\circ=W\setminus D_W$, and $\cL_W=f^*\bF|_{[W^\circ/T]}$, a $T$-equivariant local system on $W^\circ$ with structure maps $\psi_g:\cL_{W,v}\to\cL_{W,gv}$ ($g\in T$), continuous in $g$, with $\psi_{gh}=\psi_g\psi_h$. For $g\in\mu_x$, $\psi_g$ is left multiplication on the fibre $D$ by $\kappa(g)$, where $\kappa:\mu_x\to\tA$ is the stabiliser homomorphism of the $\tA$-torsor: this is the twisted torus action with finite kernel. Put $P_W=Y\times_{\mathcal B}W$, a $T$-torsor over $Y\times_{\mathcal B}[W/T]$ on which $T$ acts freely. \begin{lemma}[One centre: the equivariant diagram]\label{lem:L1} Let $\mathbb G=\G_m\times T$ act on $V'=(W\setminus0)\times\A^1_t$ by $(\tau,g)(v,t)=(\bar\lambda(\tau)gv,\tau^{-1}t)$; let $\beta(v,t)=\bar\lambda(t)v$, $\gamma(v,t)=v$, $\varepsilon_0(\tau,g)=g$, $\varepsilon_1(\tau,g)=\bar\lambda(\tau)g$. Then $\beta$ is $\varepsilon_0$-equivariant, $\gamma$ is $\varepsilon_1$-equivariant, $[V'/\mathbb G]\to[W/T]$ is the weighted blow-up at $w$, in the equivariant chart form of \cite[Cor.~5.2.6]{QR}, and on $V'^\times=W^\circ\times\C^*_t$ \[ \Psi:\ \gamma^*\cL_W\xrightarrow{\ \sim\ }\beta^*\cL_W,\qquad \Psi_{(v,t)}=\psi_{\bar\lambda(t)}, \] is an isomorphism of $\mathbb G$-equivariant local systems, where $\gamma^*\cL_W$ carries the $\varepsilon_1$-structure and $\beta^*\cL_W$ the $\varepsilon_0$-structure. Consequently, on $V'^\natural=W^\circ\times\A^1_t$, $\beta^*\cL_W|_{t\neq0}$ extends to the $\mathbb G$-equivariant local system $\gamma^*\cL_W$, and for a point $q$ of $\mathcal X$ over $(v,0)$: \begin{enumerate} \item the stabiliser of $q$ contains $\mu_{e(v)}=\{\tau:\bar\lambda(\tau)v=v\}$, acting on $t$ by $\tau^{-1}$ and on $L^{\mathrm{st}}_q=\cL_{W,v}$ by $\psi_{\bar\lambda(\tau)}$; \item the meridian of $E$ at $q$ in $U_X$ is the orbifold loop formed by the path $t\mapsto e^{2\pi i\theta/e(v)}t$ and the element $\zeta_{e(v)}$, with monodromy $\rho(\ell_v)=\psi_{\bar\lambda(\zeta_{e(v)})}$; at a generic point $e(v)=e_0$, and this is $\rho(h)=\kappa(\bar\lambda(\zeta_{e_0}))$. \end{enumerate} \end{lemma} \begin{proof} $(\tau,g)$ sends $(v,t)$ to $(\bar\lambda(\tau)gv,\tau^{-1}t)$, and $\psi_{\bar\lambda(\tau^{-1}t)}\circ\psi_{\varepsilon_1(\tau,g)}=\psi_{\bar\lambda(t)g}=\psi_{\varepsilon_0(\tau,g)}\circ\psi_{\bar\lambda(t)}$ by the cocycle rule and commutativity of $T$; so $\Psi$ is equivariant. That $\beta$ is the weighted blow-up is the chart description \cite[Prop.~5.2.2, Cor.~5.2.5--5.2.6]{QR}, compatible with flat base change \cite[Cor.~3.2.14(iii)]{QR}, applied on an atlas. The stabiliser in $\mathbb G$ of a point $(p,(v,0))$ of $P_W\times_WV'$ has $g=1$, since $T$ acts freely on $P_W$, and then $\tau\in\mu_{e(v)}$. So $\mathcal X$ is Deligne--Mumford there, and (1) holds by the $\varepsilon_1$-structure. Off $E$ the $\mathbb G$-action is free; a loop in the quotient is a path in $P'^\times$ ending at a $\mathbb G$-translate of its start, and the path displayed in (2), closed by $(\zeta_{e(v)},1)$, is the meridian. Transport of $\gamma^*\cL_W$ along it is trivial, because $\gamma^*\cL_W$ is constant in $t$, and the group element acts by $\psi_{\varepsilon_1(\zeta,1)}=\psi_{\bar\lambda(\zeta)}$. \end{proof} Here $\Psi$ is an isomorphism of local systems on the stack $[V'^\times/\mathbb G]$, not an identity of fundamental-group symbols; all monodromies and stalks are read from it, and the finite isotropy enters only through the stabiliser action in (1). % [referee report 003, R3-03; audit hodge-YAUD3] -- begin \begin{lemma}[Descent of radial extensions]\label{lem:radialdescent} Let $\mathfrak M$ be a smooth complex algebraic stack, let $j:\mathfrak O\hookrightarrow\mathfrak M$ be an open immersion whose complement has positive codimension on every smooth atlas, and let $L$ be a local system on $\mathfrak O$. Let $p:V\to\mathfrak M$ be a smooth surjective scheme atlas. Suppose a local system $L_V$ on $V$ is given together with an isomorphism $\alpha:L_V|_{V^\circ}\xrightarrow{\sim}p^*L$, where $V^\circ=V\times_{\mathfrak M}\mathfrak O$. Then there is a unique descent datum on $L_V$ extending that of $L$. It is effective, and yields a local system $\overline L$ on $\mathfrak M$ with the specified identification on $\mathfrak O$, unique up to a unique isomorphism respecting that identification. \end{lemma} \begin{proof} For a smooth complex space $Z$ and a closed analytic subset $A$ of positive complex codimension, restriction of local systems from $Z$ to $Z\setminus A$ is fully faithful: on a small ball both local systems are trivial, the complement of $A$ in the ball is connected, so a morphism on the complement is a constant matrix and extends uniquely, and these local extensions agree. This is local for smooth covers and applies to algebraic spaces. Put $V_2=V\times_{\mathfrak M}V$ with projections $p_1,p_2$. On $V_2^\circ$, using the canonical identification of the two pullbacks of $L$, define $\theta_{12}^{\circ}=(p_2^*\alpha)^{-1}(p_1^*\alpha):p_1^*L_V\to p_2^*L_V$. The space $V_2$ is smooth, so full faithfulness extends $\theta^\circ_{12}$ uniquely to $\theta_{12}$ on $V_2$, and its inverse extends too. On $V_3=V\times_{\mathfrak M}V\times_{\mathfrak M}V$ the equality $\theta_{23}\theta_{12}=\theta_{13}$ holds on $V_3^\circ$, hence everywhere by faithfulness; the identity and inverse conditions follow in the same way. Descent of local systems along the smooth atlas is effective. The whole \v{C}ech groupoid, including self-overlaps, is used; it records isotropy. No agreement, conjugacy or descent of the radial maps themselves is required: only their identified pullbacks of the sheaf on the common open enter. \end{proof} We apply Lemma~\ref{lem:radialdescent} with $\mathfrak M=\mathcal B_n^\natural$, $\mathfrak O=\mathcal U$ and $L_V=g_V^*\bF$, which is defined on the whole chart by $(I_n)$. Over the complement of all exceptional divisors the iterated equivariance maps of Lemma~\ref{lem:L1} give $\alpha_V$ to the pullback along the blow-down, a single global map, so the comparisons have a common target. The exceptional complement is dense in every chart and every overlap, since the exceptional equations are independent smooth parameters in $(I_n)$. At an intersection of $r$ exceptional divisors, away from the strict boundary, on a small chart with equations $s_1\cdots s_r=0$ both extended local systems are trivial on the polydisc; on $(\Delta^*)^r\times\Delta^{d-r}$ their comparison is one constant matrix, every generator of $\pi_1=\Z^r$ acts trivially on each and on the comparison, and there is no further triple-intersection condition. \emph{Trivial meridians here are meridians on smooth atlases of the stack, not on its coarse space:} on $[\Delta/\mu_e]$ with $t\mapsto\zeta t$, a constant local system upstairs with stabiliser representation $\chi$ extends across $t=0$, while a once-around coarse loop has monodromy $\chi(\zeta)^{\pm1}$ (for $e=2$ and the sign character, $-1$). Coarse stalks are computed by invariants, not by an assertion that coarse meridians act trivially (Lemma~\ref{lem:link}). % [referee report 003, R3-03; audit hodge-YAUD3] -- end \begin{lemma}[Extension across the exceptional loci]\label{lem:link} For every $n$ there is a local system $\bF_n$ on the open substack $\mathcal B_n^\natural=\mathcal B_n\setminus\widetilde{\mathcal D}$ (the complement of the strict transform of $\mathcal B\setminus\mathcal U$) extending $\bF$ on $\mathcal U$, with $\bF_n|_V\cong g_V^*\bF$ on every chart $V$ of $(I_n)$, compatibly on overlaps. Its pullback $L^{\mathrm{st}}$ to the Deligne--Mumford stack $\mathcal X_n^\natural=Y\times_{\mathcal B}\mathcal B_n^\natural$ extends $\bL$, and on the coarse space \[ R^0j_*\bL\,|_{X_n^\natural}\;=\;\pi_*^{\mathrm{inv}}L^{\mathrm{st}}, \] where $\pi:\mathcal X_n^\natural\to X_n^\natural$ is the coarse map and $\pi_*^{\mathrm{inv}}$ takes invariants of the finite stabilisers (exact over $\Q$). In particular $\mathcal F_q=(L^{\mathrm{st}}_q)^{H_q}=D^{I_q}$, where $H_q$ is the stabiliser of $q$ in $\mathcal X$, acting on $L^{\mathrm{st}}_q=D$ by left multiplication through $H_q\to\Stab_{\mathcal B_n}(\bar q)\to\Stab_{\mathcal B}(g_V(q))$, and $I_q$ is the image. \end{lemma} \begin{proof} By $(I_n)$ the strict transform of $\mathcal B\setminus\mathcal U$ on a chart $V$ is $g_V^{-1}(\mathcal B\setminus\mathcal U)$, so $V\cap\mathcal B_n^\natural=g_V^{-1}(\mathcal U)$ and $g_V^*\bF$ is defined there. On $V^\times=V\setminus(\text{all boundary})$ the blow-down $b_V:V\to\mathcal B$ and $g_V$ are related by the isomorphisms $\Psi$ of Lemma~\ref{lem:L1}, composed over the earlier centres; the $b_V$ glue, so the $\Psi$ give gluing isomorphisms $g_{V_1}^*\bF\cong g_{V_2}^*\bF$ on $V_{12}^\times$ satisfying the cocycle condition there. By full faithfulness of restriction of local systems on smooth charts, these extend uniquely over $V_{12}\cap\mathcal B_n^\natural$ and the cocycle identity extends, which is Lemma~\ref{lem:radialdescent}. % [referee report 003, R3-03; audit hodge-YAUD3] On the coarse space: near $q$, $\mathcal X$ is $[Z/H_q]$ with $Z$ smooth and contractible and the boundary a union of coordinate hyperplanes; $\pi_1(N_q\cap U_X)\to\pi_1^{\mathrm{orb}}([Z/H_q])=H_q$ is surjective, and $\bL$ is the restriction of $L^{\mathrm{st}}$, whose monodromy on $Z$ is trivial; so the invariants of $\pi_1(N_q\cap U_X)$ on $\bL$ are the $H_q$-invariants of $L^{\mathrm{st}}_q$. \end{proof} % [referee report 003, R3-03; audit hodge-YAUD3] -- begin \emph{Several old exceptional branches.} At a multiple crossing in the natural locus write a local DM chart as $[Z/H]$, with $Z$ a small contractible smooth neighbourhood, each exceptional branch preserved, and the coefficient system given by $\rho:H\to\operatorname{GL}(D)$. The atlas meridians act trivially, and on the complement $\pi_1(Z^\times)\to\pi_1^{\mathrm{orb}}([Z^\times/H])\to H\to1$ is exact ($Z^\times$ is connected), the coefficient representation killing the left-hand group. So the local invariant space is exactly $D^{\rho(H)}$, including at simultaneous old/new crossings, and $R^0j_*\bL=\pi_*L^{\mathrm{st}}$ with $(R^0j_*\bL)_q=(L^{\mathrm{st}}_{\tilde q})^{H_q}$, as an identification of sheaves via the restriction map. Since every nontrivial subgroup of $Q_8$ contains $z=-1$, $D^{\rho(H_q)}$ is $0$ if the image is nontrivial and $D$ otherwise. This image is the image of the actual local group, not the full stabiliser of the radial point of $\mathcal U$; the transported meridian images $c_i=\rho(h_i)$ of Lemma~\ref{lem:L1} lie in it, and no canonical inclusion of the old generic stabilisers into $H_q$ is used. % [referee report 003, R3-03; audit hodge-YAUD3] -- end % [referee report 002, R2-01, R2-04] % [referee report 005, R5-01; audit hodge-YAUD10]: stale I_q containment deleted \begin{corollary}[Stalks]\label{cor:stalks} At every $q\in\partial$ off $\widetilde D$, $\mathcal F_q=D^{I_q}$, with $I_q$ the image of the actual local group (local monodromy); it is $D$ if $I_q=1$ and $0$ otherwise. On $\widetilde D$ the stalk is $0$: at the end of the sequence a point of $\widetilde D$ maps to a codimension-one point of $\mathcal B\setminus\mathcal U$, its local image contains the original meridian $m_D$, and $D^{m_D}=0$ by cleanness. \end{corollary} \begin{lemma}[Generalised Duality Lemma]\label{lem:duality} Let $x=x_m$ with slice $W$, contracting cocharacter $\bar\lambda$ of weights $a_1,\dots,a_c>0$, and $\bar\lambda$-stable boundary $D_W$. Let $M=S_\epsilon\setminus D_W$ for a $\bar\lambda$-adapted sphere $S_\epsilon$ about $0$, let $B^\circ_x=(W\setminus(0\cup D_W))/\C^*$, an orbifold open in $\mathbb P(a)$, and let $\varpi:M\to B^\circ_x$ be the Seifert map, with regular fibre the orbit loop $h$ and exceptional fibres $\ell$ with $\ell^k=h$. Let $\mathcal M$ be a local system of $\Q$-vector spaces on $M$ (carrying, if one wishes, a compatible module structure over a semisimple $\Q$-algebra such as $D^{\mathrm{op}}$) with finite monodromy such that $H^*(M,\mathcal M)=0$, and let $\mathcal R=R^0\varpi_*\mathcal M$, with stalks $\mathcal M^{\langle h\rangle}$ generically and $\mathcal M^{\langle\ell\rangle}$ at orbifold points. Then $H^*(B^\circ_x,\mathcal R)=0$ and $H^*_c(B^\circ_x,\mathcal R)=0$, whatever the image of $h$. % [referee report 002, R2-04] \end{lemma} \begin{proof} Since $h$ is central in the local fundamental group, $\mathcal M=\mathcal M^{\bar h}\oplus\mathcal M'$ as local systems, where $\mathcal M^{\bar h}$ is the part on which $h$ acts trivially. Every Leray sheaf of $\mathcal M'$ vanishes (at an exceptional orbit, $\mathcal M'^{\ell}\subset\mathcal M'^{h}=0$), and $\mathcal M^{\bar h}$ descends to the orbifold base; $H^*(M,\mathcal M^{\bar h})$ is a summand of $H^*(M,\mathcal M)=0$. So we may assume $\mathcal M=\mathcal M^{\bar h}$, pulled back from the base. [UDEFECT-X, D3] Rationally the Leray sheaves of $\varpi$ are then $R^0=\mathcal R$ and $R^1\cong\mathcal R$: for a finite-order operator on a $\Q$-vector space invariants equal coinvariants, and the orientation of the $\C^*$-orbits trivialises $R^1\varpi_*\Q$ rationally, including at exceptional fibres, where $H^1(\ell)\to H^1(h)$ is multiplication by $k$. The Leray spectral sequence has two rows, and $d_2$ is cup product with the rational Euler class $e\in H^2(B^\circ_x,\Q)$, giving the Gysin sequence \[ \cdots\to H^{k}(B^\circ_x,\mathcal R)\to H^{k}(M,\mathcal M)\to H^{k-1}(B^\circ_x,\mathcal R)\xrightarrow{\cup e}H^{k+1}(B^\circ_x,\mathcal R)\to\cdots. \] Since $H^*(M,\mathcal M)=0$, $\cup e$ is an isomorphism $H^{k-1}\to H^{k+1}$ for all $k$, and induction from $H^{-1}=H^{-2}=0$ gives $H^*(B^\circ_x,\mathcal R)=0$. % [referee report 002, R2-04] \emph{Compact supports.} Finite monodromy gives an invariant positive definite symmetric form on $\mathcal M$ (average any one over the monodromy group; for $\mathcal M\subset\bL$ one may take $\trd(x\bar y)$, invariant under left multiplication by $\pm1,\pm i,\pm j,\pm k$, for which the splitting $\mathcal M=\mathcal M^{\bar h}\oplus\mathcal M'$ is orthogonal). Hence $\mathcal M\cong\mathcal M^\vee$, and $\mathcal M^\vee$ satisfies the hypothesis. Let $d=\dim_\C B^\circ_x$. Locally on the coarse space, $B^\circ_x=\C^d/H$ and $\mathcal R=q^H_*\mathcal E$ for a local system $\mathcal E$ on $\C^d$, with $q$ finite; at orbifold points $\mathcal R$ is a constructible sheaf, not a local system, and we \emph{define} $\mathcal R^\vee:=(q_*\mathcal E^\vee)^H$, not $\mathcal Hom(\mathcal R,\Q)$. % [R2LINK-X 1-b] The form restricts to a nondegenerate form on the invariants of every finite group, so $R^0\varpi_*(\mathcal M^\vee)\cong\mathcal R^\vee$, with stalks $(\mathcal M^\vee)^{\langle\ell\rangle}=(\mathcal M^{\langle\ell\rangle})^\vee$. Since $\mathbb D\circ q_*=q_*\circ\mathbb D$ ($q$ finite) and $H$-invariants commute with $\mathbb D$ (an exact functor), $\mathbb D_{B^\circ_x}\mathcal R\cong\mathcal R^\vee[2d]$. Verdier duality then gives $H^i_c(B^\circ_x,\mathcal R)^\vee\cong H^{2d-i}(B^\circ_x,\mathcal R^\vee)=H^{2d-i}(B^\circ_x,R^0\varpi_*\mathcal M^\vee)=0$, by the first half applied to $\mathcal M^\vee$. The image of $h$ enters only through the rank of $\mathcal R$: if $\rho(h)$ has no fixed vector (for $\Gtwo$, $z$ or any element of order four), $\mathcal R$ vanishes generically. [referee report 001, U2] \end{proof} % [referee report 003, R3-03; audit hodge-YAUD3] -- begin In the next lemma and the proof of Theorem~\ref{thm:defect}, $\mathfrak X_n$ and $\mathfrak\Sigma_k$ denote stacks and $X_n,\Sigma_k$ their coarse spaces; all blow-ups are stack-theoretic. \begin{lemma}[Exceptional fibre on a smooth refinement]\label{lem:excfibre} Let $\mathfrak M=\mathfrak X_{k-1}$ and let $I_\bullet$ be its next regular weighted centre, supported on $\mathfrak S$. Suppose that on an atlas $V\to\mathfrak M$ the filtration is $g_V^*F_\bullet$, where $g_V:V\to\mathcal U_{k-1}$ is smooth, and that $(g_V,s_1,\ldots,s_r)$ is smooth with the $s_i$ the old exceptional equations. Let $W=\A^c\to\mathcal U_{k-1}$ be the torus chart at $x_k$, with weights $a_1,\ldots,a_c>0$. Put $C=V\times_{\mathcal U_{k-1}}W$ and $C_0=C\times_W\{0\}$. After covering algebraic spaces by schemes if necessary, $C\to V$ is smooth and covers a neighbourhood of the centre, and there are Cartesian diagrams \[ \begin{array}{ccc} \widetilde C=C\times_W\operatorname{Bl}^{\mathrm{st}}_{a}(W)&\longrightarrow&\operatorname{Bl}^{\mathrm{st}}_{a}(W)\\ \downarrow&&\downarrow\\ C&\longrightarrow&W, \end{array} \qquad \begin{array}{ccc} E_C=C_0\times\PP^{\mathrm{st}}(a)&\longrightarrow&\mathfrak E_k\\ \downarrow&&\downarrow\\ C_0&\longrightarrow&\mathfrak S, \end{array} \] where $\mathfrak E_k=\Proj^{\mathrm{st}}_{\mathfrak S}\bigl(\bigoplus_{m\ge0}I_m/I_{m+1}\bigr)$ is the exceptional divisor at stage $k$. On this refinement the open complement of the transformed boundary in $\mathfrak E_k$ is $C_0\times\mathscr B_x$, with $\mathscr B_x=[W^\circ/\G_m]$ for the weighted action. The old exceptional equations remain equations on $C_0$, and the formula also holds where any subset of them vanishes. \end{lemma} \begin{proof} Both projections of $C$ are smooth, as base changes of the two smooth maps to $\mathcal U_{k-1}$, and its centre is $C_0$. Flatness identifies the pulled-back Rees algebra and its associated graded with those on $W$. In the radial presentation $y_i=t^{a_i}v_i$ with $v\ne0$; setting $t=0$ gives $y_i=0$ and the weighted quotient of $v$, hence $C_0\times\PP^{\mathrm{st}}(a)$. The boundary ideal, saturated by $t$, is the pullback of the ideal evaluated at $v$, whose complement is $W^\circ$. Smoothness of $(g_V,s)$ makes the $s_i$ independent parameters relative to $W$, so imposing $s_i=0$ does not change the calculation. The construction uses the current stack atlas, not the original chart. The torus $T_x$ supplies the contracting coordinates and the equivariance of Lemma~\ref{lem:L1}; the full stabiliser need not act linearly on $W$, and its action on the weighted normal cone (through $\operatorname{gr}_F$) can be non-linear in weighted coordinates (in weights $(1,2)$, $u\mapsto u$, $v\mapsto v+bu^2$ is degree-preserving with identity tangent action). Only the intrinsic graded algebra on the centre is used. \end{proof} % [referee report 003, R3-03; audit hodge-YAUD3] -- end % [referee report 004, R4-02; audit hodge-YAUD8] -- begin \begin{lemma}[Coarse fibre descent]\label{lem:coarsefibre} Let $f:\mathfrak A\to\mathfrak S$ be a morphism of separated finite-type Deligne--Mumford stacks over $\C$, and $L$ a rational local system on $\mathfrak A$. Write $\pi_A,\pi_S$ for the coarse maps and $\bar f:A\to S$ for the induced map. For $\sigma\in S(\C)$ choose an object $s$ above it with stabiliser $H$, and put $\mathfrak T=\mathfrak A\times_{\mathfrak S}BH$ and $\mathfrak F=\mathfrak T\times_{BH}\Spec\C$. Let $Z$ be the coarse space of $\mathfrak F$ and $\mathcal R=\pi_{F*}(L|_{\mathfrak F})$. Then on analytic spaces \[ R\Gamma_c\bigl(\bar f^{-1}(\sigma),(\pi_{A*}L)|_{\bar f^{-1}(\sigma)}\bigr)\simeq\bigl(R\Gamma_c(Z,\mathcal R)\bigr)^H, \] the action being induced by descent. No splitting of stabiliser extensions is required. \end{lemma} \begin{proof} The map $q:\mathfrak F\to\mathfrak T$ is the base change of $\Spec\C\to BH$, a representable finite \'etale $H$-torsor. Its deck transformations induce an action on $Z$ and $\mathcal R$; coherence isomorphisms of the stack action become identities on the coarse pushforward, whose stalks already take invariants of internal inertia, so this is an honest action on the pair $(Z,\mathcal R)$. If $T$ is the coarse space of $\mathfrak T$, finite descent gives $T=Z/H$ and $\pi_{T*}(L|_{\mathfrak T})=(\rho_*\mathcal R)^H$ for the finite quotient $\rho:Z\to T$; on finite-quotient charts this takes first kernel invariants and then quotient invariants, and no semidirect product is needed. The reduced fibre of $\mathfrak S\to S$ at $\sigma$ is $BH$, so $\mathfrak T$ and the fibre of $\mathfrak A\to S$ have the same reduced underlying stack; tame coarse base change \cite[Thm.~3.2(d), Cor.~3.3(a)]{AOV} identifies their coarse analytic space with $\bar f^{-1}(\sigma)$, and nilpotents do not affect analytic sheaves. Restriction of $\pi_{A*}L$ agrees with the displayed coarse pushforward: on $[U/G]\to U/G$, restriction to the invariant inverse image of a closed fibre commutes with finite proper pushforward and exact rational $G$-invariants. Finally $\rho$ is finite and proper, so its pushforward commutes with compact supports, and the averaging idempotent of $H$ commutes with derived compactly supported sections. \end{proof} % [referee report 004, R4-02; audit hodge-YAUD8] -- end % [referee report 002, R2-04] \begin{proof}[Proof of Theorem~\ref{thm:defect}] For $1\le k\le N$ let $Z_k=\bigcup_{i\ge k}E_i$ (closed, $Z_{N+1}=\emptyset$), $i_k:Z_{k+1}\hookrightarrow Z_k$ and $u_k:A^\flat_k=Z_k\setminus Z_{k+1}\hookrightarrow Z_k$. The triangles $u_{k!}u_k^*\mathcal F\to\mathcal F|_{Z_k}\to i_{k*}i_k^*\mathcal F\xrightarrow{+1}$ and, with $v:\partial\setminus Z_1\hookrightarrow\partial$, $v_!v^*\mathcal F\to\mathcal F\to\mathcal F|_{Z_1}\xrightarrow{+1}$ give, all $Z_k$ being compact, \begin{gather*} H^*_c(\partial\setminus Z_1,\mathcal F)\to H^*(\partial,\mathcal F)\to H^*(Z_1,\mathcal F)\to,\\ H^*_c(A^\flat_k,\mathcal F)\to H^*(Z_k,\mathcal F)\to H^*(Z_{k+1},\mathcal F)\to . \end{gather*} \textup{(a)} $\partial\setminus Z_1\subset\widetilde D$, where $\mathcal F=0$ (Corollary~\ref{cor:stalks}). By downward induction on $k$ the theorem reduces to $H^*_c(A^\flat_k,\mathcal F)=0$ for every $k$. \textup{(b)} $A^\flat_k=A_k\sqcup(A^\flat_k\cap\widetilde D)$, with $A_k=\{q\in A^\flat_k:\ g(q)\in\mathcal U\}$ open: if $g(q)$ is a codimension-one point of $\mathcal B\setminus\mathcal U$ then $q\in\widetilde D$ by $(I_N)$, and if $g(q)=x_m$ with $m>k$ then $q$ lies in the centre of a later blow-up, so its preimage lies in $Z_{k+1}$. Since $\mathcal F|_{A^\flat_k\cap\widetilde D}=0$, $\mathcal F|_{A^\flat_k}=w_!(\mathcal F|_{A_k})$ for $w:A_k\hookrightarrow A^\flat_k$, so $H^*_c(A^\flat_k,\mathcal F)=H^*_c(A_k,\mathcal F)$. % [referee report 003, R3-03; audit hodge-YAUD3] -- begin (replaces the former (c)--(f)) \textup{(c)} The later blow-ups are isomorphisms off their centres, and on the exceptional open where the radial image lies in $\mathcal U$ all later centre ideals are units; so $A_k$ is the coarse space of the open $\mathfrak A_k\subset\mathfrak E_k$ of Lemma~\ref{lem:excfibre}, and $\mathcal F$ near it is computed at stage $k$, since $R^0j_*$ is local. Let $f_k:\mathfrak A_k\to\mathfrak\Sigma_k$ be the restriction of the blow-down. No global torsor over $\mathfrak\Sigma_k$, no linear action of the full stabiliser on $T^1$, and no inclusion of old generic stabilisers in a chart stabiliser is used. % [referee report 004, R4-02; audit hodge-YAUD8] \textup{(d)} \emph{Coefficients on the fibres.} Let $\mathscr L_x$ be the local system on $\mathscr B_x=[W^\circ/\G_m]$ obtained by restricting the $T_x$-equivariance of $\cL_W$ along $\bar\lambda$. At $t=0$ the radial chart $P$ of Lemma~\ref{lem:globalatlas} is $C_0\times(W\setminus0)$, with natural part $C_0\times W^\circ$, on which $g_P(c,v,0)=v$ and the coefficient is $\mathrm{pr}_2^*\cL_W$ by Lemma~\ref{lem:globalatlas}(c). Compare the equivariance maps on $P^\natural\times\G_m$, where both sheaves exist: Lemma~\ref{lem:L1} proves their equality at $t\ne0$, and faithfulness extends it; the right-hand action is precisely $\varepsilon_1$. Hence \[ L_k^{\mathrm{st}}|_{C_0\times\mathscr B_x}\simeq\mathrm{pr}_2^*\mathscr L_x,\qquad\mathscr B_x=[W^\circ/\G_m], \] canonically relative to the chosen chart and its specified open comparison, and compatibly with all groupoid transitions; it is not a chart-independent trivialization over the centre. Setting any old $s_i$ to zero preserves it. Trivial atlas meridians do not imply trivial coarse meridians, so the $\varepsilon_1$-action followed by invariants remains essential. \textup{(e)} \emph{The fibre vanishes on the stack.} % [referee report 004, R4-02; audit hodge-YAUD8] The Betti form of Yun's cleanness (Theorem~\ref{thm:clean}, an $\ell$-adic statement in \cite{Yun}) at points of $\mathcal B$ is supplied by Lemma~\ref{lem:ratclean} on finite-type smooth charts; it is not proved by the weighted blow-ups. % [referee report 004, R4-03; audits hodge-YAUD7, hodge-YAUD9] Given it, cleanness at $x_k$ (by smooth base change, and the contracting weighted action, which identifies the punctured local complement with its adapted link times a contractible factor) gives $R\Gamma(W^\circ,\cL_W)=0$. The map $W^\circ\to\mathscr B_x$ is a principal $\G_m$-bundle over the stack, even where the coarse action has finite stabilisers, and $\cL_W$ is the pullback of $\mathscr L_x$; the Gysin sequence $\cdots\to H^{j-2}(\mathscr B_x,\mathscr L_x)\xrightarrow{e}H^j(\mathscr B_x,\mathscr L_x)\to H^j(W^\circ,\cL_W)\to H^{j-1}(\mathscr B_x,\mathscr L_x)\to\cdots$ then gives $H^*(\mathscr B_x,\mathscr L_x)=0$ by induction from negative degrees. Finite monodromy admits an invariant positive definite form, so the dual has the same vanishing, and duality on the smooth DM stack of dimension $c-1$ gives \[ R\Gamma_c(\mathscr B_x,\mathscr L_x)=0.\tag{$*$} \] Passing to the coarse weighted projective open takes finite isotropy invariants, with coefficient sheaf $R^0\varpi_*\cL_W|_M$; so $(*)$ has exactly the coefficients of Lemma~\ref{lem:duality}, including exceptional orbit points, and does not assume that the coarse fibre loop has trivial image. % [referee report 004, R4-02; audit hodge-YAUD8] -- begin (replaces the former (f)) \textup{(f)} \emph{Coarse vanishing.} Take $V$ an atlas of the current Deligne--Mumford stage $\mathfrak X_{k-1}=Y\times_{\mathcal B}\mathcal B_{k-1}$, obtained by smooth base change of the atlas of $(I_{k-1})$ and scheme refinement; form $C=V\times_{U_{k-1}}W$ and $C_0=C\times_W\{0\}$. These $C_0$ cover the Deligne--Mumford centre $\mathfrak S=\mathfrak\Sigma_k$, and the weighted-normal-cone calculation ($y_i=t^{a_i}v_i$, then $t=0$; old equations stay on $C_0$) gives the Cartesian square \[ \begin{array}{ccc} C_0\times\mathscr B_x&\longrightarrow&\mathfrak A_k\\ \downarrow{\scriptstyle\mathrm{pr}_1}&&\downarrow{\scriptstyle f_k}\\ C_0&\longrightarrow&\mathfrak S. \end{array} \] So the fibre over an object $s:\Spec\C\to\mathfrak S$ is the link stack $\mathscr B_x$ with the coefficient $\mathscr L_x$ of (d); the factor $Y$ is absorbed into the choice of $s$. The coarse spaces are separated and of finite type, hence Hausdorff and locally compact, and the stalk of $R\bar f_!$ is compactly supported cohomology of the fibre for maps of locally compact spaces, with no properness or manifold hypothesis \cite[Thm.~4.5.3, (4.5.4)]{Schapira}. By Lemma~\ref{lem:link}, $\mathcal F|_{A_k}=\pi_{A*}L_k^{\mathrm{st}}$. Lemma~\ref{lem:coarsefibre} and $(*)$, in the coarse form of Lemma~\ref{lem:duality} with $\mathcal R=R^0\varpi_*\cL_W|_M$, give \[ \bigl(R\bar f_{k!}(\mathcal F|_{A_k})\bigr)_\sigma\simeq R\Gamma_c\bigl(\bar f_k^{-1}(\sigma),\mathcal F\bigr)\simeq\bigl(R\Gamma_c(B^\circ_{x_k},\mathcal R)\bigr)^{H_\sigma}=0. \tag{$**$} \] Hence $R\bar f_{k!}(\mathcal F|_{A_k})=0$ and, composing proper-support direct images, $R\Gamma_c(A_k,\mathcal F)=0$. Only the existence of the link identification over one object above each point enters this vanishing; the compatibility of (d) was needed to identify the global coefficient sheaf. This is the coarse statement the defect needs; no stack six-functor statement for the nonrepresentable $f_k$ is used, and $(**)$ replaces the former stack-level formula $Rf_{k!}L^{\mathrm{st}}_k=0$. On the stage-$k$ exceptional natural open all later centre ideals are units by $(I_N)$, so that open survives unchanged, and the inverse images of later centres lie in the later exceptional union. After discarding the strict-boundary zero stalks, the largest-index pieces in (a)--(c) are exactly these surviving opens. On the final strict boundary a codimension-one transverse chart retains the original nontrivial odd meridian, so its stalks vanish. The triangles for $Z_k$ in (a) now give $H^*(\partial,\mathcal F)=0$ for the constructed sequence, without assuming Definition~\ref{def:wclean}. % [referee report 004, R4-02; audit hodge-YAUD8] -- end % [referee report 003, R3-03; audit hodge-YAUD3] -- end \end{proof} \begin{remark}[What the proof uses] The proof uses the weighted structure at each centre (the Seifert link, and the fibration of $A_k$ over $\Sigma_k$ with fibre $B^\circ_{x_k}$), the extension of the local system across the exceptional loci along $(I_n)$ (Lemmas~\ref{lem:L1} and~\ref{lem:link}), and Yun's cleanness at every point of $\mathcal B$. It uses no condition on divisors with trivial inertia or on their crossings. The $\Gtwo$ group theory (every nontrivial subgroup of $Q_8$ contains $z$) enters only in Corollary~\ref{cor:stalks}. % [referee report 002, R2-04] Purity of $H^k(R)$ (weight $k$, since $R$ is projective with quotient singularities, so $H^*(R)=IH^*(R)$) is not used in Theorem~\ref{thm:defect}; it is used only in the Hodge-theoretic statements built on it. [referee report 001, U3] \end{remark} \begin{example}[A clean local system with nonzero defect on a non-weighted resolution]\label{ex:tacnode} On $\C^2$ let $C=C_1\cup C_2$ with $C_1=\{x=0\}$ and $C_2=\{x=y^2\}$, a tacnode. Then $H_1(\C^2\setminus C)=\Z m_1\oplus\Z m_2$; let $L$ be the rank-one $\Qi$-local system with $m_1\mapsto\zeta=i$ and $m_2\mapsto\zeta^{-1}$. (As a rational local system, by restriction of scalars, it has rank two and every group below is the same $\Qi$-space regarded over $\Q$. Inside $Q_8$ take $m_1\mapsto\mathbf i$ and $m_2\mapsto\mathbf i^{-1}$; then $D\otimes\C=(L_\C\oplus\overline L_\C)^{\oplus2}$ as a left $\langle\mathbf i\rangle$-module, and rationally $D=(\operatorname{Res}_{\Qi/\Q}L)^{\oplus2}$, so the defects below are multiplied by two.) % [referee report 002, R2-01, R2-08] \emph{$L$ is clean at $0$.} The germ is quasi-homogeneous for $\mathrm{wt}(y)=1$, $\mathrm{wt}(x)=2$, both branches of weighted degree $2$; the central loop is $h=2m_1+2m_2$, so $\rho(h)=1$, and the exceptional orbit $\{y=0\}$ (isotropy $\mu_2$, off $C$) has $[\ell]=m_1+m_2$, so $\rho(\ell)=1$. On $B^\circ=\mathbb P(1,2)\setminus\{[C_1],[C_2]\}\cong\C^*$ the rank-one local system $R^0\varpi_*L$ has monodromy $\zeta\ne1$ around $[C_1]$, so its cohomology vanishes and the Gysin sequence gives $H^*(M,L)=0$. \emph{On the weighted model the defect is $0$.} The $(1,2)$-weighted blow-up has exceptional divisor $\mathbb P(1,2)$, meeting $C_1,C_2$ in two distinct points, and Lemma~\ref{lem:duality} gives $H^*_c(B^\circ,\overline L)=0$. \emph{On the ordinary resolution the defect is nonzero.} Blowing up $0$ (chart $x=yu$) makes the strict transforms $\{u=0\}$, $\{u=y\}$ and $E_1=\{y=0\}$ pass through one point transversally in pairs; blowing that point up gives $E_2$, and now $\widetilde E_1,\widetilde C_1,\widetilde C_2$ meet $E_2$ in three distinct points. Since $\mathrm{ord}_{E_1}(x)=\mathrm{ord}_{E_1}(x-y^2)=1$ and $\mathrm{ord}_{E_2}(x)=\mathrm{ord}_{E_2}(x-y^2)=2$, both $E_1$ and $E_2$ have trivial inertia, and they cross. The bad locus is $B^\circ=\widetilde E_1\cup(E_2\setminus\{2\text{ points}\})$; $\widetilde E_1\cong\PP^1$ is closed in it with $\overline L$ trivial there, and its complement $O=E_2\setminus\{3\text{ points}\}$ carries $\overline L$ with monodromies $\zeta,\zeta^{-1},1$. Then $H^*_c(O,\overline L)$ is $\Qi$ in degree $1$ and $0$ otherwise, and the exact sequence \[ 0=H^2_c(O,\overline L)\to H^2_c(B^\circ,\overline L)\to H^2(\PP^1,\Qi)\to H^3_c(O,\overline L)=0 \] gives $H^2_c(B^\circ,\overline L)\cong \Qi(-1)\ne0$ (that is, $\Q(-1)^2$ as a rational Hodge structure); in degrees $0$ and $1$ the connecting map from $H^0(\widetilde E_1,\Qi)$ is an isomorphism, so $H^0_c$ and $H^1_c$ vanish. [outside referee hodge-PREF1, section 4.2] So the same clean local system has zero defect on the weighted model and nonzero defect on an ordinary simple normal crossing resolution: on the latter, $E_1$ is not the exceptional divisor of a weighted structure at a stratum and no Seifert link attaches to it. [referee report 001, U4] % [referee report 002, R2-03] Definition~\ref{def:good} is stated only for $(Y,D_x)$, so this example is not literally an instance of it, and there is no contradiction with any statement of this paper. Transplanted to $(\C^2,C)$, however, the ordinary two-step resolution satisfies every clause of that definition (smooth with trivial stabilisers; smooth, individually labelled components meeting as coordinate hyperplanes; an isomorphism off $C$). So the implication ``good model $\Rightarrow$ zero defect'' is false as a general principle, and Example~\ref{ex:blowupB} shows that it also fails for $(Y,D_x)$ itself. \end{example} % [referee report 002, R2-03] \begin{example}[A good model of $(Y,D_x)$ with nonzero defect]\label{ex:blowupB} % [R2LINK-X 2-b: unconditional construction] \emph{Step 1: a divisor with trivial inertia.} Fix a centre $x_m$ and let $S=\widetilde D\cap E_m$ on the weighted clean model; $S\ne\emptyset$, since it is the weighted projectivisation of the cone $D_W$, of dimension $c-2\ge0$. At a generic point of $S$ the local group is $\Z^2$, generated by the meridians of $\widetilde D$ and $E_m$, with commuting images $a=\rho(m_D)\ne1$ and $b=c_m$. Abelian subgroups of $Q_8$ are cyclic, so $a=g^\alpha$, $b=g^\beta$ for some $g$ of order $n\mid4$. Iterated blow-ups of codimension-two crossing strata over $S$ (a regular subdivision of the fan of $\Z_{\ge0}^2$) produce, for every primitive $(p,q)\in\Z^2_{>0}$, a divisor $E'$ whose meridian has image $a^pb^q$. The congruence $\alpha p+\beta q\equiv0\pmod n$ has a coprime solution with $p,q\ge1$: if $\gcd(\beta,n)\mid\alpha$ take $p=1$; if $\gcd(\alpha,n)\mid\beta$ take $q=1$; otherwise $n=4$, $\gcd(\alpha,4)=\gcd(\beta,4)=2$, and $(p,q)=(1,1)$ works. Then $I=1$ at generic points of $E'$. Each step keeps a good model: the centres are smooth strata of the boundary, the map stays an isomorphism over $B\setminus D_x$, the components stay labelled, and the stabilisers fix branches. \emph{Step 2: the point blow-up.} Choose $q_0\in E'$ generic, off the other components, so $I_{q_0}=1$, and blow up $q_0$ (or its residual gerbe, if it has a stabiliser $H$, which acts trivially on the cohomology of the exceptional $\PP^5$). The result $\mathcal X'$ is again a good model: it is smooth Deligne--Mumford, and the new divisor is one labelled branch transverse to the old ones. Since $I_{q_0}=1$, the cover $T_x$ is trivial over $N_{q_0}\cap U_X$, so $R\to X$ is \'etale over $N_{q_0}$ and $p^{-1}(q_0)$ is a free $\tA$-orbit. Then $R'=\mathrm{Bl}_{p^{-1}(q_0)}R$ and \[ H^*(R')=H^*(R)\oplus\Q[\tA]\otimes\textstyle\bigoplus_{j=1}^{5}\Q(-j)[-2j], \] so $H^*(R')_{\odd}=H^*(R)_{\odd}\oplus D\otimes\bigoplus_{j=1}^5\Q(-j)[-2j]$: the defect of $\mathcal X'$ is nonzero, while $H^*_c(B\setminus D_x,\bL)$ is unchanged. So the defect clause fails for some good model of $(Y,D_x)$, unconditionally. Merely creating trivial-inertia points does not create defect. Blowing up a crossing whose meridian images are $z,z$ gives a divisor in $B^\circ$ at its generic points, but its open part is a $\C^*$-bundle with fibre monodromy $z$, so its compactly supported cohomology vanishes. The nonzero defect comes from the \emph{closed} $\PP^5\subset B^\circ$ on which $\overline{\bL}$ is trivial, just as the closed $\widetilde E_1\cong\PP^1$ does in Example~\ref{ex:tacnode}. \end{example} \section{A canonical weighted resolution}\label{sec:G} We construct a good model by blowing up $\Bun$, not $Y$, and pulling back along the smooth map $\omega$. \subsection{Hypotheses} Let $\mathcal B$ be the open substack of $\Bun$ where the stabiliser dimension is at most six; by smoothness of $\omega$ and $\dim Y=6$, $\omega(Y)\subset\mathcal B$. A point $y$ of $\mathcal B$ is a residual gerbe $[\mathrm{pt}/\cG_y]$; since $\Bun$ is smooth of dimension zero, its codimension is $c=\dim\cG_y$, and the deformation space $T^1_y$ has dimension $c$. \begin{hypothesis}\label{hyp:G} \begin{enumerate} \item[(G0)] $\mathcal B$ has finitely many points. \item[(G1)] For every point $y$ of $\mathcal B$ with $c\ge 1$ there is a maximal torus $T_y\subset\cG_y^\circ$ and a cocharacter $\lambda_y$ of $T_y$ whose weights on $T^1_y$ are all positive. \item[(G2)] The weights of $\lambda_y$ on $\Lie(\cG_y)/\Lie(T_y)$ are all $\le 0$. \end{enumerate} \end{hypothesis} Section~\ref{sec:census} verifies these hypotheses. \subsection{Charts and the averaged cocharacter} Let $\Gamma_y$ be the image of $N_{\cG_y}(T_y)$ in $\Aut X_*(T_y)$; it is finite, and we set $\bar\lambda_y=\sum_{\gamma\in\Gamma_y}\gamma\lambda_y$. The normaliser permutes the weights on $T^1_y$ and on $\Lie\cG_y/\Lie T_y$, so $\bar\lambda_y$ still satisfies (G1) and (G2). (In our case $\Gamma_y=1$; see Proposition~\ref{prop:gamma}.) [ledger 4] By the local structure theorem of Alper--Hall--Rydh \cite[Thm.~1.1]{AHR} applied to the linearly reductive subgroup $T_y\subset\cG_y$ there is a smooth morphism $f:([W/T_y],w)\to(\Bun,y)$ with $W$ affine, $w$ fixed, and $f^{-1}(B\cG_y)=BT_y$; in the minimal chart $\dim W=c$. [ledger 2] Because $\bar\lambda_y$ contracts the tangent space, after shrinking $W$ is $T_y$-equivariantly isomorphic to $\A^c$ with linear action: the coordinate ring is positively graded with degree-zero part $\C$, and lifts of an eigenbasis of $\mathfrak m_w/\mathfrak m_w^2$ generate it by graded Nakayama (equivalently, the attracting cell of the isolated fixed point $w$ for a cocharacter with positive weights is $T_y$-equivariantly $\A^c$ \cite{BB73}). [ledger 3] % [R2LINK-X 3-b] Let $F_k\subset\cO_W$ be the ideal of functions of $\bar\lambda_y$-weight $\ge k$; it is generated by monomials and has finite colength. Let $R=W\times_{\Bun}W$ with source and target $s,t:R\to W$, both smooth. All pullbacks of ideals below are image ideals in the target structure sheaf. % [referee report 003, R3-04; audit hodge-YAUD4] -- begin \begin{lemma}[The closed fibre of the relation]\label{lem:BNfibre} Let $u:W\to\Bun$ be the smooth chart above, with $W\times_{\Bun}B\cG_y=\{w\}$, and $R=W\times_{\Bun}W$. Then on all of $R$ \[ Z=s^{-1}(w)=t^{-1}(w)\simeq\cG_y,\qquad s^*\mathfrak m_w\cdot\cO_R=t^*\mathfrak m_w\cdot\cO_R=I_Z, \] and for $D_N=\Spec(\cO_{W,w}/\mathfrak m_w^N)$, $R\times_{s,W}D_N=V(I_Z^N)=R\times_{t,W}D_N$. \end{lemma} \begin{proof} The condition $f^{-1}(B\cG_y)=BT_y$ pulls back along $W\to[W/T_y]$ to $W\times_{\Bun}B\cG_y=\{w\}$. So an arrow with source $w$ has target $w$, and conversely by inversion; the two closed fibres have the same support. Both are smooth, since $s$ and $t$ are, hence reduced, so they are the same reduced closed subscheme of $R$, namely the isomorphism scheme $\cG_y$ of $u(w)$. Equality of the defining ideals is global on $R$; take powers. Since $W=\Spec\C[y_1,\dots,y_c]$, $D_N\hookrightarrow W$ is a closed immersion and $V(I_Z^N)$ is the whole thickening: no completion and no open neighbourhood depending on $N$ occur. \end{proof} Fix $N\ge1$ and put $R_N=R\times_{s,W}D_N=V(I_Z^N)$, the $N$-th neighbourhood of $Z$ (Lemma~\ref{lem:BNfibre}). \begin{lemma}[Relative section lifting and evaluation]\label{lem:BNtyped} Let $A$ be a finite-dimensional local $\C$-algebra with residue field $\C$, $D=\Spec A$, and $E=\Spec P\to D$ affine, smooth and of finite presentation. Let $B=\operatorname{Res}_{D/\C}(E)$ be the scheme of $D$-sections after base change. Then $B$ is affine of finite presentation, and evaluation $\ev:B\times D\to E$ is smooth and surjective, hence faithfully flat; source and target are flat over $D$. Explicitly, if $C'\twoheadrightarrow C$ is a surjection of $\C$-algebras with square-zero kernel $J$, $p'\in D(C')$, $e'\in E(C')$ lies over $p'$, and $\sigma\in B(C)$ satisfies $\sigma(p'\bmod J)=e'\bmod J$, then there is $\sigma'\in B(C')$ lifting $\sigma$ with $\sigma'(p')=e'$. \end{lemma} \begin{proof} Choosing a basis of $A$ and a finite presentation of $P$ over $A$ and expanding the relations gives $B$ as an affine scheme of finite presentation. For lifting put $Q=A\otimes C'$, $L=A\otimes J$, $\epsilon_{p'}:Q\to C'$, $a\otimes c\mapsto p'(a)c$, and $H=\ker\epsilon_{p'}$. Then $Q/L=A\otimes C$, $Q/H=C'$, $Q/(L+H)=C$ and $Q/(L\cap H)\simeq(A\otimes C)\times_CC'$. The old section $P\to A\otimes C$ and $e':P\to C'$ agree in $C$, so give an $A$-algebra map $P\to Q/(L\cap H)$; since $(L\cap H)^2=0$, smoothness of $P/A$ lifts it to $P\to Q$, which is $\sigma'$. This is formal smoothness of $\ev$ for every affine square-zero test, including non-reduced parameter rings and moving evaluation points; $\ev$ is locally of finite presentation, so it is smooth by the infinitesimal lifting criterion \cite[Tag~02H6]{Stacks}. For surjectivity, a geometric point $e:\Spec\Omega\to E$ lies over the residue point of $D$ (the maximal ideal of $A$ is nilpotent), and $P\to\Omega$ lifts through powers of that ideal to $P\to A\otimes\Omega$ by smoothness, an element of $B(\Omega)$ evaluating to $e$; as $D$ has one point, this is global surjectivity. $E\to D$ is smooth, hence flat, and $B\times D\to D$ is a base change of $B\to\Spec\C$. \end{proof} Applied with $A=\cO_{W,w}/\mathfrak m_w^N$ and $E=R_N$, the section scheme is $B_N$ below and evaluation is the map $\Phi$ of Proposition~\ref{prop:descent}. Source flatness does not depend on first knowing that $B_N$ is a smooth group: base change from the field gives it. Every section is a bisection, over non-reduced bases too: by Lemma~\ref{lem:BNfibre} the target endomorphism $\alpha=t\sigma$ of $D_{N,C}$ preserves the closed subscheme $w_C$; for $N\ge2$ smoothness of $s,t$ gives conormal isomorphisms $s_Z^*(\mathfrak m_w/\mathfrak m_w^2)\cong I_Z/I_Z^2\cong t_Z^*(\mathfrak m_w/\mathfrak m_w^2)$, so $\alpha^*$ is invertible on $(\mathfrak m_w/\mathfrak m_w^2)\otimes C$, hence on every graded piece of the finite $\mathfrak m_w$-adic filtration (W smooth), hence an automorphism of $A\otimes C$. Composition and inversion therefore define the group functor on all test rings. % [referee report 003, R3-04; audit hodge-YAUD4] -- end % [referee report 002, R2-05] \begin{lemma}[Truncated bisections]\label{lem:BN} Let $B_N=\operatorname{Res}_{D_N/\C}(R\times_{s,W}D_N)$, the functor of sections $\sigma$ of $s$ over $D_N$. Then: \begin{enumerate} \item $B_N$ is an affine group scheme of finite type, smooth (characteristic zero), with product $(\sigma\sigma')(p)=\sigma(t\sigma'(p))\circ\sigma'(p)$; every section has value in $s^{-1}(w)=\cG_y$ at the closed point, and $t\circ\sigma\in\Aut(D_N)$; \item $\ev_N:B_N\to\cG_y$ is a smooth surjective homomorphism, and $K_N=\ker\ev_N$ is connected unipotent; \item $T_y\hookrightarrow B_N$, $g\mapsto\sigma_g$, is a maximal torus of $B_N^\circ$, and $\ev_N$ restricts to the given inclusion $T_y\subset\cG_y$; \item $\Lie B_N=\Gamma(D_N,\mathcal A|_{D_N})$ for the Lie algebroid $\mathcal A=\ker(ds)|_{\mathrm{identity}}$, acting on $\cO_{D_N}$ through the anchor $a$. \end{enumerate} \end{lemma} \begin{proof} (1) Since $\Bun$ has affine diagonal, $R$ is affine, and the Weil restriction of the affine finite-type $R\times_WD_N$ along the finite flat $D_N\to\Spec\C$ is affine of finite type \cite[\S7.6]{BLR}. Because $f^{-1}(B\cG_y)=BT_y$, every arrow out of $w$ ends at $w$, so $s^{-1}(w)=\mathrm{Isom}(w,w)=\cG_y$. The map $t\circ\sigma$ fixes the closed point, and its differential there is the invertible isotropy action, since the anchor vanishes at $w$; so it is an automorphism of the Artin scheme $D_N$. The product is the bisection product and is algebraic, and a group scheme of finite type in characteristic zero is smooth. (2) Over a section $\sigma$, the lifts along $\operatorname{Res}_{D_{j+1}}\to\operatorname{Res}_{D_j}$ form a torsor under $\Hom(\sigma^*\Omega_{R/W},\mathfrak m^j/\mathfrak m^{j+1}\otimes C)$, by smoothness of $s$: a vector-bundle torsor over the preceding parameter scheme, which on the kernel over the identity is the vector group of $\Lie(\cG_y)\otimes\mathfrak m^j/\mathfrak m^{j+1}$. % [referee report 003, R3-04; audit hodge-YAUD4] So $\ev_N$, which is $B_N\to B_1=\cG_y$, is smooth and surjective with affine-space fibres. The congruence subgroups $K^{(j)}=\ker(B_N\to B_j)$ are normal and $K^{(j)}/K^{(j+1)}$ is a vector group: commutators of elements congruent to the identity modulo $\mathfrak m^j$ vanish modulo $\mathfrak m^{j+1}$, because the anchor vanishes at $w$. Hence $K_N=K^{(1)}$ is connected unipotent, and $\ev_N^{-1}(\cG_y^\circ)$ is connected. (3) The $T_y$-action on $W$ gives $W\times T_y\to R$, whose sections $\sigma_g$ preserve $D_N$ because $T_y$ fixes $w$. Since $\ev_N(B_N^\circ)=\cG_y^\circ$ with unipotent kernel, every torus of $B_N^\circ$ maps injectively onto a torus of $\cG_y^\circ$, so $T_y$ is maximal. (4) This is standard for Weil restriction: the action of $\sigma$ on $\cO_{D_N}$ is pullback by $t\circ\sigma$, whose derivative is the anchor. Every element of $\Lie B_N$ is $\xi=\sum_jf_je_j$ with $f_j\in\cO_{D_N}$ \emph{arbitrary}, for a $T_y$-homogeneous frame $(e_j)$ of $\mathcal A$. Such a frame exists: $\mathcal A$ is a $T_y$-equivariant bundle on $W\cong\A^c$, whose coordinate ring is positively graded, and graded projective modules over it are free with homogeneous basis (graded Nakayama). \end{proof} \begin{lemma}[Identity component]\label{lem:BNcirc} $B_N^\circ$ preserves $F_k\bmod\mathfrak m_w^N$ for every $k$. \end{lemma} % [referee report 002, R2-05] \begin{proof} Grade $\cO_W$ by $\deg y_i=a_i>0$, using (G1); a vector field is homogeneous of degree $m$ if it maps $\cO_b$ into $\cO_{b+m}$. In the convention in which $T^1_y$ has positive $\bar\lambda_y$-weights, a vector field of weight $\nu$ has degree $-\nu$ (for instance $\partial_i$ has weight $a_i$ and degree $-a_i$). By $T_y$-equivariance of the anchor, $a(e_j)$ has the weight $\nu_j$ of $e_j(w)\in\Lie\cG_y$, and $\nu_j\le0$ by (G2), with $\nu=0$ on $\Lie T_y$. So $\deg a(e_j)=-\nu_j\ge0$, and $a(e_j)$ maps $F_k=\bigoplus_{b\ge k}\cO_b$ into itself. For arbitrary coefficients, $f_ja(e_j)$ maps $F_k$ into $f_jF_k\subset F_k$, since $F_k$ is an ideal, and $\xi$ preserves $\mathfrak m^N$ because $a(e_j)(w)=0$. So $\Lie B_N$ preserves $F_k\bmod\mathfrak m^N$. The group $B_N$ acts algebraically on the finite-dimensional $\cO_{D_N}$, and in characteristic zero a subspace of a finite-dimensional algebraic representation of a connected group that is stable under the Lie algebra is stable under the group. \end{proof} \begin{lemma}[All components]\label{lem:BNall} $B_N$ preserves $F_k\bmod\mathfrak m_w^N$ for every $k$. \end{lemma} \begin{proof} For $b\in B_N$ the tori $bT_yb^{-1}$ and $T_y$ are maximal in $B_N^\circ$ (Lemma~\ref{lem:BN}), so there is $h\in B_N^\circ$ with $m=hb$ normalising $T_y$. Since $\ev_N$ is injective on $T_y$, the conjugation action of $m$ on $T_y$ is that of $\ev_N(m)\in N_{\cG_y}(T_y)$, i.e.\ an element of $\Gamma_y$. So $m$ fixes $\bar\lambda_y$, centralises $\bar\lambda_y(\G_m)$, preserves the $\bar\lambda_y$-grading of $\cO_{D_N}$ and hence each $F_k$. By Lemma~\ref{lem:BNcirc}, $h^{-1}$ preserves $F_k$, so $b=h^{-1}m$ does. No assumption on $\pi_0(\cG_y)$ is used. \end{proof} % [referee report 002, R2-05] \begin{proposition}[Descent]\label{prop:descent} For every $k$, $s^*F_k=t^*F_k$ as ideals of $\cO_R$. Hence the ideals $F_k$ descend to coherent ideals on the open image $f(W)\subset\Bun$, all on the same open, and the Rees algebra $\bigoplus_kF_ku^k$ descends to a finitely generated graded algebra there. \end{proposition} % [referee report 003, R3-04; audit hodge-YAUD4] -- begin (replacement proof) \begin{proof} Fix $k\ge0$ and choose $N$ with $\mathfrak m_w^N\subset F_k$ (for instance $N=\max(1,\lceil k/a_{\min}\rceil)$, $a_{\min}$ the least weight). By Lemma~\ref{lem:BNfibre}, on all of $R$ we have $s^{-1}(\mathfrak m_w)\cO_R=t^{-1}(\mathfrak m_w)\cO_R=I_Z$ and $R_N=V(I_Z^N)$. Lemma~\ref{lem:BNtyped}, applied to $R_N\to D_N$, makes the evaluation map $\Phi:B_N\times D_N\to R_N$ smooth and surjective, hence faithfully flat, with $s\Phi=\mathrm{pr}_2$ and $t\Phi=a_N$, the target action of the bisection group. Write $J=F_k/\mathfrak m_w^N\subset\cO(D_N)$. Lemma~\ref{lem:BNall} makes $J$ invariant under all complex points of $B_N$; since $B_N$ is a reduced affine group of finite type in characteristic zero, the matrix coefficients of $J\to\cO(B_N)\otimes(\cO(D_N)/J)$ vanish, so $J$ is a sub-comodule. This gives $a_N^*J\subset\mathrm{pr}_2^*J$ as image ideals, and the same inclusion for the inverse action (or for the automorphism $(b,p)\mapsto(b,bp)$ and its inverse) gives equality. Hence $\Phi^*(s^*J)=\mathrm{pr}_2^*J=a_N^*J=\Phi^*(t^*J)$, and faithful flatness gives $s^*J=t^*J$ on $R_N$. Both $s^*F_k$ and $t^*F_k$ contain $I_Z^N$ on all of $R$, and ideals containing $I_Z^N$ are determined by their images in $\cO_R/I_Z^N$; so $s^*F_k=t^*F_k$ on all of $R$, with no shrinking depending on $k$. The common image ideal is a descent datum for $F_k$ along the smooth surjection $u:W\to u(W)$ with relation $R$ (its cocycle is automatic, being an equality of submodules on the triple fibre product), and effective flat descent gives a coherent ideal on $u(W)$. These descents respect multiplication, so the graded Rees algebra descends; on $W$ it is generated by $y_iu^j$, $1\le j\le a_i$, hence of finite type, and finite type is smooth-local \cite[IV, 2.7.1]{EGA}. [ledger 4; referee report 001, G1] \end{proof} % [referee report 003, R3-04; audit hodge-YAUD4] -- end \subsection{The radial lemma} Let $a_1,\dots,a_c>0$ be the $\bar\lambda_y$-weights of the coordinates $y_i$ on $W=\A^c$. The stack-theoretic weighted blow-up of $W$ at the origin \cite[Def.~3.2.1]{QR} has, since $y_1,\dots,y_c$ is a regular sequence, the charts of \cite[Cor.~5.2.5]{QR}, namely % [Quek--Rydh checked at source] charts $V=(W\setminus 0)\times\A^1_s$ modulo $\G_m$ acting by $(y,s)\mapsto(\bar\lambda_y(\tau)y,\tau^{-1}s)$, with blow-down $\beta(y,s)=(s^{a_i}y_i)$ and exceptional divisor $\{s=0\}$. Let $\gamma(y,s)=y$ be the radial projection. [ledger 5] \begin{lemma}[Radial lemma]\label{lem:R} Let $I$ be an ideal of $\cO_{\Bun}$ near $y$ and $J=f^*I$. Then $(\beta^*J:s^\infty)=\gamma^*J$, degree by degree for filtrations, and $(f\gamma,s):V\to\Bun\times\A^1$ is smooth with image avoiding $y$. \end{lemma} \begin{proof} $J$ is $T_y$-stable, hence $\bar\lambda_y$-stable, so on $s\ne0$ the action identifies $\beta^*J$ with $\gamma^*J$. On an affine open $A$ of $W\setminus0$, $\cO/\gamma^*J=(A/J)[s]$ has no $s$-torsion, so $\gamma^*J$ is $s$-saturated and is the contraction of its localisation. Smoothness holds because $(f\gamma,s)$ is $f|_{W\setminus0}\times\mathrm{id}$ after the product identification, and $f^{-1}(B\cG_y)$ is the origin. [ledger 3] \end{proof} \subsection{The construction} By (G0) and the dimension argument of Proposition~\ref{prop:frontier}, ordering the points of $\mathcal B$ with $c\ge 2$ by decreasing $c$ (ties arbitrary) as $x_1,x_2,\dots$ makes each $x_m$ closed in $U_{m-1}=\mathcal B\setminus\{x_1,\dots,x_{m-1}\}$. For each $m$ let $F^{(0)}_{m,\bullet}$ be the contraction to $\mathcal B$ of the descended filtration of $x_m$ extended by the unit ideal on $U_{m-1}\setminus\{x_m\}$: \[ F^{(0)}_{m,k}=\ker\bigl(\cO_{\mathcal B}\to j_{m*}(\cO_{U_{m-1}}/F_{m,k})\bigr). \] These are coherent ideals, compatible with flat base change, supported on the closure of $x_m$, and equal to the weighted ideals near $x_m$; no claim is made that they are unit ideals at deeper points. [ledger 3] \begin{theorem}\label{thm:G} Assume Hypothesis~\ref{hyp:G}. Blowing up the points $x_1,x_2,\dots$ in order, each with its weighted filtration (the stack-theoretic weighted blow-up of $\mathcal B_n$ along the intrinsic transform $I^{(n)}_\bullet$ of Definition~\ref{def:globaldata}, whose support is closed in $\mathcal B_n$; Lemma~\ref{lem:intrinsic}) % [referee report 004, R4-02; audit hodge-YAUD8] , gives a smooth stack $\mathcal B_N\to\mathcal B$ with finite relative inertia such that the total transform of $\mathcal B\setminus U$ is a divisor with smooth, individually labelled components meeting as coordinate hyperplanes in smooth charts. Consequently $\mathcal X_x$, obtained by gluing $Y\times_{\mathcal B}\mathcal B_N$ to a neighbourhood of $E$ in $B$ along $Y\setminus D_x$ (the modification is an isomorphism there, and $D_x$ is closed and disjoint from $\star$), is a good model of $(Y,D_x)$ for every $x\in S$. \end{theorem} % [referee report 004, R4-02; audit hodge-YAUD8] -- begin (global atlas compatibility for $(I_n)$) \begin{definition}[Intrinsic transforms]\label{def:globaldata} Put $\mathcal B_0=\mathcal B$ and let $b_n:\mathcal B_n\to\mathcal B_{n-1}$ be the blow-ups constructed so far, with exceptional Cartier divisor $E_n$ \cite[Def.~3.2.2]{QR}. For a tracked ideal $J$ (degree by degree for a filtration) its \emph{intrinsic transform} is $J^{(0)}=J$ and \[ J^{(n)}=\bigl(b_n^{-1}J^{(n-1)}\cdot\cO_{\mathcal B_n}:\ \mathcal I_{E_n}^{\infty}\bigr), \] an ideal on the smooth site of $\mathcal B_n$. The older divisors are the total transforms $E_i^{(n)}=b_n^{-1}E_i^{(n-1)}$, $in$. \emph{Statement $(I_n)$.} $\mathcal B_n$ has a smooth atlas by schemes $V$ (charts of algebraic-space presentations), each with a smooth morphism $(g_V,s):V\to U_n\times\A^r$, where $s=(s_1,\dots,s_r)$ are local equations of the labelled exceptional divisors through $V$, such that every tracked ideal $J$ restricts on $V$ to the image ideal $g_V^*J$, degree by degree, and these identifications are compatible on overlaps of charts. For $n=0$ take the charts $f:W\to\mathcal B$ of Section~5.2 with $r=0$. \emph{Step 1: the next centre.} Let $x'=x_{n+1}$. A point of $\overline{\{x'\}}\setminus\{x'\}$ has larger stabiliser dimension, hence is among $x_1,\dots,x_n$, hence is not in $U_n$; so $x'$ is closed in $U_n$. On a chart $V$, $g_V^*F^{(0)}_{n+1,\bullet}$ is therefore, locally, either the smooth pullback of the regular weighted filtration at $x'$ (where $g_V$ meets $x'$) or the unit filtration. On the chart $W$ of $x'$ the filtration is the monomial one $(y_1^{1/a_1},\dots,y_c^{1/a_c})$, a regular weighted centre in the sense of \cite[Def.~5.2.7, Def.~5.3.1]{QR}. Regularity is smooth-local by those definitions, and it is stable under smooth pullback, since flat pullback preserves regular sequences and smooth pullback preserves regularity of the reduced centre. Its Rees algebra is the pullback of a finitely generated one. The centre is the intrinsic transform $I^{(n)}_\bullet$, a regular weighted centre with finitely generated Rees algebra by Lemma~\ref{lem:intrinsic}. % [referee report 004, R4-02; audit hodge-YAUD8] We blow it up stack-theoretically \cite[\S1.2, Def.~3.2.1]{QR}. The result is regular \cite[Cor.~5.3.2]{QR}, its charts are those of \cite[Prop.~5.2.2, Cor.~5.2.5--5.2.6]{QR}, and the blow-up commutes with flat base change \cite[Cor.~3.2.14(iii)]{QR} (stated for schemes; applied on an atlas). % [referee report 002, R2-05; R2LINK-X 3-c] \emph{Step 2: charts and smoothness.} Let $V'$ be a radial chart of Lemma~\ref{lem:R} at $x'$, built inside $U_n$, with blow-down $\beta:V'\to U_n$, radial map $\gamma:V'\to U_{n+1}$ and new coordinate $t$, and let $P=V\times_{g_V,U_n,\beta}V'$ (a $2$-fibre product; pass to scheme covers if needed). The structure map used to form $P$ is $\beta$; the new invariant map is $g_P=\gamma\circ\mathrm{pr}_2$. The morphism $(g_P,s\circ\mathrm{pr}_1,t\circ\mathrm{pr}_2)$ factors as \[ P\xrightarrow{(\mathrm{pr}_2,\,s\circ\mathrm{pr}_1)}V'\times\A^r\xrightarrow{(\gamma,\,\mathrm{id},\,t)}U_{n+1}\times\A^{r+1}, \] the first arrow being the base change of the smooth $(g_V,s)$ along $(\beta,\mathrm{id})$ and the second the product of the smooth $(\gamma,t)$ with $\mathrm{id}_{\A^r}$. So it is smooth; no equality $\beta=\gamma$ is used. \emph{Step 3: the tracked ideals.} For a tracked ideal $J$, the fibre square gives $\mathrm{pr}_1^*g_V^*J=\mathrm{pr}_2^*\beta^*J$; ideals on a stack pull back to the same ideal along $2$-isomorphic maps, so this uses only $g_V\circ\mathrm{pr}_1\cong\beta\circ\mathrm{pr}_2$. % [referee report 002, R2-05] Saturating along $t=0$ and using Lemma~\ref{lem:R} together with flatness of $\mathrm{pr}_2$ (a base change of $g_V$), the strict transform is $\mathrm{pr}_2^*\gamma^*J=g_P^*J$. By the first arrow of Step~2 the old coordinates $s_i$ are independent relative to $V'$, so $\cO_P/g_P^*J$ is locally smooth over $(\cO_{V'}/\gamma^*J)[s_1,\dots,s_r]$ and has no $s_i$-torsion: the transform is saturated along every old exceptional divisor, not only the newest. The old divisors have total transform $(s_i)$, equal to their strict transform; they remain smooth, individually labelled and transverse to $t=0$. On overlaps both sides are restrictions of the intrinsic transform $J^{(n+1)}$, so the identifications agree (Lemma~\ref{lem:globalatlas}); being pullbacks of one ideal of $\mathcal B$ along two different radial maps would not suffice, and the radial maps are not compared. % [referee report 004, R4-02; audit hodge-YAUD8] This proves $(I_{n+1})$, in the global form preceding Lemma~\ref{lem:chartsquare}. [ledger 5; referee report 001, G2] \emph{Step 4: disjointness.} Two distinct strata of equal codimension have closures meeting only in deeper strata (Proposition~\ref{prop:frontier}). On a chart their strict transforms have supports $g_V^{-1}$ of their closures, and $g_V$ avoids the processed points, so these supports are disjoint once the deeper strata have been processed. \emph{Step 5: termination.} After the finite list of points with $c\ge2$ is processed, $g_V$ lands in the open point or in a point with $c=1$, where the minimal transverse chart is a smooth curve and the boundary is its origin. So the strict boundary is locally a smooth Cartier divisor or empty, and by smoothness of $(g_V,s)$ it is transverse to the exceptional coordinates: the total boundary consists of smooth labelled components meeting as coordinate hyperplanes in smooth charts. \emph{Step 6: finite inertia and branches.} Each weighted blow-up has finite relative inertia, and so do composites (nonsplit finite extensions are allowed). The stack $\mathcal B_N$ need not be Deligne--Mumford, but $\mathcal X_x$ is: its stabilisers are kernels of the maps to stabilisers in $\mathcal B$, since $Y$ is a scheme, so it has finite inertia, and it is smooth because $\omega$ and $\mathcal B_N$ are. Étale locally at a geometric point it is a finite stabiliser quotient of a smooth chart in which the boundary is a union of coordinate hyperplanes; each $E_i$ is a globally labelled smooth Cartier divisor with one local branch, and the strict boundary is smooth with at most one branch, so the stabiliser preserves each branch, and after averaging the local equations may be taken to be eigenvectors for the normal characters. [ledger 6] \end{proof} \section{The census of points of $\Bun$}\label{sec:census} \subsection{Bundle types} A point of $\mathcal B$ is a pair $(\mathcal P_{\tilde w},b)$ up to $\Aut(\mathcal P_{\tilde w})$. Put $H_{\tilde w}=\Aut(\mathcal P_{\tilde w})$ and let $j(\tilde w)=\dim H_{\tilde w}-\dim K$ be the codimension of the bundle-type locus. The stabiliser of $b$ is $S_b=\ev_1^{-1}(B_b)$, of dimension $c=\dim H_{\tilde w}-\dim(H_{\tilde w}\cdot b)$, so $c\ge j$. \begin{lemma}\label{lem:types} The bundle types with $j\le 6$ are exactly seven: one for each $j\in\{0,1,3,4,5\}$ and two with $j=6$, with $\dim H_{\tilde w}=6,7,9,10,11,12,12$; representatives are $\star=t^{(-2,-2)}s_1s_2s_1$, $B_1=t^{(-2,-2)}s_1s_2s_1s_2$, $B_3=t^{(-3,-3)}s_1s_2s_1s_2$, $B_4=t^{(-2,-3)}s_1s_2s_1$, $B_5=t^{(-3,-3)}s_1s_2s_1$, $B_6=t^{(-2,-3)}s_1s_2$ and $C_6=t^{(-2,-2)}s_1$. % [blind recheck 2026-10-19: v0.7 listed six types and omitted C_6 (hodge-VYUN2B; LAB YUN-61; countertheorist YUNADJ)] \end{lemma} \begin{proof} With $v=\rho^\vee/2$ and $\nu=\mu+v-wv$, \eqref{eq:interval} gives $N_\alpha=\max(\langle\alpha,\nu\rangle+1-(\varepsilon(\operatorname{ht}\alpha)+\varepsilon(\operatorname{ht}w^{-1}\alpha))/2,0)$, $\varepsilon$ the parity, so $N_\alpha+N_{-\alpha}\ge|\langle\alpha,\nu\rangle|$ and $j=\sum_\alpha N_\alpha-4\le 6$ forces $\sum_{\alpha>0}|\langle\alpha,\nu\rangle|\le 10$. Writing $\nu=(x,y)$, the six positive pairings are $2x-y$, $-3x+2y$, $-x+y$, $x$, $3x-y$, $y$; the sum of the first, third, fourth, fifth and sixth is $5x$, and the sum of the second to the sixth is $3y$, so $5|x|\le10$ and $3|y|\le10$, and since $\nu\in\frac12\Z^2$ this gives $|x|\le2$, $|y|\le3$. As $wv-v$ varies by at most $3$ in the first coordinate and $5$ in the second, $|\mu_1|\le5$ and $|\mu_2|\le8$; we enumerate the larger box $|\mu_1|\le15$, $|\mu_2|\le13$. % [referee report 007, R7-01] An exact enumeration over this box and all $w\in W$, with both parahoric Weyl groups the stabilisers of $-\rho^\vee/2$ (the only choice for which $\dim H_{\tilde w}$ is constant on double cosets), gives the list; $C_6$ and $B_6$ lie in distinct double cosets. [Certificate: exp/yun61/census.py, which replaces exp/enum6/e1.py and enum.py; the earlier script missed $C_6$.] \end{proof} \begin{proposition}\label{prop:evimage} The image $\ev_1(H_{\tilde w})$ is the connected subgroup with Lie algebra $\mathfrak t\oplus\bigoplus_{N_\alpha\ge1}\mathfrak g_\alpha$: \begin{itemize} \item for $\star$, the symmetric subgroup $K$; \item for $B_6$, a Borel subgroup $B'$; \item for $C_6$, a Borel subgroup $B''$ (the roots with $N_\alpha\ge1$ are $\alpha_2$, $-\alpha_1$, $-(\alpha_1+\alpha_2)$, $-(2\alpha_1+\alpha_2)$, $-(3\alpha_1+\alpha_2)$, $-(3\alpha_1+2\alpha_2)$, a positive system); \item for $B_5$, a parabolic subgroup $P'$ with Levi of type $\mathrm{GL}_2$ (roots $\pm(\alpha_1+\alpha_2)$); \item for $B_1$, $B_3$, $B_4$, the subgroup $T\cdot U_0$ where $U_0$ is the product of the positive root groups of a Borel $B'$ other than one simple root $\gamma$ (respectively $\gamma=-\alpha_1$, $-(3\alpha_1+2\alpha_2)$, $-(2\alpha_1+\alpha_2)$). \end{itemize} The kernel of $\ev_1$ on $H_{\tilde w}$ is unipotent and meets every torus trivially. \end{proposition} \begin{proof} The root sets are read off from \eqref{eq:interval}. For the solvable types, if $\ev_1(tu)=1$ then $\ev_1(t)=\ev_1(u)^{-1}$ is semisimple and unipotent, hence $t=1$; for $B_5$ evaluation identifies the Levi with the root Levi of $P'$ and the unipotent part maps into its unipotent radical; for $\star$ evaluation is the inclusion of $K$. \end{proof} \begin{proposition}\label{prop:gamma} For every point $b$ of every type and every maximal torus $T_S\subset S_b^\circ$, the normaliser $N_{S_b}(T_S)$ centralises $T_S$. In particular $\Gamma_y=1$ and $\bar\lambda_y=\lambda_y$. \end{proposition} \begin{proof} In a connected solvable group $B$ the projection $B\to B/R_u(B)$ is injective on every torus and has commutative target, so if $n$ normalises a torus $T_0$ then $ntn^{-1}$ and $t$ have the same image, hence are equal. Apply this inside $B_b$ to $\ev_1(T_S)$ and use Proposition~\ref{prop:evimage}. [ledger 7: C-GROUP] \end{proof} \subsection{Orbit classification} \begin{proposition}\label{prop:orbits} The $H_{\tilde w}$-orbits on $G/B$ are finite in number for each type: $12$ Bruhat cells for each of $B_6$ and $C_6$; $6$ orbits for $B_5$; $18$ orbits for each of $B_1,B_3,B_4$, namely one orbit $vB$ for each $v$ with $\gamma\in v\Phi^+$ and two orbits $vB$, $u_\gamma(1)vB$ for each $v$ with $\gamma\notin v\Phi^+$; and $10$ orbits for $\star$. Every orbit with $c\ge1$ has a representative fixed by $T$ or by $(\ker\gamma)^\circ$ for a root $\gamma$; in particular no orbit of positive codimension has a stabiliser without a torus. \end{proposition} \begin{proof} For $B_6$, $C_6$ and $B_5$ this is the Bruhat decomposition. For $B_1,B_3,B_4$, $U_0$ is normal in $B'$ and $B'/U_0=T\ltimes U_\gamma$ with image of $H'=T\cdot U_0$ equal to $T$. With $Q_v=B'\cap vBv^{-1}$, the image of $Q_v$ in $T\ltimes U_\gamma$ is $T$ if $\gamma\notin v\Phi^+$ and all of $T\ltimes U_\gamma$ otherwise; in the first case $T\backslash(T\ltimes\G_a)/T$ has the two geometric double cosets $0$ and $1$, since $T$ scales the coordinate through the nontrivial character $\gamma$. [ledger 7: C-BRUHAT] For $\star$, $K$ is connected split of type $A_1\times A_1$ with Weyl group of order four. The twelve $T$-fixed flags form three closed orbits $K/B_K$ (two fixed flags lie in one $K$-orbit iff their Weyl labels differ by $W_K$). The six flags $u_\gamma(1)vB$ with $\gamma$ in Table~\ref{tab:K} have stabiliser dimensions $c=1,2,3$ (computed exactly over $\Q$), maximal stabiliser torus $(\ker\gamma)^\circ$ of rank one, and unipotent radical of dimension $c-1$; the two in each row are not $K$-conjugate (see \emph{Distinctness over $\C$} below). The flag $u_0$ has $c=0$ and lies in the open orbit. These ten orbits are pairwise distinct. [ledger 7: C-DISTINCT] They exhaust $G/B$: for a geometric orbit $O=L/S$ with a rational point, $S^\circ$ solvable with split torus quotient of rank $r$, unipotent radical of dimension $u$, and component group acting trivially on that torus quotient (Proposition~\ref{prop:gamma}), Lang's theorem gives \[ |O(\F_q)|=|L(\F_q)|/\bigl((q-1)^rq^u\bigr), \] because the rational orbits are indexed by twisted conjugacy classes in $\pi_0(S)$ and $\sum_{[a]}1/|C(aF)|=1$; no knowledge of $\pi_0(S)$ is needed. [ledger 7: C-MASS] With $|K(\F_q)|=q^2(q^2-1)^2$ the ten counts sum to $(q+1)^2\bigl(q^2(q-1)^2+2(q-1)(q^2+q+1)+3\bigr)=(q+1)^2(q^4+q^2+1)=|G/B(\F_q)|$ for every $q$ prime to a fixed $N$. Spreading out the orbit maps over $\Z[1/N]$, the complement of the ten orbits has no $\F_{p^n}$-points for all good $p$ and all $n$, hence is empty over $\overline\Q$. [ledger 7: C-SPREAD] \emph{Distinctness over $\C$.} In characteristic $0$ stabilisers are smooth, so $\operatorname{Lie}K_x=\mathfrak k\cap\mathfrak b_x=:\mathfrak s_x$, and $y=k\cdot x$ implies $\mathfrak s_y=\operatorname{Ad}(k)\mathfrak s_x$. For each representative with $\dim\mathfrak s_x\in\{1,2,3\}$ the certificate's stabiliser table gives $\mathfrak s_x=\C h\ltimes\mathfrak n$ with $h\in\mathfrak t$, $\mathfrak n$ abelian and spanned by $\operatorname{ad}h$-eigenvectors, each a root vector or, at the $c=2$ short-root orbit, the combination $E_{\alpha_1+\alpha_2}+3E_{-(3\alpha_1+\alpha_2)}$ (the roots involved, $\alpha_1+\alpha_2$ and $\pm(3\alpha_1+\alpha_2)$, have no root among their sums), and $\operatorname{ad}h$ invertible on $\mathfrak n$. Hence $ah+n=\operatorname{Ad}(\exp m)(ah)$ with $m=-a^{-1}(\operatorname{ad}h|_{\mathfrak n})^{-1}n$ for $a\ne0$, while $\mathfrak n$ consists of nilpotent elements; so the nonzero semisimple elements of $\mathfrak s_x$ are exactly the $\operatorname{Ad}(K_x^\circ)$-conjugates of nonzero multiples of $h$. Therefore \[ \rho(x):=\max|\operatorname{spec}\operatorname{ad}h|\,\big/\,\min|\operatorname{spec}\operatorname{ad}h\setminus0| \] (on $\mathfrak g_2$) is an invariant of the orbit $K\cdot x$, indeed of its $\operatorname{Ad}(\Gtwo)$-class. If $h$ is proportional to the coroot of a long (resp.\ short) root, the values $\gamma(h)$, $\gamma\in\Phi$, are proportional to $\{0,\pm1,\pm2\}$ (resp.\ $\{0,\pm1,\pm2,\pm3\}$), so $\rho=2$ (resp.\ $3$). With $\alpha_1$ short and $E(a,b)$ the root vector of $a\alpha_1+b\alpha_2$, $h_2=\alpha_2^\vee$ and $h_1+2h_2=(3\alpha_1+2\alpha_2)^\vee$ are long coroots, while $h_1=\alpha_1^\vee$ and $h_1+\tfrac32h_2=\tfrac12(2\alpha_1+\alpha_2)^\vee$ are short ones. Thus, in the certificate's labels, $\rho=2$ for $c_1$\,A, $c_2\,(1,2)$, $c_3\,(1,1,2)$ and $\rho=3$ for $c_1$\,B, $c_2\,(2,3)$, $c_3\,\Sigma$: each codimension in Table~\ref{tab:K} has one orbit of each value, so its two orbits are distinct. The three closed orbits are distinct because a closed orbit $K\cdot wB\cong K/B_K$ meets $(G/B)^T$ exactly in $\{vB:v\in W_Kw\}$, and $W_K\backslash W$ has $12/4=3$ elements. Orbits of different dimension are distinct, so the ten orbits are pairwise distinct. [referee report 001, C2; C-DIST] \emph{Completeness over $\C$.} By Matsuki \cite{Matsuki79} and Richardson--Springer \cite{RS90}, $K\backslash G/B$ is in bijection with $\bigsqcup_H W(G,H)/W_K(H)$ over the $K$-classes of $\theta$-stable Cartan subgroups. By Sugiura \cite[Thm.~6, Def.~9, Thm.~7]{Sugiura59} (Theorem~7 as corrected in \cite{Sugiura71}), the conjugacy classes of Cartan subalgebras of the split real form correspond to the classes of admissible (strongly orthogonal) sets of noncompact roots, each realised by the iterated Cayley transform $\rho_{\alpha_l}\circ\dots\circ\rho_{\alpha_1}$ from the compact Cartan subalgebra; for split $G_2$ there are exactly four classes \cite[p.~429]{Sugiura59}, which the 1971 correction does not affect. % [referee report 004, R4-05; librarian hodge-LIB7] Cayley transforms from $T$ give these four classes: compact, one mixed class with a real short root and one with a real long root ($W_K$ is transitive, up to sign, on the noncompact imaginary roots of each length), and split. Their contributions are $12/|W_K|=3$, $12/4=3$, $12/4=3$ (for a mixed Cartan $H$ with real root $\alpha$, $W_K(H)=\{1,s_\alpha,s_\beta,-1\}$ with $\beta\perp\alpha$ noncompact imaginary and $s_\beta$ realised in $K$; this is the full centraliser of $s_\alpha$ in $W\cong D_6$, of order four), and $12/12=1$. The total is $10$, so the list is complete; $\rho=2$ and $\rho=3$ label the two mixed Cartans, three orbits each. The point counts above are a check. % [referee report 002, R2-06] The general parametrisation is that of Matsuki \cite{Matsuki79} and Richardson--Springer \cite{RS90}; its specialisation to $\Gtwo/SO_4$ is checked by the certificate. The certificate reproduces the count $3+3+3+1$ independently, from explicit $\theta$-stable Cartan data, exactly over $\Q$ (Cayley transforms in a Chevalley basis of $\mathfrak g_2$; $\theta=\operatorname{Ad}(s)$, $s\in T$). It finds: \begin{itemize} \item four $K$-classes of $\theta$-stable Cartans: compact, short-root Cayley, long-root Cayley and split, with pairwise distinct root-type patterns; \item $|W|/|W_K(H)|=3,3,3,1$, with $|W_K(H)|=4,4,4,12$ in characteristic $0$: the upper bound ($W_K$ for the compact class, $Z_W(\theta_S)$ otherwise) is attained, because every one of its elements $w$ is realised by an explicit $k=C_S\,n_w\,D(y)\,C_S^{-1}\in K$, with $C_S$ the rational Cayley element, $n_w$ the rational $N_G(T)$-representative along a reduced word of $w$ and $y$ a point of $T$, each $k$ defined over $\Q$ or $\Q(i)$ and checked exactly ($\theta(k)=k$, $\operatorname{Ad}(k)\mathfrak h_S=\mathfrak h_S$, and $\operatorname{Ad}(k)$ permutes the root lines of $\mathfrak h_S$ as $w$ does on all twelve roots); since $K=\Gtwo^\theta$ is connected (Steinberg), $\theta$-fixed in $\operatorname{Aut}(\mathfrak g_2)=\Gtwo$ means $k\in K$; % [referee report 003, R3-01; lab MATSUKI-C0] \item the ten orbits of Table~\ref{tab:K} matched to the classes by the rank of their stabiliser torus ($2$, $1$, $1$, $0$) and, in rank one, by $\rho$ ($2$ for the short-root Cayley class, $3$ for the long-root one). \end{itemize} The same code returns the known counts $3$ for $SL_2/SO_2$ and $6$ for $SL_3/S(GL_2\times GL_1)$, with the normaliser elements again realised in characteristic $0$ (for $SL_2$ the split class needs an element over $\Q(i)$, which is absent from $K(\F_7)$; this is why the earlier comparison over finite fields was not by itself a proof). Scripts \texttt{matsuki.sage}, \texttt{matsuki\_c0.sage} and \texttt{match.sage}, with their logs and the exact representatives, are in \texttt{certificate/matsuki/} of the bundle. \end{proof} \begin{table}[ht] \caption{Representatives $u_\gamma(1)vB$ of the $K$-orbits of codimension $1,2,3$. Stabiliser Lie algebras, exact over $\Q$, and point counts over $\mathbb F_5,\mathbb F_7$ for all ten orbits are in the certificate (\texttt{K-ORBIT-TABLE.md}, script \texttt{korb\_table.sage}).}\label{tab:K} \begin{tabular}{ccc} \toprule $c$ & short root $\gamma$ & long root $\gamma$\\ \midrule $1$ & $-(2\alpha_1+\alpha_2)$, $v=e$ & $-(3\alpha_1+2\alpha_2)$, $v=e$\\ $2$ & $-(2\alpha_1+\alpha_2)$, $v=s_1$ & $-\alpha_2$, $v=s_1$\\ $3$ & $-\alpha_1$, $v=e$ & $-\alpha_2$, $v=e$\\ \bottomrule \end{tabular} \end{table} \begin{proposition}[Frontier order]\label{prop:frontier} $\mathcal B$ has finitely many points, and a point in the closure of a distinct point $y$ has strictly larger stabiliser dimension than $y$. \end{proposition} \begin{proof} Finiteness follows from Lemma~\ref{lem:types} and Proposition~\ref{prop:orbits}. On a smooth atlas of pure relative dimension $d$ over the smooth dimension-zero stack, the inverse image of a residual gerbe with stabiliser dimension $c$ is smooth of pure dimension $d-c$ and open in its closure, so a distinct stratum in its boundary has smaller dimension, i.e.\ larger $c$; this compares strata of different bundle types as well. [ledger 7: C-FRONTIER] \end{proof} \subsection{Verification of (G1) and (G2)} At a $T$-fixed point $(\mathcal P_{\tilde w},vB)$ the stabiliser dimension, the weights of $T$ on $T^1$ and on $\Lie\cG/\Lie T$ are given by the affine root counts: \[ c=2+\sum_{\alpha\in v\Phi^+}N_\alpha+\sum_{\alpha\in v\Phi^-}(N_\alpha-1)_+ , \] the $T^1$-weights are $\{\alpha\in v\Phi^-:N_\alpha=0\}$ together with each $\alpha$ repeated $(-d_\alpha-1)_+$ times ($d_\alpha=h_\alpha-\ell_\alpha$), and the stabiliser weights are $\alpha\in v\Phi^+$ with multiplicity $N_\alpha$ and $\alpha\in v\Phi^-$ with multiplicity $N_\alpha-1$. At a point $u_\gamma(1)vB$ the stabiliser torus is $(\ker\gamma)^\circ$ and its weights are computed from a semisimple element of the stabiliser acting on $T^1=\operatorname{coker}(\ev_1(\Lie H)+\mathfrak b)\oplus H^1$. Both computations were carried out exactly over $\Q$ at every orbit of Proposition~\ref{prop:orbits} with $1\le c\le 6$, with an explicit integral cocharacter in each case. \begin{proposition}\label{prop:H} Hypothesis~\ref{hyp:G} holds. \end{proposition} \begin{proof} (G0) is Proposition~\ref{prop:frontier}. For (G1) and (G2), Table~\ref{tab:weights} lists, for each orbit with $1\le c\le 6$, an integral cocharacter with all $T^1$-weights positive and all stabiliser weights $\le 0$. [Certificate: certificate/blind-weights/ (every row, including $C_6$ and the rank-one $c=6$ orbits of $B_1$, $B_3$, $B_4$); earlier, exp/enum6/sweepQ.sage, tcurve.sage and exp/korb, and for $C_6$ certificate/census-repair/check.py.] % [referee report 008, R8-01] \end{proof} \begin{table}[ht] \caption{Contracting cocharacters $\lambda=m_1\alpha_1^\vee+m_2\alpha_2^\vee$ (selected rows). The full list is in \texttt{certificate/blind-weights/} (\texttt{weights.sage}, log \texttt{yun-blind5.log}, values \texttt{mine.json}): a re-implementation made from Section~\ref{sec:census} alone, which tests $57$ representative rows ($38$ $T$-fixed flags and $19$ rank-one flags) covering all $46$ distinct orbits with $1\le c\le6$ over the seven types (several $T$-fixed flags represent the same orbit, e.g.\ twelve over $\star$ represent three closed $K$-orbits). % [referee report 008, R8-02] The earlier reference log \texttt{reference-logs/sweepQ.log}, produced by \texttt{run\_all.sh} (the same content as v0.1's \texttt{logs/e6Q.log}), agrees with it on every row it contains, but has no row for $C_6$ (whose row is also in \texttt{certificate/census-repair/}) nor for the three rank-one orbits with $c=6$ of $B_1$, $B_3$ and $B_4$. % [referee report 007, R7-05, R7-06; blind re-implementation 2026-10-20] Rank is the rank of the stabiliser torus.}\label{tab:weights} \small \begin{tabular}{llcp{3.2cm}p{4.3cm}} \toprule type & $c$ & rank & $\lambda$ & $T^1$-weights\\ \midrule $\star$ & 1,2,3 & 1 & $(0,-1)$, $(-2,-3)$, $(-1,-2)$ & $(2)$; $(1,2)$, $(2,3)$; $(1,1,2)$, $(1,2,3)$\\ $\star$ & 4 & 2 & $(1,1)$ & $(1,1,1,1)$\\ $B_1$ & 1--6 & 1 & $(-1,-2)$ & $(2)$, \dots, $(1,1,1,1,2,2)$\\ $B_1$ & 3,4,5,6 & 2 & $(-5,-12)$, $(-11,-12)$ & $(2,9,12)$, $(2,7,9,12)$, \dots\\ $B_3$ & 3--6 & 1 & $(1,0)$ & $(2,3,3)$, $(1,2,3,3)$, \dots\\ $B_3$ & 5,6 & 2 & $(1,1)$, $(-3,-12)$ & $(1,1,1,1,2)$, $(3,6,9,12,15,15)$\\ $B_4$ & 4--6 & 1 & $(0,1)$ & $(1,1,2,2)$, \dots\\ $B_4$ & 5,6 & 2 & $(-12,1)$ & $(1,12,25,38,38)$, \dots\\ $B_5$ & 5,6 & 2 & $(1,1)$ & $(1,1,1,1,2)$, $(1,1,1,1,1,2)$\\ $B_6$ & 6 & 2 & $(1,2)$ & $(1,1,1,1,2,2)$\\ $C_6$ & 6 & 2 & $(1,1)$ & $(1,1,1,1,2,2)$\\ \bottomrule \end{tabular} \end{table} \begin{remark} The weights $(1,2,3)$ at the $\star$-orbit of codimension three are the values $\langle\alpha,\beta^\vee\rangle$ of the positive $\Gtwo$-roots on the coroot of a short root $\beta$. Independently of the census, the weighted local models at the codimension-three and codimension-four strata of $D_x$ were also computed directly from the equations of $D_x$ (first in chart $C_0$), with the same weights; they serve as a check. \end{remark} % [referee report 002, R2-07: Section 7 replaced by ref-BCH1-final.tex (after referee BCHR1), retyped to the paper's conventions] \section{Algebraic projectors through quotient singularities}\label{sec:transfer} Throughout this section the ground field is $\C$, a \emph{scheme} is a separated scheme of finite type over $\C$, $\mathrm{CH}_*(-)_\Q=\mathrm{CH}_*(-)\otimes\Q$ is the Chow group \cite[Ch.~1]{Fulton}, and for a morphism $f:T\to Y$, $\mathrm{CH}^*(T\xrightarrow{f}Y)_\Q$ is Fulton's bivariant (operational) group with rational coefficients \cite[Ch.~17]{Fulton}, with cap product $\cap$, pullback along arbitrary morphisms, proper pushforward and product \cite[\S17.1]{Fulton}; $\mathrm{CH}^*(Y)_\Q=\mathrm{CH}^*(Y\to Y)_\Q$. Bivariant degrees may be negative. $H^{BM}_*$ is rational Borel--Moore homology (equal to $H_*(-;\Q)$ on compact spaces) and $\cl:\mathrm{CH}_k(T)_\Q\to H^{BM}_{2k}(T)$ is the cycle map, compatible with proper pushforward, exterior product and refined Gysin maps of regular embeddings \cite[\S\S19.1--19.2]{Fulton}. \tg{S-lit} When $T$ is a compact oriented rational homology manifold of real dimension $2n$ we also write \[ \clH:=\PD_T^{-1}\circ\cl:\ \mathrm{CH}_k(T)_\Q\longrightarrow H^{2n-2k}(T;\Q) \] for the cohomological cycle class; all pullbacks $q^*$ below act on $\clH$, all pushforwards and caps on $\cl$. \subsection{Setting}\label{ssec:setting} We fix the following. \begin{enumerate} \item[(T1)] $R$ is the variety of Theorem~\ref{thm:U} \emph{for the weighted clean model constructed in Theorem~\ref{thm:G}}: a normal projective scheme of pure dimension $d$ with quotient singularities. Projectivity is a property of this model, not of an arbitrary proper good model (Remark~\ref{rem:Rproj}). \tg{D} \item[(T2)] $q:\overline X\to R$ is a projective birational morphism from a smooth projective variety, an isomorphism over the smooth locus $R^{\mathrm{sm}}$; $\Gamma_q\subset\overline X\times R$ is its graph and ${}^t\Gamma_q\subset R\times\overline X$ the transpose. \item[(T3)] A finite group $A$ acts on $R$ and on $\overline X$, and $q$ is $A$-equivariant. \item[(T4)] $p\in\mathrm{CH}_d(R\times R)_\Q$ is an algebraic correspondence. The abstract results (Definition~\ref{def:comp} to Proposition~\ref{prop:transfer}) hold for \emph{every} such $p$. Its realisation in this paper is $p=\varepsilon P_\star$ of Remark~\ref{rem:realise}. \end{enumerate} A \emph{test space} is a product $R^a\times\overline X^{\,b}$ ($a,b\ge0$). A product of two quotient singularities $S_1/H_1\times S_2/H_2=(S_1\times S_2)/(H_1\times H_2)$ is a quotient singularity, so every test space has quotient singularities and is equidimensional. \tg{D} % [referee report 003, R3-06; audit hodge-YAUD6] -- begin \paragraph{Stagewise gluing over $B$.} The open orbit $Y=\Gr_{x,\lambda}$ is not proper; the projective base is $B=\operatorname{Bl}_\star\Gr_{x,\le\lambda}$. Glue at every stage: put $\mathcal X_0=B$, $V_B=B\setminus D_x$, and $\mathcal X_i|_{B\setminus E}=Y\times_{\mathcal B}\mathcal B_i$, $\mathcal X_i|_{V_B}=V_B$. On the overlap $Y\setminus D_x$ the weighted filtration is the unit filtration, whose Rees algebra is canonically $\cO[t]$; glue the pulled-back Rees algebra to $\cO_{V_B}[t]$ degree by degree, and call the result $\mathcal A_i$ on $\mathcal X_{i-1}$ (finitely generated, being so locally, with $(\mathcal A_i)_0=\cO$). Then $b_i:\mathcal X_i=\Proj^{\mathrm{st}}_{\mathcal X_{i-1}}(\mathcal A_i)\to\mathcal X_{i-1}$ with $L_i=\cO_{\mathcal X_i}(1)$, the section $t$ trivialising $L_i$ on the unchanged locus. On the $Y$-part flat base change applies, $Y\times_{\mathcal B}\mathcal B_{i-1}\to\mathcal B_{i-1}$ being smooth \cite[Cor.~3.2.14(iii)]{QR}. Inductively each $\mathcal X_i$ is of finite type, proper over $B$, with finite absolute diagonal, hence a tame Deligne--Mumford stack with a separated proper coarse space $C_i$ ($C_0=B$). No coarse-space assertion about any $\mathcal B_i$ is made. \begin{lemma}[Projectivity of the coarse weighted sequence]\label{lem:coarse-weighted-projective} Let $B$ be a projective scheme over $\C$, $\mathcal X_0=B$, and for $1\le i\le N$ let $b_i:\mathcal X_i=\Proj^{\mathrm{st}}_{\mathcal X_{i-1}}(\mathcal A_i)\to\mathcal X_{i-1}$ with $\mathcal A_i$ a finitely generated non-negatively graded quasi-coherent algebra, $(\mathcal A_i)_0=\cO$, and $L_i=\cO_{\mathcal X_i}(1)$. Then the absolute coarse spaces $c_i:\mathcal X_i\to C_i$ exist, each $C_i$ is a projective scheme, and $f_i:C_i\to C_{i-1}$ is projective; there are $n_i>0$ and an $f_i$-ample line bundle $M_i$ with $M_i=(c_i)_*L_i^{\otimes n_i}$ and $c_i^*M_i\simeq L_i^{\otimes n_i}$. Consequently the normalisation of $C_N$ in a finite extension of its function field is projective. \end{lemma} \begin{proof} By \cite[Prop.~1.6.1(i)--(ii)]{QR} the $\mathcal X_i$ are proper of finite type with finite diagonal, hence tame Deligne--Mumford with proper coarse spaces. Choose $d_i$ with $\mathcal A_i^{(d_i)}$ generated in degree one \cite[Prop.~1.6.3]{QR}, and by \cite[Setup~1.7.2]{QR} applied to $\mathcal X_i\to\Spec\C$ choose $n_i$, divisible by $d_i$, such that $L_i^{\otimes n_i}$ descends to $M_i$ on $C_i$ with the adjunction isomorphism; this kills the characters of all absolute stabilisers, including inherited ones. To see that $M_i$ is $f_i$-ample, choose \'etale locally on $C_{i-1}$ an affine $V$ with $\mathcal X_{i-1}\times_{C_{i-1}}V=[U/G]$, $U$ affine and finite over $V$ and $G$ finite constant \cite[Thm.~3.2, Cor.~3.3]{AOV}. Put $\mathcal P=\Proj^{\mathrm{st}}_U(\mathcal A_i|_U)$, $P=\Proj_U((\mathcal A_i|_U)^{(d_i)})$ and $H=\cO_P(1)$; $P$ is the relative coarse space of $\mathcal P$ over $U$, with $p^*H=\cO_{\mathcal P}(d_i)$, the construction is $G$-equivariant, and $C_i\times_{C_{i-1}}V=P/G$ by compatibility of coarse spaces with \'etale base change (this does not identify an absolute coarse space with a relative one over a stack). The quotient $P/G$ is a scheme ($H$ is ample and $G$-linearised; invariant affine opens from products over $G$ of homogeneous sections), with finite surjective $\rho:P\to P/G$ and $\rho^*M_{i,V}\simeq H^{\otimes n_i/d_i}$, which is ample as $U$ is affine. Both $P$ and $P/G$ are proper over the noetherian affine $V$, so finite-surjective descent of ampleness \cite[Lemma~30.17.2]{Stacks} makes $M_{i,V}$ ample, and fpqc-locality of ampleness of a given line bundle \cite[Lemma~74.13.1]{Stacks} makes $M_i$ $f_i$-ample; in particular $f_i$ is representable by schemes \cite[\S71.14]{Stacks}. If $H_{i-1}$ is ample on the projective $C_{i-1}$, then $M_i^{\otimes a_i}\otimes f_i^*H_{i-1}^{\otimes b_i}$ is ample for suitable $a_i,b_i>0$, so $C_i$ is projective and $f_i$ is projective. Finite-type $\C$-schemes are excellent, so normalisation in a finite function-field extension is finite, and the pullback of $H_N$ is ample. \end{proof} The global coarse square $c_{i-1}\circ b_i=f_i\circ c_i$ is commutative, not asserted Cartesian, and $\rho$ may be ramified: finite-surjective descent is used, not flatness. A Veronese (changing the grading, giving the relative coarse space $P$) and finite-group invariants (giving $P/G$) are different operations. % [referee report 004, R4-02; audit hodge-YAUD8] \begin{remark}[The global graded algebras]\label{rem:projconditional} Lemma~\ref{lem:coarse-weighted-projective} is conditional on global finitely generated graded algebras $\mathcal A_i$, and Theorem~\ref{thm:G} supplies them. Lemmas~\ref{lem:intrinsic} and~\ref{lem:globalatlas} give global finitely generated $I^{(i-1)}_\bullet$ on $\mathcal B_{i-1}$. Pull back along the smooth current $Y$-stage. The centre is disjoint from the unchanged $\mathcal U$-locus (at old exceptional crossings the radial image and the original blow-down image differ, so it is this disjointness, not a statement that the original image is only $x_i$, that is used), and there every graded ideal is canonically $\cO$, so the algebra is $\cO[u]$; glue it to $\cO[u]$ over $V_B=B\setminus D_x$. The gluing preserves multiplication and grading, with no line-bundle twisting ambiguity, and gives $\mathcal A_i$ on the glued $\mathcal X_{i-1}$ with degree-zero part $\cO$, finitely generated smooth-locally and hence globally. Lemma~\ref{lem:chartsquare} identifies its $\Proj^{\mathrm{st}}$ with the prescribed stage. The initial glued base is the projective $B$; each $\mathcal X_i$ is proper over $B$ with finite absolute diagonal. So $R$ is projective given Theorem~\ref{thm:G} as proved with Lemma~\ref{lem:globalatlas}; no new projectivity hypothesis is required. \end{remark} % [referee report 003, R3-06; audit hodge-YAUD6] -- end \begin{remark}[projectivity of $R$, and the choice of $q$]\label{rem:Rproj} Definition~\ref{def:good} asks only for a \emph{proper} good model, and properness does not give projectivity. For the model of Theorem~\ref{thm:G}, glue the stages over the projective $B$ as above and apply Lemma~\ref{lem:coarse-weighted-projective}: the coarse space $C_N$ (the $X$ of Section~\ref{sec:U}) is projective over $B$, hence projective, and the finite normalisation $R$ is projective, not merely proper. \tg{D, conditional on the global construction of Theorem~\ref{thm:G}, supplied by Lemma~\ref{lem:globalatlas} and Remark~\ref{rem:projconditional}} % [referee report 003, R3-06; audit hodge-YAUD6] In the transfer calculation itself, properness of $R$ suffices; projectivity of $R$ is what makes a projective strong resolution by blow-ups a smooth \emph{projective} variety $\overline X$ (projectivity of $q$ alone would not). (T2) and (T3) can be met together. A functorial resolution \cite[Thm.~1.1]{BM97} is a finite sequence of blow-ups with smooth centres, an isomorphism over the regular locus, and functorial for automorphisms. It is therefore projective and lifts the action of $A$, which already acts on the normalisation $R$ by functoriality. \tg{S + D} The neighbourhood of $E$ is smooth, so it is untouched. The abstract statements below need neither equivariance nor an isomorphism over all of $R^{\mathrm{sm}}$: a proper map of degree one between these rational homology manifolds already gives $q_*q^*=1$. The stronger choices serve the group and $E$ constructions. \tg{D} \end{remark} \subsection{The formalism}\label{ssec:formalism} \begin{proposition}[Alexander property]\label{prop:alex} Let $Y$ be a test space, $n=\dim Y$, and $f:T\to Y$ any morphism of schemes. Then \[ \ev_Y:\mathrm{CH}^r(T\xrightarrow{f}Y)_\Q\longrightarrow \mathrm{CH}_{n-r}(T)_\Q,\qquad \kappa\longmapsto \kappa\cap[Y], \] is an isomorphism. For $Y$ smooth its inverse sends $w\in\mathrm{CH}_*(T)_\Q$ to the class $\kappa_w$ with $\kappa_w\cap z=\gamma^!(z\times w)$ for $z\in\mathrm{CH}_*(Y')_\Q$, $Y'\to Y$. Here $\gamma$ is the graph (equivalently the diagonal $\Delta_Y$), a regular embedding. \end{proposition} \begin{proof} In characteristic $0$, schemes with quotient singularities are Alexander schemes \cite[Thm.~5.1]{Vistoli-Compositio}, \cite[Cor.~6.4]{Vistoli-Invent}. By \cite[Def.~2.1]{Vistoli-Compositio}, this means that $\ev_Y$ is an isomorphism for \emph{every} $T\to Y$ and the commutativity condition holds. (\cite[Def.~6.2]{Vistoli-Invent} is the \emph{weak} Alexander notion in the stack-compatible bivariant theory; it agrees with the Alexander notion when a resolution exists, as here \cite[p.~657]{Vistoli-Invent}.) \tg{S} For $Y$ smooth this is \cite[Props.~17.3.1, 17.4.2]{Fulton}, as quoted in \cite[Introduction and Prop.~2.3]{Vistoli-Compositio}, and the inverse is the formula in the proof of \cite[Prop.~4]{EG}. \tg{S} \end{proof} \begin{definition}[products and the composition through $R$]\label{def:comp} Let $Y$ be a test space, $f:T\to Y$ and $g:Y'\to Y$ morphisms, $x\in\mathrm{CH}_*(T)_\Q$, $y\in\mathrm{CH}_*(Y')_\Q$. Following \cite[Note~2.4]{Vistoli-Compositio} and \cite[p.~658]{Vistoli-Invent} put \[ x\cdot_Y y\;:=\;\ev_Y^{-1}(x)\cap y\;\in\;\mathrm{CH}_*(T\times_YY')_\Q . \] For test spaces $Q_1,Q_2$ and $x\in\mathrm{CH}_a(Q_1\times R)_\Q$, $y\in\mathrm{CH}_b(R\times Q_2)_\Q$, take $Y=R$, $T=Q_1\times R$ and $Y'=R\times Q_2$, both mapping to $R$ by projection. Then $T\times_RY'=Q_1\times R\times Q_2$, and $\kappa=\ev_R^{-1}(x)$ has relative degree $d-a$. Define \[ y\circ_R x\;:=\;p_{13*}\bigl(x\cdot_R y\bigr)\;\in\;\mathrm{CH}_{a+b-d}(Q_1\times Q_2)_\Q , \] where $p_{13}$ is proper because $R$ is proper. The transpose of $x\in\mathrm{CH}_*(Q_1\times Q_2)_\Q$ is ${}^tx=\sigma_*x$, with $\sigma$ the exchange of factors. Replacing $R$ by the smooth $\overline X$ gives Fulton's composition of correspondences \cite[Def.~16.1]{Fulton}, by the second half of Proposition~\ref{prop:alex}. \tg{S-lit} Replacing $R$ by $R^m$ gives $\circ_{R^m}$. For a correspondence $c\in\mathrm{CH}_*(Q_1\times Q_2)_\Q$ through a smooth $Q_1$ (or one of the form above) we write $c_*(w)=c\circ w$ for $w\in\mathrm{CH}_*(Q_1)_\Q$, regarded as a correspondence from a point. \end{definition} The product $\cdot_Y$ has the formal properties of \cite[Prop.~8.1.1]{Fulton}. In particular $\mathrm{CH}_*(Y)_\Q$ is a commutative ring and the projection formula of \cite[Prop.~8.3]{Fulton} holds \cite[Note~2.4]{Vistoli-Compositio}, \cite[p.~658]{Vistoli-Invent}. \tg{S} So all three of $R\times R$, $\overline X\times R$ and $\overline X\times\overline X$ carry intersection products. The first two do by Definition~\ref{def:comp}, being Alexander schemes; the third does by \cite[Ch.~8]{Fulton}, and on a smooth test space the two agree (Proposition~\ref{prop:alex}). On a test space $Y=Z/G$ with $Z$ smooth, $\cap[Y]:\mathrm{CH}^*(Y)_\Q\to\mathrm{CH}_*(Y)_\Q$ is an isomorphism, and the resulting ring is the equivariant one \cite[Thm.~4]{EG}, which agrees with Vistoli's and Gillet's products on the quotient \emph{stack} \cite[Prop.~15]{EG}. \tg{S} Proposition~15 is not a Betti comparison on coarse-space products; that comparison is Proposition~\ref{prop:cc}. \emph{No flatness or lci property of $q$ is used anywhere: $q$ enters only through the cycle $\Gamma_q$, and the diagonal of the singular $R$ is never claimed to be a regular embedding.} \tg{D} \paragraph{Presentations and mixed spaces.} By \cite[Prop.~2.8]{Vistoli-Invent} the normal scheme $R$ with quotient singularities is the moduli space of a smooth Deligne--Mumford stack $\mathcal R$ with generically trivial stabiliser, as recorded in \cite[before Cor.~2.20]{EHKV}. By \cite[Thm.~2.18]{EHKV}, $\mathcal R\cong[Z/G]$ with $Z$ a smooth algebraic space and $G$ a linear algebraic group of dimension $g$, so $R=Z/G$ \cite[Cor.~2.20]{EHKV}. \tg{S} The stabiliser in $G$ of a point of $Z$ is the automorphism group of its image in $\mathcal R$, so the generic stabiliser is trivial. \tg{S-lit} Fix a representation $V$ of dimension $l$ and an open $U_V\subset V$ on which $G$ acts freely, with $\operatorname{codim}(V\setminus U_V)\ge c$. Put $Z_U=(Z\times U_V)/G$, a smooth algebraic space of dimension $d+l$ \cite[Prop.~22, \S2.2]{EG}. \tg{S} For a scheme $S$ on which $G$ acts trivially, $S\times Z\to S\times R$ is a geometric quotient, since geometric quotients are stable under the flat base change $S\times R\to R$ \cite[Ch.~0, \S2]{GIT}. \tg{S-lit} Write \begin{gather*} \rho=\rho_S:\;S\times Z_U\longrightarrow S\times R,\\ \hat\pi=\hat\pi_S:\;\mathrm{CH}_k(S\times R)_\Q\xrightarrow{\ \sim\ }\mathrm{CH}^G_{k+g}(S\times Z)_\Q=\mathrm{CH}_{k+l}(S\times Z_U)_\Q , \end{gather*} the isomorphism of \cite[Thm.~3(a)]{EG}, valid for $k>\dim(S\times R)-c$. On cycles it is $[W]\mapsto e_W\,i_W^{-1}[(\pi^{-1}W)_{\mathrm{red}}]$ \cite[Remark~(6) after Thm.~4]{EG}, where $e_W$ is the order of the generic stabiliser \emph{in $G$} of $\pi^{-1}W$, and $i_W=1$ in characteristic $0$ \cite[\S4.2, before Lemma~5]{EG}. In particular $\hat\pi[S\times R]=[S\times Z_U]$. \tg{S} The map $\rho$ factors as $S\times Z_U\to S\times\mathcal R\to S\times R$: smooth of relative dimension $l$, then a bijection on points. So $\rho^{-1}$ of a pure $k$-dimensional closed set is pure of dimension $k+l$. \tg{D} % [referee report 003, R3-05; audit hodge-YAUD5] -- begin \paragraph{Category and local properness.} The mixed spaces $Z_U$ are algebraic spaces, so the bivariant operations are used in their extension to algebraic spaces: \cite[\S6.1]{EG} supplies the Chow operations, and \cite[Prop.~6.1 and the discussion after Def.~6.2]{Vistoli-Invent} the comparison with the stack-compatible bivariant theory for scheme endpoints. The class $\kappa$ of Proposition~\ref{prop:alex} is used on these intermediate algebraic spaces in that sense, and the smooth-target evaluation formula is the diagonal formula there. \tg{S} The locally proper action required by \cite[Thm.~3, Prop.~10]{EG} holds: take $\mathcal R$ the separated smooth canonical stack of $R$; its atlas relation is the normalisation of $V_a\times_RV_b$, finite over $V_a\times V_b$ since $R$ is separated, so the diagonal of $\mathcal R$ is finite, and for $\mathcal R=[Z/G]$ the action map $G\times Z\to Z\times Z$ is a base change of that diagonal, hence proper. Generic inertia is trivial by the choice of $\mathcal R$. For a separated finite-type scheme $S$ with trivial $G$-action the projection $S\times R\to R$ is flat (a base change of $S\to\Spec\C$), whatever the singularities of $S$, so the action stays locally proper on $S\times Z$. \tg{D} % [referee report 003, R3-05; audit hodge-YAUD5] -- end \paragraph{Topology.} A complex variety with quotient singularities is an oriented rational homology manifold. At the image of a fixed point of a finite linear group $H\subset GL_n(\C)$, transfer \cite[Ch.~III, \S7]{BredonCTG} gives $H^*(\C^n/H,\C^n/H\setminus0;\Q)=H^*(\C^n,\C^n\setminus0;\Q)^H=\Q[-2n]$, because $H$ preserves the complex orientation. \tg{S-lit + D} Hence every test space, and every $S\times Z_U$ with $S$ a test space, satisfies Poincar\'e and Alexander duality over $\Q$: $\PD:H^{j}_{T}(M)\xrightarrow{\sim}H^{BM}_{\dim_\R M-j}(T)$ for $T\subset M$ closed \cite[Ch.~V]{BredonSheaf}, natural for closed inclusions and open restriction. \tg{S-lit} For compact oriented rational homology manifolds $Q_1,Q_2$ and $\gamma\in H_*(Q_1\times Q_2)$ of even degree, write \[ \gamma_*:H^*(Q_1)\to H^*(Q_2),\qquad \gamma_*\xi=\PD_{Q_2}^{-1}\,\mathrm{pr}_{2*}\bigl(\mathrm{pr}_1^*\xi\cap\gamma\bigr). \] For an algebraic class $x$ we write $x_*:=\cl(x)_*$. For a generically free finite quotient $\varphi:A'\to A'/K'$ of oriented rational homology manifolds, $\varphi^!:=\PD\circ\varphi^*\circ\PD^{-1}$, with supports, satisfies $\varphi_*\varphi^!=|K'|$. \tg{D} \subsection{The cycle class on quotient-singular test spaces} % [referee report 003, R3-05; audit hodge-YAUD5] -- begin \begin{lemma}[Supported quotient multiplicities]\label{lem:suppmult} Let $a:M'\to M=M'/L$ be a generically free finite quotient of complex oriented rational homology manifolds of the same dimension, with $L$ acting effectively. For a closed $W\subset M$ define $a_W^!:H_j^{BM}(W;\Q)\to H_j^{BM}(a^{-1}W;\Q)$ by ambient duality with supports and pullback in cohomology with supports. Then $(a|_{a^{-1}W})_*a_W^!=|L|\,\mathrm{id}$. If $W$ is integral and the generic point stabiliser above $W$ has order $e$, then $a_W^![W]=e\,[(a^{-1}W)_{\mathrm{red}}]$, the bracket summing all reduced components dominating $W$. \end{lemma} \begin{proof} For $\alpha\in H^*_W(M;\Q)$ the supported projection formula gives $a_*(a^*\alpha\cap[M'])=\alpha\cap a_*[M']=|L|\,\alpha\cap[M]$, an equality in the homology of $W$, not merely after pushing to $M$. The class $a_W^![W]$ lies in top Borel--Moore degree on the reduced inverse image and is $L$-invariant; $L$ permutes the components over $W$ transitively, so their coefficients agree, and their total generic pushforward degree is $|L|/e$. The projection formula forces the common coefficient to be $e$. In analytic neighbourhoods one argues branch by branch. \end{proof} % [referee report 003, R3-05; audit hodge-YAUD5] -- end \begin{lemma}\label{lem:V1} Let $S$ be a test space, $Q=S\times R$, $n=\dim Q$, $Q_U=S\times Z_U$, and define \[ c_{\mathrm{top}}=\PD_{Q_U}\circ\rho^*\circ\PD_Q^{-1}:\;H^{BM}_m(Q)\longrightarrow H^{BM}_{m+2l}(Q_U). \] For $W\subset Q$ closed, with $W_U=\rho^{-1}W$, define its supported version \[ c^W_{\mathrm{top}}:\ H^{BM}_m(W)\cong H^{2n-m}_W(Q)\xrightarrow{\ \rho^*\ }H^{2n-m}_{W_U}(Q_U)\cong H^{BM}_{m+2l}(W_U), \] which is compatible with pushforward to $Q$ and $Q_U$. \begin{enumerate} \item[(a)] $c_{\mathrm{top}}$ is an isomorphism for $m>2(n-c)+1$. \item[(b)] For $\beta\in\mathrm{CH}_k(Q)_\Q$, $\cl(\hat\pi\beta)=c_{\mathrm{top}}(\cl\beta)$, and the same holds with supports. \end{enumerate} \end{lemma} \begin{proof} (a) Put $E_Q=S\times(Z\times V)/G\xrightarrow{p}Q$ and $j:Q_U\hookrightarrow E_Q$, so that $\rho=p\circ j$. Scalar multiplication on $V$ retracts $E_Q$ onto the zero section, so $p$ is a homotopy equivalence. $E_Q$ and $Q$ have quotient singularities, so $\PD_{E_Q}\circ p^*\circ\PD_Q^{-1}$ is an isomorphism; we use no Vietoris--Begle theorem for the non-proper $p$, and no smoothness of $S$. The complement $E_Q\setminus Q_U=S\times(Z\times(V\setminus U_V))/G$ has complex dimension at most $n+l-c$. By the localisation sequence, $j^*:H^{BM}_{m+2l}(E_Q)\to H^{BM}_{m+2l}(Q_U)$ is therefore an isomorphism for $m+2l>2(n+l-c)+1$. \tg{D} (b) Both sides are linear, so take $\beta=[W]$ with $W\subset Q$ closed integral of dimension $k$. Both $\cl(\hat\pi[W])=e_W[W_U]$ and $c^W_{\mathrm{top}}(\cl[W])$ lie in $H^{BM}_{2(k+l)}(W_U)$, the top degree of the pure $(k+l)$-dimensional $W_U$. This group is the $\Q$-vector space on the irreducible components of $W_U$, and it injects into the same group of any dense open subset \cite[\S19.1]{Fulton}. \tg{S-lit} Since $c^W_{\mathrm{top}}$ commutes with restriction to opens, it suffices to compare the two classes near a general point of each component. No Poincar\'e duality on the possibly singular $W$ is used. \emph{Keep the two groups distinct.} Near a general point of $W$, analytically locally, $S=S'/K'$ with $K'$ finite acting effectively and generically freely on the smooth $S'$, and $\mathcal R=[\Omega/H]$ with $H$ finite, $\Omega$ smooth and trivial generic stabiliser \cite[proof of Prop.~2.8]{Vistoli-Invent}. \tg{S} Let $\Omega_U=\Omega\times_{\mathcal R}Z_U$. This is a smooth $U_V$-fibration over $\Omega$, and $H$ acts \emph{freely} on it, because $\Omega_U\to Z_U$ is the base change of the $H$-torsor $\Omega\to[\Omega/H]$. The factor $K'$ need not act freely: on $S=\mathbb A^2/\mu_2$, with $R$ a point, the chart $\mathbb A^2\times U_V\to S\times U_V$ stays ramified along $\{0\}\times U_V$. Put \begin{gather*} A'=S'\times\Omega,\qquad A'_U=S'\times\Omega_U,\\ \varphi:A'\to Q,\qquad \varphi_U:A'_U\to Q_U,\qquad \mathrm{pr}:A'_U\to A', \end{gather*} where $\varphi$ and $\varphi_U$ are both quotient maps by $K'\times H$, $\varphi_U$ need not be \'etale, and $\mathrm{pr}$ is smooth of relative dimension $l$. Shrink further around a general point of a component, so that the generic stabiliser orders of the inverse image of $W$ are $s$ in $K'$ and $r$ in $H$. The Edidin--Graham number is $e_W=r$, since $G$ acts only through the $R$-factor. Downstairs in $A'$ the generic stabiliser of $A'_W=\varphi^{-1}(W)_{\mathrm{red}}$ is of order $sr$. Upstairs in $A'_U$ the generic stabiliser of $A'_{U,W}=\varphi_U^{-1}(W_U)_{\mathrm{red}}$ is of order $s$, since $H$ acts freely there. The supported pullbacks $\varphi^![W]$ and $\varphi_U^![W_U]$ are invariant top-degree classes on these reduced inverse images, whose components the group permutes transitively over the irreducible $W$. Lemma~\ref{lem:suppmult}, applied to $\varphi$ (point stabiliser of order $sr$) and to $\varphi_U$ (order $s$), with $\varphi_*\varphi^!=|K'||H|=\varphi_{U*}\varphi_U^!$, gives % [referee report 003, R3-05; audit hodge-YAUD5] \[ \varphi^![W]=sr\,[A'_W],\qquad \varphi_U^![W_U]=s\,[A'_{U,W}],\qquad \mathrm{pr}^![A'_W]=[A'_{U,W}], \] with brackets summing reduced components; the last identity holds because $\mathrm{pr}$ is smooth. From $\rho\circ\varphi_U=\varphi\circ\mathrm{pr}$ and functoriality of supported pullback, \[ \varphi_U^!\bigl(c^W_{\mathrm{top}}\cl[W]\bigr)=\mathrm{pr}^!\varphi^![W]=sr\,[A'_{U,W}]=\varphi_U^!\bigl(r\,[W_U]\bigr). \] Since $\varphi_{U*}\varphi_U^!=|K'||H|$, the map $\varphi_U^!$ is injective over $\Q$. Hence $c^W_{\mathrm{top}}\cl[W]=r[W_U]=e_W[W_U]=\cl(\hat\pi[W])$ near the chosen point, and therefore globally by the top-degree injectivity above. \tg{D, after ref-BCHR1~\S3} The factor is the stabiliser factor of \cite[Remark~(6) after Thm.~4]{EG}, not its inverse. Generic freeness and effectivity of both charts are essential to the degree formulas. \end{proof} \begin{lemma}[topological composition]\label{lem:topcomp} Let $Q_1,Q_2,Q_3$ be compact oriented rational homology manifolds, $N=Q_1\times Q_2\times Q_3$, and $\gamma\in H_*(Q_1\times Q_2)$, $\delta\in H_*(Q_2\times Q_3)$ of even degrees. Put \[ \gamma\cdot^{\mathrm{top}}_{Q_2}\delta=\PD_N\bigl(\mathrm{pr}_{12}^*\PD^{-1}\gamma\cup\mathrm{pr}_{23}^*\PD^{-1}\delta\bigr)\in H_*(N),\qquad \delta\circledast\gamma=p_{13*}(\gamma\cdot^{\mathrm{top}}_{Q_2}\delta). \] Then $(\delta\circledast\gamma)_*=\delta_*\circ\gamma_*$. For a morphism $f:Q_1\to Q_2$ with graph class $\Gamma_f=(1,f)_*[Q_1]$, we have $(\Gamma_f)_*=f_*:=\PD^{-1}_{Q_2}f_*\PD_{Q_1}$ and $({}^t\Gamma_f)_*=f^*$. Finally, $({}^t\gamma)_*$ is the adjoint of $\gamma_*$ for the Poincar\'e pairings. \end{lemma} \begin{proof} Let $\xi\in H^*(Q_1)$. Use $\mathrm{pr}_{12}^*\theta\cap[N]=(\theta\cap[Q_1\times Q_2])\times[Q_3]$, the projection formula for $\mathrm{pr}_{23}$, and even degrees to suppress signs: \[ \mathrm{pr}_{23*}\bigl(\mathrm{pr}_1^*\xi\cap(\gamma\cdot^{\mathrm{top}}\delta)\bigr) =\PD^{-1}\delta\cap\bigl(\mathrm{pr}_{2*}(\mathrm{pr}_1^*\xi\cap\gamma)\times[Q_3]\bigr) =\mathrm{pr}_2^*(\gamma_*\xi)\cap\delta . \] Applying $\mathrm{pr}_{3*}$ and $\PD^{-1}_{Q_3}$ gives $\delta_*\gamma_*\xi$. For the graph, $\mathrm{pr}_1^*\xi\cap(1,f)_*[Q_1]=(1,f)_*(\xi\cap[Q_1])$ and $\mathrm{pr}_2\circ(1,f)=f$. For the transpose ${}^t\Gamma_f=(f,1)_*[Q_1]\in H_*(Q_2\times Q_1)$ and $\eta\in H^*(Q_2)$, $\mathrm{pr}_1^*\eta\cap(f,1)_*[Q_1]=(f,1)_*(f^*\eta\cap[Q_1])$, and $\mathrm{pr}_2\circ(f,1)=\mathrm{id}_{Q_1}$. Adjointness is the symmetry of $\int_{Q_1\times Q_2}\mathrm{pr}_1^*\xi\cup\mathrm{pr}_2^*\eta\cap\gamma$. Only cup and cap products, Poincar\'e duality and the projection formula over $\Q$ are used, all valid on compact oriented rational homology manifolds \cite[Ch.~V]{BredonSheaf}. \tg{D + S-lit} \end{proof} % [referee report 003, R3-05; audit hodge-YAUD5] -- begin \begin{lemma}[Bivariant compatibility with a coarse quotient]\label{lem:bivcoarse} Work with rational Chow groups and their bivariant extension to algebraic spaces. Let $\mathcal R=[Z/G]$ be a separated smooth Deligne--Mumford stack with trivial generic inertia and coarse space $R$, $U_V\subset V$ a free open in a representation, $\dim V=l$, $Z_U=(Z\times U_V)/G$ and $\rho:Z_U\to R$. For separated finite-type schemes $A,B$ over $\C$, every $\kappa\in\mathrm{CH}^t(A\times R\to R)_\Q$ and $y\in\mathrm{CH}_b(R\times B)_\Q$ satisfy, in the stable range, \[ \hat\pi_{A\times B}(\kappa\cap y)=\rho^*\kappa\cap\hat\pi_B(y)\quad\text{in }\mathrm{CH}_{b-t+l}(A\times Z_U\times B)_\Q . \] No smoothness of $A$ or $B$ is required. \end{lemma} \begin{proof} The action is proper, its action map being a base change of the proper diagonal of $\mathcal R$. Choose the finite surjective diagram of \cite[Prop.~10]{EG}: $g:Z'\to Z$ over $f:R'\to R$, with $Z'\to R'$ a principal $G$-bundle. For every inert scheme $S$ its product with $S$ is still a diagram of geometric quotients, as $S\times R\to R$ (and the analogue over $R'$) is flat. Write $g_U:Z'_U\to Z_U$ and $\rho':Z'_U\to R'$; then $\rho g_U=f\rho'$, a square not required to be Cartesian. Put $f_B=f\times1_B$, $g_B=g_U\times1_B$, $F=1_A\times f_B$, $G_U=1_A\times g_B$. Choose $y'$ with $(f_B)_*y'=y$ (rational pushforward along a finite surjective map is surjective) and let $\kappa'=f^*\kappa$. The smooth map $\rho'\times1_B$ is flat of relative dimension $l$ and the principal-bundle comparison is its flat pullback, so $\hat\pi'_{A\times B}(\kappa'\cap y')=\rho'^*\kappa'\cap\hat\pi'_B(y')$. Every square used for the bivariant proper or flat axiom is the Cartesian square \[ \begin{array}{ccc} A\times T'&\xrightarrow{1_A\times h}&A\times T\\ \downarrow&&\downarrow\\ T'&\xrightarrow{h}&T, \end{array} \] with $h=f_B$, $g_B$ (proper) or $\rho'\times1_B$ (flat). Functoriality gives $g_U^*\rho^*\kappa=\rho'^*f^*\kappa$. Applying $(G_U)_*$, \cite[Prop.~11(a)]{EG} ($g_*\hat\pi'=\hat\pi f_*$) and the proper axiom, \[ \begin{aligned} \hat\pi_{A\times B}(\kappa\cap y)&=\hat\pi_{A\times B}F_*(\kappa'\cap y')=(G_U)_*\hat\pi'_{A\times B}(\kappa'\cap y')\\ &=(G_U)_*(\rho'^*\kappa'\cap\hat\pi'_By')=\rho^*\kappa\cap(g_B)_*\hat\pi'_By'=\rho^*\kappa\cap\hat\pi_By. \end{aligned} \] Choose the representation with complement codimension large enough for the finitely many degrees occurring. \end{proof} % [referee report 003, R3-05; audit hodge-YAUD5] -- end \begin{proposition}[cycle-class compatibility of $\circ_R$]\label{prop:cc} Let $Q_1,Q_2$ be test spaces, $x\in\mathrm{CH}_a(Q_1\times R)_\Q$, $y\in\mathrm{CH}_b(R\times Q_2)_\Q$. Then in $H_*(Q_1\times R\times Q_2)$ \[ \cl(x\cdot_R y)=\cl x\cdot^{\mathrm{top}}_R\cl y , \] hence $\cl(y\circ_R x)=\cl y\circledast\cl x$ and $(y\circ_R x)_*=y_*\circ x_*$. The same holds with $R$ replaced by $R^m$, using the presentation $Z^m/G^m$. With $Q_1=Q_2=\mathrm{pt}$ this says that $\clH:(\mathrm{CH}_*(R)_\Q,\cdot_R)\to(H^{*}(R),\cup)$ is a ring homomorphism. \end{proposition} \begin{proof} Let $N=Q_1\times R\times Q_2$, $N_U=Q_1\times Z_U\times Q_2$ and $\kappa=\ev_R^{-1}(x)\in\mathrm{CH}^{d-a}(Q_1\times R\to R)_\Q$ (Proposition~\ref{prop:alex}), so that $x\cdot_Ry=\kappa\cap y$. All the maps $\hat\pi$ below are those of \S\ref{ssec:formalism} for the inert factors shown. Taking the codimension bound $c$ large, we may assume all degrees are in the range of Lemma~\ref{lem:V1}(a). \emph{Step 1 (the quotient map intertwines the bivariant class).} Let $\rho^*\kappa\in\mathrm{CH}^*(Q_1\times Z_U\to Z_U)_\Q$ be the bivariant pullback along $\rho:Z_U\to R$ \cite[\S17.1]{Fulton}. We claim \[ \hat\pi_{Q_1\times Q_2}(\kappa\cap y)=\rho^*\kappa\cap\hat\pi_{Q_2}(y)\quad\text{in }\mathrm{CH}_*(N_U)_\Q .\tag{$\ast$} \] This is Lemma~\ref{lem:bivcoarse} with $A=Q_1$, $B=Q_2$, which extends \cite[Lemma~6]{EG} from $\kappa\in\mathrm{CH}^*(R)$ to bivariant classes with source $Q_1\times R\to R$, inert factors possibly singular. Every square in which a cap product is formed is the Cartesian product square with an inert factor; the finite principal-bundle diagram need not be Cartesian, and no flatness of $\rho$, $f$ or $q$ is used. \tg{D from S; after ref-BCHR1~\S2} % [referee report 003, R3-05; audit hodge-YAUD5] Taking $y=[R]$ in $(\ast)$, and using $\hat\pi[R]=[Z_U]$, gives $\rho^*\kappa\cap[Z_U]=\hat\pi_{Q_1}(x)$. \emph{Step 2 (on the smooth $Z_U$ the class is a refined intersection).} Since $Z_U$ is smooth, Proposition~\ref{prop:alex} gives $\rho^*\kappa\cap z=\Delta_{Z_U}^!\bigl(\hat\pi_{Q_1}(x)\times z\bigr)$ for every $z\in\mathrm{CH}_*(Z_U\times Q_2)_\Q$. Here $\Delta^!_{Z_U}:\mathrm{CH}_*(Q_1\times Z_U\times Z_U\times Q_2)\to\mathrm{CH}_*(N_U)$ is the refined Gysin map of the regular embedding $\Delta_{Z_U}$ \cite[\S6.2]{Fulton}. Combined with Step 1, \[ \hat\pi_{Q_1\times Q_2}(x\cdot_Ry)=\Delta^!_{Z_U}\bigl(\hat\pi_{Q_1}x\times\hat\pi_{Q_2}y\bigr). \] \emph{Step 3 (cycle class upstairs).} By \cite[\S\S19.1--19.2]{Fulton}, extended to algebraic spaces as in \cite[\S\S2.8, 6.1]{EG}, $\cl$ commutes with exterior products and with $\Delta^!_{Z_U}$. \tg{S-lit} The topological $\Delta^!$ is cap product with the orientation class $u\in H^{2(d+l)}(Z_U^2,Z_U^2\setminus\Delta)$, pulled back to $\widetilde N_U=Q_1\times Z_U\times Z_U\times Q_2$; the inert factors need only be rational homology manifolds, because $u$ comes from the smooth middle factors. In the rational homology manifold $\widetilde N_U$, $\PD(\alpha\times\beta)$ is the exterior product of the duals. Moreover $u\cap\PD_{\widetilde N_U}(\theta)=\PD_{N_U}(\iota^*\theta)$ for the diagonal inclusion $\iota:N_U\hookrightarrow\widetilde N_U$, because $u\cap[\widetilde N_U]=[N_U]$, and $\iota^*(\theta_1\times\theta_2)=\mathrm{pr}_{12}^*\theta_1\cup\mathrm{pr}_{23}^*\theta_2$. Hence \tg{D} \[ \cl\bigl(\Delta^!_{Z_U}(\hat\pi x\times\hat\pi y)\bigr)=\PD_{N_U}\bigl(\mathrm{pr}_{12}^*\PD^{-1}\cl(\hat\pi x)\cup\mathrm{pr}_{23}^*\PD^{-1}\cl(\hat\pi y)\bigr). \] \emph{Step 4 (descend).} Apply Lemma~\ref{lem:V1}(b) on $Q_1\times R$, on $R\times Q_2$ and on $N$, with inert factors $Q_1$, $Q_2$ and $Q_1\times Q_2$. This gives $\cl(\hat\pi x)=c_{\mathrm{top}}\cl x$, $\cl(\hat\pi y)=c_{\mathrm{top}}\cl y$ and $\cl(\hat\pi(x\cdot_Ry))=c_{\mathrm{top}}\cl(x\cdot_Ry)$. Since $c_{\mathrm{top}}=\PD\circ\rho^*\circ\PD^{-1}$, and $\rho_N=1\times\rho\times1$ restricts to $\rho_{Q_1\times R}$ and $\rho_{R\times Q_2}$ on the factors, naturality of the cup product gives \[ c_{\mathrm{top}}\bigl(\cl x\cdot^{\mathrm{top}}_R\cl y\bigr)=\PD_{N_U}\bigl(\mathrm{pr}_{12}^*\PD^{-1}c_{\mathrm{top}}\cl x\cup\mathrm{pr}_{23}^*\PD^{-1}c_{\mathrm{top}}\cl y\bigr). \] Steps 2--4 together give $c_{\mathrm{top}}\cl(x\cdot_Ry)=c_{\mathrm{top}}(\cl x\cdot^{\mathrm{top}}_R\cl y)$, and $c_{\mathrm{top}}$ is injective in this degree by Lemma~\ref{lem:V1}(a). Finally, $\cl$ commutes with the proper pushforward $p_{13*}$ \cite[\S19.1]{Fulton}, and Lemma~\ref{lem:topcomp} gives the action. \tg{D} For $R^m$, use $Z^m$, $G^m$, $V^m$ and $U_V^m$; the complement of $U_V^m$ has codimension at least $c$, and the generic stabiliser is still trivial. \tg{D} \end{proof} \subsection{Graphs, and the transferred projector} \begin{lemma}[graph conventions]\label{lem:graphs} The classes $[\Gamma_q]\in\mathrm{CH}_d(\overline X\times R)_\Q$, $[{}^t\Gamma_q]\in\mathrm{CH}_d(R\times\overline X)_\Q$ and $[\Gamma_g]\in\mathrm{CH}_d(R\times R)_\Q$, for $g\in A$, act on cohomology by \[ (\Gamma_q)_*=q_*=\PD_R^{-1}q_*\PD_{\overline X},\qquad({}^t\Gamma_q)_*=q^*,\qquad(\Gamma_g)_*=g_*=(g^{-1})^* . \] All three have degree $0$, because $\dim\overline X=\dim R=d$, and $q_*q^*=\mathrm{id}_{H^*(R)}$. A class $(i\times i)_*\zeta$ with $\zeta$ on $E\times E$, $E\subset R^{\mathrm{sm}}$, acts by the smooth-variety formula on a neighbourhood of $E$. \end{lemma} \begin{proof} The three formulas are Lemma~\ref{lem:topcomp} for $f=q$ and $f=g$, together with $g_*=(g^{-1})^*$ for an automorphism. For $q_*q^*$: $q_*(q^*\eta\cap[\overline X])=\eta\cap q_*[\overline X]=\eta\cap[R]$, because $q$ has degree one, so $q_*[\overline X]=[R]$. \tg{D} For the last claim, $\PD^{-1}\cl((i\times i)_*\zeta)$ is supported on $E\times E$, so the action formula only involves the open neighbourhood $R^{\mathrm{sm}}$ of $E$, where $R$ is smooth. \tg{D} \end{proof} \begin{proposition}\label{prop:transfer} Assume (T1)--(T3) and let $p\in\mathrm{CH}_d(R\times R)_\Q$ be arbitrary. \begin{enumerate} \item[(i)] $P_X:={}^t\Gamma_q\circ_R\bigl(p\circ_R\Gamma_q\bigr)$ is a well-defined class in $\mathrm{CH}_d(\overline X\times\overline X)_\Q$, independent of any presentation of $R$. \item[(ii)] $(P_X)_*=q^*\,p_*\,q_*$ on $H^k(\overline X)$ for every $k$, where $p_*$ is the action of $\cl(p)$ on $H^*(R)$ through Poincar\'e duality on the rational homology manifold $R$. \item[(iii)] If $p_*$ is idempotent on $H^*(R)$, then $(P_X)_*$ is idempotent, with image $q^*\operatorname{im}(p_*)\cong\operatorname{im}(p_*)$. \item[(iv)] (Products.) Let $n\ge1$. Applying (i)--(iii) to $(R^n,\overline X^{\,n},q^n,p^{\times n})$, where $p^{\times n}\in\mathrm{CH}_{nd}(R^n\times R^n)_\Q$ is the exterior power with factors reordered, gives \begin{gather*} P^{(n)}_X:={}^t\Gamma_{q^n}\circ_{R^n}\bigl(p^{\times n}\circ_{R^n}\Gamma_{q^n}\bigr)\in\mathrm{CH}_{nd}(\overline X^{\,n}\times\overline X^{\,n})_\Q,\\ (P^{(n)}_X)_*=(q^*p_*q_*)^{\otimes n} \end{gather*} under K\"unneth. For $W\in\mathrm{CH}_k(R^n)_\Q$, regarded as a correspondence from a point, put $(q^n)^!W:={}^t\Gamma_{q^n}\circ_{R^n}W\in\mathrm{CH}_k(\overline X^{\,n})_\Q$. Then \begin{gather*} \clH\bigl((q^n)^!W\bigr)=(q^n)^*\clH(W),\\ \clH\bigl((P^{(n)}_X)_*((q^n)^!W)\bigr)=(q^n)^*\,p_*^{\otimes n}\,\clH(W), \end{gather*} where $(P^{(n)}_X)_*$ on Chow is correspondence action through the smooth $\overline X^{\,n}$ \cite[Def.~16.1]{Fulton}. \end{enumerate} \end{proposition} \begin{proof} (i) By Definition~\ref{def:comp}, $p\circ_R\Gamma_q\in\mathrm{CH}_{d+d-d}(\overline X\times R)_\Q$ (take $Q_1=\overline X$, $Q_2=R$), and then $P_X\in\mathrm{CH}_d(\overline X\times\overline X)_\Q$ (take $Q_1=Q_2=\overline X$). The maps $\ev_R$ and $\cap$ are intrinsic to $R$; presentations enter only the proof of Proposition~\ref{prop:cc}. \tg{D} (ii) Apply Proposition~\ref{prop:cc} twice and Lemma~\ref{lem:graphs}: $(P_X)_*=({}^t\Gamma_q)_*\circ p_*\circ(\Gamma_q)_*=q^*p_*q_*$, degree by degree. \tg{D} (iii) By Lemma~\ref{lem:graphs}, $(q^*p_*q_*)^2=q^*p_*(q_*q^*)p_*q_*=q^*p_*^2q_*=q^*p_*q_*$. The image is $q^*\operatorname{im}(p_*)$ because $q_*$ is onto, and $q^*$ is injective because $q_*q^*=1$. \tg{D} (iv) $R^n$ has quotient singularities, and its smooth locus is $(R^{\mathrm{sm}})^n$. $\overline X^{\,n}$ is smooth projective, and $q^n$ is a projective resolution that is an isomorphism over $(R^{\mathrm{sm}})^n$, so (T1)--(T2) hold. $\Gamma_{q^n}$ is the reordered exterior product of $n$ copies of $\Gamma_q$. Cross products are compatible with $\cl$, $\cap$, $\PD$ and pushforward, so under K\"unneth over $\Q$ we get $\cl(p^{\times n})_*=p_*^{\otimes n}$, $(q^n)_*=q_*^{\otimes n}$ and $(q^n)^*=(q^*)^{\otimes n}$. This is K\"unneth for vector spaces; no algebraic K\"unneth projector is involved. \tg{D + S-lit} For $W$, Proposition~\ref{prop:cc} with $Q_1=\mathrm{pt}$, middle factor $R^n$ and $Q_2=\overline X^{\,n}$, together with Lemma~\ref{lem:graphs}, gives $\clH((q^n)^!W)=(q^n)^*\clH(W)$. The last identity then follows from (ii) and $(q^n)_*(q^n)^*=1$, since $(P^{(n)}_X)_*(q^n)^*=(q^n)^*p_*^{\otimes n}(q^n)_*(q^n)^*=(q^n)^*p_*^{\otimes n}$. \tg{D} \end{proof} \begin{remark}[the realisation of $p$]\label{rem:realise} In this paper $p=\varepsilon P_\star$, with the rational quaternionic convention of Section~\ref{sec:setup}: $D=e_-\Q[\tA]=(-1,-1)_\Q$, the two-sided block $D\otimes D^{\mathrm{op}}\cong M_4(\Q)$, and $\varepsilon=\tfrac1{32}\sum_{a_1,a_2\in\tA}\phi_\chi(a_1a_2^{-1})\,a_1\otimes a_2$, a primitive idempotent of $M_4(\Q)$ that acts on $D$ by $x\mapsto\frac12\trd(x)$. As a correspondence, $\varepsilon$ is a $\Q$-combination (coefficients $\pm\frac1{16}$ and $0$) of graphs of the automorphisms of $R$ given by elements of $\tA(1)\times\tA(2)$; $P_\star=[\Delta_R]-\Pi_E$ with $\Pi_E$ built from classes $(i\times i)_*\zeta$ on $E\times E$ (Section~\ref{sec:A}). The abstract Proposition~\ref{prop:transfer} is coefficient-neutral: its conclusion is the same for any already realised $p\in\mathrm{CH}_d(R\times R)_\Q$. \tg{D} The realisation is supplied elsewhere in the paper: \begin{itemize} \item both commuting actions $\tA(1)$ and $\tA(2)$ are realised by automorphisms of $R$ (Section~\ref{sec:setup}, by functoriality of normalisation), and lifted to $\overline X$ by Remark~\ref{rem:Rproj}; \item $(\varepsilon P_\star)_*$ is idempotent on $H^*(R)$, with image $M_x\subset H^6(R)$ of rank $7$, for the weighted clean model (Theorem~\ref{thm:U}(b)). \end{itemize} A one-sided central isotypic projector would not do this. Proposition~\ref{prop:transfer}(iii) therefore gives a cohomological projector $P_X$ of rank $7$ on $H^*(\overline X)$ with image $q^*M_x\cong M_x$. \end{remark} \begin{example}[$\C^2/\mu_2$ and its blow-up; checking the graph convention]\label{ex:A1} \emph{Local model (a divisor calculation).} Let $R_0=\C^2/\pm1=\operatorname{Spec}\C[a,b,c]/(ac-b^2)$, with $a=u^2$, $b=uv$, $c=v^2$, and let $q_0:X_0\to R_0$ be the blow-up of the vertex. Then $X_0=\operatorname{Tot}\mathcal O_{\mathbb P^1}(-2)$, with exceptional curve $C$, $C^2=-2$. In the chart $v=ut$ the invariants are $y=u^2$ and $t$, and $q_0^*a=y$. In the chart $u=sv$ the invariants are $s$ and $z=v^2$, and $q_0^*a=s^2z$. Hence $\operatorname{div}(q_0^*a)=C+2F$, where $F$ is the strict transform of the ruling $\ell_0=\{a=b=0\}$. Since $2\ell_0=\operatorname{div}(a)$ is Cartier, the operational rational pullback is $q_0^![\ell_0]=F+\tfrac12C$. This is a divisor calculation, not a flat pullback. \tg{C} Locally both sides are rationally zero: $\ell_0=0$ in $\mathrm{CH}_1(R_0)_\Q$ because $2\ell_0$ is principal, and $F+\frac12C=0$ in $\mathrm{CH}_1(X_0)_\Q$. Topologically, $H^{BM}_2(R_0)=H_1(\mathbb{RP}^3;\Q)=0$, and $C=-2F$ in $H^{BM}_2(X_0)$. So the local identity is consistent but cannot detect a non-zero transferred class. Moreover, $\circ_{R_0}$ applies over the affine $R_0$ only to correspondences with proper support. The non-trivial test is the compact model. \tg{D} \emph{Compact model.} The same singularity sits in $\overline R=\mathbb P(1,1,2)$, resolved by the Hirzebruch surface $\overline X_0=\mathbb F_2\xrightarrow{q}\overline R$. The map $q$ contracts the $(-2)$-section $C$ and sends a fibre $f$ to a line $\ell$ through the vertex; here $\ell\cdot\ell=\frac12$ and $2\ell$ is Cartier. Work in the basis $(f,C)$, with $f^2=0$, $f\cdot C=1$, $C^2=-2$. The Chow side is the operational pullback $q^![\ell]=f+\tfrac12C$. The cohomology side is the class $\Theta$ with $\Theta\cdot f=\langle\ell,q_*f\rangle=\frac12$ and $\Theta\cdot C=\langle\ell,q_*C\rangle=0$, namely $q^*\ell=f+\tfrac12C$. The two sides agree, and $\Theta^2=\frac12=\ell^2$. The graph acts by $q_*f=\ell$ and $q_*C=0$, and $q_*q^*=1$. The composite $C_q={}^t\Gamma_q\circ_{\overline R}\Gamma_q$ acts by $f\mapsto f+\frac12C$, $C\mapsto0$ on $H_2$, and as the identity on $H_0$ and $H_4$. In the basis $(f,C)$, $(q^*q_*)|_{H^2}=\left(\begin{smallmatrix}1&0\\ \frac12&0\end{smallmatrix}\right)$. \tg{C} The projective bundle formula for $\mathbb F_2$ and for $\mathbb F_2\times\mathbb F_2$ gives a cellular Chow basis, so $\cl$ is an isomorphism on $\mathbb F_2\times\mathbb F_2$ \cite[\S19.1]{Fulton}, and a correspondence is determined by its action (K\"unneth and $\PD$). Hence \[ C_q=[\Delta_{\overline X_0}]+\tfrac12[C\times C]\quad\text{holds in }\mathrm{CH}_2(\overline X_0\times\overline X_0)_\Q , \] not only in cohomology. \tg{D + C} With $E_0=[C\times C]$, exact composition gives $E_0\circ E_0=(C\cdot C)E_0=-2E_0$, so $(\Delta+\lambda E_0)^{\circ2}=\Delta+(2\lambda-2\lambda^2)E_0$. At $\lambda=\frac12$ this is a Chow idempotent. The reduced fibre product $\overline X_0\times_{\overline R}\overline X_0=\Delta\cup(C\times C)$ has naive fundamental cycle $\Delta+E_0$, which acts by $C\mapsto-C$, satisfies $(\Delta+E_0)^{\circ2}=\Delta$, and is not idempotent. The coefficient $\frac12$ is the rational multiplicity that the bivariant composite supplies, and it is why rational coefficients are essential. \tg{C} \end{example} \begin{remark}[idempotence]\label{rem:chowidem} Proposition~\ref{prop:transfer}(iii) is an idempotence statement \emph{in cohomology}; Theorem~B uses nothing more. \tg{D} Chow-level idempotence $P_X\circ_{\overline X}P_X=P_X$ would need three further facts. \begin{enumerate} \item[(a)] Associativity of mixed compositions through $\overline X$ and $R$. The argument of \cite[Prop.~16.1.1(a)]{Fulton} uses only the formal properties of \cite[Prop.~8.1.1]{Fulton}, which $\cdot_R$ has. \tg{D, sketch only} \item[(b)] $\Gamma_q\circ_{\overline X}{}^t\Gamma_q=[\Delta_R]$, which does hold in Chow: the intersection is transverse, with $(q\times q)_*[\Delta_{\overline X}]=[\Delta_R]$ since $q$ has degree one. Moreover $[\Delta_R]$ is a unit for $\circ_R$, because $\ev_R^{-1}[\Delta_R]=\delta_*1$. \tg{D} \item[(c)] $p\circ_Rp=p$ in $\mathrm{CH}_d(R\times R)_\Q$. This holds for the group-algebra part, since $\Gamma_g\circ_R\Gamma_h=\Gamma_{gh}$. For $P_\star$ it holds if $P_\star=[\Delta_R]-\Pi_E$ with $\Pi_E$ $A$-equivariant and of Gram form on the cellular $E\times E$. \tg{D, conditional on the construction of $P_\star$} \end{enumerate} The example shows the phenomenon is consistent, since $C_q$ is a Chow idempotent. We do not claim, and do not need, that $P_X$ is a Chow motive. \end{remark} \begin{remark}[scope]\label{rem:scope} \begin{enumerate} \item[(1)] $R$ is a rational homology manifold, so on $R$ ordinary rational cohomology equals intersection cohomology. The obstruction exhibited by a cubic threefold with ordinary double points (which is not a rational homology manifold, and whose ordinary $H^3$ has a weight-two part) therefore does not arise here. \tg{D} \item[(2)] The proposition transfers cycles \emph{already given} on $R^n$, with their classes, to $\overline X^{\,n}$. That the specific closure $\overline W_u$ projects to the intended tensor, with its normalisation and independently of the closure, is proved in Section~\ref{ssec:localize} (Lemma~\ref{lem:closure}, Proposition~\ref{prop:cyclevalue}), not here. \tg{D} \item[(3)] No algebraic K\"unneth projector, Lefschetz-type conjecture or Chow--K\"unneth decomposition is used. \tg{D} \end{enumerate} \end{remark} \section{Proof of Theorem B}\label{sec:proofB} Let $\overline X_x=\overline X$ be the smooth projective variety of Proposition~\ref{prop:transfer} for the weighted clean model of Theorem~\ref{thm:G} (which exists for every $x\in S$ by Proposition~\ref{prop:H}). % [referee report 002, R2-03] \begin{proof}[Proof of Theorem B] (1) For an invariant $u\in(V_7^{\otimes n})^{\Gtwo}$ let $W_u$ be the cycle of Section~\ref{sec:A} on $\widetilde Y_x^{\,n}$ and $\overline W_u$ its closure in $R^n$. % [referee report 002, R2-02, R2-07: BCH1 Section 8(1) patch] By Theorem~\ref{thm:U}(b) and Remark~\ref{rem:realise}, $\varepsilon P_\star$ acts on $H^*(R)$ as a projector of rank $7$ with image $M_x$. By Lemma~\ref{lem:closure} and Proposition~\ref{prop:cyclevalue}, independently of the closure, \[ (\varepsilon P_\star)^{\otimes n}\,\clH(\overline W_u)=\frac{2^{1-n}}4\,\tau_u(-3n), \] with $\tau_u=\iota^{\otimes n}\varphi_n^{-1}E(s_u)_x(1_{\rm Yun})$ the transported tensor on Yun's tensor unit (Proposition~\ref{prop:square}); in particular $(\varepsilon P_\star)^{\otimes 3}\,\clH(\overline W_t)=\frac1{16}\,\tau_t(-9)\neq0$ for the cubic, and $\frac14\tau_t(-9)$ for $4\overline W_t$. % [referee report 003, R3-02; audits YAUD2, YAUD2R, YAUD2R2] Let $p=\varepsilon P_\star$ and consider the cycle \[ \mathcal Z_u:=(P^{(n)}_X)_*\bigl((q^n)^![\overline W_u]\bigr)\in\mathrm{CH}_*(\overline X^{\,n})_\Q . \] By Proposition~\ref{prop:transfer}(iv), \[ \clH(\mathcal Z_u)=(q^n)^*p_*^{\otimes n}\clH(\overline W_u)=\frac{2^{1-n}}4(q^n)^*\tau_u(-3n), \] and $(q^n)^*$ is injective. So $4\cdot2^{n-1}\mathcal Z_u$ (equivalently, $2^{n-1}$ times the transported cycle built from $4\overline W_u$) is an algebraic cycle on $\overline X^{\,n}$ whose class is the transported tensor $(q^n)^*\tau_u(-3n)$; for the cubic the multiplier is $16$. % [referee report 003, R3-02; audits YAUD2, YAUD2R, YAUD2R2] The same projector is used in every factor; only the one-factor model is needed, and rational smoothness of $C_n$ supplies the top-component generators for every $n$. (2) The pairing $b$ and the cubic $t$ are algebraic Hodge classes by (1), so $\MT(N_x)\subseteq\Aut(b,t)=\Gtwo$. At a Hodge-generic $x$ the connected monodromy of the Betti local system $N_S$ is contained in $\MT(N_x)$ and is $\Gtwo$ by Lemma~\ref{lem:dense} (a statement for the local system $N_S$ of Lemma~\ref{lem:NS}, not for stalks), so $\MT(N_x)=\Gtwo$, % [referee report 004, R4-03; audits hodge-YAUD7, hodge-YAUD9] and the Hodge classes in $T^{a,b}(N_x)$ are exactly the $\Gtwo$-invariants. Duals are handled by the pairing and coevaluation, which are the $n=2$ invariants; symmetric and exterior summands by rational permutation idempotents; the twists are those forced by weight. All are algebraic by (1). \end{proof} \begin{remark}\label{rem:special} At special $x$ the Mumford--Tate group may be a proper subgroup of $\Gtwo$ with more invariant tensors; such points form a countable set. The construction gives algebraicity of the transported $\Gtwo$-invariants there, but not of the additional invariants. We do not know whether such points occur at interior parameters. \end{remark} \begin{remark}\label{rem:DR} Yun's local system is the same as that of Dettweiler--Reiter \cite[Thm.~6.1.5 and the paragraph after Thm.~6.2.3]{YunCDM}. The corresponding statement for their family therefore reduces to the algebraicity of the Hodge class in $\Hom(M_x,M^{\mathrm{DR}}_x)$ identifying the two motives; this class is motivated if and only if the determinant class of the Dettweiler--Reiter motive is, and we do not address it. [referee report 002, R2-09] \end{remark} \begin{remark} The method uses rigidity of the Hecke eigensheaf in an essential way: the odd sheaf is a representation of the finite group $\tA$ on the open point of a zero-dimensional moduli stack, defined without reference to the local system, and the Hecke operators produce $M_x$ from it. Yun's other rigid data (types $A_1$, $D_{2n}$, $E_7$, $E_8$) have the same shape; for $A_1$ the construction reproduces the known algebraicity of the Hodge tensors of $\operatorname{Sym}^2$ of a weight-one $\Q(i)$-Hodge structure of abelian type. For $D_4$, $E_7$ and $E_8$ the analogue of Theorem~\ref{thm:U} needs a meridian-by-meridian analysis, since not every nontrivial subgroup of $\tA$ contains the central element. \end{remark} \section*{Computational certificates} The finite computations used above (the enumeration of bundle types in Lemma~\ref{lem:types}; the exact stabiliser dimensions, stabiliser tori and cocharacters behind Table~\ref{tab:weights}; the stabiliser data of Table~\ref{tab:K}) were done in exact rational arithmetic in Sage. The orbit counts over finite fields at $q=5,7$ were used only as a cross-check of Proposition~\ref{prop:orbits}, whose proof does not depend on them. The scripts, a generator for every input, a runner \texttt{run\_all.sh} and a reference run with input, output and log hashes are deposited as the certificate bundle \texttt{notes/yun-g2-algebraic-review/certificate/} of the repository \texttt{benzsevern/hodge-fleet} (commit \texttt{3f7a05ed}; tarball SHA-256 \texttt{f598645060d69b34\-03a0ccee7561f410\-dc5e3123323214176\-aac083ab8e76614}); the reference run used SageMath~10.7. That tarball predates the characteristic-zero certificate for Proposition~\ref{prop:orbits}: the script \texttt{matsuki\_c0.sage}, its logs and its exact representatives are in \texttt{certificate/matsuki/} (added at commit \texttt{e2f38848}). It is not called by \texttt{run\_all.sh} and is run separately, from that directory, as \texttt{CT=G2 EPS=-1,-1 sage matsuki\_c0.sage} (and with \texttt{CT=A1 EPS=-1}, \texttt{CT=A2 EPS=1,-1} for the controls); a pinned rerun with checksums of every script, log and representative file is recorded in the bundle. That rerun (\texttt{certificate/matsuki/RERUN-2026-10-19.md}) was made from a clean checkout of commit \texttt{e2f38848b3f5d5d49cd20ffabf97661f75632b18} with SageMath~10.7 and records the exact commands, the environment and the SHA-256 hash of every input and output; the script \texttt{matsuki\_c0\_canonical.sage} selects points by an exact canonical key, and two consecutive runs of it give byte-identical logs and representatives (\texttt{certificate/matsuki/canon/SHA256SUMS}). These are runs made by the author's computing setup, deposited for independent re-execution; the outside reviews of this paper inspected them but did not rerun them. Likewise, where the text cites a source by theorem or section number, the number was read at source by the author's programme; the outside audits named in the review record checked some of these citations, not all, and an audit's positive verdict is not an independent check of every supporting proof it relies on. % [referee report 006, R6-03, R6-04] The corrected census of Lemma~\ref{lem:types} (seven bundle types; v0.7 listed six) and the check of Hypothesis~\ref{hyp:G} at the added type $C_6$ are in \texttt{certificate/census-repair/}: \texttt{census.py} enumerates the double cosets with both parahoric Weyl groups the stabilisers of $-\rho^\vee/2$, and \texttt{check.py} recomputes, from fresh code using only \eqref{eq:interval} and the formulas of Section~\ref{sec:census}, the evaluation image, the stabiliser dimensions, the $T^1$- and stabiliser weights and a contracting cocharacter at $C_6$, with the $B_6$ row of Table~\ref{tab:weights} as control; their logs and SHA-256 sums are deposited with them. The omission was found by a blind reconstruction of the paper from its statements alone, made by a model family different from the author's, and confirmed by the exact enumeration. % [blind recheck 2026-10-19, r1364-r1365] The complete evidence for the finite computations of Section~\ref{sec:census} is therefore the union of the bundle \texttt{certificate/} described above (whose \texttt{run\_all.sh} and \texttt{reference-logs/sweepQ.log} cover the six types of the earlier census and are kept unchanged as the record of the original computation, including its omission) and \texttt{certificate/census-repair/} (\texttt{census.py}, \texttt{check.py}, their logs \texttt{yun61.log}, \texttt{yun61b.log} and \texttt{SHA256SUMS}). The census and the $C_6$ weights were also recomputed by the outside referee of round seven with a separate implementation that does not run the author's code (\texttt{review/checks/round7-independent-census.py}); the other rows of Table~\ref{tab:weights}, the non-$T$-fixed stabiliser computations and the stabilisers of Table~\ref{tab:K} were then recomputed by a separate implementation written from Section~\ref{sec:census} alone, without access to the earlier scripts or logs until the comparison (\texttt{certificate/blind-weights/}, with a README and \texttt{SHA256SUMS}): it reproduces all ten stabilisers of Table~\ref{tab:K} and every earlier row, and supplies the three $c=6$ rows absent from \texttt{sweepQ.log}. Its first, deliberately over-inclusive comparison (\texttt{yun-cmp2.log}) ends with five provisional mismatches: three are those new rows, and two are $T$-fixed rows over $\star$ where several flags share a $T^1$-multiset at the given $\lambda$; the bipartite matching that follows (\texttt{yun-match.log}), restricted to sign-valid flags, pairs the $37$ earlier $T$-fixed points bijectively with its own, so there is no numerical disagreement on the matched entries. The three new rows were also recomputed independently by the outside referee of round eight (report 008, R8-03). % [referee report 008, R8-02, R8-03] It was made by the author's computing setup, so it is a second implementation, not an outside one; the Matsuki certificate for Proposition~\ref{prop:orbits} has not been re-executed. % [referee report 007, R7-05, R7-06; blind re-implementation 2026-10-20] % Acknowledgements: to be written by the author. \begin{thebibliography}{99} \bibitem{AHR} J. Alper, J. Hall, D. Rydh, \emph{A Luna \'etale slice theorem for algebraic stacks}, Ann. of Math. 191 (2020), 675--738; arXiv:1504.06467. \bibitem{Andre96} Y. Andr\'e, \emph{Pour une th\'eorie inconditionnelle des motifs}, Publ. Math. IH\'ES 83 (1996), 5--49. \bibitem{DR10} M. Dettweiler, S. Reiter, \emph{Rigid local systems and motives of type $G_2$}, Compos. Math. 146 (2010), 929--963. \bibitem{EG} D. Edidin, W. Graham, \emph{Equivariant intersection theory}, Invent. Math. 131 (1998), 595--634; arXiv:alg-geom/9609018. \bibitem{EHKV} D. Edidin, B. Hassett, A. Kresch, A. Vistoli, \emph{Brauer groups and quotient stacks}, Amer. J. Math. 123 (2001), 761--777. \bibitem{Faergeman} J. F{\ae}rgeman, \emph{Motivic realization of rigid $G$-local systems on curves and tamely ramified geometric Langlands}, arXiv:2405.18268v2 (2024). \bibitem{HNY} J. Heinloth, B.C. Ng\^o, Z. Yun, \emph{Kloosterman sheaves for reductive groups}, Ann. of Math. 177 (2013), 241--310. \bibitem{Sugiura59} M. Sugiura, \emph{Conjugate classes of Cartan subalgebras in real semisimple Lie algebras}, J. Math. Soc. Japan 11 (1959), 374--434. \bibitem{Sugiura71} M. Sugiura, \emph{Correction to my paper: Conjugate classes of Cartan subalgebras in real semisimple Lie algebras}, J. Math. Soc. Japan 23 (1971), 379--383. \bibitem{Matsuki79} T. Matsuki, \emph{The orbits of affine symmetric spaces under the action of minimal parabolic subgroups}, J. Math. Soc. Japan 31 (1979), 331--357. \bibitem{RS90} R. W. Richardson, T. A. Springer, \emph{The Bruhat order on symmetric varieties}, Geom. Dedicata 35 (1990), 389--436. \bibitem{Fulton} W. Fulton, \emph{Intersection theory}, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, 1998. \bibitem{BredonSheaf} G. E. Bredon, \emph{Sheaf theory}, 2nd ed., Grad. Texts in Math. 170, Springer, 1997. \bibitem{BredonCTG} G. E. Bredon, \emph{Introduction to compact transformation groups}, Academic Press, 1972. \bibitem{GIT} D. Mumford, J. Fogarty, F. Kirwan, \emph{Geometric invariant theory}, 3rd ed., Springer, 1994. \bibitem{BM97} E. Bierstone, P. D. Milman, \emph{Functoriality in resolution of singularities}, Publ. RIMS 44 (2008), 609--639. \bibitem{BB73} A. Bia\l{}ynicki-Birula, \emph{Some theorems on actions of algebraic groups}, Ann. of Math. 98 (1973), 480--497. \bibitem{BLR} S. Bosch, W. L\"utkebohmert, M. Raynaud, \emph{N\'eron models}, Ergeb. Math. Grenzgeb. 21, Springer, 1990. \bibitem{EGA} A. Grothendieck, J. Dieudonn\'e, \emph{\'El\'ements de g\'eom\'etrie alg\'ebrique IV, deuxi\`eme partie}, Publ. Math. IH\'ES 24 (1965). \bibitem{AOV} D. Abramovich, M. Olsson, A. Vistoli, \emph{Tame stacks in positive characteristic}, Ann. Inst. Fourier 58 (2008), 1057--1091. \bibitem{Schapira} P. Schapira, \emph{Sheaves on manifolds}, lecture notes, \url{https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/SHV.pdf}, Thm.~4.5.3 and (4.5.4). % [referee report 004, R4-02; audit hodge-YAUD8] \bibitem{SGA4} M. Artin, A. Grothendieck, J.-L. Verdier, \emph{Th\'eorie des topos et cohomologie \'etale des sch\'emas} (SGA~4), Tome~3, Lecture Notes in Math.~305, Springer, 1973: Exp.~XVI (M.~Artin, Th\'eor\`eme de comparaison) and Exp.~XVII (P.~Deligne). \bibitem{SGA4half} P. Deligne, \emph{Cohomologie \'etale} (SGA~4$\tfrac12$), Lecture Notes in Math.~569, Springer, 1977; chapter ``Cohomologie \'etale~: les points de d\'epart'' (Arcata, r\'edig\'e par J.-F.~Boutot), Th.~4.6.3 and Th.~5.3.6. \bibitem{MilneLEC} J. S. Milne, \emph{Lectures on \'etale cohomology}, v2.21 (2013), \url{https://www.jmilne.org/math/CourseNotes/LEC.pdf}, \S23. \bibitem{Stacks} The Stacks Project Authors, \emph{The Stacks Project}, \url{https://stacks.math.columbia.edu} (Tags 02H6, 01XO, 0D3B, 0D30). \bibitem{BBD} A. Beilinson, J. Bernstein, P. Deligne, \emph{Faisceaux pervers}, Ast\'erisque 100 (1982). \bibitem{MV} I. Mirkovi\'c, K. Vilonen, \emph{Geometric Langlands duality and representations of algebraic groups over commutative rings}, Ann. of Math. 166 (2007), 95--143; numbering as in arXiv:math/0401222v5. \bibitem{QR} M. H. Quek, D. Rydh, \emph{Weighted blow-ups}, preprint (2022). Draft of 29 March 2022 (https://people.kth.se/~dary/weighted-blowups20220329.pdf), cited at source: \S1.2 (stack-theoretic Proj); Prop.~1.6.1(iii) (relative coarse space of the stack-theoretic Proj is the usual Proj); Prop.~1.6.3 (Veronese subalgebra generated in degree one); Setup~1.7.2 (a power of a line bundle descends to the relative coarse space); Def.~3.2.1 (weighted blow-up); Cor.~3.2.14(iii) (for $f$ flat, $\iota$ is an isomorphism: flat base change, stated for schemes); Prop.~5.2.2 (Proj presentation for an $H_1$-regular sequence); Cor.~5.2.5--5.2.6 (charts for blow-ups along regular weighted embeddings, the latter with a $D(A)$-action); Def.~5.2.7 (regular weighted embedding, smooth-locally $(x_1,d_1)+\dots+(x_m,d_m)$); Def.~5.3.1 (regular weighted centre); Cor.~5.3.2 (the deformation to the weighted normal cone and the stack-theoretic blow-up of a regular centre are regular). \bibitem{Vistoli-Compositio} A. Vistoli, \emph{Alexander duality in intersection theory}, Compos. Math. 70 (1989), 199--225. \bibitem{Vistoli-Invent} A. Vistoli, \emph{Intersection theory on algebraic stacks and on their moduli spaces}, Invent. Math. 97 (1989), 613--670. \bibitem{Yun} Z. Yun, \emph{Motives with exceptional Galois groups and the inverse Galois problem}, Invent. Math. 196 (2014), 267--337; arXiv:1112.2434. Numbered statements are cited in the numbering of arXiv:1112.2434v1. \bibitem{YunCDM} Z. Yun, \emph{Rigidity in automorphic representations and local systems}, in: Current Developments in Mathematics 2013, Int. Press, Somerville, MA, 2014. % [referee report 005, R5-01; audit hodge-YAUD10] -- locators verified at arXiv source (ref-LIB9); printed theorem ordinals not checked against the journal \bibitem{LO08} Y. Laszlo, M. Olsson, \emph{The six operations for sheaves on Artin stacks I: Finite coefficients}, Publ. Math. IH\'ES 107 (2008), 109--168, arXiv:math/0512097; \emph{II: Adic coefficients}, ibid., 169--210, arXiv:math/0603680. Cited as [I, \S5.1] (smooth base change), [I, \S5.2, Prop.~f!=f*propre] ($Rf_!\cong Rf_*$ for proper representable $f$), [I, \S5.5] (base change for $Rf_!$) and [II, \S9, Thm.~basechangethm] (its adic form), by arXiv section and label. \bibitem{Gaitsgory01} D. Gaitsgory, \emph{Construction of central elements in the affine Hecke algebra via nearby cycles}, Invent. Math. 144 (2001), 253--280, arXiv:math/9912074; \S1.1 (convolution as twisted product), \S\S2.1.2--2.1.3 (spreading of $P_{G(\cO)}(\Gr)$ over the curve, perverse normalisation $\Q_\ell[1]$), \S4.1 (finite-level spread), Appendix A.3--A.4 (nice actions, finite-level equivariant categories), as cited in \cite[\S3.2.3]{Yun}; section numbers from the arXiv source. \end{thebibliography} \end{document}