Ordered elliptic product stability and actual diagonal symmetries

Categorical Hilbert basis calculation. For a noetherian C-linear abelian category A, define graded $S=Sym(V)$ objects by components $M_n$ in A with associative S action and finite-type objects as quotients of finite sums $S\otimes A_i$ shifted in degree. Every subobject of S tensor A is finite type. Here is the actual induction: write $S=B[x]$, filter a submodule by x-degree, and project its ith associated piece to B tensor A to obtain leading-coefficient modules $L_i$. Multiplication by x gives $L_i$ subset $L_{i+1}$. By induction in dim V, B tensor A is noetherian, so this chain stabilizes at $i=N$. Each $L_i$ for $i\le N$ is finitely generated over B, in degrees bounded by M. The multiplication $S_1\otimes M_n\to M_{n+1}$ is onto for $n\ge N+M$: on x-degree $i\ge N$ use $L_i=L_{i+1}$; for $i<N$ use generation of $L_{i+1}$ by $B_1$. Lifting through the finite x-degree filtration proves surjectivity. Thus the submodule is generated in finitely many degrees. For an increasing chain $M_i$ in S tensor A, take the degreewise union; it exists because each finite-degree component is noetherian. The union is finite type by the previous paragraph, so its finitely many generating components already lie in one $M_i$. The chain stabilizes. Every finite-type object is a quotient of a finite-type free object, hence noetherian; kernels and images are finite type by the submodule result. This proves an abelian noetherian category of finite-type graded A-valued S modules. No projective generator of A is required.

The projective moving windows. On $P^r$ the diagonal is the zero locus of the universal tautological map $O(-1)\to Q$; it is a regular rank-r zero locus. Its Koszul complex resolves $O_{Delta}$. Taking its exterior powers gives the explicit diagonal resolution whose terms are O(−i) external-tensor $Omeg{a}^i(i)$. Cech cohomology of O(m) (group monomials by their negative exponents) gives $H^{>0}(O(m))=0$ for $m\ge 0$ and all cohomology zero for $-r\le m<0$. Tensoring/pushing the diagonal complex therefore gives the relative decomposition into pDbCoh(X)(−r),...,pDbCoh(X), including its projection triangles. For each integer n glue the base t-structures using those finite projection triangles. Its window is exactly E in D_n^[a,b] iff Rp_*E(n),...,Rp_*E(n+r) in D_X^[a,b]. Indeed apply Rp_(−(j)) to a projection triangle: earlier summands contribute only H0(O(j−i)) tensor the corresponding base object. Induction through the r+1 triangles gives both orthogonality and truncations. Boundedness is finite gluing of bounded t-structures. The required resolution has variable multiplicities at EVERY term, including the first: 0->V0 tensor O(m)->V1 tensor O(m+1)->...->Vr tensor O(m+r)->O(n+r)->0 for $n\ge m$. Put $k=n-m+r$. The dual form of the diagonal Koszul resolution above, applied to O(k), gives these terms, with $dim(V_j)=binom(k,j)binom(k-j-1,r-j)$. To compute them, resolve each exterior power of the tautological quotient by its exterior Euler sequence and group the resulting homogeneous monomials by degree; the Koszul complex of the coordinate variables is exact in every positive total degree. This also shows that no positive cohomology remains in those coefficient complexes for $k\ge r$, so the finite diagonal convolution is the displayed exact vector-bundle resolution, rather than a complex with omitted higher terms. Here binom(a,b)=0 for $0\le a<b$ and binom(a,0)=1. In particular $k=r$ gives $V_r=C$ and $V_j=0$ for $j<r$, so for $n=m$ the augmentation is the identity. Every O(l) with $l\ge m$ has such a resolution of length at most r by O(m),...,O(m+r): for $m\le l\le m+r$ use the identity, and otherwise use $k=l-m$ above. Thus applying Rp_(E tensor −) shows $D_m^{\le 0}$ subset $D_n^{\le 0}$ and $D_m^{\ge 0}$ subset $D_n^{\ge -r}$. Orthogonality of t-structures gives the other two inclusions. It implies D_n^>=0 subset D_m^>=0 subset D_n^>=−r, D_m^<=0 subset D_n^<=0 subset D_m^<=r. Thus every fixed E has one finite uniform range of moving-window cohomology.

Separate original cohomology from the next truncation of its top piece. Write $\mathrm{\Phi }F=(M_0,\dots ,M_r)$ for a window-heart object, with $M_j=R{p}_{\ast }F(j)$, and let $V=H^0({\mathbb{P}}^r,\mathcal{O}(1))$, $S=\mathrm{Sym}V$. The inverse window functor is the diagonal convolution $[M_0⊠{\mathrm{\Omega }}^r\to M_1⊠{\mathrm{\Omega }}^{r-1}(-1)\to \cdots \to M_r⊠\mathcal{O}(-r)]$, placed in degrees $[-r,0]$. Its differential squares to zero because the coordinate actions commute and the exterior maps anticommute. The diagonal resolution in step 2.1 identifies both composites with the identity, by tensoring it with the object and applying the finite projection triangles. Define $T_i(M)=\mathrm{\Phi }{H}_0^i(F(1))$. The coordinate Koszul sequence gives $T_0(M)=(M_1,\dots ,M_r,\mathrm{coker}({\bigwedge }^{2}V\otimes M_{r-1}\to V\otimes M_r))$, while for $i<0$, $T_i(M)$ has only its last component nonzero, equal to $H^i(K(M))$. Here $K(M)=[{\bigwedge }^{r+1}V\otimes M_0\to {\bigwedge }^{r}V\otimes M_1\to \cdots \to V\otimes M_r]$ lies in degrees $[-r,0]$. Choosing a volume element identifies its first coefficient with $M_0$; keeping the determinant factor makes the formula canonical.

For a fixed $F\in D_0^{\le 0}$ put $M=\mathrm{\Phi }{H}_0^0(F)$. Tensoring by $\mathcal{O}(k)$ preserves $D_0^{\le 0}$ for $k\ge 0$ by step 2.1, so the triangle of $F$ shows $H_0^0(F(k))=H_0^0(H_0^0(F)(k))$. Applying the length-$r$ line-bundle resolution of step 2.1 with last term $\mathcal{O}(n+r+j)$ shows that the $j$th component of $\mathrm{\Phi }{H}_0^0(F(n+r))$ is $N_{n+j}$, where $N=\mathrm{coker}(S(-1)\otimes {\bigwedge }^{2}V\otimes M_{r-1}\to S\otimes M_r)$. Indeed its last coefficient is $S^{n+j}V$, and its penultimate coefficient is $\ker (S^{n+j}V\otimes V\to S^{n+j+1}V)$, the image of $S^{n+j-1}V\otimes {\bigwedge }^{2}V$ by the coordinate Koszul identity. All earlier terms have negative convolution degree and cannot change this cokernel. This computes the moving top truncation, not the negative cohomology of the original $F$.

For any finite-type graded $S$-object $Q$ define ${\tilde{K}}_n(Q)=[{\bigwedge }^{r+1}V\otimes Q_n\to {\bigwedge }^{r}V\otimes Q_{n+1}\to \cdots \to V\otimes Q_{n+r}\to Q_{n+r+1}]$ in degrees $[0,r+1]$. This complex is exact for all sufficiently large $n$. For a finite sum of free objects $S(-d)\otimes A$ this follows degree by degree from the polynomial Koszul complex: multiplication by the coordinate variables has the usual contraction away from its single degree-zero residual, and tensoring finite-dimensional vector spaces with $A$ is exact. In general choose a finite free-object surjection $P\to Q$. Its kernel $Q^{\mathrm{\prime }}$ is finite type by step 1.1. Descending induction on the cohomological degree $j$, starting above $r+1$, and the exact sequence $H^j({\tilde{K}}_n(P))\to H^j({\tilde{K}}_n(Q))\to H^{j+1}({\tilde{K}}_n(Q^{\mathrm{\prime }}))$ prove eventual vanishing in each degree. This induction is uniform as a statement about all finite-type objects at each degree; it does not assume a finite projective resolution or projectives in the heart.

Apply step 5.1 to $N$ of step 4.1. Its exact augmented complex implies $H^{<0}(K((N_n,\dots ,N_{n+r})))=0$. By step 3.1 this says precisely that $H_0^0(F(k))(1)$ belongs to the zero-window heart for $k\gg 0$. Equivalently $H_k^0(F)$ belongs also to the $(k+1)$st window heart. The natural map $H_k^0(F)\to H_{k+1}^0(F)$ is its $(k+1)$st nonnegative truncation: $D_k^{\le -1}\subseteq D_{k+1}^{\le -1}$ makes the negative part of $F$ disappear under that truncation. Hence this map is an isomorphism eventually. There is no conclusion that $H_k^{<0}(F)$ vanishes. For example $F={\mathcal{O}}_{{\mathbb{P}}^2}(-3)[2]$ has $H_k^0(F)=0$ eventually but retains $H_k^{-2}(F)=F[-2]$.

A general fixed object has one finite uniform window-cohomology range $[a,b]$ by step 2.1. Shift to reduce stabilization of its top cohomology to step 6.1. Once the top-cohomology transition is an isomorphism, the compatible truncation triangles identify all its lower truncations; applying the same argument to that fixed lower object shortens the range by one. Finite induction stabilizes every cohomology object and every truncation triangle. Therefore $D_C^{\le 0}={\bigcup }_k{D}_k^{\le 0}$ and $D_C^{\ge 0}={\bigcap }_k{D}_k^{\ge 0}$ form a bounded t-structure: orthogonality follows from the inclusions, and a stabilized truncation triangle has its two terms in the indicated union and intersection. Its heart is exactly the large-twist heart used below.

Noetherianity and restriction to opens. For E in the resulting projective heart, $M(E{)}_n=H{0}_{A}R{p}_{\ast }E(n),n>0$, is finite type. The Euler sequence and its exterior-power successors show that $Sy{m}^{1}V\otimes M(E{)}_n\to M(E{)}_{n+1}$ is onto for n large: the successive obstruction groups reduce to Rp_E(n−r−1), which is already in A. Exact sequences in the heart give exact sequences of these modules in all sufficiently high degrees. Thus the tail functor into the Serre quotient of finite-type graded modules by finite-degree modules is exact. It is faithful: if every large Rp_E(n) vanishes, the standard coherent cohomology sheaves of E vanish by ordinary large-twist Serre vanishing/evaluation (which itself follows from the same polynomial-module presentation). A chain of subobjects therefore stabilizes because its graded-tail image is in a noetherian category. Hence the projective heart is noetherian. Tensoring by O(1) is t-exact by construction. On the complement of a divisor $f=0$, morphisms are the colimit of Hom(E,F(md)) under multiplication by f; clear denominators in finite coherent Cech representatives to prove this formula. The subcategory supported on $f=0$ is stable under truncation: a sufficiently large f-power kills the cohomology and then the bounded object, and functorial truncation preserves that zero scalar map, hence its support. The t-structure descends to the Verdier quotient on the open complement. Its heart is the corresponding abelian Serre quotient and is noetherian. Finite affine covers glue these local t-structures: the truncation triangles agree on intersections by their orthogonality/universal property, and the resulting negative Ext vanishings make the Cech object/morphism descent unique. This proves actual locality, not just compatibility of a membership formula.

Restricting to a smooth closed base: the essential-image issue. Embed the smooth curve C into $P^r$ and work locally on the ambient base where it is a succession of smooth principal divisors. One cannot just assume ambient truncations are pushforwards. The needed divisor descent is as follows. If f acts by zero on a bounded coherent object F on a smooth affine ambient base, then F is a pushforward from $f=0$. The Koszul triangle makes F a retract of i_LiF. Represent all finitely many objects and roofs on a sufficiently high f-thickening. This assertion is obtained explicitly for a roof $P\to F$: choose a bounded torsion-free resolution, let K be its acyclic kernel, and quotient P by fK; f² kills the resulting representative and its map still is a quasi-isomorphism. Iterate this finite construction for the finitely many roofs of the retract. Smoothness of the affine zero locus provides a retraction of that finite thickening, by lifting its etale coordinates through the nilpotent ideal. Push the retract diagram along that finite retraction. Its two compositions remain the split identities, giving $F=i_{\ast }F0$ for an actual bounded coherent F0 on $f=0$. If $Ho{m}^j(i_{\ast }F,i_{\ast }G)=0$ for $j<0$, the Koszul triangle $F[1]\to Li\ast i_{\ast }F\to F$ implies by induction in j that $Ho{m}^j(F,G)=0$ for $j<0$ and Hom(F,G)=Hom(i_F,i_G). Thus the descended truncation morphism is unique. For an ambient truncation of i_E, scalar f still acts by zero by functoriality, so the preceding paragraph descends the object and its map. Their cones shorten its bounded cohomological range. Induction produces both truncations on the closed base. Negative Ext vanishings glue these local constructions. The heart embeds exactly and conservatively into the ambient heart under i_, so is noetherian. Derived pullback is right t-exact by adjunction, and its amplitude along a principal divisor is [−1,0], by its two-term Koszul resolution.

Independence of ample line and base-line locality. For two projective embeddings use their diagonal embedding into $P^r$ times $P^s$. Iterating the moving-window construction in either order gives the same t-structure: if a fixed object is in the iterated heart, choose n0 so the pushforward in the second factor lies in the first-factor heart; choose finitely many first-factor bounds for n0,...,n0+s; the second-factor Koszul sequence then propagates them to all $n\ge n0$. Hence there is ONE bound M so every double twist (m,n), $m,n\ge M$, has base pushforward in A. The resulting symmetric characterization proves order-independence of the CONSTANT (un-tilted) constructions. A graph/finite projection and the same local trivialization identify its restriction with either chosen embedding, proving independence of the ample line. Replacing a very ample power by its ample root uses the finitely many residues of powers and local triviality of the corresponding line twists. In particular tensoring by ANY line pulled back from C is t-exact: on a trivializing open cover it is identity, and locality detects the heart. This proves the precise Pic0/naturality input needed later. It does NOT prove that the Liu TILTED hearts commute in two factor orders; constant-heart order independence and tilted-heart order independence are different statements.

Relative torsion and flat-family tests. A maximal subobject supported on a closed divisor exists by noetherianity, since sums of such subobjects are again supported there. The torsion-free quotient has injective multiplication by its local parameter, so its derived restriction is in A by the two-term Koszul triangle. On a curve, removing the maximal relative torsion therefore gives an actual t-flat family. The open-heart/Nakayama test is also concrete. A noetherian object M on a local base satisfying $M\mathrm{/}tM=0$ has surjective t; a surjective endomorphism of a noetherian object is injective, since the increasing kernels $ker(t^n)$ stabilize and a lift under $t^n$ kills ker(t). Thus t is invertible, and derived Nakayama on its finite coherent representative makes it vanish near the zero fiber. Apply this first to the highest nonzero t-cohomology object of a bounded family and then to the lowest. It proves that membership of a derived fiber in a specified constant heart persists on a neighborhood; equivalently a family whose every fiber is in the original heart lies in the constant heart. The same argument applies to rational rotations of the base stability heart, whenever that rotated heart is noetherian.

Countable finite-type perfect-object and morphism charts in this exact X. Use the ACTUAL installed supplier lem-gts-morimura-ample-power-perfect-generator, whose complete current Statement and proof were read. For a chosen projective embedding $i:X\to P^r$ with $M=i\ast O(1)$, it proves that $G={⨁}_{j=0}^r{M}^{-j}$ compactly generates $D_{qc}(X)$, and every perfect object is a finite cone/shift/retract of G. Its proof uses the coordinate Koszul complex, localization on standard affines and an explicit semifree dg-module construction; no smoothness is required for that supplier. It applies in particular to the present smooth projective X. We do NOT use the generally invalid shortcut that every F is a direct summand of Lii_F for an arbitrary complete-intersection embedding: the derived self-intersection action can obstruct that splitting. The actual installed ample-generator theorem supplies exactly the needed finite generator without that claim. Take an algebraic Cech dg endomorphism model for G. Its cohomology $A=Ext\ast (G,G)$ is finite-dimensional and bounded. Choose vector-space complements to cycles/boundaries and transfer to its minimal A-infinity model. A fixed finite upper triangular twisted complex of shifts of G is specified by finitely many matrix coefficients delta, with the finite polynomial Maurer–Cartan equations: every word of delta length at least the matrix size vanishes. Its closed degree-zero endomorphism p and a homotopy h satisfying $p^2-p=dh$ are also finite coefficient data and polynomial equations (the twisted products use only finitely many delta insertions). The homotopy image of p exists algebraically: form the telescope Cone(1−shift_p on the algebraic direct sum of copies), whose maps to/from the original complex split p in the derived module category. It is a retract of a finite twisted complex, hence perfect. This construction is valid over the parameter coordinate ring as well; it is a derived retract, not a pointwise assertion of cohomology dimensions. The A-infinity realization in the Cech enhancement carries this universal perfect module to an actual perfect complex on X times that finite-type parameter scheme. Every perfect F occurs because G split-generates. Matrix sizes and shifts are countable choices, so there are COUNTABLY many finite-type algebraic charts covering all perfect objects. A minimal-model structure constant can be an arbitrary complex number; it is still a legitimate coefficient of these finite polynomial equations, and there is no convergence or formal power series in a parameter involved. For a family F on one chart and the fixed family E over C, the relative Hom complex Rp_*RHom(F,E) is perfect locally on the parameter base. One can verify this directly on each smooth affine stratum: resolve both perfect complexes by bounded vector bundles, apply the actual proper-flat finite cohomology complex to each vector-bundle term, and use the finite total complex. Terms are flat over that base; the same Cech construction and finite free syzygy replacement make the comparison compatible with its morphisms and coefficient change. A closed degree-zero morphism is then parametrized by a finite free degree-zero coordinate vector with its linear closedness equations. Its cone is a perfect family. Stratify reduced parameter schemes by smooth locally closed strata (successively remove singular loci in characteristic zero). Refining these countable charts gives countably many SMOOTH finite-type charts of perfect pairs $F\to E_s$ with their cone families. Numerical classes are constant on each connected stratum: all Euler pairings are alternating ranks of those relative finite free complexes. This supplies the precise finite-type parameterization needed below. It imports no moduli space of stable objects, boundedness of semistable objects, or openness of stability. The actual published geometric interfaces used are thm-chevalley-constructible-image-varieties, thm-generic-smoothness-characteristic-zero, and lem-proper-flat-cohomology-perfect-complex (applied termwise to base-flat vector bundles). Their exact Statements were read; no whole proof audit is claimed for those geometric suppliers. The complete actual ample-generator proof was read and its exact finite-cone/retract assertion is the supplier used in this section.

Turn a destabilizing point into an OPEN family-heart chart. First suppose $c>0$. A nonzero relative C-torsion subobject would have zero leading charge and weak phase1, strictly larger than the leading phase $\varphi =\arg (a+ic)\mathrm{/}\pi$. Hence E has no such subobject. The AP divisor/Koszul calculation therefore makes every $E_s$ belong to A, and its numerical charge is a+i c, independent of s. At an unstable closed fiber choose its first ordinary HN factor $F\to E_s$ of phase $\theta >\varphi$. Choose a rational-phase cut alpha with $\varphi <\alpha <\theta$, meaning its rotation matrix can be taken rational after a positive scaling. Let $B_{\alpha }=P(\alpha ,\alpha +1]$; it is a noetherian rational rotation of the base heart. Then F belongs to BOTH A and $B_{\alpha }$, and its cone belongs to A. The chart in section1 containing this morphism has these three properties at its selected point. The AP OPEN-HEART theorem puts the FAMILY F in $A_T$ and $(B_{\alpha }{)}_T$, and its FAMILY cone in $A_T$, on an open neighborhood in the smooth chart T. This is exactly what is used: we do NOT claim that all closed fibers on that neighborhood are in these hearts or semistable. Let $V_T$ be the union of open neighborhoods where those family properties hold. It is an open finite-type subset of T. On it the family map is an actual monomorphism in $A_T$, with F in the intersection $T_{\alpha }=A_T\cap (B_{\alpha }{)}_T$. The source E pulled back from C is in $A_T$: its fibers are in A, and the open-heart covering argument gives this family statement. There are countably many charts/cuts/connected components. If every image $V_T\to C$ is non-dense, Chevalley's actual constructible-image theorem says each such image is finite, since a non-dense constructible subset of an irreducible algebraic curve is finite. Thus all unstable fibers form at most a countable set. Since C(C) is uncountable, a very general closed point is semistable. It remains to rule out a DOMINANT chart, which we now do constructively.

Reduce a dominant chart to a curve, without a generic-flatness assertion. Assume one smooth irreducible chart V dominates C. The families F, E and its cone are in the required constant hearts; F is also in the rationally rotated heart. For EACH of these finitely many heart objects, take its maximal subobject supported over a proper closed subset of V. It exists by noetherianity: finite sums still have proper support on an irreducible base, and the chain of their sums stabilizes. Remove the finitely many resulting proper supports from V. On the remaining dense open, every selected object is torsion-free over V in every heart in which it was specified. Now a nonzero smooth local divisor equation is injective on each such object. Its two-term Koszul triangle therefore restricts that object to the corresponding heart on the divisor. Choose a smooth divisor still dominant over C and repeat. Existence with the correct dimension is explicit at the generic fiber: after generic smoothness choose etale coordinates on an open of that smooth fiber, impose one general coordinate linear equation there, and clear its finitely many denominators over C. Its smooth open and a component containing the chosen generic point give the desired divisor. Repeat until the relative dimension is zero. This is the ordinary smooth multisection construction; no irreducibility/nonemptiness conclusion is falsely attributed to Bertini. At every cut remove the finitely many new relative torsion supports first, so all the required heart memberships and the monomorphism survive derived restriction. The resulting smooth curve C' is generically finite over C. Normalize C in its function field and pass to a finite Galois closure; shrink C to an open where this is a finite etale Galois map. Finite pushforward is t-exact for all these constant hearts: use the large-twist formula with the pullback of an ample line on C. Finite flat pullback is t-exact as well: its push-pull is tensor by a finite locally free base algebra, which is locally a finite direct sum, and conservative t-exact finite pushforward reflects heart membership. Thus we obtain on that cover an actual AP subobject F' of $E_C^{\mathrm{\prime }}$, belonging to the high-cut torsion class $T_{\alpha }$.

Galois saturation produces a genuine destabilizer downstairs. On the common etale open take the sum of all Galois conjugates of F' INSIDE $E_C^{\mathrm{\prime }}$. It is nonzero, invariant, and still in $T_{\alpha }$: that class is a torsion class in the AP heart, closed under finite sums and quotients. Its canonical equivariant structure comes from its inclusion in $E_C^{\mathrm{\prime }}$, not arbitrary chosen isomorphisms. Exact invariants of its finite pushforward give a descended subobject of E; averaging is an idempotent because characteristic is zero. Finite etale descent on the underlying coherent complexes identifies its pullback with the saturated sum; the usual descent equalizer works termwise, with negative Ext vanishing making the heart/subobject descent unique. Extend this subobject across the finitely many missing points as follows: extend its bounded coherent representative across the missing points, take its AP H0 (which restricts to the same heart object on the open), and twist by a sufficiently negative base divisor to clear denominators in its map to E. Base line twists are t-exact. Its IMAGE in the AP heart is the required global subobject and restricts to the original subobject on the open. This changes no generic numerical class; no nonexistent maximal torsion quotient is used. Thus it is an ACTUAL subobject of E on X times C. On a dense open upstairs the sum belongs to $T_{\alpha }$. Hence its large-twist pushforwards have phases strictly above alpha, and its nonzero leading charge has phase at least alpha. Nonzero leading charge follows from relative torsion-freeness and a nonzero generic fiber in A. Descent scales that leading vector by the positive cover degree, so its phase downstairs is also at least $\alpha >\varphi$. It therefore destabilizes the assumed leading-semistable E. This contradiction excludes every dominant chart in section2. We have proved that $E_s$ is sigma-semistable at a very general closed point. In particular its numerical class v1(E) is the class of an actual sigma-semistable object, as needed for the original support quadratic form. This conclusion used no extension of sigma to an unproved function-field stability condition.

One good fiber implies the determinant inequality globally. Choose a semistable fiber of phase phi. Because the charge is rational, the rotated heart $B=P(\varphi -1,\varphi ]$ is noetherian (rotate and positively rescale by a rational real 2-by-2 matrix). The open-heart property gives $E_U$ in $B_U$ on a nonempty open U of C. The constant-heart construction preserves inclusions of aisles. The original $A_C$ and $B_C$ therefore differ by the corresponding one-step tilt, with torsion pair T=A_C intersect B_C[1], F=A_C intersect B_C, B_C=<F,T[−1]>. This torsion pair is derived directly from the two truncation triangles: the inclusion of their aisles has amplitude one, so the two cohomology terms give the stated short exact sequence in $A_C$. Let $T_E$ be the high-cut subobject of E. Its restriction to U is zero, since $E_U$ is in the low part $F_U$. Thus it is supported over finitely many points. But E was relatively torsion-free in $A_C$, so $T_E=0$. Consequently E belongs to $B_C$ on ALL of C. Rotate Z by $\theta =pi(1-\varphi )$, making a+i c negative real. For every large n the rotated large-twist charge lies in the upper half-plane or negative real line, since E is in $B_C$. Its linear coefficient is real, so Im(exp(i theta)(b+i d))>=0. As exp(i theta) is a positive multiple of (−a+i c), this inequality is exactly b c−a $d\ge 0$. If $c=0$, the AP positivity inequalities give $a\le 0,d\ge 0$, and b c−a $d=-ad\ge 0$ immediately. This completes the required determinant inequality in every leading-semistable case.

Two elementary heart facts, with the chain arguments supplied. A rational Bridgeland heart is noetherian. Indeed for an increasing chain $F_i$ in E, its imaginary charges stabilize in the finite lattice interval [0,Im Z(E)]. Separate the maximal phase1 HN subobject E1 of E. Its subobjects have real charge between Re Z(E1) and zero, so a chain of them stabilizes by discreteness. After removing this stabilized part, all HN phases of the $F_i$ are bounded above by a fixed $a<1$. With their imaginary charges now constant, their decreasing real charges have a lower bound from that phase bound. They stabilize, so each successive quotient has charge zero and is zero. This proves the noetherian input for the rational heart and every rational rotation used in the AP construction. Here is also the HN lemma used twice below. Let B be noetherian, $I:B\to hZ$ be nonnegative, $R:B\to h^{\mathrm{\prime }}Z$ additive, with $R\le 0$ when $I=0$. Set mu=−R/I, infinity at $I=0$ (including zero weak charge). Every object has a finite weak HN filtration. First take its maximal $I=0$ subobject, which exists since these subobjects are closed under finite sums and B is noetherian. Work with the remaining object E, which has no nonzero $I=0$ subobject. Induct on I(E)/h. If positive-I subobjects had unbounded slope, choose the smallest positive r at which such unboundedness occurs, and fix one F of I(F)=r. For a subobject $F_j$ of that same r with $R(F_j)\to -\mathrm{\infty }$, put $K_j=F\cap F_j$ and $J_j$ its image in E/F. If $I(K_j)<r$, its real charge is bounded below by minimality of r; $J_j$'s real charge is bounded below by the inductively existing HN filtration of E/F. To verify this last bound, $\cap J_j$ with that finite HN filtration: each positive-I semistable piece bounds the subpiece's slope, and the phase1 piece has subcharge bounded below by its own charge. Thus $R(F_j)$ would be bounded, contradiction. Consequently along an unbounded subsequence $I(K_j)=r$. Then $S_j=F+F_j$ has $I=r$ and $R(S_j)\le R(F_j)$, since $F\mathrm{/}{K}_j$ has $I=0$. All finite sums of these $S_j$ still have $I=r$, because their intersections contain F. Their ascending union stabilizes by noetherianity at S. But $S\mathrm{/}{S}_j$ has $I=0$, giving $R(S)\le R(S_j)\to -\mathrm{\infty }$, impossible. Hence slopes are bounded above. Discreteness and the finite possible positive I-values give a maximum slope. Sums of maximum-slope subobjects have that same slope: their intersection has no larger slope, and additivity forces the sum's slope at least the maximum. Noetherianity gives a maximal such subobject, which is semistable. Its quotient has no $I=0$ subobject, since its preimage would have larger slope or the same slope with a larger maximal subobject. The quotient's maximum slope is strictly smaller and its positive I is strictly smaller. Induction completes the finite HN filtration. Orthogonality and uniqueness follow from the seesaw inequality on a nonzero image. This proves the lemma without using the slogan 'noetherian plus discrete implies HN' as an unproved step.

The polynomial coefficients and their two weak functions. For the degree-one base line, the divisor triangle shows v(Rp_E(n))−v(Rp_E(n−1)) is independent of n, equal to the numerical fiber class v1(E). Thus the large-twist numerical charge is (a+i c)n+(b+i d). It lies in the upper half-plane or negative real axis for every sufficiently large n when E is a nonzero object of the AP heart $B=A_C$. To justify the leading boundary sign, first remove its maximal relative-torsion subobject as in step 11.1. That subobject has zero leading numerical class. The remaining t-flat family has every derived fibre in A, so its leading charge a+i c is the charge of an actual object of A, independent numerically of the fibre. It is therefore either zero, in the strict upper half-plane, or negative real. In particular $c=0$ forces $a\le 0$; this conclusion does not follow from eventual polynomial positivity alone. Eventual polynomial positivity also gives $c\ge 0$ and $d\ge 0$ if $c=0$. Combining these facts gives c>=0; c=0 implies d>=0 and a<=0; c=a=d=0 implies b<0. (1) The leading charge a+i c is a rational weak stability function and has HN by section1. The weak function Z_t=t a−d+i t c, t positive rational, nu_t=(−t a+d)/(t c), infinity at c=0, likewise has HN by section1 and (1). Its torsion pair $T={\nu }_{min}>0,F={\nu }_{max}\le 0$ gives the actual tilted bounded heart $B^t=<T,F[1]>$: the truncation of an object with respect to B has only the two possible pieces, and the T/F exact sequence separates them; orthogonality of the two pieces and their universal triangles verify the bounded t-structure directly. The positivity argument above proves that for every leading-semistable object Q, b_Q c_Q−a_Q d_Q>=0, and, when $c_Q>0,w_Q=v1(Q)$ is the numerical class of an actual base-sigma semistable object. Write the base support form as Q_X(w)=C0² |Z(w)|²−||w||², with C0 chosen from the base norm support bound.

The weighted-variance inequality for nu_t-semistables. Let E be nu_t-semistable, with c(E)>0. Take its leading-charge HN factors $Q_i$, ordered so $x_i=a_i\mathrm{/}{c}_i$ strictly increases. All $c_i>0$: a first factor with $c=0$ would be a nu_t-infinity subobject and would destabilize E. Let $C=\sum c_i,A=\sum a_i,D=\sum d_i$, and prefix sums $C_j,A_j,D_j$. Semistability gives D_j−t A_j <= ((D−t A)/C) C_j. Leading positivity gives $b_i\ge a_i{d}_i\mathrm{/}{c}_i$. Abel summation, with $x_{j+1}-x_j>0$, yields b(E) >= sum_i x_i d_i =x_l D−sum_{j<l} D_j(x_(j+1)−x_j) >=t sum_i a_i²/c_i + ((D−t A)/C) A. Multiplying by C and subtracting A D proves bc−ad >= t sum_{i<j} c_i c_j (x_i−x_j)². (2) Let $w_i=v1(Q_i)$. Each is base semistable, so $Q_X(w_i)\ge 0$ and $\mathrm{\mid }\mathrm{\mid }{w}_i\mathrm{\mid }\mathrm{\mid }\le C0sqrt(a_i^2+c_i^2)$. Polarization/Cauchy–Schwarz give 2Q_X(w_i,w_j) >=2C0²(a_i a_j+c_i c_j −sqrt(a_i²+c_i²)sqrt(a_j²+c_j²)). The elementary inequality (x−y)²>=2(sqrt(1+x²)sqrt(1+y²)−xy−1) is just $(1+x^2)+(1+y^2)\ge 2sqrt((1+x^2)(1+y^2))$. Combining it with (2) proves bc−ad+eta Q_X(v1(E))>=0, 0<=eta<=t/C0². (3) If $c=0,(1)$ gives $bc-ad=-ad\ge 0$, and leading semistability is automatic at its infinity slope; remove the maximal relative torsion from E; its t-flat quotient has the same v1. Its fiber is in A with charge a+i $c=a$. If $a<0$ that fiber is automatically semistable of phase1, since every HN factor has nonnegative imaginary charge and their sum has zero imaginary part. If $a=0$ its fiber has zero charge and is zero, so v1=0. Thus $Q_X(v1)\ge 0$, and (3) follows in this case as well. This is the actual generalization of the Bogomolov determinant inequality used next.

Strict positivity of the final charge. Set Z^{s,t}=s c+b+i(d−t a), s,t positive rational. On $B^t$ its imaginary part is nonnegative by the definition of T,F. For T in T, if $d-ta=0$ then its nu-HN factors have $c=0$, hence $a=d=0$ by (1); therefore its real charge b is strictly negative. For nonzero F in F, c(F)>0. If $d-ta=0$, every nu-HN factor has slope zero, so F is nu-semistable. The Abel estimate gives $b(F)\ge ta(F{)}^2\mathrm{/}c(F)\ge 0$. Thus the real charge of F[1] is −s c(F)−b(F)<0. Every object of $B^t$ is an extension of these two sorts; its boundary real charge is consequently strictly negative. Thus $Z^{s,t}$ is an actual stability function, not a weak one.

The tilted heart is noetherian: supply the omitted saturation induction. Write $I=d-ta$. In a quotient chain $E\to E_i\to E_{i+1}$ in $B^t$ the nonnegative discrete $I(E_i)$ eventually stabilizes. REBASE E at one quotient in that tail, and put $F_i=ker(E\to E_i)$. Now, and only after this replacement, $I(F_i)=0$. AP noetherianity stabilizes $H_B^0(E_i)$, a quotient chain of $H_B^0(E)$, and $H_B^{-1}(F_i)$, an increasing subobject chain in H^(−1)_B(E). Put $K_i=F_{i+1}\mathrm{/}{F}_i$. Its B-cohomologies $G_i=H_B^{-1}(K_i)$ and $T_i=H_B^0(K_i)$ satisfy $I(G_i)=I(T_i)=0$, since $G_i\in F,T_i\in T$ and both contributions to $I(K_i)=-I(G_i)+I(T_i)$ are nonnegative. Every nonzero $G_i$ is weak-semistable of slope zero, hence $c(G_i)>0$ by the boundary argument of step 20.1. Also $T_i\in C0=a=c=d=0$: its positive/infinite weak HN factors can have total I zero only if each has $c=0$, when (1) forces $a=d=0$. After $H_B^{-1}(F_i)$ stabilizes, the cohomology sequence gives $0\to G_i\to H_B^0(F_i)\to H_B^0(F_{i+1})\to T_i\to 0$. Consequently $c(H_B^0(F_{i+1}))=c(H_B^0(F_i))-c(G_i)$. These nonnegative discrete values cannot strictly decrease infinitely often. Thus $G_i=0$ eventually and $K_i=T_i\in C0$. The same sequence for $0\to K_i\to E_i\to E_{i+1}\to 0$ then gives injections $Q_i=H_B^{-1}(E_i)\to Q_{i+1}$ whose cokernels lie in C0. This category is Serre in B: zero total c forces both $c=0$, and $a\le 0,d\ge 0$ then force $a=d=0$ for both subobject and quotient. Every nonzero C0 object has $b<0$. Use the following generalized induction: a chain in F containing a fixed reference U with C0 cokernels terminates, by induction on the weak HN length of U. Here $U=Q_1$. Take its first weak HN subobject A1⊂U. In $Q_i\mathrm{/}A1$ take its largest C0 subobject and let $P_i$ be its preimage. These exist by AP noetherianity and form an increasing chain with $P_i\mathrm{/}A1\in C0$. Every subobject of $P_i$ has the same weak charge as its intersection with A1; a C0 subobject cannot lie in $Q_i\in F$. Thus $P_i$ is weak-semistable, with the same a,c,d as A1 and $c>0$. Inequality (2) bounds $b(P_i)$ below by the fixed ad/c. Every proper increase has a nonzero C0 quotient, so decreases b by a positive discrete amount. Hence $P_i$ stabilizes. The quotient $Q_1\mathrm{/}A1$ is in F and has no C0 subobject, so $P_i\cap Q_1=A1$. Thus $Q_1\mathrm{/}A1$ embeds into $Q_i\mathrm{/}{P}_i$ with C0 cokernel. The latter has no C0 subobject by maximality of $P_i$; every subobject intersects $Q_1\mathrm{/}A1$ with the same weak charge. Its maximal weak slope is therefore at most the next HN slope of $Q_1$. These quotients lie in F, and their fixed reference $Q_1\mathrm{/}A1$ has HN length one smaller. The generalized induction applied to that reference makes their chain stabilize (the length-one case has $Q_i=P_i$). Thus $Q_i$ stabilizes, and the cohomology sequences make $E_i$ stabilize. This proves $B^t$ noetherian. The previously proved HN criterion applies to its strict function $Z^{s,t}$, giving actual finite HN filtrations.

Large-s behavior is explicit. Fix t and suppose E is $Z^{s,t}$-semistable for arbitrarily large rational s. Write $0\to F[1]\to E\to T\to 0$ in the tilted heart. If $F=0$, the leading slopes −c/(d−ta) force T to be nu_t-semistable: its first nu-HN subobject otherwise has strictly larger large-s slope and destabilizes. If $F\mathrm{\ne }0$, compare its subobject F[1] with E. Its leading slope is c(F)/(ta(F)−d(F)), while T contributes nonnegative d−ta and $c(T)\ge 0$. Semistability for unbounded s forces c(T)=a(T)=d(T)=0. Comparing the first nu-HN subobject of F[1] then forces F nu_t-semistable. The endpoint $d-ta=0$ is phase1, and the same conclusion follows by its strict negative real boundary charge. Consequently (3) gives bc−ad+eta Q_X(v1(E))>=0, 0<=eta<=t/C0² for large-s semistables: in the second case b(T)<0 and c(F)>0 add the nonnegative term −b(T)c(F), while v1(T)=0.

Wall induction proves support at EVERY rational s,t. Put $Q_{\eta }=bc-ad+\eta Q_X(v1)$. On $ker{Z}^{s,t},b=-sc$ and $d=ta$, so Q_eta=−s c²−t a²+eta Q_X(v1) <=−(s−eta C0²)c²−(t−eta C0²)a²−eta||v1||². For $0<\eta <min(s,t)\mathrm{/}C{0}^2$ this is negative definite on the lattice Lambda_new=coimage of (v1, v2 mod ker Z_X). Indeed equality forces $v1=a=c=0$, then $b=d=0$ makes the second quotient class zero. The lattice is finite-rank and numerical: pushforward/fiber maps preserve numerical equivalence by their Euler adjunction pairings. On any fixed positive Z-ray, the cone $Q_{\eta }\ge 0$ is convex. To verify rather than assume this familiar Lorentz fact, decompose by the positive ray coordinate; the restriction to the kernel is a negative definite norm, completing the square makes the cone an epigraph of that norm. Its triangle inequality proves closure under sums on that ray. Induct on the discrete nonnegative I(E)=d−ta for $Z^{s,t}$-semistables. The phase1 case and the minimal positive case cannot cross a proper same-ray wall with smaller positive I, and follow from section6. For larger I, strict semistable objects reduce to their finite stable factors, whose positive I are smaller; phase1 factors cannot have the same ray as an object of positive I. Hence assume E stable at s0. It has no $I=0$ subobject in the fixed heart, independently of s. At any later s1, its finite HN filtration therefore has maximum $\text{phase}<1$ and bounds the slopes of all its subobjects. Subobjects destabilizing at s1 have $0<I(F)\le I(E)$ and both upper/lower real-charge bounds, so their $Z^{s1,t}$ values form a FINITE subset of the rational lattice. For each such value choose a representative with maximal slope at s0; it exists because that slope is bounded above by stability of E and is discrete when I and $Z_{s1}$ are fixed. Differences of subobject/total slopes are affine functions of s. Each crosses zero at a rational wall; for a fixed s1 value the chosen maximal initial slope gives the earliest crossing. The minimum among finitely many such walls is an actual first wall $s^{\mathrm{\prime }}>s0$: no other subobject can cross earlier, including one not destabilizing at s1, since an affine difference negative at both endpoints cannot be positive between. At s' all stable factors have smaller positive I. By induction they satisfy $Q_{\eta }\ge 0$ for $\eta <min(s0,t)\mathrm{/}C{0}^2$, also permitted at s'. Their same-ray sum satisfies it by the preceding norm argument. If no later destabilization occurs, section6 gives it. This proves the inequality for all rational s,t, including $\eta =0$ by taking limits. Negative definiteness on the kernel and nonnegativity on semistable vectors imply norm support: the intersection of $Q_{\eta }\ge 0$ with the unit sphere is compact and has no zero of Z, so |Z| has a positive minimum there. Thus $\mathrm{\mid }\mathrm{\mid }{v}_{new}(E)\mathrm{\mid }\mathrm{\mid }\le C_{new}\mathrm{\mid }Z(E)\mathrm{\mid }$ for every semistable E. Finally local finiteness is not omitted. A finite-rank lattice has a positive minimum norm on its nonzero vectors, so support gives a positive lower bound on masses of all nonzero semistables. In a phase interval of width less than one, projection to its central ray is at least a fixed positive cosine times mass. Each nonzero strict quotient in its quasi-abelian extension category consumes at least that fixed amount of this additive bounded projection. Therefore both ascending and descending strict chains terminate. This proves the required local finiteness of the slicing.

Iterate and state exactly what is obtained. For the empty product $n=0$, use $X=\mathrm{Spec}\mathbb{C}$, the empty divisor sum $H=0$, and $e^{-iH}=1$. Thus the stated formula is $Z(F)=-\chi (F)$, and on the heart ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^{\mathrm{f}\mathrm{d}}$ it is $-\dim F$. Define the slicing by $\mathcal{P}(1+k)={\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^{\mathrm{f}\mathrm{d}}[k]$ for $k\in \mathbb{Z}$ and zero at all other phases. Every bounded complex splits into its finitely many cohomology spaces, giving the finite HN filtration; the phase pieces have finite length. The rank-one lattice $K_0(X)=\mathbb{Z}$ with norm $\mathrm{\mid }r\mathrm{\mid }$ has support constant one. The point sheaf is simple of phase one. The factorwise symmetry group and ${\mathrm{Pic}}^0(X)$ are trivial, so every claimed geometric symmetry acts identically and this support lattice is equivariant. This is the zero-step ordered construction; no AP extension or tilt is needed. On an elliptic curve start with Coh(E) and Z=−degree+i rank. Its finite HN property follows from section1; numerical rank/degree give the norm support immediately. At every step take the rational choices $s=t=1$ and degree-one base line. Sections2–7 construct an actual rational noetherian support-property Bridgeland condition on the next product, retaining bounded coherent amplitude because a constant heart and one tilt have finite amplitude. Induction yields such a condition on EVERY ordered product E1 times ... times En. Its central charge is −integral exp(−i H)ch, as computed in the naturality proof below. All skyscrapers are stable of phase1: they have $a=c=d=0,b=-1$, and in a tilted-heart subobject sequence $0\to K\to E\to Q\to 0,H^{-1}(K)=0$ and zero total imaginary charge force K and H0(Q) into C0. The cohomology sequence then gives c(H^(−1)(Q))=0, so H^(−1)(Q)=0 because it lies in F. Hence K,Q both lie in C0 in the original heart. Their nonzero b-values are negative integers, so additivity of b(E)=−1 forces either K or Q to be zero. This is the explicit point simplicity argument, not a general geometric-stability theorem. The naturality proof below then proves factorwise automorphism and Pic0-twist invariance of these ACTUAL conditions, and constructs a finite-group-equivariant support lattice. In particular the diagonal order2/order3 groups needed for the Cynk–Hulek construction are covered without the forthcoming uniqueness theorem. Permuting induction order still requires a separate proof; the symmetric charge does not supply it. Thus the wreath-product $S_n$ interface for the other named families remains an explicit next root, not erased from the original theorem.

The ordered step and its numerical convention. Let sigma=(A,Z) be the already constructed rational numerical stability condition on X. Let C be an elliptic curve, p:X times $C\to X$, q:X times $C\to C$, and choose a degree-one line L on C. The actual AP heart is A_C={E:Rp_*(E tensor q*L^n) belongs to A for all n sufficiently large}. Its local construction makes it a noetherian sheaf of hearts over C, and it is independent of the ample line. Write Z(Rp_*(E tensor q*L^n))=(a+i c)n+(b+i d). Use precisely the $\beta =0$ rational Liu step: weak charge t a−d+i t c, HN slope nu=(−t a+d)/(t c), infinity if $c=0$, torsion pair $T={\nu }_{min}>0,F={\nu }_{max}\le 0$, and tilted heart $A_C^t=<T,F[1]>$. The final charge is Z_C^{s,t}=s c+b+i(d−t a), s,t>0 rational. No sign ambiguity from Cheng's beta formula is needed for the existence/invariance cases: $s=t=1$ at each step suffices. Starting on an elliptic curve with Coh(C) and Z=−degree+i rank, this ordered induction has the symmetric numerical charge Z_n(E)=−integral_{E1 times ... times En} exp(−i H) ch(E), H=sum_j pr_j*c1(O_Ej(point)). Indeed td(Ej)=1 and the degree-one divisor exact triangle gives the leading class v1 and constant class v2; expanding $exp(-i{H}_{last})=1-i{H}_{last}$ yields $Z_{new}=Z(v2)-iZ(v1)=b+c+i(d-a)$. This is an identity of numerical homomorphisms; it is not an equality of two differently ordered hearts.

Factor automorphisms commute with the actual heart and tilt. Suppose Phi is an exact autoequivalence on D(X) with Phi(A)=A and $ZPhi=Z$, and $g:C\to C$ is an automorphism. For the applications Phi is pullback by a previously constructed product automorphism, possibly combined with a flat twist. Let $U=Phi$ times g*. Projection formula/base change give Rp_*(U(E) tensor q*L^n) =Phi Rp_*(E tensor q*((g^{-1})*L)^n). The line $(g^{-1})\ast L$ is again ample of degree one. Independence of the AP heart from the ample line therefore proves $U(A_C)=A_C$. This argument does not require gL isomorphic to L; translations need not preserve a chosen degree-one divisor as a line bundle. The coefficients a,b,c,d are unchanged. To see this without an invalid Kunneth assertion, use the divisor triangle on C. For a numerical class map v, v(Rp_*(E(n)))−v(Rp_*(E(n−1)))=v(Li_x*E) for a degree-one divisor x, with the harmless one-dimensional line fiber omitted in K0. This class is independent of n. Its NUMERICAL value is independent of x in the connected curve: equivalently pair with any perfect test on X and use constancy of the Euler characteristic of the derived family over C, or the finite coherent-complex computation of that Euler characteristic. Thus the leading class and the final numerical charge are independent of which degree-one line g pulls back. The constant coefficient is fixed by evaluating the same polynomial numerically. Alternatively numerical Riemann–Roch gives this directly since degree(gL)=degree L and td(C)=1. Consequently U preserves the weak charge and the order on its HN factors. Uniqueness of HN filtrations implies U(T)=T and U(F)=F. It therefore preserves the actual tilted heart $A_C^t$ and the final charge. A semistable object is defined by the inequalities for all subobjects in that heart, and an exact heart autoequivalence preserves precisely those inequalities. Thus U preserves the full slicing, not only Z. By induction, the ORDERED product stability is invariant under every product of elliptic-curve automorphisms. In particular it is invariant under factorwise translations, negation, the order-three automorphisms in the Cynk–Hulek case, and every subgroup of their product for which the geometric family is defined. This uses no injectivity theorem for the central-charge map.

Every degree-zero line twist is also an actual slicing symmetry. For M in Pic0(C), tensoring by qM preserves $A_C$ because this heart is local over C: on a trivializing open cover it is literally the identity functor on the heart, and the sheaf-of-t-structures membership criterion glues those local statements. This is exact locality, not an unsupported appeal to numerical triviality. The coefficients a,b,c,d are unchanged because the polynomial is numerical and ch(M)=1; the degree-one divisor difference and the constant Euler pairing give the same conclusion without asserting a K0 equality for arbitrary families. Hence the torsion pair, tilted heart, stability inequalities and slicing are all preserved, exactly as in step 2. Twisting by pN for a flat line N on X is covered inductively by Phi. Since Pic0(E1 times ... times En)=product_j Pic0(Ej), all degree-zero twists on the product preserve the ordered stability. The equality of Pic0 here is concrete: a flat unitary character of the product lattice is the product of its restrictions to the factor lattices; the exponential-sequence description identifies the same holomorphic degree-zero bundle. In particular ALL finite-order flat twists needed after the Hilbert/finite-cover comparisons are covered.

Make the support lattice equivariant instead of assuming it was. Liu's support lattice is formed at each step from Lambda plus Lambda/ker Z and can depend on the ordering. Even for a slicing symmetry group one must not simply declare that lattice equivariant. Let finite G preserve the stability just constructed. If $\mathrm{\mid }\mathrm{\mid }v(E)\mathrm{\mid }\mathrm{\mid }\le C$ |Z(E)| for every semistable E, define v_G(E)=(v(g*E))_{g in G}, and let Lambda_G be the coimage lattice, a subgroup of the finite direct sum of copies of Lambda (modulo any torsion). G permutes the components, so $v_G$ is genuinely equivariant. Every g*E is again semistable, with the same central-charge magnitude, hence ||v_G(E)|| <=sqrt(|G|) C |Z(E)|. This is the norm form of the support property. The charge factors through $v_G$ by its identity component. It gives an explicit G-equivariant finite-rank NUMERICAL support lattice for finite equivariant induction; no order-independence of the original Liu lattice is assumed. This applies immediately to the finite diagonal groups $G_n$ in the $m=2$ and $m=3$ Cynk–Hulek cases, once the product existence part is supplied. It also applies to the finite factorwise groups used before wreath-product equivariantization.