Stability under specified smooth CY3 orbifolding

Fix any selected target fiber $V$ in the Statement. By the stated family assumptions it is a smooth projective complex CY3. Apply F7 directly to $V$; all its hypotheses are smoothness, projectivity over $\mathbb{C}$, and AC. It gives a locally finite Bridgeland stability condition with finite HN filtrations and finite-rank support on $D^b(\mathrm{Coh}V)$ in precisely the sense of F2. This application neither transfers a condition from $X$ to a different source fiber nor uses an unstated invariant source condition.

Retain the equivariant construction under its actual additional premises. Suppose a selected target has been identified with a smooth projective crepant resolution $V\to Q=M\mathrm{/}G$, where $M$ is a smooth source CY3 fiber, $G$ is finite faithful, and $M$ carries an actual geometric BMT stability condition with admissible $G$-invariant parameters $(\omega ,B)$. These premises are conditions on this alternative construction and are not inferred from existence on the single input $X$. If the fiberwise quotient is obtained from a reduced disjoint orbit of smooth source fibers, restriction to one component identifies the invariant functions with the invariants of its stabilizer; replacing that stabilizer by its effective quotient does not change the coarse quotient. The actual reduced fiber identification must be checked in the construction before this conditional route is applied.

The choice of $V$ was arbitrary among the selected smooth projective targets. Thus every such target admits the required conditions, including the specified quintic-mirror members. This is a separate existence assertion for each member, with no claim of a common relative slicing or of BMT geometric form on it. It establishes the complete unchanged Statement.

Use the actual G-invariant parameters (ω,B) in the conditional premise of step 1.2. To check actual invariance, pullback by g preserves coherent sheaves, their exact sequences and their Chern characters, and $g\ast \omega =\omega ,g\ast B=B$. Each numerical ${\int }_M{\omega }^{3-j}c{h}_j^B(E)$ is therefore unchanged: g is a holomorphic automorphism of degree one and preserves the complex orientation. Hence the central charge and every cohomological slope defining the source torsion pairs are unchanged, including the zero-denominator convention. Pullback takes a slope HN filtration to one with the same factors' slopes, so uniqueness of that filtration makes both torsion classes invariant. The construction of the tilted heart as complexes with the two specified coherent cohomology pieces is functorial, and the same reasoning applies at the next tilt. Thus this source heart and central charge are G-invariant. The slicing is determined by the heart's semistable objects and their shifts, so this full locally finite support-property condition σ is G-invariant. This uses the conditional invariant source parameters; it makes no averaging operation on a stability condition.

Make the charge G-stable explicitly: replace its finite-rank charge v by $v^{\mathrm{\prime }}(E)=(v(g\ast E){)}_{g\in G}$, with Γ′ the image inside the finite free direct sum of |G| copies of Γ. Pullback permutes these components. By σ-invariance their central charges agree on Γ′; use any component to define Z′. Every g*E is semistable whenever E is, so the product norm has support constant at most √|G| times the original constant. Relabel this actual G-stable charge as v. Apply F2 to σ on Db(Coh M), which is the compact-support category because M is projective. It gives ${\sigma }_G=(Z_G,P_G)$ on $D^b({\mathrm{Coh}}_{G}M)$, with $P_G(\phi )$ the equivariant objects whose underlying objects lie in P(φ), and $Z_G=Z$ composed with the forgotten charge. Its finite-rank charge group is ${\mathrm{\Gamma }}_G=\mathrm{im}(v\mathrm{Forget})$, with the inherited norm. The supplied proof gives actual equivariant HN filtrations and the same local-finiteness interval; its support inequality is inherited on this image. These are exactly the existence requirements of F2, without a claim of support on an unspecified larger equivariant lattice.

Apply F4 to (M,G,V→Q). It constructs a smooth projective crepant CY G-Hilbert resolution T→Q and an actual equivalence $E:D^b(\mathrm{Coh}T)\to D^b({\mathrm{Coh}}_{G}M)$. The selected resolution V is a separate given resolution. Pull ${\sigma }_G$ back along E: its slicing is $E^{-1}{P}_G$ and its charge is $Z_G$ composed with the induced K-group isomorphism. Exact equivalence carries all HN triangles, phase-Hom vanishing and strict exact sequences in the local-finiteness subcategories bijectively. The norm and support inequality use the same finite group ${\mathrm{\Gamma }}_G$. Thus this is a genuine locally finite support-property stability condition on Db(Coh T).

Q is normal projective Gorenstein, and T,V are its actual smooth projective crepant resolutions with trivial canonical line. The full hypotheses of F5 hold even if Q itself is nonterminal. Its finite directed-flop chain gives an actual triangulated equivalence J:Db(Coh V)→Db(Coh T), obtained by composing the supplied global perverse-point kernel equivalences (and inverses in the required direction). Pull the condition from step 4.1 back along J. The preceding exact-equivalence argument again verifies the complete slicing, finite HN, local-finiteness and finite-rank support axioms. Therefore the specified selected V admits Bridgeland stability conditions. This conditional construction applies to each selected member for which the additional source data in step 1.2 have been supplied.

The source quintic-mirror example is within these inputs. Its smooth source members can be written ${\sum }_{i=0}^4{x}_i^5-5\psi {\prod }_{i=0}^4{x}_i=0$ in $P^4$, with ${\psi }^5\mathrm{\ne }1$. The group $\{\mathrm{diag}(\zeta 0,\dots ,\zeta 4):{\zeta }_i^5=1,\prod {\zeta }_i=1\}\mathrm{/}{\mu }_5$ acts on each member, preserves the hyperplane class, and has order125. Whenever that source member carries the admissible geometric BMT family, taking parameters proportional to the preserved hyperplane class supplies the conditional invariant choices in step 1.2; such availability is a separate fact and is not deduced from the original input X alone. The specified projective crepant mirror resolutions are precisely selected source orbifolding targets in the Statement. Step 5.1 applies whenever the conditional source data have been supplied. The smoothness restriction follows directly from the Jacobian: a singular point cannot have a zero coordinate, and the five derivative equations then multiply to ${\psi }^5=1$. This describes the conditional equivariant route. The unconditional existence conclusion for every specified smooth target, including these mirror resolutions, already follows from steps 1.1–2.1.