Homological mirror symmetry for a quartic K3

Use the Fermat relative pair with the source symplectic scaling. The finite category and all its actual higher operations exist by [F2]. By [F3] its finite graph collection has the computed exterior/smash cohomology, central primary product and nonzero Fermat first-order class. The compact grading is the actual branched/object phase grading; no fixed auxiliary integer-weight grading is carried upstairs unchanged.

The full compact graph collection is split-generating: [F4] identifies it, in the actual q and coefficient categories, with the named vanishing collection, while [F5] constructs the global graded monodromy, uniformly negative iterates off invariant neighborhoods and the dimension-specific Hamiltonian avoidance. Its compact rational evaluation triangles and finite cone argument apply to all admissible source rational probes, including those meeting the divisor. Thus this is full compact generation, not only exact affine or central-fibre generation.

Let $A_q$ be the total finite geometric endomorphism algebra over $R=C\mathrm{⟦}q\mathrm{⟧}$ and let $S_q$ be the actual coherent Čech endomorphism model of [F6]. The finite-free replacements and actual strict finite symmetry admit the equivariant contractions of [F7]. The central primitive-Hom computation of [F3], used in [F2], has no adjacent cohomological degrees in a fixed summand; since q has degree zero, its transferred differential is zero. On the coherent side [F6] proves that multiplication by the hypersurface equation induces the zero map on ambient cohomology, so its explicit Koszul contraction has literally zero differential. These are the actual minimality inputs of the transfer in [F7]; free underlying chain groups alone would not suffice. All higher operations and their comparison maps are retained by the explicit tree identities and q-adic completion.

Their common binary algebra and primary class identify the TWO full central A-infinity structures: the binary Koszul degree-two group has no arity above N, so the first remaining arity-d difference is $[\mu 2,h_{d-1}]$ and is successively removed without changing smaller arities. This is the all-arity central normalization in [F7]. Both actual formal deformations have nonzero even-coordinate invariant Fermat class. Apply the constructive one-parameter comparison with the coherent deformation as reference to obtain a strictly unital object-preserving A-infinity quasi-isomorphism $A_q$ equivalent to ${\psi }^{\ast }{S}_q$, where $\psi (q)=c1$ q+... and c1 is nonzero. The recursive formal inverse makes psi an automorphism, not merely an endomorphism.

The category presentations are taken in a fixed ambient set universe with AC. The small dg models are those specified in [F1] and constructed for the coherent generator in [F6]; finite words of objects and shifts, finite matrices of morphisms, and their idempotents form sets, so their finite twisted and split completions are set presentations. We use the supplied enhancement equivalences when identifying these with the named geometric and coherent categories. This is an ambient presentation convention and retains every brane and coherent object covered by those supplied equivalences; it assumes no global class choice. Extend along the actual rational action lines to the source field $Lambd{a}_Q$ and the chosen lift of psi. The finite morphism quasi-isomorphisms extend over the field; their tensor, bar and perfect-module constructions commute with this specified scalar extension by [F6, F7]. They give an exact fully faithful functor between the split-complete finite derived categories, carrying the geometric generator to the coherent one. This compares all higher products, rather than substituting a binary cohomology-algebra isomorphism.

The coherent collection thickly generates the exact mirror category by [F6], and the geometric collection thickly generates the full source compact category by step 2.1. The generator criterion [F8] therefore gives the desired equivalence for the Fermat model after the specified formal pullback. The quartic uses the actual minimal-resolution McKay bridge, and the quintic uses the source quotient-stack category; neither is replaced by a different target.

Finally connect the Fermat polynomial to the given smooth degree-N hypersurface in the complement of the discriminant. The smooth proper projective family has complete symplectic transport. Its residue-phase homotopy, prequantum connection and transported brane data retain the source symplectic scaling, gradings, spin and rational action-line convention, as constructed in [F2, F5]. Transport the resulting compact equivalence to the given X. The coefficient automorphism remains the constructed unknown psi; no Picard–Fuchs identification or broader Calabi–Yau assertion is used. This proves precisely the original Statement.