In the Fock–Hecke prime-address model, PUSH and POP first appear as algebraic operations. PUSH inserts a prime into a noncommutative address word, while POP tests and removes a prime from that address. In the Fock-space realisation, these become creation and annihilation operators. However, when the same structure is interpreted through complex differential geometry, PUSH and POP acquire a geometric meaning. A PUSH operation may be interpreted as a holomorphic thickening of the object. It adds a new prime-address direction to the state, increasing its local geometric complexity. In complex-geometric language, PUSH behaves like a deformation that moves an object into a larger holomorphic configuration space. It is similar to adding a new local branch, section, or divisor-like direction to the object. A POP operation is the dual process. It tests whether a given prime-address direction is present and then contracts, projects, or removes that direction. Geometrically, POP behaves like a residue, contraction, or adjoint projection. It asks whether the object contains a certain holomorphic component and, if so, extracts or collapses it back to a lower-complexity state. Thus, in complex differential geometry, PUSH and POP can be read as a pair of dual operations: PUSH creates a holomorphic address direction. POP detects and contracts a holomorphic address direction. Inside a Calabi–Yau register, this pair must be compatible with the Ricci-flat metric. This is the key point. A PUSH operation may deform the object, and a POP operation may collapse part of the object, but the resulting object must remain inside the Yau-readable geometric window. That means the deformation should not escape the controlled field of view, should not cause metric collapse, and should not create uncontrolled caustics or singularities. This matches the paper’s interpretation of Yau’s estimates as lens-resolution and caustic-control bounds. From the Yoneda-lens viewpoint, PUSH and POP are not merely internal operations on an address word. They change how the object is seen by all test objects. A PUSH changes the object’s representable functor by adding a new visible direction. A POP tests whether that direction is actually visible and removable. Therefore, the PUSH/POP pair becomes a geometric mechanism for controlling how prime-address information appears under the Yoneda projection. The Ricci-flat condition then imposes a zero-bias constraint. The lens should not systematically favour the PUSH branch over the POP branch, or the POP branch over the PUSH branch. In complex-geometric terms, this means that the holomorphic expansion direction and the dual contraction direction must balance at the level of Yoneda-visible refractive distortion. This is why the analytically normalised Fock–Hecke operator has the form of a balanced creation–annihilation pair. The PUSH coefficient and the POP coefficient are dual under the transformation from one side of the critical strip to the other. Requiring zero refractive bias forces these two branches to have equal magnitude. That balance selects the critical line as the self-dual axis.
Jeong Min Yeon (Thu,) studied this question.