ChronoMembrane H; Geometric Sufficient Conditions for Quantum Polynomial Time on ChronoMembrane

This paper establishes a geometric sufficient framework for the quantum polynomial
time class BQP on compact spin manifolds. The lattice skeleton carries legal
computation paths, the fermionic Fock space on the lattice spin fibres carries the
quantum register, and the Jordan–Wigner correspondence maps fermionic modes to
qubits. The core observation is that the Jordan–Wigner non-locality of single-mode
fermion operators degenerates into constant-depth circuit blocks under edge-local
couplings on the skeleton, thereby preserving compatibility with the locality of the
standard quantum circuit model. On this basis, we establish quantitative rela
tions among three geometric control quantities: a positive scalar curvature lower
bound fixes the Dirac spectral gap via the Friedrich constant; the spin connection
curvature, as an infinitesimal quantum gate, determines the universality of the
gate set through its Lie algebra generation property; the Dirac spectral gap and
bounded curvature together control the resource consumption of adiabatic initial
state preparation. Furthermore, under the spin condition, polynomial spectral
gap, polynomial transmission depth, holonomic universality, readout resolution,
and a compilation axiom, we prove that the corresponding decision problem be
longs to BQP. The converse direction proves that any polynomial-size quantum
circuit family can be constructively embedded as a family of spin computation
manifolds satisfying the above conditions. Quantum phase estimation is explicitly
geometrically realised as a positive example, while Grover’s unstructured search
is excluded by the structural constraint of the polynomial depth condition. The
paper does not assume P= BQP, does not assume NP ⊂ BQP, and does not claim
any new complexity class separation.