Field-Theoretic Computer: A Novel Computing Paradigm Based on Continuous Field Theory and Null-Operators with Optical Quadrature Vectors

This is a preprint (not peer-reviewed). Corresponding author: Shuntian Deng, mm.stdeng23@gzu.edu.cn. [Objective] To address the discretization error and the ontological fragmentation faced by the traditional Turing machine and von Neumann architecture, this paper proposes a novel computing paradigm — the field-theoretic computer — based on continuous field theory and null-operators. The paradigm takes "field is matter" as its axiomatic starting point and, within the fiber-bundle framework, redefines the wave function as a field-theoretic section. [Methods] The basic computational unit consists of an XY plane and a Z axis. On the XY plane, field A and field B are coupled orthogonally, and the mismatch of their symmetric and antisymmetric structures generates the null-operator a_ij = 0. Field C projects the null-operator along the Z axis onto a storage plate, where a second complement duality forms the stored photon quantum. What the null-operator carries is not a numerical value but the topology of a complement — the hole structure of the complement of the computed object inside a primitive cell of phase space. Complement duality is an involution, so the identical operation serves both inter-layer stacking and storage delivery. The probability measure is induced by the geometry of the section, and in the quadratic case its marginal distribution reproduces Born statistics; probability is therefore not a separate postulate, and the four premises of the Copenhagen interpretation are not mutually independent assumptions but four separate consequences of the single condition that the defining function of the section be of degree two. Stacked extension is realized by tensor-product stacking and contraction readout. The null-operator acts as a topological logic gate, so computation is intrinsically bidirectional and reversible and enjoys topological protection against decoherence. For physical implementation, continuous-variable quantum computing (CVQC) is adopted: the symplectic phase space and the Wigner function describe geometric storage, Gaussian operations correspond to single-layer computation, the nullifiers of CV cluster states realize the null-operator, and non-Gaussian operations (the cubic phase gate) read out the topological information. [Results] The complement count of a single layer is β0(Γ∖R) = β1(R) + 1, and stacking n layers gives a logical capacity n(h+1), where h is the number of holes contained in a single-layer region. The capacity is an integer and is invariant under symplectic transformations, so its protection does not rely on error-correcting codes; capacity is preserved under inter-layer coupling if and only if that coupling is non-degenerate; the per-layer fault-tolerance threshold reduces formally to a critical condition on the determinant of the covariance matrix. The computational stages are linked by a six-stage temporal chain whose realizability requires a bounded activation window for field C — this is the dynamical reason why field C must be of a different kind from fields A and B. The capacity and threshold structure has been tested at model level by a lattice simulation: within the parameter window fixed by the ordering condition, the single-hole system exhibits a two-level staircase (2→1→0) while the simply connected system exhibits a single step (1→0), with threshold positions matching the unified-caliber closed form. [Limitations] The threshold expressions are model-level analytical results whose coefficients have been calibrated by the lattice simulation under a unified caliber, but the relation between the area criterion and the semi-axis criterion remains undetermined. The closure mechanism of the inner hole must be stated explicitly — an expanding fill front is adopted here — and the existence of the inner-hole level depends on the parameter regime. Mainstream implementation of non-Gaussian operations remains difficult; the engineering of the cubic phase gate and of gate teleportation is still to be studied; cases with boundary degree higher than quadratic are not developed here; and the bidirectional coupling and calibration layer required for a real-time twin remain to be built. [Conclusion] This paradigm is complementary to the von Neumann architecture and is suited to continuous-field simulation and reversible computation. For the three long-standing obstacles of physical computation — no memory, no iteration, no error correction — the present architecture provides three corresponding mechanisms: hole-structure storage, involutive reversible read/write, and protection by symplectic invariance. It therefore offers recyclable physical computation rather than a one-off physical demonstration. Three directions follow: field-computing-field, field twins, and field-theoretic artificial intelligence.