We present a rigorous head-to-head analytical comparison between the KAN-DAM (KolmogorovArnold Network + Dense Associative Memory) architecture operating on a xed nite spectral manifold (N = 861) and conventional tensor-network (TN) methods. Tensor networks suer exponential scaling with bond dimension due to volume-law entanglement generation, and their exact contraction in 2D/3D is strictly #P-hard, forcing systematic topological destruction via Singular Value Decomposition (SVD) truncation. Conversely, KAN-DAMmaintains a xed dimensionality N, achieves a bounded per-step complexity of O(N2) and total cost O(N2T), and suppresses non-physical errors globally via Dense Associative Memory geometric projection rather than truncation. The Kolmogorov-Arnold representation guarantees dense functional expressivity without requiring explicit, localized entanglement management. This establishes KAN-DAM as a bounded-cost, geometrically stabilized computational paradigm ideally suited for non-perturbative problems whose physics is constrained to a low-dimensional topological surface.
We present a rigorous head-to-head analytical comparison between the KAN-DAM (KolmogorovArnold Network + Dense Associative Memory) architecture operating on a xed nite spectral manifold (N = 861) and conventional tensor-network (TN) methods. Tensor networks suer exponential scaling with bond dimension due to volume-law entanglement generation, and their exact contraction in 2D/3D is strictly #P-hard, forcing systematic topological destruction via Singular Value Decomposition (SVD) truncation. Conversely, KAN-DAMmaintains a xed dimensionality N, achieves a bounded per-step complexity of O(N2) and total cost O(N2T), and suppresses non-physical errors globally via Dense Associative Memory geometric projection rather than truncation. The Kolmogorov-Arnold representation guarantees dense functional expressivity without requiring explicit, localized entanglement management. This establishes KAN-DAM as a bounded-cost, geometrically stabilized computational paradigm ideally suited for non-perturbative problems whose physics is constrained to a low-dimensional topological surface.
The KAN-DAM computing architecture establishes a deterministic quantum-computational model hosted on a finite 861-node spectral manifold, replacing isolated physical qubits and discrete gate sequences with globally distributed, non-local state representations. Information processing advances through an alternating cycle of linear functional mode mixing, non-linear dense associative coupling, and immediate geometric projection back onto the constrained manifold. The system integrates global Chebyshev spectral polynomial expansions with localized adaptive spline representations, coupling macro-level coherence with localized gradient absorption to eliminate oscillatory instabilities. The non-linear stage generates many-body interactions and effective entanglement in a single transformation without requiring deep sequences of two-qubit operations. Error suppression operates as an automatic geometric filter rather than an active algorithmic routine: perturbations that drift outside the rigid manifold constraints are discarded by the projection step, eliminating the need for ancilla qubits or syndrome measurements. This design circumvents the exponential entanglement scaling limits of classical tensor networks and the critical gap-closing slowdowns of adiabatic quantum annealing, providing an inherently stable platform that can be simulated classically or implemented across photonic, superconducting, and cold-atom hardware.
We present a unied architecture combining KolmogorovArnold Networks and Dense Associative Memory on discrete spectral graphs, in which computation is conned to a protected low-frequency subspace by a rigid spectral projectorreplacing active gate-based error correction with passive, hardware-enforced mode suppression. Four results carry the framework. (I)The graph Laplacian generates the native unitary evolution U = e−iL∆t, and an orthogonal spectral projector P=Pk≤M |φk⟩⟨φk| truncates the Hilbert space to the protected span. (II)The Projection Stability Theorem: under xn+1 = P F(xn) for any nonlinearity F, the dynamicsremain conned with exactly zero leakageby idempotency, (I −P)xn+1 = (I −P)P F(xn) = 0,an algebraic identity, not an approximation. (III) Graph-KAN layers place trainable B-splinefunctions on the edges rather than static weights on the nodes, and their local support suppliesa decisive inductive bias for localized graph signals. (IV) DAM dynamics with a bounded-belowenergy functional give dE/dt ≤ 0 and unconditional convergence to a topological attractor.Version 2 replaces the arithmetic rationale for the projector rank with the correct and strongerprinciple: the rank must sit at a spectral gap, because eigenvalue separationand nothing elsecontrols perturbative mode mixing. We prove the Perturbative Firewall Theorem, which bounds the leakage of the projected dynamics under perturbation V by ∥(I − P)δx∥ ≤ ∥V ∥/∆ with ∆ the gap at the cut, converting the previously sampled stability into a theorem with a computable constant; and we exhibit the exact gap structure of the 861-node triangular Laplacian, whose protected cuts in the working region are M = 62, 110, 126, 143, 159, 215, 237, 256, with M = 215 (∆ = 0.094) the natural rewall for a quarter-spectrum truncation. Hardwarerealizations photonic bandgap, microwave-ltered superconducting, Rydberg-dressed neutral atomimplement the projector as a physical mode cuto placed at the gap.
Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for two-dimensional systems and to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. Surprisingly, it turns out that, for large systems in D>1 spatial dimensions, MPS simulations of low-energy states are nevertheless more efficient than TTNS simulations. With a focus on D=2 and 3, the scaling of computational costs for different boundary conditions is determined under the assumption that the system obeys an entanglement (log-)area law, implying that bond dimensions scale exponentially in the surface area of the associated subsystems.
We present an explicit algorithmic realization of quantum computation on a finitespectral manifold of dimension N = 861 using the KAN–DAM architecture. Logical states areencoded nonlocally across all modes, and evolution proceeds through repeated nonlineartransformations followed by a deterministic projection onto the constrained manifold. Thismechanism supplies intrinsic error suppression and state-independent dynamics withoutrequiring external error-correcting codes or precise gate calibration. We detail the update rule,prove universality through functional approximation and nonlinear coupling, compare theapproach to tensor-network methods and adiabatic quantum computing, and provide a minimalsimulation protocol that can be implemented immediately on classical hardware. The result is acomputational paradigm in which logic, error correction, and thermodynamic consistency allemerge naturally from projection onto a finite-capacity spectral structure.
We present a complete cryptographic trapdoor primitive constructed by embeddingKolmogorov–Arnold Networks (KAN) with Dense Associative Memory (DAM) layersonto the DSM-861 Deterministic Spectral Manifold — a rigid triangular-lattice 3-toruswith exactly N = 861 nodes, 72-fold gear-lock symmetry, topological defect δ = -1/24,and intrinsic Casimir residue |ε| = 1/12 (derived via Euler-Maclaurin regularization ofδ). The private key is an exact initial spectral signature ψ0. Forward trapdoor evaluation (private key → public key) is an exact algebraic spectral mapping of the Kolmogorov–Arnold decomposition executed via discrete Lax pairs satisfying the zero-curvature condition [∂/∂t − A, L] = 0. The Dense Associative Memory layerresolves superpositions deterministically. Inversion is rendered computationallyintractable by the non-periodic, non-Abelian geometry of the closed manifold,topological phase locking, and intrinsic Casimir stabilization. This constructiondelivers deterministic, drift-free signature generation and verification with zeroprobabilistic failure. We provide full mathematical definitions, explicit constructions,rigorous security reductions, quantitative resource comparisons with Babbush et al.(2026), implementation considerations, and integration with intrinsic topologicalquantum error correction. The primitive offers a practical, quantum-resistantalternative to lattice-based post-quantum cryptography with dramatically reducedoverhead and no reliance on heuristic hardness assumptions.