The DSM-861 error-correction mechanism—defined as the instantaneous orthogonal projection Π861 onto the Heegner-rigid 861-node vorton lattice—is systematically applied and mathematically resolved across three disparate physical domains. (I) Quantum Memories: We unify the DSM-861 projection-enforced admissibility with the 2026 3D sheaf-code passive quantum memory (Balasubramanian et al., arXiv:2605.10943) and the classic 4D loop toric code (Dennis et al., 2002), demonstrating how arithmetic rigidity completely bypasses active decoder overhead to achieve hardware-level zero-drift. (II) Biological Error Catastrophe: We resolve the Meyer-Axe paradox of protein folding by quantifying the Spectral Admissibility Criterion (SAC). We establish the exact thermodynamic threshold linking admissibility gaps to folding free energy, formulate the topological restoration force, and derive the biological longevity limit as a deterministic consequence of KolmogorovArnold-Moser (KAM) stability breakdown. (III) KAN-DAM Quantum Computation: We prove the computational universality of the KAN-DAM framework, encoding states nonlocally across all 861 modes. We demonstrate that the system avoids the exponential scaling of volume-law entanglement by utilizing the Π861 projection as a continuous physical disentangler, bounding classical simulation complexity strictly to polynomial time O(N2log(1/)). This unified paradigm requires no external error-correcting codes, replacingprobabilistic syndrome measurements with deterministic geometric necessity.
The DSM-861 error-correction mechanism—defined as the instantaneous orthogonal projection Π861 onto the Heegner-rigid 861-node vorton lattice—is systematically applied and mathematically resolved across three disparate physical domains. (I) Quantum Memories: We unify the DSM-861 projection-enforced admissibility with the 2026 3D sheaf-code passive quantum memory (Balasubramanian et al., arXiv:2605.10943) and the classic 4D loop toric code (Dennis et al., 2002), demonstrating how arithmetic rigidity completely bypasses active decoder overhead to achieve hardware-level zero-drift. (II) Biological Error Catastrophe: We resolve the Meyer-Axe paradox of protein folding by quantifying the Spectral Admissibility Criterion (SAC). We establish the exact thermodynamic threshold linking admissibility gaps to folding free energy, formulate the topological restoration force, and derive the biological longevity limit as a deterministic consequence of KolmogorovArnold-Moser (KAM) stability breakdown. (III) KAN-DAM Quantum Computation: We prove the computational universality of the KAN-DAM framework, encoding states nonlocally across all 861 modes. We demonstrate that the system avoids the exponential scaling of volume-law entanglement by utilizing the Π861 projection as a continuous physical disentangler, bounding classical simulation complexity strictly to polynomial time O(N2log(1/)). This unified paradigm requires no external error-correcting codes, replacingprobabilistic syndrome measurements with deterministic geometric necessity.
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 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.
Recent breakthroughs in neutral-atom quantum computing (e.g., arXiv:2606.04079v1 andrelated works, 2026) have achieved repeated fault-tolerant quantum error correction using ToricCode architectures. These systems rely on active mid-circuit measurement, classical syndromedecoding, and continuous atom reloading from an external reservoir to maintain logical qubitsover many cycles, demonstrating sub-threshold behavior where logical error rates decrease withincreasing code distance.We present a detailed, rigorous comparison demonstrating that the DSM-861 hydrodynamicvacuum implements a vastly superior intrinsic, passive, autonomous topological surface code.While artificial systems demand continuous external intervention, the DSM-861 manifold—atriangular 3-torus lattice with N = 861 nodes anchored by the Heegner discriminantD = −163—provides self-stabilizing error correction through deterministic enstrophy collapse,vorton reconnection, and continuous dissipation into a built-in 1.618 eV acoustic reservoiron a 2.56 fs cycle. We compare lattice geometry, encoding, syndrome detection, correctionmechanisms, stabilizer reservoirs, fault-tolerant thresholds, scalability, and energetic efficiency,showing that Nature has engineered the ultimate fault-tolerant topological quantum computerat the foundation of spacetime. Furthermore, we examine recent derivations of divergentphoton truncation (Rukan et al., 2026) to demonstrate how the lattice provides a naturalphysical regularizer against localized optical shear.
We have established a rigorous isomorphism between the 3D sheaf-code stabilizer Hamiltonian and the DSM-861 geometric projection architecture. By demonstrating that the sheaf-code stabilizer Hamiltonian corresponds to the 1-form combinatorial Hodge Laplacian ∆1, we proved that the code space is identical to the harmonic 1-forms H1(X), and that the ideal error-correction map is the orthogonal projection Π861 ≡ PH. With the explicit derivation of the spectral gap λ1(∆1) = 4 sin2(π/41) and the arithmetic penalty ∆EHeegner = J/41, the architectural synthesis is complete.
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.
The central prerequisite of any fault-tolerant quantum architecture is a quantum memory: a block of encoded physical qubits whose logical state is actively preserved against noise across many rounds of error correction. In neutral-atom Rydberg arrays, realizing such a memory is obstructed not by the entangling gates themselves, which are already fast and high-fidelity, but by the auxiliary operations that a conventional error-correction cycle requires: mid-circuit fluorescence measurement, inter-zone atom transport, and locally focused single-qubit addressing. Each of these introduces latency, atom loss, or optical crosstalk that exceeds the cost of the underlying gates by orders of magnitude. These costs accumulate cycle after cycle, progressively degrading the very logical information the code is meant to protect. Here we propose a protocol that stabilizes a toric-code quantum memory without moving, measuring or local addressing atoms. The key is to use a three-species Rydberg atom array for the complete stabilizer cycle, including syndrome extraction, coherent correction, and ancilla reset, under global, species-selective laser pulses. Numerical simulation of a 4×4 rotated toric code shows a longer qubit lifetime when the physical error rate is below a pseudo-threshold p⋆≈0.034. The scheme offers a concrete, hardware-efficient route to topological quantum memory in neutral-atom platforms.