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.
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 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.
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.
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.
DSM–861 is best understood as a deterministic spectral computing architecture built from a rigid finite graph, a toroidal Hodge geometry, nonlinear KAN transport, vorton-inspired skew coupling, and associative-memory snapping. The arithmetic anchor is 861 = 42,2= T41 = 3 · 7 · 41, while the interaction topology is exactly J(42, 2) with degree 80 and spectrum {80, 38, −2}. The toroidal layer supplies a genuine discrete differential calculus. The finite-state theorem fixes the exact nine-qubit boundary of a single 861-dimensional core. The controllability theorem converts universal computation into a concrete Lie-algebra calculation. The snapping theorem supplies a mathematically correct route to non-accumulating numerical error through a deliberately constructed quadratic attractor.The resulting program is therefore executable as mathematics and engineering. Build the graph. Build the Hodge operators. Construct the control generators. Compute their Lie closure. Implement the nonlinear snap. Measure the ϵ2 law. Synthesize the FPGA. Then benchmark complete circuits. If those tests succeed, DSM–861 becomes a demonstrated deterministic spectral processor rather than merely a conceptual analogy to quantum computing.
This addendum provides targeted technical extensions and mathematical clarifications to themain monograph “Deterministic Spectral Manifold for Post-Quantum Cryptography” (ResearchGate, 2026) without altering its core structural results. Five core claims are updated with June 2026 technical precision to support rigorous intellectual property auditing. Section 1 formalizes the UMetric (arithmetic firewall) and corrects the characterization of ε0 = |δ| = 1/24 as a configurable system sensitivity parameter rather than an absolute topological lower bound. Section 2 evaluates the 220-dimensional protected subspace: the nf −nb = 220 mode count is explicitly reconciled, and the Grover security level is constrained to 2110, identifying the precise structural adjustments required for absolute NIST Level 1 compliance. Section 3 reframes the 72-phase entropy harvester, demonstrating that the structural δ = −1/24 defect bounds the deterministic grid while physical thermal jitter across boundary transitions provides the true randomness. Section 4 replaces the probabilistic “zero-knowledge” framing with a mathematically rigorous “computationally private projection”protocol. Section 5 scales back the side-channel evaluation from “immune” to “reduced attack surface,” detailing the benefits of fixed-N computational metrics. All extensions derive directly from the rigid DSM-861 baseline: Heegner discriminant D = −163, projection matrix P, and the 72-fold synchronization clock.
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.