{"id":"296949b1-f23a-4f1a-bae6-d2217b267cf3","arxiv_id":"2603.27063","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Wick rotation of p-adic hierarchical CNNs produces nonlinear p-adic Schrödinger QNNs with graph discretizations, local solutions, and simulations of open-system pulse response and habituation.","lead":"The paper builds quantum neural networks whose states solve p-adic Schrödinger equations obtained by Wick-rotating hierarchical cellular neural network dynamics. The construction yields graph discretizations, local existence proofs, and simulations of non-unitary evolution and habituation.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"The open-system (Lindblad-type) reading of (1.1) is only phenomenological; the paper never derives complete positivity or a Kraus/Lindblad generator from the nonlinear term.","rationale":"The Reader correctly isolates the unsupported Lindblad identification as the weakest assumption. The rest of the paper is internally consistent: the Wick rotation is formal, the discretization to graphs is standard, and Theorem 8.1 is a routine application of abstract semilinear evolution theory. Because the open-system reading is only phenomenological, the paper’s strongest claim is overstated, justifying the CONDITIONAL verdict already given. No stronger objection (e.g., mathematical contradiction or circularity) appears, so the verdict remains unchanged.","tokens_in":21933,"tokens_out":452,"duration_ms":5147,"concrete_test":"Take the finite-dimensional system (4.12) or (1.2) with a concrete W (e.g., the cat-cortex matrix of Simulation 3.2) and check whether the map ρ ↦ Ψ(t)Ψ(t)* generated by the nonlinear ODE is completely positive for small t. If the Choi matrix of the linearized generator has a negative eigenvalue, the Lindblad-type claim fails and the open-system interpretation must be withdrawn.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central interpretive claim (Introduction and §1) is that the Wick-rotated equation (1.1) with nonzero W is a Lindblad-type master equation for an open quantum network. The only evidence offered is that numerical solutions of the discretized system fail to conserve the L2-norm when W or Z is nonzero (Simulations 2–5). No microscopic system-bath Hamiltonian is written, no completely-positive map is exhibited, and the nonlinear integral term involving ϕ is not shown to arise from a Lindblad generator. Consequently the strongest claim that the new objects are genuine open QNNs rests on an unproven identification; the mathematical constructions (discretization to (1.2)/(5.2), local existence Theorem 8.1) remain valid as nonlinear Schrödinger equations, but the open-system narrative is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces a class of p-adic quantum neural networks whose states satisfy the nonlinear Schrödinger equation (1.1), obtained by Wick rotation of the state equations of the authors’ earlier p-adic cellular neural networks. The free part of the Hamiltonian is the convolution operator associated with a probability kernel J, so that the linear equation is a continuous-time quantum Markov chain; the nonlinear integral term involving a weight kernel W and an activation function ϕ is interpreted as an interaction that produces non-unitary evolution. Section 4 carries out an explicit discretization onto the finite group G_l = Z_p / p^l Z_p, yielding the matrix system (4.12). Specializing the kernel to the adjacency matrix of a simple graph produces the graph QNNs (1.2)/(5.2). The Appendix proves local (and, under a boundedness assumption on ϕ, global) existence of mild solutions in L^{2}(Z_p) by standard semigroup arguments. Extensive numerical experiments on trees of depth 6 illustrate unitary free evolution, non-conservation of the L^{2}-norm when W or Z is nonzero, and a habituation-like response when the interaction is taken from a p-adic approximation of the cat-cortex connectivity matrix.","tokens_in":22258,"tokens_out":957,"duration_ms":9391,"significance":"If the constructions are accepted, the work supplies a mathematically coherent bridge between hierarchical p-adic neural models, continuous-time quantum walks on graphs, and a family of nonlinear Schrödinger equations that can be simulated on ordinary computers. The discretization calculations are fully explicit, the free Hamiltonian is self-adjoint by Fourier analysis, and the local-existence theorem is standard once the nonlinearity is Lipschitz and bounded. These ingredients give a concrete, reproducible platform for exploring quantum analogues of Wilson–Cowan dynamics and for generating new continuous-time quantum walks with interaction terms. The open-system (Lindblad-type) reading remains interpretive rather than derived, yet the mathematical objects themselves—nonlinear p-adic Schrödinger equations, their graph discretizations, and the associated numerical phenomenology—are new and potentially useful for quantum-algorithm and quantum-cognition research.","major_comments":[{"comment":"Introduction and §1: the claim that (1.1) with W ≠ 0 is a “Lindblad-type master equation describing an open quantum network” is supported only by the numerical observation that ∥Ψ(·,t)∥₂ is not conserved (Simulations 2–5). No microscopic system-bath Hamiltonian, completely-positive map, or Kraus/Lindblad generator is exhibited. The mathematical constructions (discretization, existence) remain valid as nonlinear Schrödinger equations, but the open-system narrative should either be derived or clearly labeled as a phenomenological interpretation.","section":null},{"comment":"§6 and Figures 6–15: the “habituation” interpretation of the decay of ∥Ψ∥₂ under constant or pulsed drive is suggestive but not quantified. No comparison with a classical Wilson–Cowan system, no definition of a habituation index, and no systematic scan of the free parameters (α, scale of W, pulse amplitudes) are provided. Without such controls the claim that the networks exhibit a biologically meaningful learning phenomenon remains anecdotal.","section":null}],"minor_comments":[{"comment":"Several figures (especially 5–15) contain garbled axis labels and missing units; the captions should state the precise values of p, l, α, W and the support of Z used in each panel.","section":null},{"comment":"The activation function ϕ(s) = ½(|s+1| + |s-1|) is introduced only in §6; it should be stated once in the general equation (1.1) or (3.3) so that the existence theory applies to the same nonlinearity used in the simulations.","section":null},{"comment":"Typographical inconsistencies appear in the author list (Z´U˜NIGA vs. Zúñiga) and in several equation references (e.g., “discretizations of (6.4)” while the displayed equation is (1.1)).","section":null},{"comment":"The forthcoming-work remark on traveling waves at the end of §1 is unnecessary in a research article and can be moved to the discussion.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a natural continuation of the authors’ earlier p-adic CNN and free Schrödinger papers; the novelty is real but incremental. The Lindblad claim is the only load-bearing interpretive over-reach; once it is softened the paper is publishable. Fit for a quant-ph or mathematical-physics journal is good; a pure neuroscience venue would require stronger biological validation."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real addition here is the nonlinear interaction term obtained by Wick-rotating the p-adic CNN potential, the clean discretization that produces QNNs on ordinary graphs, a standard local-existence theorem in L2, and the numerics that show non-unitary evolution plus habituation when W is nonzero. The free (W=0) case and the continuous-time quantum walks already sat in their earlier papers; this is a genuine, self-contained extension rather than a re-packaging.\n\nWhat they do well is explicit. Sections 4–5 walk through the discretization carefully: characteristic functions of balls, the averaged kernel, the resulting matrix J(l) that reduces to a scaled graph Laplacian when the adjacency matrix is plugged in. The free Hamiltonian is self-adjoint by the usual Fourier argument, so Stone gives the unitary group. The Appendix then treats the nonlinear term as a Lipschitz map on L2 (once φ is bounded and Lipschitz) and gets local mild solutions by the standard Cazenave–Haraux theory; when φ is also bounded they get global existence. That part is clean and correctly cited. The simulations are illustrative rather than exhaustive, but they consistently show norm growth or decay once W or Z is switched on, and the habituation patterns when the cat-cortex matrix is used are at least suggestive.\n\nThe soft spot is the interpretive claim that nonzero W turns (1.1) into a Lindblad-type master equation for an open quantum network. That is only phenomenological: they observe that the L2-norm is not conserved and stop there. No system-bath Hamiltonian, no completely-positive map, no Kraus or Lindblad generator is derived from the nonlinear integral. So the strongest “open QNN” language is unsupported; what remains is a well-defined family of nonlinear p-adic Schrödinger equations whose discretizations are ordinary ODEs on graphs. Free parameters (α, scale of W, pulse shapes of Z, tree depth) are numerous, and no code is shipped, so the numerics cannot be re-run. Those are real but secondary limitations; they do not break the math.\n\nThis is for people already working on p-adic models, continuous-time quantum walks, or quantum-like neural dynamics. A serious referee should see it. I would accept it for peer review and would cite the discretization and existence results if I needed hierarchical QNNs on graphs.","headline":"Solid math extension of the authors’ p-adic CNN/Schrödinger line; the Lindblad reading is only phenomenological, but the constructions themselves hold up.","tokens_in":22847,"tokens_out":609,"would_cite":true,"duration_ms":7943,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q35","35Q55","68T07","11S80"],"pacs":["03.67.Lx","87.19.lj","05.45.-a"],"model":"grok-4.5","headline":"New quantum neural networks arise by Wick-rotating p-adic cellular neural networks into nonlinear Schrödinger equations whose discretizations live on graphs and show open-system dynamics.","keywords":["p-adic quantum neural networks","Wick rotation","cellular neural networks","continuous-time quantum walks","hierarchical neural networks","nonlinear Schrödinger equation","open quantum systems","Wilson-Cowan model"],"falsifier":"Derive (or rigorously disprove) that the nonlinear evolution generated by equation (1.1) with nonzero W is completely positive and trace-preserving for the associated density operator; alternatively, exhibit a microscopic system-bath model whose reduced dynamics exactly recover (1.1).","tokens_in":22839,"feed_emoji":"⚛️","tokens_out":709,"duration_ms":6393,"temperature":0.7,"pith_summary":"The paper constructs a new class of quantum neural networks by taking continuous hierarchical cellular neural networks over the p-adic numbers, which are themselves continuous limits of discrete tree-like neural networks bio-inspired by the Wilson–Cowan model of large neural populations, and Wick-rotating their state equations. The resulting p-adic nonlinear Schrödinger equations have a free convolution term plus a nonlocal interaction potential that couples neurons. When that potential vanishes the evolution is unitary and recovers continuous-time quantum Markov chains and quantum walks; when it is present the evolution is non-unitary and the authors interpret the network as an open quantum system. Rigorous discretizations of the same equations produce finite quantum networks on ordinary simple graphs. Local existence of solutions in L² is proved, and extensive numerical experiments illustrate pattern formation, pulse response, and habituation once the interaction kernel is switched on.","feed_headline":"Wick rotation turns p-adic neural nets into quantum networks on graphs","feed_subtitle":"Nonlocal interaction makes the evolution non-unitary and produces habituation in simulations","key_machinery":"The p-adic nonlinear Schrödinger equation (1.1) (and its matrix discretization (1.2)/(4.12)/(5.2)) obtained by Wick-rotating a hierarchical cellular-neural-network state equation; the free convolution operator generates unitary quantum walks while the nonlocal potential W supplies the neuron-to-neuron coupling that drives non-unitary evolution.","core_discovery":"States of a new family of quantum neural networks are solutions of the p-adic nonlinear Schrödinger equation obtained by Wick rotation of the state equation of a p-adic cellular neural network; the free part generates continuous-time quantum Markov chains while a nonzero interaction kernel produces non-unitary evolution that the authors regard as open-system dynamics, and discretizations of the same equation yield concrete quantum networks on simple graphs for which local L² solutions exist.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Wick rotation turns p-adic CNNs into quantum neural nets on graphs","p-Adic Schrödinger equations yield hierarchical quantum neural networks","Nonlocal kernels make p-adic quantum nets non-unitary with habituation","Discretized p-adic Schrödinger equations build QNNs on simple graphs","Wick-rotated hierarchical CNNs produce open quantum neural dynamics"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The claim that the Wick-rotated equation with nonzero interaction is a Lindblad-type master equation for an open quantum network rests only on the observation that the L²-norm is not conserved in simulations, not on a derivation from a system-bath Hamiltonian or a completely-positive map.","fun_headline_variants_meta":{"raw":{"variants":["Wick rotation turns p-adic CNNs into quantum neural nets on graphs","p-Adic Schrödinger equations yield hierarchical quantum neural networks","Nonlocal kernels make p-adic quantum nets non-unitary with habituation","Discretized p-adic Schrödinger equations build QNNs on simple graphs","Wick-rotated hierarchical CNNs produce open quantum neural dynamics"]},"model":"grok-4.5","effort":"low","cost_usd":0.003642,"raw_usage":{"total_tokens":1089,"prompt_tokens":723,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":36420000,"prompt_tokens_details":{"text_tokens":723,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":289,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":723,"tokens_out":77,"duration_ms":4089,"temperature":1.0,"reasoning_tokens":289,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T17:13:39.532196+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Derive (or rigorously disprove) that the nonlinear evolution generated by equation (1.1) with nonzero W is completely positive and trace-preserving for the associated density operator; alternatively, exhibit a microscopic system-bath model whose reduced dynamics exactly recover (1.1).","supporting_citations":[],"review_version":1}