{"id":"e25d939d-d909-4ff1-8adf-a81863139d93","arxiv_id":"2507.00461","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Two new activation functions quantize magnitude and phase in complex-valued Hopfield networks, enlarging the state space and showing empirical convergence in small trials.","lead":"The paper defines two complex-valued Hopfield neural networks that quantize both phase and magnitude, creating far larger discrete state spaces than phase-only or magnitude-only models. It shows small numerical trials where energy decreases and states converge, but the convergence guarantee remains a conjecture, not a proof.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence claim for CoCeil- and CoSign-CvHNNs is unproved; Eq. (16) is not a valid real Lyapunov energy for Hermitian W, so the experiments do not establish stability.","rationale":"The architectures and state-count calculations are sound: coceilQ,R and CoSignQ,R,K are well-defined, their image sets have (Q+1)^2 and QK states per neuron, and the paper correctly notes that these exceed previous 4- or K-state models. The problem is that the headline benefit, stable associative memories with more states, depends on a convergence theorem, and none is provided. Section IV states the weight condition as a conjecture, and the empirical check (N = 10, five initial states, W_ii = 0, one parameter setting) is far too small to establish a universal statement. More importantly, the proposed Lyapunov function Eq. (16) is not real-valued for the complex states and Hermitian weights used in the experiments; a concrete example gives E = -2i, so monotone decrease is undefined. The known theorems for csign and split-sign are proved with real-valued energies (typically using conjugates or a real-coordinate decomposition), and transferring them to the new activations requires an analogous argument, which is absent. This supports the reader's CONDITIONAL verdict: the definitions and state counts are a valid contribution, but the convergence/memorization claim needs a proof, a counterexample, or substantially narrower claims plus more decisive experiments. I agree with the reader's weakest-assumption analysis and would not change the verdict.","tokens_in":9744,"tokens_out":15827,"duration_ms":192452,"concrete_test":"Enumerate all Hermitian weight matrices with entries in {-1,0,1} + i{-1,0,1} and diagonal in {0,1} for N = 2, and serially update CoCeil- and CoSign-CvHNNs (Q = 3, R = 2, K = 4) from every initial state in their image sets, keeping a neuron fixed when its argument lands on an undefined phase boundary; run each trajectory for up to 100N updates and check whether every one reaches a fixed point. If any run enters a cycle of length greater than 1, the conjecture is false; if none does, the missing theorem remains unproved but the conjecture gains support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that both new networks converge to stable states under Hermitian weights with nonnegative real diagonal, enabling a larger associative memory. Section IV explicitly states that no convergence theorems are proven and proposes this as a conjecture. The only theoretical bridge offered is the energy E(S) = -1/2 sum_i sum_j S_i W_ij S_j in Eq. (16), said to be the same energy used for Theorems 1 and 2. But for complex states and Hermitian W, this expression is not generally real-valued: with W_12 = 1+i, W_21 = 1-i, W_ii = 0, and S = (1+i, 1+i), we get E = -2i, so 'decreasing energy' is not even defined. Valid Lyapunov functions for the known complex-signum and split-sign theorems rely on conjugates or a real-coordinate decomposition; no analogous function is supplied for the new activations. The experiments also set W_ii = 0 (Eq. 13), test only N = 10 with five random initial states and one parameter setting, so they do not cover the conjectured condition or rule out cycles. Thus the memorization claim rests on an unproved conjecture and an invalid-looking energy, not on a demonstrated argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two complex-valued Hopfield neural networks, CoCeil-CvHNN and CoSign-CvHNN, which quantize both magnitude and phase through ceiling-type activation functions in Cartesian and polar coordinates, respectively. The authors claim that these networks increase the number of possible states compared to existing CvHNNs and conjecture that, under Hermitian synaptic weights with nonnegative real diagonal, they converge to stable states. Section IV presents an energy function in Eq. (16) and computational experiments with five random initial states and a single parameter setting to support the conjecture. The paper also proposes that the increased state space enables these networks to memorize more stable states than previous models.","tokens_in":10038,"tokens_out":6321,"duration_ms":68728,"significance":"The proposed activation functions are well-defined and the state-space counts, (Q+1)^(2N) for CoCeil-CvHNN and (QK)^N for CoSign-CvHNN, are correct and represent a genuine enlargement of the discrete state spaces relative to the complex-signum and split-sign models. If the convergence conjecture were established, these models could be useful extensions for complex-valued associative memory. The paper is clearly written and honestly states that no convergence theorems are proved. However, the theoretical bridge offered, the energy function in Eq. (16), is not generally real-valued for complex states, and the empirical evidence is far too limited to confirm the conjecture. Thus the central claim remains unsupported.","major_comments":[{"comment":"The energy function E(S) = -1/2 Σ_i Σ_j S_i W_ij S_j is not real-valued for Hermitian W and complex states. For example, with N=2, W_11=W_22=0, W_12=1+i, W_21=1-i, and S=(1+i,1+i), one obtains E = -2i. Since a Lyapunov function must be real-valued and bounded below, Eq. (16) cannot serve as an energy function for the new activation functions, and the statement that 'this energy is also used to prove Theorems 1 and 2' is incorrect in this setting.","section":"Section IV, Eq. (16)"},{"comment":"The experiments set W_ii=0 for all i, so they do not test the conjectured condition W_ii ≥ 0 with nonzero diagonal entries. Moreover, the experiments use only N=10, five random initial states, and one parameter setting (Q=3, R=2, K=4). This sample is too small to rule out cycles or non-convergent trajectories, and it cannot 'confirm' a conjecture about the dynamics for general Hermitian weights with nonnegative real diagonal.","section":"Section IV, Eq. (13) and experiments"},{"comment":"The claim that the new networks 'can memorize more stable states' is not supported by any storage or retrieval experiment. The paper does not store patterns, measure the number of stable states, or evaluate retrieval success; the energy curves in Figure 5 only show that the particular random initial states reach a stationary point. Associative memory capacity requires a demonstration that stored patterns are stable and recoverable from noisy inputs.","section":"Section IV"},{"comment":"The conjecture is based on Theorems 1 and 2, but those theorems apply to activation functions with unit-magnitude states (complex signum) or split-sign states on the hypercube {±1±i}. The new activation functions produce different state spaces: CoCeil outputs nonnegative integer coordinates and CoSign outputs states with variable magnitude. The analogy to the cited theorems is therefore not automatic, and no proof or sufficient numerical study is provided to bridge this gap.","section":"Section IV, conjecture paragraph"},{"comment":"The text states that both networks 'reached a stationary state when employing synchronous update mode,' but the caption of Figure 5 says 'serial update mode.' This discrepancy should be resolved, as the convergence behavior can differ between serial and parallel update modes.","section":"Section IV and Figure 5"}],"minor_comments":[{"comment":"The phrase 'θK = π/K is know as phase quanta' contains a grammatical error; it should read 'is known as the phase quantum.'","section":"Section II-B"},{"comment":"The domain notation 'D ∈ C' should be 'D ⊂ C' to indicate that the activation function's domain is a subset of the complex plane.","section":"Section II-A, Eq. (2)"},{"comment":"The parameter ordering in the caption, 'CoSignQ,K,R', is inconsistent with the definition CoSignQ,R,K used in Eq. (11) and the surrounding text; this should be unified.","section":"Section III-B, Figure 4 caption"},{"comment":"The initial states for CoCeil are obtained by applying coceilQ,R to samples from U(-3,7), but the activation function maps to {0,...,Q}; clarify whether the initial states are always elements of the image set (they are, by construction), and avoid the impression that states outside the image set are used.","section":"Section IV"},{"comment":"Reference [47] is cited as 'submitted for publication'; if the dynamics discussion relies on this work, a preprint or a published version should be made available to the reader.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's main constructive contribution is the two activation functions and the observation that their state spaces are larger. However, the theoretical claim of convergence is supported neither by a valid Lyapunov function nor by adequate experiments, and the associative-memory claim is untested. The authors should either provide a convergence proof (for example, via a real-coordinate decomposition of the dynamics) or substantially expand the experimental study with multiple parameter settings, more initial states, and explicit storage/retrieval tests. The reliance on an unpublished reference [47] is also a concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe short version: the paper defines two new complex-valued Hopfield activations that combine phase and magnitude quantization, and the state-space counting is correct. What it does not do is prove the convergence needed for associative memory. The authors admit the proof is missing, but then the experiments are too thin to carry the empirical load, and the energy function they plot (Eq. 16) is not even real-valued in general.\n\nThe genuinely new piece is the pair of activation functions: coceilQ,R, a split ceiling on Cartesian coordinates, and CoSignQ,R,K, a ceiling on the magnitude followed by phase quantization. The formulas are clean, the image sets are precisely characterized, and the counts (Q+1)^(2N) and (QK)^N are right. The paper is also honest about what is missing: it says plainly that there is no convergence theorem and states the Hermitian-plus-nonnegative-diagonal condition as a conjecture. That is the right way to write up a preliminary result.\n\nThe soft spot is the theoretical bridge. The paper invokes Eq. (16) as the energy used in Theorems 1 and 2, but that energy, E(S) = -1/2 sum_i sum_j S_i W_ij S_j, is not real-valued for complex states and Hermitian W. For a concrete example, take W_12 = 1+i, W_21 = 1-i, W_ii=0, and S = (1+i, 1+i). Then E = -2i. So 'decreasing energy' is ill-defined. The known convergence proofs for the complex signum and split-sign networks use a real-valued energy with conjugates (or a real-coordinate decomposition); no such function is supplied for the new activations. The experiments do not fill the gap: N=10, five random starts, one parameter setting, and the text says the runs are serial mode in one place and synchronous mode in another. There is also no actual associative-memory test—no stored patterns, no recall experiment—so the claim that the larger state space lets the networks \"memorize more stable states\" is not directly demonstrated.\n\nThis is a modest but real contribution that is not ready in its current form. I would send it to review, because the definitions and the conjecture are worth putting on record, but I would expect the referee to demand either a convergence proof (or a counterexample), or a substantial narrowing of the claims to what the experiments support.","headline":"New phase-and-magnitude quantized Hopfield activations with honest conjectures, but no convergence proof and an energy function that isn't real-valued; worth refereeing but not citable yet.","tokens_in":10522,"tokens_out":2951,"would_cite":false,"duration_ms":32160,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"By quantizing both phase and magnitude at each neuron, two new complex-valued Hopfield networks obtain state spaces of $(Q+1)^{2N}$ and $(QK)^N$, and experiments confirm that they converge to stable states under Hermitian weights.","keywords":["complex-valued Hopfield neural networks","phase quantization","magnitude quantization","ceiling activation function","associative memory","complex signum function","state space","convergence conjecture"],"falsifier":"Simulate a small CoCeil- or CoSign-CvHNN, say $N=2$ or $3$ with $Q=2$, $R=1$, $K=4$, using a weight matrix with $W_{ij}=\\overline{W_{ji}}$ and positive real diagonal entries such as $W_{11}=1$; if the value of $E(S)$ in (16) is ever non-real or increases between consecutive serial updates, or if the network enters a cycle of length greater than one from some initial state, the conjecture is false.","tokens_in":9531,"feed_emoji":"🧠","tokens_out":9004,"duration_ms":90917,"temperature":0.7,"pith_summary":"Two new complex-valued Hopfield neural networks are proposed, differing only in their activation functions: one applies a ceiling quantizer separately to the real and imaginary parts of the net input (CoCeil-CvHNN), and the other quantizes magnitude and phase in polar coordinates (CoSign-CvHNN). The paper's central claim is that this phase-and-magnitude quantization multiplies the number of states a Hopfield network can in principle store, from $4^N$ to $(Q+1)^{2N}$ in the Cartesian model and from $K^N$ to $(QK)^N$ in the polar model. Since a Hopfield network is only useful as associative memory if it settles, the paper further conjectures that both networks converge under the same weight conditions used for earlier models, namely $W_{ij}=\\overline{W_{ji}}$ with real nonnegative diagonal entries. The experiments with random weights and initial states show the energy $E(S)=-\\frac{1}{2}\\sum_{i,j} S_i W_{ij} S_j$ decreasing to a stationary value, which the authors read as empirical confirmation of the conjecture. A sympathetic reader cares because, if the conjecture holds, these networks provide a straightforward way to enlarge the memory capacity of complex-valued associative memories without changing the number of neurons.","feed_headline":"Phase-magnitude quantization multiplies Hopfield memory states","feed_subtitle":"Ceiling activations expand the state space of complex-valued associative memories, and experiments show the networks settle.","key_machinery":"The load-bearing objects are the two quantizing activation functions. $\\operatorname{ceil}_{Q,R}$ is a ceiling-type step quantizer that maps a real number to one of $Q+1$ integer levels; applied coordinate-wise it gives $\\operatorname{coceil}_{Q,R}$, whose image has $(Q+1)^2$ complex states, and applied to the magnitude before the phase quantizer $\\operatorname{csign}_K$ it gives $\\operatorname{CoSign}_{Q,R,K}$, whose image has $QK$ states. The convergence argument is carried, by conjecture, by the same quadratic Lyapunov energy $E(S)=-\\frac{1}{2}\\sum_{i,j}S_i W_{ij}S_j$ used in the convergence proofs of the earlier sign-based networks, together with the weight condition $W_{ij}=\\overline{W_{ji}}$, $W_{ii}\\ge 0$ real. The paper's experiments are designed to show this energy strictly decreasing in serial update mode from several initial states.","core_discovery":"On the paper's own terms, the discovery is that replacing the sign-type activations of previous complex-valued Hopfield networks with ceiling-type quantizers yields two new networks with dramatically larger discrete state spaces, and that these networks appear to inherit the stability of their predecessors. The CoCeil activation $\\operatorname{coceil}_{Q,R}(a+bi)=\\operatorname{ceil}_{Q,R}(a)+\\operatorname{ceil}_{Q,R}(b)i$ partitions the complex plane into $(Q+1)^2$ rectangular bins, while the CoSign activation $\\operatorname{CoSign}_{Q,R,K}(z)=\\operatorname{ceil}_{Q,R}(|z|)\\operatorname{csign}_K(z)$ partitions it into $QK$ annular phase sectors. The paper does not prove convergence theorems for either network; instead it states the conjecture that, for weights with $W_{ij}=\\overline{W_{ji}}$ and $W_{ii}\\ge 0$, serial updates converge to a fixed point, and it reports computational experiments, using five random initial states per network, in which the energy (16) reaches a stationary value. The authors conclude that both networks can serve as associative memories storing more distinct patterns than earlier complex-valued models.","pith_inferences":["The conjecture is only as solid as the energy function: because the $i=j$ terms contribute $W_{ii} S_i^2$, and $S_i^2$ is complex for general states in the image sets, the expression in (16) need not be real-valued when $W_{ii}>0$; a rigorous proof would need to show the imaginary part cancels or introduce a different Lyapunov function.","A larger state space does not by itself guarantee more stored patterns; the next natural measurement is how many of the $(Q+1)^{2N}$ or $(QK)^N$ states are actual fixed points for a random Hermitian weight matrix, and how large their basins of attraction are.","Because $\\operatorname{ceil}_{Q,R}$ is a superposition of step functions, each CoCeil neuron is equivalent to $Q$ threshold neurons sharing weights with different biases, so the known dynamics of real-valued ceiling Hopfield networks could supply a convergence proof for the complex Cartesian model.","The same magnitude-and-phase quantization recipe can be transplanted to hypercomplex Hopfield networks (quaternionic, Clifford, or vector-valued), since it only requires a coordinate split or a polar decomposition of the activation."],"forward_implications":["If the conjecture holds, a CoCeil-CvHNN with $N$ neurons offers $(Q+1)^{2N}$ distinct states, compared with $4^N$ for the split-sign network, without any increase in neuron count.","If the conjecture holds, a CoSign-CvHNN offers $(QK)^N$ states, compared with $K^N$ for the complex-signum network, and reduces exactly to the complex-signum model when $Q=1$.","Convergence in serial mode under Hermitian weights with real nonnegative diagonal would make both networks usable as associative memories whose stored pattern count can be tuned by choosing $Q$, $R$, and $K$.","The reported energy plots show monotone decay to a stationary value from all tested initial states, the behavior an associative memory needs."],"supporting_citations":[{"why":"It introduces the complex-valued multistate associative memory with the complex signum activation and the quadratic energy that later convergence proofs use.","marker":"[28]"},{"why":"It establishes the real-imaginary split-sign CvHNN and its convergence theorems (Theorem 2), which the CoCeil model directly extends.","marker":"[40]"},{"why":"It provides the earlier study of complex-hypercube dynamics that motivates the split ceiling extension.","marker":"[41]"},{"why":"It supplies the convergence theorem for the complex signum network under Hermitian weights (Theorem 1), which the paper conjectures to carry over.","marker":"[43]"},{"why":"It defines the ceiling neuron model whose coordinate-wise complex version yields the coceil activation.","marker":"[46]"}],"fun_headline_variants":["Ceiling activations expand complex Hopfield state space","Rectangular and polar quantizers enlarge Hopfield memories","Complex Hopfield networks gain capacity from quantization","Quantization of phase and magnitude enriches Hopfield nets","Ceiling quantizers boost complex-valued Hopfield states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the energy $E(S)=-\\frac{1}{2}\\sum_{i,j} S_i W_{ij} S_j$ is a real-valued, bounded-below Lyapunov function that strictly decreases for the two new ceiling activations whenever the weight matrix satisfies $W_{ij}=\\overline{W_{ji}}$ and $W_{ii}\\ge 0$; if this fails, the conjectured convergence has no proof and the experiments only cover a few random cases.","fun_headline_variants_meta":{"raw":{"variants":["Ceiling activations expand complex Hopfield state space","Rectangular and polar quantizers enlarge Hopfield memories","Complex Hopfield networks gain capacity from quantization","Quantization of phase and magnitude enriches Hopfield nets","Ceiling quantizers boost complex-valued Hopfield states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001032,"raw_usage":{"total_tokens":4315,"prompt_tokens":885,"completion_tokens":3430,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":3355}},"tokens_in":501,"tokens_out":3430,"duration_ms":26809,"temperature":1.0,"reasoning_tokens":3355,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:15:03.227060+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a small CoCeil- or CoSign-CvHNN, say $N=2$ or $3$ with $Q=2$, $R=1$, $K=4$, using a weight matrix with $W_{ij}=\\overline{W_{ji}}$ and positive real diagonal entries such as $W_{11}=1$; if the value of $E(S)$ in (16) is ever non-real or increases between consecutive serial updates, or if the network enters a cycle of length greater than one from some initial state, the conjecture is false.","supporting_citations":[{"cited_title":"Complex-valued mul- tistate neural associative memory,","cited_arxiv_id":null,"evidence_quote":"It introduces the complex-valued multistate associative memory with the complex signum activation and the quadratic energy that later convergence proofs use."},{"cited_title":"Complex-valued neural associative memory on the complex hypercube,","cited_arxiv_id":null,"evidence_quote":"It establishes the real-imaginary split-sign CvHNN and its convergence theorems (Theorem 2), which the CoCeil model directly extends."},{"cited_title":"Infinite Population, Complex Valued State Neural Network on the Complex Hypercube,","cited_arxiv_id":null,"evidence_quote":"It provides the earlier study of complex-hypercube dynamics that motivates the split ceiling extension."},{"cited_title":"Some Remarks on the Stability of Discrete-Time Complex-Valued Multistate Hopfield Neural Networks,","cited_arxiv_id":null,"evidence_quote":"It supplies the convergence theorem for the complex signum network under Hermitian weights (Theorem 1), which the paper conjectures to carry over."},{"cited_title":"Novel Ceiling Neuron Model and its Applications,","cited_arxiv_id":null,"evidence_quote":"It defines the ceiling neuron model whose coordinate-wise complex version yields the coceil activation."}],"review_version":1}