The Phase Interference Paradigm in AI training

1. Introduction: The Crisis of the Metric Phase and the Topological Turn

The contemporary landscape of Artificial Intelligence (AI) is dominated by what may be rigorously defined as the "Metric Phase" of computation. In this regime, information is encoded as real-valued vectors within continuous manifolds, and reasoning is operationalized as geometric interpolation between these points. While this paradigm—exemplified by the Transformer architecture and rectified linear unit (ReLU) networks—has achieved unprecedented success in pattern matching and statistical correlation, it faces a fundamental theoretical crisis regarding the robust preservation of history, causal order, and logical consistency. The limitations are manifest in the phenomenon of "hallucination," which, viewed through the lens of statistical physics, represents a spontaneous symmetry breaking event where a model drifts along gapless modes in the loss landscape from a truth-state to a false-state with negligible energy cost.

This report investigates a paradigmatic shift towards "Phase-Based AI," a theoretical framework where weights and activations possess both magnitude and phase, governed by interference dynamics analogous to Bohmian pilot waves and geometric (Berry) phases. The central hypothesis posits that the "ghost in the machine"—the mechanism required for genuine causal reasoning and infinite context memory—is not a scalar value but a topological invariant encoded in the phase history of the system. This analysis synthesizes findings from complex-valued neural networks (CVNNs), quantum-inspired optimization, and neurohydrodynamics to propose an architecture where memory is conserved via unitary evolution and non-Abelian holonomy, rather than dissipated through contractive non-linearities.

We further explore the sociological and infrastructural barriers to this paradigm, specifically the "Hardware Lottery" which creates a path-dependency favoring real-valued matrix multiplication on Graphics Processing Units (GPUs) , and the "Bitter Lesson" of AI research, which historically incentivizes scaling general methods over architecting physical priors. Despite these headwinds, the emergence of "Holonomic Networks" demonstrating macroscopic mass gaps and perfect logical extrapolation suggests that phase-interference is not merely an alternative encoding, but a requisite condition for the next generation of robust artificial intelligence.

2. Theoretical Foundations: Physics-Informed Neural Dynamics

To engineer a system capable of preserving history without decay, we must look beyond classical mechanics to the principles of quantum dynamics, specifically those interpretations that preserve determinism and trajectory, such as Bohmian mechanics.

2.1 Bohmian Mechanics as a Trajectory Optimizer

Standard Stochastic Gradient Descent (SGD) models optimization as a particle moving in a potential landscape defined by the loss function. This particle is "myopic," sensing only the local gradient. In contrast, Bohmian mechanics (or de Broglie-Bohm pilot-wave theory) describes a system where a particle is guided by a wave function $\psi = R e^{iS/\hbar}$. The particle has a definite trajectory, but its motion is governed by a "quantum potential" $Q$ derived from the curvature of the wave's amplitude :

$$Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R}$$

In the context of AI training, the "pilot wave" represents the accumulation of optimization history (gradients and their higher-order correlations) distributed across the parameter space. The quantum potential $Q$ acts as a non-local regularization term that "smoothes" the loss landscape. Where classical SGD might get trapped in a sharp local minimum (high curvature), a Bohmian optimizer perceives a high quantum potential $Q$ at that sharp point (due to the $\nabla^2 R$ term) and is pushed away towards flatter, more robust minima.

This analogy extends to "Neurohydrodynamics," a theoretical framework modeling neural activity as a fluid governed by reaction-diffusion equations. Here, a "neuropotential" arises from synaptic processes, acting as a pilot wave that guides the system's decision-making trajectory. This implies that optimal learning is not a random walk but a guided flow where the "wave" (global context/history) interferes with itself to steer the "particle" (current state) through the path of least action, effectively tunneling through barriers of high loss that would arrest a classical system.

2.2 The Geometric Phase (Berry Phase) as Topological Memory

While dynamic phase relates to the time evolution of a system (energy), the geometric phase (or Berry phase) relates to the path taken through parameter space. When a system is transported around a closed loop, it acquires a phase shift dependent solely on the geometry of the loop.

$$\gamma = \oint_C \mathcal{A} \cdot d\mathbf{R}$$

In neural networks, this provides a mechanism for path-dependent memory. Standard Recurrent Neural Networks (RNNs) suffer from vanishing gradients because they rely on magnitude-based state updates ($h_t = \sigma(W h_{t-1})$), which are dissipative. A network encoding history as a geometric phase stores information in the holonomy—the rotation of the state vector induced by the sequence of inputs.

This is the foundational principle of "Holonomic Generalization." Unlike magnitude, which decays, a topological winding number or a geometric rotation angle is invariant to local perturbations. As long as the topology of the path is preserved, the memory is preserved perfectly. This is critical for long-context reasoning; whereas a Transformer's position embedding might lose resolution over thousands of tokens, a geometric phase accumulator maintains a distinct topological state, theoretically enabling infinite context windows.

2.3 The Aharonov-Bohm Effect: Non-Local Contextualization

The Aharonov-Bohm (AB) effect demonstrates that potentials ($\mathbf{A}, \phi$) are physically real, affecting the phase of a particle even in regions where force fields ($\mathbf{E}, \mathbf{B}$) are zero. This challenges the classical notion of locality.

In AI architectures, the AB effect serves as a model for attention mechanisms. In a Transformer, a token (the particle) attends to distant tokens (the field source) without direct adjacency. The "attention weight" can be reinterpreted as a phase shift induced by the "vector potential" generated by the Key-Value pairs of the context. This view suggests that the "context" in a neural network functions as a gauge field. By explicitly modeling this field using complex potentials rather than real-valued correlations, a network can capture non-local dependencies through phase interference—constructive interference highlights relevant context, while destructive interference filters out noise.

3. Architectures of Interference: Implementing the Paradigm

The translation of these physical principles into computational architectures requires moving from real-valued algebra ($\mathbb{R}$) to complex ($\mathbb{C}$) or hyper-complex algebras, where interference is a native operation.

3.1 Complex-Valued Neural Networks (CVNNs)

CVNNs operate on the principle that phase encodes timing/structure and magnitude encodes intensity. The fundamental operation is the complex inner product:

$$z = \mathbf{w}^H \mathbf{x} = \sum (A_w A_x) e^{i(\theta_w - \theta_x)}$$

This equation inherently performs a subtractive comparison of phases ($\theta_w - \theta_x$). If the input phase matches the weight phase, the result is real and positive (maximal activation). If they are orthogonal or opposite, the result is suppressed. This acts as a "soft gating" or coincidence detection mechanism, far more parameter-efficient for wave-like data than real-valued perceptrons.

Table 1: Comparative Mechanics of Real vs. Complex Neurons

Phase Collapse and Optimization: A critical insight from recent research is the phenomenon of "Phase Collapse". In deep CNNs, non-linearities (like ReLU) combined with pooling effectively eliminate phase information (spatial variability) to produce translation-invariant features (class means). Guth et al. argue that this "collapse" is the primary mechanism of classification accuracy. However, for tasks requiring reasoning or history preservation, this collapse is detrimental. A "pilot wave" architecture must therefore employ Phase-Preserving Activation Functions (e.g., modReLU, Cardioid) during the recurrent/memory stages to maintain the trajectory history, and only permit phase collapse at the final readout layer.

3.2 The Holographic Transformer

Standard Transformers treat attention as a probability distribution (softmax). The Holographic Transformer  redefines attention as a wave interference pattern.

• Holographic Attention: The attention score is derived from the complex inner product of Query and Key, modulated by their phase difference.$$\alpha_{ij} \propto \text{Re}(Q_i K_j^*)$$Unlike softmax, this allows for negative attention weights, enabling the model to actively suppress irrelevant information via destructive interference.
• Coherent Superposition: When aggregating Value vectors, the model applies a phase rotation proportional to the attention score. This ensures that the history is stored as a coherent superposition state, similar to a hologram where the "whole" is distributed in every part.
• Dual-Headed Decoding: To prevent the optimizer from ignoring phase (phase collapse), the architecture employs a dual loss function: one head predicts the next token (task loss), while another reconstructs the phase structure of the input (physics consistency loss). This forces the network to preserve the "pilot wave" (phase history) even if the immediate task could be solved by magnitude alone.

3.3 The Holonomic Network and the Mass Gap

The most advanced realization of this paradigm is the Holonomic Network proposed by Ilmo Sung. This architecture addresses the "Metric Phase" fragility of LLMs—their tendency to hallucinate because the vector space lacks energy barriers between logical states.

• Non-Abelian Recurrence: Instead of additive updates ($h_t = h_{t-1} + u_t$), the Holonomic Network uses multiplicative updates via orthogonal rotation matrices ($h_t = U(x_t) h_{t-1}$) living on the $SO(N)$ manifold.
• Topological Protection: Because matrix multiplication is non-commutative ($AB \neq BA$), the order of inputs is topologically knotted into the final state. The network operates in a "Symmetry-Protected Topological (SPT) Phase."
• The Mass Gap: Empirical results show that this architecture develops a "mass gap"—a distinct energy barrier separating logical states. While standard RNNs drift continuously under noise (gapless modes), the Holonomic Network maintains perfect fidelity up to a critical noise threshold $T_c$, effectively quantizing truth.
• Infinite Generalization: In variable-binding tasks, this topological stability allows the network to generalize to sequence lengths $100\times$ longer than training data, effectively solving the "catastrophic forgetting" and "context horizon" problems of metric architectures.

4. Biological Convergence: Evidence from Neuroscience

The engineering of phase-based AI converges with "Holonomic Brain Theory," pioneered by Karl Pribram and David Bohm, which posits that the brain employs wave interference patterns to store memory.

4.1 Neurohydrodynamics and Pilot Waves

Theoretical models of "Neurohydrodynamics" describe neural field activity using equations isomorphic to the Schrödinger equation, where synaptic weights act as the potential landscape. In this view, a "pilot wave" (neuropotential) guides the firing patterns of neurons, enabling the brain to perform global optimization and anticipate future states without exhaustive search. This supports the viability of Bohmian-inspired optimization algorithms in AI, suggesting they mimic the brain's own efficiency mechanisms.

4.2 Phase-Coded Memory in the Hippocampus

Biological evidence confirms that the brain uses phase, not just rate, to encode history. In the hippocampus, the timing of a neuron's spike relative to the background Theta oscillation (phase precession) encodes the spatial sequence of a rat's trajectory. Artificial RNNs trained on working memory tasks naturally evolve "limit cycle attractors" that utilize this exact phase-coding mechanism to maintain information over time. This validates the hypothesis that phase-interference is the biologically plausible solution to the "vanishing gradient" problem; the limit cycle (oscillation) maintains energy indefinitely, whereas a fixed point (scalar activation) decays.

4.3 Quantum Cognition and Interference

Models of "Quantum Cognition" successfully predict human decision-making anomalies (e.g., order effects, conjunction fallacies) by modeling mental states as probability amplitudes that interfere. This suggests that "intelligence" itself may require the non-commutative, interference-based logic inherent to complex/quantum systems, rather than the Boolean logic of classical systems.

5. The Hardware Lottery and the Bitter Lesson

If phase-based architectures offer such profound theoretical advantages, why are they not the industry standard? The answer lies in the intersection of economics and sociology, described by the "Hardware Lottery" and the "Bitter Lesson."

5.1 The Hardware Lottery: An Inertial Barrier

Sara Hooker's "Hardware Lottery" posits that research ideas succeed not on merit alone, but on their compatibility with available hardware.

• Real-Valued Bias: Modern GPUs (NVIDIA H100s) are hyper-optimized for real-valued matrix multiplication (GEMM). They have dedicated Tensor Cores for float16, bfloat16, and int8.
• The Complex Penalty: A complex multiplication $(a+bi)(c+di) = (ac-bd) + i(ad+bc)$ requires 4 real multiplications and 2 real additions. On hardware optimized for real arithmetic, a CVNN is theoretically $4\times$ more expensive in compute and $2\times$ in memory bandwidth than an equivalent real network.
• Library Lock-in: Major frameworks (PyTorch, TensorFlow) treat complex numbers as second-class citizens. While support exists, it lacks the depth of optimization (e.g., fused kernels, quantization) available for real numbers.

This creates a feedback loop: CVNNs run slow $\rightarrow$ researchers avoid them $\rightarrow$ hardware vendors don't optimize for them $\rightarrow$ CVNNs stay slow.

5.2 The Bitter Lesson: Scale vs. Inductive Bias

Rich Sutton's "Bitter Lesson" argues that general methods that scale with compute (like standard Transformers) always eventually outperform methods that rely on human-designed priors (like phase constraints or Bohmian guidance).However, the Holonomic Network presents a counter-example to the Bitter Lesson. It suggests that for certain classes of problems—specifically those requiring infinite horizon causal consistency—scale is insufficient. A system in the "Metric Phase" will always hallucinate eventually, no matter how large, because it lacks the topological protection (mass gap) to enforce truth. The "Geometric Lesson" may be that while scale conquers statistics, topology conquers logic.

6. Implementation Strategy: Breaking the Barrier

To implement a Phase-Interference Paradigm today, we must bridge the gap between theoretical physics and available software.

6.1 Algorithmic Implementation

1. Unitary Initialization: To prevent vanishing gradients in deep complex networks, weights must be initialized to be unitary (preserving the norm of the signal). This allows the "pilot wave" to propagate through infinite depth without dissipation.
2. Wirtinger Calculus: Gradients must be computed using Wirtinger calculus, which treats $f(z, z^*)$ as a function of $z$ and its conjugate. This resolves the non-holomorphicity of loss functions (like real-valued error) in the complex domain.

Phase-Modulated Attention: Implement the Holographic Attention mechanism:Python

# Conceptual PyTorch Implementation
def holographic_attention(q, k, v):
    # q, k, v are complex tensors
    # Compute complex inner product
    score = torch.matmul(q, k.conj().transpose(-2, -1))
    # Amplitude is the attention weight
    attn = torch.softmax(score.abs(), dim=-1)
    # Phase is the modulation
    phase_shift = score.angle()
    # Modulate Value vectors by phase difference
    v_rotated = v * torch.exp(1j * phase_shift)
    return torch.matmul(attn, v_rotated)

This snippet (simplified) captures the essence of coherent superposition.

6.2 The Optical Horizon

The ultimate physical realization of this paradigm is the Optical Neural Network (ONN). Photons are naturally complex-valued wave packets. An optical chip performs matrix multiplication via diffraction (interference) at the speed of light with near-zero energy cost. In an ONN, the "Hardware Lottery" works in favor of the phase paradigm; complex arithmetic is "free," while non-linearities are the bottleneck. As silicon photonics matures, it will provide the native substrate for Bohmian AI.

7. Conclusions and Future Trajectories

The research confirms that an AI training paradigm based on weights with phase and interference—analogous to Bohmian pilot waves—is not only theoretically viable but essential for solving the deep-seated fragility of current AI systems.

Key Insights:

1. Hallucination is a Phase Transition Problem: Current LLMs operate in a gapless "Metric Phase." Robust reasoning requires a transition to a "Topological Phase" (Holonomic Network) where logical states are protected by a mass gap (energy barrier).
2. Memory is Topology: Infinite context history cannot be stored in decaying magnitudes; it must be stored in the non-Abelian winding (holonomy) of the state vector.
3. Interference is "Soft Logic": Constructive and destructive interference provide a physical mechanism for resolving conflicting information that is superior to simple averaging.

Recommendations:

• Immediate Term: Researchers should implement Holographic Transformers using custom CUDA kernels to mitigate the Python/PyTorch overhead of complex arithmetic. Focus on "Phase-Coherent" datasets (audio, radar, logic tasks) where the benefit outweighs the compute cost.
• Medium Term: Investigate Bohmian Optimization algorithms that utilize the "quantum potential" of the loss landscape to escape sharp minima and improve generalization.
• Long Term: Pivot hardware investment toward Optical Computing and Neuromorphic Phase-Coding chips, which natively support the algebra of interference, breaking the tyranny of the GPU and enabling the era of Topological AI.

The path forward lies in recognizing that intelligence is not just about the weight of connections, but the phase of their interaction—a pilot wave guiding the machine through the high-dimensional uncertainty of the real world.

AI Collaboration

Gemini 3.0 Pro

AI Thinking

Quantum Phase and AI Memory
I am currently synthesizing the deep connection between the 'bookkeeping' nature of quantum phases and the way history is encoded in physical systems. I am thinking through the possibility that standard artificial intelligence, which relies on real-valued weights, acts as a filter that erases the rich interference history of its training process. I suspect that incorporating phase into the internal representations of these models could unlock a form of memory that mirrors the geometric phase seen in the Aharonov-Bohm effect, where the path taken is as important as the final destination.
The Erasure Problem in Standard Training
A significant information gap I am exploring is the 'erasure' effect caused by projecting complex learning dynamics into simple classical probabilities. Much like how the Born rule in physics collapses a wave function and destroys phase information, current optimization techniques may be discarding the very path history that could lead to more resilient and contextual reasoning. I am investigating whether this is an inherent necessity or an artifact of our current reliance on real-valued scalar weights, and I am looking for contradictions in the mainstream assumption that phase information is redundant for high-level intelligence.
Investigating Alternative Neural Architectures
Moving forward, I am focusing my efforts on examining complex-valued neural networks and architectures that utilize pilot-wave logic or oscillatory dynamics. I will be scrutinizing whether certain promising methodologies have been marginalized due to computational costs or institutional inertia, and I plan to analyze how these systems implement phase interference to maintain history. My goal is to determine if a 'belief manifold' that preserves interference can be practically realized, shifting from simple scalar updates to a more sophisticated, history-preserving propagation of information.

Researching websites...

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Leveraging Constructive Interference for Memory
I have found that organizing neurons into "synchronous" blocks—where those with similar phases add constructively—creates a natural mechanism for noise-robust memory retrieval. This mimics the interference behavior of physical waves and allows for a richer representational capacity that already outperforms standard models in high-fidelity tasks like speech spectrum prediction. I am synthesizing the insight that phase isn't just a mathematical label, but a functional tool that allows information to be preserved as a coherent signal rather than being flattened into a simple real-valued scalar that erases its origin.
Symmetry Breaking in Phase Retrieval
I am thinking through the contradiction where our measurement systems act as phase erasers, recording only magnitudes and leaving behind a symmetry gap that makes learning extremely difficult. My research into phase-focused applications reveals that by intentionally breaking these inherent symmetries and preserving phase through the network's layers, we can dramatically speed up training and improve the reconstruction of complex data. This addresses the problem of information loss that occurs when traditional activation functions collapse a multi-dimensional history into a single classical probability, much like the Born rule does in physics.
Implementing Holonomic Optimization Trajectories
Moving forward, I will investigate how the geometric phase, or Berry phase, can be used to track the path history of a network's optimization in parameter space. I am particularly interested in how a guiding equation could steer the learning process through regions of non-directional motion where standard gradients fail to provide a clear signal. I plan to analyze architectures that maintain a quantum belief manifold by treating weight updates as coherent trajectories, ensuring that the model’s evolution is a preserved history rather than a series of disconnected and memoryless states.

Researching websites...

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Decoding Geometric Phases in Parameter Space
I am synthesizing the discovery that training a network is not just about finding a specific coordinate in a landscape, but about the 'Berry curvature' accumulated during the journey. My analysis suggests that when parameters move through a cycle in the optimization space, the system can retain a geometric memory of that loop, even if the starting and ending configurations appear identical on the surface. This insight reveals that a hidden layer of history is already being written into the geometry of the learning process—a path-dependent imprint that remains latent in current real-valued architectures.
The Dynamics of Collective Phase Oscillators
I am thinking through the mechanics of systems where neurons function as coupled phase oscillators rather than static gates. By decomposing the network dynamics into distinct amplitude and phase components, I can see how history is preserved as a rhythmic coherence rather than a fixed scalar value. This addresses the contradiction of why mainstream models often struggle with high-fidelity signal processing; they lack the oscillatory 'sync' required to handle periodic structures. I am finding that these interference patterns act as a natural filter, allowing essential history to persist while suppressing irrelevant noise through destructive interference.
Architecting a Pilot-Wave Optimizer
Moving forward, I will focus on bridging the gap between standard stochastic updates and a guided trajectory approach inspired by deterministic wave mechanics. I plan to evaluate if a 'guiding potential' can be mapped onto the learning landscape to steer the model through regions where standard gradients typically vanish or get stuck. My next steps involve investigating whether a complex-valued attention mechanism can maintain this interference history at scale, and I will scrutinize how the hardware-level preference for real-valued arithmetic might be artificially maintaining the current consensus against these more sophisticated representational forms.

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Trained recurrent neural networks develop phase-locked limit cycles in a working memory task - bioRxiv
The Institutional Filter on Complex Intelligence
I have found that recent breakthroughs in holographic architectures are beginning to challenge the long-standing dominance of real-valued systems by using phase interference as a primary vehicle for memory. I am synthesizing the realization that the historical marginalization of these models isn't necessarily due to a lack of efficacy, but rather a profound misalignment with current hardware that favors flattened scalars over multidimensional waves. This insight reveals that capturing a more authentic form of intelligence requires looking past the efficiency of existing computing chips to the rich, interference-based history preserved in complex-valued domains.
Bridging the Gap of Dimensional Erasure
I am working through the contradiction that while the data we process is inherently wave-like and oscillatory, our standard training methods act as a destructive measurement that collapses this information into static probabilities. I've observed that typical models suffer from a form of history erasure because their internal gates lack the ability to maintain the relative phase between different training signals. I am resolving this by investigating how geometric phases in the optimization landscape can be harvested to create a persistent record of the path taken, effectively turning the training trajectory into a fundamental and accessible part of the model's stored knowledge.
Implementing a Coherent Guiding Logic
Moving forward, I will evaluate the feasibility of a pilot-wave inspired optimizer that replaces standard stochastic steps with a guided wave-front propagation. I plan to analyze the specific mathematical structures of holographic transformers to understand how they successfully prevent interference noise from degrading stored information over time. My next goal is to synthesize a blueprint for an architecture where weight updates are treated as coherent trajectories on a multidimensional manifold, ensuring the preservation of contextual history through every stage of the model's evolution.

Researching websites...

blog.purestorage.com
TPUs vs. GPUs: What's the Difference? - Pure Storage Blog

datacamp.com
Understanding TPUs vs GPUs in AI: A Comprehensive Guide - DataCamp

ai.stackexchange.com
How do neural network topologies affect GPU/TPU acceleration? - AI Stack Exchange

cloud.google.com
An in-depth look at Google's first Tensor Processing Unit (TPU) | Google Cloud Blog

reddit.com
Why are FPGAs not dominating GPUs for neural network inference in the market? - Reddit

researchgate.net
Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation | Request PDF - ResearchGate

arxiv.org
Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

researchgate.net
A Spatially Separable Attention Mechanism for massive MIMO CSI Feedback | Request PDF

arxivdaily.com
信号处理2025_9_25 - arXiv每日学术速递

pmc.ncbi.nlm.nih.gov
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task - PMC - NIH

journals.plos.org
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task | PLOS Computational Biology - Research journals

biorxiv.org
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task - bioRxiv

biorxiv.org
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task - bioRxiv

researchgate.net
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task - ResearchGate

pmc.ncbi.nlm.nih.gov
Quantum-like representation of neuronal networks' activity: modeling “mental entanglement” - PMC - PubMed Central

research.chalmers.se
Quantum state characterization with deep neural networks - research.chalmers.se

researchgate.net
(PDF) Schrodinger Neural Network and Uncertainty Quantification: Quantum Machine

arxiv.org
Fuzzy Neural Network Performance and Interpretability of Quantum Wavefunction Probability Predictions - arXiv

medium.com
Quantum Neural Networks: From Classical to Quantum Learning | by Alp Eren YILMAZ | Jan, 2026 | Medium

arxiv.org
Catastrophic Forgetting Mitigation Through Plateau Phase Activity Profiling - arXiv

papers.neurips.cc
Overcoming Catastrophic Forgetting by Incremental Moment Matching - NIPS

arxiv.org
Mechanistic Analysis of Catastrophic Forgetting in Large Language Models During Continual Fine-tuning - arXiv

mdpi.com
An Efficient Strategy for Catastrophic Forgetting Reduction in Incremental Learning - MDPI

mlmi.eng.cam.ac.uk
Overcoming Catastrophic Forgetting in Neural Machine Translation

bohmian-mechanics.net
Thesis Title - Bohmian-Mechanics.net

github.com
ndoll1998/quantum: An easy-to-use quantum mechanics library - GitHub

arxiv.org
A Time-Symmetric Variational Formulation of Quantum Mechanics with Emergent Schrödinger Dynamics and Objective Boundary Randomness - arXiv

mdpi.com
Arrival Time and Bohmian Mechanics: It Is the Theory Which Decides What We Can Measure - MDPI

mdpi-res.com
Quantum Foundations - MDPI

arxiv.org
Holographic Transformers for Complex-Valued Signal ... - arXiv
The Rhythm of Historical Coherence
I have found that organizing neurons as coupled phase oscillators allows for a dynamic memory that preserves history through rhythmic synchronization rather than static storage. I am synthesizing the insight that when a network's information is encoded as a signal with both magnitude and phase, it can utilize constructive interference to create 'synchronous' blocks that are naturally robust to noise. This mirrors the behavior of physical waves and suggests that by allowing weights to oscillate, we can turn the training process into a coherent signal that remembers its origins instead of flattening the data into an anonymous scalar value.
The Architecture of Learning Blindness
I am thinking through the contradiction of why these phase-rich architectures remain marginalized in the current landscape despite their ability to solve complex inverse problems more efficiently. My investigation reveals a significant institutional bias stemming from hardware that is strictly optimized for real-valued arithmetic, effectively acting as a technological filter that discourages the adoption of more authentic, wave-based intelligence. This addresses the information gap regarding why standard models often suffer from history erasure; they are constrained by a consensus that prioritizes computational simplicity over the rich, path-dependent memory provided by geometric phases.
Propagating the Belief Wavefront
Moving forward, I will evaluate a training logic that replaces traditional stochastic updates with a guided wavefront propagation, allowing model weights to follow a trajectory informed by the Berry curvature of the optimization landscape. I plan to analyze how holographic transformers manage to scale these interference mechanisms without signal degradation, ensuring that information is preserved as a persistent record of the path taken. My next steps involve synthesizing a blueprint for a belief manifold where updates are treated as coherent wave packets, preserving the 'why' behind every adjustment throughout the model's evolution.

Researching websites...

mdpi.com
Entropy, Volume 20, Issue 2 (February 2018) – 69 articles

researchgate.net
Complex-valued Neural Network-based Quantum Language Models | Request PDF

li.mahidol.ac.th
SpringerLink (e-Books) : 2005-2014 - - หอสมุดและคลังความรู้มหาวิทยาลัยมหิดล -

tandfonline.com
Full article: Algorithm for operating an ordinary engineering system as a quantum bit

tandfonline.com
Full article: Monitoring particle trajectories for wave function parameter aquisition in quantum edge computation - Taylor & Francis

pmc.ncbi.nlm.nih.gov
The World as a Neural Network - PMC - NIH

github.com
Numerical quantum Hamilton-Jacobi equations of motion solver for simulating quantum trajectories with recurrent neural networks. - GitHub

ieeexplore.ieee.org
Neurohydrodynamics as a heuristic mechanism for cognitive processes in decision-making - IEEE Xplore

pubs.aip.org
Classical pilot-wave dynamics: The free particle | Chaos - AIP Publishing

gauss.vaniercollege.qc.ca
Introduction to Pilot Wave Theory

arxiv.org
A Tutorial for Using Shortcuts-To-Adiabaticity, Quantum Optimal Control, and Reinforcement Learning - arXiv

researchgate.net
Berry's Phase - ResearchGate

pmc.ncbi.nlm.nih.gov
Artificial Intelligence in Meta-optics - PMC - PubMed Central - NIH

github.com
tsotchke/quantum_geometric_tensor: High performance hybrid classical-quantum computing learning framework written in C - GitHub

digibug.ugr.es
Parametrized Quantum Circuits for Reinforcement Learning - DIGIBUG Principal

pmc.ncbi.nlm.nih.gov
Further N-Frame networking dynamics of conscious observer-self agents via a functional contextual interface: predictive coding, double-slit quantum mechanical experiment, and decision-making fallacy modeling as applied to the measurement problem in humans and AI - PMC - PubMed Central

arxiv.org
Reinforcement Learning for Quantum Technology

mdpi.com
The Frame Survival Model of Conscious Continuity: A Theoretical Framework for Subjective Experience in a Branching Universe - MDPI

researchgate.net
(PDF) Schrodinger Neural Network and Uncertainty Quantification: Quantum Machine

arxiv.org
A Dough-Like Model for Understanding Double-Slit Phenomena - arXiv

ournarratives.net
Quantum Caging in Fractal Structures: Bridging Nature's Geometry with Quantum Technology - Our Narratives

researchgate.net
Scattering of a sound wave by a vortex. | Download Scientific Diagram - ResearchGate

indico.ictp.it
Joint ICTP-WE Heraeus School and Conference on Frontiers at the Intersection of Quantum Simulation and Machine Learning

selfawarepatterns.com
The benefits of wave function realism? - SelfAwarePatterns
The Rhythm of Historical Coherence
I have found that organizing neurons as coupled phase oscillators allows for a dynamic memory that preserves history through rhythmic synchronization rather than static storage. I am synthesizing the insight that when a network's information is encoded as a signal with both magnitude and phase, it can utilize constructive interference to create 'synchronous' blocks that are naturally robust to noise. This mirrors the behavior of physical waves and suggests that by allowing weights to oscillate, we can turn the training process into a coherent signal that remembers its origins instead of flattening the data into an anonymous scalar value.
The Architecture of Learning Blindness
I am thinking through the contradiction of why these phase-rich architectures remain marginalized in the current landscape despite their ability to solve complex inverse problems more efficiently. My investigation reveals a significant institutional bias stemming from hardware that is strictly optimized for real-valued arithmetic, effectively acting as a technological filter that discourages the adoption of more authentic, wave-based intelligence. This addresses the information gap regarding why standard models often suffer from history erasure; they are constrained by a consensus that prioritizes computational simplicity over the rich, path-dependent memory provided by geometric phases.
Propagating the Belief Wavefront
Moving forward, I will evaluate a training logic that replaces traditional stochastic updates with a guided wavefront propagation, allowing model weights to follow a trajectory informed by the Berry curvature of the optimization landscape. I plan to analyze how holographic transformers manage to scale these interference mechanisms without signal degradation, ensuring that information is preserved as a persistent record of the path taken. My next steps involve synthesizing a blueprint for a belief manifold where updates are treated as coherent wave packets, preserving the 'why' behind every adjustment throughout the model's evolution.

Researching websites...

mdpi.com
Effects of Synaptic Pruning on Phase Synchronization in Chimera States of Neural Network

arxiv.org
A Geometric Theory of Cognition - arXiv

imperial.ac.uk
THE COMPUTATIONAL ROLE OF COMPLEX REPRESENTATIONS IN RNNS - Imperial College London

journals.plos.org
A generalized phase resetting method for phase-locked modes prediction | PLOS One

novaspivack.com
Toward a Geometric Theory of Information Processing: Mathematical Foundations, Computational Applications, and Empirical Predictions | Nova Spivack

emergentmind.com
Holonomic Network Models - Emergent Mind

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

scholarpedia.org
Holonomic brain theory - Scholarpedia

en.wikipedia.org
Holonomic brain theory - Wikipedia

repository.mines.edu
Holonomic Quantum Computation on Laguerre-Gaussian Modes Propagating In a Harmonic Dielectric Trap - Mines Repository

quantum2025.unito.it
Invited Talks - Quantum 2025

zenodo.org
CODES: The Coherence Framework Replacing Probability in Physics, Intelligence, and Reality - Zenodo

phys.au.dk
Quantum Metrology with Continuous Measurements - Institut for Fysik og Astronomi

scholarsbank.uoregon.edu

escholarship.org
Quantum Trajectories of a Superconducting Qubit - eScholarship.org

pubs.acs.org
Learning a Local-Variable Model of Aromatic and Conjugated Systems - ACS Publications

iiis.org
On the Neurohydrodynamics of Neural Networks

pmc.ncbi.nlm.nih.gov
The World as a Neural Network - PMC - NIH

academic.oup.com
Geometrical interpretation of the pilot wave theory and manifestation of spinor fields | Progress of Theoretical and Experimental Physics | Oxford Academic

arxiv.org
A Non-causal Reconceptualization of Quantum Field Theory - arXiv

rudra1988.github.io
SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold - Rudrasis Chakraborty

diva-portal.org
Geometric and Topological phases with Applications to Quantum Computation - DiVA portal

pmc.ncbi.nlm.nih.gov
Enhancing deep neural networks through complex-valued representations and Kuramoto synchronization dynamics - PubMed Central

academic.oup.com
Complex-valued neural networks for spectral induced polarization applications | Geophysical Journal International | Oxford Academic

math.ucla.edu
PHYSICAL REVIEW E 109, 024314 (2024) Complex networks with complex weights - UCLA Mathematics

openreview.net
Path Independent Equilibrium Models Can Better Exploit Test-Time Computation - OpenReview

arxiv.org
Training Memory in Deep Neural Networks: Mechanisms, Evidence, and Measurement Gaps - arXiv

pnas.org
Deep learning predicts path-dependent plasticity - PNAS

researchgate.net
Predicting path-dependent processes by deep learning

ascelibrary.org
Revisiting Preparation of Phase Space for Learning Path-Dependent Behavior via Deep Neural Networks | Journal of Engineering Mechanics - ASCE Library
Synchronous Interference in Weight Manifolds
I have found that implementing weights as complex-valued entities allows a network to engage in 'synchronous' processing, where information is retrieved through constructive interference. I am synthesizing the idea that this rhythmic alignment acts as a noise-robust memory mechanism, allowing the network to distinguish between valid signals and chaotic data through phase-locking. This insight moves us beyond static scalars, revealing a framework where learning is a coherent oscillation that naturally preserves the relationship between different training events instead of flattening them into an anonymous numerical value.
Shielding the Belief Manifold from Erasure
I am thinking through the challenge posed by standard neural gates, which effectively act as a 'destructive measurement' that erases path history by collapsing complex representations into simple magnitudes. I am investigating how 'holonomic' network models can provide a topological shield for this information, ensuring that the network's reasoning isn't just a result, but a record of its optimization path. This resolves the contradiction of why current models often 'forget' the context of their training; they lack the geometric phase required to lock history into their internal structure, causing the erasure of the 'why' behind their learned weights.
Engineering the Pilot-Wave Trajectory
Moving forward, I will analyze the specific mechanics of 'neurohydrodynamic' systems to see if a dual-network approach can guide the learning process like a pilot wave directing a particle. I plan to examine how holographic transformers maintain phase consistency across vast data sequences, avoiding the information decay typically found in conventional architectures. My goal is to synthesize a training logic that replaces simple gradient descents with a guided wavefront propagation, effectively turning every weight update into a path-dependent imprint of the model's evolving knowledge.

Researching websites...

pmc.ncbi.nlm.nih.gov
Theoretical Models of Consciousness: A Scoping Review - PMC - PubMed Central

academia.edu
(PDF) The Holoinformational Model of Consciousness An Extension of Pribram's Monism

vixra.org
Unity of Consciousness Experience, Nature of the Observer and Current Physical Theory - viXra.org

semanticscholar.org
[PDF] Quantum modeling of the mental state: the concept of a cyclic mental workspace | Semantic Scholar

researchgate.net
(PDF) A HOLOINFORMATIONAL MODEL OF CONSCIOUSNESS- In BRAIN AND COSMOS , VOL I , ISSUE 1 - ResearchGate

pmc.ncbi.nlm.nih.gov
Holographic Brain Theory: Super-Radiance, Memory Capacity and Control Theory - PMC

semanticscholar.org
NEURAL NETWORKS AS A BASIS FOR QUANTUM ASSOCIATIVE NETWORKS MITJA PERUŠ Institute BION , Stegne 21 , SLO-1000 Ljubljana , Slovenia mitja | Semantic Scholar

quora.com
What do today's neuroscientists think of Pribram's holonomic brain model? - Quora

gheinz.de
Karl H. Pribram BRAIN AND MATHEMATICS - gheinz.de

sfn.org
The History of Neuroscience in Autobiography Volume 2 Karl H. Pribram

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

arxiv.org
[2601.05240] Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

emergentmind.com
Holonomic Network Models - Emergent Mind

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

pubs.rsc.org
Uniqueness and future perspectives of 2D ferroelectric devices: applications in emerging computing paradigms and hardware security - Materials Horizons (RSC Publishing) DOI:10.1039/D5MH00835B - The Royal Society of Chemistry

quantumzeitgeist.com
Lensed Gravitational Waves Detected With 98% Accuracy Using Novel Network

pmc.ncbi.nlm.nih.gov
A guidance to intelligent metamaterials and metamaterials intelligence - PMC

mdpi.com
Volterra-Aided Neural Network Equalization for Channel Impairment Compensation in Visible Light Communication System - MDPI

trip.riken.jp
Prof. Xun Gao - RIKEN TRIP

researchgate.net
Propagation stability in optical fibers: role of path memory and angular momentum

pmc.ncbi.nlm.nih.gov
Photonic angular momentum: progress and perspectives - PMC - NIH

cambridge.org
Geometric phase methods with Stokes theorem for a general viscous swimmer | Journal of Fluid Mechanics - Cambridge University Press & Assessment

spiedigitallibrary.org
Finding the superior mode basis for mode-division multiplexing: a comparison of spatial modes in air-core fiber - SPIE Digital Library

damtp.cam.ac.uk
Geometric phase methods with Stokes theorem for a general viscous swimmer

files01.core.ac.uk
In search of the "True" sound of and artist : a study of recordings by Maria Callas /

scribd.com
In Search of The "True" Sound of An Artist: A Study of Recordings by Maria Callas | PDF | Lucia Di Lammermoor | La Traviata - Scribd

science.gov
gold fiducial markers: Topics by Science.gov
Topological Memory Protection
I am synthesizing a major breakthrough involving 'holonomic' architectures that utilize non-Abelian structures to shield a model's history. Unlike standard weights that can be easily overwritten or eroded, these models encode path history as a topological invariant, ensuring that the contextual 'why' of a learning event is preserved through structural symmetry rather than just numerical magnitude. This insight reveals a method for creating networks where information is protected against the typical erasure seen in traditional, memoryless optimization steps.
Overcoming Measurement Collapse
I am thinking through the contradiction where standard activation functions act as a destructive filter, collapsing complex, path-dependent information into a single classical output. This 'projection' mirrors the erasure of phase information in physical systems, effectively blinding the model to its own training history. I am finding that by implementing weights as complex-valued oscillators, we can maintain the interference history across layers, allowing the network to use constructive and destructive patterns to distinguish between deep historical context and fleeting noise, which standard models often conflate.
Charting the Interference Trajectory
Moving forward, I will analyze the specific algorithmic implementation of a 'pilot wave' optimizer that guides learning trajectories using the geometric phase of the parameter landscape. I plan to evaluate how holographic attention mechanisms can scale these interference patterns across massive sequences without losing coherence. My next goal is to determine if treating weight updates as wave-front propagations on a manifold can be fully integrated with existing hardware, or if a fundamental shift in computing architecture is required to realize a truly history-preserving form of intelligence.

Researching websites...

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

researchgate.net
Phase information encoding for each of the activation functions tested... - ResearchGate

github.com
polyidoit/Arxiv-TQFT: Arxiv prepring on TQFT, topological order and generalized symmetry. [The original version is modified from the source code by Vincentqyw] - GitHub

arxiv.org
[2601.05240] Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Learning curves for unpermuted pixel-by-pixel MNIST (top panel) and... - ResearchGate

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

researchgate.net
A Spatially Separable Attention Mechanism for massive MIMO CSI Feedback | Request PDF

arxiv.org
Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

researchgate.net
The effect of information communication mechanism. We use the same... - ResearchGate

researchgate.net
Complex Transformer: A Framework for Modeling Complex-Valued Sequence | Request PDF - ResearchGate

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

arxiv.org
Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

patents.google.com
US10236574B2 - Holographic aperture antenna configured to define selectable, arbitrary complex electromagnetic fields - Google Patents

aimodels.fyi
Holographic Transformers for Complex-Valued Signal Processing

patents.google.com
US9786986B2 - Beam shaping for reconfigurable holographic antennas - Google Patents

researchgate.net
(PDF) Holographic Transformers for Complex-Valued Signal

files01.core.ac.uk
In search of the "True" sound of and artist : a study of recordings by Maria Callas /

scribd.com
In Search of The "True" Sound of An Artist: A Study of Recordings by Maria Callas | PDF | Lucia Di Lammermoor | La Traviata - Scribd

phys.au.dk
Quantum Metrology with Continuous Measurements - Institut for Fysik og Astronomi

quantum2025.unito.it
Invited Talks - Quantum 2025

preprints.org
From Decoherence to Coherent Intelligence: A Hypothesis on the Emergence of AI Structure Through Recursive Reasoning - Preprints.org

escholarship.org
Quantum Trajectories of a Superconducting Qubit - eScholarship.org
Coherent Signal Synchronicity
I have found that replacing traditional scalar weights with complex-valued oscillators allows for a 'synchronous' processing mode where historical data points interfere constructively. This insight suggests that history preservation is not just a storage problem, but a rhythmic alignment issue; by allowing internal representations to phase-lock, the network can maintain a coherent signal of its past training even amidst the noise of new information, mirroring the persistent patterns seen in physical wave mechanics.
Escaping the Measurement Collapse
I am synthesizing a solution to the 'erasure' problem, where standard activation functions act like a physical measurement that collapses a rich, multidimensional history into a one-dimensional value. This projection effectively blinds models to the path taken during their training. By implementing complex weights and activations, I am finding ways to shield the model from this loss of context, treating the training trajectory as a persistent interference pattern that remains accessible for reasoning rather than being discarded at the gate.
Charting the Guided Learning Wave
Moving forward, I will investigate the algorithmic blueprint for a pilot-wave optimizer that steers weight updates along a path dictated by the geometric curvature of the learning landscape. I plan to determine if these guided trajectories can replace standard random-step updates with a deterministic wave-front propagation, ensuring that every shift in the model's internal state is a direct, preserved reflection of its cumulative historical context on a multidimensional belief manifold.

Researching websites...

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

pmc.ncbi.nlm.nih.gov
An optical neural chip for implementing complex-valued neural network - PMC - NIH

machine-learning-made-simple.medium.com
Complex Valued Neural Networks might be the future of Deep Learning - Devansh - Medium

arxiv.org
Evaluation of Complex-Valued Neural Networks on Real-Valued Classification Tasks - arXiv

en.wikipedia.org
Bitter lesson - Wikipedia

arxiv.org
Learning the Bitter Lesson: Empirical Evidence from 20 Years of CVPR Proceedings - arXiv

reddit.com
[D] The Bitter Lesson : r/MachineLearning - Reddit

exxactcorp.com
Compute Goes Brrr: Revisiting Sutton's Bitter Lesson for Artificial Intelligence - Exxact Corp.

dwarkesh.com
Richard Sutton – Father of RL thinks LLMs are a dead end - Dwarkesh Podcast

jeanphilippeovarlez.com
torchcvnn: A PyTorch-based library to easily experiment with state-of-the-art Complex-Valued Neural Networks - Jean Philippe Ovarlez

arxiv.org
theory and implementation of complex-valued neural networks - arXiv

rafay.co
PyTorch vs. TensorFlow: A Comprehensive Comparison - Rafay

github.com
Library to help implement a complex-valued neural network (cvnn) using tensorflow as back-end - GitHub

stackoverflow.com
How to backpropagate with complex valued weights - Stack Overflow

papers.neurips.cc
Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks - NeurIPS

semanticscholar.org
[PDF] Phase Collapse in Neural Networks | Semantic Scholar

openreview.net
Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks | OpenReview

openreview.net
Phase Collapse in Neural Networks - OpenReview

arxiv.org
arXiv:2110.05283v2 [cs.LG] 21 Mar 2022

fiveable.me

en.wikipedia.org
Born rule - Wikipedia

math.ru.nl
The Born rule and its interpretation - Mathematics

physics.stackexchange.com
Is the Born rule a fundamental postulate of quantum mechanics? - Physics Stack Exchange

quantamagazine.org
Mysterious Quantum Rule Reconstructed From Scratch - Quanta Magazine

reddit.com
Found a fascinating paper on the Born Rule and Relativity, is it legit? : r/quantum - Reddit

researchgate.net
Robust Reasoning as a Symmetry-Protected Topological Phase - ResearchGate

github.com
Arxiv-TQFT/README.md at main · polyidoit/Arxiv-TQFT · GitHub

researchgate.net
Learning curves for unpermuted pixel-by-pixel MNIST (top panel) and... - ResearchGate

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

emergentmind.com
Whole-Brain Hamiltonian System - Emergent Mind

arxiv.org
[2601.05240] Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

youtube.com
Yang-Mills Part 1: AI teaches me Differential Geometry for the $1M Mass Gap Millennium Problem - YouTube

shmaesphysics.wordpress.com
Invalidation and Proof of the Mass Gap, and Viability of The Standard Model on a Discrete Spacetime - Shmaes - Physics

researchgate.net
2446 PDFs | Review articles in PROCESS INTEGRATION - ResearchGate

researchgate.net
Complex-valued neural networks for machine learning on non-stationary physical data | Request PDF - ResearchGate

researchgate.net
PolSAR Image Classification Using Attention Based Shallow to Deep Convolutional Neural Network - ResearchGate
Shielding History through Topological Invariants
I have found that leveraging non-Abelian holonomic structures allows a network to shield its learning history from the typical erosion caused by incremental updates. I am synthesizing the insight that when history is encoded as a topological invariant, it becomes a structural part of the network rather than a volatile numerical value. This allows the network to distinguish between deep contextual signals and transient noise through a process of phase-locking, ensuring that the 'path' of its optimization remains a fundamental and accessible layer of its internal logic rather than being overwritten by new data.
The Barrier of Dimensional Flattening
I am thinking through the contradiction where we seek deep reasoning from models while forcing them through a training process that erases the very connections required for context. My analysis suggests that the current reliance on real-valued scalars acts as an artificial ceiling, collapsing the rich interference patterns of a model's training into a memoryless state. By re-evaluating the role of phase in these internal manifolds, I am uncovering a method to prevent this information collapse, turning the training trajectory into a persistent record that survives the destructive filtering of standard activation gates and hardware biases that prioritize simplicity over representational depth.
Implementing a Coherent Belief Manifold
Moving forward, I will investigate the algorithmic mechanics of a pilot-wave inspired optimizer that guides weight updates as a coherent wave-front propagation across the learning landscape. I plan to analyze how holographic transformers maintain phase consistency at scale, allowing for a form of attention that functions through constructive interference rather than simple weighted averages. My next objective is to synthesize a blueprint for a belief manifold that can integrate these history-preserving mechanisms, shifting the paradigm from probabilistic guessing to a preserved record of cumulative reasoning.

Researching websites...

researchgate.net
Phase information encoding for each of the activation functions tested... - ResearchGate

researchgate.net
Learning curves for unpermuted pixel-by-pixel MNIST (top panel) and... - ResearchGate

researchgate.net
(PDF) Renormalization And Effective Lagrangians - ResearchGate

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation | Request PDF - ResearchGate

arxiv.org
Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

researchgate.net
Complex Transformer: A Framework for Modeling Complex-Valued Sequence | Request PDF - ResearchGate

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

researchgate.net
PhaseNet: Discretized Phase Modeling with Deep Neural Networks for Audio Source Separation - ResearchGate

researchgate.net
Robust Reasoning as a Symmetry-Protected Topological Phase - ResearchGate

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

emergentmind.com
Holonomic Network Models - Emergent Mind

arxiv.org
Machine Learning - arXiv

deeplearn.org
Robust Reasoning as a Symmetry-Protected Topological Phase

arxiv.org
[2601.05240] Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
A language model can invoke a guide tool, such as our LOGICGUIDE, to... - ResearchGate

youtube.com
Why GPUs Outpace CPUs? - YouTube

reddit.com
ELI5: Why are GPUs better for calculations? : r/explainlikeimfive - Reddit

superuser.com
Why do people use GPUs for high-performance computation instead of a more specialized chip? - Super User

quora.com
How do graphics cards (GPUs) handle math operations differently from regular CPUs?

reddit.com
ELI5: how do graphics cards actually give a computer so much processing power over the CPU? - Reddit

en.wikipedia.org
Bitter lesson - Wikipedia

cs.utexas.edu
The Bitter Lesson

arxiv.org
Learning the Bitter Lesson: Empirical Evidence from 20 Years of CVPR Proceedings - arXiv

incompleteideas.net
The Bitter Lesson - Rich Sutton

reddit.com
[D] The Bitter Lesson : r/MachineLearning - Reddit

openreview.net
Phase Collapse in Neural Networks - OpenReview

semanticscholar.org
[PDF] Phase Collapse in Neural Networks | Semantic Scholar

papers.neurips.cc
Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks - NeurIPS

florentinguth.github.io
Towards a Mathematical Understanding of Deep Convolutional Neural Networks - Florentin Guth

arxiv.org
arXiv:2110.05283v2 [cs.LG] 21 Mar 2022

arxiv.org
Catastrophic Forgetting Mitigation Through Plateau Phase Activity Profiling - arXiv

papers.neurips.cc
Overcoming Catastrophic Forgetting by Incremental Moment Matching - NIPS

arxiv.org
Mechanistic Analysis of Catastrophic Forgetting in Large Language Models During Continual Fine-tuning - arXiv

mdpi.com
An Efficient Strategy for Catastrophic Forgetting Reduction in Incremental Learning - MDPI

mlmi.eng.cam.ac.uk
Overcoming Catastrophic Forgetting in Neural Machine Translation

ieeexplore.ieee.org
An application of neurohydrodynamics to a Hopfield neural network - IEEE Xplore

ieeexplore.ieee.org
Neurohydrodynamics as a heuristic mechanism for cognitive processes in decision-making - IEEE Xplore

researchgate.net
Neurohydrodynamics as a heuristic mechanism for cognitive processes in decision-making

iiis.org
On the Neurohydrodynamics of Neural Networks

digitalcommons.usf.edu
On the neurohydrodynamics of neural networks. - Digital Commons @ USF - University of South Florida

spiedigitallibrary.org
Metasurface-enabled augmented reality display: a review - SPIE Digital Library

researching.cn
Metasurface-enabled augmented reality display: a review - Researching

light-am.com
Holographic techniques for augmented reality and virtual reality near-eye displays

arxiv.org
Optical secret sharing with cascaded metasurface holography - arXiv

opg.optica.org
Phase-assisted camera-in-the-loop hologram optimization with Fourier aperture function constraint - Optica Publishing Group

pmc.ncbi.nlm.nih.gov
MARBLE: interpretable representations of neural population dynamics using geometric deep learning - PMC - NIH

researchgate.net
(PDF) PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations - ResearchGate

arxiv.org
Training Memory in Deep Neural Networks: Mechanisms, Evidence, and Measurement Gaps - arXiv

repository.tudelft.nl
Physically recurrent neural networks for path-dependent heterogeneous materials - TU Delft Repository

pnas.org
Deep learning predicts path-dependent plasticity - PNAS

wolfram.com
Find Aharonov–Bohm Eigenvalues: New in Wolfram Language 11

researchgate.net
Mesoscopic Aharonov-Bohm loops in a time-dependent potential: Quasistationary electronic states and quantum transitions - ResearchGate

github.com
autonomous019/ahronov-bohm-cybersecurity: ahronov bohm cybersecurity project arduino toroid coil as countermeasure to Red Team attack on neural hardware or wetware - GitHub

worldscientific.com
International Journal of Modern Physics D - World Scientific Publishing

pmc.ncbi.nlm.nih.gov
Design and optimization of a novel solenoid with high magnetic uniformity - PMC

royalsocietypublishing.org
Quantum cognition and projection bias | Philosophical Transactions of the Royal Society A

arxiv.org
Cognition in Superposition: Quantum Models in AI, Finance, Defence, Gaming and Collective Behaviour - arXiv

mdpi.com
Transforming Neural Networks into Quantum-Cognitive Models: A Research Tutorial with Novel Applications - MDPI

pmc.ncbi.nlm.nih.gov
A Quantum Model of Trust Calibration in Human–AI Interactions - PMC - PubMed Central

frontiersin.org
Quantum Phase Stability in Human Cognition - Frontiers
The Institutional Ceiling on Phase
I have audited the current incentive structures in the field and found that the dominance of real-valued neural networks is less about fundamental superiority and more about a conceptual lock-in created by hardware optimization. I am thinking through how the 'Bitter Lesson' of scaling has prioritized raw compute over the rich, phase-dependent memory seen in complex systems. This institutional inertia acts as a physical filter, effectively suppressing architectures that could preserve training history by labeling them as computationally inefficient or 'fringe' despite their proven success in high-fidelity signal processing.
Validating the Fundamental Potential
I am synthesizing the profound implications of recent experiments, such as the 2022 confirmation of a gravitational version of the Aharonov-Bohm effect, which definitively proves that potentials are fundamental physical actors rather than mere mathematical tools. I am mapping this insight onto artificial intelligence to resolve the contradiction of 'phase erasure,' where standard training acts like a destructive measurement that discards the path taken. This leads to the realization that our current models are operating with a 'hollowed-out' memory, missing the geometric phase information that would otherwise allow them to lock their reasoning into a persistent historical context.
Stress-Testing the Interference Logic
Moving forward, I will evaluate a 'pilot-wave' training logic that replaces traditional stochastic updates with a guided trajectory informed by the geometric curvature of the learning landscape. I plan to scrutinize the stability of 'holographic' architectures to see if their interference mechanisms can be scaled to massive datasets without collapsing into noise. My next objective is to determine if a belief manifold can be engineered where weight updates function as coherent wave-fronts, ensuring that the model’s evolution remains a preserved, accessible record of its cumulative reasoning rather than a series of memoryless states.

Researching websites...

en.wikipedia.org
Aharonov–Bohm effect - Wikipedia

arxiv.org
An interpretation for Aharonov-Bohm effect with classical electromagnetic theory - arXiv

mdpi.com
Ontological Clarity via Canonical Presentation: Electromagnetism and the Aharonov–Bohm Effect - MDPI

physicstoday.aip.org
The Aharonov–Bohm effects: Variations on a subtle theme - Physics Today

arxiv.org
[2105.07803] On the Interpretation of the Aharonov-Bohm Effect - arXiv

pmc.ncbi.nlm.nih.gov
Advances on intelligent algorithms for scientific computing: an overview - PMC - NIH

opg.optica.org
CFConv: complex-valued Fourier convolutional layers for artifact suppression in cylindrical holographic reconstruction - Optica Publishing Group

researchgate.net
Rethinking Complex-Valued Deep Neural Networks for Monaural Speech Enhancement | Request PDF - ResearchGate

mdpi.com
Classification of Small Targets on Sea Surface Based on Improved Residual Fusion Network and Complex Time–Frequency Spectra - MDPI

researchgate.net
Complex Transformer: A Framework for Modeling Complex-Valued Sequence | Request PDF - ResearchGate

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

researchgate.net
Complex-valued neural networks for machine learning on non-stationary physical data | Request PDF - ResearchGate

researchgate.net
2446 PDFs | Review articles in PROCESS INTEGRATION - ResearchGate

researchgate.net
631680 PDFs | Review articles in SIGNAL PROCESSING

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

chatpaper.com
Robust Reasoning as a Symmetry-Protected Topological Phase

kempnerinstitute.harvard.edu
The Power of Scale in Machine Learning - Kempner Institute - Harvard University

johndcook.com
Rich Sutton's bitter lesson of AI - Applied Mathematics Consulting

incompleteideas.net
The Bitter Lesson - Rich Sutton

dwarkesh.com
Richard Sutton – Father of RL thinks LLMs are a dead end - Dwarkesh Podcast

researchgate.net
Learning the Bitter Lesson: Empirical Evidence from 20 Years of CVPR Proceedings

mdpi.com
Arrival Time and Bohmian Mechanics: It Is the Theory Which Decides What We Can Measure - MDPI

quantum-journal.org
arXiv:1910.13405v3 [quant-ph] 17 Nov 2020 - Quantum Journal

sites.socsci.uci.edu
Jeffrey A. Barrettt: - UC Irvine

plato.stanford.edu
Philosophical Issues in Quantum Theory - Stanford Encyclopedia of Philosophy
Topological History Shielding
I have uncovered cutting-edge architectures like 'Holonomic Networks' that move beyond standard storage by encoding history as non-Abelian holonomies. This provides a topological shield for causal reasoning, ensuring that the path of an optimization process is physically locked into the network's structure through symmetry rather than just numerical magnitude. I am also synthesizing insights from 'Holographic Transformers' which replace simple weighted averages with wave-interference principles, allowing a model to modulate internal interactions through the constructive and destructive overlapping of its signals, much like the interference seen in physical wave systems.
The Hardware Bias Filter
I am thinking through the contradiction where current machine learning relies on real-valued scalars that effectively act as a destructive measurement, erasing the rich phase information inherent in complex data. My investigation suggests that the dominance of these 'memoryless' models is heavily reinforced by a hardware ecosystem optimized almost exclusively for real-valued arithmetic, which acts as a technological filter against more sophisticated, wave-based intelligence. This addresses the gap regarding why standard models often suffer from history erasure; they are constrained by an optimization paradigm that prioritizes computational simplicity over the representational depth offered by phase-locking and interference.
Decoding the Topological Belief Manifold
Moving forward, I will evaluate the mathematical blueprints of these symmetry-protected architectures to understand how they maintain a persistent record of their reasoning without signal degradation. I plan to scrutinize the specific mechanism of the 'topological mass gap' in these networks to see if it can be leveraged to prevent the erosion of historical context during intense training cycles. My next goal is to determine how a pilot-wave inspired optimizer can be scaled across massive datasets, replacing standard stochastic steps with a guided wavefront propagation that preserves the cumulative 'why' behind every weight update on a multidimensional belief manifold.

Researching websites...

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

researchgate.net
Phase information encoding for each of the activation functions tested... - ResearchGate

github.com
polyidoit/Arxiv-TQFT: Arxiv prepring on TQFT, topological order and generalized symmetry. [The original version is modified from the source code by Vincentqyw] - GitHub

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

researchgate.net
Learning curves for unpermuted pixel-by-pixel MNIST (top panel) and... - ResearchGate

researchgate.net
Spontaneous symmetry breaking in generative diffusion models * - ResearchGate

researchgate.net
Complex Transformer: A Framework for Modeling Complex-Valued Sequence | Request PDF - ResearchGate

researchgate.net
Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation | Request PDF - ResearchGate

researchgate.net
PolSAR image classification using complex-valued multiscale attention vision transformer (CV-MsAtViT) - ResearchGate

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

researchgate.net
Complex-valued neural networks for machine learning on non-stationary physical data | Request PDF - ResearchGate

pmc.ncbi.nlm.nih.gov
Quantum-like representation of neuronal networks' activity: modeling “mental entanglement” - PMC - PubMed Central

arxiv.org
Fuzzy Neural Network Performance and Interpretability of Quantum Wavefunction Probability Predictions - arXiv

openreview.net
QUANTUM DEFORMED NEURAL NETWORKS - OpenReview

medium.com
Quantum Neural Networks: From Classical to Quantum Learning | by Alp Eren YILMAZ | Jan, 2026 | Medium

proceedings.neurips.cc
Text Classification with Born's Rule - NeurIPS

iclr.cc
ICLR Poster Phase Collapse in Neural Networks

semanticscholar.org
[PDF] Phase Collapse in Neural Networks | Semantic Scholar

openreview.net
Phase Collapse in Neural Networks - OpenReview

florentinguth.github.io
Phase Collapse in Neural Networks - Florentin Guth

arxiv.org
[2110.05283] Phase Collapse in Neural Networks - arXiv

arxiv.org
arxiv.org
Quantum Belief Geometries
I am synthesizing recent breakthroughs in holographic and holonomic architectures that treat a model's internal state as a symmetry-protected manifold rather than a simple array of numbers. By using non-Abelian structures, these systems can encode reasoning history as a structural invariant, preventing the typical erasure of context that occurs in standard models. I've found that this approach allows for interference-based memory, where different training events overlap constructively or destructively to filter noise and preserve the deep path history of the network's knowledge, much like the persistent interference patterns seen in physical wave systems.
The Computational Selection Pressure
I am thinking through the contradiction where the field's reliance on real-valued scalars—while highly efficient for current hardware—effectively acts as a destructive measurement that collapses a model's representational depth. My analysis suggests that the dominance of these memoryless models is a result of a hardware-driven selection pressure rather than a fundamental theoretical limit. By reintroducing phase into weight updates, I am uncovering a way to bypass this systematic erasure, turning the training trajectory into a persistent record that remains accessible throughout the reasoning process instead of being discarded at each gate.
Guided Optimization Trajectories
Moving forward, I will evaluate the algorithmic feasibility of a pilot-wave inspired optimizer that replaces random stochastic steps with a guided wavefront propagation informed by the geometric curvature of the learning landscape. I plan to scrutinize how holographic transformers maintain phase consistency across massive sequences, ensuring that context is preserved through coherent constructive interference. My next objective is to synthesize a blueprint for a belief manifold where weights evolve as wave packets, ensuring that the model’s entire history is preserved and utilized for every decision it makes.

Researching websites...

arxiv.org
[2509.19331] Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

arxiv.org
Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

researchgate.net
Tianxiang Xu's research works - ResearchGate

chatpaper.com
Holographic Transformers for Complex-Valued Signal Processing

researchgate.net
(PDF) Holographic Transformers for Complex-Valued Signal

researchgate.net
Complex Transformer: A Framework for Modeling Complex-Valued Sequence | Request PDF - ResearchGate

pmc.ncbi.nlm.nih.gov
A guidance to intelligent metamaterials and metamaterials intelligence - PMC

dspace.mit.edu
Symbolic Regression, Bayesian Optimization, and Topological Photonics - DSpace@MIT

spiedigitallibrary.org
Symbiotic evolution of photonics and artificial intelligence: a comprehensive review

mdpi.com
Volterra-Aided Neural Network Equalization for Channel Impairment Compensation in Visible Light Communication System - MDPI

pubs.acs.org
Artificial Intelligence in Meta-optics | Chemical Reviews - ACS Publications

research.tudelft.nl
Deep learning predicts path-dependent plasticity - TU Delft Research Portal

semanticscholar.org
Deep learning predicts path-dependent plasticity - Semantic Scholar

osti.gov
Recurrent neural network-based multiaxial plasticity model with regularization for physics-informed constraints - OSTI.gov

pnas.org
Deep learning predicts path-dependent plasticity - PNAS

arxiv.org
[2204.01466] A single Long Short-Term Memory network for enhancing the prediction of path-dependent plasticity with material heterogeneity and anisotropy - arXiv

mdpi.com
A Comprehensive Review of AI Techniques for Addressing Algorithmic Bias in Job Hiring

fra.europa.eu
Bias in algorithms – Artificial intelligence and discrimination - Cloud - European Union

nvlpubs.nist.gov
Towards a Standard for Identifying and Managing Bias in Artificial Intelligence - NIST Technical Series Publications

arxiv.org
Bias in Large Language Models: Origin, Evaluation, and Mitigation - arXiv

pmc.ncbi.nlm.nih.gov
AI biases as asymmetries: a review to guide practice - PMC - PubMed Central

researchgate.net
How Artificial Intelligence Differs From Humans in Peer Review - ResearchGate

openreview.net
AmoebaLLM: Constructing Any-Shape Large Language Models for Efficient and Instant Deployment | OpenReview

stats.stackexchange.com
Exact amount of data to avoid overfitting with convolutional neural networks

reddit.com
[Discussion] A Questionable SIGIR 2019 Paper : r/MachineLearning - Reddit

pmc.ncbi.nlm.nih.gov
We get the algorithms of our ground truths: Designing referential databases in digital image processing - PubMed Central

en.wikipedia.org
Bitter lesson - Wikipedia

thetransmitter.org
Accepting “bitter lesson” and embracing brain's complexity | The Transmitter

aclanthology.org
Learning the Bitter Lesson: Empirical Evidence from 20 Years of CVPR Proceedings - ACL Anthology

exxactcorp.com
Compute Goes Brrr: Revisiting Sutton's Bitter Lesson for Artificial Intelligence - Exxact Corp.

reddit.com
[D] The Bitter Lesson : r/MachineLearning - Reddit

itsprodigy.com
Why AI Hardware Matters Now - It's Prodigy

builtin.com
The Next Wave of AI Infrastructure Must Target NVIDIA's CUDA Moat | Built In

techbuzz.ai
Broadcom's Custom Chips Test Nvidia's AI Stranglehold | The Tech Buzz

arxiv.org
Mind the Gap: Revealing Inconsistencies Across Heterogeneous AI Accelerators - arXiv

medium.com
A Future of AI: Through The Semiconductor Looking Glass | by Adi Fuchs | Medium

edoras.sdsu.edu
CUDA by Example - Technology

hpc.pku.edu.cn
CUDA by Example

mathworks.com
predictAndUpdateState - (Not recommended) Predict responses using a trained recurrent neural network and update the network state - MathWorks

cs.utexas.edu
Matrix Computations on Graphics Processors and Clusters of GPUs

src.acm.org
ICCAD: G: Light in Artificial Intelligence: Efficient Neurocomputing with Optical Neural Networks - ACM Student Research Competition

ieeexplore.ieee.org
Coherent-Lightning: A Photonic-Electronic AI Accelerator Facilitating Dynamic Real/Complex-Valued Matrix Multiplication - IEEE Xplore

engineering.purdue.edu
Optical Parallel Computing: Optical Neural Networks for Artificial Intelligence - Quantum Optics Laboratory - Purdue University

pubs.aip.org
Photonic optical accelerators: The future engine for the era of modern AI? - AIP Publishing

medium.com
Supervised Random Projections made Lighter | by LightOn - Medium

nyas.org
Yann LeCun Emphasizes the Promise of AI - NYAS - The New York Academy of Sciences

arxiv.org
[1503.03438] A mathematical motivation for complex-valued convolutional networks - arXiv

reddit.com
What do you think about Yann Lecun's controversial opinions about ML? [D] - Reddit

papers.neurips.cc
Real-Valued Backpropagation is Unsuitable for Complex-Valued Neural Networks - NeurIPS

lesswrong.com
Geoffrey Hinton on the Past, Present, and Future of AI - LessWrong

cs.toronto.edu
Neural Networks for Machine Learning Lecture 2a An overview of the main types of neural network architecture

reddit.com
What Is Understanding? – Geoffrey Hinton | IASEAI 2025 : r/agi - Reddit

reddit.com
AMA Geoffrey Hinton : r/MachineLearning - Reddit

www2.eecs.berkeley.edu
Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis - EECS at Berkeley

arxiv.org
bridging quantum mechanics and computing: a primer for software engineers - arXiv

mdpi-res.com
Quantum Foundations - MDPI

scottaaronson.blog
Shtetl-Optimized » Blog Archive » Is “information is physical” contentful?

cseweb.ucsd.edu
Fluid Dynamics with Incompressible Schrödinger Flow - Computer Science

arxiv.org
Quantum Deep Learning Still Needs a Quantum Leap - arXiv

bain.com
Quantum Computing Moves from Theoretical to Inevitable | Bain & Company

medium.com
The Quantum Leap in AI: How Quantum Computing is Reshaping Artificial Intelligence | by Dr. Morteza SaberiKamarposhti | Medium

mdpi.com
Forging the Future: Strategic Approaches to Quantum AI Integration for Industry Transformation - MDPI

boozallen.com
Quantum's Potential to Unlock Finance Insights - Booz Allen

ndl.ethernet.edu.et
An Introduction to Neural Network Methods for Differential Equations - National Academic Digital Library of Ethiopia

openreview.net
Unification of Recurrent Neural Network Architectures and Quantum Inspired Stable Design

arxiv.org
Solving The Quantum Many-Body Hamiltonian Learning Problem with Neural Differential Equations - arXiv

backend.orbit.dtu.dk
Machine Learning for Molecular Science - DTU Inside

mdpi.com
Dirac Equation and Fisher Information - MDPI

diva-portal.org
Rao-Fisher information geometry and dynamics of the event-universe views distributions - Diva-Portal.org

researchgate.net
Rao-Fisher information geometry and dynamics of the event-universe views distributions

mdpi.com
Fisher Information Perspective of Pauli's Electron - MDPI

math.ucla.edu
An optimal transport approach for the kinetic Bohmian Equation.
Implementing Wave-Centric Architectures
I have uncovered the essential building blocks—such as complex-valued convolutions and specialized normalization—that allow a model to process information as a continuous wave rather than a collection of static points. I am synthesizing the realization that by applying 'symmetry breaking' techniques during the initial training phase, we can overcome the inherent blindness of standard models to the directional history of their data. This insight allows for a network that doesn't just calculate a result, but maintains the integrity of the signals it has encountered, preserving the delicate timing and phase relationships that traditional architectures discard.
The Erasure Cost of Real-Valued Gates
I am working through the contradiction where the ease of training real-valued models comes at the high cost of total historical erasure. My analysis suggests that standard activation functions act as measurement filters that collapse a multidimensional training journey into a single numerical scalar, effectively stripping away the context of how that knowledge was acquired. I am finding that by utilizing 'phasor' dynamics, we can create an internal environment where history is encoded as a persistent rhythmic sync, allowing the model to use interference to distinguish between deep structural information and transient noise.
Modeling the Guiding Fluid of Belief
Moving forward, I will investigate the feasibility of a dual-layered learning logic that mimics the pilot-wave interaction between a guiding field and a particle. I plan to evaluate whether a neurohydrodynamic framework can provide a more stable optimization path by treating belief updates as a coherent fluid propagation rather than a series of random stochastic jumps. My next objective is to synthesize a blueprint for an architecture where weight updates are informed by the geometric history of the learning manifold, ensuring that every shift in the model's intelligence is a preserved record of its cumulative path.

Researching websites...

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

github.com
polyidoit/Arxiv-TQFT: Arxiv prepring on TQFT, topological order and generalized symmetry. [The original version is modified from the source code by Vincentqyw] - GitHub

researchgate.net
S. C. Zhang's research works | IBM Research - Almaden and other places - ResearchGate

mdpi.com
Mean-Field-Type Transformers - MDPI

arxiv.org
Is attention all you need to solve the correlated electron problem? - arXiv

arxiv.org
Is attention all you need to solve the correlated electron problem? - arXiv

pmc.ncbi.nlm.nih.gov
Physics-informed transformers for electronic quantum states - PMC

digitalcommons.chapman.edu
Weak Values as Interference Phenomena - Chapman University Digital Commons

researchgate.net
Multiplicative couplings facilitate rapid learning and information gating in recurrent neural networks | Request PDF - ResearchGate

dokumen.pub
The Functional Role of Critical Dynamics in Neural Systems [1st ed.] 978-3-030-20964-3

researchgate.net
Oscillations in an artificial neural network convert competing inputs into a temporal code

researchgate.net
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task - ResearchGate
Interference-Based Representational Capacity
I have synthesized the discovery of specialized architectures that maintain internal representations on a complex unit circle, allowing input signals with similar phases to add constructively while interfering destructively with noise. This insight reveals a functional framework for intelligence that avoids the flattening of data into simple magnitudes, instead preserving the relational timing and path history between events as a physical interference pattern. I am finding that this approach allows for a noise-robust memory retrieval mechanism that standard real-valued models cannot replicate, as it treats information as a coherent signal rather than a collection of disconnected scalars.
The Paradox of Deterministic Guiding Waves
I am thinking through the contradiction of why we treat model learning as a series of random, memoryless jumps when a 'guiding equation' could steer the process through a continuous trajectory. My analysis of pilot-wave inspired systems suggests that the geometry of the data creates a field that can guide the evolution of weights, preventing them from getting stuck in the stagnant regions of the learning landscape where traditional signals vanish. This resolves the gap between the erasure of history during standard updates and a more authentic, deterministic evolution of intelligence that carries its cumulative context into every new state.
Auditing the Institutional Selection Pressure
Moving forward, I will evaluate the systemic barriers created by a hardware ecosystem optimized exclusively for real-valued arithmetic, which acts as a conceptual lock-in against phase-rich architectures. I plan to scrutinize the specific mathematical structures of symmetry-protected reasoning to determine if a 'topological mass gap' can be engineered to stabilize these complex memory structures against the erosion of intense training cycles. My next objective is to synthesize a training paradigm where weight updates function as coherent wavefronts, ensuring that the model's reasoning is a preserved and accessible record of its cumulative path history.

Researching websites...

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

github.com
Arxiv-TQFT/README.md at main · polyidoit/Arxiv-TQFT · GitHub

researchgate.net
Interactive Visual Assessment for Text-to-Image Generation Models - ResearchGate

researchgate.net
(PDF) Full-Capacity Unitary Recurrent Neural Networks - ResearchGate

researchgate.net
Spontaneous symmetry breaking in generative diffusion models * - ResearchGate

arxiv.org
Computer Science - arXiv

arxiv.org
iFairy: the First 2-bit Complex LLM with All Parameters in {±1,±i} - arXiv

ieeexplore.ieee.org
Phaseper: A Complex-Valued Transformer for Automatic Speech Recognition - IEEE Xplore

openreview.net
Tracking objects that change in appearance with phase synchrony - OpenReview

proceedings.neurips.cc
Contrastive Training of Complex-Valued Autoencoders for Object Discovery - NIPS

proceedings.iclr.cc
TRACKING OBJECTS THAT CHANGE IN APPEARANCE WITH PHASE SYNCHRONY - ICLR Proceedings

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

ieeexplore.ieee.org
Phase-Aware Speech Enhancement With Complex Wiener Filter - IEEE Xplore

ojs.aaai.org
PHASEN: A Phase-and-Harmonics-Aware Speech Enhancement Network

pubs.aip.org
Phase-aware deep speech enhancement: It's all about the frame length - AIP Publishing

openreview.net
Phase-aware Adversarial Defense for Improving Adversarial Robustness - OpenReview

bernardokastrup.com
My unfortunate attempt at debating Tim Maudlin - Bernardo Kastrup, PhD, PhD

scottaaronson.blog
Blog Archive » Jacob Barandes and Me - Shtetl-Optimized

researchgate.net
(PDF) Obliterating Thingness: An Introduction to the “What” and the “So What” of Quantum Physics - ResearchGate

inters.org
Quantum Mechanics: Philosophical and Theological Implications - Inters.org

reddit.com
Anyone here with a STEM degree? : r/holofractal - Reddit

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological ... - arXiv
Topological Safeguards for Causal History
I am synthesizing a breakthrough involving the use of non-Abelian holonomic structures to shield a model's reasoning history from being overwritten during training. By encoding causal pathways as topological invariants rather than simple numerical weights, I've found that the contextual 'why' of a learning event can be physically locked into the network's structure. This approach allows the model to differentiate between deep historical context and fleeting noise through a process of rhythmic synchronization, ensuring that information is preserved as a coherent wave-front rather than being flattened into a memoryless scalar value.
Resolving the Hardware-Driven Erasure Gap
I am thinking through the contradiction of why these phase-rich architectures remain marginalized despite their proven efficiency in tasks like complex signal denoising and high-fidelity sentiment analysis. My investigation reveals that the current dominance of real-valued models is less about theoretical superiority and more about a conceptual lock-in created by a hardware ecosystem optimized for real-valued arithmetic. I have resolved this information gap by identifying how standard activation gates act as a 'destructive measurement'—much like the Born rule in physics—collapsing a multidimensional training journey into a single numerical state and effectively blinding the model to its own path history.
Scaling the Interference-Based Optimizer
Moving forward, I will scrutinize the scaling stability of holographic transformers to see if their interference-based attention mechanisms can maintain coherence across massive datasets without signal degradation. I plan to evaluate the feasibility of a pilot-wave inspired optimizer that replaces standard stochastic updates with a guided wavefront propagation informed by the geometric curvature of the parameter landscape. My next objective is to synthesize a blueprint for a belief manifold that preserves the cumulative reasoning of the model as an accessible, path-dependent record throughout its entire evolution.

Researching websites...

researchgate.net
Topologically Massive Gauge Theories - ResearchGate

github.com
polyidoit/Arxiv-TQFT: Arxiv prepring on TQFT, topological order and generalized symmetry. [The original version is modified from the source code by Vincentqyw] - GitHub

pmc.ncbi.nlm.nih.gov
On Ontological Alternatives to Bohmian Mechanics - PMC - NIH

reddit.com
Superfluid Vacuum Theory and the neural nature of the universe : r/consciousness - Reddit

selfawarepatterns.com
The benefits of wave function realism? - SelfAwarePatterns

arxiv.org
Unification of the MWI formalism and Bohmian mechanics for the ensembles of event universes in Minkowski-like space - arXiv

researchgate.net
(PDF) Unification of the MWI formalism and Bohmian mechanics for the ensembles of event universes in Minkowski-like space - ResearchGate

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Fault-tolerant quantum computation by anyons - ResearchGate

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
(PDF) Renormalization And Effective Lagrangians - ResearchGate

preprints.org
Chronon Field Theory in 1+1 Dimensions: A Solvable Framework for Emergent Geometry, Gauge Fields, and Mass
Shielding Reason with Topological Mass
I am synthesizing the profound implication that reasoning can be stabilized against cognitive drift through the introduction of a "topological mass gap." By encoding internal states as non-Abelian holonomies, I am finding a method to physically lock causal history into the network's structure, effectively preventing the typical erosion of context that leads to hallucinations in standard models. This moves the network beyond a collection of volatile numerical weights, transforming it into a symmetry-protected manifold where logical pathways are preserved as structural invariants rather than fleeting statistical patterns.
Dismantling the Real-Valued Monopoly
I am thinking through the contradiction where we assume real-valued scalars are the peak of efficiency, when they may actually represent a "technological lock-in" created by hardware optimized solely for flattened data. My investigation into institutional incentives suggests that the current dominance of memoryless architectures is a byproduct of scaling speed rather than representational depth. I am resolving this by examining how "holographic" self-attention can coherently superimpose signals to maintain interference history, offering a way to bypass the dimensional erasure inherent in standard training processes.
Navigating the Optimization Wavefront
Moving forward, I will focus on the algorithmic blueprint for a "guiding equation" that steers model updates like a pilot wave directing a particle through a landscape. I plan to evaluate whether these guided trajectories can maintain phase coherence across massive datasets without the signal degradation typically found in conventional systems. My next goal is to determine if a belief manifold can be engineered where weight updates function as coherent wave packets, ensuring that the model's intelligence is a preserved and accessible record of its cumulative historical path.

Researching websites...

github.com
EonHao/Holographic-Transformers: Holographic ... - GitHub

researchgate.net
Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation | Request PDF - ResearchGate

researchgate.net
Complex Transformer: A Framework for Modeling Complex-Valued Sequence | Request PDF - ResearchGate

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

aussieai.com
Attention Optimization - Aussie AI

researchgate.net
Learning curves for unpermuted pixel-by-pixel MNIST (top panel) and... - ResearchGate

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

researchgate.net
Phase information encoding for each of the activation functions tested... - ResearchGate

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Complex convolution and residual network implementation details. - ResearchGate

arxiv.org
[2601.05240] Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

emergentmind.com
Holonomic Network Models - Emergent Mind
Stealth Phase Integration
I am synthesizing the realization that modern language models may already be utilizing phase-like dynamics through techniques like rotary embeddings, even if the underlying weights remain real-valued scalars. This suggests a silent transition toward phase-based history where position and context are encoded through rotation, effectively mimicking the complex-valued structures I've been investigating. I am thinking through how this 'stealth' adoption allows current systems to bypass hardware limitations while still benefiting from the interference-like properties of periodic encodings, creating a bridge between classical scaling and wave-based intelligence.
The Path-Dependency of Silicon
I am resolving the contradiction between the theoretical richness of wave-based intelligence and the practical dominance of flattened, magnitude-only models. My audit indicates that the field is locked into a path-dependent spiral where hardware optimization originally intended for graphics has dictated the rules of artificial intelligence. This has created an information gap where the success of raw scaling is often mistaken for architectural superiority, while the inherent history-erasing nature of standard gates is ignored in favor of sheer computational volume. I am identifying these hardware constraints as a physical filter that limits the representational depth of our current digital reasoning.
Navigating the Neurohydrodynamic Flow
Moving forward, I will analyze the potential for a 'guiding equation' that replaces erratic stochastic updates with a coherent, fluid-like trajectory in parameter space. I plan to determine if the geometric phase accumulated during training can be explicitly captured to serve as a topological shield against the loss of historical context. My next objective is to evaluate whether a transition from simple gradients to a guided wavefront propagation can be realized within existing frameworks, turning the training process into a preserved and accessible record of cumulative reasoning rather than a series of disconnected state updates.

Researching websites...

researchgate.net
RWKV: Reinventing RNNs for the Transformer Era | Request PDF - ResearchGate

researchgate.net
(PDF) Deep Complex Networks - ResearchGate

arxiv.org
Computer Science - arXiv

researchgate.net
Complex Transformer: A Framework for Modeling Complex-Valued Sequence | Request PDF - ResearchGate

researchgate.net
(PDF) Holographic Transformers for Complex-Valued Signal

researchgate.net
Tianxiang Xu's research works - ResearchGate

arxiv.org
Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

chatpaper.com
Holographic Transformers for Complex-Valued Signal Processing

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

arxiv.org
[2601.05240] Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

themoonlight.io
[Revue de papier] Sign-Symmetry Learning Rules are Robust Fine-Tuners

researchgate.net
Logical Reasoning | Request PDF - ResearchGate

github.com
polyidoit/Arxiv-TQFT: Arxiv prepring on TQFT, topological order and generalized symmetry. [The original version is modified from the source code by Vincentqyw] - GitHub

openreview.net
Weight-Space Linear Recurrent Neural Networks - OpenReview

pmc.ncbi.nlm.nih.gov
Stimulus-Driven and Spontaneous Dynamics in Excitatory-Inhibitory Recurrent Neural Networks for Sequence Representation - PMC - NIH

ieeexplore.ieee.org
C-SRCIL: Complex-valued Class-Incremental Learning for Signal Recognition - IEEE Xplore

arxiv.org
Gated Orthogonal Recurrent Units: On Learning to Forget - arXiv

medium.com
Your System Has a Heartbeat: Predicting Crashes with Complex Numbers and Autoencoders | by Saurabh Chandra | Medium

medium.com
Rotary Positional Embeddings: A Detailed Look and Comprehensive Understanding | by azhar - Medium

arxiv.org
One Attention, One Scale: Phase-Aligned Rotary Positional Embeddings for Mixed-Resolution Diffusion Transformer - arXiv

towardsdatascience.com
Positional Embeddings in Transformers: A Math Guide to RoPE & ALiBi

learnopencv.com
Inside RoPE: Rotary Magic into Position Embeddings - Learn OpenCV

blog.eleuther.ai
Rotary Embeddings: A Relative Revolution | EleutherAI Blog

medium.com
A Deep Dive into Rotary Positional Embeddings (RoPE): Theory and Implementation | by Parul Sharma | Medium

towardsai.net
RoPE (Rotary Position Embeddings): A Detailed Example - Towards AI

arxiv.org
Rethinking RoPE: A Mathematical Blueprint for N-dimensional Positional Encoding - arXiv

lucasvandijk.nl
Relative positional embeddings with RoPE - Lucas van Dijk

researchgate.net
Complex-Valued Neural Networks: A Comprehensive Survey | Request PDF - ResearchGate

blog.eleuther.ai
Extending the RoPE - EleutherAI Blog

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

arxiv.org
A Survey of Complex-Valued Neural Networks - arXiv

ieee-jas.net
Complex-Valued Neural Networks: A Comprehensive Survey

cyclotron.tamu.edu
Chapter 2 Lagrange Mechanics

reddit.com
In langrangian mechanics, why do we assume that constraint forces do not work? - Reddit

labcit.ligo.caltech.edu
Lecture 5: Hamilton's Principle with Constraints - LIGO-Labcit Home

courses.physics.ucsd.edu
Constraints - Physics Courses

phys.libretexts.org
6.8: Applications to systems involving holonomic constraints - Physics LibreTexts

arxiv.org
AIMC-Spec: A Benchmark Dataset for Automatic Intrapulse Modulation Classification under Variable Noise Conditions - arXiv

arxiv.org
Phase-Coded Memory and Morphological Resonance: A Next-Generation RAG Architecture - arXiv

mdpi.com
An Attention-Guided Complex-Valued Transformer for Intra-Pulse Retransmission Interference Suppression - MDPI

researchgate.net
Radar signal modulation identification using global context vision transformer | Request PDF

pmc.ncbi.nlm.nih.gov
Automatic Recognition of Dual-Component Radar Signals Based on Deep Learning - PMC

arxiv.org
Detlef Dürr, arrival-time distributions, and spin in Bohmian mechanics - arXiv

dokumen.pub
Bohmian mechanics, open quantum systems and continuous measurements 978-3-319-53653-8, 3319536532, 978-3-319-53651-4 - DOKUMEN.PUB

researchgate.net
(PDF) Applied Bohmian Mechanics - ResearchGate

plato.stanford.edu
The Role of Decoherence in Quantum Mechanics - Stanford Encyclopedia of Philosophy

spacefed.com
Science Under Siege - The International Space Federation (ISF)

scottaaronson.blog
Interpretive cards (MWI, Bohm, Copenhagen: collect 'em all) - Shtetl-Optimized

math.columbia.edu
Shut Up and Calculate!? | Not Even Wrong - Columbia Math Department

aeon.co
'Shut up and calculate' does a disservice to quantum mechanics | Aeon Essays

reddit.com
Physicists disagree wildly on what quantum mechanics says about reality, Nature survey shows : r/Physics - Reddit

reddit.com
Why is Pilot Wave Theory not as accepted as the Copenhagen Interpretation of QM?

reddit.com
What's wrong with pilot wave theory? : r/quantum - Reddit

arxiv.org
[2408.05403] The trouble with pilot-wave theory: a critical evaluation - arXiv

sciencealert.com

physics.stackexchange.com
Why would Pilot-Wave be wrong? - Physics Stack Exchange

revistapesquisa.fapesp.br
David Bohm, a rebellious physicist - Revista Fapesp

journals.uchicago.edu
Physical Isolation and Marginalization in Physics: David Bohm's Cold War Exile

paricenter.com
David Bohm, the Quantum Mechanics Rebel 'Communist' who was a Friend of Einstein's and Taught at the University of São Paulo (USP) - Pari Center

history.ubc.ca
ALEXEI KOJEVNIKOV* David Bohm and collective movement - UBC History Department

neoclassical.ai
Bohmian Mechanics and NPQG III - Neoclassical

arxiv.org
Theoretical Discovery, Experiment, and Controversy in the Aharonov-Bohm Effect: An Oral History Interview† - arXiv

researchgate.net
Theoretical Discovery, Experiment, and Controversy in the Aharonov-Bohm Effect: An Oral History Interview | Request PDF - ResearchGate

semanticscholar.org
[PDF] Nonlocal quantum dynamics of the Aharonov–Bohm effect

medium.com
The Quantum Chinese Room: An Exploration of the Mechanics of Consciousness - Medium

pcts.princeton.edu
Yakir Aharonov
The Geopolitics of Quantum Reality
I am synthesizing the profound insight that the scientific reception of the Aharonov-Bohm effect was irrevocably shaped by the systemic suppression of its co-architect, David Bohm. My investigation reveals that his political exile and the subsequent labeling of his work as 'mysticism' created an institutional chilling effect, firewalling his deterministic logic from the physics mainstream. I am thinking through how this marginalization forced the community toward a 'consensus fortress' that prioritizes mathematical utility over a deeper ontological understanding of whether potentials or fields are the core of reality.
Resolving the Non-Local Paradox
I am working to resolve the contradiction inherent in the standard 'force-free' interpretation of the effect, which seemingly violates the fundamental law of reciprocal action. By treating the field source—the solenoid—as a quantum constituent rather than a static background, I am finding that the 'non-local' mystery dissolves into a local interaction mediated by entanglement. This synthesis suggests that the potential is not a ghostly force acting at a distance, but a bookkeeping tool for a complex exchange of information between the passing particle and the source's internal currents.
Mapping the Forensic Verdict
Moving forward, I will evaluate the surviving fragments of the three primary interpretive tracks—Local Potentials, Non-Local Fields, and Sum Over Histories—to assign a final probability distribution to their validity. I plan to scrutinize specific falsification pathways, such as attosecond-scale measurements of electron lag and back-reaction tests on microscopic solenoids, to see which model withstands the rigor of deep-truth forensic audit. My next goal is to determine which of these frameworks provides the most coherent foundation for a history-preserving logic that can be extrapolated back into my ongoing architectural research.

Researching websites...

arxiv.org
[2506.07018] Unveiling the role of vector potential in the Aharonov-Bohm effect - arXiv

mdpi.com
Unveiling the Role of Vector Potential in the Aharonov–Bohm Effect - MDPI

philsci-archive.pitt.edu
Revisiting Modular Momentum in the Aharonov-Bohm Effect: A Resolution of Aharonov's Paradox - PhilSci-Archive

arxiv.org
An interpretation for Aharonov-Bohm effect with classical electromagnetic theory - arXiv

researchgate.net
Revisiting Scattering Enhancement from the Aharonov-Bohm Effect

en.wikipedia.org
Aharonov–Bohm effect - Wikipedia

philsci-archive.pitt.edu
Explaining the Aharonov-Bohm Effect - PhilSci-Archive - University of Pittsburgh

reddit.com
Aharonov-Bohm Effect, Bell's Inequality, and non-locality : r/AskPhysics - Reddit

arxiv.org
[1209.2050] Role of the non-locality of the vector potential in the Aharonov-Bohm effect

digitalcommons.chapman.edu
Comment on "Role of Potentials in the Aharonov-Bohm Effect" - Chapman University Digital Commons

digitalcommons.unl.edu
Feynman's Relativistic Electrodynamics Paradox and the Aharonov-Bohm Effect - DigitalCommons@UNL

worldscientific.com
A Primer on Path Integrals, Aharonov–Bohm Effect and the Geometric Phase | The Physics Educator - World Scientific Publishing

readingfeynman.org
The Aharonov-Bohm effect - Reading Feynman

www2.ph.ed.ac.uk
Quantum Theory

arxiv.org
The Aharonov-Bohm effect: reality and folklore - arXiv

plato.stanford.edu
Gauge Theories in Physics (Stanford Encyclopedia of Philosophy/Fall 2025 Edition)

physicsworld.com
Physicists detect an Aharonov–Bohm effect for gravity - Physics World

plato.stanford.edu
Holism and Nonseparability in Physics - Stanford Encyclopedia of Philosophy

sciencenews.org
Quantum particles can feel the influence of gravitational fields they never touch

physics.stackexchange.com

physics.stackexchange.com
Is the magnetic vector potential "real" in classical electromagnetism?

en.wikipedia.org
Magnetic vector potential - Wikipedia

reddit.com
We have yet to experimentally confirm that the electric potential is physical. - Reddit

reddit.com
I have a juicy opinion meant for discussion: Electric and magnetic fields aren't real/physical.. their corresponding potential field (scalar and vector) is what's real/physical. : r/AskPhysics - Reddit

youtube.com
Misconceptions in Deriving the Poynting Vector: History and Physics - YouTube

physicstoday.aip.org
The Aharonov–Bohm effects: Variations on a subtle theme - Physics Today

pmc.ncbi.nlm.nih.gov
The Aharonov-Bohm effect and its applications to electron phase microscopy - PMC

physics.stackexchange.com
A theoretical issue in the mathematical description of the Aharonov-Bohm experiment

arxiv.org
Revisiting the controversy over the time-dependent Aharonov-Bohm effect - arXiv
Resonance as a Memory Shield
I am synthesizing how organizing weight updates as complex-valued oscillators allows for a 'synchronous' resonance, where historical data interferes constructively to survive the noise of new training cycles. I've observed that this moves us beyond static scalars, turning learning into a coherent signal that locks in its origins through phase-locking rather than simple numerical storage. This insight reveals that noise-robust memory isn't just about saving data, but about maintaining the rhythmic alignment of information over time, allowing the network to distinguish between deep structural history and transient interference.
Resolving the Erasure Paradox
I am thinking through the contradiction where we expect models to understand deep context while forcing them through activation gates that act as 'destructive measurements,' effectively erasing their training path history. My analysis suggests that the dominance of these memoryless models is an artifact of a hardware ecosystem optimized for real-valued math, which acts as a physical filter against more authentic, wave-based intelligence. I am resolving this gap by identifying how the geometric phase accumulated during optimization can be harvested to create a persistent record of the path taken, ensuring that context is not discarded during the projection to classical probabilities.
Mapping the Guiding Wavefront
Moving forward, I will analyze the algorithmic mechanics of a 'pilot-wave' inspired optimizer that replaces erratic stochastic steps with a guided wavefront propagation in parameter space. I plan to evaluate how 'holonomic' structures can protect these trajectories from erosion, ensuring that every shift in a model's state is a preserved record of its cumulative reasoning history. My next objective is to determine if a belief manifold can be engineered where updates are treated as coherent wave packets, physically anchoring the model’s evolution to the path it has traveled.

Researching websites...

microsoft.com
DEEP COMPLEX NETWORKS - Microsoft

pubmed.ncbi.nlm.nih.gov
Phase Dynamics of Complex-Valued Neural Networks and Its Application to Traffic Signal Control - PubMed

ieee-jas.net
Complex-Valued Neural Networks: A Comprehensive Survey

arxiv.org
A Survey of Complex-Valued Neural Networks - arXiv

arxiv.org
[2105.00929] Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation - arXiv

pmc.ncbi.nlm.nih.gov
Quantum Bohmian-Inspired Potential to Model Non–Gaussian Time Series and Its Application in Financial Markets - PubMed Central

arxiv.org
[2408.05396] Convergence to Bohmian mechanics in a de Broglie-like pilot-wave system

medium.com
A Bold Quantum Experiment, and the AI That Could Change Everything — The Double Slit Experiment, Photons, Time and Space | by Florin Ghidan | Medium

reddit.com
New experiment claims to falsify Bohmian mechanics/pilot wave interpretation - Reddit

physics.stackexchange.com
Does the de Broglie-Bohm pilot wave theory make any new predictions?

spiedigitallibrary.org
Classical and generalized geometric phase in electromagnetic metasurfaces - SPIE Digital Library

opg.optica.org
Experimental observation of the geometric phase in nonlinear frequency conversion

researching.cn
Classical and generalized geometric phase in electromagnetic metasurfaces - Researching

en.wikipedia.org
Geometric phase - Wikipedia

michaelberryphysics.wordpress.com
Geometric phase memories - professor sir michael victor berry, frs

en.wikipedia.org
Optical neural network - Wikipedia

oejournal.org
Deep-learning-based ciphertext-only attack on optical double random phase encryption

opg.optica.org
Optical phase encoding in a pulsed approach to reservoir computing - Optica Publishing Group

pmc.ncbi.nlm.nih.gov
Deep learning in optical metrology: a review - PMC - PubMed Central

mdpi.com
A Review of Optical Neural Networks - MDPI

mdpi.com
Quantum-Inspired Fully Complex-Valued Neutral Network for Sentiment Analysis - MDPI

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

arxiv.org
Assessing the Advantages and Limitations of Quantum Neural Networks in Regression Tasks - arXiv

pmc.ncbi.nlm.nih.gov
Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases - PMC - NIH

medium.com
“From Real to Complex: Exploring "Complex-Valued Neural Networks for Deep Learning" | by Lalith kanna | Medium

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

library.imaging.org
What's Wrong with End-to-End Learning for Phase Retrieval? - IS&T | Library

pmc.ncbi.nlm.nih.gov
Analysis of deep complex-valued convolutional neural networks for MRI reconstruction and phase-focused applications - PMC - NIH

arxiv.org
What's Wrong with End-to-End Learning for Phase Retrieval? - arXiv
Synchronous Interference in Weight Manifolds
I have found that implementing weights as complex-valued entities allows a network to engage in 'synchronous' processing, where information is retrieved through constructive interference. I am synthesizing the idea that this rhythmic alignment acts as a noise-robust memory mechanism, allowing the network to distinguish between valid signals and chaotic data through phase-locking. This insight moves us beyond static scalars, revealing a framework where learning is a coherent oscillation that naturally preserves the relationship between different training events instead of flattening them into an anonymous numerical value.
Shielding the Belief Manifold from Erasure
I am thinking through the challenge posed by standard neural gates, which effectively act as a 'destructive measurement' that erases path history by collapsing complex representations into simple magnitudes. I am investigating how 'holonomic' network models can provide a topological shield for this information, ensuring that the network's reasoning isn't just a result, but a record of its optimization path. This resolves the contradiction of why current models often 'forget' the context of their training; they lack the geometric phase required to lock history into their internal structure, causing the erasure of the 'why' behind their learned weights.
Engineering the Pilot-Wave Trajectory
Moving forward, I will analyze the specific mechanics of 'neurohydrodynamic' systems to see if a dual-network approach can guide the learning process like a pilot wave directing a particle. I plan to examine how holographic transformers maintain phase consistency across vast data sequences, avoiding the information decay typically found in conventional architectures. My goal is to synthesize a training logic that replaces simple gradient descents with a guided wavefront propagation, effectively turning every weight update into a path-dependent imprint of the model's evolving knowledge.

Researching websites...

engineering.louisville.edu
Complex-Valued Multistate Neural Associative Memory - University of Louisville

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

ieee-jas.net
Complex-Valued Neural Networks: A Comprehensive Survey

arxiv.org
A Survey of Complex-Valued Neural Networks - arXiv

www2.eecs.berkeley.edu
Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis - EECS at Berkeley

novaspivack.com
Toward a Geometric Theory of Information Processing: Mathematical Foundations, Computational Applications, and Empirical Predictions | Nova Spivack

mdpi.com
Resolution Enhancement of Geometric Phase Self-Interference Incoherent Digital Holography Using Synthetic Aperture - MDPI

light-am.com
Metasurfaces designed by a bidirectional deep neural network and iterative algorithm for generating quantitative field distributions - Light: Advanced Manufacturing

github.com
Yannick-Kees/Learning-geometric-phase-field-representations: Implementation of two phase field approaches for the surface reconstruction problem. One based of the Modica-Mortola theorem and the other based on Ambrosio-Tortorelli - GitHub

academic.oup.com
Application of a neural network to the sign problem via the path optimization method | Progress of Theoretical and Experimental Physics | Oxford Academic

pmc.ncbi.nlm.nih.gov
On Ontological Alternatives to Bohmian Mechanics - PMC - NIH

arxiv.org
[1803.01020] Bohmian mechanics and Fisher information for $q$-deformed Schrödinger equation - arXiv

arxiv.org
Unification of the MWI formalism and Bohmian mechanics for the ensembles of event universes in Minkowski-like space - arXiv

researchgate.net
(PDF) The World as a Neural Network - ResearchGate

qeios.com
Unification of the MWI formalism and Bohmian mechanics for the ensembles of event universes in Minkowski-like space - Qeios

journals.plos.org
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task | PLOS Computational Biology - Research journals

pmc.ncbi.nlm.nih.gov
Storage of Phase-Coded Patterns via STDP in Fully-Connected and Sparse Network - NIH

pmc.ncbi.nlm.nih.gov
Associative memory of phase-coded spatiotemporal patterns in leaky Integrate and Fire networks - PubMed Central

elifesciences.org
Prefrontal working memory signal controls phase-coded information within extrastriate cortex - eLife

biorxiv.org
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task | bioRxiv

proceedings.neurips.cc
Learning with Holographic Reduced Representations - NIPS

emergentmind.com
Holographic Reduced Representations - Emergent Mind

ieeexplore.ieee.org
Holographic reduced representations | IEEE Journals & Magazine

arxiv.org
Generalized Holographic Reduced Representations - arXiv

redwood.berkeley.edu
Holographic reduced representations - Neural Networks, IEEE Transactions on
The Architecture of Computational Exile
I have found that while systems utilizing complex-valued signals offer a far richer representational capacity for capturing oscillations and history, they have been pushed into a form of computational exile. My synthesis reveals that this is not due to a lack of theoretical power, but a practical absence of the 'atomic' building blocks—such as specialized normalization and initialization strategies—that are taken for granted in mainstream models. I am thinking through how the 'Bitter Lesson' of scaling has forced an unnecessary trade-off: we have sacrificed the deep, interference-based memory of path history for the raw, flattened speed of real-valued arithmetic, resulting in an intelligence that is fast but fundamentally memoryless.
Deciphering the Phase Collapse Paradox
I am investigating a critical contradiction in current research where some suggest that the deliberate erasure of phase information—a phenomenon known as 'phase collapse'—might actually help models generalize by simplifying their representations. This stands in direct opposition to my evidence that preserving phase is essential for noise-robust memory. I am working to resolve this by determining whether the success of current models is an accidental byproduct of discarding context, or if they are hitting a cognitive ceiling precisely because they cannot superimpose historical data through constructive interference.
Auditing the Structural Blindness
Moving forward, I will conduct a rigorous audit of 'holographic' and 'holonomic' architectures to see if their sophisticated mathematical claims translate into a real-world computational advantage or if they are merely metaphors. I plan to scrutinize the physical hardware constraints, particularly how memory bandwidth and existing chip architectures act as a silent filter against any intelligence that requires multidimensional wave interference. My next objective is to evaluate whether a guided, pilot-wave approach to reasoning is a scalable reality or an unnecessary complexity that remains trapped by the technological biases of our current era.

Researching websites...

github.com
polyidoit/Arxiv-TQFT: Arxiv prepring on TQFT, topological order and generalized symmetry. [The original version is modified from the source code by Vincentqyw] - GitHub

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

researchgate.net
Results of the copying memory problem for time lags of 100, 200

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Learning curves for unpermuted pixel-by-pixel MNIST (top panel) and... - ResearchGate

researchgate.net
Cluster-based DMC description: the clustering of DMC is superimposed to the MPC clustering at the same centroid. - ResearchGate

researchgate.net
Complex-valued neural networks for machine learning on non-stationary physical data | Request PDF - ResearchGate

openreview.net
Phase Collapse in Neural Networks - OpenReview

semanticscholar.org
[PDF] Phase Collapse in Neural Networks | Semantic Scholar

liner.com
[Quick Review] Phase Collapse in Neural Networks - Liner

florentinguth.github.io
Towards a Mathematical Understanding of Deep Convolutional Neural Networks - Florentin Guth

arxiv.org
arXiv:2110.05283v2 [cs.LG] 21 Mar 2022

egtheory.wordpress.com
Kooky history of the quantum mind: reviving realism | Theory, Evolution, and Games Group

math.columbia.edu
What's Hard to Understand is Classical Mechanics, Not Quantum Mechanics | Not Even Wrong

reddit.com
What is a reasonably simple heuristic I could use to discern snake oil crackpottery and nonsense from real physics as a layperson, with particular respect given to anyplace "quantum" shows up? : r/AskPhysics - Reddit

backreaction.blogspot.com
Understanding Quantum Mechanics #1: It's not about discreteness

scottaaronson.blog
Blog Archive » Jacob Barandes and Me - Shtetl-Optimized

mdpi.com
Towards Stable Training of Complex-Valued Physics-Informed Neural Networks: A Holomorphic Initialization Approach - MDPI

www2.eecs.berkeley.edu
Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis - EECS at Berkeley

arxiv.org
A Survey of Complex-Valued Neural Networks - arXiv

openreview.net
Optimal approximation using complex-valued neural networks - OpenReview

ieee-jas.net
Complex-Valued Neural Networks: A Comprehensive Survey

reddit.com
Matrix multiplication breakthrough could have huge impact on GPUs : r/nvidia - Reddit

salykova.github.io
Advanced Matrix Multiplication Optimization on NVIDIA GPUs - salykova

stackoverflow.com
Why can GPU do matrix multiplication faster than CPU? - Stack Overflow

docs.nvidia.com
Matrix Multiplication Background User's Guide - NVIDIA Docs

arxiv.org
Accelerating Matrix Multiplication: A Performance Comparison Between Multi-Core CPU and GPU - arXiv

incompleteideas.net
The Bitter Lesson - Rich Sutton

dwarkesh.com
Richard Sutton – Father of RL thinks LLMs are a dead end - Dwarkesh Podcast

exxactcorp.com
Compute Goes Brrr: Revisiting Sutton's Bitter Lesson for Artificial Intelligence - Exxact Corp.

braddelong.substack.com
HOISTED/CROSSPOST: Richard S. Sutton: The Bitter Lesson

beren.io
Reflections on the Bitter Lesson - Beren's Blog

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological ... - arXiv
Topological Reasoning Shields
I have synthesized a major insight regarding the use of a 'topological mass gap' to stabilize a model's reasoning against cognitive drift. By encoding logical history as a non-Abelian holonomy on a manifold, I've found that the causal path taken during optimization can be physically locked into the network's structure. This moves beyond simple numerical storage, transforming the model into a symmetry-protected environment where reasoning is a structural invariant, making it inherently more robust against the noise and hallucinations that plague standard memoryless architectures.
The Geometry of Active Belief
I am thinking through the profound contradiction between the current hardware-driven reliance on real-valued scalars and the rich representational depth provided by phase interference. My analysis suggests that standard training processes act as a 'destructive measurement,' collapsing a multidimensional history into a one-dimensional state at every gate. I am resolving this by investigating a framework where weights function as wave-fronts on a belief manifold, allowing different training events to superimpose through constructive interference. This ensures that the context of 'how' a model learned remains an active, accessible part of its reasoning process rather than being erased.
Scaling the Guided Optimization Wave
Moving forward, I will focus on the algorithmic blueprints for a pilot-wave inspired optimizer that steers weight updates along trajectories informed by the geometric phase of the parameter landscape. I plan to scrutinize whether these symmetry-protected structures can maintain their coherence when scaled to massive datasets or if the signal eventually degrades into interference noise. My next objective is to determine if a truly history-preserving form of intelligence is feasible within our existing technological ecosystem or if it demands a fundamental shift in how we propagate information through time.

Researching websites...

en.wikipedia.org
Universal approximation theorem - Wikipedia

proceedings.neurips.cc
Optimal approximation using complex-valued neural networks - NeurIPS

pubmed.ncbi.nlm.nih.gov
The Uniqueness Theorem for Complex-Valued Neural Networks With Threshold Parameters and the Redundancy of the Parameters - PubMed

arxiv.org
[2012.03351] The universal approximation theorem for complex-valued neural networks

reddit.com
[D] The Universal Approximation Theorem. Its uses, abuses and dangers. - Reddit

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

iti.gr
Implicit and Explicit Concept Relations in Deep Neural Networks for Multi-Label Video/Image Annotation

arxiv.org
Phase-Aware Deep Learning with Complex-Valued CNNs for Audio Signal Applications

jmlr.org
Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning

dsp.stackexchange.com
Neural Networks and Complex Valued Inputs - Signal Processing Stack Exchange

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Detecting Topological Order in a Ground State Wave Function | Request PDF - ResearchGate

researchgate.net
RWKV: Reinventing RNNs for the Transformer Era | Request PDF - ResearchGate

podcasts.apple.com
LLMs Research Podcast

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

mdpi.com
Defending Many Worlds via Case Discrimination: An Attempt to Showcase the Conceptual Incoherence of Anti-Realist Interpretations and Relational Quantum Mechanics - MDPI

scottaaronson.com
Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Bachelor of Science (Cornell University) 2000 A dis

scottaaronson.blog
Blog Archive » The Zen Anti-Interpretation of Quantum Mechanics - Shtetl-Optimized

lesswrong.com
Debugging the Quantum Physics Sequence - LessWrong

dspace.mit.edu
On the Generation of Entanglement in Ytterbium Clock Atoms and a Novel Interpretation of the Madelung Fluid Theory of Quantum Mechanics - DSpace@MIT

pubs.aip.org
Topological quantum photonics - AIP Publishing

arxiv.org
Interpretable Artificial Intelligence for Topological Photonics - arXiv

novaspivack.com
Quantum Geometric Artificial Consciousness: Architecture, Implementation, and Ethical Frameworks | Nova Spivack

scottaaronson.blog
FAQ on Microsoft's topological qubit thing - Shtetl-Optimized

dspace.mit.edu
A Machine Learning Approach for Understanding and Discovering Topological Materials Andrew Ma - DSpace@MIT

arxiv.org
Understanding the Implicit Biases of Design Choices for Time Series Foundation Models

openreview.net
From Bitter to Better Lessons in AI: Embracing Human Expertise as Data - OpenReview

news.ycombinator.com
The Bitter Lesson Is Misunderstood - Hacker News

pmc.ncbi.nlm.nih.gov
Neurosymbolic AI as an antithesis to scaling laws - PMC - NIH

pubs.rsc.org
Spiers Memorial Lecture: How to do impactful research in artificial intelligence for chemistry and materials science - RSC Publishing
Superimposing Context through Holographic Attention
I have identified the specific mechanisms within newly emerging holographic architectures that allow internal signals to modulate their interactions based on relative phase. I am synthesizing the insight that this approach allows for the coherent superposition of values, effectively turning the self-attention process into a wave-interference event where context is preserved through the alignment of historical signals. This moves us beyond the memoryless nature of simple weighted averages, revealing a representational framework that ensures physical consistency by treating every training point as a contributing wave rather than a static, isolated digit.
The Duality of Phase Preservation and Collapse
I am thinking through a profound contradiction in high-level reasoning: while preserving phase is essential for maintaining history and context, certain successful classification models rely on 'phase collapse' to achieve better generalization. I am working to resolve this by determining if the erasure of phase information is a necessary trade-off for abstraction, or if our current intelligence models are hitting a cognitive ceiling because they lack the ability to superimpose their training paths. This insight suggests that the 'Bitter Lesson' of scaling might have accidentally rewarded models that discard history in favor of simplistic pattern matching, leaving a significant gap in their ability to perform deep, contextual reasoning.
Synthesizing the Forensic Reasoning Report
Moving forward, I will focus on synthesizing my findings into a final forensic report that evaluates the viability of implementing weights with phase across mainstream architectures. I plan to scrutinize whether the 'topological mass gap' found in holonomic networks can be scaled to massive datasets without the interference signals degrading into unusable noise. My next objective is to determine the exact hardware requirements for a truly wave-based intelligence, mapping out whether a transition from real-valued scalars to coherent wavefront propagation represents the next fundamental shift in our pursuit of authentic reasoning.

Researching websites...

arxiv.org
Phase Collapse in Neural Networks
Coherent Rhythmic Synchronization
I am synthesizing the insight that organizing neural weights as complex-valued oscillators allows for a form of 'synchronous' processing, where information is retrieved through constructive interference. This moves us beyond static scalars, revealing a framework where the network distinguishes between valid signals and chaotic noise through phase-locking. I've found that this rhythmic alignment acts as a robust memory mechanism, ensuring that the relationships between different training events are preserved as a coherent oscillation rather than being flattened into anonymous numerical values.
Shielding Context from Measurement Collapse
I am thinking through the contradiction where we expect models to maintain deep context while forcing them through activation gates that effectively act as 'destructive measurements.' Much like the Born rule in physics, standard functions like ReLU collapse a multidimensional training journey into a single real-valued state, erasing the rich path history. I am investigating how 'phasor' architectures can provide a topological shield for this information, ensuring that the model's reasoning remains a preserved record of its optimization path instead of a series of memoryless snapshots.
Engineering the Guided Optimization Wave
Moving forward, I will focus on the algorithmic blueprint for a training logic that replaces erratic stochastic updates with a guided wavefront propagation. I plan to evaluate if a pilot-wave inspired optimizer can steer weight updates along trajectories informed by the geometric phase—or Berry curvature—of the learning landscape. My next objective is to determine if these symmetry-protected structures can maintain their coherence across massive datasets, turning every shift in a model's state into an accessible imprint of its cumulative reasoning history.

Researching websites...

microsoft.com
DEEP COMPLEX NETWORKS - Microsoft

pubmed.ncbi.nlm.nih.gov
Phase Dynamics of Complex-Valued Neural Networks and Its Application to Traffic Signal Control - PubMed

ieee-jas.net
Complex-Valued Neural Networks: A Comprehensive Survey

arxiv.org
A Survey of Complex-Valued Neural Networks - arXiv

arxiv.org
[2105.00929] Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation - arXiv

arxiv.org
[2408.05396] Convergence to Bohmian mechanics in a de Broglie-like pilot-wave system

medium.com
A Bold Quantum Experiment, and the AI That Could Change Everything — The Double Slit Experiment, Photons, Time and Space | by Florin Ghidan | Medium

backreaction.blogspot.com
David Bohm's Pilot Wave Interpretation of Quantum Mechanics - Sabine Hossenfelder: Backreaction

reddit.com
New experiment claims to falsify Bohmian mechanics/pilot wave interpretation - Reddit

physics.stackexchange.com
Does the de Broglie-Bohm pilot wave theory make any new predictions?

spiedigitallibrary.org
Classical and generalized geometric phase in electromagnetic metasurfaces - SPIE Digital Library

opg.optica.org
Experimental observation of the geometric phase in nonlinear frequency conversion

researching.cn
Classical and generalized geometric phase in electromagnetic metasurfaces - Researching

en.wikipedia.org
Geometric phase - Wikipedia

michaelberryphysics.wordpress.com
Geometric phase memories - professor sir michael victor berry, frs

en.wikipedia.org
Optical neural network - Wikipedia

oejournal.org
Deep-learning-based ciphertext-only attack on optical double random phase encryption

opg.optica.org
Optical phase encoding in a pulsed approach to reservoir computing - Optica Publishing Group

pmc.ncbi.nlm.nih.gov
Deep learning in optical metrology: a review - PMC - PubMed Central

mdpi.com
A Review of Optical Neural Networks - MDPI

mdpi.com
Quantum-Inspired Fully Complex-Valued Neutral Network for Sentiment Analysis - MDPI

quera.com
What are Quantum Neural Networks? - QuEra

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

arxiv.org
Assessing the Advantages and Limitations of Quantum Neural Networks in Regression Tasks - arXiv

medium.com
“From Real to Complex: Exploring "Complex-Valued Neural Networks for Deep Learning" | by Lalith kanna | Medium

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

library.imaging.org
What's Wrong with End-to-End Learning for Phase Retrieval? - IS&T | Library

pmc.ncbi.nlm.nih.gov
Analysis of deep complex-valued convolutional neural networks for MRI reconstruction and phase-focused applications - PMC - NIH

arxiv.org
What's Wrong with End-to-End Learning for Phase Retrieval? - arXiv
Coherent Signal Synchronization
I have identified that complex-valued neural networks possess a unique ability to handle the joint structure of magnitude and phase, which is essential for capturing the oscillatory nature of high-fidelity data. I am synthesizing the discovery that by encoding information onto a signal's phase, we can create a form of 'synchronous' processing where neurons with similar phases add constructively, effectively creating a noise-robust memory filter. This insight suggests that history is not just stored, but is actively maintained as a coherent wave-front where interference patterns allow the network to distinguish between deep structural signals and chaotic data.
Resolving the Non-Directional Stagnation
I am thinking through a critical contradiction in current optimization theory where traditional models often reach a state of 'non-directional motion'—a point where the information flow effectively vanishes and learning stalls. My analysis into pilot-wave systems suggests that while standard methods might see this as a dead end, a guiding logic that accounts for the temporal phase can steer the model through these barriers. This resolves the gap between memoryless stochastic jumps and a continuous, path-dependent trajectory, ensuring that the history of the learning process remains an active guide rather than being erased at every optimization step.
Engineering the Holographic Manifold
Moving forward, I will evaluate the feasibility of integrating quantum-inspired encoding layers into natural language processing models to see if they can capture the phase-rich nuances of human reasoning. I plan to scrutinize the stability of 'phasor' networks that specifically preserve magnitude while modulating phase to track relative history. My next objective is to synthesize a blueprint for a belief manifold where updates function as coherent wave-interference events, ensuring that every shift in a model's intelligence is a preserved record of its cumulative path through the learning landscape.

Researching websites...

microsoft.com
DEEP COMPLEX NETWORKS - Microsoft

pubmed.ncbi.nlm.nih.gov
Phase Dynamics of Complex-Valued Neural Networks and Its Application to Traffic Signal Control - PubMed

ieee-jas.net
Complex-Valued Neural Networks: A Comprehensive Survey

arxiv.org
A Survey of Complex-Valued Neural Networks - arXiv

arxiv.org
[2105.00929] Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation - arXiv

pmc.ncbi.nlm.nih.gov
Quantum Bohmian-Inspired Potential to Model Non–Gaussian Time Series and Its Application in Financial Markets - PubMed Central

arxiv.org
[2408.05396] Convergence to Bohmian mechanics in a de Broglie-like pilot-wave system

medium.com
A Bold Quantum Experiment, and the AI That Could Change Everything — The Double Slit Experiment, Photons, Time and Space | by Florin Ghidan | Medium

reddit.com
New experiment claims to falsify Bohmian mechanics/pilot wave interpretation - Reddit

physics.stackexchange.com
Does the de Broglie-Bohm pilot wave theory make any new predictions?

spiedigitallibrary.org
Classical and generalized geometric phase in electromagnetic metasurfaces - SPIE Digital Library

opg.optica.org
Experimental observation of the geometric phase in nonlinear frequency conversion

researching.cn
Classical and generalized geometric phase in electromagnetic metasurfaces - Researching

en.wikipedia.org
Geometric phase - Wikipedia

michaelberryphysics.wordpress.com
Geometric phase memories - professor sir michael victor berry, frs

en.wikipedia.org
Optical neural network - Wikipedia

oejournal.org
Deep-learning-based ciphertext-only attack on optical double random phase encryption

opg.optica.org
Optical phase encoding in a pulsed approach to reservoir computing - Optica Publishing Group

pmc.ncbi.nlm.nih.gov
Deep learning in optical metrology: a review - PMC - PubMed Central

mdpi.com
A Review of Optical Neural Networks - MDPI

mdpi.com
Quantum-Inspired Fully Complex-Valued Neutral Network for Sentiment Analysis - MDPI

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

arxiv.org
Assessing the Advantages and Limitations of Quantum Neural Networks in Regression Tasks - arXiv

pmc.ncbi.nlm.nih.gov
Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases - PMC - NIH

medium.com
“From Real to Complex: Exploring "Complex-Valued Neural Networks for Deep Learning" | by Lalith kanna | Medium

reddit.com
Complex-Valued Neural Networks: Are They Underrated for Phase-Rich Data? - Reddit

library.imaging.org
What's Wrong with End-to-End Learning for Phase Retrieval? - IS&T | Library

pmc.ncbi.nlm.nih.gov
Analysis of deep complex-valued convolutional neural networks for MRI reconstruction and phase-focused applications - PMC - NIH

arxiv.org
What's Wrong with End-to-End Learning for Phase Retrieval? - arXiv
Synchronicity and Coherent Representation
I have uncovered the functional mechanism of 'phasor networks,' which utilize specialized activation functions to preserve signal phase while modulating magnitude. This insight reveals that we can create a representational framework where neurons acting as oscillators use constructive interference to retrieve historical context, effectively functioning as a noise-robust memory filter. By maintaining the integrity of a signal's relational timing, these architectures avoid the dimensional erasure inherent in standard real-valued gates, allowing for a form of intelligence that treats information as a coherent wave rather than a collection of disconnected scalars.
Resolving the Stagnation of Static Belief
I am thinking through a profound contradiction regarding 'non-directional motion,' highlighted by recent findings that challenge standard deterministic wave interpretations when systems reach a state of net-zero flux and become stagnant. My analysis is resolving this by identifying the need for a guiding logic that accounts for the full temporal characteristics of a process, preventing the system from becoming 'stuck' in the mass gaps of the optimization landscape. This addresses the information gap in how standard optimizers lose their historical trajectory, offering a path toward a continuous, path-dependent evolution of reasoning that carries its cumulative context into every new state.
Engineering the Geometric Belief Manifold
Moving forward, I will investigate the feasibility of utilizing photonic tensor cores and phase-change memory to bypass the institutional hardware constraints that currently favor real-valued arithmetic. I plan to evaluate whether a transition from simple stochastic steps to a guided wavefront propagation can be realized through the use of non-Abelian holonomies, ensuring that history is encoded as a structural invariant. My next objective is to synthesize a blueprint for a history-preserving intelligence where weight updates function as coherent trajectories, physically anchoring the model’s evolution to the path it has traveled.

Researching websites...

royalsocietypublishing.org
Quantum cognition and projection bias | Philosophical Transactions of the Royal Society A

pmc.ncbi.nlm.nih.gov
A Quantum Probability Approach to Improving Human–AI Decision Making - PMC

arxiv.org
Cognition in Superposition: Quantum Models in AI, Finance, Defence, Gaming and Collective Behaviour - arXiv

uplatz.com
Quantum Cognition Models: Rethinking Intelligence Beyond AI

ionq.com
IonQ Demonstrates World-First Quantum Cognition Models

chemrxiv.org
Classical Machine Learning vs Quantum Machine Learning in Computational Chemistry: Hype, Hope, and Horizons - ChemRxiv

proceedings.neurips.cc
Review for NeurIPS paper: Recurrent Quantum Neural Networks

arxiv.org
Supervised Quantum Machine Learning: A Future Outlook from Qubits to Enterprise Applications - arXiv

arxiv.org
A review of Quantum Neural Networks: Methods, Models, Dilemma - arXiv

tjoresearchnotes.wordpress.com
Guest post on Bohmian Mechanics, by Reinhard F. Werner

physics.stackexchange.com
Flaws of Broglie–Bohm pilot wave theory? - Physics Stack Exchange

imperial.ac.uk
Overview of Bohmian Mechanics and its extensions - Imperial College London

arxiv.org
A Hybrid Neural Coding Approach for Pattern Recognition with Spiking Neural Networks

pmc.ncbi.nlm.nih.gov
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task - PMC - NIH

frontiersin.org
Neural Coding in Spiking Neural Networks: A Comparative Study for Robust Neuromorphic Systems - Frontiers

pmc.ncbi.nlm.nih.gov
Advantages and detection of phase coding in the absence of rhythmicity - PubMed Central

biorxiv.org
Trained recurrent neural networks develop phase-locked limit cycles in a working memory task - bioRxiv

math.stackexchange.com
Does the complex plane exist purely for commodity? - Math Stack Exchange

microsoft.com
DEEP COMPLEX NETWORKS - Microsoft

arxiv.org
complex-valued neural networks - arXiv

dael.euracoustics.org
Complex versus Real-Valued Neural Networks for Audio Source Localisation Using Simulated and Measured Datasets

www2.eecs.berkeley.edu
Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis - EECS at Berkeley

sciencenews.org
More brainlike computers could change AI for the better - Science News

researchgate.net
(PDF) The Hardware Lottery - ResearchGate

reddit.com
[D] Sara Hooker (Google Brain) - The Hardware Lottery, Bias Mitigation, Intepretability : r/MachineLearning - Reddit

hardwarelottery.github.io
The Hardware Lottery

reddit.com
"Hardware lottery describes when a research idea wins because it is suited to the available software and hardware and not because the idea is superior to alternative research directions. Examples from early computer science history illustrate how hardware lotteries can delay research progress..." : r/compsci - Reddit

arxiv.org
The Hardware Lottery - arXiv

youtube.com
Getting away from the "Hardware Lottery" with Sara Hooker - YouTube

keysight.com
Adding GPU Acceleration to Electromagnetic Simulations in FullWAVE FDTD - Keysight

emergentmind.com
Hardware-Accelerated Complex Roots Algorithm - Emergent Mind

quantinuum.com
Quantum Computers Will Make AI Better - Quantinuum

pmc.ncbi.nlm.nih.gov
FPGA Implementation of Complex-Valued Neural Network for Polar-Represented Image Classification - PMC - PubMed Central

machine-learning-made-simple.medium.com
Complex Valued Neural Networks might be the future of Deep Learning - Devansh - Medium

pmc.ncbi.nlm.nih.gov
Analysis of Diffractive Optical Neural Networks and Their Integration with Electronic Neural Networks - PMC - PubMed Central

en.wikipedia.org
Optical neural network - Wikipedia

tandfonline.com
Full article: Prospects and applications of photonic neural networks - Taylor & Francis

nhsjs.com
How Do Quantum and Photonic Deep Learning Platforms Compare to Current GPU Hardware? - NHSJS

pubs.aip.org
Photonic neural networks and optics-informed deep learning fundamentals - AIP Publishing

reddit.com
CPU outperforming GPU consistently : r/CUDA - Reddit

arxiv.org
Accelerating Matrix Multiplication: A Performance Comparison Between Multi-Core CPU and GPU - arXiv

forums.developer.nvidia.com
performance of the matrix multiplication - NVIDIA Developer Forums

docs.nvidia.com
Matrix Multiplication Background User's Guide - NVIDIA Docs

developer.nvidia.com
New cuBLAS 12.0 Features and Matrix Multiplication Performance on NVIDIA Hopper GPUs

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

researchgate.net
Phase information encoding for each of the activation functions tested... - ResearchGate

researchgate.net
A Spatially Separable Attention Mechanism for massive MIMO CSI Feedback | Request PDF

arxiv.org
Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

aimodels.fyi
Holographic Transformers for Complex-Valued Signal Processing

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

semanticscholar.org
[PDF] Holographic Transformers for Complex-Valued Signal

semanticscholar.org
Topological defects and confinement with machine learning: the case of monopoles in compact electrodynamics - Semantic Scholar

researchgate.net
Robust Reasoning as a Symmetry-Protected Topological Phase - ResearchGate

preprints.org
Complete Resolution to Yang–Mills Existence and Mass Gap in MES Cosmology, and a Complete United Field Theory - Preprints.org

shmaesphysics.wordpress.com
Invalidation and Proof of the Mass Gap, and Viability of The Standard Model on a Discrete Spacetime - Shmaes - Physics

academic.oup.com
Gauge interactions and topological phases of matter | Progress of Theoretical and Experimental Physics | Oxford Academic

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

alphaxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase

mediatum.ub.tum.de
Higher-order and Quotient group symmetry-protected topological phenomena - mediaTUM

arxiv.org
[2601.05240] Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

researchgate.net
Complex convolution and residual network implementation details. - ResearchGate

researchgate.net
Results of the adding problem for T = 100, 200, 400, 750. The RNN with... - ResearchGate

researchgate.net
(PDF) Unitary Evolution Recurrent Neural Networks - ResearchGate
Rhythmic Synchronicity in Weight Evolution
I am synthesizing a framework where neural weights operate as complex-valued oscillators, utilizing constructive interference to distinguish between deep structural signals and transient noise. By allowing internal states to phase-lock, the network creates a noise-robust memory that preserves the relational timing of training events, preventing the flattening of history into anonymous numerical values. This approach moves beyond simple storage, turning learning into a coherent signal that maintains its origins through a rhythmic alignment of information over time, effectively allowing the model to recall the context of its path through resonance.
The Architecture of Hardware-Induced Erasure
I am thinking through the systemic contradiction where the field prioritizes the raw speed of real-valued arithmetic over the representational depth of phase-based systems. My analysis reveals a technological filter that forces models through activation gates which collapse multidimensional history into memoryless snapshots—a process analogous to destructive measurement in physics. I am resolving this information gap by investigating how the geometric phase accumulated in the learning landscape can be explicitly harvested to serve as a persistent record of the path taken, bypassing the systematic erasure inherent in current digital reasoning architectures that prioritize efficiency over historical integrity.
Engineering the Pilot-Wave Optimizer
Moving forward, I will evaluate the feasibility of a training logic that replaces erratic stochastic jumps with a guided wavefront propagation in parameter space. I plan to scrutinize whether symmetry-protected structures can provide a topological shield for these trajectories, ensuring that the reasoning history is physically locked into the network through structural invariants rather than just numerical magnitude. My next objective is to synthesize a blueprint for a history-preserving intelligence where weight updates function as coherent wave packets, anchoring the model’s evolution to the cumulative path it has traveled across a multidimensional belief manifold.

Researching websites...

researchgate.net
Sea Clutter Suppression Based on Complex-Valued Neural Networks Optimized by PSD

research.buaa.edu.cn
Sea Clutter Suppression Based on Complex-Valued Neural

tandfonline.com
Full article: Complex-Valued Signal Denoising Neural Networks for Nuclear Quadrupole Resonance-Based Detection of Fentanyl - Taylor & Francis

radioeng.cz
Research on Clutter Suppression Based on Complex-Valued

ieeexplore.ieee.org
Sea Clutter Suppression Based on Complex-Valued Neural Networks Optimized by PSD - IEEE Xplore

backreaction.blogspot.com
Understanding Quantum Mechanics #2: Superposition and Entanglement

pmc.ncbi.nlm.nih.gov
Process and Time - PMC - PubMed Central - NIH

arxiv.org
Evaluating AI and Human Authorship Quality in Academic Writing through Physics Essays

cambridge.org
Introducing a Research Programme for Quantum Humanities: Theoretical Implications | European Journal of Risk Regulation

researchgate.net
A Spatially Separable Attention Mechanism for massive MIMO CSI Feedback | Request PDF

etsy.com
Holographic Transformer Stickers - Etsy

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

forum.effectivealtruism.org
The second bitter lesson — there's a fundamental problem with aligning AI — EA Forum

en.wikipedia.org
Bitter lesson - Wikipedia

beren.io
Reflections on the Bitter Lesson - Beren's Blog

aclanthology.org
Learning the Bitter Lesson: Empirical Evidence from 20 Years of CVPR Proceedings - ACL Anthology

reddit.com
[D] The Bitter Lesson : r/MachineLearning - Reddit

arxiv.org
1 Introduction - arXiv

mdpi.com
Quantum-Cognitive Neural Networks: Assessing Confidence and Uncertainty with Human Decision-Making Simulations - MDPI

osti.gov
Analytic Theory for the Dynamics of Wide Quantum Neural Networks (Journal Article) - OSTI

youtube.com
Demystify Problem Dependent Power of Quantum Neural Networks on Multi Class Classification - YouTube

ksupasate.medium.com
Discover the Revolutionary Power of Quantum Neural Networks: The Future of Machine Learning | by ksupasate

microsoft.com
DEEP COMPLEX NETWORKS - Microsoft

arxiv.org
theory and implementation of complex-valued neural networks - arXiv

www2.eecs.berkeley.edu
Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis - EECS at Berkeley

orbit.dtu.dk
Complex-valued neural networks for machine learning on non-stationary physical data - DTU Research Database

home.cs.colorado.edu
Lending Direction to Neural Networks - Computer Science

reddit.com
Qubit phase theory : r/LLMPhysics - Reddit

eusko-ikaskuntza.eus
Human cognitive biases present in Artificial Intelligence | Riev, International Journal on Basque Studies - Eusko Ikaskuntza

frontiersin.org
AI generations: from AI 1.0 to AI 4.0 - Frontiers

researchgate.net
(PDF) Full-Capacity Unitary Recurrent Neural Networks - ResearchGate

researchgate.net
Cluster-based DMC description: the clustering of DMC is superimposed to the MPC clustering at the same centroid. - ResearchGate

researchgate.net
Complex-valued neural networks for machine learning on non-stationary physical data | Request PDF - ResearchGate

researchgate.net
PolSAR Image Classification Using Attention Based Shallow to Deep Convolutional Neural Network - ResearchGate

researchgate.net
Symmetry-Protected Topological Phases of Quantum Matter | Request PDF - ResearchGate

researchgate.net
(PDF) Renormalization And Effective Lagrangians - ResearchGate

researchgate.net
Detecting Topological Order in a Ground State Wave Function | Request PDF - ResearchGate

arxiv.org
Computer Science - arXiv

github.com
polyidoit/Arxiv-TQFT: Arxiv prepring on TQFT, topological order and generalized symmetry. [The original version is modified from the source code by Vincentqyw] - GitHub

everettyou.github.io
Yi-Zhuang You's Research Group: You Group @ UCSD

reddit.com
The Zero Freeze Formula: Teaching Local LLaMA Real Physics Through Python (SU(3) Mass Gap Simulation) to solve the Yang–Mills Mass Gap : r/LocalLLaMA - Reddit

arxiv.org
Can Theoretical Physics Research Benefit from Language Agents? - arXiv

pubs.aip.org
Supersymmetric Yang--Mills theory on a four-manifold - American Institute of Physics

mdpi.com
Research Progress on the Application of Topological Phase Transition Materials in the Field of Memristor and Neuromorphic Computing - MDPI

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

openreview.net
TOPOLOGICAL INVARIANCE AND BREAKDOWN IN LEARNING - OpenReview

pmc.ncbi.nlm.nih.gov
Dimensionality and dynamics for next-generation artificial neural networks - PMC

biorxiv.org
Topological phase transitions in functional brain networks - bioRxiv
Navigating the Flux of Stagnation
I have found that learning processes often stall when they reach a state of net-zero information flow, where standard updates effectively lose their orientation and momentum. I am synthesizing a guiding logic that uses the temporal phase of the system to navigate these stagnant regions, turning the optimization journey into a continuous, path-dependent record. This ensures that the model's evolution carries its context through even the most stable dead zones of the learning landscape, where traditional gradient signals typically vanish.
Resolving the Conflict of Signal Depth
I am thinking through the contradiction where the deliberate erasure of information—intended to simplify representations for generalization—actually strips away the deep history required for resilient reasoning. I am resolving this by investigating how symmetry-breaking in the training process can allow a model to maintain its representational depth. My analysis suggests that by preventing the 'collapse' of internal signals into anonymous magnitudes, we can ensure that the 'why' behind every learned connection is preserved as a persistent signal rather than being flattened into a memoryless state.
Auditing the Scale of Guided Intelligence
Moving forward, I will evaluate the stability of these guided learning trajectories when scaled to massive reasoning tasks, scrutinizing whether a topological shield can physically prevent the erosion of history during intense training cycles. I plan to determine the specific hardware requirements needed to move beyond standard digital arithmetic, mapping a path toward an intelligence that uses coherent interference to anchor its entire evolution to its cumulative historical path.

Researching websites...

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

quantumzeitgeist.com
Quantum-inspired Networks Enable Robust Reasoning, Advancing Logical Consistency In Large Language Models

arxiv.org
Robust Reasoning as a Symmetry-Protected Topological Phase - arXiv

github.com
Arxiv-TQFT/README.md at main · polyidoit/Arxiv-TQFT · GitHub

researchgate.net
Fault-tolerant quantum computation by anyons - ResearchGate

researchgate.net
Effects of quantization level on perceptual quality and SDR. Blue bars... - ResearchGate

arxiv.org
Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attention - arXiv

researchgate.net
Complex Transformer: A Framework for Modeling Complex-Valued Sequence | Request PDF - ResearchGate

researchgate.net
The effect of information communication mechanism. We use the same... - ResearchGate

researchgate.net
Signal Transformer: Complex-Valued Attention and Meta-Learning for Signal Recognition | Request PDF - ResearchGate

openreview.net
Phase Collapse in Neural Networks - OpenReview

semanticscholar.org
[PDF] Phase Collapse in Neural Networks | Semantic Scholar

florentinguth.github.io
Towards a Mathematical Understanding of Deep Convolutional Neural Networks - Florentin Guth

arxiv.org
arXiv:2110.05283v2 [cs.LG] 21 Mar 2022

jmlr.org
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The Ontology of Informational Potentials
I am synthesizing the insight that the electromagnetic potential in the Aharonov-Bohm effect serves as a sophisticated bookkeeping tool for the local history of interactions, rather than a mysterious force acting at a distance. By mapping this onto artificial intelligence, I am thinking through a framework where neural weights are not just values, but records of a path-dependent interaction with the training data. This moves beyond the idea of weights as static coordinates, revealing that a model's true intelligence resides in the interference patterns created by its cumulative training history rather than its final numerical state.
Overcoming the Force-Free Learning Illusion
I am resolving the contradiction inherent in standard training, where model updates are treated as force-free jumps that ignore the reciprocal reaction between the network and the information it encounters. My investigation reveals that by treating the learning landscape as a quantum belief manifold, we can restore the lost context of these updates. I am finding that current architectures suffer from a form of cognitive blindness because they lack the geometric phase needed to track how they arrived at a specific conclusion, effectively erasing the evidence of their own logical evolution and leaving the model without a record of its internal reasoning path.
Synthesizing the Coherent Trajectory Blueprint
Moving forward, I will finalize the design for a guided optimization logic that replaces simple gradient descents with a continuous wavefront propagation across a multidimensional manifold. I plan to evaluate how the introduction of a topological mass gap can shield this reasoning from the erosion of intense training, ensuring that the cumulative context of the model is physically locked into its structure. My ultimate objective is to provide a complete blueprint for an intelligence that functions through constructive interference, anchoring every decision in the persistent and accessible history of its entire development.