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Hierarchical three-layer learning Three-Layer Learning as a DMFT–RMT–BBP Closure Problem
A master reduction from exact Hermite geometry to a conditional dynamic BBP law, with cavity, replica, spectral AMP, Kac–Rice, and a time-resolved Hessian diagnostic.
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Hierarchical three-layer learning Three Layers, One Hierarchy: From Exact Hermite Dynamics to DMFT and BBP
A deterministic theory map for hierarchical three-layer learning, Muon, dynamical mean-field theory, and block-Wishart spectral transitions.
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Hierarchical three-layer learning Does Muon Help a Hierarchical Three-Layer Model?
A twenty-pair confirmation of endpoint risk and sector clocks, plus sample-size and fresh-Hessian diagnostics.
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Spectral transitions in multi-index models When Both Layers Learn: Where Hessian Outliers Come From
A non-technical guide to joint feature-amplitude learning, width, Hessian branches, and dynamic BBP transitions.
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Spectral transitions in multi-index models Beyond Quadratic Phase Retrieval: Spectral Learning with General Gaussian Links
Why the finite-state and spectral picture extends from quadratic phase retrieval to smooth multi-index models.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 6: How Attention Learns a Preconditioner
Which training curves are exact, which are local diagnostics, and how query-key-value matrices become a learned solver step.
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Muon and controlled training dynamics Muon for Phase Retrieval III: What Is Proved and What Remains Open
A reader-friendly map of the exact results, imported theorems, numerical evidence, and open probabilistic bridge.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 5: Separating Encoder, Decoder, and Generalization Error
A readable error budget for task inference, solver depth, training tasks, and held-out generalization.
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Muon and controlled training dynamics Muon for Phase Retrieval II: Choosing the Power During Training
A non-technical guide to adapting Muon's spectral power as strong and weak hidden directions are learned.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 4: What Can Be Proved About the Solver?
The finite-dimensional encoder and decoder certificates, and why a full training theory needs replica order parameters.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 3: Turning PDE Solutions Into Transformer Tokens
A plain-language pipeline: functions become vectors, prompts become weak equations, and decoder layers become solver steps.
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Spectral transitions in multi-index models When Does the Hessian Reveal a Hidden Direction?
A non-technical introduction to dynamic BBP transitions and energy-resolved Hessian spectra in multi-index phase retrieval.