Best Practices: Human–AI Synergy in Mathematical Research
I. Framing the Interface
* LLM as Partner, Not Oracle:
Treat the model
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Structural Pre-Derivation of the Bellec–Fritz Bound
Before our knowledge of the September 2025 preprint of Pierre C. Bellec
A Sheaf-Theoretic View of Dynamic Inverse Problems
Dynamic X-ray CT isn’t just an optimization problem—it’s a gluing problem.
I reinterpret Aryeh Keating’s reconstruction framework as a sheaf: optical flow becomes a connection, EM a derived limit, Kalman filtering a pushforward. Cohomology reveals where structure fails.
Completing Mixing Implies k-Fold Mixing
We just closed the loop on one of ergodic theory’s core open chains — proving that strong mixing implies mixing of all orders, and reducing the plain mixing case to a single structural lemma. Here’s what happened.
A Contact Tensor Category for Higher Chern–Schwartz–MacPherson Classes
A new categorical framework for computing higher Chern–Schwartz–MacPherson classes. By treating multisingularities as tensor products of local algebras with contact constraints, we turn the “higher-order terms” into explicit geometric structure.
Randomness, Structure, and Efficiency: From CountSketch to Commutator-Aware Tensor Summation
CountSketch speeds up computation with randomness. CATS goes further—using commutators to decide when randomness is needed, merging tensor algebra with adaptive sketching.
Fracture as Recursion: From Etch-A-Sketch Physics to Adaptive Intelligence
1. Setting the stage
Timo Heister’s work on phase-field fracture has
Collatz Drift and the Burns Field: Why Smoothness Fails and Oscillation Wins
A small arithmetic map reveals a structural fault line in analytic number theory. The Collatz function, long treated as a curiosity, exposes why the Riemann Hypothesis’s smoothness cannot sustain deterministic convergence—and how Burns Law’s modular oscillation restores it.
Two Model Constructions for Stratified Vector Bundles on Singular Spaces
This post gives two concrete examples of stratified vector bundles — objects that
Stability and the Geometry of Unstable Modes
This note extends the Chen–Moutinho framework on the asymptotic stability of
Nonlinear Krylov Geometry: Tangent and Jet Embeddings for Closure and Convergence
Classical Krylov subspace methods such as GMRES and GCR rely on linear