Chapters
Every chapter, grouped by Part. Use the card metadata to calibrate how much trust to place in a chapter's specific claims.
Foundations
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Linear ODEs as state-space systems
Continuous-time linear systems, the matrix exponential as fundamental solution, the damped oscillator as canonical example, and the eigenvalue classification of phase portraits.
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Stability theory: Lyapunov, A-stability, BIBO
Three distinct notions of stability — Lyapunov (eigenvalue criterion + QR-based computation), A-stability (regions of absolute stability for ODE integrators), and BIBO (bounded-input bounded-output for LTI systems).
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Linear algebra for sequence models: structured matrices and conditioning
Eigenvalue and SVD decompositions, condition number, the four structured-matrix families (Toeplitz, Vandermonde, Cauchy, semiseparable), low-rank updates, and a Krylov-subspace primer — the linear-algebra vocabulary later chapters assume.
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Discretization theory: ZOH, bilinear, exponential families
One-step discretizations of linear ODEs — zero-order hold, bilinear (Tustin), and exponential-trapezoidal — with the Lax equivalence theorem as the unifying convergence criterion.
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Stability regions, Butcher tableau, and order conditions
Runge–Kutta theory, the Butcher tableau formalism, order conditions, stability regions in the complex plane, A-stability and L-stability, and the Dahlquist barrier — applied to the Chapter 4 discretization atlas.
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Implicit methods, stiff systems, symplectic integration
When explicit methods fail — backward Euler, BDF, DIRK as A-stable alternatives; exponential integrators as the family of choice for selective SSMs; Hamiltonian mechanics and symplectic integration as the natural geometric language for hidden-state dynamics.
SSM Core
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HiPPO theory: orthogonal-basis projection operators
Online function approximation by projection onto orthogonal polynomial bases — the HiPPO-LegS operator, its lower-triangular structure and exact spectrum, discretization to the S4 recurrence, and the basis-conditioning question.
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LTI SSMs: S4, S4D, S5 as discretized continuous systems
The S4 architecture as a discretized HiPPO LTI system — the convolution–recurrence duality and its FFT kernel, the diagonal-plus-low-rank structure behind S4's fast kernel, S4D's diagonal restriction with by-construction stability, and S5's parallel associative scan.
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Selective SSMs: Mamba-1, Mamba-2, and the SSD framework
Making the SSM parameters input-dependent — selectivity breaks linear time-invariance, collapses the convolution–recurrence duality, and leaves the associative scan as the only route; the structured state-space duality then re-reads the selective scan as a semiseparable matrix and, for scalar state, as masked linear attention.
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Mamba-3 and the exponential-trapezoidal integrator
How Mamba-3 keeps selectivity and changes two things — a second-order exponential-trapezoidal integrator and a complex state — turning the dynamical-systems lens of Chapters 4-6 onto a production sequence model.
Beyond SSMs
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Linear attention and Hyena: long convolution and gated recurrence
Linear attention read as a matrix-state linear recurrence — the dual of Chapter 9's scan, entered from the attention side — then gated into the LTV/selectivity face (GLA, RetNet), pivoted to Hyena's implicit LTI long convolution, and landed honestly on the associative-recall capacity limit of any finite state.
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Delta-rule lineage: DeltaNet, Gated DeltaNet, Kimi Linear
The delta rule read as one gradient step on an associative-recall loss — DeltaNet (explicit Euler) and Longhorn (backward Euler) as the textbook explicit/implicit discretization pair, the stability dichotomy that follows, chunkwise parallelization via the WY trick, and gated forgetting from Gated DeltaNet to Kimi Linear.
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Exponential gates and matrix memory: xLSTM and RWKV-7
One generalized linear recurrence whose transition is diagonal-plus-rank-one — RWKV-7's generalized delta rule (rank-one removal in a learned direction, not the key) and xLSTM's matrix memory with exponential gating (and the log-domain stabilizer state that overflow forces) — extending Chapter 12's delta-rule lineage to matrix memories with their own stability questions.
Integration
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Hybrid architectures and gating mechanisms
Hybrids read through the two-timescale lens — attention as the boundary layer (exact on fast, token-local structure), the decaying state as the slow manifold (compressed carried context), and composition and gate granularity as the design variables; with the production lineup as of May 2026, the MAD methodology, and the two-timescale benchmark seed for pilot B.
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Counter-evidence and diagnostic tools: where SSMs fail
The prosecution's file — what fixed-state and SSM-heavy designs provably cannot do (the TC⁰ ceiling, the illusion of state, the copying separation) and the diagnostic toolkit that measures stability and capacity on real systems, from an architecture-agnostic information-counting bound through the Benettin–QR Lyapunov estimator and its resolution limit to effective state size as a regime diagnostic, all validated against known ground truth.
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Empirical methodology: benchmark protocols and evaluation
Benchmarks as measurement instruments — the discriminative-regime principle for synthetic probes (tokenized MQAR with exact readers), length stress that pads with distractors rather than blanks (L90/AUC), the statistics of honest comparison (paired SEs, selection inflation), and the two-timescale protocol — composite predictor and probe signature — where pilot B's book-side prerequisites close.