Phase 0: Foundations of High School & Elementary Mathematics

• Elementary Algebra & Functions: Polynomials • Factoring • Rational expressions • Exponential and logarithmic functions • Inequalities • Absolute value equations • Systems of linear/non-linear equations.
• [/] Trigonometry & Polar Coordinates: Unit circle • Trigonometric identities • Inverse functions • Polar coordinate system • Polar form of complex numbers • De Moivre's formula.
• [/] Single-Variable Calculus: Limits • Continuity • Derivative definitions • Differentiation rules • Mean Value Theorem • L'Hôpital's Rule • Indefinite/definite integrals • Fundamental Theorem of Calculus • Integration techniques • Improper integrals.
• Multivariable & Vector Calculus: Vectors • Dot/cross products • Partial derivatives • Gradient • Directional derivatives • Jacobian matrix • Chain rule in multiple variables • Multiple integrals • Cylindrical and spherical coordinates.
• Introductory Linear Algebra: Vector spaces • Subspaces • Linear independence • Span • Basis • Dimension • Matrix operations • Determinants • Gaussian elimination • Inverse matrices • Eigenvalues and eigenvectors.
• [/] Elementary Ordinary Differential Equations (ODEs): First-order separable and linear ODEs • Second-order linear ODEs with constant coefficients • Method of undetermined coefficients • Variation of parameters.

Phase 1: Real, Complex Analysis & Linear Algebra

• Topology of Metric Spaces: Continuity definitions • Open and closed sets • Compactness (Heine-Borel, sequential) • Connectedness • Completeness • Cauchy sequences • Baire Category Theorem.
• Real Analysis: Uniform vs. pointwise convergence • Arzelà-Ascoli theorem • Stone-Weierstrass theorem • Riemann-Stieltjes integration • Power series • Radius of convergence.
• Complex Analysis: Cauchy-Riemann equations • Cauchy's Integral Theorem and Formula • Laurent series • Residue Theorem • Conformal mappings • Analytic continuation.
• Linear Operators & Canonical Forms: Spectral Theorem for normal operators • Jordan Normal Form • Schur Decomposition • Singular Value Decomposition (SVD) • Low-rank approximations.
• Matrix Analysis: Matrix norms (operator, Schatten, Lp,q) • Courant-Fischer Min-Max Theorem • Gershgorin Circle Theorem • Perron-Frobenius Theorem • Condition numbers • Matrix stability.
• Differential Forms & Vector Calculus: Exterior algebra • Wedge product • Exterior derivative • Differential k-forms • Pullbacks • Generalized Stokes' Theorem on Manifolds.

Phase 2: Abstract Algebra, Galois Theory & Number Theory

• Group Theory: Normal subgroups • Quotient groups • Isomorphism Theorems • Group actions • Sylow Theorems • Solvable groups • Symmetric or alternating groups.
• Ring & Ideal Theory: Commutative rings • Prime and maximal ideals • Principal Ideal Domains (PID) • Unique Factorization Domains (UFD) • Euclidean domains • Quotient rings.
• Field Extensions & Galois Theory: Algebraic vs. transcendental extensions • Splitting fields • Finite fields (Galois Fields) • Fundamental Theorem of Galois Theory • Cyclotomic polynomials.
• Computational Number Theory: Extended Euclidean Algorithm • Chinese Remainder Theorem • Euler's Totient function • Discrete Logarithm Problem • Quadratic Reciprocity • Miller-Rabin primality testing • Pollard's rho algorithm.

Phase 3: Measure-Theoretic Probability & Stochastic Analysis

• Measure Theory: Sigma-algebras • Borel sets • Outer measures • Carathéodory Extension Theorem • Lebesgue measure construction • Vitali non-measurable sets.
• Lebesgue Integration: Measurable and simple functions • Monotone Convergence Theorem • Fatou's Lemma • Dominated Convergence Theorem • Lp spaces • Riesz-Fischer Theorem.
• Product Measures & Absolute Continuity: Product sigma-algebras • Fubini-Tonelli Theorems • Radon-Nikodym Theorem and derivatives • Lebesgue Decomposition Theorem.
• Probability Foundations: Probability spaces • Random variables as measurable mappings • Independence • Borel-Cantelli Lemmas • Kolmogorov's 0-1 Law.
• Martingale Theory: Filtrations • Conditional expectation as L2 projection • Super/sub-martingales • Stopping times • Doob's Optional Stopping Theorem • Doob's Convergence Theorem • Azuma-Hoeffding inequality.
• Stochastic Calculus: Brownian motion (Wiener process) • Itô integral • Itô's Lemma • Stochastic Differential Equations (SDEs) • Fokker-Planck equation • Ornstein-Uhlenbeck processes.

Phase 4: Convex, Non-Convex & High-Dimensional Optimization

• Convex Analysis: Convex sets • Supporting Hyperplane Theorem • Proper/closed/lower-semicontinuous functions • Fenchel Conjugate • Subgradients and subdifferentials.
• Optimality & Duality: Karush-Kuhn-Tucker (KKT) conditions • Slater's constraint qualification • Lagrangian duality • Strong duality • Dual ascent • Primal-dual algorithms.
• First-Order Optimization: Nesterov Accelerated Gradient Descent • Proximal Gradient Descent (ISTA/FISTA) • ADMM • Mirror Descent • Frank-Wolfe algorithms.
• Stochastic & Non-Convex Optimization: SGD convergence under smoothness • Polyak-Łojasiewicz inequality • Variance reduction (SVRG, SAGA) • Escaping saddle points • Adam/RMSProp stability limits.
• Manifold Optimization: Riemannian Gradient Descent • Retractions • Vector transport • Optimization on Grassmannian and Stiefel manifolds.

Phase 5: Differential Geometry & Geometric Mechanics

• Smooth Manifolds & Tensors: Smooth maps • Immersions • Submersions • Tangent and cotangent spaces • Tangent bundles • Vector fields • Lie brackets • Exterior algebra.
• Riemannian Geometry: Riemannian metrics • Levi-Civita connection • Parallel transport • Geodesics • Exponential maps • Riemann curvature tensor • Ricci and scalar curvature.
• Lie Groups & Lie Algebras: SO(3), SE(3), SU(N) groups • Exponential mapping from Lie algebras to groups • Adjoint representations • Lie algebra actions on manifolds.

Phase 0: Foundations of Discrete Mathematics & Logic

• Mathematical Logic & Proofs: Propositional/predicate logic • Truth tables • Direct proofs • Contraposition • Proof by contradiction • Mathematical and strong induction.
• Set Theory & Relations: Sets • Subsets • Power sets • Cartesian products • Equivalence relations and classes • Partial orderings • Hasse diagrams.
• Basic Number Theory: Divisibility • Prime numbers • Greatest Common Divisor (GCD) • Euclidean Algorithm • Modular Arithmetic • Fermat's Little Theorem.
• Combinatorics & Graph Theory: Permutations • Combinations • Pigeonhole Principle • Inclusion-Exclusion Principle • Eulerian and Hamiltonian paths • Trees • Graph colorings.
• Elementary Automata & Formal Languages: Deterministic Finite Automata (DFA) • Nondeterministic Finite Automata (NFA) • Regular expressions • Context-Free Grammars (CFG).

Phase 1: Computational Complexity & Information Theory

• Structural Complexity: Turing machines • Classes P, NP, coNP • Polynomial Hierarchy • PSPACE • Ladner's Theorem • Baker-Gill-Solovay Relativization barrier.
• Space & Circuit Complexity: Savitch's Theorem • Immerman-Szelepcsényi Theorem (NL = coNL) • NC and AC circuit classes • Razborov-Smirnov Natural Proofs barrier.
• Interactive Proofs & Zero-Knowledge: IP = PSPACE Theorem • Interactive Proof Systems • Arthur-Merlin games • Zero-knowledge definitions • PCP Theorem • Hardness of Approximation.
• Quantum Complexity: Postulates of Quantum Mechanics • Quantum circuits • Class BQP • Shor's factoring algorithm • Grover's search • Quantum Supremacy frontiers.
• Information Theory: Shannon entropy • Joint and conditional entropy • Mutual information • KL divergence • Rényi entropy • Channel capacity • Fano's Inequality • Data Processing Inequality.

Phase 2: Lattice-Based Cryptography, PQC & Cryptanalysis

• Lattice Geometry: Geometry of numbers • Full-rank lattices • Determinant • Dual lattice • Fundamental parallelotope • Successive minima • Minkowski's Theorems.
• Lattice Reduction Algorithms: Gram-Schmidt Orthogonalization • LLL reduction algorithm • Hermite factor • BKZ reduction • Babai's Closest Vector algorithms.
• Lattice Hard Problems: Shortest Vector Problem (SVP) • Closest Vector Problem (CVP) • Bounded Distance Decoding (BDD) • SIVP • Worst-case to average-case reductions.
• Learning With Errors (LWE) Mechanics: LWE formulation • Search-LWE to Decision-LWE reduction • Ring-LWE • Module-LWE.
• PQC Standards & Implementations: NIST PQC Standards (ML-KEM/Kyber, ML-DSA/Dilithium) • Number Theoretic Transform (NTT) • Constant-time side-channel mitigations • SPA/DPA defenses.
• Zero-Knowledge Proof Construction: Rank-1 Constraint Systems (R1CS) • Algebraic Intermediate Representation (AIR) • Polynomial Commitment Schemes (KZG, IPA, FRI) • SNARKs • STARKs.

Phase 3: Type Theory, Formal Semantics & Interactive Theorem Proving

• Lambda Calculus: Untyped lambda calculus • Simply Typed lambda calculus • System F • System F-omega.
• Dependent Type Theory: Calculus of Constructions (CoC) • Pure Type Systems • Dependent types • Martin-Löf Type Theory (MLTT) • Homotopy Type Theory (HoTT) & Univalence Axiom.
• Curry-Howard Isomorphism: Propositions-as-Types • Proofs-as-Programs • Classical vs. intuitionistic logic • Constructive mathematics.
• Formal Program Semantics: Operational semantics • Denotational semantics • Axiomatic semantics (Hoare logic, invariants, weakest preconditions) • Separation Logic.
• Verification Toolchains & Solvers: SMT solving algorithms • Abstract Interpretation • Theorem Provers (Coq / Lean 4 proof automation).

Phase 0: Elementary Probability, Statistics & Machine Learning Fundamentals

• Basic Probability: Sample spaces • Axioms of probability • Conditional probability • Bayes' Theorem • Discrete/continuous random variables • Expectation, variance, covariance.
• Standard Distributions: Uniform • Binomial • Poisson • Gaussian • Exponential • Beta • Gamma • Multivariate Normal.
• Classical Statistics: Maximum Likelihood Estimation (MLE) • Maximum A Posteriori (MAP) • Confidence intervals • Hypothesis testing • Linear/logistic regression.
• Foundational Machine Learning: Supervised/unsupervised learning • Train/val/test splits • Bias-variance tradeoff • Overfitting/underfitting • L1/L2 Regularization • Decision trees • k-NN • SVMs.
• Neural Network Basics: Perceptrons • Multilayer perceptrons (MLP) • Activation functions • Loss functions • Forward pass • Backpropagation via chain rule.

Phase 1: Statistical Learning Theory & Infinite-Width Regime

• PAC Learning Framework: PAC learning • Agnostic PAC learning • Sample Complexity bounds • Empirical Risk Minimization (ERM).
• Combinatorial & Function Class Complexity: Growth function • Sauer-Shelah Lemma • VC-Dimension • Vapnik-Chervonenkis bounds • Rademacher and Gaussian complexities • Fat-shattering dimension.
• Concentration of Measure: Sub-Gaussian/Sub-Exponential random variables • Hoeffding's Lemma • Chernoff bounds • McDiarmid's Inequality • Bernstein's Inequality • Hansen-Wright Inequality.
• Overparameterization Mechanics: Double Descent curve • Benign overfitting • Neural Tangent Kernel (NTK) regime • Lazy training vs. feature learning (muP parametrization).

Phase 2: Geometric Deep Learning & Symmetry

• Symmetry and Equivariance: Invariance and equivariance principles • Group Equivariant CNNs (G-CNNs) • Spherical CNNs • Gauge Equivariant Mesh CNNs.
• Graph & Mesh Theory: Spectral Graph Theory • Message Passing Neural Networks (MPNNs) • Oversmoothing and oversquashing.

Phase 3: Adversarial Robustness, Certified Bounds & Distributional Drift

• Adversarial Threat Formalization: L-infinity, L2, L1 threat models • First-order attacks (FGSM, PGD, C&W, AutoAttack) • Black-box attacks • Backdoor/data poisoning.
• Certified Defense Mechanics: Randomized smoothing • Interval Bound Propagation (IBP) • Bound propagation via abstract interpretation (alpha/beta-CROWN, LiRPA) • SMT/MILP verification.
• Distributional Robustness & Domain Generalization: Wasserstein DRO • Optimal Transport distance • Invariant Risk Minimization (IRM) • Domain Adversarial Neural Networks • OOD detection.

Phase 4: Alignment Mathematics, Game Theory & Agent Safety

• Preference Learning: Bradley-Terry & Plackett-Luce models • RLHF objective formulation (PPO with KL-penalty) • Direct Preference Optimization (DPO) • KTO • Identity-PO.
• Control-Theoretic Safety: Control Barrier Functions (CBF) • Control Lyapunov Functions (CLF) • Safe Reinforcement Learning • Constrained MDPs.
• AI Agent Security & Red Teaming: Direct/Indirect prompt injection • Representation Engineering • Jailbreak mechanics (GCG) • Guardrail system design • Multi-agent red-teaming.

Phase 0: Foundations of Computer Architecture & C/C++ Systems Programming

• Basic Physics & Circuits: Ohm's & Kirchhoff's Laws • RLC circuits • Semiconductor basics • Logic gates • Truth tables.
• Number Representations: Binary, octal, hex • Two's Complement • IEEE 754 Floating-Point • Bitwise operations.
• Basic Assembly & Machine Code: RISC vs CISC • Register sets • Stack frames • Calling conventions • x86-64 / ARM assembly • Memory layout (Text, Data, BSS, Heap, Stack).
• C/C++ Systems Programming: Pointers & arithmetic • Manual memory allocation (malloc/free, new/delete) • RAII • References • Struct alignment/padding • File I/O.
• Basic Data Structures & Algorithms: Arrays, linked lists, stacks, queues, hash tables, BSTs • Sorting (QuickSort, MergeSort) • Big-O notation.

Phase 1: Silicon Foundations & Digital Microarchitecture

• Semiconductor Physics & MOSFETs: P-N junctions • Bandgap mechanics • NMOS/PMOS operation • CMOS logic gates • Parasitic capacitance • Leakage current • Dynamic power.
• Digital Logic & Sequential Circuitry: Boolean minimization (Karnaugh maps) • Setup/hold times • Clock jitter/skew • Metastability • Synchronizers • Finite State Machines.
• Computer Organization & RTL: RTL design • ALU architectures • Carry-lookahead adders • Wallace tree multipliers • Pipelining hazards • Forwarding • Stall units • Branch predictors.
• Memory Subsystem Microarchitecture: SRAM topology vs DRAM cell • Cache line architecture • Direct-mapped vs set-associative • MESI/MOESI cache coherence • Write-buffers.

Phase 2: Hardware Microarchitecture & Parallel Compute Engines

• Advanced CPU Microarchitecture: Out-of-Order execution • Reorder Buffer • Reservation Stations • Register Renaming • Branch Target Buffers • SIMD vectorization • Cache bottlenecks.
• GPU Architecture & Compute Pipelines: SIMT execution model • Streaming Multiprocessors • Warps • Warp divergence • Register pressure • Shared memory bank conflicts • Tensor Cores.
• Roofline Performance Modeling: Arithmetic intensity • Operational boundaries • Memory-bound vs compute-bound classification • Latency hiding via concurrency.
• Custom Kernel Engineering (CUDA & Triton): Shared memory tiling • Coalesced global memory access • Warp shuffle primitives • Double buffering • Triton compiler IR • PTX/SASS inspection.

Phase 3: Compiler Architecture, Polyhedral Model & MLIR

• Compiler Frontend & Middle-End: Abstract Syntax Trees (AST) • Control Flow Graphs (CFG) • Static Single Assignment (SSA) • Dominance frontiers • Dominator trees.
• Optimization Passes: Loop Invariant Code Motion (LICM) • Dead Code Elimination (DCE) • Common Subexpression Elimination (CSE) • Alias analysis • Inlining • Vectorization.
• Polyhedral Compilation Model: Polyhedral representation of nested loops • Affine transformations • Loop tiling, skewing, interchange, fusion • Pluto Algorithm.
• MLIR Infrastructure: Dialect Ecosystem • Operation Definition Specification (ODS) • Declarative Rewrite Rules (DRR) • Passes & lowering pipelines • JIT compilation via LLVM ORC.
• Advanced Runtimes & Memory Management: Garbage collection algorithms • Lock-free memory reclamation (Epoch-Based Reclamation, Hazard Pointers, RCU).

Phase 4: OS Kernel Subsystems, Concurrency & Low-Level Systems

• Virtual Memory Subsystem: Multi-level page tables • Page walks • Translation Lookaside Buffer (TLB) • TLB shootdowns • HugePages • Memory-Mapped I/O • Page fault handling • KPTI.
• Kernel Memory Allocation: Buddy Allocator • Slab/Slub/Slob allocators • Memory overcommit • OOM killer internals.
• Async I/O Subsystems: epoll architecture • io_uring ring-buffer design • Zero-copy networking • Direct I/O (O_DIRECT).
• eBPF: Bytecode verification • JIT compilation to native machine code • kprobes, uprobes, tracepoints • XDP packet processing at NIC driver level.
• Microarchitectural Vulnerabilities & Hardware Security: Transient execution attacks (Spectre, Meltdown) • Rowhammer DRAM bit-flips • Fault injection • Hardware enclaves (SGX, SEV, CCA).

Phase 0: Foundations of Computer Networks & Operating Systems

• Network Stack & Protocols: OSI Model vs TCP/IP stack • Ethernet • IP addressing/subnetting • ICMP • TCP 3-way handshake • UDP • DNS basics.
• Socket Programming: Client-server architecture • TCP/UDP sockets • Blocking vs. non-blocking I/O • Select/poll primitives • HTTP protocol structure.
• Operating System Basics: Process management • Threads • Context switching • Processes vs. threads • IPC (pipes, shared memory, sockets).
• Concurrency Fundamentals: Race conditions • Critical sections • Mutual exclusion • Locks/mutexes • Semaphores • Condition variables • Deadlocks/livelocks.
• Basic Storage Systems: POSIX I/O • Inodes • Directory trees • File descriptors • HDD vs SSD access patterns.

Phase 1: Theoretical Distributed Systems & Formal Specifications

• Fundamental Theorems: FLP Impossibility Theorem • CAP Theorem • PACELC Theorem • Synchronous vs. asynchronous network bounds.
• Logical Time & Ordering: Lamport Timestamps • Vector Clocks • Matrix Clocks • Causal consistency • Total order broadcast • Chandy-Lamport snapshot algorithm.
• Formal Specification & Verification: TLA+ (Temporal Logic of Actions) • PlusCal • Model checking via TLC • Verifying safety invariants and liveness properties • Refinement mapping.

Phase 2: Asynchronous Consensus, BFT Mechanics & Distributed Data

• Crash Fault Tolerant (CFT) Consensus: Paxos (Single-decree, Multi-Paxos) • Raft • Viewstamped Replication (VR) • State Machine Replication (SMR).
• Byzantine Fault Tolerant (BFT) Consensus: PBFT • HotStuff • Narwhal & Tusk • Async BFT (HoneyBadgerBFT) • DAG-based consensus mechanics.
• Distributed Data Structures & Transactions: Consistent hashing • Distributed Hash Tables (Kademlia) • Two-Phase Commit (2PC) • Three-Phase Commit (3PC) • SAGA Pattern • Spanner architecture.