Course · Foundations
Foundations
Einstein’s completion program and the mathematics it actually uses, in that order. Sets, then functions as the apparatus, then the algebra of this analog, then derivatives and their inverse integrals, then ordinary differential equations, uniqueness, Fourier remainder, and probability only after the remainder is unread. Every lesson returns to S and T*. University courses sit at the bottom of each lesson as the full treatment.
After this path, open The math of T* for the analog walkthrough.
Book 00 · A program, then the math it needs
Why this course
You will know what Einstein asked for, which objects this site uses, and how the later books exist to make those objects computable.
3 lessons
Book 01 · The backbone
Sets
You will read braces, membership, and infimum without fear, and you will see T* as the infimum of an event set.
5 lessons
Book 02 · The primary apparatus
Functions
You will treat T*, q(t), and the drive as functions — a rule from a complete input to a unique output.
4 lessons
Book 03 · The well and the drive, written
Algebra
You will expand polynomials, evaluate sines, and write the analog force without calculus yet — then see why calculus is about to enter.
4 lessons
Book 04 · Rates of change
Derivatives
You will read dq/dt as velocity, dV/dq as slope, and F = −dV/dq as “force is minus the slope of the well.”
5 lessons
Book 05 · The underneath of a derivative
Integrals
You will treat integration as the inverse of differentiation, recover V from F, and see why Picard’s existence proof is an integral equation.
4 lessons
Book 06 · The law itself
Differential equations
You will know what an ODE is, why this analog is an ODE not a PDE, and how to write Newton as a first-order system.
4 lessons
Book 07 · Uniqueness, then a method
Solving
You will know why two copies of S cannot give two T*, and you will be able to describe one RK4 step as a practical approximation of the integral equation.
5 lessons
Book 08 · A definite background
Fourier remainder
You will see a finite Fourier sum as a realization, not an ensemble, and the eight phases as S_⊥.
4 lessons
Book 09 · After the law, not inside it
Probability of the unread
You will use probability only as the theory of an unread remainder: measure, pushforward, variance, and the law of total variance as A6.
5 lessons
Book 10 · Nothing skipped, nothing invented
Map and double-pass
You will see every theorem we used, every theorem we did not claim, and the next honest courses toward the field ceiling.
4 lessons