Every authored skill, laid out left to right by prerequisite depth: a skill’s column is the length of the longest chain of prerequisites ending at it, and it means that for every node in the picture. Column six means that chain is six skills long. That is a depth, not a workload — the skills behind a node form a set, which is larger than any one chain through it, and the ones still to do depend on what you have already mastered. Select a node and the panel counts those.
The distinction is not pedantry. docs/08-curriculum-map.md measures it on this graph: the longest chain to the Maxwell payoff is 10, its full prerequisite closure is 41, and the number a reader partway along actually has left is 22. The column is the first of those. Making a distant destination reachable means making it a countable number of named steps, so a locked node here always names which prerequisite is short and by how much, and a skill whose lesson is not yet written says so rather than being hidden to make the distance look shorter.
There are three readings of the same graph and the tabs below switch between them. The canvas is the one described above. The line is the core track top to bottom in curriculum order with the branches folded in where they attach — it drops the columns and keeps the sequence, which is the reading that survives a phone, so below 640px the page opens on it. The table is every encoding as a word or a number. The filters above apply to all three and remove the same skills from each. The search applies to all three as well, and what counts as a match is one rule, but what a match does differs by what the view can carry: the canvas and the table mark the matches and keep every other skill on screen, while the line — which is read by scrolling rather than at a glance — is drawn over the matches alone. Each view says which it is doing.
141 skills, 331 prerequisite edges, across 29 columns. Counted over every skill in this reading of the graph; nothing here paginates. These are figures for the whole graph and they do not move when you filter — the figure below carries its own caption, and that one counts what is actually drawn.
The structure below is the authored curriculum. Whether there is a signed-in reader whose progress can be drawn on it is still being asked.
Searching title, slug, statement, strand, and track. Matches are marked where they are; nothing is hidden.
141 skills
algebra5 min
Substituting a value for a letter
definition
depth 0#1
algebra30 min
The real numbers as a field
definition
depth 0#2
algebra5 min
Collecting like terms
theorem · proved here
depth 1#3
algebra30 min
Order axioms, and why a negative reverses an inequality
theorem · proved here
depth 1#5
algebra5 min
What xⁿ abbreviates
definition
depth 1#7
algebra5 min
A product is zero only when a factor is
theorem · proved here
depth 1#21
algebra5 min
Solving ax + b = c
theorem · proved here
depth 2#4
algebra30 min
Absolute value as distance
theorem · proved here
depth 2#6
algebra35 min
Exponent laws, integer then rational
theorem · proved here
depth 2#8
algebra25 min
Which axiom licenses which move
definition
depth 2#12
algebra5 min
Dividing powers of the same base
theorem · proved here
depth 3#9
algebra6 min
Making a letter the subject
exercise
depth 3#13
algebra6 min
Clearing denominators, safely
theorem · proved here
depth 3#14
algebra5 min
Distributing over one bracket
theorem · proved here
depth 3#15
polynomials25 min
Polynomials: degree, leading coefficient, standard form
definition
depth 3#30
algebra5 min
Why x⁰ = 1 is forced
theorem · proved here
depth 4#10
algebra6 min
(x + p)(x + q)
theorem · proved here
depth 4#16
algebra6 min
The highest common factor of two monomials
theorem · proved here
depth 4#18
algebra35 min
Linear systems: none, one, or infinitely many
theorem · proved here
depth 4#28
polynomials30 min
Sums, products, and the degree of a product
theorem · proved here
depth 4#31
algebra35 min
√(x²) = |x|, not x
theorem · proved here
depth 5#11
algebra6 min
The square and the difference of two squares
theorem · proved here
depth 5#17
polynomials35 min
The division algorithm for polynomials
theorem · proved here
depth 5#32
polynomials30 min
The leading term governs behaviour at ±∞
theorem · proved here
depth 5#37
algebra30 min
Common factor, difference of squares, grouping
theorem · proved here
depth 6#19
algebra6 min
Which constant completes the square
theorem · proved here
depth 6#23
polynomials25 min
The remainder theorem
theorem · proved here
depth 6#33
algebra30 min
Factoring a quadratic over ℚ
theorem · proved here
depth 7#20
algebra30 min
Every quadratic is a shifted, scaled square
theorem · proved here
depth 7#24
algebra30 min
Domain restrictions are part of the expression
definition
depth 7#27
polynomials25 min
The factor theorem
theorem · proved here
depth 7#34
algebra6 min
From a factorisation to the roots
theorem · proved here
depth 8#22
algebra35 min
Sign analysis and interval notation
theorem · proved here
depth 8#29
polynomials30 min
The rational root theorem
theorem · proved here
depth 8#35
polynomials30 min
The fundamental theorem of algebra
theorem · cited
depth 8#36
functions30 min
A function is a map, not a formula
definition
depth 8#39
algebra35 min
The quadratic formula and the discriminant
theorem · proved here
depth 9#25
functions25 min
Domain is part of the function's identity
definition
depth 9#40
functions20 min
Even and odd functions
definition
depth 9#43
functions30 min
Exponential functions
definition
depth 9#45
trigonometry6 min
sin, cos and tan of an acute angle
definition
depth 9#50
algebra6 min
Counting real roots without finding them
theorem · proved here
depth 10#26
functions25 min
Composition and where it is defined
definition
depth 10#41
functions30 min
Shifts, scalings, reflections
theorem · proved here
depth 10#44
trigonometry5 min
Radians, arc length and sector area
definition
depth 10#51
functions25 min
Piecewise definitions and well-definedness
definition
depth 10#59
series5 min
A sequence is a function on the indices
definition
depth 10#60
polynomials30 min
Multiplicity and the shape at a root
theorem · proved here
depth 11#38
functions35 min
An inverse exists iff the function is a bijection
theorem · proved here
depth 11#42
functions30 min
Trigonometric functions from the unit circle
definition
depth 11#52
series5 min
What Σ abbreviates
definition
depth 11#61
limits40 min
The ε–N definition of a sequence limit
definition
depth 11#63
functions30 min
Logarithms: laws and domain
theorem · proved here
depth 12#46
trigonometry6 min
The exact values, and which sign to attach
exercise
depth 12#53
trigonometry5 min
Period 2π — and why tan is different
theorem · proved here
depth 12#55
series6 min
Σi and Σi², in closed form
theorem · proved here
depth 12#62
limits40 min
The ε–δ definition of a function limit
definition
depth 12#64
algebra35 min
The complex numbers
definition
depth 12
functions6 min
The number e, and the natural logarithm
definition
depth 13#47
functions35 min
Pythagorean and angle-addition identities
theorem · proved here
depth 13#54
trigonometry5 min
What the three graphs look like
observation
depth 13#56
limits40 min
Sum, product and quotient laws for limits
theorem · proved here
depth 13#65
limits25 min
One-sided limits
definition
depth 13#66
analysis35 min
Functions of a complex variable
definition
depth 13
functions6 min
Solving an equation with the unknown in an exponent
theorem · proved here
depth 14#48
trigonometry6 min
Reading A sin(Bx + C) + D
theorem · proved here
depth 14#57
trigonometry6 min
arcsin, arccos, arctan and the branch you chose
definition
depth 14#58
limits40 min
Continuity, ε–δ
definition
depth 14#67
limits30 min
The squeeze theorem
theorem · proved here
depth 14#69
limits35 min
Limits at infinity and asymptotes
theorem · proved here
depth 14#71
linear algebra35 min
Vectors and the dot product
definition
depth 14
functions6 min
Doubling time and half-life
theorem · proved here
depth 15#49
limits45 min
Uniform continuity: the quantifier order is the content
definition
depth 15#68
limits35 min
The intermediate value theorem
theorem · cited
depth 15#70
limits25 min
What 0/0 does and does not mean
observation
depth 15#72
calculus40 min
The derivative as a limit of difference quotients
definition
depth 15#73
calculus35 min
Riemann and Darboux sums
definition
depth 15#91
linear algebra35 min
The cross product
definition
depth 15
calculus30 min
Differentiable implies continuous; the converse is false
theorem · proved here
depth 16#74
calculus40 min
The power rule, with its hypotheses
theorem · proved here
depth 16#75
calculus25 min
Linearity of differentiation
theorem · proved here
depth 16#76
calculus40 min
The chain rule
theorem · proved here
depth 16#79
calculus40 min
Derivatives of sine and cosine
theorem · proved here
depth 16#82
calculus30 min
Tangent lines and linearisation
theorem · proved here
depth 16#85
calculus40 min
The Riemann integral
definition
depth 16#92
analysis45 min
Holomorphic functions
definition
depth 16
calculus40 min
Partial derivatives
definition
depth 16
calculus30 min
The product rule
theorem · proved here
depth 17#77
calculus35 min
Derivatives of exponentials, and why e
theorem · proved here
depth 17#80
calculus35 min
Implicit differentiation
theorem · proved here
depth 17#83
calculus25 min
Higher derivatives
definition
depth 17#84
analysis40 min
Contour integrals
definition
depth 17
calculus40 min
The multivariable chain rule
theorem · proved here
depth 17
vector calculus30 min
Vector fields
definition
depth 17
calculus30 min
The quotient rule
theorem · proved here
depth 18#78
calculus35 min
Derivatives of logarithms
theorem · proved here
depth 18#81
calculus40 min
Critical points and the derivative tests
theorem · proved here
depth 18#86
calculus40 min
Related rates
exercise
depth 18#89
analysis45 min
Cauchy's integral theorem
theorem · cited
depth 18
vector calculus40 min
The gradient and steepest ascent
theorem · proved here
depth 18
vector calculus40 min
Parametrised curves and surfaces
definition
depth 18
calculus40 min
The mean value theorem
theorem · cited
depth 19#87
analysis45 min
Analytic continuation and the identity theorem
theorem · cited
depth 19
vector calculus35 min
Curl as local circulation
definition
depth 19
vector calculus35 min
Divergence as local source density
definition
depth 19
vector calculus40 min
Line integrals of vector fields
definition
depth 19
calculus45 min
Optimisation problems
exercise
depth 20#88
calculus40 min
L'Hôpital's rule, and where it does not apply
theorem · cited
depth 20#90
calculus45 min
The fundamental theorem of calculus
theorem · cited
depth 20#93
vector calculus45 min
Surface integrals and flux
definition
depth 20
calculus35 min
Integration by substitution
theorem · proved here
depth 21#94
calculus35 min
Integration by parts
theorem · proved here
depth 21#95
calculus40 min
Improper integrals
definition
depth 21#96
vector calculus45 min
The divergence theorem
theorem · cited
depth 21
vector calculus45 min
Stokes' theorem
theorem · cited
depth 21
fields40 min
The Ampère–Maxwell law
theorem · cited
depth 22
series40 min
Dirichlet series
definition
depth 22
fields35 min
Faraday's law of induction
theorem · cited
depth 22
fields35 min
Gauss's law for the electric field
theorem · cited
depth 22
fields30 min
Gauss's law for magnetism
theorem · cited
depth 22
probability45 min
Probability spaces
definition
depth 22
fields45 min
Maxwell's equations, integral form
theorem · cited
depth 23
probability35 min
Random variables and distributions
definition
depth 23
analysis45 min
The Riemann zeta function
definition
depth 23
analysis40 min
Ergodicity
definition
depth 24
probability40 min
Expectation
definition
depth 24
analysis50 min
The functional equation
theorem · cited
depth 24
fields50 min
Maxwell's equations, differential form
theorem · cited
depth 24
analysis45 min
The critical strip
theorem · cited
depth 25
probability40 min
Heavy tails and regular variation
definition
depth 25
analysis40 min
Jensen's inequality
theorem · proved here
depth 25
probability40 min
The law of large numbers
theorem · cited
depth 25
probability45 min
The central limit theorem
theorem · cited
depth 26
probability35 min
Convex payoff and dispersion
observation
depth 26
probability45 min
Limit laws for the maximum
theorem · cited
depth 26
analysis40 min
The Riemann hypothesis
conjecture
depth 26
probability40 min
Time average against ensemble average
observation
depth 26
probability40 min
The Berry–Esseen rate
theorem · cited
depth 27
analysis45 min
Birkhoff's pointwise ergodic theorem
theorem · cited
depth 27
probability35 min
What a finite sample variance does not tell you
observation
depth 28
analysis45 min
Lyapunov exponents
definition
depth 28
corethe spine
maxwellto the field equations
talebheavy tails, ergodicity
zetacomplex analysis
A heavier, curved edge with an arrowhead is a bridge — a prerequisite crossing out of its strand. Nothing here is drawn as locked or mastered: that needs a reader’s record and none is being read.
Columns are prerequisite depth. Click selects; double-click opens the skill.
Press enter or space to select a node. You can then use the arrow keys to move the node around. Press delete to remove it and escape to cancel.
Press enter or space to select an edge. You can then press delete to remove it or escape to cancel.
The whole authored graph. Columns are prerequisite depth over the whole graph: 29 columns, 331 edges drawn. No mastery and no lock is drawn: those are readings of a reader’s record, and no record is being read. They are absent rather than shown at zero — a bar at zero is a reading too. What is on screen is the authored structure, which is the same for everyone.
edge
prerequisite— the skill at the tail must be known before the one at the head
claim status
definition
theorem (proved here)
theorem (cited)
conjecture
observation
exercise
A node’s colour is its claim status and nothing else — it does not change when you filter, search, or sort. Strand is written on the node as a word. Track is a hue on bridge edges only, where three branches take the three colour slots docs/04-plotting.md validates and the core spine takes a text token; the key to it is on the canvas, beside the marks it explains, and a bridge is already distinguished by its curve, its weight, and its arrowhead before any colour is read. Nothing here keys mastery, a lock, or whether a lesson exists, because no reader’s record is being read and those are all readings of one. The claim status, the strand, the track and the prerequisite structure are properties of the mathematics and are drawn in full.
The line view above is this graph in curriculum order; what to sit down and do next is on Learn; the mastery record and the attempt log are on Progress.