Oxford Math, as Blueprints¶

A seed prototype for compressing the Oxford notes into something feelable. Three layers: PRIMITIVES (a small coined vocabulary of moves + structures, grounded in what actually recurs across 113 courses — completion 95%, closure 90%, span 88%, limit 86%), COMPOSITION (theorems are built by combining primitives with one operator algebra — refine ∩, compose ∘, generalize, transport ≅ — per STATEMENT-COMPOSITION), and BLUEPRINTS (one feelable real-life scene that carries SEVERAL processes at once and metaphor-translates to other subjects — the transport ≅ made physical). The quotient move alone runs identically across quotient-group ×53, quotient-map ×51, quotient-space ×11, quotient-module ×7 in the notes: one scene (fold & glue), four subjects. This is the planning-phase prototype on a few notes — it grows a few notes at a time, not all 502 at once.

🌱 seedling tended 2026-06-02 S714 plan mathematics blueprints compression primitives composition metaphor feelable oxford notes

flowchart LR
  prim["PRIMITIVES<br/>coined moves + structures<br/>(empirically the recurring ones)"] --> comp["COMPOSITION<br/>combine via one operator algebra<br/>∩ ∘ define generalize ≅"]
  comp --> blue["BLUEPRINT<br/>one feelable scene<br/>carrying many processes"]
  blue -->|"transport ≅"| trans["metaphor-translate<br/>to other subjects"]

L0 — the model in one breath¶

Three layers, bottom-up:

Primitives — the recurring atoms. Structures (the carriers: set, map, sequence, vector space, group, metric space…) and moves (the verbs: complete, close, span, quotient, project, bound, take-a-limit…). We are free to coin the names (Can's call) — pick the most feelable word, not the textbook one — but we ground which atoms matter in what actually recurs across the corpus (below).
Composition — a theorem is built from primitives with one small operator algebra (from STATEMENT-COMPOSITION): refine ∩ (stack conditions), compose ∘ (do one move then another), define (name a reusable bundle), generalize (subsume N), transport ≅ (carry it across an isomorphism). A theorem is an expression over primitives.
Blueprints — one feelable scene that carries several moves at once and metaphor-translates to other subjects. The translation is just the transport ≅ operator made physical: once a process lives in a scene, the same scene maps onto economics, biology, code. This is what makes the compression accessible and portable.

flowchart LR
  notes["130 courses<br/>502 PDFs"] -->|"mine what recurs"| prim["~20 moves + ~20 structures<br/>(primitives)"]
  prim -->|"operator algebra<br/>∩ ∘ define generalize ≅"| comp["theorems as expressions"]
  comp -->|"anchor several moves<br/>in one scene"| blue["~13 feelable blueprints"]
  blue -->|"transport ≅"| world["maths · physics · finance · biology · code"]

L1 — the primitives, grounded in the corpus¶

We text-mined 113 of the 130 undergraduate courses (PyMuPDF; the rest are scanned/handwritten). Document frequency = in how many courses the move/structure actually appears. The point of the number is honesty: these are not the primitives we like, they are the ones the notes use.

Top moves (operations) — the verbs that recur everywhere:

Move	Coined feelable name	Courses
completion	fill the cracks	95%
extension	carry it further	94%
image / range	what survives	93%
closure	shrink-wrap	90%
bound / estimate	fence it	89%
span / generated-by	reach	88%
inverse	undo	88%
take-a-limit	where the steps head	86%
quotient	fold & glue	58%
projection	cast a shadow	72%
induction	climb the staircase	69%
diagonalize	find the grain	32%

Top structures (carriers) — the nouns ideas get represented as:

Structure	Coined feelable name	Courses
set	the bag	100%
function / map	the arrow	99%
sequence / series	the trail of steps	94%
basis / dimension	the set of base colours	83%
linear map / matrix	the stretch-and-rotate	81%
group	the symmetry deck	67%
metric space	the world with a ruler	64%
vector space	the mixing studio	70%
continuous map	the unbroken thread	94%
compact set	the corner-able crowd	61%
topology / homeomorphism	rubber-sheet sameness	55%

These are starting names — the test of a good coin is whether a 15-year-old feels it and a mathematician agrees it's exact. We refine them as we add notes.

L2 — the blueprints (the heart)¶

A blueprint is one scene that satisfies three rules (this is the upgrade over the old ad-hoc anchor-an-abstraction-in-an-object habit): it must (a) be feelable — you can see or touch it; (b) carry ≥2 processes at once — one scene, many moves; (c) translate to ≥1 other subject. Thirteen to start, grounded in a few notes — and the set grows as we add courses.

① The Staircase / Domino Run¶

Scene. A staircase where each step is only built once the step below it exists. Or a line of dominoes: knock the first, the rest fall.
Processes carried. climb the staircase (induction: base + step), recursion, the trail of steps heading somewhere (a sequence and its limit), telescoping sums, and guessing the next step (a recurrence / ansatz).
The maths, exact. \(P(0)\) and \(P(n)\Rightarrow P(n{+}1)\) give \(\forall n\,P(n)\); \(a_n \to L\); \(\sum_{k}(b_{k+1}-b_k)\) collapses to its endpoints.
Translation. dominoes (physics), compound interest and generations (finance, biology), "true for one ⇒ true for all" (logic).

② Water Finding Its Level (filling the cracks)¶

Scene. Pour water into a rough container. It seeps into every gap and settles flat — no point sits higher than the average of its neighbours.
Processes carried. fill the cracks (completion — the corpus's #1 move, 95%), shrink-wrap (closure), limit, equilibrium / fixed point, and the Laplacian ("a value equals the average around it" ⇒ harmonic = perfectly settled). "Solve the PDE" = let it relax to that level.
The maths, exact. \(\mathbb{Q}\hookrightarrow\mathbb{R}\) completes the rationals; \(\Delta u = 0 \iff u(x)=\text{mean of }u\) on small spheres; the heat equation relaxes to the steady state.
Translation. heat spreading, consensus / opinion-averaging (social), price equilibrium (economics), diffusion (biology).

③ Folding & Gluing the Map¶

Scene. Take a paper map and fold it so every pair of points you've declared "the same" touches and fuses. The folded sheet is a new, smaller object — and it is a perfect copy of the picture you get by just looking at where everything lands.
Processes carried. quotient (fold & glue by an equivalence relation), equivalence classes / cosets, the First Isomorphism Theorem (folded object \(\cong\) image), and homotopy (can I deform one path into another without tearing the sheet?).
The maths, exact. \(G/\ker f \;\cong\; \operatorname{im} f\) — the same statement in groups, vector spaces, rings, modules. The notes prove this is one move, not four: across the corpus it appears as quotient-group ×53, quotient-map ×51, quotient-space ×11, quotient-module ×7.
Translation. zip codes and citizenship (sorting people into equivalence classes), a clock (arithmetic mod 12), "the same up to relabeling."

④ Casting a Shadow (nearest wall)¶

Scene. Shine a light on an object; its shadow on the wall is the closest flat stand-in. Equivalently: the nearest point on a wall to where you're standing.
Processes carried. cast a shadow (projection), orthogonality, best approximation / least squares, "keep the gist, drop what doesn't fit" (lossy compression), and conditional expectation (best guess given partial info).
The maths, exact. \(\hat x=\arg\min_{v\in V}\lVert x-v\rVert\), with the clean split \(x=\hat x+(x-\hat x)\) where the remainder is orthogonal to the wall.
Translation. a sundial, JPEG (the gist of an image), linear regression (statistics), summarizing a document.

⑤ The Mixing Studio (a few base colours)¶

Scene. A few base paints on a table. Everything you can make is exactly what you can mix from them; a recipe you can scale up or down.
Processes carried. span (reach — 88%), linear combination, base colours (basis & dimension — the smallest mixing set), change of basis (same colour, new recipe), adding scalars / scaling, and guessing a solution's format (ansatz: assume \(y=e^{\lambda t}\), then mix).
The maths, exact. \(\operatorname{span}\{v_i\}\), \(\dim V\); a linear ODE solved by guessing \(e^{\lambda t}\) and taking linear combinations; Fourier — any sound is a mix of pure tones.
Translation. colour mixing, ingredient ratios, map coordinates, a music chord.

⑥ The Squeeze (two fences closing in)¶

Scene. A value trapped between two fences that creep toward each other until they pinch it onto a single spot.
Processes carried. fence it (bound / estimate — 89%), the squeeze / sandwich theorem, ε–δ ("get within any tolerance I name"), inequalities, convergence-by-trapping.
The maths, exact. if \(a_n\le x_n\le b_n\) and \(a_n,b_n\to L\) then \(x_n\to L\); \(|f(x)-L|<\varepsilon\) whenever \(|x-a|<\delta\).
Translation. error bars (measurement), bisection / binary search (code), "between a rock and a hard place."

⑦ The Unbroken Thread (draw without lifting the pen)¶

Scene. A line drawn without lifting the pen — no teleporting, no gaps; a thread you can't cut without leaving a hole.
Processes carried. continuity (the unbroken thread — 94%), the Intermediate Value Theorem ("you must cross"), connectedness ("one piece"), path-connectedness.
The maths, exact. \(f\) continuous on \([a,b]\) with \(f(a)<0<f(b)\) ⇒ \(\exists c\in(a,b),\,f(c)=0\); a connected set admits no split into two non-empty disjoint open pieces.
Translation. a road with no breaks, a story with no plot holes, "to get from below to above you must pass through the middle."

⑧ Cornering the Crowd (an infinite crowd in a closed pen)¶

Scene. Pack infinitely many people into a closed, bounded pen — they must bunch up somewhere; and a finite handful of nets covers the whole pen.
Processes carried. compactness (the corner-able crowd — 61%), Bolzano–Weierstrass ("a bounded infinite set clusters"), finite subcover, the extreme value theorem ("a continuous function on a compact set attains its max").
The maths, exact. every sequence in a compact set has a convergent subsequence; every open cover has a finite subcover.
Translation. pigeonhole ("more pigeons than holes ⇒ two share"), a packed room, statistical sampling.

⑨ Finding the Grain (turn the plank till it splits clean)¶

Scene. Turn a plank until you see the wood grain — along the grain it splits cleanly. Rotate an object until it sits on its natural axes.
Processes carried. diagonalize (find the grain), eigenvalues / eigenvectors (the grain directions), the spectral theorem, principal axes, change of basis to "pure stretch."
The maths, exact. \(A=PDP^{-1}\); along an eigenvector \(Av=\lambda v\) — pure scaling, no twist.
Translation. principal component analysis (data), resonant modes — a drum's natural tones (physics), "find the angle where it's simple."

⑩ The Symmetry Deck (the moves you can do to a thing)¶

Scene. The deck of all moves that leave an object looking the same — the rotations and flips of a tile, the twists of a Rubik's cube. Do two in a row → still in the deck; every move can be undone.
Processes carried. group (the symmetry deck — 67%), group action (a move applied to the object), orbit (where a point can be sent), stabilizer (the moves that pin it), the Cayley table; every permutation is a product of disjoint cycles.
The maths, exact. the square's symmetries form \(D_4\); orbit–stabilizer: \(|\text{orbit}|\cdot|\text{stab}|=|G|\).
Translation. a Rubik's cube, dance steps that return home, a snowflake's symmetry, error-correcting codes.

⑪ Rubber-Sheet Sameness (stretch, don't tear)¶

Scene. Two shapes are "the same" if you can stretch or bend one into the other without cutting or gluing — the coffee mug is the donut.
Processes carried. homeomorphism (rubber-sheet same — 55%), topology (what survives stretching), isometry (same, with the ruler kept), equivalent metrics — the structure side of transport ≅.
The maths, exact. a homeomorphism is a continuous bijection with continuous inverse; connectedness, compactness and "number of holes" are preserved.
Translation. map vs territory (distances distort, connections survive), genre vs plot, "same skeleton, different skin."

⑫ The Log Ruler (slide rule)¶

Scene. A slide rule: numbers spaced by their logarithm, so multiplying two of them becomes sliding and adding their lengths. Put the world on a log ruler and products turn into sums.
Processes carried. the logarithm (multiply → add), log-differentiation ("log-diff": take logs to turn a product or power into a sum, then differentiate each piece), exponential growth as a straight line on log paper, orders of magnitude.
The maths, exact. \(\log(ab)=\log a+\log b\); \((\ln f)'=f'/f\), so \(f'=f\,(\ln f)'\) turns \(\frac{d}{dx}(uvw)\) into \(uvw\big(\tfrac{u'}{u}+\tfrac{v'}{v}+\tfrac{w'}{w}\big)\).
Translation. decibels · pH · the Richter scale (science), compound growth as a straight line (finance), Weber–Fechner "feels-like" perception (psychology).

⑬ The Tree (split until trivial)¶

Scene. A branching tree: split a problem into smaller copies of itself until each leaf is trivial; or lay two trees side by side and read off where they differ (tree-diff).
Processes carried. recursion, divide & conquer, structural induction (induct over how a thing was built), case-trees (exhaustive branches), parsing, and tree-diff (the minimal edits between two structures).
The maths, exact. \(T(n)=2\,T(n/2)+O(n)\Rightarrow O(n\log n)\); a proof by cases is a finite branch-tree; structural induction over a recursively-defined set.
Translation. a family tree, an org chart, nested folders, a decision tree, git merging two histories (tree-diff).

⑫ and ⑬ are coined to cover moves you flagged (log-diff, recursion/tree) that the frequency scan under-counts — the blueprint set is allowed to grow past the measured top primitives whenever a recurring move deserves a feelable home.

Composition, worked on a few real notes¶

Two theorems pulled straight from the downloaded notes, written as expressions over primitives — showing the compression isn't lossy:

First Isomorphism Theorem (M1 Groups · A0/M1 Linear Algebra · A3 Rings & Modules · B2 Representation Theory):

\[\underbrace{\operatorname{im} f}_{\text{what survives}} \;\cong\; \underbrace{\text{Domain}}_{\text{the bag}}\,/\,\underbrace{\ker f}_{\text{what collapses}}\]

As composition: arrow (homomorphism) → take what-collapses (kernel) → fold & glue (quotient) by it → the result equals what-survives (image).
As blueprint: ③ Folding & Gluing. One scene explains the theorem in four courses (transport ≅).

Rank–Nullity (the linear shadow of the same fact):

\[\dim\ker f + \dim\operatorname{im} f = \dim V \quad\text{— "what you crush + what survives = what you started with."}\]

Same blueprint (③), now measured by dimension. The two theorems are one move seen at two resolutions — which is precisely the compression we're after.

How this evolves (planning phase)¶

This page is a seed, deliberately built from a few notes. The loop to grow it:

flowchart LR
  pick["pick 2–3 related notes"] --> extract["extract the moves<br/>+ structures used"]
  extract --> coin["coin / reuse a feelable name"]
  coin --> scene["assign each cluster<br/>to ONE blueprint scene"]
  scene --> measure["measure: how many<br/>theorems collapse onto it?"]
  measure -->|"good ⇒ keep, translate"| pick
  measure -->|"weak ⇒ recut the scene"| scene

Next step (one session): take M1 Linear Algebra I + A2.1 Metric Spaces + Prelims Groups, fill in blueprints ②③⑤ fully, and count how many of each course's named theorems reduce to a blueprint. That count is the falsifiable measure — if most theorems don't collapse onto a small scene-set, the compression is too lossy and we recut.
Guardrail. A blueprint must stay exact: every scene links to the precise statement it carries, and the σ-metric stops us from gluing two genuinely-different ideas into one scene just because the metaphor is pretty.