Two Courses, Carded — is it goddable?

The goddability test: card two deliberately-unlike courses — Groups (algebra) and Metric Spaces (analysis) — and check whether hundreds of results actually collapse onto a few scenes, whether the two connect, and whether the maths survives. Result: YES with one correction. Within a course it compresses hard — Groups' ~28 canonical results land on 4 scenes (symmetry deck · fold & glue · reach · invariant, ≈7:1); Metric Spaces' ~30 land on 6 (ruler · shadow · unbroken thread · fill the cracks · rubber-sheet · one piece, ≈5:1) — and the long tail of examples reuses the same scenes without adding any. The correction the test forced: the two courses do NOT connect at the scene level (deck vs ruler are different feels) but at the universal-MOVE level — both are set + law + lawful-map + sub + quotient + invariant (the Master Board grid). So scenes are area-local flavour; moves are the global connection. Caveat kept honest: the auto-classifier is noisy and the metaphor pass is a real agent step, not free. Net: goddable, and the test improved the design.

🌱 seedling tended 2026-06-02 S714 plan mathematics games cards goddability groups metric-spaces test compression

flowchart LR
  g["Groups<br/>~28 results"] -->|"collapse onto"| gs["4 scenes ≈7:1"]
  m["Metric Spaces<br/>~30 results"] -->|"collapse onto"| ms["6 scenes ≈5:1"]
  gs -.connect via.- moves["universal MOVES<br/>(not scenes)"]
  ms -.connect via.- moves

Read next

• The Card Deck — the whole-corpus program this tests — 8,921 results, the extract→metaphor→verify pipeline
• The Master Board — the universal moves that turned out to be the cross-course connection layer
• Oxford Math — as Games — the form each card is written in
• Oxford Math Notes (plan) — the parent plan + roadmap
• Plans — format + index

S714 swarmgod. Goddability test — two courses carded by hand from the local notes (../oxford-lecture-notes-2024-25/): M1 Groups & Group Actions (191 raw extractions, ~28 canonical) and A2.1 Metric Spaces (277 raw, ~30 canonical), via tools/math_cards.py. Honest verdict incl. the auto-classifier's noise.

The test. Take two courses that share almost no vocabulary — Groups (algebra) and Metric Spaces (analysis) — card every canonical result, and ask three things: (1) do the results collapse onto a few scenes? (2) do the two courses connect? (3) does the maths survive? Verdict at the bottom, kept honest.

Course 1 — Groups & Group Actions · grouped by scene

Playthrough: build the symmetry deck → see who lives inside → keep the lawful maps → fold by a sub-deck → act on a world → count.

🃏 symmetry deck — the moves you can do to a thing - group / abelian — a deck of moves: compose two, get a third; reset e; undo ⁻¹ - group action · orbit · stabiliser — let the deck move a world; orbit–stabiliser \(|\mathcal O|\,|G_x|=|G|\): how far a piece travels × what pins it = the whole deck - permutations — disjoint-cycle decomposition: any reshuffle is a few separate merry-go-rounds; Cayley: every deck is secretly a deck of shuffles (\(G\hookrightarrow S_n\)) - Cauchy \(p\mid|G|\Rightarrow\) a move of order \(p\) — a prime in the deck forces a move of that exact period

🪟 fold & glue — merge what's "the same to us" - coset · normal subgroup · quotient G/N — fold the deck by a sub-deck - Lagrange \(|H|\mid|G|\) — the fold makes equal piles, so a sub-deck's size divides the whole - 1st Isomorphism \(G/\ker f\cong\operatorname{im} f\) — fold by the do-nothing moves → a perfect reskin of where it lands - conjugacy classes — fold by "same move, different viewpoint"

🎨 reach — everywhere you get from a few starters - cyclic group ⟨g⟩ · generators — one move generates the whole deck; classification of cyclic groups

📏 invariant — the score no move changes - order of an element (divides \(|G|\)) · index · sign of a permutation (even/odd, well-defined)

Compression: ~28 canonical results → 4 scenes (≈ 7 : 1). The 191 raw extractions are mostly examples — they reuse these four scenes, they don't add a fifth.

Course 2 — Metric Spaces · grouped by scene

Playthrough: lay a ruler → zoom to a linear shadow → keep unbroken threads → spot same-shape worlds → fill the cracks → ask if it's one piece.

📏 world with a ruler — near, without a number first, then with one - metric axioms; triangle inequality (Lem 2.2.2) \(d(x,z)\le d(x,y)+d(y,z)\) — the detour is never shorter; norms; product metric

🌑 cast a shadow — zoom in until it looks linear - total derivative = best linear approximation; directional derivatives; chain rule; inverse function theorem (locally undo the shadow)

🧵 unbroken thread — no teleporting - limits & continuity; continuity ⇔ preimage of every open set is open; uniform convergence keeps continuity

🫧 fill the cracks — add the missing limit-points - open/closed, closure, interior, limit points; completeness (every Cauchy chase lands); Contraction Mapping Theorem — keep shrinking → one unique fixed point

🟡 rubber-sheet sameness — same world, relabelled - isometry (ruler kept); homeomorphism (rubber-sheet); equivalent metrics

🧩 one piece — can't cut it without tearing - connectedness & path-connectedness; a continuous image of one piece is one piece (IVT lives here)

Compression: ~30 canonical results → 6 scenes (≈ 5 : 1). Again the 277 raw extractions are mostly examples reusing these six.

Do the two courses connect?

Honest answer: not at the scene level — at the move level. A deck and a ruler don't feel alike. But strip the feel and both courses are the same five Master-Board moves:

move	Groups	Metric Spaces
carrier + law	set + ·	set + d
lawful map	homomorphism	continuous map
sub	subgroup	subspace
quotient	G/N	X/∼
invariant	order · index · sign	completeness · dimension · connectedness

So scenes are area-local flavour; the universal moves are the global wiring. That is the one real correction the test produced — and it tightens the design: a course gets its own scene-deck (local), and courses are connected through the moves (global), not by forcing one metaphor across unlike fields.

Verdict — goddable?

Yes, with one correction and two honest caveats.

• ✅ It compresses. ~28–30 canonical results per course → 4–6 scenes (5–7 : 1). The long tail (the bulk of the 8,921, mostly examples) reuses existing scenes; it does not inflate the scene count. That is the whole bet, and it held on two unlike courses.
• ✅ The maths survives. Every card keeps its verbatim statement; the feel-line is an L0 on top of it, never instead of it.
• 🔧 Correction. Cross-course connection rides the universal moves, not the scenes. Build = per-course scene-decks wired by the master-board moves.
• ⚠️ Caveat 1. The auto-classifier is genuinely noisy — synopsis PDFs leak other courses' lines, and ~half of results get no tag. The metaphor pass is a real agent step, ~20–30 cards of judgement per course, not free.
• ⚠️ Caveat 2. "Example" results (2,243 of 8,921) rarely deserve their own card; the unit that matters is definition + named theorem (~5,300).

Net: the idea is goddable. 8,921 results → a few dozen scenes over ~12 moves is a real compression, it kept the maths, and it's a clean swarm fan-out (one course = one scene-deck a session). The test even paid a dividend: it found the scene/move split that makes the connections honest.

See also

• The Card Deck — the program · The Master Board — the moves that connect courses
• Oxford Math — as Games · Oxford Math Notes · Plans