deckreference

PH-DECK Expanded Worked Cases

These are slower versions of the card examples. The deck should test recall; this page should teach the moves.

1. Triangle Filtration: H0 and H1 Ledger

Filtration: vertex a, vertex b, edge ab, vertex c, edges ac and bc, triangle abc.
01At the first vertex, there is one connected component. H0 class born at 0.
02At the second vertex, there are two components. A second H0 class is born at 1.
03When edge ab appears, the two components merge. Elder rule: the younger component dies. This gives H0 bar [1,2).
04Vertex c creates a new component at 3. When c connects to the existing component at 4, that younger class dies, giving [3,4).
05The oldest component never dies in ordinary persistence. It gives [0,∞).
06H1 is zero until the three edges form a closed loop. At that moment the 1-cycle is born.
07When the 2-simplex abc appears, its boundary is exactly the loop, so the loop becomes a boundary and dies.
Conclusion: H0 = [0,∞), [1,2), [3,4). H1 = [4,5).

2. Bottleneck Matching

D1 = {(1,5), (2,3)} and D2 = {(1.5,4.5)}.
01Match the long bar to the long bar: max(|1-1.5|, |5-4.5|) = 0.5.
02The remaining short bar (2,3) can be sent to the diagonal. Diagonal cost is half its persistence: (3-2)/2 = 0.5.
03The matching cost is the maximum cost among all pairings, not the sum. max(0.5,0.5)=0.5.
04Why optimal? The point (1,5) cannot disappear cheaply: diagonal cost is 2. Its only good protection is the point (1.5,4.5), costing 0.5. Therefore every matching costs at least 0.5.

3. Hexagon Rips H1

01Six points on the unit circle have three chord lengths: side 1, skip-one chord sqrt(3), and diameter 2.
02At threshold 1, adjacent edges appear. The graph is a hexagon. It has one visible loop and no triangles, so an H1 class is born.
03Nothing changes until threshold sqrt(3).
04At sqrt(3), skip-one edges appear. Because Rips is a flag complex, every 3-clique becomes a filled triangle automatically.
05Those triangles provide a 2-chain whose boundary is the original hexagon loop, so the H1 class dies.
Conclusion: the H1 bar is [1, sqrt(3)).

How to Use This With the Deck

  1. Read the reference term before attempting a card if the statement feels opaque.
  2. Walk the expanded worked case once without trying to memorize it.
  3. Return to the deck and answer the self-check aloud before reveal.
  4. Only mark settled if you can explain the move, not just recognize the answer.

project=ph-deck · repo=/Users/uprootiny/Tamuz/orchies · branch=main · commit=4c943e3 · session=/Users/uprootiny/.codex/sessions/2026/07/06/rollout-2026-07-06T18-40-56-019f3816-f670-71b3-8125-733a6255ff7a.jsonl