Day 88 — Capstone outline
Day 88 — Capstone outline
Stage VIII · capstone dossier (planning day)
Goal: Select every required dossier item; write precise problem statements; list definitions and theorems you will use; schedule Day 89–90 writing; no new theory dump beyond what you need to plan.
Why this matters
A capstone is not “more exercises.” It is a portable portfolio of mathematics you can defend: definitions, proofs, counting, graphs, modular arithmetic, complexity, and a retrospective. Today you design the dossier so Day 89 is drafting and Day 90 is polish — not panic topic selection.
Format: Written mathematics only. No labs, no coding projects, no repo deliverables. Pseudocode fragments are allowed only inside the complexity item as static text to analyze.
Required dossier contents (canonical)
Your dossier must include:
- Key definitions (logic, sets, graphs, \(O\)-notation — and ideally one NT definition you use).
- Three proofs
- one direct
- one contrapositive or contradiction
- one induction (weak or strong)
- one direct
- Two counting solutions with full justification (product/sum, bijection, inclusion-exclusion, etc.).
- One graph model of a CS-ish situation (vertices, edges, question answered).
- One modular computation (+ inverse if it exists, or CRT/Euclid as appropriate).
- One complexity argument for a small static code fragment (sums → \(\Theta\)).
- Retrospective + 30-day continued plan.
Optional strong extras (not required): Bayes update, Master theorem classification, hashing collision expectation, coupon collector.
Planning principles
| Principle | Practice |
|---|---|
| Prefer problems you can grade yourself | Known numerical checks |
| Prefer problems that reuse Stage tools | Not random internet contests |
| Write the claim first | Theorem/problem in one sentence |
| List lemmas before proof | Avoid mid-proof invention |
| One page per major item target | Depth over sprawl |
| Honest difficulty | Stretch one item, not all |
| Capstone is yours | Do not copy solution manuals verbatim |
Section A — Inventory of strengths and gaps
Complete honestly (copy into your notes):
| Area | Confidence (1–5) | Evidence (gate day / exercise) | Gap to close before Day 90 |
|---|---|---|---|
| Direct proof | |||
| Contrapositive / contradiction | |||
| Induction | |||
| Counting | |||
| Graphs | |||
| Modular / Euclid | |||
| Big-O / loops | |||
| Probability (optional) |
Minimum: any area at confidence \(\le 2\) must either (a) be avoided as a required showcase if possible, or (b) get a 30-minute repair drill today from the matching day.
Gate scores log (fill):
| Gate | Score | Weak sections |
|---|---|---|
| III (logic/proof) | ||
| IV (sets) | ||
| V (counting) | ||
| VI (graphs) | ||
| VII (NT for CS) |
Section B — Choose concrete items (fill every blank)
B1. Definitions packet (8–12 definitions)
List the exact terms you will define formally (symbol + English). Template:
- Proposition / logical connectives you need
- Valid argument / proof method you use
- Set operations (\(\cup,\cap,\setminus,\) complement)
- Function: injective / surjective / bijective (as needed)
- Graph \(G=(V,E)\); path or tree as needed
- \(a\mid b\) or \(a\equiv b\pmod n\) (pick what modular item uses)
- \(f=O(g)\) with quantifiers
- …
- …
- …
For each definition later: include one example and one non-example.
Selection guidance: definitions that appear in your three proofs and modular/complexity items first; do not define unused jargon.
B2. Proof 1 — Direct
- Claim (one sentence): …
- Source day / variant of: e.g. Day 28 / Day 71 linear combination
- Key definitions used: …
- Lemma checklist: …
- Self-check plan: special case \(n=2\) / numeric instance
Suggested pool (pick one or adapt):
| Claim idea | Flavor |
|---|---|
| If \(a\mid b\) and \(a\mid c\) then \(a\mid(bx+cy)\) | NT direct |
| Handshaking: sum degrees \(=2|E|\) | Graph direct |
| \((A\cup B)^c=A^c\cap B^c\) | Set algebra direct |
| If \(f,g\) injective then \(g\circ f\) injective | Functions |
B3. Proof 2 — Contrapositive or contradiction
- Claim: …
- Method chosen: contrapositive / contradiction
- Why this method fits: …
- Negation of conclusion written carefully: …
Suggested pool:
| Claim idea | Method |
|---|---|
| If \(n^2\) even then \(n\) even | Contra / contrapositive |
| \(\sqrt{2}\) irrational | Contradiction + FTA |
| If \(p\) prime and \(p\mid ab\) then \(p\mid a\) or \(p\mid b\) | Contradiction via Bézout |
| Infinite primes | Contradiction (Euclid) |
| If \(f=O(1)\) and unbounded… | pick carefully |
B4. Proof 3 — Induction
- Predicate \(P(n)\): …
- Universe of \(n\): \(n\ge n_0\)
- Base case(s): …
- IH: …
- Strong vs weak: …
Suggested pool:
| Claim idea | Notes |
|---|---|
| \(\sum_{i=1}^n i=n(n+1)/2\) | Classic weak |
| \(2^n>n\) for \(n\ge 1\) or tighter | Easy |
| FTA existence of prime factorization | Strong induction |
| Number of subsets \(2^n\) | Weak |
| Tree with \(n\) vertices has \(n-1\) edges | Strong / structural |
B5. Counting A
- Problem statement: …
- \(\Omega\) or combinatorial object: …
- Method: product / sum / bijection / IE / stars-bars
- Answer target form: integer or formula in \(n\)
- Second-method check: …
Suggested pool: passwords with constraints; injections \(A\to B\); bit-strings avoiding \(00\); committees with officers; IE “at least one of each type.”
B6. Counting B
- Problem statement (different method family from A if possible): …
- Method: …
- Answer: …
- Small-\(n\) brute force check: for \(n=3\) or \(4\), list
B7. Graph model
Use the Day 69 writeup template:
| Field | Your plan |
|---|---|
| CS-ish story (2–4 sentences) | |
| \(V=\) | |
| \(E=\) | |
| Directed? Weighted? | |
| Question (math) | |
| Tool (BFS / topo / bipartite / paths / coloring / …) | |
| Conclusion (story language) |
Suggested situations: package dependencies; course prerequisites; network reachability; conflict graph for scheduling; git commit DAG lite; user–permission bipartite.
B8. Modular computation
- Problem: e.g. compute \(a^{-1}\bmod m\); solve \(ax\equiv b\); CRT system; \(a^k\bmod m\) via Fermat/Euler
- Parameters (numbers): …
- Must include: inverse computation or explicit justification that inverse fails + alternative path
- Verification: multiply back / plug into each congruence
Minimum bar: extended Euclid inverse or CRT with verification or large exponent reduction with inverse application.
B9. Complexity argument
- Static fragment (write 5–15 lines of pseudocode): …
- Elementary operation counted: …
- Sum or recurrence: …
- Bound: \(\Theta(\ldots)\) preferred; honest \(O\) if not tight
- Best/worst note if relevant: …
Suggested fragments: triangular double loop; binary search; mergesort recurrence citation + Master; doubling loop with geometric \(j\).
B10. Retrospective + 30-day plan (structure only today)
Outline headings you will write fully on Day 89–90:
- Journey map (stages I–VIII)
- Three conceptual shifts
- Proof craft reflection
- CS connections (3)
- Honest gaps
- Weekly themes for 30 days
- Micro-habits
- Gate retest calendar dates
Section C — Dossier skeleton (copy into your document)
Maths for CS — Capstone Dossier
Author: ________ Date: ________ Volume: 90DaysOfX / 03-maths
§1 Definitions (8–12)
§2 Proof A — Direct
§3 Proof B — Contrapositive/Contradiction
§4 Proof C — Induction
§5 Counting A
§6 Counting B
§7 Graph model
§8 Modular computation
§9 Complexity argument
§10 Retrospective
§11 30-day plan
Appendix (optional): error log, gate scoresSection D — Selection guidance by prior days
| Need | Strong source days |
|---|---|
| Direct proof | 28, 35–36, 59, 71 |
| Contra / contradiction | 29–30, 71 (√2, Euclid primes), 74–75 |
| Induction | 31–32, 54–55, FTA existence |
| Counting | 47–52, 53 |
| Graph model | 59–69 especially 69 |
| Modular + inverse | 72–75, 78, Gate VII |
| Complexity | 81–84 |
| Probability optional | 85–87 |
Balance rule: at least two different stages represented among the three proofs; modular item from Stage VII; complexity from Stage VIII; graph from Stage VI.
Section E — Rubric (how Day 90 will judge you)
| Component | Points | Pass bar |
|---|---|---|
| Definitions quality | 10 | ≥ 7 |
| Direct proof | 12 | ≥ 8 |
| Contra/contradiction proof | 12 | ≥ 8 |
| Induction proof | 12 | ≥ 8 |
| Counting A | 10 | ≥ 7 |
| Counting B | 10 | ≥ 7 |
| Graph model | 10 | ≥ 7 |
| Modular + inverse | 10 | ≥ 7 |
| Complexity | 8 | ≥ 5 |
| Retrospective + 30-day plan | 6 | ≥ 4 |
| Total | 100 | ≥ 75, no major item blank |
Major item blank = any of §2–§9 missing or only a title.
Section F — Day 89–90 schedule
| Day | Focus | Output |
|---|---|---|
| 88 (today) | This outline fully filled | No empty “…” in Section B |
| 89 | Full draft of §1–§9; start §10 | Complete draft, ¿? marks OK |
| 90 | Verify, polish, finish §10–§11, archive | Final dossier |
Today’s exit criteria:
- Strength/gap table filled
- All B1–B10 choices concrete (numbers chosen for modular/counting)
- Skeleton document created
- 30 min repair scheduled for any confidence \(\le 2\) area
- Rubric understood
Section G — Anti-patterns
| Anti-pattern | Replace with |
|---|---|
| “Prove something about primes” (vague) | Exact claim sentence |
| Counting without justification | Method name + bijection/product sentences |
| Graph doodle without \(V,E\) sets | Explicit finite sets |
| Modular without verification | \(ax\equiv 1\) check line |
| \(\Theta\) by vibe | Sum or Master case |
| Retrospective as diary only | Structured prompts Day 90 |
| Coding project “to illustrate” | Forbidden as deliverable |
Exercises (planning exercises — all required)
- Fill Section A completely.
- Write final claim sentences for all three proofs.
- Write full problem statements for both counting items.
- Complete the graph model table.
- Fix modular parameters and compute a preview inverse today (scratch work; final write-up Day 89).
- Paste pseudocode for the complexity item.
- List 10 definitions for B1.
- Draft calendar dates for four gate retests in the next 30 days.
- Identify one optional extra you will not do (scope control).
- Peer check (if available): can another student understand your graph story question?
Checkpoint
- Required contents 1–7 mapped to concrete problems
- No labs planned as deliverables
- Rubric pass plan realistic
- Day 89 start time scheduled
- Weak topics remediated or avoided consciously
Two planning takeaways:
- …
- …
Tomorrow
Day 89 — Capstone draft (write everything).