Day 90 — Capstone final
Day 90 — Capstone final and retrospective
Stage VIII · capstone dossier (final day)
Goal: Produce a defensible final dossier: verify every claim, close all ¿? gaps, polish exposition, complete the retrospective and 30-day plan. Celebrate finishing the volume — then schedule maintenance.
Why this matters
A final is not a rewrite of the whole course. It is quality control under a fixed outline: the mathematics should be correct, readable, and yours. After today you own a portable record of proofs, counting, graphs, modular arithmetic, and complexity literacy.
Still no labs. Final means polished writing. Optional arithmetic checks only after hand work stands alone.
Final-day schedule (recommended)
| Block | Duration | Task |
|---|---|---|
| 0 | 10 min | Inventory ¿? and failed self-checks from Day 89 |
| 1 | 40–60 min | Repair proofs first (highest risk) |
| 2 | 30–40 min | Counting + modular verification pass |
| 3 | 20–30 min | Graph clarity + complexity \(\Theta\) witnesses |
| 4 | 20 min | Definitions consistency sweep |
| 5 | 40–50 min | Retrospective + 30-day plan (full prose) |
| 6 | 20 min | Read-aloud pass + packaging |
| 7 | 10 min | Completion statement + archive path |
Section A — Verification pass (do this before cosmetic edits)
A1. Proof audit checklist (each of 3 proofs)
For each proof, mark:
- Claim copied exactly; conclusion matches claim
- Every variable introduced
- Definitions expanded at first use
- No “obviously” without a reason
- Quantifier order correct
- Induction: base true; IH applied to legal smaller case
- Contradiction: falsehood is genuine (e.g. \(0=1\), \(p\mid 1\))
- Independent check: special case \(n=2\) or numeric instance
Fix protocol: If a step is wrong, rewrite the whole paragraph containing it — do not patch with a footnote only.
A2. Counting audit
- \(\Omega\) or combinatorial object defined
- Overcount argument or “why not overcount” sentence
- Answer recomputed via second method or small-\(n\) brute force
- Units/meaning of the answer stated
A3. Graph audit
- \(V\) and \(E\) reconstructible
- Diagram matches sets
- Theorem/algorithm named correctly
- Story conclusion = math conclusion
A4. Modular audit
- \(\gcd\) computed where needed
- Inverse verified \(ax\equiv 1\pmod m\)
- CRT solution checked in each congruence
- No Euler without coprimality
A5. Complexity audit
- Elementary operation defined
- Sum or recurrence displayed
- \(O\) vs \(\Theta\) honest
- Sequential vs nested costs correct
- Spot-check \(n=4\) hand count vs formula
A6. Definitions audit
- Every term used in proofs appears in §1 (or standard primitive)
- \(O\) has \(C,n_0\)
- Examples not reused as definitions
- Notation consistent (\(n\) vs \(N\), \(G=(V,E)\))
Section B — Polish standards
| Aspect | Standard |
|---|---|
| Notation | Consistent \(n\), \(G=(V,E)\), \(\equiv\pmod m\) |
| Structure | Numbered sections matching Day 88 skeleton |
| Sentences | Prefer short full sentences in proofs |
| Displays | Important equalities on their own lines |
| Figures | Labeled vertices; caption one sentence |
| Honesty | “I used notes for X” if true — better than fake fluency |
Remove: motivational fluff, lab instructions, code repos, unrelated autobiography.
Keep: one brief “why this item” sentence per major section if it helps a reader.
Title page: name, date, “Maths for CS — Capstone Dossier — 90DaysOfX Volume 3”.
Section C — Final dossier exercise checklist
Complete all items; tick in your document header.
C1. Completeness (required contents)
- Definitions packet \(\ge 8\) entries with examples/non-examples
- Direct proof complete
- Contrapositive or contradiction proof complete
- Induction proof complete with \(P(n)\) stated
- Counting solution A complete
- Counting solution B complete
- Graph model with \(V,E\), question, answer
- Modular item with verification (+ inverse as required by your problem)
- Complexity item with sum/recurrence and \(\Theta\) or honest \(O\)
- Retrospective (Section D) complete
- 30-day plan (Section E) complete
- No programming lab included as a graded artifact
C2. Correctness spot-checks (recompute today)
- Recomputed one modular arithmetic line cold
- Re-did induction step on a blank page
- Special-case check for one counting answer
- Complexity sum evaluated for \(n=4\) by hand vs formula
- Graph tool re-run (e.g. topo order regenerated)
C3. Communication
- Read entire dossier aloud (or subvocally) once
- Peer test: could a Stage-IV student follow definitions?
- Archived: PDF/photo/notes folder path written here:
________
Section D — Retrospective (full prose required)
Write 4–8 pages handwritten equivalent (or ~800–1500 words typed) covering all prompts:
D1. Journey map
Which stages felt hardest (I–VIII)? Where did gates expose false confidence? Cite specific days (e.g. “Day 52 IE”, “Day 84 Master Case 3”).
D2. Three conceptual shifts
For each: old belief → new belief → one concrete example (proof, systems literacy, bug-thinking).
Example seed (write your own): “I thought \(O\) meant ‘exactly this slow’” → “\(O\) is an upper bound class” → “\(n=O(n^2)\) is true but loose.”
D3. Proof craft
Which style is most natural (direct / contra / induction)? Which will you deliberately practice in the next month? Quote one proof from the dossier you are proud of and one you barely trust.
D4. CS connections
Name three places outside this book where you will recognize: modular thinking, graph modeling, asymptotic honesty. Be specific (hash buckets, dependency install order, API timeout retries, …).
D5. Honesty about gaps
List topics still weak (e.g. strong induction, IE, Master Case 3, Bayes base rates, CRT non-coprime). No shame — precision only. Each gap should appear in the 30-day plan.
D6. About “no labs”
How did learning theory-first change (or not change) how you want to write code later? 1–2 paragraphs. Avoid empty praise of the format; be concrete.
D7. Capstone process meta
What would you do differently on Days 88–90 if you repeated the capstone next year? (Topic selection, verification order, time boxes.)
Section E — 30-day continued plan (detailed)
Build a plan with measurable drills. Fill every cell.
E1. Weekly themes
| Week | Theme | Primary resources (book days) | Success metric |
|---|---|---|---|
| 1 | Proof repair + Gate III retest | Days 28–34, Gate III | Score ≥ 80 or time ↓ 20% |
| 2 | Counting + graphs | Days 47–53, 59–69, Gate V/VI items | 6 problems closed-book |
| 3 | NT modular fluency | Days 71–80, Gate VII | Inverse + CRT cold, timed |
| 4 | Asymptotics + probability mix | Days 81–87 | 5 recurrences classified; 1 Bayes |
E2. Daily micro-habits (choose realistic; tick what you commit)
- 15 min: redefine 3 terms from memory
- 20 min: one proof rewrite
- 15 min: one modular or sum exercise
- Weekly: full timed mini-gate (60–90 min)
- Biweekly: re-read one capstone proof without notes
E3. Topics to revisit (priority order — edit to match your gaps)
- …
- …
- …
- …
- …
E4. Optional track (pick at most one for month 1)
| Track | First steps | Why later |
|---|---|---|
| Matrices lite | \(2\times 2\) mult, identity, inverse idea, linear systems | Graphics, ML literacy, Markov chains |
| Generating functions | Ordinary GF for Fibonacci / subsets | Advanced counting, asymptotics of coeffs |
| Automata | DFA definition, regex ↔︎ DFA culture | Compilers, protocols |
| Deeper probability | Conditional independence, Chernoff awareness | Randomized algorithms |
| Production crypto course | Separate engineering path; libraries | Beyond toy RSA |
Write which track (or “none — maintenance only”): ________
E5. Review calendar
Write four calendar dates:
- Gate III retest: ________
- Gate V (or counting set) retest: ________
- Gate VII retest: ________
- Full mixed 90-min: ________
E6. Resource list (optional)
- This volume days: ________
- Rosen / Lehman–Leighton–Meyer / other: chapter ________
- Peer review date: ________
Section F — Optional oral defense script (10 minutes)
If you want a higher bar, record yourself answering:
- State \(f=O(g)\) and prove \(3n+5=O(n)\) with witnesses.
- Prove one of your three proofs live from the claim alone.
- Compute \(\gcd\) and an inverse of your modular item.
- Explain your graph model and conclusion.
- What would you improve in the dossier with one more day?
Section G — Packaging the dossier
G1. Order of materials
- Title page
- Table of contents (section list)
- §1–§9 mathematics
- §10 Retrospective
- §11 30-day plan
- Appendix: gate scores, error log (optional)
- Completion statement (below)
G2. Archive formats
- Digital: PDF or markdown export of the write-up
- Analog: photographed notebook pages in order
- Path/URL/folder:
________
- Backup:
________
G3. Naming
YYYY-MM-DD-maths-cs-capstone-lastname.pdf (or equivalent)
Section H — Volume completion statement
Copy and sign:
I completed Volume 3 (Maths for CS) of 90DaysOfX with a capstone dossier
containing:
(1) key definitions,
(2) three proofs (direct; contrapositive or contradiction; induction),
(3) two counting solutions with justification,
(4) one graph model,
(5) one modular computation (with inverse as applicable),
(6) one complexity argument,
(7) a retrospective and a 30-day continued study plan.
The work is written mathematics without programming labs as the primary
deliverable. I verified the mathematics to the best of my ability on Day 90.
Signature / name: ____________
Date: ____________
Archive location: ____________Section I — Celebration without fluff
You finished a theory spine built for programmers:
arithmetic → algebra → logic/proofs → sets → counting → graphs → number theory for CS → asymptotics & discrete probability → capstone.
That is real mathematical maturity, not a participation sticker. The dossier is the artifact; the habits (definitions first, check inverses, label worst-case, demand quantifiers) are the residual value.
Do today: sign the statement, archive the file, put Week 1 of the 30-day plan on a calendar.
Do not today: start three optional tracks, rewrite Stage I, or open a coding project “to celebrate.” Maintenance begins with one old gate problem twice a week.
Common final-day pitfalls
| Pitfall | Fix |
|---|---|
| Rewriting topics instead of verifying | Lock claims; only fix errors |
| Cosmetic edits before math repair | Section A first |
| Vague 30-day plan | Metrics + dates |
| Inflating \(\Theta\) claims | Downgrade to \(O\) if needed |
| Skipping archive | Photo/PDF now |
| Shame spiral on gaps | Put gaps in the plan; ship the dossier |
| Adding a lab “bonus” | Out of scope for this volume’s capstone |
Checkpoint — Volume 3 complete when
- Section A audits passed
- Checklist C all ticked
- Retrospective D written in full prose
- 30-day plan E filled with dates and optional track choice
- Dossier packaged and archived
- Completion statement signed
Final personal takeaways (three)
- …
- …
- …
After Day 90
Default maintenance: one old gate problem twice a week keeps the dossier honest.
Next directions (optional, out of spine): matrix algebra lite, generating functions, automata, deeper probability, production crypto courses (separate engineering path), algorithm textbooks with proofs (CLRS chapters matching Days 81–84).
Standalone libraries in this monorepo: deeper Maths, Go, NixOS books remain available when you want breadth beyond the 90-day volume.
Congratulations — Stage VIII and the Maths for CS volume are complete.