Day 30 — Capstone Dossier & Retrospective

Updated

July 30, 2025

Day 30 — Capstone Dossier & Retrospective

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.

Important

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:

  1. Key definitions (logic, sets, graphs, \(O\)-notation — and ideally one NT definition you use).
  2. Three proofs
    • one direct
    • one contrapositive or contradiction
    • one induction (weak or strong)
  3. Two counting solutions with full justification (product/sum, bijection, inclusion-exclusion, etc.).
  4. One graph model of a CS-ish situation (vertices, edges, question answered).
  5. One modular computation (+ inverse if it exists, or CRT/Euclid as appropriate).
  6. One complexity argument for a small static code fragment (sums → \(\Theta\)).
  7. 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:

  1. Proposition / logical connectives you need
  2. Valid argument / proof method you use
  3. Set operations (\(\cup,\cap,\setminus,\) complement)
  4. Function: injective / surjective / bijective (as needed)
  5. Graph \(G=(V,E)\); path or tree as needed
  6. \(a\mid b\) or \(a\equiv b\pmod n\) (pick what modular item uses)
  7. \(f=O(g)\) with quantifiers
  8. …
  9. …
  10. …

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 scores

Section 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)

  1. Fill Section A completely.
  2. Write final claim sentences for all three proofs.
  3. Write full problem statements for both counting items.
  4. Complete the graph model table.
  5. Fix modular parameters and compute a preview inverse today (scratch work; final write-up Day 89).
  6. Paste pseudocode for the complexity item.
  7. List 10 definitions for B1.
  8. Draft calendar dates for four gate retests in the next 30 days.
  9. Identify one optional extra you will not do (scope control).
  10. 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:

  1. …
  2. …

Tomorrow

Day 89 — Capstone draft (write everything).

Day 89 — Capstone draft

Stage VIII · capstone dossier (writing day)
Goal: Produce a complete first draft of every required dossier section from the Day 88 outline. Prefer finished prose over perfection; leave marked ¿? only for minor checks.

Why this matters

Drafts create the raw material Day 90 polishes. An outline is not a dossier. Today you write proofs in full sentences, expand counting justifications, draw the graph, and finish the modular and complexity write-ups by hand in your notebook or document.

Important

No labs / no coding projects. Static pseudocode only in §9. Do not open a programming environment except optionally to check arithmetic after the hand solution exists.

Draft protocol

  1. Lock topics from Day 88 — no substitutions unless a claim is false.
  2. Write in the skeleton order (definitions → proofs → …).
  3. After each major item, run its self-check below before moving on.
  4. Mark uncertain lines with ¿? rather than deleting structure.
  5. Time-box: if stuck \(>15\) minutes on a proof step, write the gap explicitly and continue; return in a second pass.
  6. Target: complete draft today, not perfect prose.

Recommended time blocks (adjust to your day):

Block Minutes Section
1 40 §1 Definitions
2 50 §2–§3 two proofs
3 40 §4 induction
4 40 §5–§6 counting
5 40 §7 graph
6 40 §8 modular
7 40 §9 complexity
8 30 §10–§11 draft starts
9 20 Self-check sweep

Section 1 — Definitions packet (draft)

Writing guide

For each of 8–12 terms:

Term (symbol):
Formal definition:
Example:
Non-example:
Used later in: §___

Quality bar: a Stage-IV reader can understand each definition without the rest of the book.

Self-check rubric (§1)

  • ≥ 8 definitions
  • Each has example and non-example
  • \(O\) / \(\mid\) / graph / set / logic as needed by later sections
  • Notation consistent with §2–§9
  • No circular definitions

Sample skeleton (teaching scaffold — not your submission)

\(a\mid b\): \(\exists k\in\mathbb{Z},\ b=ak\). Example \(3\mid 12\). Non-example \(3\nmid 10\).
\(f=O(g)\): \(\exists C>0\exists n_0\forall n\ge n_0,\ |f(n)|\le C|g(n)|\). Example \(3n+1=O(n)\). Non-example \(n^2\neq O(n)\) (state as “does not satisfy”).
Simple undirected graph: \(G=(V,E)\) with \(E\subseteq\binom{V}{2}\). …

Your definitions must be yours; expand fully.

Common failure modes (§1)

Failure Fix
Definitions only as symbols Write English quantifiers
Example but no non-example Add deliberate non-example
Defining 20 unused terms Stick to 8–12 that you use

Section 2 — Direct proof (draft)

Writing guide

  1. Restate claim with quantifiers.
  2. Assume hypotheses.
  3. Expand definitions.
  4. Chain equalities/implications with reasons.
  5. Conclude by matching claim.

Template:

Claim. …
Proof. Let … be given such that [hypotheses].
By definition of …, we have ….
Therefore ….
Hence [conclusion]. ∎

Self-check rubric (§2)

  • Claim matches Day 88 exactly
  • Every variable introduced
  • No “clearly” without expansion
  • Conclusion line present
  • Special-case numeric check in margin

Sample scaffold (structure only)

Claim: If \(a\mid b\) and \(a\mid c\) then \(a\mid(bx+cy)\) for all \(x,y\in\mathbb{Z}\).
Proof idea: write \(b=ak\), \(c=a\ell\), factor \(a(kx+\ell y)\).
Write in full sentences yourself.

Common failure modes

Failure Fix
Proof by example Keep general \(a,b,c\)
Missing quantifiers on \(x,y\) “for all integers \(x,y\)”
Using later theorems unstated List in Day 88 lemmas

Section 3 — Contrapositive or contradiction (draft)

Writing guide — contrapositive

To prove \(P\Rightarrow Q\), prove \(\neg Q\Rightarrow \neg P\).
1. State you use contrapositive.
2. Assume \(\neg Q\).
3. Derive \(\neg P\).

Writing guide — contradiction

  1. Assume claim false (negate carefully).
  2. Derive absurdity (\(0=1\), \(p\mid 1\), empty set membership, etc.).
  3. Conclude claim true.

Self-check rubric (§3)

  • Method named in first line
  • Negation is logically correct
  • Absurdity is genuine (not “surprising”)
  • No hidden direct proof labeled as contradiction

Sample scaffold

Claim: If \(n^2\) is even then \(n\) is even.
Contrapositive: if \(n\) odd then \(n^2\) odd; write \(n=2k+1\), expand.
Or \(\sqrt{2}\) irrational via even exponents / FTA.

Common failure modes

Failure Fix
Negating incorrectly Write \(\neg(P\Rightarrow Q)\equiv P\wedge\neg Q\) when needed
Contradiction that only shows “hard” Need logical falsehood
Circular use of claim Forbidden

Section 4 — Induction (draft)

Writing guide

Let P(n) be: …
Base case: Prove P(n0).
Inductive hypothesis: Assume P(k) for some k ≥ n0.
[or strong: assume P(n0),…,P(k)]
Inductive step: Prove P(k+1) using IH.
Conclusion: By induction, P(n) for all n ≥ n0.

Self-check rubric (§4)

  • \(P(n)\) stated explicitly
  • Base verified numerically
  • IH applied to a legal smaller instance
  • Algebra of step checked
  • Strong vs weak choice justified if strong

Sample scaffold

\(P(n):\sum_{i=1}^n i=n(n+1)/2\). Base \(n=1\): \(1=1\). Step: sum to \(k+1\) = sum to \(k\) + \((k+1)\) = …
Complete algebra yourself.

Common failure modes

Failure Fix
IH used on \(k+1\) Only on \(k\) (or \(\le k\))
Base missing Check \(n_0\)
\(P(n)\) vague “works for \(n\)” Exact equation/inequality

Section 5 & 6 — Two counting solutions (draft)

Writing guide (each)

  1. Define what is being counted in one sentence.
  2. Name the method.
  3. Justify each factor/summand (“why not overcount”).
  4. Box the answer.
  5. Second check: alternate method or small-\(n\) list.

Self-check rubric (each counting item)

  • Object defined
  • Method named
  • Overcount argument present
  • Answer simplified
  • Verification present

Sample scaffolds (not full student work)

A. Number of injective functions \([k]\to[n]\) for \(k\le n\): \(P(n,k)=n!/(n-k)!\). Justification: ordered distinct images.
B. Bit-strings of length \(n\) with no two consecutive \(1\)s: Fibonacci recurrence setup + closed count \(F_{n+2}\).

Choose your Day 88 problems and write complete solutions.

Common failure modes

Failure Fix
Answer only Method paragraph required
Double counting unnoticed Two methods or listing
IE missing intersection Draw Venn for two sets

Section 7 — Graph model (draft)

Writing guide (Day 69 template)

  1. Story (CS-ish).
  2. Explicit finite \(V\), \(E\) (list if \(\le 12\) edges).
  3. Diagram.
  4. Mathematical question.
  5. Tool + work (path, topo order, bipartite check, …).
  6. Answer in story language and math language.

Self-check rubric (§7)

  • \(V,E\) reconstructible without the diagram
  • Diagram matches sets
  • Question answerable from the graph
  • Conclusion not stronger than the math

Sample scaffold

Story: five packages; dependencies \(A\to C\), \(B\to C\), \(C\to D\), \(A\to E\).
Question: install order / detect cycle.
Tool: Kahn topological sort; show DAG and one order.

Common failure modes

Failure Fix
Only a picture Write \(V,E\)
Tool named, not applied Show steps
Story conclusion free of math Tie to theorem

Section 8 — Modular computation (draft)

Writing guide

  1. State problem with parameters.
  2. If inverse: extended Euclid table + verification \(ax\equiv 1\pmod m\).
  3. If CRT: construction + plug-in check each congruence.
  4. If exponent: reduce via Fermat/Euler with coprimality check.
  5. Box final answer in standard residue system.

Self-check rubric (§8)

  • \(\gcd\) computed where needed
  • Inverse verified by multiplication
  • CRT checked on each modulus
  • No Euler without \(\gcd=1\)

Sample scaffold

Compute \(17^{-1}\bmod 100\) via extended Euclid; verify \(17\cdot 53=901\equiv 1\).
Then solve \(17x\equiv 5\pmod{100}\) by \(x\equiv 5\cdot 53\pmod{100}\).

Common failure modes

Failure Fix
Sign error on inverse Always verify
CRT unique mod wrong number Product when coprime
Exponent reduced mod \(n\) Use \(\varphi(n)\)

Section 9 — Complexity argument (draft)

Writing guide

  1. Paste static pseudocode.
  2. Define elementary operation.
  3. Translate to sum or recurrence.
  4. Evaluate sum / apply Master / geometric series.
  5. State \(\Theta\) or honest \(O\) with one sentence on tightness.

Self-check rubric (§9)

  • Operation defined
  • Sum/recurrence displayed
  • Algebra correct for \(n=4\) spot-check
  • \(O\) vs \(\Theta\) honest

Sample scaffold

for i = 1..n:
  for j = 1..i:
    op

\(T(n)=\sum_{i=1}^n i=\Theta(n^2)\).

Common failure modes

Failure Fix
“Nested loops so \(n^2\)” only Write the sum
Master mis-cased Compute \(\log_b a\) vs \(f\)
Amortized claimed without total cost Show aggregate bound

Section 10 — Retrospective (draft start)

Write at least bullet answers today; expand to full prose on Day 90:

  • Hardest stage and why
  • Three conceptual shifts (old → new → example)
  • Preferred proof style + one to practice
  • Three CS connections
  • Gaps remaining
  • Effect of “no labs” pedagogy (honest)

Self-check

  • All six bullets have content
  • Not only “it was hard” — concrete math references

Section 11 — 30-day plan (draft start)

Fill the table structure:

Week Theme Days/resources Metric
1
2
3
4

Add four retest dates for gates III, V, VII, mixed.

Self-check

  • Measurable metrics
  • Realistic weekly load
  • Includes weak topics from Section A Day 88

End-of-day draft checklist

  1. §1 complete draft
  2. §2 complete draft
  3. §3 complete draft
  4. §4 complete draft
  5. §5 complete draft
  6. §6 complete draft
  7. §7 complete draft
  8. §8 complete draft
  9. §9 complete draft
  10. §10 bullets present
  11. §11 table present
  12. All ¿? listed on a cover sticky note for Day 90

If any of §2–§9 is blank: that is the first Day 90 emergency — better to finish a rough correct proof tonight than polish definitions.

Common global failure modes

Failure Fix
Rewriting theory notes instead of solving chosen problems Stick to Day 88 claims
Switching topics mid-day Lock outline
Perfect first sentence, missing later sections Time-box; move on
Unverified modular arithmetic Multiply check now
Graph without a question Add a question that the math answers

Checkpoint

  • Complete draft of §1–§9
  • §10–§11 started
  • Self-checks run per section
  • ¿? list ready for Day 90
  • No programming lab deliverable

Two draft-day takeaways:

  1. …
  2. …

Tomorrow

Day 90 — Capstone final: verify, polish, retrospective, archive.

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.

Important

Still no labs. Final means polished writing. Optional arithmetic checks only after hand work stands alone.

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)

  1. Definitions packet \(\ge 8\) entries with examples/non-examples
  2. Direct proof complete
  3. Contrapositive or contradiction proof complete
  4. Induction proof complete with \(P(n)\) stated
  5. Counting solution A complete
  6. Counting solution B complete
  7. Graph model with \(V,E\), question, answer
  8. Modular item with verification (+ inverse as required by your problem)
  9. Complexity item with sum/recurrence and \(\Theta\) or honest \(O\)
  10. Retrospective (Section D) complete
  11. 30-day plan (Section E) complete
  12. No programming lab included as a graded artifact

C2. Correctness spot-checks (recompute today)

  1. Recomputed one modular arithmetic line cold
  2. Re-did induction step on a blank page
  3. Special-case check for one counting answer
  4. Complexity sum evaluated for \(n=4\) by hand vs formula
  5. Graph tool re-run (e.g. topo order regenerated)

C3. Communication

  1. Read entire dossier aloud (or subvocally) once
  2. Peer test: could a Stage-IV student follow definitions?
  3. 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)

  1. …
  2. …
  3. …
  4. …
  5. …

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:

  1. Gate III retest: ________
  2. Gate V (or counting set) retest: ________
  3. Gate VII retest: ________
  4. 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:

  1. State \(f=O(g)\) and prove \(3n+5=O(n)\) with witnesses.
  2. Prove one of your three proofs live from the claim alone.
  3. Compute \(\gcd\) and an inverse of your modular item.
  4. Explain your graph model and conclusion.
  5. What would you improve in the dossier with one more day?

Section G — Packaging the dossier

G1. Order of materials

  1. Title page
  2. Table of contents (section list)
  3. §1–§9 mathematics
  4. §10 Retrospective
  5. §11 30-day plan
  6. Appendix: gate scores, error log (optional)
  7. 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)

  1. …
  2. …
  3. …

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.