Day 89 — Capstone draft

Updated

July 30, 2026

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

Lock topics from Day 88 — no substitutions unless a claim is false.
Write in the skeleton order (definitions → proofs → …).
After each major item, run its self-check below before moving on.
Mark uncertain lines with ¿? rather than deleting structure.
Time-box: if stuck \(>15\) minutes on a proof step, write the gap explicitly and continue; return in a second pass.
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

Restate claim with quantifiers.
Assume hypotheses.
Expand definitions.
Chain equalities/implications with reasons.
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

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

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

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)

Story (CS-ish).
Explicit finite \(V\), \(E\) (list if \(\le 12\) edges).
Diagram.
Mathematical question.
Tool + work (path, topo order, bipartite check, …).
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

State problem with parameters.
If inverse: extended Euclid table + verification \(ax\equiv 1\pmod m\).
If CRT: construction + plug-in check each congruence.
If exponent: reduce via Fermat/Euler with coprimality check.
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

Paste static pseudocode.
Define elementary operation.
Translate to sum or recurrence.
Evaluate sum / apply Master / geometric series.
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

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

Two draft-day takeaways:

Tomorrow

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

Day 89 — Capstone draft

Why this matters

Draft protocol

Section 1 — Definitions packet (draft)

Writing guide

Self-check rubric (§1)

Sample skeleton (teaching scaffold — not your submission)

Common failure modes (§1)

Section 2 — Direct proof (draft)

Writing guide

Self-check rubric (§2)

Sample scaffold (structure only)

Common failure modes

Section 3 — Contrapositive or contradiction (draft)

Writing guide — contrapositive

Writing guide — contradiction

Self-check rubric (§3)

Sample scaffold

Common failure modes

Section 4 — Induction (draft)

Writing guide

Self-check rubric (§4)

Sample scaffold

Common failure modes

Section 5 & 6 — Two counting solutions (draft)

Writing guide (each)

Self-check rubric (each counting item)

Sample scaffolds (not full student work)

Common failure modes

Section 7 — Graph model (draft)

Writing guide (Day 69 template)

Self-check rubric (§7)

Sample scaffold

Common failure modes

Section 8 — Modular computation (draft)

Writing guide

Self-check rubric (§8)

Sample scaffold

Common failure modes

Section 9 — Complexity argument (draft)

Writing guide

Self-check rubric (§9)

Sample scaffold

Common failure modes

Section 10 — Retrospective (draft start)

Self-check

Section 11 — 30-day plan (draft start)

Self-check

End-of-day draft checklist

Common global failure modes

Checkpoint

Tomorrow