Week 13 of 28 · Toolkit · due 2026-08-22 · 30–36 hrs

Full Toolkit Audit + Phase 3 Gate

Full Toolkit Audit + Phase 3 Gate gate complete

This week's lesson · 7 min read

not started · read first, then train the gates

The toolkit audit: proving to yourself that thirteen weeks stuck

This is the phase gate. Thirteen weeks built a toolkit; this week you audit it cold and take a three-problem mini-mock in the real ninety-minute format. The audit's job is ruthless: any technique you cannot produce from a blank page — statement, hypotheses, one worked instance — gets flagged and re-drilled before the slot-mastery phase, because the next fifteen weeks ASSUME the toolkit and train exam craft on top of it.

A toolkit you cannot reproduce cold is inventory you do not own — the audit converts 'I covered that' into 'I can use that' or into a named repair.

Key ideas — the week on one card

A complete audit entry is a quadruple: statement WITH hypotheses, proof idea, counterexample showing the hypotheses matter, and the problem type it kills.
Recognition is not recall: only what you can produce from a blank page survives the exam room.
The minute-30 checkpoint: no nameable next step means park, write the partial cleanly, and move — finished problems carry the score.
Calibration beats raw score: track the gap between predicted and actual on every mock from here to December.

1The blank-page audit

The format: one blank page per track, thirty minutes, book closed. Write every technique the track taught — statement WITH hypotheses, the one-line proof idea, and a problem type it kills. Score each entry green (complete, confident), yellow (statement fine, shaky on when to deploy), red (could not produce it). The checklist gate this week gives you the master list to diff against AFTER you write; diffing first turns an audit into a copying exercise.

Expect the yellows to cluster in deployment, not statements — knowing Chebyshev's sum inequality is common; remembering that it wants SAME-SORTED sequences and fires on 'sum of products vs product of averages' is rarer. Yellow repairs are recognition reps: five problems where the technique is the known answer, focusing purely on what the problem looked like BEFORE you knew.

The audit's second product is your triage instinct baseline. Before the mini-mock, privately predict which of the three problems you will solve and in what order. Comparing prediction to outcome measures self-knowledge — the skill that decides, in December, whether you spend your third half-hour on the right problem.

Worked example

A red-flag repair, worked: your blank page for analysis is missing 'uniform convergence preserves integrals'. Rebuild the entry.

Nudge 1
State the theorem WITH its hypotheses — which convergence mode does the swap require?
Reveal step 1
Statement with hypotheses: if $f_n \to f$ UNIFORMLY on $[a,b]$, then $\int_a^b f_n \to \int_a^b f$. The hypothesis that matters is uniform (pointwise fails: mass can escape — the moving-bump $f_n = n \cdot \mathbf{1}_{(0, 1/n)}$ has every $\int f_n = 1$ but $f = 0$).
Nudge 2
The proof is one inequality: bound the integral difference by the sup difference.
Reveal step 2
One-line proof idea: $\left|\int f_n - \int f\right| \le (b-a) \sup |f_n - f| \to 0$ — the interval's finite length converts uniform smallness into integral smallness.
Nudge 3
Attach the counterexample that shows pointwise is not enough — the moving bump.
Reveal step 3
Problem type it kills: any 'compute $\lim_n \int$' where you want to swap limit and integral — and the counterexample bump is the WHY behind the hypothesis, which is what makes the entry stick. Statement, mechanism, counterexample, use case: that quadruple is one complete audit entry.
Answer
A complete entry = statement + hypotheses + proof idea + counterexample showing the hypotheses matter + the problem type it kills.

Pitfall. Auditing with the checklist open. The gap between recognition ('yes, I know that one') and recall (producing it blank) is precisely what the exam room exposes — protect the blank page.

2Reading your mini-mock like an examiner

Three problems, ninety minutes — the real 2026 session shape. Grade yourself on the official 0-10 rubric mentality: 0-2 for exploration that never becomes an argument, and the cliff between 7 and 8+ is COMPLETENESS — every claim justified, every case closed, equality conditions stated. Most self-graded 8s are real-world 4s because the writer knows what they meant; grade what is ON THE PAGE, a day later, as a stranger.

Debrief in three timescales. Immediately after: log the timeline — when you committed to each problem, when you abandoned, where the minutes actually went (memory of time use decays within hours). Next day: re-grade cold and re-solve what you missed WITHOUT looking anything up, since a miss you can repair unaided was a pressure failure, not a knowledge failure — different fix. End of week: fold both into the audit — did failures land in red/yellow zones (toolkit gaps) or in green zones (exam craft gaps)? That split decides what the slot-mastery phase emphasizes for you.

Calibration matters more than the score. A 12/30 with accurate self-prediction beats an 18/30 you thought was a 27 — the second student will bleed points in December on problems they wrongly believe they closed. Track the prediction gap as its own statistic from now through every mock.

Worked example

Mock timeline analysis: you spent 55 of 90 minutes on problem 2 and got 3/10; problems 1 and 3 got 25 and 10 minutes for 8 and 1. What is the repair?

Nudge 1
Name the failure mode before the fix: where did the 55 minutes actually go?
Reveal step 1
Name the failure mode: SUNK-COST LOCK. By minute 30 on problem 2 you had no main-line argument — the rule of thumb is that a problem with no viable plan at the half-hour mark gets parked, because points come from FINISHED problems ($8 + 1 + 3 = 12$, while $8 + \text{finished } 3 + \text{parked } 2$ plausibly lands 16-19).
Nudge 2
Test the counterfactual: was the neglected problem solvable for you? Prove it untimed.
Reveal step 2
Check the counterfactual honestly: was problem 3 actually solvable for you? Re-solve it untimed the next day. If yes (say, 35 clean minutes), the mock cost you real points to a scheduling error, not a math error.
Nudge 3
Install a mechanism that survives adrenaline, not an intention.
Reveal step 3
Install the mechanism, not the intention: a hard checkpoint at minute 30 — 'do I have a plan whose next step I can name?' If no, park, and spend two minutes writing partial progress cleanly (rubric points live there). Intentions fail under adrenaline; checkpoints survive it.
Answer
Sunk-cost lock; the repair is a hard minute-30 checkpoint per problem, verified against the untimed re-solve.

Pitfall. Grading your intentions instead of your ink. If a step 'was obvious', it needed one sentence; the rubric pays for sentences, not for what you could have written.

Before you open the gates

Blank page first, checklist second — recognition is not recall, and only recall survives the exam room.
Audit entries are quadruples: statement + hypotheses, proof idea, counterexample, use case.
Predict your mock outcome before taking it; the prediction gap is a tracked statistic from here to December.
Minute-30 checkpoint: no nameable next step → park, write partials cleanly, move.
Sort mock failures into toolkit gaps vs exam-craft gaps — the split personalizes the entire next phase.

Check yourself

1. A complete blank-page audit entry contains statement, proof idea, use case, and:

2. The minute-30 checkpoint question on a mock problem is:

3. Failures on the mini-mock that land in your audit's GREEN zones indicate:

Practice ladder — three rungs, rising

Each rung: attempt cold, one hint if stuck, worked resolution only after a real try.

Rung 1 (foundational proof, cold). Prove that $\sqrt 2$ is irrational.

One hint

Assume $\sqrt 2 = a/b$ in lowest terms and force $2$ to divide both $a$ and $b$.

Worked resolution

Suppose $\sqrt 2 = a/b$ with $a, b$ integers, $b \ne 0$, and $\gcd(a,b) = 1$. Squaring gives $a^2 = 2b^2$, so $2 \mid a^2$; since $2$ is prime, $2 \mid a$, say $a = 2c$. Then $4c^2 = 2b^2$, i.e. $b^2 = 2c^2$, so $2 \mid b^2$ and hence $2 \mid b$. But then $2 \mid \gcd(a,b) = 1$, a contradiction. Therefore no such $a,b$ exist and $\sqrt 2$ is irrational. [Source: this week's blank-page audit — the CMU 01-Intro cold re-derivation of the $\sqrt 2$-irrationality proof.]

Rung 2 (foundational proof, cold). Prove that the interval $[0,1]$ is uncountable.

One hint

Assume an enumeration $x_1, x_2, \dots$ and build a number differing from $x_i$ in its $i$-th decimal digit, choosing digits that avoid the $0.999\ldots = 1.000\ldots$ ambiguity.

Worked resolution

Suppose $[0,1]$ were countable, listed as $x_1, x_2, x_3, \dots$ with decimal expansions $x_i = 0.d_{i1} d_{i2} d_{i3}\dots$. Define $y = 0.e_1 e_2 e_3 \dots$ by $e_i = 5$ if $d_{ii} \ne 5$ and $e_i = 4$ if $d_{ii} = 5$. Every digit $e_i \in \{4,5\}$, so $y$ has no trailing $0$s or $9$s and hence a unique decimal expansion; thus $y \in [0,1]$, yet $y \ne x_i$ for every $i$ because they differ in the $i$-th digit. This contradicts the list being exhaustive, so no enumeration exists and $[0,1]$ is uncountable. [Source: this week's blank-page audit — the CMU 01-Intro cold re-derivation of the uncountability proof.]

Rung 3 (toolkit staple, done solo). Prove Fermat's little theorem: for a prime $p$ and any integer $a$, $a^p \equiv a \pmod p$.

One hint

Induct on $a \ge 0$ using the binomial theorem, and use that $p \mid \binom{p}{k}$ for $0 < k < p$.

Worked resolution

It suffices to prove $a^p \equiv a \pmod p$ for integers $a \ge 0$, since both sides depend only on $a \bmod p$. Induct on $a$. Base: $0^p = 0 \equiv 0$. Step: assume $a^p \equiv a$. By the binomial theorem, $(a+1)^p = \sum_{k=0}^{p} \binom{p}{k} a^k = a^p + 1 + \sum_{k=1}^{p-1}\binom{p}{k}a^k$. For $0 < k < p$ the coefficient $\binom{p}{k} = \frac{p!}{k!\,(p-k)!}$ carries the prime $p$ in its numerator with nothing in the denominator to cancel it (as $k, p-k < p$), so $p \mid \binom{p}{k}$ and that middle sum vanishes mod $p$. Hence $(a+1)^p \equiv a^p + 1 \equiv a + 1 \pmod p$, completing the induction. [Source: this week's Full Toolbox Checklist — the number-theory entry FLT ($a^p \equiv a \bmod p$), reproduced cold.]

Prove it — constructed response

Rebuild the analysis-toolkit entry 'uniform convergence lets you pass the limit through the integral' as a full proof. State the theorem with its hypothesis, prove that if $f_n \to f$ uniformly on $[a,b]$ (each $f_n$ Riemann integrable) then $\int_a^b f_n \to \int_a^b f$, and exhibit a pointwise-convergent counterexample proving the uniform hypothesis cannot be dropped.

Self-grade against the rubric — completion requires the judgment, not the text

States the theorem with the UNIFORM hypothesis and proves the convergence via the sup bound: $\left|\int_a^b f_n - \int_a^b f\right| \le \int_a^b |f_n - f| \le (b-a)\sup_{[a,b]}|f_n - f| \to 0$.
Addresses that the limit $f$ is itself integrable (a uniform limit of integrable functions is integrable), so $\int_a^b f$ is even defined — e.g. via the Riemann criterion $U(f,P) - L(f,P) \le [U(f_n,P) - L(f_n,P)] + 2(b-a)\sup|f_n - f|$.
Gives a pointwise-but-not-uniform counterexample where the integral is NOT preserved, e.g. $f_n = n\,\mathbf{1}_{(0,1/n)}$ on $[0,1]$: $f_n \to 0$ pointwise yet $\int_0^1 f_n = 1 \not\to 0$, and names $\sup|f_n| = n \to \infty$ as the failure of uniformity.

Model proof (compare AFTER grading yourself)

Theorem. If $f_n \to f$ uniformly on $[a,b]$ and each $f_n$ is Riemann integrable, then $f$ is Riemann integrable and $\int_a^b f_n \to \int_a^b f$. (Integrability of $f$.) Let $\varepsilon_n = \sup_{[a,b]}|f_n - f|$, so $\varepsilon_n \to 0$ by uniform convergence. Given $\varepsilon > 0$, pick $n$ with $\varepsilon_n < \varepsilon$; as $f_n$ is integrable, choose a partition $P$ with $U(f_n,P) - L(f_n,P) < \varepsilon$. Since $|f - f_n| \le \varepsilon_n$ everywhere, on each subinterval the sup and inf of $f$ differ from those of $f_n$ by at most $\varepsilon_n$, so $U(f,P) - L(f,P) \le [U(f_n,P) - L(f_n,P)] + 2(b-a)\varepsilon_n < \varepsilon + 2(b-a)\varepsilon$. As $\varepsilon$ was arbitrary, $f$ meets the Riemann criterion and is integrable. (Convergence of integrals.) Then $\left|\int_a^b f_n - \int_a^b f\right| = \left|\int_a^b (f_n - f)\right| \le \int_a^b |f_n - f| \le (b-a)\varepsilon_n \to 0$, so $\int_a^b f_n \to \int_a^b f$. (Necessity of uniformity.) On $[0,1]$ let $f_n = n\,\mathbf{1}_{(0,1/n)}$, equal to $n$ on $(0,1/n)$ and $0$ elsewhere. For each fixed $x$: at $x = 0$ every $f_n(x) = 0$, and for $x > 0$ we have $f_n(x) = 0$ once $n > 1/x$, so $f_n \to 0$ pointwise. Yet $\int_0^1 f_n = n \cdot \frac1n = 1$ for every $n$, so $\int_0^1 f_n = 1 \not\to 0 = \int_0^1 \lim_n f_n$. The convergence is not uniform — $\sup_{[0,1]}|f_n - 0| = n \to \infty$ — which is exactly the hypothesis the theorem requires. $\blacksquare$

Confidence in your proof (0–5): 2

The gates

Toolkit Maturity Review — proof + method consolidation (study with the checklist)

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Teaching/maturity support so this problem-heavy week has the resource quota: learn/consolidate the methods before (and after) the problems.

sources & assignments (4)

Source Michael Penn — Proof Writing (playlist) — Proof-writing series — tighten exposition/maturity before the cold toolkit re-examination Watch-for: begins with set-theory foundations (sets, Cartesian products, operations) before the proof-technique videos.
Source Michael Penn — Putnam Exam Solutions (playlist) — Putnam walkthroughs — see full clean solutions modeling the maturity the toolkit week expects
Source CMU 01-Intro (Po-Shen Loh Putnam seminar) — CMU 01-intro — re-derive the √2-irrationality and uncountability proofs cold (proof-method self-check)
Source Supplemental — Lupu TTU MATH 4000, Recitation 5 (direct) — Writeup companion for “Toolkit Maturity Review — proof + method consolidation (study with the checklist)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Toolkit Audit — Technique Catalog Review

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- Review gate (depth): Toolkit Audit — Technique Catalog Review

why this gate: the lesson's §1 The blank-page audit is what it trains

sources & assignments (11)

Source Putnam and Beyond — PnB §2–§6 — for each technique header, state it from memory; for any you blank on, do 1 problem from that section (target ≥6 techniques)
Source Engel + CMU seminar handouts — Engel contents + CMU seminar index — self-quiz invariants/GF/valuations/trace-det/Feynman/roots-of-unity; do 1 Engel problem per gap
Source Larson — Problem-Solving Through Problems — Larson Ch.1–8 section headers — self-quiz each; do 1 problem from any chapter you cannot summarize in one sentence
Source CMU 21-295 Putnam Seminar (full session recordings) — One full seminar session as the audit's external benchmark: solve along, and log every technique the room uses that is missing from your catalog.
Source Kin Y. Li — Mathematical Excalibur — Kin Li *Mathematical Excalibur* — pick 2 issues matching your weak topics; do 1 featured problem from each
Source Evan Chen — OTIS Excerpts (olympiad handouts) — Evan Chen *OTIS Excerpts* — functional-equations chapter problems 1–2 + inequalities chapter problems 1–2
Source Michael Penn — Putnam Exam Solutions (playlist) — Watch 2 solutions tagged with this gate's topic; pre-commit your first move before each reveal.
Source Cezar Lupu — TTU MATH 4000, Lecture 13 — Self-contained problem session — before each solution is revealed, name which toolkit item unlocks the problem. Score yourself out of the session's problem count.
Source Mathematical Circles (Russian Experience) — Aligned circle companion: Ch. 8 Problems for the First Year + Ch. 17 Problems for the Second Year. Use as mixed toolkit audit texture; select only problems matching this week's weakest topics.

Problems Putnam archive — default fallback — Default audit set (use when topicAccuracy has no data) — 4 problems spanning the remaining families, cold, 25 min each: (1) functional equations — Putnam 2008 A1, (2) combinatorics — Putnam 2009 A1, (3) number theory — Putnam 2014 B1, (4) linear algebra — Putnam 2017 B1. Any family below 8/10 goes on the Phase 3 watch list.Putnam · archive: Putnam 2008 A1 (functional eqs)Putnam 2009 A1 (combinatorics)Putnam 2014 B1 (number theory)Putnam 2017 B1 (linear algebra)
Problems Putnam archive — Audit sprint: solve the 4 fallback problems listed above under 25-min caps in one sitting (100 min block), then grade each family 0-10 in the toolkit catalog — the audit is not complete until all four grades are written.Putnam

AMC/AIME Speed Warmup

AMC/AIME Speed Warmup (build pattern recognition + speed)

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- AMC/AIME speed warmup — fast pattern-recognition reps; not full Putnam difficulty.

sources & assignments (5)

Source CMU 01-Intro (Po-Shen Loh Putnam seminar) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here.
Source Michael Penn — Proof Writing (playlist) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here. Watch-for: begins with set-theory foundations (sets, Cartesian products, operations) before the proof-technique videos.
Source Michael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Recitation 5 (direct) — Writeup companion for “AMC/AIME Speed Warmup (build pattern recognition + speed)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive (hand-picked) — 3 AMC/AIME combinatorics speed reps (open each below) · ~8 min eachAMC/AIME · archive: AMC 12 A AMC 12 A AMC 10 A

Toolkit Integration

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- Archive: 1 Putnam-band problem per topic area · timed 20 min each: - Analysis · Polynomials · Inequalities+FE · Combinatorics · NT · Linear Algebra · Geometry · Probability+Abstract

sources & assignments (5)

Source Putnam and Beyond — Drill reference — this gate trains the material of “Toolkit Audit — Technique Catalog Review”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source CMU 21-295 Putnam Seminar (full session recordings) — Drill reference — this gate trains the material of “Toolkit Audit — Technique Catalog Review”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Michael Penn — Putnam Exam Solutions (playlist) — Drill reference — this gate trains the material of “Toolkit Audit — Technique Catalog Review”. Stuck mid-drill? The tool lives here; go back, find the move, return and finish in writing.
Source Supplemental — Lupu TTU MATH 4000, Recitation 5 (direct) — Writeup companion for “Toolkit Integration” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive Browser — Putnam 2023 B1, 2020 B1, 2016 B1, 2015 B1 · Analysis · 20 min eachPutnam

archive pull: 4 problems · Analysis · Standard · 15 min →

Cold Re-Examine

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- Do: 3 problems each from your 2 weakest checklist areas (from Week 10 log)

sources & assignments (5)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — The canonical source for every real Putnam problem and official solution referenced by this gate.
Source Supplemental — Lupu TTU MATH 4000, Recitation 5 (direct) — Strategy companion for “Cold Re-Examine” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
Source Maths 505 — Feynman's technique is the greatest integration method — Pacing companion for “Cold Re-Examine” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Maths 505 — This integral taught me Feynman's technique — Writeup companion for “Cold Re-Examine” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems 3 — 3 problems each from your 2 weakest checklist areas (from Week 10 log)Putnam

Full Toolbox Checklist

Full Toolbox Checklist (fill cold — if you can't, that topic is not solid)

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**Analysis:** 5 convergence tests (name + when) · EVT/IVT/Rolle/MVT hypotheses · Taylor: eˣ, sin, cos, ln(1+x), (1+x)^α to 3–4 terms · FTC both forms · Feynman trick template · Convexity: f'' ≥ 0 ↔ Jensen · Leibniz rule · Uniform convergence: when ∫ and d/dx commute with limit · Auxiliary function template · Abel summation formula **Algebra — Polynomials:** Vieta · Factor theorem · Integer-valued poly (p(a)−p(b) div by a−b) · Root-of-unity filter · Newton's identities · Lagrange interpolation · Finite differences: Δᵈ[degree-d] = const · Newton forward difference formula **Algebra — Inequalities + FE:** AM-GM equality case · Cauchy-Schwarz both forms · Jensen · Smoothing · Homogenization · FE substitution sequence · ODE recognition template **Combinatorics:** Bijection (state inverse explicitly) · IE formula · Pigeonhole (name objects + boxes) · Double counting · GF: OGF moves · Recurrence: 6 methods · Extremal object first · Game: P/N / pairing / invariant / mod-cycle · Graph: Euler circuit / bipartite / tree · LA in combinatorics: dimension argument / incidence matrix · Probabilistic method (existence) **NT:** CRT · FLT (aᵖ ≡ a mod p) · Euler's theorem · Sophie Germain factorization · Infinite descent · vₚ(n!) Legendre · LTE (statement) · Integer polynomials · Valuations **Linear Algebra:** rank + nullity = dim V · det(AB) = det(A)det(B) · tr = Σλ · Cayley-Hamilton · Gram matrix PSD · Spectral theorem · LA in combinatorics moves **Geometry:** Shoelace · Dot product · z = re^{iθ} · Pick's theorem · Graph modeling: vertices/edges first **Probability:** Linearity of expectation · Indicator variables · Law of total probability · Symmetry · Probabilistic method **Abstract Algebra:** Lagrange · Cosets partition G · Burnside: # orbits = avg fixed points **Hidden Tools (all):** ODE card · Feynman card · Root-of-unity filter · Auxiliary function template · Finite differences · Valuations · Abel summation · LA in combinatorics · Graph modeling · Invariants/monovariants · Compactness/extremal · Convexity/Jensen · Generating functions

Ritual: Fill the full toolbox checklist cold from memory (see the complete list in the source text below). Every item you cannot state without looking marks a topic that is not solid — send it to the repair queue before moving on.

sources & assignments (4)

Source CMU 01-Intro (Po-Shen Loh Putnam seminar) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here.
Source Michael Penn — Proof Writing (playlist) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here. Watch-for: begins with set-theory foundations (sets, Cartesian products, operations) before the proof-technique videos.
Source Michael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here.
Source Supplemental — Lupu TTU MATH 4000, Recitation 5 (direct) — Writeup companion for “Full Toolbox Checklist (fill cold — if you can't, that topic is not solid)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Diagnostic Mini-Mock — 3 problems, 90 minutes

Diagnostic Mini-Mock — 3 problems, 90 minutes (end of toolkit phase)

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Ritual: Postmortem is the gate: 3 lines, one per problem — gap class + the first move you should have made. This is the earliest honest signal of conversion problems; do not skip it because the score stings.

sources & assignments (5)

Source Kedlaya — Putnam Archive (problems + official solutions, 1985–present) — The canonical source for every real Putnam problem and official solution referenced by this gate.
Source Supplemental — Lupu TTU MATH 4000, Recitation 5 (direct) — Strategy companion for “Diagnostic Mini-Mock — 3 problems, 90 minutes (end of toolkit phase)” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
Source Maths 505 — Feynman's technique is the greatest integration method — Pacing companion for “Diagnostic Mini-Mock — 3 problems, 90 minutes (end of toolkit phase)” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
Source Maths 505 — This integral taught me Feynman's technique — Writeup companion for “Diagnostic Mini-Mock — 3 problems, 90 minutes (end of toolkit phase)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.

Problems Archive Browser — Pull 3 unseen problems via the archive button (one A1-band, one B1-band, one A2-band), exam conditions: 90 minutes total, no notes, full writeups. Score like a grader (0/10 each — Putnam gives almost no middle scores: a writeup is ~10 or ~0–2). Log the result in the mock logger below so your projected score gets its first real calibration. Postmortem: for each non-10, name the gap class (classification / theorem gap / proof gap / time / algebra slip / no idea).Putnam

archive pull: 3 problems · Analysis/Algebra/Combinatorics · A1/B1/A2 · Putnam · 90 min →

FE Micro-Spine — closed-book pattern card

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Added because FE cannot be a one-week topic; pattern exposure must recur after the W6 load.

Ritual: Write the substitution table before any solution check. Every FE rep must name the first move and the trap it avoided.

sources & assignments (2)

Source 100 Functional Equations / Putnam FE repair spine — Micro-rep source for repeated FE pattern exposure: special values, symmetry, injective/surjective forcing, Cauchy/Jensen, iteration, and polynomial-degree comparison.

Problems Functional equations micro-rep — Closed-book card: for f(x+y), f(xy), f(f(x)), bounded/continuous Cauchy, and polynomial FE, write first three moves plus one failure mode.Putnam

Reach Bridge — optional A3 in toolkit audit

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Makes the transition into W14 systematic A3–A6 practice smoother.

Ritual: If this causes fatigue, stop after the first-move/lemma artifact; do not cannibalize the A1/B1/A2 diagnostic.

why this gate: the lesson's §1 The blank-page audit is what it trains

sources & assignments (2)

Source Phase 3 reach entry ticket — Add one optional A3 after the diagnostic mini-mock. This is not a score base; it decides which reach family enters Phase 3 first.

Problems Phase 3 reach entry — After the W13 mini-mock, attempt Putnam 1993 B3 for 25 minutes. Output: first move, lemma attempt, and topic-family tag.Nightmare

Problem-Solving Reps — weekly homework

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Standing weekly homework — the problem-solving book stack, every week. Solve ALL listed; write ONE full clean solution (the rest may stay scratch). Up to 2 due re-solves from your review queue surface first.

sources & assignments (4)

Problems PnB — PnB Ch. 6 probs 1–6 (default spine; rotate to your weakest-area chapter)Putnam
Problems MOC — Mathematical Olympiad Challenges — Number Theory ch. probs 1–2Putnam
Problems Kedlaya — Putnam 1994 A1,A2,B1Putnam
Problems Putnam Full Past Exam — Full Past Exam 04: Putnam 1998 A1, Putnam 1998 A2, Putnam 1998 A3, Putnam 1998 A4, Putnam 1998 A5, Putnam 1998 A6, Putnam 1998 B1, Putnam 1998 B2, Putnam 1998 B3, Putnam 1998 B4, Putnam 1998 B5, Putnam 1998 B6 - Scout protocol: Day 1 A1-A2-B1-B2, 80 minutes total; Day 2 A3-B3, 60 minutes lemma hunt; Day 3 A4-A6-B4-B6, 60 minutes first-move scan; Day 4 score all 12 slots and write three miss tags; Day 5 do one same-family repair pull before the next paper.Putnam

Reflect

Reflect: reconstruct and log the pattern accelerated only

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- Reflect: notes complete — one line per area - Verify: hardest toolkit-audit solve — find one step you skipped - **MILESTONE: FULL TOOLKIT + AXLER + RUDIN COMPLETE.** Log 2 weakest checklist areas for Phase 3 spiral priority. --- ## Phase 3 — Mastery (Weeks 14–28) Archive-driven. No parallel tracks. Every week: archive block + spiral reinforcement + reflect + verify. A1/A2/B1/B2 practice continues every week even during nightmare weeks — these are your scoring problems. ---

Ritual: Audit-week reconstruction: pick the toolkit-audit problem that felt least automatic. Close all notes, rewrite the full solution from memory, then write one line per audited family: "[family]: my first move is [move], grade N/10." Those lines ARE the Phase 3 entry ticket.

sources & assignments (4)

Source CMU 01-Intro (Po-Shen Loh Putnam seminar) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here.
Source Michael Penn — Proof Writing (playlist) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here. Watch-for: begins with set-theory foundations (sets, Cartesian products, operations) before the proof-technique videos.
Source Michael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — Intuition companion for “Reflect: reconstruct and log the pattern” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Verify: audit one proof before closing accelerated only

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unlocks after: Reflect: reconstruct and log the pattern

Ritual: Pick your hardest toolkit-audit solve. Re-check for: (1) a false claim, (2) a missing case, (3) a wrong bound — and find the one step you skipped because it felt obvious; write it out in full.

why this gate: the lesson's §1 The blank-page audit is what it trains

sources & assignments (4)

Source CMU 01-Intro (Po-Shen Loh Putnam seminar) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here.
Source Michael Penn — Proof Writing (playlist) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here. Watch-for: begins with set-theory foundations (sets, Cartesian products, operations) before the proof-technique videos.
Source Michael Penn — Putnam Exam Solutions (playlist) — Week reference — this gate applies this week's lead material (“Toolkit Maturity Review — proof + method consolidation (study with the checklist)”). If an attempt stalls, the repair source is here.
Source 3Blue1Brown — Essence of Linear Algebra — Intuition companion for “Verify: audit one proof before closing” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.

Exit contract — Week 13

Verification remaining

reading progress…

Carry-forward repairs

reading queue…

Next week opens with

W14 · A1/B1 Slot Mastery
the exam follows the final taper week

Train Week 13 in the trainer →