The second mock has a different job than the first: Mock #1 discovered your failure pattern; Mock #2 tests whether the repairs took. Same full-fidelity protocol, but the autopsy now reads DIFFERENTIALLY — every Mock-#1 failure bin gets a verdict: fixed, improved, or persistent. Persistent bins with two mocks of evidence become the gap-closing agenda, and they are the last major training decisions of the cycle.
One mock is a sample; two mocks are a trend line — and a repair that survives a second mock under full pressure is the only kind you can count on in December.
Key ideas — the week on one card
The differential autopsy gives every prior-mock bin a verdict: fixed (reuse the repair style), improved (maintain), persistent (re-diagnose UPSTREAM).
A repair that failed its targeted week was aimed at the wrong level — recurring 'slips' are usually recognition gaps in disguise.
Containment plans convert accepted weaknesses into bounded, rehearsed procedures: trigger, hard cap, scoped partial-credit menu, pre-written exit artifact.
After this week, training changes how reliably you DELIVER, not what you know — the boundary is a design decision.
1The differential autopsy
Run the full three-pass autopsy from Week 25, then add the fourth pass: line up the classification bins against Mock #1's. FIXED (bin emptied — e.g. the verification reps killed the execution slips): log what worked; that repair style is now proven for you and gets reused on whatever remains. IMPROVED (bin shrank): continue the same repair at maintenance dose. PERSISTENT (bin unchanged after a targeted week): the repair was wrong for the defect, and the diagnosis needs revisiting — usually the bin was mislabeled (an 'execution slip' that recurs is often a recognition gap in disguise: you keep dropping THAT case because you never really see it).
Also compare the meta-statistics, which matter as much as the bins: calibration error (self-score minus cold regrade — shrinking?), triage accuracy (did your bank/fight/park tags match reality better?), and time discipline (fewer post-checkpoint overstays on the timeline?). These craft statistics respond to rehearsal faster than mathematical gaps do, and improvements there carry directly to December because they are content-independent.
Two-mock evidence also settles your SLOT STRATEGY: with sessions of three problems, the data now says whether you are a two-solid-banks-plus-partial player or should swing at a third. Top-200 arithmetic tolerates both — what it punishes is not knowing which player you are and improvising on the day.
Worked example
Differential read, worked: Mock #1 had three recognition gaps (two probability, one valuation); Mock #2 shows the probability pair fixed but a NEW recognition gap in a geometry setup, plus the valuation gap persisting. Read it.
Nudge 1
Give each old bin a verdict before planning anything new.
Reveal step 1
Probability fixed: the five-problem burst worked — file 'burst' as your proven repair for recognition gaps and apply it to the geometry gap this week without further deliberation.
Nudge 2
For the persistent bin, re-diagnose upstream: is the missing piece the technique or its TRIGGER?
Reveal step 2
Valuation persistent after a burst: re-diagnose rather than repeat. Pull both mock attempts — if the same first move was missing twice, the gap is upstream: not 'valuations' but the trigger for them ('divisibility of a difference of powers' never fires LTE for you). Re-drill the TRIGGER: cover the technique name, classify ten mixed problem statements, check which cues fire and which stay silent.
Nudge 3
Scale the response to the evidence: two mocks structural, one mock cheap.
Reveal step 3
New geometry gap: one occurrence is a sample, not a pattern — burst it, but do not restructure the final weeks around a single data point. The differential read's discipline is exactly this: two-mock evidence gets structural responses, one-mock evidence gets cheap responses.
Answer
Fixed → reuse the proven repair style; persistent → re-diagnose upstream (usually a trigger, not a technique); new-and-single → cheap response only.
Pitfall. Repeating a failed repair harder. A bin that survived its targeted week does not need more dosage; it needs a different diagnosis — most often the defect lives one level upstream of where you treated it.
2Gap closing: the last structural week
This is the final week that changes WHAT you know rather than how reliably you deliver it — after this, Weeks 27-28 are consolidation and taper by design, and late structural cramming is negative-EV under every study of exam performance. So the gap-closing block gets the persistent bins and nothing else: each gets a diagnosis-driven drill with a defined end state ('classify 10 valuation triggers at 90%', 'three clean geometry setups from cold'), not an open-ended 'get better at X'.
Scope discipline is the week's virtue: the list of things you will NOT fix is written explicitly (the Week 25 routing decisions, plus whatever Mock #2 newly revealed as fringe), and each gets a one-line containment plan — 'if a lattice-geometry A5 appears, hunt partial credit via Pick and move on'. Containment plans convert known weaknesses from anxiety sources into handled scenarios; the December version of you should never meet a weakness without a pre-decided response.
End the week by drafting the exam-day runbook (finalized in Week 28): wake time, food, arrival, the triage ritual verbatim, the between-session reset routine, the checkpoint rules, and the containment plans. Writing it now, while mock experience is fresh, is the point — the runbook is Mock #2's distilled operational lessons, and by exam day it should feel less like a plan than a habit transcript.
Worked example
Containment plan, worked: two-mock evidence says hard FE problems (A4+ tier) reliably eat 20+ minutes for 0-2 points. Write the plan.
Nudge 1
Write the decision rule so it executes under adrenaline: trigger, cap, scoped menu.
Reveal step 1
Decision rule, written for adrenaline: 'FE tagged fight-or-worse at triage → hard cap 15 minutes, and the attempt is SCOPED: substitutions $x=y=0$, symmetry checks, injectivity/surjectivity from the usual moves — the 1-3 point menu — then out.'
Nudge 2
Pre-write the exit artifact — the two rubric points are in the template.
Reveal step 2
Pre-commit the exit artifact: partial write-up template — state what was derived ($f(0) = 0$, $f$ odd, $f$ injective on...), state the natural conjecture, stop. Two rubric points, eight saved minutes, zero decision fatigue.
Nudge 3
Rehearse the plan once on a real problem; unrehearsed plans lose to adrenaline.
Reveal step 3
Rehearse it once this week on a real archive FE under the timer: run the scoped attack, write the exit artifact, leave. A containment plan executed once in training runs automatically on exam day; one that exists only on paper competes with adrenaline and loses.
Answer
Hard cap + scoped attack menu + pre-written exit artifact + one rehearsal — a weakness converted into a bounded, rehearsed cost.
Pitfall. Letting gap-closing week sprawl into general study. Every hour here has an owner (a persistent bin) and an end state; hours without owners drift to comfortable topics, which are by definition not where the points are.
Before you open the gates
Fourth autopsy pass: every Mock-#1 bin gets a verdict — fixed (reuse the repair style), improved (maintain), persistent (re-diagnose upstream).
Track the craft statistics — calibration error, triage accuracy, checkpoint discipline — they improve fastest and transfer whole.
Gap-closing drills have owners and end states; the not-fixing list is written, and each entry gets a containment plan.
Draft the exam-day runbook now, while the mock lessons are fresh; rehearse each containment plan once on a real problem.
Check yourself
1. A failure bin unchanged after its targeted repair week most often means:
Repairs fail by treating the wrong level. Recurring 'slips' in one area are recognition gaps in disguise; re-diagnose before re-dosing.Repair: revisit §1 The differential autopsy ↑
Each rung: attempt cold, one hint if stuck, worked resolution only after a real try.
Rung 1 (abstract algebra, direct). Let $G$ be a group and $a, b \in G$. Prove that $ab$ and $ba$ have the same order.
One hint
Show $ba$ is conjugate to $ab$ by computing $a^{-1}(ab)a$. Conjugate elements have equal order.
Worked resolution
Compute $a^{-1}(ab)a = (a^{-1}a)(ba) = ba$, so $ba = a^{-1}(ab)a$ is conjugate to $ab$. For every integer $k$, $\bigl(a^{-1}(ab)a\bigr)^k = a^{-1}(ab)^k a$, so $(ba)^k = e \iff (ab)^k = e$. Hence $ab$ and $ba$ have exactly the same order (both infinite, or both equal to the same least positive $k$). [Source: this week's gap-closing block on the Abstract Algebra topic (the Michael Penn Abstract Algebra gate).]
✓ rung 1 done
Rung 2 (abstract algebra into number theory). Let $p$ be prime and $a$ an integer with $p \nmid a$. Using Lagrange's theorem in the group $(\mathbb{Z}/p\mathbb{Z})^\times$, prove Fermat's little theorem $a^{p-1} \equiv 1 \pmod p$.
One hint
The nonzero residues mod $p$ form a group of order $p-1$ under multiplication. What does Lagrange say about the order of the element $a$?
Worked resolution
Since $p$ is prime, $(\mathbb{Z}/p\mathbb{Z})^\times$ is a group of order $p - 1$ under multiplication, and $p \nmid a$ places $\bar a$ in this group. Let $d$ be the order of $\bar a$; by Lagrange's theorem $d \mid p - 1$. Then $\bar a^{\,p-1} = (\bar a^{\,d})^{(p-1)/d} = \bar 1$, i.e. $a^{p-1} \equiv 1 \pmod p$. [Source: this week's Abstract Algebra gap-closing — Lagrange's theorem, applied across the algebra/number-theory seam.]
✓ rung 2 done
Rung 3 (abstract algebra, the classic solo). Prove that every group $G$ of even order contains an element of order $2$.
One hint
Pair each element with its inverse. Elements that are NOT their own inverse come in disjoint pairs; count parities.
Worked resolution
Pair each $g \in G$ with $g^{-1}$. Elements with $g \ne g^{-1}$ fall into disjoint pairs $\{g, g^{-1}\}$, contributing an even total. Let $T = \{g : g = g^{-1}\} = \{g : g^2 = e\}$; then $|G| = |T| + (\text{an even number})$, so $|T|$ has the same parity as $|G|$. As $|G|$ is even, $|T|$ is even; and $e \in T$, so $|T| \ge 2$. Hence there is some $g \ne e$ with $g^2 = e$ — an element of order exactly $2$. [Source: this week's Abstract Algebra gap-closing block.]
✓ rung 3 done
Prove it — constructed response
An Abstract Algebra gap-closing re-solve. Let $G$ be a group in which every element satisfies $x^2 = e$. Prove that $G$ is abelian. Write it exam-clean: state what $x^2 = e$ gives for each element, apply the hypothesis to the product $ab$, and finish.
Self-grade against the rubric — completion requires the judgment, not the text
Model proof (compare AFTER grading yourself)
Let $a, b \in G$ be arbitrary. Applying the hypothesis to each element, $x^2 = e$ gives $x = x^{-1}$ for every $x \in G$; in particular $a^{-1} = a$ and $b^{-1} = b$. Now apply the hypothesis to the element $ab$: from $(ab)^2 = e$ we get $ab = (ab)^{-1} = b^{-1}a^{-1}$. Substituting $b^{-1} = b$ and $a^{-1} = a$ gives $ab = ba$. Since $a$ and $b$ were arbitrary, every pair of elements commutes, so $G$ is abelian. $\blacksquare$
The gates
Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)
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Learning support for this week's reinforcing IMO/Putnam problems. Read/watch these before (and after) attempting the full mock #2 + gap closing — review resources problems so each attempt has a source to learn the method and to check your write-up against.
SourceMichael Penn — Abstract Algebra (playlist) — Michael Penn — targeted technique videos for whichever topic the mock exposed Watch-for: this playlist opens with set-theory / proof-writing videos — skip to the 'Abstract Algebra |' entries for the group/ring/field content this gate needs.
SourceCMU 21-295 Putnam Seminar (full session recordings) — Writeup companion for “Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Mock Exam
Mock Exam
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- Do: second 6-hour mock (different exam year)
- Score + autopsy: same protocol as Week 25
Ritual: - Do: second 6-hour mock (different exam year)
- Score + autopsy: same protocol as Week 25
SourceCMU 21-295 Putnam Seminar (full session recordings) — Strategy companion for “Mock Exam” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
SourceCezar Lupu — TTU MATH 4000, Recitation 13 — Pacing companion for “Mock Exam” — note when the presenter abandons a line; compare with your own bail rule before starting this gate's timed work.
ProblemsMAA archive — Second full 6-hour mock — a DIFFERENT un-drilled recent year than Week 25. Same scoring (0–2 / 8–10, no in-betweens) and structured miss-autopsy protocol as Week 25.Putnam
Gap Closing
Gap Closing
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- Do: Default: Putnam 2017 A1, Putnam 2018 B1, Putnam 2019 A1. If a topic appears in BOTH mock-miss lists, add 3 Archive Putnam-band problems for that topic.
SourceKedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Abstract Algebra (playlist) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here. Watch-for: this playlist opens with set-theory / proof-writing videos — skip to the 'Abstract Algebra |' entries for the group/ring/field content this gate needs.
SourcePo-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceCMU 21-295 Putnam Seminar (full session recordings) — Writeup companion for “Gap Closing” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Problemsmock-miss overlap repair — Default: Putnam 2017 A1, Putnam 2018 B1, Putnam 2019 A1. If a topic appears in BOTH mock-miss lists, add 3 Archive Putnam-band problems for that topic.Putnam
Problem-Solving Reps — weekly homework
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)
ProblemsMAA archive — One full recent paper from a DIFFERENT un-drilled year than Week 25, 6 hrs timedPutnam
ProblemsPutnam Full Past Exam — Full Past Exam 29: Putnam 2023 A1, Putnam 2023 A2, Putnam 2023 A3, Putnam 2023 A4, Putnam 2023 A5, Putnam 2023 A6, Putnam 2023 B1, Putnam 2023 B2, Putnam 2023 B3, Putnam 2023 B4, Putnam 2023 B5, Putnam 2023 B6 - Calibration mock protocol: strict A-session A1-A6 for 180 minutes and strict B-session B1-B6 for 180 minutes under exam rules; next day score all 12 slots, write a mock autopsy, tag the three largest point leaks, and complete one repair pull before any new paper.Putnam
ProblemsPutnam Full Past Exam — Full Past Exam 30: Putnam 2024 A1, Putnam 2024 A2, Putnam 2024 A3, Putnam 2024 A4, Putnam 2024 A5, Putnam 2024 A6, Putnam 2024 B1, Putnam 2024 B2, Putnam 2024 B3, Putnam 2024 B4, Putnam 2024 B5, Putnam 2024 B6 - Calibration mock protocol: strict A-session A1-A6 for 180 minutes and strict B-session B1-B6 for 180 minutes under exam rules; next day score all 12 slots, write a mock autopsy, tag the three largest point leaks, and complete one repair pull before any new paper.Putnam
Reflect
Reflect: reconstruct and log the pattern
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- Reflect: identify your 3 most reliable topics — these are your confidence set for exam day
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Ritual: identify your 3 most reliable topics — these are your confidence set for exam day
sources & assignments (4)
SourceKedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Abstract Algebra (playlist) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here. Watch-for: this playlist opens with set-theory / proof-writing videos — skip to the 'Abstract Algebra |' entries for the group/ring/field content this gate needs.
SourcePo-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Newton's Sums — 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: fatal-slip audit before closing
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unlocks after: Reflect: reconstruct and log the pattern
Mandatory Verify gate from AGENTS.md for every week 3–28.
SourceKedlaya — Putnam Archive (problems + official solutions, 1985–present) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Abstract Algebra (playlist) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here. Watch-for: this playlist opens with set-theory / proof-writing videos — skip to the 'Abstract Algebra |' entries for the group/ring/field content this gate needs.
SourcePo-Shen Loh — CMU 21-738 Extremal Combinatorics (full course) — Week reference — this gate applies this week's lead material (“Technique Resources — full mock #2 + gap closing — review resources (study before the reinforcing problems)”). If an attempt stalls, the repair source is here.
SourceMichael Penn — Newton's Sums — Intuition companion for “Verify: fatal-slip audit before closing” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
Exit contract — Week 26
Verification remaining
reading progress…
Carry-forward repairs
reading queue…
Next week opens with
W27 · Cold Re-Solves + Memory Locking the exam follows the final taper week
