Topic Brief: Proof By Contradiction

Provenance: compiled 2026-09-29. Sources checked directly for this document: research/veridian/WMA14-verified-facts.md §5.1 in full (all six dated citations for spec 1.1) and §7 (confirms this is the richest-evidenced spec point in the bank); signal-frequency-and-boundaries.md §1 (the existing frequency row and its 7-series denominator) and §3 (flagged gaps, reused rather than re-derived); forge-exam-technique-guide.md §1 (the three confirmed real mark tariffs) and its proof-by-contradiction checklist; codex-deep-dive-proof-by-contradiction.md, read in full specifically to confirm this document does not restate its formal-logic and mechanism-by-mechanism synthesis; the full header and body of content/veridian/curriculum/wma14/proof-by-contradiction.ts (both worked chains, the marked-solution, the method-comparison block, and every trap-taxonomy item) to identify what is already taught and avoid duplicating its worked exemplars; and content/veridian/question-bank/wma14.ts, checked directly — it holds one item for this spec point (wma14-oct24-q2-proof-by-contradiction-quartic), built on a citation already counted below, contributing no new sighting. No new primary Pearson PDF was fetched this session; every citation below reuses sourcing the facts bank has already independently verified.

#SIGNAL Topic Master Brief — Proof by Contradiction (spec 1.1)

#Frequency: seen in 6 of 7 series checked

Of the 7 WMA14 series with real, question-by-question mining (signal-frequency-and-boundaries.md §1: January 2021, October 2021, June 2022, October 2022, January 2024, October 2024, January 2026), six carry a confirmed citation for spec 1.1 — the highest count of any spec point in the whole facts bank. The six sightings:

October 2022 is the one series, of the seven mined, with no spec-1.1 citation captured in this course's review — read as a gap in what this review's own document search caught, per the standing rule that a topic's absence from a series' row is not evidence the series omitted it, not as a confirmed skip.

#Mark-tariff pattern: consistently small, and structurally identical when confirmed

Only three of the six sightings carry a mark total confirmed anywhere in this facts bank: June 2022 Q9 (4 marks), October 2024 Q2 (4 marks), and January 2026 Q8 (5 marks, split 1 for a factorisation + 4 for the proof). No mark total is recorded for January 2021 Q3, October 2021 Q10 or January 2024 Q8 — each is cited via examiner-report commentary, not a mark-scheme total this review captured. On the three confirmed totals, this is the smallest, most self-contained item type on the paper (forge-exam-technique-guide.md §1) — it never carries the double-digit tariffs the paper's combined parametric, vectors or volume-of-revolution items do.

Where the scheme is confirmed, the ladder shape repeats: a standalone opening mark for stating the assumption (M1 or B1 — "no need for explicit statement of assumption... accept if just a suitable equation is set up," per the June 2022 scheme's own generosity note), independent of everything after it; one or two accuracy/method marks for the derivation; then a final mark always dependent on every mark before it (dM1 → A1* on June 2022 Q9 and January 2026 Q8; a plain dependent A1 on October 2024 Q2). No confirmed scheme awards the closing mark independently of the working before it — a correct-looking closing sentence cannot rescue broken algebra.

#Context: no repeated statement, but two confirmed recurring angles

The actual claim being proven is different every single time — a fact about consecutive integers, a two-part cube/square parity chain, a divisibility statement, a stationary-point claim, a curve-intersection claim, a Diophantine-style equation — so "context" here is not a repeated scenario the way it is for, say, the paper's parametric-area question. Two narrower patterns are genuinely confirmed across independent series, though:

• Non-algebraic shortcuts are never credited, in every series where this was checked. January 2024 Q8's examiner report explicitly names the non-scoring routes candidates tried — "attempting the 'discriminant', using a graph only, testing values of x or attempting to use small angle approximations" — and October 2024 Q2's report separately records candidates using a calculator to state the equation's complex roots "without justification," also uncredited. Two independent series confirm the same rule: the proof must be carried in visible algebra, never read off a graph, a calculator screen, or tested values.
• The topic's position in the paper is genuinely unpredictable. The six sightings span Q2 through Q10 — twice relatively early (October 2024 Q2, January 2021 Q3), four times in the back half (Q8–Q10, including the final question of June 2022's nine-question paper). With a full question count confirmed for only three of the papers this topic has appeared on, no safe claim can be made about where to expect it beyond "anywhere."

One further pattern rests on only two data points, so it is flagged as an observation, not a trend: the two most recent sightings before January 2026 (January 2024, October 2024) moved away from the pure number-theory statements of the earlier sightings (√2, n² − 2) toward calculus/coordinate-geometry contexts (a cubic's stationary points; two curves not intersecting), before January 2026 returned to a pure integer equation carrying a genuinely new exhaustion-based mechanism (codex-deep-dive-proof-by-contradiction.md §2.3). Two points is too few to call this a direction; it is stated only as what the record shows.

#What is, and isn't, on record

No grade-boundary data exists anywhere in this course's WMA14 source review, so none is implied here. January 2026 has no published examiner's report yet, so nothing is known about where real candidates lost marks on that series' Q8 — only what its mark scheme rewards. Half of the six sightings (January 2021, October 2021, January 2024) have no mark-scheme total confirmed in this facts bank at all, only examiner-report commentary — a real, stated gap, not an oversight in this document's own search.

#If this topic appears: what to expect

On the six-sighting record: expect a short, self-contained item, not a chained multi-technique question — the three confirmed totals never exceed 5 marks, well below this paper's double-digit combined questions. Expect the specific statement to be unfamiliar rather than one of the spec's two named examples (√2 and the infinitude of primes sit in the lesson's own teaching material, not in any of the six real sightings above) — the spec's own guidance extends to "application to unfamiliar proofs," and every real sighting bears that out. Whatever the claim, budget marks for the sentences either side of the algebra: state the assumption as an explicit sentence (earned independently of the rest, and lost by candidates who skip straight to equations), and close with a sentence naming the specific impossibility reached and tying it back to the original claim — the confirmed record shows marks lost at exactly these two points far more consistently than inside the derivation itself. Never substitute a graph, a calculator value, or tested cases for the algebraic derivation; two independent series confirm none of those earn credit.

#Sources used

research/veridian/WMA14-verified-facts.md §5.1 (all six dated citations), §7 (confirms this is the richest-evidenced spec point in the bank). signal-frequency-and-boundaries.md §1 (7-series denominator and the existing frequency row) and §3 (flagged gaps, reused rather than re-derived). forge-exam-technique-guide.md §1 (the three confirmed real mark tariffs) and its proof-by-contradiction technique checklist. codex-deep-dive-proof-by-contradiction.md (read in full to confirm no overlap with its formal-logic and mechanism-synthesis content). content/veridian/curriculum/wma14/proof-by-contradiction.ts (full file, read to identify what is already taught and avoid duplicating worked exemplars). content/veridian/question-bank/wma14.ts (checked directly — holds one item for this spec point, contributing no new sighting beyond what §5.1 already establishes).

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