Pure Mathematics 2

Topic Brief: Proof

Provenance (2026-09-29). Read in full for this pass: research/veridian/WMA12-verified-facts.md §2 (spec ToC), §5 (archive scope — 20 series with question paper + mark scheme reviewed, 17 with a published examiner report), §6.1, §7, and both proof.ts audit-log entries in §9 (2026-09-14, PRs #448/#449). Also read content/veridian/curriculum/wma12/proof.ts (grepped for every series citation, to confirm this brief adds none the lesson doesn't already cite) and content/veridian/technique-hub/wma12/codex-deep-dive-proof.md, codex-formula-and-method-bank.md §1, forge-exam-technique-guide.md (grepped for "proof"), and signal-frequency-and-boundaries.md (this topic's existing row). Checked content/veridian/question-bank/wma12.ts — zero matches for "proof", no worked past-paper item exists there yet. Not affiliated with or endorsed by Pearson Edexcel. Every quotation below is short (under ~15 words) and attributed to a named document; no full question or mark-scheme extract is reproduced at length.

#SIGNAL Topic Master Brief — Proof (WMA12, spec 1.1–1.3)

#What this is, and isn't

A single-topic zoom-in on the frequency/expectation layer for spec item 1. codex-deep-dive- proof.md already covers the theory, the three-technique map, and nine illustrative micro-examples; proof.ts carries the full worked exemplars and trap taxonomy; codex-formula-and-method-bank.md §1 gives the condensed method steps. This document repeats none of that. It exists to answer three narrower questions — which real series have tested this topic, what the real mark tariff and paper position look like across them, and what a candidate should expect walking in — and to reconcile a count that has drifted between two existing documents.

#Real frequency: seen in 8 of 20 series checked

The archive has 20 series with a question paper and mark scheme read (June 2019–January 2026) and 17 with a published examiner report (WMA12-verified-facts.md §5). Individually-cited real Proof anchors across that archive: 8 series, 10 confirmed question-parts.

Series	Q	Technique	Marks	Real angle	Source
June 2019	3(i)+(ii)	General proof, then counter-example	3+1=4	$(x-4)^2\geqslant2x-9$ ; then " $2^n+1$ prime" disproved	`WMA12-verified-facts.md` §6.1; `proof.ts`
Oct 2021	9(a)+(b)	General proof, then counter-example on a restricted domain	3+1=4	Same AM–GM-shape inequality proved true, then disproved once $x,y<0$	`WMA12-verified-facts.md` §9; `codex-deep-dive-proof.md`
Jan 2022	10	Counter-example, then exhaustion	1+4=5	Two-part, disproof first	`WMA12-verified-facts.md` §9 (second audit pass); `proof.ts`
June 2022	3(i)	Counter-example	2 (bundled M1/A1)	" $(n+1)^3-n^3$ prime for all $n$ " — false	`WMA12-verified-facts.md` §6.1, §9
Oct 2022	1	Exhaustion	3	$c=b+2$ , $a+b+c=10$ ; product always even	`WMA12-verified-facts.md` §6.1, §9
Oct 2023	1	Exhaustion	3	$c=3a+1$ , $a+b+c=15$ ; product multiple of 4	`WMA12-verified-facts.md` §6.1, §9
June 2023	8	Counter-example, then exhaustion	2+3=5	Two-part, disproof first	`WMA12-verified-facts.md` §9 (second audit pass)
June 2025	10	Exhaustion	not transcribed	Last question of the paper, not Q1	`forge-exam-technique-guide.md`; `signal-frequency-and-boundaries.md`'s FORGE table

Reconciling a discrepancy. signal-frequency-and-boundaries.md's own Proof row currently reads "seen in 6," omitting the January 2022 and June 2023 two-part anchors that a later, independent audit pass on proof.ts (§9's second entry) confirmed directly against the primary question papers. That row predates that audit. 8, not 6, is the count this brief uses, and that table should be updated to match.

#Real mark-tariff pattern

Two distinct shapes, not one flat tariff. Single-technique questions run 2–3 marks: counter- example alone is a bundled 2-mark M1/A1 (June 2022); exhaustion alone is a flat 3-mark B1→M1→ conclusion, confirmed identically on two anchors (Oct 2022, Oct 2023) with June 2025's instance unconfirmed but the same technique. Two-technique questions run 4–5 marks: proving-then- disproving the same statement costs 4 (June 2019, Oct 2021); disproving-then-proving two separate statements costs 5 (Jan 2022, June 2023). No anchor in this archive has Proof worth more than 5 marks or fewer than 2.

#Real position: not a reliable Q1 anchor

The facts bank's own §6.1 and §7 describe Proof as anchoring "Question 1... in almost every series reviewed" and "consistently the paper's opener." Checked directly against the 8 named anchors above, that holds for only 2 of 8: Oct 2022 Q1 and Oct 2023 Q1. The other six sit at Q3 (June 2019, June 2022), Q8 (June 2023), Q9 (Oct 2021), and Q10 — the paper's last question, twice (Jan 2022, June 2025). forge-exam-technique-guide.md independently confirms the June 2025 instance from the examiner report itself. Read across all 8, not just the two most recent series in the Q1-anchored set, Proof's real position ranges across the entire paper — a candidate should not treat "not Q1" as a signal that something else is wrong.

#Real recurring context/angle across the series checked

Exhaustion's domain is generated from a stated linear relation between two variables plus a sum bound, not a bare range — confirmed identically in both flat 3-mark anchors (Oct 2022: $c=b+2$ ; Oct 2023: $c=3a+1$ ), both closing on a claim about the product of three positive integers.
Counter-example targets a "prime for all $n$ " or similarly-phrased algebraic expression over an unbounded domain (June 2019, June 2022) rather than an inequality — the one anchor disproving an inequality (Oct 2021(b)) does so only because part (a) had just proved it true on the unrestricted domain, a domain-flip structure unique to that question in the archive.
The algebra-vs-exhaustion mismatch is the most independently-documented recurring error, confirmed in both directions across three separate series: candidates reaching for algebra on a question built for a short case list (Oct 2022, Oct 2023), and candidates treating a handful of checked values as a finished proof over an infinite domain (Oct 2021) — the latter the single worst-performing question in the whole archive, at 47% scoring zero.

#If this topic appears: what to expect

Expect 2–5 marks, never more — bundled if disproof-only, a flat three-line chain if exhaustion-only, additive if the question has two parts.
Do not expect it at Q1 by default — confirmed as early as Q1 and as late as the paper's final question, roughly evenly split across the archive.
If the domain is a small, explicitly bounded set of integers generated from a stated relation, list and check it — every confirmed exhaustion anchor rewards exactly that, and penalises algebra reached for instead.
If the domain is every real number (or every positive number), no quantity of checked examples is a proof — this is the archive's single most costly documented mistake.
If two parts appear, expect either the same statement proved then disproved on a restricted domain, or two separate statements in counter-example-then-exhaustion order.

#What this document deliberately leaves out

No proof of the underlying logic (conjunction/disproof asymmetry), no three-technique method map, and no illustrative micro-examples — all in codex-deep-dive-proof.md. No worked exemplars or trap taxonomy — proof.ts. No condensed method steps — codex-formula-and-method-bank.md §1. No percentage, likelihood, or grade-boundary claim of any kind — frequency is stated only as "seen in N of M series checked," per this course's standing discipline.

#Sources used

research/veridian/WMA12-verified-facts.md — §2, §5, §6.1, §7, and both proof.ts audit-log entries in §9, read in full.
content/veridian/curriculum/wma12/proof.ts — grepped for every series citation.
codex-deep-dive-proof.md, codex-formula-and-method-bank.md §1, forge-exam-technique-guide.md (grepped for "proof"), signal-frequency-and-boundaries.md (this topic's row) — read to confirm nothing above restates them, and to source the June 2025 position finding and the frequency discrepancy this brief reconciles.
content/veridian/question-bank/wma12.ts — grepped for "proof"; zero matches.