Deep Dive: Proof

#CODEX Deep-Dive: Proof (WMA12 · Pure Mathematics 2)

Provenance (2026-09-29): read research/veridian/WMA12-verified-facts.md §2 (spec ToC), §6.1 (Proof), §7, §8 and §9 (the two proof.ts audit-log entries) in full; read content/veridian/curriculum/wma12/proof.ts (712 lines) in full; read content/veridian/technique-hub/wma12/codex-formula-and-method-bank.md, forge-exam-technique-guide.md and signal-frequency-and-boundaries.md for existing coverage; confirmed content/veridian/question-bank/wma12.ts currently has no Proof items. No primary PDF was independently re-opened for this document — every citation below traces to the facts bank's own transcription, which is itself matched against the primary Pearson PDFs (see that file's own header). Re-checked same day: forge-exam-technique-guide.md's own per-question mark table (independently extracted from wma12-01-rms-20250814.pdf) does in fact transcribe June 2025 Q10's total as 4 marks — the paper's lightest question — so §3.1/§3.2/§4 below have been corrected from an earlier draft of this document that had missed that table and called the mark value unconfirmed. Not affiliated with or endorsed by Pearson Edexcel.

#Scope, and what this adds

Spec item 1, "Proof," is three flat sub-points — 1.1 (general proof structure), 1.2 (proof by exhaustion), 1.3 (disproof by counter-example) — and it is small: no formula, three techniques, and (per the facts bank) ten confirmed real question-parts across the whole reviewed archive. Two documents already teach and drill it in depth: content/veridian/curriculum/wma12/proof.ts (three full worked examples, nine named traps, five MCQs, all mark-scheme-cited) and codex-formula-and-method-bank.md (a condensed four-step method card per technique). This deep-dive does not re-teach either. Its job is the piece neither does: pull every confirmed real anchor into one place, classify them by the shape of question they actually take (not just list them), and give short, clearly-labelled illustrative micro-examples for the sub-cases the existing worked examples don't already cover with a different one.

Explicit non-scope: this is spec item 1 only. The proof of the arithmetic/geometric series sum formula (spec 4.2/4.4, "the proof of the sum formula should be known") is a bookwork-memorisation task under a completely different spec item, sharing only the word "proof" — it is not covered here, and confusing the two is itself a real trap this document flags in §4.

#1. The real theory

#1.1 The logical foundation

A claim of the form "[property] holds for every [n] in [domain]" is a conjunction: one sub-statement per element of the domain, all asserted simultaneously. Two facts about a conjunction decide everything else in this topic. A conjunction is true only if every one of its parts is true — so proving it needs every part established, which is only tractable by direct listing if the domain is finite. A conjunction is false the moment one part is false, regardless of how many other parts hold — so disproving it needs exactly one failing part, whatever the domain's size. That asymmetry, not a convention or a lower evidentiary bar, is why disproof is structurally cheaper than proof for this shape of claim.

#1.2 The three-technique map

1.1 is easy to misread as pure framing for 1.2/1.3. It isn't: it is itself a proof technique, and it is the only one of the three that can prove (not merely disprove) a claim the other two cannot even attempt — a "for all real x" or "for all positive x, y" claim, where there is no last case to finish checking. The facts bank's own audit log records this as a confirmed correction to an earlier draft of proof.ts, which had stated a claim of that shape "is not attempted on this paper at all" before two real anchors (below) showed otherwise.

#2. Sub-cases, with micro-examples

Each micro-example below is original and illustrative, built to show the sub-case cleanly with different numbers from every worked example already in proof.ts — none are past-paper reproductions.

#2.1 Exhaustion: a two-variable domain generated by a linear relation

The domain is rarely "just" a range — real anchors generate it from a stated relation between two variables plus a sum or sum-bound (October 2022 Q1: c = b + 2, a+b+c = 10; October 2023 Q1: c = 3a+1, a+b+c = 15). The relation's direction is part of the domain definition, not cosmetic — applying it backwards silently generates a different, wrong case set from the first line (a confirmed error, October 2022: "using $c = b - 2$ instead of $c = b+2$").

Illustrative micro-example (not from a real series): positive integers $d, e$ with $d = e+3$ and $d+e \leqslant 11$. Substituting, $2e+3 \leqslant 11 \Rightarrow e \leqslant 4$, so $e \in \{1,2,3,4\}$, giving $(d,e) = (4,1),(5,2),(6,3),(7,4)$. Checking $de$: $4,10,18,28$ — all even, and every valid pair has been checked, so $de$ is even for every such pair.

#2.2 Counter-example: unrestricted domain, and domain-restricted variant

The plain form — search small values of an unbounded variable until one fails — is what June 2019 (ii) and June 2022 Q3(i) both test. A distinct, real variant restricts the search to a sub-domain of the original claim's variables: October 2021 Q9(b) asks candidates to disprove the same inequality tested true in part (a), specifically when both variables are negative — the domain restriction is the whole content of the question, not incidental.

Illustrative micro-example, unrestricted: disprove "$3^n - n$ is prime for every positive integer $n$." $n=1$: $2$ (prime). $n=2$: $7$ (prime). $n=3$: $24 = 2^3 \times 3$, not prime — disproved, search stops.

Illustrative micro-example, restricted sub-domain (mirroring the October 2021 Q9 shape): the inequality "$x + \tfrac1x \geqslant 2$" holds for all positive real $x$ (§2.3 below proves it) but fails once $x$ is allowed to be negative: at $x=-1$, $x+\tfrac1x = -2 \not\geqslant 2$ — one exhibited value in the restricted domain disproves the claim there, exactly as the real anchor's part (b) does for its own inequality.

#2.3 General/direct proof over an infinite or continuous domain

Rearrange to a form compared with zero; reach an expression whose sign is guaranteed for every case in the domain (almost always a squared term, since a real number squared is never negative); state that guarantee explicitly; close by referring back to the original claim. The real mark scheme for this shape (June 2019 Q3(i)) requires all three of correct algebra, the explicit squared-number explanation, and the reference back — withholding the final mark if any one is missing, even with correct working otherwise.

Illustrative micro-example (coefficient ≠ 1, distinct from proof.ts's $x^2+9\geqslant6x$): prove $2x^2+3 \geqslant 4x$ for all real $x$. Rearrange: $2x^2-4x+3 \geqslant 0$. Complete the square: $2(x-1)^2+1 \geqslant 0$. Since $(x-1)^2 \geqslant 0$ for every real $x$, $2(x-1)^2+1 \geqslant 1 > 0$, so the original inequality holds for all real $x$.

#2.4 Proof by contradiction — the accepted alternative, not a fourth spec point

Not separately numbered; confirmed as a credited alternative route to 2.3's direct algebra on the same real anchor ("Alt to (i) via contradiction — Proof by contradiction is acceptable," June 2019 mark scheme). A full dedicated treatment, with the spec's own two named contradiction examples ($\sqrt2$ irrational; infinitely many primes), already exists for a different paper at content/veridian/curriculum/wma14/proof-by-contradiction.ts — those two examples are not WMA12 anchors and are not reproduced here.

Illustrative micro-example: prove $x^2+1>0$ for all real $x$, by contradiction. Assume the negation: there exists a real $x$ with $x^2+1 \leqslant 0$, i.e. $x^2 \leqslant -1$. But $x^2 \geqslant 0$ for every real $x$, so $x^2 \leqslant -1$ is impossible — contradiction. The assumption is false, so $x^2+1>0$ for all real $x$.

#3. Synthesis: every real exam angle confirmed for this topic

#3.1 The anchor catalogue

Ten confirmed question-parts, drawn from the facts bank's own citations. This is not a claim that these are the only ten that have ever been set — six or seven of the twenty reviewed series were only keyword-grepped rather than read start-to-finish (facts bank §8), so further real Proof anchors in those series are plausible but not verified here, and are not claimed.

#3.2 Question-shape taxonomy

The ten anchors fall into four distinct real shapes, not a random scatter:

1. Single-technique, exhaustion only — Oct 2022 Q1 and Oct 2023 Q1 (three marks each, B1→M1→conclusion), plus June 2025 Q10 (four marks total — one more than the other two; the internal mark split is not independently transcribed, so whether it follows the same three-stage shape with one extra mark, or a different structure entirely, is not confirmed here).
2. Single-technique, counter-example only — June 2022 Q3(i). Two marks, bundled M1/A1.
3. Two independent statements, two techniques — this itself splits by part order, confirmed both ways: counter-example-then-exhaustion (Jan 2022 Q10, June 2023 Q8) and general-proof-then- counter-example (June 2019 Q3, where the two parts test different statements, unlike the next shape).
4. One statement, two complementary domains, technique flips with the domain — October 2021 Q9 only: the identical inequality is proved true by general algebra on the stated (positive) domain in (a), then disproved by counter-example on the negated (negative) domain in (b). This is the richest real anchor for the domain-decides-the-technique principle in §1, because the same expression changes truth value when the domain is restricted — nothing else in the archive demonstrates that specific point as directly.

proof.ts's own marked-solution question is modelled on shape 3 (counter-example-then-exhaustion, matching Jan 2022/June 2023's real order), and its worked-chain algebra example is modelled on shape 1/2's technique in isolation (June 2019 Q3(i)'s method, not its full two-part structure) — so shape 4 (Oct 2021 Q9's domain-flip) is the one confirmed real shape with no dedicated worked example anywhere in the existing lesson content; §2.2 and §2.3 above are this document's illustrative stand-in for it, not a claim that the gap has been filled with new lesson content.

#3.3 Positioning and origin

Proof is confirmed "new to the IAL specification" at its first sitting ("tested 'proof' which is new to the IAL specification," June 2019 examiner report) and the facts bank states it "anchors Question 1... in almost every series reviewed." June 2025 is the one confirmed exception in this archive — Proof by exhaustion sits at Q10, the paper's last question, not Q1 (independently re-confirmed this session against the facts bank's own citation of that series' examiner report, not re-opened from a raw PDF here).

#4. What has not been seen

Stated plainly, since padding a gap with invented content is exactly what this document must not do:

• Proof by induction is not part of this spec item's toolkit. It is only mentioned in the facts bank in connection with the arithmetic-series sum-formula proof (spec 4.2, a different topic), where it was attempted by a few candidates with little success — not a credited method for spec item 1 in any anchor reviewed.
• The spec's own two guidance examples — "$x,y$ odd integers less than 7, sum divisible by 2" (1.2) and "$n^2-n+1$ prime for all $n$" (1.3) — have not appeared verbatim in any real series checked in this project's archive to date; the facts bank states this explicitly after correcting an earlier, wrong transcription that had claimed June 2022 used the 1.3 example.
• No question-bank entries exist yet for this topic in content/veridian/question-bank/wma12.ts — a real, currently open gap, noted here as an observation, not addressed by this document.
• The internal mark-scheme breakdown for June 2025 Q10 (which lines earn B1 vs. M1 vs. the closing mark) is not transcribed in any source read for this document — only the question's 4-mark total is confirmed (via forge-exam-technique-guide.md's per-question mark table), so §3.1/§3.2 state the total but not the internal structure.

#5. Cross-references

• Full worked exemplars, trap-taxonomy (9 named items) and drilled MCQs: proof.ts.
• Condensed method steps per technique: codex-formula-and-method-bank.md §1.
• Paper-position and frequency table: signal-frequency-and-boundaries.md.
• Full contradiction treatment (different paper, spec's own two named contradiction examples): content/veridian/curriculum/wma14/proof-by-contradiction.ts.

#Sources

All citations above trace to research/veridian/WMA12-verified-facts.md §2, §6.1, §7, §8 and the two proof.ts audit-log entries in §9 (2026-09-14, PRs #448/#449), which themselves record verification against the primary Pearson question papers, mark schemes and examiner reports for June 2019, October 2021, October 2022, June 2022, January 2022, October 2023 and June 2023, plus forge-exam-technique-guide.md's own direct read of the June 2025 examiner report for the Q10 position claim, and that same document's independently-extracted per-question mark table (from wma12-01-rms-20250814.pdf) for the Q10 = 4 marks total. No primary PDF was independently re-opened for this document.