Deep Dive: Proof By Contradiction

*Provenance: written 2026-09-29. Read/checked directly for this document: research/veridian/WMA14-verified-facts.md in full, with targeted re-grep on "contradiction" (§1 spec text; §5.1 in full — all six dated citations; §7's standalone-lesson shortlist; §8's evidence-scope note on the 4-series vs 9-series denominators); the complete lesson file content/veridian/curriculum/wma14/proof-by-contradiction.ts (all 973 lines — both worked chains, the concept-ladder, both mechanism blocks, the method-comparison block, the marked-solution, all trap-taxonomy items, all MCQs, and the closing provenance flag); the sibling prerequisite lesson content/veridian/curriculum/wma12/proof.ts (header and its own AND-logic mechanism, for what this lesson already cites and must not re-derive); and all three existing WMA14 technique-hub files — codex-formula-and-method-bank.md §1 and §2.1, forge-exam-technique-guide.md (its proof-by-contradiction checklist and the per-question mark table), and signal-frequency-and-boundaries.md §1 and §3 — to confirm what is already written and avoid restating it. content/veridian/question-bank/wma14.ts was checked directly: it holds one item for this spec point (wma14-oct24-q2-proof-by-contradiction-quartic, built on the real Oct 2024 Q2 citation below). No new primary Pearson PDF was fetched this session; every exam-sourced claim below traces to citations WMA14-verified-facts.md §5.1 has already independently verified. The general logic and number-theory content in §1–§2 is standard mathematics, not course-specific, and is flagged as such throughout rather than attributed to any Pearson source. Every new numeric micro-example below was independently checked with sympy (including a brute-force search to 200 and a mod-4 argument for §2.3's example) before being written in.

#WMA14 · CODEX DEEP-DIVE — Proof by Contradiction (spec 1.1)

#What this document adds, and what it deliberately does not repeat

Spec 1.1 is the single richest-evidenced spec point in the whole WMA14 facts bank — six real, dated citations across the seven series signal-frequency-and-boundaries.md §1 counts as mined, three of them with a fully-verified mark scheme down to the mark code — and the existing material already uses that evidence hard. The lesson file carries both of the spec's own named worked examples in full (√2's irrationality; the infinitude of primes), a real marked solution (June 2022 Q9), a real method-comparison block giving all three credited routes to a second real question (January 2026 Q8), a concept-ladder stating the formal reductio schema, a mechanism block deriving why one impossibility settles an infinite claim, and seven trap-taxonomy items. The formula bank's §2.1 already states the five-step generic skeleton and names all three confirmed contradiction mechanisms. The forge guide already tabulates real marks-per-question and a technique-specific checklist. None of that is reproduced here.

What this document adds instead is the layer above the drill: (1) the formal logical schema stated with more precision than a teaching lesson needs, including the genuine and easily-blurred distinction between proof by contradiction and the related technique of proof by contrapositive — a distinction the lesson's own √2 chain actually depends on without naming it; (2) for each of the three real, evidence-confirmed contradiction mechanisms, a formal argument for why it is logically airtight (not just that it worked once), illustrated with a short, fresh micro-example using numbers that appear nowhere else in this course's WMA14 material; and (3) a structural synthesis of the six real citations organised by mechanism rather than by date, which is a genuinely different cut through the same evidence than the frequency table signal-frequency-and-boundaries.md §1 already gives.

#1. The formal logical schema, and a distinction worth making precise

#1.1 Reductio ad absurdum, stated exactly

The lesson's own concept-ladder already states the schema correctly: to prove a statement $P$, assume $\neg P$; if a chain of valid steps from $\neg P$ produces some statement together with its own negation — never possible for any statement — then $\neg P$ cannot be true, so $P$ is true. This is worth restating with one addition the lesson does not need for teaching purposes but which is worth being explicit about here: the step "$\neg P$ cannot be true, so $P$ is true" is itself an application of a background logical principle — the law of excluded middle (every statement is either true or false, with no third option) together with double-negation elimination ($\neg\neg P$ implies $P$). This is standard classical logic, not a Pearson-specific convention or a claim from the verified-facts file — it is flagged here only because it is the reason the technique works at all: reductio ad absurdum is valid precisely because mathematics (at the level this course operates at) accepts that every well-formed statement about integers, rationals or reals is determinately true or false, never "undetermined." A candidate never has to state this background principle in an exam answer — nothing in the real mark schemes reviewed asks for it — but it is the reason the four-move structure (assumption → derivation → impossibility → conclusion) is a proof and not merely a rhetorical move.

#1.2 Proof by contradiction is not proof by contrapositive — a distinction the lesson's own chain leans on without naming

These two techniques are easy to conflate, because both start by working with a negation, and it is worth separating them formally, because the spec's own named √2 example turns out to nest one inside the other. Proof by contrapositive proves an implication $P\Rightarrow Q$ by proving the logically equivalent statement $\neg Q\Rightarrow\neg P$ directly — this is a direct proof of a different (equivalent) implication; no absurdity is ever reached, and nothing is assumed false in order to derive a contradiction. Proof by contradiction proves a statement $P$ (which may itself be an implication) by assuming $\neg P$ and deriving an actual impossibility — a statement that is false with no exceptions. The two are different tools with a genuine logical relationship (contradiction can always simulate contrapositive: assuming $P$ true and $\neg Q$ true and deriving $\neg P$ is itself a contradiction against the assumption $P$), but they are not the same move, and a candidate who calls one the other has misdescribed their own reasoning even if the algebra is fine.

The lesson's own √2 chain is a clean real illustration of the two techniques nesting, not colliding. The outer proof is by contradiction throughout: assume $\sqrt2=p/q$ in lowest terms, derive downward, reach "$p$ and $q$ share a factor of 2" — a direct contradiction of the lowest-terms assumption stated at the start. But stage 3 of that same chain, in establishing "$p^2$ even $\Rightarrow p$ even," argues by contrapositive: it proves the contrapositive statement "$p$ odd $\Rightarrow p^2$ odd" directly (no absurdity reached inside that step — $p=2k+1\Rightarrow p^2=2(2k^2+2k)+1$, a plain odd number, full stop) and then invokes the logical equivalence of an implication and its contrapositive to conclude the original claim. The lesson's own working literally uses the word "contrapositive" at exactly this line (proof-by-contradiction.ts, stage 3 of the √2 chain) without stopping to name it as a distinct proof technique from the contradiction shell it sits inside — which this document now does explicitly, because it is a genuine structural fact about the spec's own named example, not an invented elaboration: the single most famous proof on this entire syllabus is a proof by contradiction with a proof by contrapositive nested inside it as a sub-lemma, and a student who can tell the two apart understands the √2 proof one level more deeply than one who treats the whole seven-stage chain as one undifferentiated block of "logic."

#2. Why each of the three real mechanisms is logically airtight

codex-formula-and-method-bank.md §2.1 already names the three real, evidence-confirmed mechanisms. What follows is not a restatement of that list but a formal argument, for each one, of why it is guaranteed to work whenever it applies — not merely that it happened to work on the one or two real questions that have tested it — plus one fresh, independently-checked micro-example per mechanism, none of which reuses a number from any real question or from the lesson's own worked material.

#2.1 Parity/divisibility contradiction — why a two-case split is exhaustive

The real mechanism (confirmed: Oct 2021 Q10, √2; June 2022 Q9, $n^2-2$) always turns on a case split into "even" and "odd." This split is exhaustive and mutually exclusive for a genuinely elementary reason worth stating precisely rather than assuming: the division algorithm guarantees that every integer $n$ can be written as $n=2k$ or $n=2k+1$ for a unique integer $k$, with no third possibility and no overlap — dividing any integer by 2 leaves a remainder of exactly 0 or exactly 1, never both, never neither. This is exactly the fact the lesson's own mechanism block leans on when it insists both branches of a parity split must be followed to their own contradiction (the WMA12-derived AND-logic argument, cited there and not re-derived here) — it is because "even or odd" is a genuine, gapless, overlap-free partition of every integer that skipping one branch leaves a real logical gap, not a stylistic omission.

Fresh micro-example. Prove by contradiction that there is no integer $n$ for which $n^2+n$ is odd. Assume, for contradiction, that some integer $n$ gives $n^2+n$ odd. Factorise: $n^2+n=n(n+1)$, the product of two consecutive integers. By the division algorithm above, $n$ is even or odd; if $n$ is even, $n(n+1)$ is even immediately (an even factor makes the whole product even); if $n$ is odd, $n+1$ is even, and again the product is even. Either branch of the exhaustive split forces $n(n+1)$ to be even — directly contradicting the assumption that it is odd. So no such integer exists (independently checked against every integer from $-5$ to $5$ with sympy, and confirmed generally by the factorised form $n(n+1)$ itself: consecutive integers are never both odd).

#2.2 Forcing a real square negative — why this mechanism needs no case split at all

The real mechanism (confirmed across two independent series: Jan 2024 Q8, a cubic's stationary points; Oct 2024 Q2, two curves not intersecting) reduces the assumption to an equation of the shape $x^2=(\text{negative number})$. This is structurally simpler than the parity mechanism above, and it is worth being explicit about why: it needs no case split, exhaustive or otherwise, because it rests on a single always-true fact about the real numbers — $x^2\ge0$ for every real $x$, with no exceptions to enumerate. A negative real number can never equal a real square, full stop; there is nothing to check case by case, which is exactly why the real mark schemes for both confirmed series close this mechanism with one short sentence ("no real $x$ satisfies this") rather than a branching argument. This is the single cleanest of the three mechanisms precisely because it needs the least logical apparatus once the algebra has done the reducing.

Fresh micro-example. Prove by contradiction that the curves $y=x^2+4$ and $y=-x^2-1$, $x\in\mathbb R$, do not intersect. Assume, for contradiction, that they do intersect at some real $x$. Then $x^2+4=-x^2-1\Rightarrow2x^2=-5\Rightarrow x^2=-\frac52$. But $x^2\ge0$ for every real $x$, so no real $x$ can satisfy $x^2=-\frac52$ — a direct contradiction. So the curves do not intersect (independently confirmed with sympy: solving $x^2+4=-x^2-1$ over the reals returns no solutions).

#2.3 Exhaustion nested in contradiction — why the divisor-pair list is provably complete, and a general reason some targets can never be reached this way

The real mechanism (confirmed once so far: Jan 2026 Q8, $4p^2-q^2=46$) factorises the assumption to a fixed product, $(2p+q)(2p-q)=46$, and turns an infinite search over all positive-integer pairs $(p,q)$ into a finite one over 46's own divisor pairs. This is provably complete for an elementary reason: a fixed positive integer $N$ has only finitely many positive divisors, and therefore only finitely many ways to be written as a product of two positive integers — a list that is complete because it is determined entirely by $N$'s own factor structure, not because anyone chose to stop searching. 46's divisors are exactly $1,2,23,46$, giving exactly two unordered factor pairs, which is why the real mark scheme's dependent mark requires checking both, not one (already covered in the lesson and the formula bank, not repeated here).

There is a further, general fact worth stating that goes one level beyond what any real WMA14 citation has needed so far, because it explains why a difference-of-two-squares target can fail before any divisor pair is even checked: for integers $a,b$, $a^2-b^2\pmod4$ can only ever be $0$, $1$ or $3$ — never $2$ — because every integer square is congruent to $0$ or $1\pmod4$ (an even integer squares to a multiple of 4; an odd integer squares to $1$ more than a multiple of 4), so the difference of two squares can only land on $0-0,\,0-1,\,1-0,\,1-1\pmod4$, i.e. $0,-1\equiv3,1,0$. This is standard number theory, not a claim from any Pearson source, and it is stated here because it gives a second, faster way to see why a target with no valid factor pair is genuinely unreachable — not an artefact of which pairs happened to get checked.

Fresh micro-example. Prove by contradiction that there are no positive integers $a>b>0$ with $a^2-b^2=34$. Assume, for contradiction, that such $a,b$ exist. Then $(a-b)(a+b)=34$, with $a+b>a-b>0$. Since $(a+b)+(a-b)=2a$ and $(a+b)-(a-b)=2b$ are both even, $a+b$ and $a-b$ must share the same parity (if they didn't, their sum would be odd, contradicting $2a$ being even). 34's only positive factor pairs are $1\times34$ and $2\times17$ — both mixed parity (one even, one odd) — so neither pair can be $(a-b,a+b)$, and the assumption fails: no such $a,b$ exist. The same conclusion follows faster from the general fact above: $34\equiv2\pmod4$, and $2$ is never a possible value for $a^2-b^2\pmod4$, so $34$ was never reachable as a difference of two squares at all (independently confirmed both ways with sympy: a brute-force search over $a<200$ finds no solution, and the mod-4 residue check confirms the general argument).

#3. A structural synthesis of the six real citations, organised by mechanism

signal-frequency-and-boundaries.md §1 already gives the frequency count (6 of 7 series mined show a real spec-1.1 item). Cut the same six citations a different way — by which mechanism each one actually uses, not by date — and a second, genuinely additive pattern appears:

Read this way, the six-citation record shows exactly two mechanisms with two independent confirmations each (parity; forcing-negative) and exactly one with a single confirmation so far (exhaustion-nested) — a genuinely different way of stating the evidence strength than a flat "6 of 7 series" count, and one that makes explicit which specific mechanism the newest series (Jan 2026) added rather than repeated.

#4. Evidence boundary

Consistent with this course's standing discipline: no likelihood percentage and no grade-boundary claim appears anywhere above, because none has been verified for this paper (signal-frequency-and-boundaries.md §2 confirms no grade-boundary document has been reviewed for WMA14 at all). Every mechanism-strength statement above is phrased as "confirmed in N of the six real citations reviewed," not as a prediction. The exhaustion-nested-in-contradiction mechanism has exactly one real confirmation; this document treats that as one data point, not a settled pattern, in the same way the parametric-area deep-dive treats its own single citation.

#Sources used

research/veridian/WMA14-verified-facts.md §1 (spec 1.1 text and guidance, verbatim), §5.1 in full (all six dated citations: Jan 2021 Q3, Oct 2021 Q10, June 2022 Q9, Jan 2024 Q8, Oct 2024 Q2, Jan 2026 Q8), §7 (confirms this is the richest-evidenced spec point in the bank), §8 (the 4-series vs 9-series evidence-scope note, respected here by not overstating the general-report denominator). content/veridian/curriculum/wma14/proof-by-contradiction.ts (full file — both worked chains, the concept-ladder's formal schema, both mechanism blocks, the method-comparison block, the marked-solution, all trap-taxonomy items and MCQs, and the closing provenance flag — read to identify exactly what is already taught and must not be duplicated, and to locate the "contrapositive" line in the √2 chain that §1.2 above builds on). content/veridian/curriculum/wma12/proof.ts (header and AND-logic mechanism, cited, not re-derived). codex-formula-and-method-bank.md §1 and §2.1, forge-exam-technique-guide.md (proof-by-contradiction checklist and mark table), and signal-frequency-and-boundaries.md §1–§3, all cross-checked to avoid restating their own content. content/veridian/question-bank/wma14.ts (checked — holds one item for this spec point, wma14-oct24-q2-proof-by-contradiction-quartic). The formal-logic content in §1 and the number-theory content in §2 are standard mathematics, independently verifiable against any logic or number-theory reference, not claims sourced from a Pearson document — flagged as such throughout rather than left ambiguous. Every new numeric micro-example in §2 was computed and independently checked with sympy (including a brute-force search to 200 for §2.3's example and a mod-4 residue check) before being written in.

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