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Diagnostic Intake Key

Use this key to grade diagnostic-intake.md Mode A answers. It is not a placement test for the whole foundations course. Its only purpose is to decide how to start the first live foundation session.

Scoring:

  • strong: answer is correct enough to compress or skip the corresponding material.
  • partial: answer has the right direction but needs cleanup during the lesson.
  • weak: answer is missing, incorrect, or too vague to use as evidence.

Do not mark Lesson 01 as skipped by diagnostic unless questions 1-5 are all strong.

Question 1

Prompt: In GF(2), why is \(1+1=0\), and why does that matter for parity checks?

Strong answer:

  • GF(2) arithmetic is modulo 2, so two ones cancel.
  • Parity checks count whether the number of selected 1 entries is even or odd.
  • A parity-check equation is satisfied when the selected sum is 0 mod 2.

Partial answer:

  • Says "mod 2" but does not connect it to parity checks.

Weak answer:

  • Uses ordinary integer arithmetic;
  • says only "because binary" without explaining cancellation;
  • cannot connect to parity.

Routing:

  • If weak, teach Lesson 01 from the GF(2) arithmetic section.

Question 2

Prompt: For a parity-check row \(h\), what does \(h c^T=0\) mean?

Strong answer:

  • It means the dot product over GF(2) between the check row and codeword is zero.
  • Equivalently, the coordinates selected by \(h\) have even parity.
  • For a parity-check matrix \(H\), a codeword satisfies all rows, so \(Hc^T=0\).

Partial answer:

  • Correctly says "it is a parity check" but does not mention GF(2) or selected coordinates.

Weak answer:

  • Treats it as a real-valued orthogonality statement without mod 2 context;
  • cannot identify what the row checks.

Routing:

  • If weak, teach Lesson 01 slowly and include explicit dot-product examples.

Question 3

Prompt: In the \([7,4,3]\) Hamming code, what information does a single-bit-error syndrome give you?

Strong answer:

  • For a standard Hamming parity-check matrix whose columns are the nonzero binary triples, a single-bit error at position \(i\) gives syndrome equal to column \(i\).
  • The syndrome identifies the error location.

Partial answer:

  • Says "it tells you the error" but does not explain the column-label mechanism.

Weak answer:

  • Confuses syndrome with codeword;
  • says it only detects but cannot locate a single-bit error;
  • cannot state what the syndrome indexes.

Routing:

  • If weak, start Lesson 01 from the Hamming worked example rather than skipping.

Question 4

Prompt: What is the difference between a generator matrix and a parity-check matrix?

Strong answer:

  • A generator matrix \(G\) spans the code by linear combinations of message rows.
  • A parity-check matrix \(H\) defines the code as the kernel:
\[ C=\{c:Hc^T=0\}. \]
  • In the full-rank classical setting, the row space of \(H\) is the dual code \(C^\perp\).

Partial answer:

  • Says "one generates, one checks" but cannot state span versus kernel.

Weak answer:

  • Treats \(G\) and \(H\) as interchangeable;
  • cannot describe either span or kernel.

Routing:

  • If questions 1-3 are strong but question 4 is weak, compress Lesson 01 and move quickly into Lesson 02.

Question 5

Prompt: In one sentence, why does the Conjecture 3 course begin with classical linear codes rather than QLDPC codes directly?

Strong answer:

  • QLDPC and CSS constructions are built from binary parity-check matrices, Tanner graphs, duality, and code parameters, so the classical layer provides the algebraic and graph language needed for the Conjecture 3 barrier.

Partial answer:

  • Says classical codes are prerequisites but does not name the specific objects that carry into QLDPC.

Weak answer:

  • Gives a generic "basics first" answer with no research-map connection;
  • says classical codes are unrelated and only historical background.

Routing:

  • If weak while questions 1-4 are strong, use the research-orientation part of Lesson 01 but compress the calculations.

Question 6

Prompt: Which part feels least automatic: GF(2) arithmetic, syndrome calculation, matrix row spaces, or research-map motivation?

Strong answer:

  • Identifies a concrete weak area and gives a usable reason or example.

Partial answer:

  • Selects a weak area but gives no detail.

Weak answer:

  • Does not answer, or says everything is fine while earlier answers show a gap.

Routing:

  • Use this answer to choose emphasis. It should not by itself pass or fail Lesson 01.

Mode A Routing Summary

Use this routing after grading questions 1-6.

Evidence pattern Route
Any of questions 1-3 weak Start Lesson 01 from the beginning
Questions 1-3 strong, question 4 weak or partial Compress Lesson 01 calculations, then emphasize Lesson 02
Questions 1-4 strong, question 5 weak Teach the research-orientation part of Lesson 01, then continue to Lesson 02
Questions 1-5 strong Propose marking Lesson 01 as skipped by diagnostic, but update progress only after learner confirmation

Recording Rule

After grading, propose a concise update:

  • Mode A result;
  • question-level strengths and gaps;
  • recommended next lesson;
  • whether Lesson 01 should remain not started, become in progress, or be marked skipped by diagnostic.

Record the result in session-log.md and the foundation progress tracker only after learner confirmation, following session-protocol.md.