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Conjecture 3 Frontier Assessment Rubric

Use this rubric after Lessons 01-09. It is the grading layer for Lesson 10, not a substitute for teaching.

Passing standard: the learner answers at least 8 of 10 questions cleanly and can identify the missing premise in any answer they cannot complete.

Scoring:

  • 2: correct, precise, and scoped to the current graph.
  • 1: directionally correct but missing a key condition, model assumption, or open-gap distinction.
  • 0: wrong, unsupported, or collapses a solved theorem with an open frontier.

Recommended pass threshold: at least 16/20, with no 0 on questions 1, 3, 5, or 10.

Question 1

Prompt: State the minimal static 2D result and give the one-line parameter derivation.

Expected answer:

  • For theorem-level Quantum Tanner good families with \(k,d=\Theta(n)\), any 2D-local syndrome-extraction circuit using \(m=\Theta(n)\) qubits has depth \(\Omega(\sqrt n)\).
  • The parameter theorem gives:
\[ \Delta=\Omega\left(\frac{k\sqrt d}{m}\right) = \Omega\left(\frac{n\sqrt n}{n}\right) = \Omega(\sqrt n). \]

Common mistakes:

  • saying the static theorem is still open;
  • deriving the result from a chosen Tanner graph when the answer asks for the presentation-invariant route;
  • omitting the linear-space condition.

Primary sources:

Question 2

Prompt: Explain why the remaining frontier is not the static theorem itself.

Expected answer:

  • The static near-square 2D lower bound is already theorem-backed for good Quantum Tanner families.
  • The remaining work is to refine or extend the statement toward compiler-native CD(T_n,G) semantics, explicit deterministic family packaging, and intrinsic cut-rank mechanisms.

Common mistakes:

  • treating "prove any 2D lower bound" as the current frontier;
  • ignoring the distinction between static 2D local circuits and broader compiler semantics.

Primary sources:

Question 3

Prompt: Define \(\chi_L(\mathcal S)\) and give the rank formula.

Expected answer:

  • For a stabilizer space \(\mathcal S\) on qubits \(Q=L\sqcup R\), let \(\mathcal S_L\) and \(\mathcal S_R\) be the subspaces supported entirely on \(L\) and \(R\).
  • The cross-cut stabilizer rank is:
\[ \chi_L(\mathcal S) = \dim \frac{\mathcal S}{\mathcal S_L+\mathcal S_R}. \]
  • For a full-row-rank support matrix \(H=[H_L\mid H_R]\):
\[ \chi_L(\mathcal S) = \operatorname{rank}(H_L)+\operatorname{rank}(H_R)-\operatorname{rank}(H). \]

Common mistakes:

  • counting a chosen set of crossing generators instead of the quotient dimension;
  • forgetting that this is generator-invariant;
  • treating it as ordinary graph edge capacity.

Primary sources:

Question 4

Prompt: Explain why token crossing is false but cut-edge service remains useful.

Expected answer:

  • A cross-cut stabilizer can be served by cross-cut gates without requiring a data qubit token to move permanently across the cut.
  • Therefore token traversal is not the correct primitive.
  • The circuit still must provide cross-cut interaction service proportional to intrinsic cross-cut stabilizer demand in the SWAP-only measurement-free model.

Common mistakes:

  • concluding that the lower-bound route dies because token crossing is false;
  • treating cross-cut service as identical to packet routing.

Primary sources:

Question 5

Prompt: Define \(T_{\mathrm{sub}}\) and \(CD_{\mathrm{sub}}\).

Expected answer:

  • The canonical submodular demand object is:
\[ T_{\mathrm{sub}}=(Q,\lambda_{\mathcal S}), \qquad \lambda_{\mathcal S}(L)=\chi_L(\mathcal S). \]
  • Against hardware \(G_{\mathrm{hw}}\), the submodular cut congestion is:
\[ CD_{\mathrm{sub}}(T_{\mathrm{sub}},G_{\mathrm{hw}}) = \max_L \frac{\lambda_{\mathcal S}(L)}{|\partial_{G_{\mathrm{hw}}}L|}. \]
  • This lower-bounds SWAP-only service depth up to constants in the stated model.

Common mistakes:

  • calling this an ordinary guest-graph congestion without qualification;
  • leaving out the hardware boundary denominator;
  • forgetting that \(\lambda_{\mathcal S}\) is a submodular cut function, not a packet list.

Primary sources:

Question 6

Prompt: Explain why ordinary graph cuts fail to represent stabilizer cut rank.

Expected answer:

  • Stabilizer cut rank is a symmetric submodular function, but it is not generally an ordinary nonnegative weighted graph cut on the physical qubit terminals.
  • The local graph contains explicit small stabilizer examples showing that ordinary graph-cut representation fails.

Common mistakes:

  • confusing graph cut-rank over GF(2) with edge-cut capacity;
  • claiming no graph-theoretic representation exists at all.

Primary source:

Question 7

Prompt: Explain the fundamental graph cut-rank positive result and why it still is not routing.

Expected answer:

  • For a binary matroid and a chosen basis, the fundamental graph realizes matroid connectivity as graph cut-rank:
\[ \lambda_M(X)=\operatorname{cutrk}_{G_{\mathrm{fund}}(M,B)}(X)+1. \]
  • This is exact and important, but cut-rank is algebraic over GF(2), not nonnegative edge capacity.
  • Ordinary edge-cut readings are basis-unstable and do not supply a basis-independent packet-routing semantics.

Common mistakes:

  • treating the fundamental graph as a demand graph;
  • ignoring basis dependence;
  • saying the theorem contradicts ordinary graph-cut nonrepresentability.

Primary sources:

Question 8

Prompt: State dense tangle breadth as a sufficient intrinsic target.

Expected answer:

  • A sufficient target is dense original-matroid concentration: the original qubit parity-check matroid has a high-order tangle with breadth dense enough to force linear connectivity on every hardware-balanced cut.
  • If this holds, then \(\chi_L(\mathcal S_n)=\Omega(n)\) on balanced cuts, feeding the stabilizer cut-rank and submodular CD lower-bound route.

Common mistakes:

  • claiming dense tangle breadth has already been proved for Quantum Tanner codes;
  • describing generic tangle order without the density or original-matroid requirement.

Primary sources:

Question 9

Prompt: Explain local quotient-image accumulation and the two losses.

Expected answer:

  • For local row blocks \(S(v)\) and cut quotient \(S/(S_L+S_R)\), define:
\[ W_v(L)=\frac{S(v)+S_L+S_R}{S_L+S_R}. \]
  • The global cut rank is:
\[ \chi_L(S) = \dim\left(\sum_v W_v(L)\right). \]
  • The two losses are survival loss, where local cross-cut classes die in the global quotient, and overlap loss, where surviving local images fail to be independent.

Common mistakes:

  • saying many crossed neighborhoods automatically imply large global cut rank;
  • discussing only support overlap and not quotient-image independence.

Primary sources:

Question 10

Prompt: Explain why Route D is now a semantic frontier.

Expected answer:

  • Hidden-vertex or auxiliary-variable realizations can represent functions beyond terminal graph or hypergraph cuts.
  • But exact auxiliary representability does not automatically provide physical compiler meaning, packet routing, or terminal service semantics.
  • The current graph therefore treats Route D as a semantic question: whether any auxiliary realization can be converted into a compiler-meaningful theorem, or whether submodular CD is the right stopping point for now.

Common mistakes:

  • equating hidden-vertex representability with routing;
  • treating Route D as only a search for the first counterexample;
  • ignoring the connected witness separation.

Primary sources:

Recording Results

After grading, update learner-progress.md with:

  • date of assessment;
  • score out of 20;
  • questions missed or partially answered;
  • next lesson or research target;
  • whether the learner is cleared for Target A, Target B, or Target C from Lesson 10.