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Quantum Tanner Left-Right Cayley Source Package Stops At Tester-Side Structure

Claim/Theorem

Let the currently sourced target-family package consist of the following exact family theorems:

  1. Leverrier--Zemor, Lemma 4 (10.1109/FOCS54457.2022.00117, p. 12), which gives spectral expansion of the diagonal graphs of the left-right Cayley complex.
  2. Leverrier--Zemor, Theorem 17 (10.1109/FOCS54457.2022.00117, pp. 23-24), which gives good-parameter quantum Tanner codes under the paper's distance and dual-tensor robustness hypotheses.
  3. Dinur--Evra--Livne--Lubotzky--Mozes, Theorem 4.5 (10.1145/3519935.3520024, p. 16), which proves that the left-right-Cayley square code is locally testable from local agreement testability plus two-direction Cayley expansion.
  4. Dinur--Evra--Livne--Lubotzky--Mozes, Theorem 1.1 (10.1145/3519935.3520024, pp. 2, 22-23), which instantiates this into an explicit c3-LTC family.

Then the present graph supports the following family-contact no-go statement:

these sourced target-family theorems still do not imply any dense intrinsic connectivity statement on the original qubit parity-check matroid, such as

  • dense tangle breadth,
  • a dense \(k\)-connected set,
  • or a dense large-rank lean bag.

What they do imply is strictly tester-side or chosen-presentation structure:

  • via [[quantum-tanner-theorem17-parity-expander.md]], the Leverrier--Zemor package yields a bounded-degree local-expander statement for a chosen parity Tanner graph;
  • via [[left-right-cayley-ltc-from-local-agreement-plus-expansion.md]], the Dinur--Evra--Livne--Lubotzky--Mozes package yields local testability of the adjacent square code under its natural tester.

The argument stops exactly when one tries to pass from these tester-side objects to the intrinsic quantity

\[ \chi_L(\mathcal S)=\lambda_{M_{\mathcal S}}(L), \]

or to any dense connected object in the original qubit matroid. No currently sourced family theorem identifies:

  1. a sparse cut or expansion statement in a chosen tester / Tanner graph, with
  2. an exact low-connectivity or high-connectivity statement in the column matroid on original qubits.

The existing graph then shows that the obvious generic substitutes are insufficient:

  • [[good-code-parameters-do-not-imply-cut-rank.md]] rules out using only [n,k,d];
  • [[good-ltc-does-not-imply-balanced-cut-rank.md]] rules out using LTC alone;
  • [[ltc-sparse-cut-product-decomposition.md]] gives only approximate tester-side product structure;
  • [[cut-rank-is-interface-state-dimension.md]] shows that the intrinsic frontier needs exact interface dimension, not approximate tester decomposition.

Therefore the current family-contact bottleneck is not another abstract implication among intrinsic notions. It is the absence of one new qubit-side structural hypothesis for the actual family.

The sharpest such hypothesis currently on disk is:

\[ H_{\mathrm{dense}}:\quad \text{the original qubit parity-check matroid }M_n \text{ of the target family has a tangle of order }k_n=\Omega(n) \text{ and breadth }t_n>(1-\beta)n+k_n-2. \]

Under \(H_{\mathrm{dense}}\), [[dense-tangle-breadth-forces-balanced-cut-rank.md]] immediately gives linear balanced stabilizer cut rank, hence linear interface-state dimension across every \(\beta\)-balanced cut and the corresponding SWAP-only cut-congestion lower bound.

So the present source package stops one level below the true intrinsic target: it reaches tester-side expansion and local agreement, but not qubit-side dense matroidal connectivity.

Dependencies

  • [[quantum-tanner-theorem17-parity-expander.md]]
  • [[left-right-cayley-ltc-from-local-agreement-plus-expansion.md]]
  • [[ltc-sparse-cut-product-decomposition.md]]
  • [[strong-ltc-constraint-graph-small-set-expander.md]]
  • [[good-code-parameters-do-not-imply-cut-rank.md]]
  • [[good-ltc-does-not-imply-balanced-cut-rank.md]]
  • [[cut-rank-is-interface-state-dimension.md]]
  • [[dense-tangle-breadth-forces-balanced-cut-rank.md]]
  • [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]]

Conflicts/Gaps

  • This is a source-grounded no-go theorem about the currently loaded family package. It does not prove that no future theorem from the same family can imply dense intrinsic connectivity.
  • The hypothesis \(H_{\mathrm{dense}}\) is intentionally strong and qubit-side. A weaker qubit-side hypothesis might also suffice, but it is not yet on the graph in a source-grounded family-contact form.
  • The node does not identify any finite explicit counterexample inside the target family. It isolates the exact missing theorem needed to convert the present source package into the dense-breadth route.

Sources

  • 10.1109/FOCS54457.2022.00117
  • 10.1145/3519935.3520024
  • 10.37236/12467
  • 10.48550/arXiv.2109.14599
  • 10.48550/arXiv.0711.1383