Tester-Side Irreducibility Still Stops Before Original-Qubit Matroid Connectivity¶
Claim/Theorem¶
Keep the notation of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]], [[quantum-tanner-constituent-ltc.md]], [[balanced-linear-cut-rank-from-trellis-width.md]], [[ltc-sparse-cut-product-decomposition.md]], [[strong-ltc-constraint-graph-small-set-expander.md]], and [[cut-rank-is-interface-state-dimension.md]].
After stress-testing the current Route C package, the strongest sourced tester-side irreducibility statements still stop strictly before any intrinsic theorem on the original qubit parity-check matroid.
More precisely:
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The currently sourced left-right-Cayley package already gives a strong auxiliary classical object. By [[quantum-tanner-constituent-ltc.md]], the square-code
\[ C_{\mathrm{LTC}}=T(G^\square,C_A\otimes C_B) \]adjacent to the Quantum Tanner construction is an asymptotically good constant-query LTC.
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Because \(C_{\mathrm{LTC}}\) is asymptotically good, [[balanced-linear-cut-rank-from-trellis-width.md]] and [[cut-rank-is-interface-state-dimension.md]] imply that every ordering of its square coordinates has a balanced prefix cut carrying linear exact interface dimension. So Route C does produce a theorem-sized interface statement, but only for an auxiliary code on the square set and only along ordering cuts.
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The tester-graph irreducibility package is also strong on its own terms. [[ltc-sparse-cut-product-decomposition.md]] and [[strong-ltc-constraint-graph-small-set-expander.md]] show that, for a chosen tester graph, a sparse tester cut on a linear-size set is impossible unless the code approximately factors across that tester cut.
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However, no currently sourced theorem identifies any of the following pairs:
- the square-coordinate code \(C_{\mathrm{LTC}}\) and the original qubit parity-check matroid \(M_n\);
- a sparse cut or expansion statement in a chosen tester graph and an exact lower bound for \(\lambda_{M_n}(L)\) on qubit cuts;
- balanced prefix cuts in an ordering of the auxiliary square code and all hardware-balanced separators on the original qubit set.
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The obvious generic repairs are already blocked on the graph:
- [[qldpc-css-constituent-codes-not-good.md]] rules out importing good-code trellis/pathwidth theorems directly into the CSS constituent codes of the quantum family;
- [[good-ltc-does-not-imply-balanced-cut-rank.md]] rules out deducing intrinsic balanced cut rank from LTC alone;
- [[cayley-ltc-characterization-insufficient-for-balanced-cut-rank.md]] rules out any theorem that uses only properties equivalent to smooth LTC.
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Therefore the exact missing theorem on Route C is now:
\[ H_{\mathrm{tester}\to\mathrm{matroid}}(\beta): \quad \text{the left-right-Cayley / local-agreement / tester-irreducibility package forces } \lambda_{M_n}(L)=\Omega(n) \text{ on every }\beta\text{-balanced qubit cut }L, \]or at least forces dense tangle breadth or a sufficiently dense linear \(k_n\)-connected set in the original qubit parity-check matroid.
Therefore Route C is no longer merely "tester-side only" in a vague sense. Its exact failure mode is sharper:
the current source package reaches linear interface complexity for an auxiliary square code and irreducibility for chosen tester graphs, but still lacks a presentation-invariant lift to exact original-qubit matroid connectivity.
Dependencies¶
- [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
- [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]]
- [[quantum-tanner-constituent-ltc.md]]
- [[balanced-linear-cut-rank-from-trellis-width.md]]
- [[ltc-sparse-cut-product-decomposition.md]]
- [[strong-ltc-constraint-graph-small-set-expander.md]]
- [[cut-rank-is-interface-state-dimension.md]]
- [[qldpc-css-constituent-codes-not-good.md]]
- [[good-ltc-does-not-imply-balanced-cut-rank.md]]
- [[cayley-ltc-characterization-insufficient-for-balanced-cut-rank.md]]
Conflicts/Gaps¶
- This node does not prove that \(H_{\mathrm{tester}\to\mathrm{matroid}}(\beta)\) is false. It isolates that lift as the exact currently missing theorem.
- It does not rule out that the same left-right-Cayley geometry secretly enforces dense original-matroid concentration by a new family-specific argument.
- The positive balanced-prefix-cut theorem for \(C_{\mathrm{LTC}}\) may still become useful later if a future source-grounded theorem identifies those auxiliary cuts with intrinsic qubit cuts or with dense connected mass in the original matroid.
Sources¶
10.1109/FOCS54457.2022.0011710.1145/3519935.352002410.48550/arXiv.2005.01045dinurLocallyTestableCodes10.48550/arXiv.0805.219910.48550/arXiv.0711.1383