Nu Saturation Yields Adapted Triangular Basis

Claim/Theorem

Keep the notation of [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]] and [[fixed-quotient-basis-pivot-coverage-is-too-rigid.md]]. For a qubit cut \(L \sqcup R = Q\), define the basis-adaptive pivot-cover number

\[ \widehat\pi_H(L) := \max_{\mathcal Q_L} \pi_H^{\mathcal Q}(L), \]

where the maximum runs over all quotient bases \(\mathcal Q_L\) of \(S/B(L)\).

Then:

1. One always has

\[ \nu_H(L)\le \widehat\pi_H(L)\le \lambda_{M(H)}(L). \]

1. Consequently, whenever the packed surrogate saturates the quotient,

\[ \nu_H(L)=\lambda_{M(H)}(L), \]

there exists an adapted quotient basis \(\mathcal Q_L^\ast\) such that

\[ \pi_H^{\mathcal Q^\ast}(L)=\lambda_{M(H)}(L). \]

Equivalently, a full triangular witness family exists across the cut.

1. Therefore the live frontier is no longer the bare existence of an adapted basis on individual cuts. Existence already follows from \nu=\lambda. The unresolved construction-level object is a rule that selects such a saturating direct-sum family, or equivalently such an adapted basis, uniformly from the visible local quotient-image data of the Quantum Tanner construction.

The exact missing invariant can now be stated as

\[ H_{\mathrm{rule}}(\beta): \]

for every \(\beta\)-balanced cut \(L\), there is a construction-controlled procedure that extracts a direct-sum family of local quotient images of total dimension \(\Omega(|Q|)\), or directly outputs an adapted quotient basis with pivot cover \(\Omega(|Q|)\).

Under \(H_{\mathrm{rule}}(\beta)\), one immediately gets

\[ \lambda_{M(H)}(L)\ge \widehat\pi_H(L)=\Omega(|Q|), \]

and hence linear balanced intrinsic cut rank.

Proof:

1. The upper bound \(\widehat\pi_H(L)\le \lambda_{M(H)}(L)\) is the fixed-basis inequality from [[fixed-quotient-basis-pivot-coverage-is-too-rigid.md]], maximized over all quotient bases.
2. For the lower bound, let \(F\) be a direct-sum family of local quotient images achieving \(\nu_H(L)\):

\[ \sum_{v\in F}\dim W_v(L)=\nu_H(L), \qquad \sum_{v\in F} W_v(L)\ \text{direct}. \]

Choose a basis \(B_v\) of each \(W_v(L)\) for \(v\in F\). Because the sum is direct, the concatenation

\[ \mathcal B_F:=\bigsqcup_{v\in F} B_v \]

is linearly independent in \(S/B(L)\). Extend \(\mathcal B_F\) to a full quotient basis \(\mathcal Q_L^\ast\). 3. In the basis \(\mathcal Q_L^\ast\), each \(W_v(L)\) for \(v\in F\) is spanned by its own designated coordinates coming from \(B_v\). Hence the row-reduced matrix of \(W_v(L)\) has exactly \(\dim W_v(L)\) pivot columns supported in those coordinates, and distinct blocks in \(F\) occupy disjoint pivot sets. Therefore

\[ \pi_H^{\mathcal Q^\ast}(L)\ge \sum_{v\in F}\dim W_v(L)=\nu_H(L). \]

Maximizing over bases gives \(\widehat\pi_H(L)\ge \nu_H(L)\). 4. If \(\nu_H(L)=\lambda_{M(H)}(L)\), then the two-sided bound collapses to

\[ \widehat\pi_H(L)=\lambda_{M(H)}(L), \]

which proves existence of a full adapted triangular witness basis.

Construction-level evidence on the current low-lambda balanced witnesses:

• D_4 / [36,8,3]: [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]] gives

\[ \nu=\lambda=5, \]

so an adapted basis with full pivot cover exists. One exact saturating family is

\[ \{g=sr,\ g=sr^3,\ h=e,\ h=r\}, \]

with dimensions (1,1,2,1).

• D_6 / [54,11,4]: one has

\[ \nu=\lambda=12, \]

so an adapted basis with full pivot cover exists even though the natural basis had pivot cover only 7. One exact saturating family is

\[ \{g=sr^4,\ h=s,\ h=sr,\ h=sr^2,\ h=sr^3,\ h=sr^5\}, \]

each of dimension 2.

• D_8 / [72,14,4]: one has

\[ \nu=\lambda=12, \]

so again a full adapted basis exists although the natural basis had pivot cover only 11. One exact saturating family is

\[ \{g=r,\ g=r^3,\ g=r^5,\ g=sr^5,\ h=e,\ h=r^7,\ h=sr^7\}, \]

with dimensions (2,2,1,2,2,1,2).

These examples make the key distinction exact:

• adapted basis existence is already certified on these cuts by the theorem above, because \nu=\lambda;
• but the basis was obtained by solving the global direct-sum packing problem first.

No current construction-level invariant on disk explains why these particular families should be selected from the local Quantum Tanner geometry alone, nor why an analogous choice should exist uniformly on every balanced cut in the family.

So the live obstruction has moved again: it is not the existence of a triangular witness basis, but the lack of a construction-controlled adapted-basis rule.

Dependencies

• [[fixed-quotient-basis-pivot-coverage-is-too-rigid.md]]
• [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]]
• [[local-quotient-image-span-controls-rank-accumulation.md]]
• [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]
• [[quantum-tanner-local-generator-blowup.md]]

Conflicts/Gaps

• The theorem is existential once \nu_H(L) is known; it does not provide a local or geometric recipe for finding the saturating family.
• The explicit witness families above come from exact optimization on small instances, not from a uniform structural rule visible in the construction.
• The node therefore does not close the family theorem. It isolates the exact remaining issue: selecting the adapted basis without solving the whole global packing problem anew for each cut.
• What remains missing is a source-grounded or construction-grounded invariant that predicts or enforces these saturating direct-sum families.

Sources

• 10.48550/arXiv.2206.07571
• 10.48550/arXiv.2508.05095