Explicit Quotient-Block Witness Branch Is Boundary Not Primary Frontier¶

Claim/Theorem¶

Keep the notation of [[local-quotient-image-span-controls-rank-accumulation.md]], [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]], [[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.md]], [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]], [[exact-low-deficiency-subfamilies-do-not-share-transferable-visible-template.md]], and [[exact-low-deficiency-witnesses-have-no-transferable-core-or-local-repair.md]].

The explicit small-instance quotient-block program has now established a sharp boundary package, but not an active selector theorem route.

More precisely:

On the current low-lambda balanced cuts of the explicit D_4, D_6, and D_8 Quantum Tanner instances, one has

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

so exact low-deficiency witnesses do exist on those cuts.
The exact optimization problem behind those witnesses is already structurally identified: by [[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.md]], it is represented-matroid block packing on every cut and weighted linear matroid parity on the current explicit low cuts.
The branch has also now ruled out the currently natural construction-controlled selector classes:
- pairwise proxies fail, by [[pairwise-skew-block-statistics-do-not-control-parity-value.md]];
- visible side-count and local-dimension templates fail, by [[exact-low-deficiency-subfamilies-do-not-share-transferable-visible-template.md]];
- common-core and local-repair explanations fail, by [[exact-low-deficiency-witnesses-have-no-transferable-core-or-local-repair.md]].
The remaining quantitative statement is real but conditional: [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]] proves that any construction-controlled block subfamily X with

\[ \rho_H(X;L)>\frac{1}{2}w_H(X;L) \]

would force a linear lower bound on \nu_H(L), and hence on balanced intrinsic cut rank. But the current explicit Quantum Tanner evidence only exhibits such subfamilies existentially, after exact global optimization.

Therefore the current branch supports the following synthesis claim:

The explicit quotient-block witness program presently provides falsification evidence against selector-style heuristics and construction-controlled witness templates, but it does not currently provide a surviving transferable selector class. Its mathematically supported output is: exact optimal witnesses exist on the sampled low cuts, the optimization problem is structurally identified, and several natural selector classes already fail. This is boundary evidence, not the primary active theorem route.

Equivalently, the exact missing object can be stated as

\[ H_{\mathrm{glob}}^{\mathrm{def}}(\beta): \]

for every \beta-balanced cut L, there is a construction-controlled global invariant of the quotient-block arrangement that identifies or certifies a block subfamily X with

\[ w_H(X;L)=\Omega(|Q|) \]

and

\[ \rho_H(X;L)>\frac{1}{2}w_H(X;L), \]

without solving the exact global optimization problem for \nu_H(L).

No such invariant is currently visible on disk. So future effort should return to the intrinsic balanced-cut-rank route unless a genuinely new selector class or family-level invariant is proposed.

Graph Maintenance Recommendation¶

For graph maintenance, the explicit quotient-block witness line should now be split into two roles.

Canonical active spine:

keep the intrinsic balanced-cut-rank route primary, especially
- [[stabilizer-cut-rank-functional.md]];
- [[cross-cut-stabilizer-rank-rank-formula.md]];
- [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]];
- [[local-quotient-image-span-controls-rank-accumulation.md]];
- [[dense-tangle-breadth-forces-balanced-cut-rank.md]];
- [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]];
- [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]].
Secondary boundary package:

group the explicit quotient-block witness nodes as one falsification package:
- [[explicit-quotient-block-witness-branch-is-boundary-not-primary-frontier.md]];
- [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]];
- [[fixed-quotient-basis-pivot-coverage-is-too-rigid.md]];
- [[nu-saturation-yields-adapted-triangular-basis.md]];
- [[no-transferable-selection-rule-is-visible-for-adapted-bases.md]];
- [[exact-optimal-quotient-families-do-not-share-transferable-shelling-law.md]];
- [[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.md]];
- [[pairwise-skew-block-statistics-do-not-control-parity-value.md]];
- [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]];
- [[exact-low-deficiency-subfamilies-do-not-share-transferable-visible-template.md]];
- [[exact-low-deficiency-witnesses-have-no-transferable-core-or-local-repair.md]].
Future prompt guidance:

future prompts should refer to this package as the "explicit quotient-block witness boundary branch" and load it only when:
- checking whether a newly proposed selector class is already falsified by the current explicit data, or
- needing the exact boundary statement that only existential optimized witnesses are presently supported.
Future prompts should not continue witness-pattern mining by default.

Dependencies¶

[[local-quotient-image-span-controls-rank-accumulation.md]]
[[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]]
[[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.md]]
[[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]]
[[exact-low-deficiency-subfamilies-do-not-share-transferable-visible-template.md]]
[[exact-low-deficiency-witnesses-have-no-transferable-core-or-local-repair.md]]
[[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]

Conflicts/Gaps¶

This node is a synthesis and reprioritization result, not a new family theorem on balanced cut rank.
It does not prove that the explicit quotient-block branch can never return to active status; it proves only that no surviving transferable selector class is currently supported on disk.
The recommendation to demote the branch is therefore evidence-based and local to the present graph state.
If a future run produces a genuinely new family-level invariant, this branch can be re-promoted.

Sources¶

10.48550/arXiv.2206.07571
10.48550/arXiv.2508.05095
10.1016/0095-8956(80)90066-0
10.1016/j.dam.2019.09.023