Quotient-Packing Route Is Demoted Before Construction-Controlled Low-Deficiency Theorem

Claim/Theorem

Keep the notation of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[local-quotient-image-span-controls-rank-accumulation.md]], [[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.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]], [[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]], [[explicit-quotient-block-witness-branch-is-boundary-not-primary-frontier.md]], [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]], and [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]].

At the current graph state, the quotient-packing branch should now be demoted as a primary subordinate route toward the canonical dense-tangle-breadth target.

More precisely:

1. The currently tested quotient-side candidate invariants have already been separated into four levels.

   • Fixed-basis pivot coverage is too rigid by [[fixed-quotient-basis-pivot-coverage-is-too-rigid.md]].
   • Basis-adaptive triangularization exists existentially once \nu_H(L)=\lambda_{M(H)}(L) by [[nu-saturation-yields-adapted-triangular-basis.md]], but [[no-transferable-selection-rule-is-visible-for-adapted-bases.md]] and [[exact-optimal-quotient-families-do-not-share-transferable-shelling-law.md]] show that no currently visible construction-controlled shelling or selector rule survives.
   • The represented-matroid block-packing / parity reduction of [[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.md]] identifies the exact optimization class, but does not itself provide a family-level invariant. It amounts to solving the global packing problem on each cut.
   • The strongest remaining theorem-sized bridge is therefore [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]].

2. Among those candidates, the low-deficiency union-rank criterion is the strongest mathematically supported one.

   Writing

   \[ w_H(X;L):=\sum_{v\in X}\dim W_v(L), \qquad \rho_H(X;L):=\dim\Big(\sum_{v\in X}W_v(L)\Big), \]

   the strongest surviving conditional route is:

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

   for every \beta-balanced cut L, there exists a construction-controlled block subfamily X(L) with

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

   and

   \[ \rho_H(X(L);L)\ge \Big(\frac{1}{2}+\varepsilon\Big)w_H(X(L);L) \]

   for some fixed \varepsilon>0.

   By [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]], this would imply

   \[ \nu_H(L)=\Omega(|Q|), \]

   hence linear balanced intrinsic cut rank.

3. However, the current source-grounded graph still stops before H_{\mathrm{def}}(\beta).

   The present nodes show only:

   • the exact quotient span formula and the exact block-packing objective;
   • existential optimized witnesses on sampled low cuts;
   • and several failure modes for natural selector classes.

   They do not produce a source-grounded theorem selecting a linear-weight low-deficiency subfamily on arbitrary balanced cuts of the target Quantum Tanner family.

4. The explicit low-cut evidence is already strong enough to show why weaker quotient-packing ideas are exhausted.

   • Taking X to be the whole visible block family is numerically too crude, as [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]] records.
   • Fixed-basis pivot exposure undercounts saturated quotient directions on sampled cuts, by [[fixed-quotient-basis-pivot-coverage-is-too-rigid.md]].
   • Visible root-side profiles, dimension profiles, common-core rules, and local-repair rules all fail to transfer, by [[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 current exact witness branch is therefore already correctly summarized as boundary evidence in [[explicit-quotient-block-witness-branch-is-boundary-not-primary-frontier.md]].

5. Therefore the quotient-packing route beyond the diagnostic node [[local-quotient-image-span-controls-rank-accumulation.md]] is no longer a primary active subordinate route.

   It should now be treated as a secondary boundary package whose role is:

   • to falsify newly proposed quotient-side selector or packing invariants against the current explicit data, or
   • to host a genuinely new construction-controlled low-deficiency / union-rank invariant if one is later proposed.

6. This is a demotion of a subordinate mechanism, not of the canonical frontier.

   [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]] remains the live intrinsic target. The quotient-packing route stays mathematically useful only as a sharply localized stop-point:

   the strongest remaining quotient-side bridge is the low-deficiency union-rank criterion, but the current source package still does not reach it construction-controlledly.

Equivalently, the exact missing theorem on this branch is no longer "find some packing invariant." It is:

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

for every \beta-balanced cut L, the sourced Quantum Tanner geometry itself reveals a block subfamily X(L) with linear weight and union-rank density strictly above 1/2.

No such theorem is currently supported on disk. So the quotient-packing branch should now be loaded only on demand, not as a default subordinate frontier.

Dependencies

• [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
• [[local-quotient-image-span-controls-rank-accumulation.md]]
• [[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.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]]
• [[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]]
• [[explicit-quotient-block-witness-branch-is-boundary-not-primary-frontier.md]]
• [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]]
• [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]]

Conflicts/Gaps

• This node does not prove that H_{\mathrm{def}}(\beta) is false. It records only that no source-grounded theorem currently on disk reaches it.
• It does not demote the diagnostic quotient-span formula of [[local-quotient-image-span-controls-rank-accumulation.md]], which remains load-bearing. The demotion concerns the stronger quotient-packing branch as a primary subordinate proof route.
• It does not rule out a future reopening of this branch. But that reopening must introduce a genuinely new construction-controlled low-deficiency or union-rank invariant, not another selector-style reformulation already falsified by the current explicit data.

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