Plain Incidence/Local-Geometry Route Is Exhausted Before Minority-Load Anti-Concentration¶

Claim/Theorem¶

Keep the notation of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md]], [[local-geometry-stops-before-uniform-linear-light-side-mass.md]], [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]], [[local-quotient-image-span-controls-rank-accumulation.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 plain incidence/local-geometry route should be demoted as a primary subordinate route toward the canonical dense-tangle-breadth target.

More precisely:

The direct local-block route already has an exact conditional closure criterion by [[lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md]]. Its first missing ingredient is a theorem of the form

\[ H_{\mathrm{anti}}^\beta(\delta,c): \quad \forall\,L\subseteq Q,\ |L|\in[\beta|Q|,(1-\beta)|Q|], \ \exists\,F(L) \text{ such that } m_v(L)<\delta\Delta \ \forall v\in F(L), \quad \sum_{v\in F(L)} m_v(L)\ge c|Q|. \]

This is a minority-load anti-concentration statement on balanced qubit cuts.
[[local-geometry-stops-before-uniform-linear-light-side-mass.md]] already isolates the exact failure of the current sourced incidence/local-neighborhood package:

incidence expansion, diagonal expansion, constant-size root neighborhoods, and bounded local overlap do not by themselves control the oriented minority-load profile

\[ v\longmapsto m_v(L) \]

below the small-side threshold \delta\Delta.
No current sourced node on the graph inserts a stronger qubit-side load-distribution invariant between that geometry package and H_{\mathrm{anti}}^\beta(\delta,c).

The graph still has only:
- aggregate incidence interaction;
- constant-size rooted local blocks;
- bounded local overlap; and
- the exact quotient-image packing criterion for what would suffice afterward.
It does not yet have a theorem governing anti-concentration of minority load across balanced qubit cuts.
Therefore re-testing the plain incidence/local-geometry package in isolation is exhausted as a primary subordinate route.

Any future reopening of the direct local-block branch must introduce a genuinely new qubit-side load-distribution invariant stronger than the current incidence/overlap package, rather than another reformulation of the same geometry.
This is a demotion of one subordinate mechanism, not of the canonical frontier.

[[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]] remains the live intrinsic target. The direct local-block branch remains admissible only if a new minority-load anti-concentration invariant is added to the graph and shown to feed the first hypothesis of [[lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md]].

Equivalently, the current source-grounded route stops before any theorem saying that every hardware-balanced qubit cut carries linear minority mass on lightly crossed local neighborhoods. So plain incidence/local geometry should now be treated as boundary evidence only, not as a primary active branch.

Dependencies¶

[[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
[[lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md]]
[[local-geometry-stops-before-uniform-linear-light-side-mass.md]]
[[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]
[[local-quotient-image-span-controls-rank-accumulation.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{anti}}^\beta(\delta,c) is false. It records only that the current sourced graph still stops before such a theorem.
It does not demote the entire direct local-block route. It demotes only the plain incidence/local-geometry attempt to supply the first ingredient of that route.
It does not preclude a future reopening of this branch. But that reopening must pass through a genuinely new qubit-side load-distribution invariant, not another isolated use of incidence expansion plus bounded overlap.

Sources¶

10.48550/arXiv.2202.13641
10.48550/arXiv.2206.07571
10.48550/arXiv.2508.05095
10.48550/arXiv.2109.14599
10.1007/BF02579166