Local Geometry Stops Before Uniform Linear Light-Side Mass¶
Claim/Theorem¶
Keep the notation of [[lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md]], [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]], [[local-quotient-image-span-controls-rank-accumulation.md]], [[small-side-local-cut-gives-full-local-cross-rank.md]], [[quantum-tanner-local-generator-blowup.md]], [[quantum-tanner-incidence-spectral-gap.md]], [[regular-graph-expansion-to-incidence-expansion.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]].
Fix the subordinate direct local-block ingredient:
every hardware-balanced cut should contain a family \(F(L)\) of lightly crossed local blocks with
\[ \sum_{v\in F(L)} m_v(L)=\Omega(|Q|), \qquad m_v(L):=\min\{|L\cap Q(v)|,\ |R\cap Q(v)|\}, \]and \(m_v(L)<\delta\Delta\) for all \(v\in F(L)\).
At the current graph state, the sourced Quantum Tanner local-geometry package stops before this uniform linear light-side-mass statement.
More precisely:
-
The currently sourced local geometry does identify the relevant combinatorial objects:
- by [[quantum-tanner-local-generator-blowup.md]], each local block is supported on one root neighborhood \(Q(v)\) of size at most \(\Delta^2\);
- each generator has weight at most \(\Delta^2\);
- each qubit belongs to at most \(\Delta^2\) rooted local blocks.
So the construction gives a bounded-overlap family of constant-size root neighborhoods on the qubit set.
-
The current source package also gives auxiliary expansion on the underlying incidence geometry:
- [[quantum-tanner-diagonal-expansion-structure.md]] gives spectral expansion of the diagonal graphs;
- [[quantum-tanner-incidence-spectral-gap.md]] upgrades this to a constant spectral gap for the one-parity square-vertex incidence graph;
- [[regular-graph-expansion-to-incidence-expansion.md]] shows how regular-graph expansion transfers to incidence-graph expansion.
So the sourced geometry does support the heuristic that balanced qubit sets should interact nontrivially with many root neighborhoods.
-
However, none of those statements control the local minority-load profile
\[ v\longmapsto m_v(L) \]for an arbitrary balanced qubit cut \(L\).
They do not prove a theorem of the form
\[ H_{\mathrm{light}}(\beta): \quad \forall\,L\subseteq Q,\ |L|\in[\beta|Q|,(1-\beta)|Q|], \ \exists\,F(L) \text{ with } m_v(L)<\delta\Delta \text{ and } \sum_{v\in F(L)}m_v(L)=\Omega(|Q|). \] -
The exact gap is not merely “one needs more crossed neighborhoods.”
The new local theorem [[small-side-local-cut-gives-full-local-cross-rank.md]] applies only when
\[ m_v(L)<\delta\Delta, \]while a full root neighborhood has size \(\Delta^2\).
Therefore linearly many crossed neighborhoods, or even linear total local crossing mass without further distribution control, would still be insufficient for this ingredient unless one also proves that a linear share of that mass occurs on the light side of the threshold
\delta\Delta. -
Incidence expansion does not by itself force that light-side distribution.
Expansion of the incidence graph can at best control aggregate boundary interaction between a qubit set and the rooted neighborhoods. But the functional
\[ \sum_v m_v(L) \]is an oriented minority-load quantity, and the statement needed here is sharper still:
one must lower-bound that quantity on the restricted subfamily with
\[ m_v(L)<\delta\Delta. \]No sourced theorem currently on the graph converts incidence expansion, diagonal expansion, or bounded local overlap into that restricted minority-load lower bound for every balanced cut.
-
Therefore the exact theorem-level gap for this ingredient is:
a local load-profile theorem for balanced qubit cuts in the Quantum Tanner root-neighborhood geometry, forcing linear total minority mass on neighborhoods whose minority side is below the small-side threshold
\delta\Delta.
Equivalently, the present sourced package still stops before the first ingredient of [[lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md]]. It gives bounded local neighborhoods and auxiliary expansion, but not a theorem saying that every balanced qubit cut contains \Omega(|Q|) light minority mass on lightly crossed root neighborhoods.
Dependencies¶
- [[lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md]]
- [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]
- [[local-quotient-image-span-controls-rank-accumulation.md]]
- [[small-side-local-cut-gives-full-local-cross-rank.md]]
- [[quantum-tanner-local-generator-blowup.md]]
- [[quantum-tanner-incidence-spectral-gap.md]]
- [[regular-graph-expansion-to-incidence-expansion.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{light}}(\beta)is false. It isolates only the exact missing theorem not yet on the graph. - It does not claim that incidence expansion is irrelevant. The claim is narrower: current sourced expansion statements do not yet control the minority-load distribution needed here.
- It also does not rule out a different direct local-block route that bypasses light-side mass entirely. The statement is only about this one chosen ingredient of the current conditional criterion.
Sources¶
10.48550/arXiv.2202.1364110.48550/arXiv.2206.0757110.48550/arXiv.2508.0509510.48550/arXiv.2109.1459910.1007/BF02579166