Lesson 08 - Local-Block Rank Accumulation

Goal

Understand the most concrete Quantum Tanner intrinsic bottleneck: linearly many crossed local blocks must survive as independent quotient images in the global stabilizer quotient.

Prerequisites And Diagnostic Checks

1. What is a local generator block \(H(v)\)?
2. What does it mean for a local block to be crossed by a qubit cut?
3. What is the quotient \(S/(S_L+S_R)\)?
4. Why can many local objects fail to give many independent global objects?

Concrete Motivation

The Quantum Tanner construction has local tensor-code blocks around root neighborhoods. A balanced qubit cut should cross many of these neighborhoods. It is tempting to conclude:

many crossed local neighborhoods imply large global cut rank.

The local graph says this is not automatic. The right question is not only whether a block is crossed. It is whether its cross-cut stabilizer class survives in the global quotient and remains independent of other surviving classes.

Worked Example Before Abstraction

Suppose ten local blocks each produce one cross-cut class. If all ten classes become the same vector in the quotient \(S/(S_L+S_R)\), the global cut rank contribution is \(1\), not \(10\).

There are two losses:

1. survival loss: a local class may die in the global quotient;
2. overlap loss: surviving local classes may span a smaller space than their count.

This is exactly why bounded overlap of neighborhoods is not enough by itself.

Formal Objects

The key nodes are:

• quantum-tanner-needs-balanced-local-block-rank-accumulation.md
• local-quotient-image-span-controls-rank-accumulation.md
• lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md
• local-geometry-stops-before-uniform-linear-light-side-mass.md
• plain-incidence-local-geometry-route-is-exhausted-before-minority-load-anti-concentration.md

Let \(S=\operatorname{rowspan}H\) and \(S(v)=\operatorname{rowspan}H(v)\). For a cut \(L\sqcup R=Q\), set:

\[ B(L)=S_L+S_R. \]

The local quotient image is:

\[ W_v(L)=\frac{S(v)+B(L)}{B(L)} \le \frac{S}{B(L)}. \]

The global cut rank is exactly:

\[ \lambda_{M(H)}(L) = \chi_L(S) = \dim\left(\sum_v W_v(L)\right). \]

So the true bottleneck is the dimension of the span of the \(W_v(L)\), not the raw number of crossed neighborhoods.

Local Mass Versus Global Rank

The graph distinguishes:

• local crossing mass \(\mu_H(L)\);
• surviving image mass \(\sigma_H(L)\);
• packed direct-sum surrogate \(\nu_H(L)\);
• actual global rank \(\lambda_{M(H)}(L)\).

The desired theorem would control both survival and overlap:

\[ \lambda_{M(H)}(L)\ge c\,\mu_H(L) \]

for every balanced cut.

The current graph does not have this theorem.

Conjecture Linkage

This is the most concrete bridge from Quantum Tanner construction data to the intrinsic cut-rank route. If it closed, the path would be:

1. balanced cuts cross many local blocks;
2. many crossed blocks survive independently in \(S/(S_L+S_R)\);
3. \(\chi_L=\Omega(n)\) for every balanced cut;
4. stabilizer cut-rank functional yields the hardware lower bound;
5. submodular CD becomes large for the family.

What This Does And Does Not Prove

This proves:

• the exact quotient-image formula for local-to-global accumulation;
• the right decomposition of the bottleneck into survival and overlap losses;
• why raw local geometry is insufficient.

This does not prove:

• linear local light-side mass for every balanced cut;
• linear independence of quotient images;
• dense tangle breadth;
• full CD(T_n,G) semantics.

The computational small-instance evidence in the nodes is useful for intuition, but it is not a family theorem.

Active Recall

1. Define \(W_v(L)\) in words.
2. What are survival loss and overlap loss?
3. Why is \(\mu_H(L)=\Omega(n)\) not enough?
4. What would \(\nu_H(L)=\Omega(n)\) mean?
5. How would a local-block theorem connect to Conjecture 3?

Next-Step Handoff

Next lesson: switch from intrinsic matroid closure to the compiler-semantics frontier, where hidden auxiliary variables can represent some functions but still fail to mean routing.