lemmaroutine

Kernel of block restriction map

Restriction to

U_A

vanishes only on the line

A=Fz

under a quadratic condition

bounds rank loss per level in the detection proof

Lemma·lem:latent-block-kernel·§3.1 The quadratic block

Statement (paper)

Let

S\le B^*

and let

\gamma:S\to X^*

be binary linear. For each line

A

, define

\kappa_{A,\gamma}:S\longrightarrow U_A^*, \qquad \kappa_{A,\gamma}(z)=(\gamma_z,z)|_{U_A}.

Under the trace identification, write

\gamma_z(x)=\operatorname{Tr}(g(z)\cdot x)

for the associated binary linear map

g:S\to F^3

. For every nonzero

z\in S

\kappa_{A,\gamma}(z)=0 \quad\Longleftrightarrow\quad A=Fz \quad\text{and}\quad z\cdot g(z)=\sqrt{z_1z_2z_3}.

In plain English

For a binary linear map

\gamma:S\to X^*

with

S\le B^*

, the restriction

\kappa_{A,\gamma}(z)=(\gamma_z,z)|_{U_A}

vanishes at a nonzero

z\in S

exactly when the line is

A=Fz

and

z\cdot g(z)=\sqrt{z_1z_2z_3}

, where

g:S\to F^3

represents

\gamma

via the trace.

Why it's here

This pins down exactly when a pair

(\gamma_z,z)

dies on a block

U_A

: only on the single line

A=Fz

and only when a quadratic condition on

g(z)

holds. Using the block structure from the previous lemma (the trace annihilator of

D_A

A

and

D

vanishes on

A

), it lets later arguments count or detect the kernel of the restriction to each block, which feeds the rank-drop estimates that use it.

Classified routine:

A direct computation from the block structure in lem-latent-block: since the trace annihilator of D_A is A and D vanishes on A, the kernel condition reduces to A=Fz plus a quadratic identity. This is linear-algebra bookkeeping inside the paper's own construction.

Uses

lemStructure of the line blocks U_A

Used by

claimExpected harmonic rank drop per level

Sources

PaperOpen

Section: §3.1 The quadratic block
Page: —
Label: lem:latent-block-kernel
Line: sections/02-latent.tex:176

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