  Fix a prime $p$ and a positive integer $R$. We study the property testing of
functions $\mathbb F_p^n\to[R]$. We say that a property is testable if there
exists an oblivious tester for this property with one-sided error and constant
query complexity. Furthermore, a property is proximity oblivious-testable
(PO-testable) if the test is also independent of the proximity parameter
$\epsilon$. It is known that a number of natural properties such as linearity
and being a low degree polynomial are PO-testable. These properties are
examples of linear-invariant properties, meaning that they are preserved under
linear automorphisms of the domain. Following work of Kaufman and Sudan, the
study of linear-invariant properties has been an important problem in
arithmetic property testing.
  A central conjecture in this field, proposed by Bhattacharyya, Grigorescu,
and Shapira, is that a linear-invariant property is testable if and only if it
is semi subspace-hereditary. We prove two results, the first resolves this
conjecture and the second classifies PO-testable properties.
  (1) A linear-invariant property is testable if and only if it is semi
subspace-hereditary.
  (2) A linear-invariant property is PO-testable if and only if it is locally
characterized.
  Our innovations are two-fold. We give a more powerful version of the
compactness argument first introduced by Alon and Shapira. This relies on a new
strong arithmetic regularity lemma in which one mixes different levels of
Gowers uniformity. This allows us to extend the work of Bhattacharyya, Fischer,
Hatami, Hatami, and Lovett by removing the bounded complexity restriction in
their work. Our second innovation is a novel recoloring technique called
patching. This Ramsey-theoretic technique is critical for working in the
linear-invariant setting and allows us to remove the translation-invariant
restriction present in previous work.
