Exact Cubic Dictionary Capacity
Rank-One Symmetric Cubics over Dyadic Rings

A Sharp Capacity Theorem

Let Rₘ=Z/(2ᵐ) and consider symmetric rank-one cubic atoms v^{⊗3} with v∈Rₘⁿ. This chapter determines the maximum size of a free Rₘ-linearly independent dictionary made entirely from such atoms:

The upper bound comes from reduction modulo two: every cubic evaluation on a binary vector collapses to a square-free monomial whose support has size one, two, or three. The lower bound is attained by an explicit family indexed by those same supports. For n=8 the exact capacity is

This is a dictionary-capacity theorem, not a claim that every cubic tensor has rank 92 or that arbitrary 92-component decompositions are identifiable.

The Question

Because Rₘ is not a field, ordinary vector-space dimension is not the right language for a family of atoms. We call atoms C₁,…,C_r free independent if

Define C₃(n,m) as the largest possible cardinality of a free independent set of atoms v^{⊗3}. The theorem below shows that the answer is independent of m.

Cubics Become Square-Free Supports

Reduce a rank-one atom modulo two. For x_i∈F₂,

Therefore a symmetric cubic coordinate depends only on the set of distinct indices appearing in it. A degree-three monomial can have support size one, two, or three. The entire parity image of all cubic atoms is contained in a binary space of dimension at most

Why N₃(n)+1 Cannot Survive

Take more than N₃(n) cubic atoms. Their reductions modulo two are linearly dependent, so there is a nonzero α∈F₂ʳ with

The sum on the left has only even entries. Multiply it by 2^{m−1}:

The coefficient vector is nonzero in Rₘ. Hence the original atoms are not free independent. Therefore

The Bound Is Sharp

Index atoms by every nonempty subset S⊆{1,…,n} of size at most three. For m≥2 define the dense vector

For each support R of size one, two, or three, inspect one cubic coordinate carrying exactly support R. Modulo two the resulting square matrix is

Order the subsets by size. Then R⊆S implies |R|≤|S|, and equal sizes force R=S. Thus Z is triangular with diagonal one. Its determinant is odd, so the corresponding minor is a unit in every Z/(2ᵐ). The cubic atoms are therefore free independent.

Exact Cubic Dictionary Capacity

The 92 Result

The result is an 11.5× free overcomplete dictionary relative to the base vector dimension:

This number is a capacity of a chosen rank-one dictionary, not the symmetric rank of a generic cubic.

Dependence Is 2-Adic

The upper bound gives a dependency for every set of 93 atoms. There are also compact explicit witnesses. Fix four coordinates Q. Consider the 15 nonempty subsets S⊆Q and their binary support atoms. Every square-free monomial of support at most three is contained in an even number of those subsets, so

Thus torsion produces exact ring dependencies that ordinary real-rank intuition would miss.

Capacity and Identifiability Are Different

Chapter 23 proves global recovery for a highly structured r=n+1 simplex family. This chapter asks only whether a fixed dictionary of rank-one cubics can be free independent. The capacity can be much larger than an identifiable unknown-atom family:

Exact Integer Regression

The accompanying verifier constructs the explicit subset family, checks the binary zeta minor for n=1,…,8, verifies its full rank, and confirms the n=8 capacity 92. It also constructs exact parity dependencies that lift to Z₂₅₆ by multiplication by 128.

What Is Earned