Semistability and extraction for repeated simplexes

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This proof is recorded by Plasma AI. Its inherited Boolean-compression, weak stable-lattice and finite phase-entropy arguments are consumed at their author-recorded standing. No independent verification is asserted. The statement and proofs of these inherited interfaces were read; the classical source results identified inside them are used as stated, not reproved or independently verified here.

Let $S_q$ be the $q$ columns of $[I_{q-1}\mid-\mathbf1]$, and let $E_{q,k}=S_q^{\otimes k}$. Write

$$N=q^k,\qquad R=(q-1)^k,\qquad m=N-R, \qquad K_{q,k}=\ker(S_q^{\otimes k})\cap\mathbb Z^N. \tag{1}$$

For $q\ge3$ its indexed columns are distinct, nonzero and pairwise non-antipodal: equality or proportionality of nonzero pure tensors would require proportional factors, and no two columns of $S_q$ are proportional. Thus the usual finite integer realization applies.

1. Semistability for power-of-two sides

Theorem. If $q=2^b$, $b\ge1$, and $k\ge1$, every nonzero sublattice $M\subseteq K_{q,k}$ satisfies

$$\det(M)^{1/\operatorname{rank}M} \ge \det(K_{q,k})^{1/m} =q^{\,k(q-1)^{k-1}/(2m)}. \tag{2}$$

Determinants are Euclidean covolumes in the respective real spans. For $q=2$ this is a lattice statement about indexed columns; the configuration used for extraction has $q\ge4$.

Identify the $q$ symbols of each coordinate with a Boolean block of $b$ bits, giving $d=bk$ bits altogether. The map $S_q$ kills exactly the constant coefficient functions on its block. Its tensor product kills the Walsh frequencies belonging to

$$\mathcal F=\{S\subseteq[d]: S\cap B_i=\varnothing\text{ for at least one block }B_i\}. \tag{3}$$

Thus $K_{q,k}$ is the integer lattice in the real span of these Walsh characters. The frequency family $\mathcal F$ is a downset and has size $m$.

We use the exact identities (2), (7), and (8) in the inherited Boolean spectral compression lemma. Their proofs apply to arbitrary spectral families: compressing a family $\mathcal H$ downwards preserves its size and does not increase the covolume of its saturated integer spectral lattice. A downset $\mathcal D$ has

$$\log_2\det K_{\mathcal D} =\sum_{S\in\mathcal D}(d/2-|S|). \tag{4}$$

The restriction to a degree ball in the later part of that lemma is not used here. Since $\mathcal F$ is a downset, compressions starting inside $\mathcal F$ remain inside it.

Mean-degree bound. Every nonempty downset $\mathcal D\subseteq\mathcal F$ has mean cardinality at most that of $\mathcal F$.

To prove this, include each of the $d$ bits independently with probability $t\in(0,1)$, conditional on belonging to $\mathcal F$. The set $T$ of empty blocks is a Bernoulli subset of $[k]$ with parameter $(1-t)^b$, conditional on being nonempty. As $t$ increases, $T$ can be coupled to decrease. Indeed a binomial random variable conditioned to be positive is stochastically increasing in its parameter: the ratio of its probability masses at two parameters is increasing in the size. Couple these sizes monotonically and take initial segments of one uniform permutation of the blocks.

Conditional on $T$, the remaining blocks are independent Bernoulli bit sets of parameter $t$, conditioned to be nonempty. Each of these can be coupled to increase with $t$, by the same conditional-binomial argument and a uniform permutation of the bits. A block leaving $T$ changes from empty to nonempty and also increases. Independent auxiliary randomness in each block gives the required conditional law at every parameter. Consequently the entire conditional bit set increases stochastically with $t$.

It follows that the probability of the downset $\mathcal D$, conditional on $\mathcal F$, decreases with $t$. Differentiating this finite probability at $t=1/2$ gives

$$0\ge 4\Pr_{\mathcal F}(\mathcal D) \left(\mathbb E_{\mathcal D}|S| -\mathbb E_{\mathcal F}|S|\right), \tag{5}$$

where the two expectations are uniform on their indicated families. This proves the bound.

To pass from spectral subspaces to all sublattices, use the minimum-slope argument in Section 3 of the same inherited lemma. The least normalized covolume is attained, since squared integral Gram determinants are positive integers and there are finitely many ranks. Saturation lowers covolume. The determinant inequality for sums and intersections shows that there is a unique saturated minimizer of maximal rank. Cube translations preserve $K_{q,k}$, so they preserve this minimizer. Their Walsh projections show that its real span is some spectral family $\mathcal H\subseteq\mathcal F$.

Compress $\mathcal H$ to a downset $\mathcal D\subseteq\mathcal F$. Equations (4)--(5) give

$$\frac{\log_2\det K_{\mathcal H}}{|\mathcal H|} \ge \frac{\log_2\det K_{\mathcal D}}{|\mathcal D|} \ge \frac{\log_2\det K_{\mathcal F}}{|\mathcal F|}. \tag{6}$$

The full lattice is itself eligible, proving semistability. Finally the complement of $\mathcal F$ consists of frequencies nonempty in every block. Its size is $R$, and its total degree is $k b q(q-1)^{k-1}/2$. The sum of $d/2-|S|$ over the whole Boolean cube is zero. Equation (4) therefore yields

$$\log_2\det K_{q,k}=\frac{k b(q-1)^{k-1}}2, \tag{7}$$

which proves the value in (2).

2. One extraction constant on the critical scale

Fix $c\in\mathbb R$. Let $q$ tend to infinity through powers of two, and, for sufficiently large $q$, put

$$k=\left\lfloor q\bigl(\log\log q+ \log\log\log q+c\bigr)\right\rfloor. \tag{8}$$

All logarithms in (8) are natural. Then

$$\frac RN\longrightarrow0, \qquad \sigma_{q,k}:=\det(K_{q,k})^{1/m} \longrightarrow \exp(e^{-c}/2)>1. \tag{9}$$

Indeed, with $t=k/q$, one has $R/N=e^{-t+o(1)}$ and $t\log q\,e^{-t}\longrightarrow e^{-c}$. Inserting these in (2) proves (9); the floor changes $t$ by less than $1/q$.

Extraction theorem. There is a $\delta(c)>0$ such that every subset of every sufficiently large configuration in (8) contains a dissociated subset of at least $\delta(c)$ times its size.

Here are the exact inherited analytic interfaces. The weak stable-lattice Gaussian theorem states, uniformly in rank, that for a stable rank-$m$ lattice $L$, $\theta_L(u)\le u^{-m}\exp(\epsilon(1/u)m)$ at sufficiently small $u>0$, with $\epsilon(T)\to0$, where $\theta_L(u)=\sum_{x\in L}e^{-\pi u^2\|x\|^2}$. The finite phase-entropy criterion turns a fixed bound $\log s+N^{-1}\log\theta_K(s)\le-a<0$ into a dissociated subset of positive relative size, uniform in $N$. These are consumed at their author-recorded standing and with their stated source premises.

In particular the latter criterion uses Rodríguez-Piazza's single-set bound as stated in Lefèvre--Li--Queffélec--Rodríguez-Piazza, Theorem 3.4, equation (3.3). The source's introduction, following (1.3), states the equivalence of the Rademacher and Gaussian norms; this is the interface permitting the Gaussian-width formulation. Those statements and definitions were read in the held transcription. Their quoted source proofs are not independently verified here.

Apply the criterion to the characters of $\mathbb R^R/\mathbb Z^R$ whose integer frequencies are the columns of $S_q^{\otimes k}$. Their phase image is $\exp(2\pi i V)$, with $V$ the rational row space of that matrix, and its integer annihilator is exactly $K_{q,k}$. The columns are distinct and nonzero, as required by the single-set theorem.

Choose $1<\sigma_0<\exp(e^{-c}/2)$ and a fixed sufficiently large $T$ with $\epsilon(T)\le(\log\sigma_0)/2$. Put $s=1/(\sigma_0 T)$. By (2), the lattice $K_{q,k}/\sigma_{q,k}$ is stable. Monotonicity of theta, followed by the preceding bound, gives

$$\log s+\frac1N\log\theta_{K_{q,k}}(s) \le \frac RN\log s -\frac mN\bigl(\log\sigma_0-\epsilon(T)\bigr) \le-\frac14\log\sigma_0 \tag{10}$$

for all sufficiently large $q$. The criterion supplies one dissociated set $D$ of size at least $\delta N$. Every permutation of the $q$ columns of $S_q$ lifts to an invertible linear map: their only rational relation is their total sum, and this relation is permutation-invariant. Tensor products of these maps act transitively on $E_{q,k}$ and preserve ternary relations. For every $A\subseteq E_{q,k}$, averaging $|A\cap gD|$ gives $|A||D|/N\ge\delta|A|$. Some such intersection is the required dissociated subset. This proves the hereditary assertion with the same constant throughout the family. A finite prefix may be absorbed by decreasing the constant.