Uniform full-dissociation extraction at the moment boundary

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Result and standing

Let $d$ run through sufficiently large even integers, and put

$$k_d=\left\lfloor\frac d2-\frac12\sqrt{d\log d}\right\rfloor.$$

Let $E_{d,k_d}$ be the integer column vectors of all squarefree degree-at-most-$k_d$ moments of points of $\{0,1\}^d$.

Theorem. There is a single $\delta>0$ such that every subset $B$ of every one of these configurations contains a fully dissociated subset of size at least $\delta|B|$.

This settles the extraction side of this family. It does not prove that $\chi_1(E_{d,k_d})$ is unbounded. The previously proved eventual lower bound six remains only a fixed lower bound. No negative solution of E0774 follows.

1. The exact lattice data

Write

$$n=2^d,\quad r=\sum_{i=0}^{k_d}\binom di,\quad m=n-r,\qquad D=(d-k_d)\binom d{k_d}.$$

The integer relation lattice $K_d$ has rank $m$, determinant $2^{D/2}$, and is isometric to the integer Boolean polynomial lattice of degree at most $d-k_d-1$. The semistability theorem proves that

$$L_d=\sigma_d^{-1}K_d,\qquad \sigma_d=(\det K_d)^{1/m}=2^{D/(2m)},$$

is stable against all sublattices. The checked boundary asymptotics give

$$r/n\longrightarrow0,\quad D/n\longrightarrow(2\pi)^{-1/2},\quad \sigma_d\longrightarrow \sigma_*=2^{1/(2\sqrt{2\pi})}>1. \tag{1}$$

In particular, the normalized determinant stays uniformly above one.

2. A single fixed-scale theta estimate

Choose any fixed $1<\sigma_0<\sigma_*$. By the weak stable-lattice theorem, choose a fixed $T$ so large that

$$\epsilon(T)\leq\tfrac12\log\sigma_0, \qquad T\geq T_0.$$

Set $s=1/(\sigma_0T)<1$. For every sufficiently large $d$, $\sigma_d\geq\sigma_0$, and monotonicity followed by the theorem gives

$$\theta_{K_d}(s) =\theta_{L_d}(s\sigma_d) \leq\theta_{L_d}(s\sigma_0) \leq(s\sigma_0)^{-m}e^{\epsilon(T)m}.$$

Consequently

$$\begin{aligned} \log s+\frac1n\log\theta_{K_d}(s) &\leq \frac rn\log s -\frac mn(\log\sigma_0-\epsilon(T))\\ &\leq-\frac{m}{2n}\log\sigma_0\\ &\leq-\tfrac14\log\sigma_0=:-b<0. \end{aligned} \tag{2}$$

The sign of the discarded first term uses $s<1$. Both $s$ and $b$ are fixed independently of $d$, of the subset to be extracted, and of any finite prefix.

Thus the former exact cancellation target is now satisfied. The growing-modulus formulation remains an alternative arithmetic description; no separate coding estimate is needed for (2).

3. From theta to hereditary full dissociation

The finite phase-entropy argument proves

$$\mu_T(B_\varepsilon)\leq e^{\pi n\varepsilon^2/s^2}s^n\theta_{K_d}(s),$$

for the actual character-image torus $T$. Choose fixed $\varepsilon>0$ with $\pi\varepsilon^2/s^2\leq b/2$. Equation (2) gives exponential fixed-scale phase packing. The finite Gaussian comparison on that page gives $W(E_{d,k_d})\geq cn$. The single-set inequality

$$\alpha(E_{d,k_d})\geq W(E_{d,k_d})^2/(Cn)$$

then supplies a dissociated subset $F_d$ of size at least $\delta n$, for one positive $\delta$. This precise inequality and the full quasi-independence definition were rechecked in Lefèvre--Li--Queffélec--Rodríguez-Piazza, Theorem 3.4, equation (3.3) (arXiv:0902.2625, printed page 10).

Cube translations permute the columns transitively and preserve every integer relation: substituting $x_i\mapsto1-x_i$ is an invertible integer change on the degree-at-most-$k_d$ moment rows. For any $B\subseteq E_{d,k_d}$, average $|B\cap gF_d|$ over cube translations $g$. The average is $|B||F_d|/n\geq\delta|B|$. Some such intersection has that size and is fully dissociated. There is no restriction on relation length.

Any finitely many earlier admissible stages can be included by decreasing $\delta$ to the minimum of this constant and their positive singleton fractions $1/|E_{d,k_d}|$. Alternatively, start the family at the uniform threshold. Neither choice creates blockwise decay.

4. The remaining required obligation

The exact integer encoding and separated-block assembly already apply to these finite integer-vector sets. To turn them into the requested counterexample, it still suffices—and is still necessary for this proposed family—to prove

$$\sup_{d\ {\rm even}}\chi_1(E_{d,k_d})=\infty.$$

No such theorem is proved here. Affine syndrome classes and fixed checksum palettes have known monochromatic trades, but general colors do not obey those restrictions. The extraction proof does not turn those restricted results into arbitrary-color growth.

The stable-theta argument, its external inputs, and this application chain received separate internal reviews. No new numerical experiment was run.