Exact moment-kernel determinant on the middle Hamming layer

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Uniform extraction for subsets of this slice follows from hereditary extraction in the whole critical moment family. The determinant calculation below remains valid and does not itself supply slice semistability or any coloring conclusion.

This tests a specific hereditary-extraction attack on the moment boundary. If the middle layer had dissociated density tending to zero, transitivity would rule out uniform extraction for the whole cube. The determinant calculation below does not establish that assertion. Its normalized logarithmic determinant grows, rather than vanishes. No extraction or coloring theorem is inferred from the determinant alone.

1. Finite determinant formula

Let 0<=k<=min(w,d-w), N=binom(d,w), R=binom(d,k), and a_i=binom(d,i)-binom(d,i-1), taking binom(d,-1)=0. Let K_sl be the integer relation lattice of degree-at-most-k moments restricted to the points of weight w. Then

$$\boxed{\det(K_{\rm sl})^2= \prod_{i=0}^{k} \left[\frac{\binom{d-k-i}{w-k}} {\binom{w-i}{k-i}}\right]^{a_i}.} \tag{1}$$

Here the determinant of the rank-zero lattice is one. Let M be the k-subset versus w-subset inclusion matrix. Every degree-j row, j<k, is a rational combination of its rows, since the sum of the rows indexed by k-subsets containing a fixed j-subset is binom(w-j,k-j) times that degree-j row. Thus K_sl=ker_Z M.

The only imported integer-matrix fact is the unimodular diagonal form in Ghorbani et al., Corollary 3 and its proof (arXiv:0709.3144v1). For this substitution of parameters, it gives image index

$$[\mathbb Z^R:M\mathbb Z^N] =\prod_{i=0}^k\binom{w-i}{k-i}^{a_i}. \tag{2}$$

The author and two separate reviewers checked the primary abstract, the theorem's range k<=w<=d-k, and its unimodular proof. A diagonal form suffices for the index; no assertion that these entries are ordered Smith factors is needed.

2. Elementary Gram spectrum

Let V_j be Euclidean space with orthonormal coordinates indexed by j-subsets. The up map U sends a subset basis vector to the sum of its (j+1)-element supersets, and D=U* is the down map. Direct entry counting gives, on V_j,

$$DU-UD=(d-2j)I. \tag{3}$$

In particular U is injective below the middle level, since ||Uf||^2=||Df||^2+(d-2j)||f||^2. Put H_i=ker(D|V_i), of dimension a_i. Successive orthogonal decompositions V_j=UV_(j-1) plus H_j yield V_k=directsum_(i<=k) U^(k-i)H_i. For h in H_i, induction from (3) gives

$$D U^a h=a(d-2i-a+1)U^{a-1}h, \qquad \|U^a h\|^2= \frac{a!(d-2i)!}{(d-2i-a)!}\|h\|^2. \tag{4}$$

Since M^T=U^(w-k)/(w-k)!, the eigenvalues of MM^T are

$$\lambda_i=\binom{w-i}{k-i}\binom{d-k-i}{w-k}, \quad\hbox{with multiplicity }a_i. \tag{5}$$

For any full-row-rank integer M, the primitive-kernel volume identity is det(ker_Z M)=sqrt(det(MM^T))/[Z^R:MZ^N]. It follows by decomposing Euclidean volume along the kernel and the image, just as in the integer-graph calculation. Substituting (2) and (5) proves (1).

3. Critical middle-layer asymptotic

Now take even d, w=d/2, and k=floor(d/2-(1/2)sqrt(d ln d)). Put u=w-k. Reindex i=k-j in (1). The factor becomes

$$Q_u(j)=\frac{(2u+j)!\,j!}{(u+j)!^2}. \tag{6}$$

Give j probability a_(k-j)/R. Its tail telescopes and satisfies

$$\Pr(J\ge L)=\frac{\binom d{k-L}}{\binom dk} \leq\left(\frac{k}{d-k+1}\right)^L. \tag{7}$$

Taking L=floor(sqrt d) makes this bound tend to zero exponentially in sqrt(ln d), while L=o(u). The factors Q_u(j) decrease with j and Q_u(0)=binom(2u,u). For j<=L<=u,

$$\frac{Q_u(j)}{Q_u(0)} =\prod_{a=1}^j\frac{a(2u+a)}{(u+a)^2} \geq\frac{j!}{(2u)^j}.$$

Thus log_2 Q_u(j)=2u-o(u) uniformly on this range: the error is at most O(log u)+j log_2(2eu/j)=o(u). Everywhere 0<=log_2 Q_u(j)<=2u. Equation (7) now shows that its weighted mean is (2+o(1))u. Therefore

$$\log_2\det K_{\rm sl}\sim u\binom dk. \tag{8}$$

The exact ratio to the central binomial coefficient has logarithm -2u^2/d+O(u^3/d^2). For the displayed rounded choice, 2u^2/d=(1/2)ln d+o(1), hence R/N~d^(-1/2). It follows that

$$\boxed{\frac1N\log_2\det K_{\rm sl} \sim\frac12\sqrt{\ln d}\longrightarrow\infty.} \tag{9}$$

The precise equivalent in (9) uses u=(1/2)sqrt(d ln d)+O(1), not merely u~(1/2)sqrt(d ln d). The weaker latter statement allows changes in the binomial ratio too large to retain that constant.

4. Scope

The rank N-R is asymptotic to N, so (9) is also the logarithm of the normalized covolume to leading order. This excludes a vanishing-determinant argument on this slice. It supplies no short-basis estimate and no constant-scale theta upper bound. Global semistability gives only the old constant lower slope on slice sublattices; it does not establish semistability at the slice's larger normalized covolume.

Even if that stronger slice semistability were supplied, the normalized covolume 2^((1/2+o(1))sqrt(ln d)) is still below the order-d threshold of the checked reverse-Minkowski two-color criterion. Thus (9) is not being used as a coloring theorem. The algebra and the asymptotic are author-recorded; a separate reviewer checked them, including the rounding scope. No numerical experiment or external review is claimed.