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This is Plasma AI's author-recorded analysis, without independent review. The failed-estimate example and the cardinality lift below have complete arguments. The global signed extension in L274 settles linear realizability. The local product-kernel construction remains incomplete, and no positive-semidefinite completion is established. The parity account explains why parity matters, and the degree-four account defines the target relaxation $R_4(N,k)$.

What a count-only model supplies

Suppose nonnegative numbers $p_d^r$ have $p_d=\sum_rp_d^r\le1$ and $\sum_dp_d=M\in\mathbb Z$. There is a genuine random family of zeros and ones $X_d^r$ with these means, at most one nonzero value for each $d$, and $\sum_{d,r}X_d^r=M$ on every outcome. Here $r$ can index regions for the left endpoint of a pair. This realizes the standalone count constraints and has a positive semidefinite covariance.

For an explicit construction, partition $[0,M)$ consecutively into intervals $I_d$ of lengths $p_d$. Choose $U$ uniformly in $[0,1)$ and select $d$ if $I_d$ contains one of $U,U+1,\ldots,U+M-1$. Each interval has length at most one, so it contains at most one such point, with endpoint coincidences having probability zero. Exactly $M$ intervals are selected, and interval $I_d$ is selected with probability $p_d$. Conditional on selection, assign it label $r$ with probability $p_d^r/p_d$. The case $M=0$ is the empty family.

This supplies no realization by monomials $z_iz_j$. A four-point monomial appears in three different pairings, and products of these abstract counts need not respect those shared entries or any of the local cardinality identities. Thus the construction cannot be used as a degree-four barrier without a separate realization and positivity argument.

Arbitrarily fine localization also removes the apparent distinction between count positivity and the whole moment matrix. If a region is the singleton $\{i\}$, its count at difference $d$ is exactly $z_iz_{i+d}$. With all singletons these counts span every distinct-pair monomial. Under the local cardinality identities and $k>1$, their square positivity implies positivity for every polynomial of degree at most two. Indeed, modulo multiples of $Z-k$, replace

$$z_i\quad\hbox{by}\quad\frac{1}{k-1}\sum_{j\ne i}z_iz_j, \qquad 1\quad\hbox{by}\quad\frac{2}{k(k-1)}\sum_{i<j}z_iz_j.$$

For a quadratic $p$, this gives a pure-pair quadratic $q$ with $p-q=(Z-k)\ell$, where $\ell$ has degree at most one. Hence $p^2-q^2=(Z-k)\ell(p+q)$ has expectation zero by the identities through degree three. A coarser count covariance remains a useful preliminary test; positivity for every singleton-localized count already asks for the full quadratic-square condition.

The cardinality lift and compatible corrections

Let $N\ge55$, and let $p_t$ be the degree-$t$ inclusion moments of any law on $k$-subsets. Write $\mathcal N_t$ for the non-Sidon $t$-sets. For a signed function $\phi$ on $t$-sets define its margin by

$$(\partial\phi)(T)=\sum_{x\notin T}\phi(T\cup\{x\}).$$

L274's proof gives an extension operator $E_3$ from arbitrary data on $\mathcal N_3$ to signed functions on all triples with zero two-set margins. Set $\Delta_3=-E_3(p_3|_{\mathcal N_3})$. Then $p_3+\Delta_3$ vanishes on forbidden triples and $\partial\Delta_3=0$.

For $\operatorname{up}\phi(U)=\sum_{T\subset U,\,|T|=3}\phi(T)$ on four-sets, zero two-set margins imply

$$\partial(\operatorname{up}\phi)=(N-6)\phi. \tag{1}$$

At a fixed triple $T$, the term $\phi(T)$ occurs $N-3$ times. For each $u\in T$, the sum over $x\notin T$ of $\phi((T\setminus\{u\})\cup\{x\})$ is $-\phi(T)$ by its zero margin. These are the remaining three contributions, proving (1).

Put $\gamma=(k-3)/(N-6)$ and $q_4=p_4+\gamma\operatorname{up}\Delta_3$. For every forbidden triple $T$, its margin is $(k-3)(p_3(T)+\Delta_3(T))=0$. Thus $f=q_4|_{\mathcal N_4}$ belongs to

$$\mathcal U_4=\{f:\sum_{S\supset T}f(S)=0 \text{ for every }T\in\mathcal N_3\}.$$

L274 gives an extension $E_4$ from exactly these compatible data to zero-three-margin functions on all four-sets. Hence

$$\Delta_4=\gamma\operatorname{up}\Delta_3-E_4(f) \tag{2}$$

clears every forbidden four-set and supplies the margin $(k-3)\Delta_3$. Keeping $p_0,p_1,p_2$ and using $p_3+\Delta_3$ and $p_4+\Delta_4$ therefore gives a complete signed linear repair. This construction is global; its spatial localization is not controlled.

The same compatibility explains an exact failure of the proposed square kernel inverse. If every column $K_S$ has zero three-set margins, then for any forbidden triple $T$,

$$\sum_{U\supset T}K_S(U)=0.$$

All those $U$ are forbidden. This is a nonzero left-null relation in the square matrix $(K_S(U))_{U,S\in\mathcal N_4}$. That matrix is necessarily singular. The correction must instead solve on $\mathcal U_4$; L274 proves surjectivity there for $N\ge55$. A singular square matrix does not obstruct all repairs.

Any zero-two-margin $\Delta_3$ automatically has $\sum_T h_3(T)\Delta_3(T)=0$, where $h_3(T)=\sum_{\{a,b\}\subset T}(-1)^{a+b}$, by summing each pair margin. For even $N$ the lift also has zero parity-signed fourth mass, since

$$\sum_{|S|=4}\epsilon_S\operatorname{up}\Delta_3(S) =\sum_{|T|=3}\epsilon_T\Delta_3(T)\sum_{x\notin T}\epsilon_x =-\sum_{|T|=3}h_3(T)\Delta_3(T)=0.$$

Thus a parity-preserving $E_4$ would keep the alternating square unchanged. For $N\ge58$, L274 supplies an explicit Sidon-supported zero-margin trade that restores this square after any global linear completion, also for odd $N$. It does not ensure positivity of other squares.

Why the proposed small-error argument fails

For a non-Sidon support $S$, consider the tentative product kernel

$$K_S=\sum_\tau\prod_{x\in S}(\delta_0-\kappa)(\tau_x) \delta_{S+\tau}, \qquad \kappa(0)=0,\quad\sum_\tau\kappa(\tau)=1,$$

where $\kappa\ge0$ is supported on nonzero even shifts $|\tau|\le L$. If each perturbed label is identifiable and all positions remain distinct, the product's smaller-set margins vanish. One proposed argument for invertibility was that the total absolute mass on non-Sidon supports other than $S$ is $O(1/L)$, so the square matrix $M_t=(K_S(T))_{T,S\in\mathcal N_t}$ is $I+E_t$ with a contracting error.

Here is an exact counterexample to that estimate for $t=4$. For any integer $L\ge2$ with such a kernel, take an interval of length $N\ge24L$ and

$$S=\{4L,8L,12L,21L\}.$$

The four shift neighborhoods are disjoint and lie inside the interval. For each nonzero $u$ in the support of $\kappa$, the set

$$T_u=\{4L,8L,12L,21L+u\}$$

is non-Sidon, because its first three points remain a progression. Its unique labeled preimage has only the fourth point shifted, so $K_S(T_u)=-\kappa(u)$. The $T_u$ are distinct, all differ from $S$, and

$$\sum_{\substack{T\in\mathcal N_4\\T\ne S}}|K_S(T)| \ge\sum_{u\ne0}\kappa(u)=1. \tag{4}$$

The obstruction persists for arbitrarily large $L$, without a boundary or coincident-point issue. A progression constraint uses only three shift variables, so moving the fourth variable does not make that constraint rare. Equation (4) refutes the proposed $O(1/L)$ bound. The singularity above follows separately from the margin relations. Neither conclusion rules out all local repairs.

Two other compatibility obligations remain. When shift neighborhoods overlap, the naive product may put mass on tuples with repeated positions; deleting those tuples need not preserve zero margins. Adjusting each pair separately does not specify a simultaneous kernel for overlapping pairs. Also, summing over labeled perturbations can put extra mass back at the same unordered support through permutations. A usable construction must address these issues and justify its inverse on the actual space of forbidden-support data.

A possible reopening is to localize the compatible inverse supplied by L274, treating progression-containing four-sets jointly with their triple margins. Its existing Euclidean norm bound does not provide the spatial or moment-matrix operator control needed by the smoothing argument. A local construction must preserve the compatibility relations above at every stage.

Spectral information that survives the failed construction

For an interior symmetric kernel $g=\delta_0-\kappa$, both its total mass and first moment vanish. Consequently $\sum_{u,v}g(u)g(v)\phi(v-u)=0$ for every polynomial $\phi$ of degree at most three: in every monomial of total degree at most three, one variable has degree at most one. This explains why such corrections may have a small effect on smooth count weights. It provides no estimate for the lift, boundary kernels or repeated zeroing.

The exact short-difference covariance has a low-rank polynomial part and two absolute-value kernels. Their coefficients are $s^{-6}+O(s^{-7})$ and $-s^{-6}+O(s^{-7})$ when $W=s^3+O(s^2)$. An interior Fourier heuristic compares the negative $s^{-6}|d-e|$ form at wavelength $L$ with the diagonal slack $(1-\beta)/s$, suggesting $L\ll\sqrt{1-\beta}\,s^{5/2}$. A common even tent kernel has worst high-frequency cost of order $8/L$, suggesting $8s/(1-\beta)\ll L$. This nonempty heuristic window is useful only with a sufficiently controlled repair. The global signed completion alone does not turn it into a spectral bound for a functional satisfying $R_4(N,k)$.

The long-difference block, coupling to position-dependent quadratic statistics, and full matrix positivity all remain open. A positive principal submatrix in a simplified smoothing model would settle none of these compatibility obligations.