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This is Plasma AI's author-recorded calculation, without independent review. It concerns the genuine, unrotated block law and does not construct Sidon pseudo-moments. Its role in the local-repair question is to identify the covariance terms that any proposed high-frequency estimate must control.

Take $s\ge2$ blocks of length $W\ge\max(s,4)$, independently choosing a uniform $s$-subset in each. Put $N=sW$ and $f_j=(s)_j/(W)_j$ for $1\le j\le4$. For $1\le d<W$, write $P_d=\sum_b U_b(d)+\sum_b V_b(d)$, where $U_b(d)$ counts pairs at distance $d$ inside block $b$, and $V_b(d)$ counts those crossing from block $b$ to block $b+1$. There are $s$ terms of the first kind and $s-1$ of the second. Their means are $u_d=(W-d)f_2$ and $v_d=df_1^2$.

For distinct $d,e<W$ define

$$\begin{split} n_{de}&=2(W-\max(d,e))+2(W-d-e)_+,\\ r_{de}&=(e-d)_++\min(W-d,e),\qquad t_{de}=(d+e-W)_+,\\ G_{de}&=((W-d)(W-e)-n_{de})f_4+n_{de}f_3-u_du_e,\\ H_{de}&=2\min(d,e)f_1f_2+(de-2\min(d,e))f_2^2-v_dv_e,\\ J_{de}&=f_1\{r_{de}f_2+((W-d)e-r_{de})f_3\}-u_dv_e,\\ Q_{de}&=f_1^2\{t_{de}f_1+(de-t_{de})f_2\}-v_dv_e. \end{split} \tag{1}$$

Then the covariance is exactly

$$C(d,e)=sG_{de}+(s-1)H_{de} +2(s-1)(J_{de}+J_{ed})+(s-2)(Q_{de}+Q_{ed}). \tag{2}$$

For the diagonal use the same $J,Q$ and replace $G,H$ by

$$\begin{split} G_{dd}={}&(W-d)f_2+2(W-2d)_+f_3\\ &+((W-d)^2-(W-d)-2(W-2d)_+)f_4-u_d^2,\\ H_{dd}={}&df_1^2+(d^2-d)f_2^2-v_d^2. \tag{3} \end{split}$$

To verify (1), two internal edges of distinct lengths have a common vertex in $n_{de}$ placements: equal left or right endpoints contribute $2(W-\max(d,e))$, and a right endpoint equaling the other's left contributes $2(W-d-e)_+$. The other placements have four distinct points. Two edges crossing the same boundary have a common endpoint in $2\min(d,e)$ placements; their other placements use two points in each block. An internal $d$-edge and a crossing $e$-edge with an endpoint in its block share one of its vertices in $r_{de}$ placements. Finally two edges crossing adjacent boundaries have a common endpoint in $t_{de}$ placements in their shared block. Subtract the product of means in each case. On the diagonal, identical edges give the extra terms in (3).

Independence eliminates all disjoint-block covariances. There are $s$ internal blocks, $s-1$ boundaries, two sides of each boundary for an internal-crossing pair, and $s-2$ adjacent boundary pairs. Including both orders gives (2). Reflection of a block makes the internal-crossing covariance identical on its two sides. This proves the formulas.

The two absolute-value terms

For $d\ne e$, expanding the maxima, minima and positive parts in (1) gives

$$C(d,e)=a_0+a_1(d+e)+a_2de +K_D|d-e|+K_A|d+e-W|, \tag{4}$$

for constants $a_0,a_1,a_2$ depending only on $s,W$, with

$$\begin{split} K_D={}&-s(f_3-f_4)-(s-1)(f_1f_2-f_2^2) +2(s-1)f_1(f_2-f_3),\\ K_A={}& s(f_3-f_4)-2(s-1)f_1(f_2-f_3) +(s-2)f_1^2(f_1-f_2). \end{split} \tag{5}$$

For example, the coefficients of $|d-e|,|d+e-W|$ in $n_{de}$ are $-1,1$; those in $r_{de}+r_{ed}$ are $1,-1$. These give the first and third terms of (5); the other two contributions come directly from $\min(d,e)$ and $t_{de}$. Every remaining expression in (1) is a linear combination of $1,d+e,de$. Thus the polynomial part of the matrix has rank at most two, on the span of the constant and coordinate vectors. Equation (4) concerns off-diagonal entries; a diagonal correction from (3) remains.

If $W=s^3+O(s^2)$, then

$$K_D=s^{-6}+O(s^{-7}),\qquad K_A=-s^{-6}+O(s^{-7}). \tag{6}$$

For a direct expansion, write $y=s/W=s^{-2}(1+O(s^{-1}))$. Then $f_2=y^2(1-s^{-1}+O(s^{-3}))$, $f_3=y^3(1-3s^{-1}+2s^{-2}+O(s^{-3}))$ and $f_4=y^4(1+O(s^{-1}))$. In $K_D$ the terms of order $sy^3$ cancel, leaving $y^3(1+O(s^{-1}))+O(sy^4)$; in $K_A$ they leave $-y^3(1+O(s^{-1}))+O(sy^4)$. This proves (6).

The negative effect of the first kernel has an exact finite-interval form. For real $b_1,\ldots,b_{W-1}$ with $\sum_d b_d=0$,

$$\sum_{d,e}b_db_e|d-e| =-2\sum_{r=1}^{W-2}\left(\sum_{d\le r}b_d\right)^2. \tag{7}$$

Express $|d-e|$ as the number of cuts between $d$ and $e$; each cut contributes $2(\sum_{d\le r}b_d)(\sum_{e>r}b_e)$, proving (7). The second kernel in (4) is the first composed with reflection $e\mapsto W-e$. Since reflection commutes with the first kernel, its quadratic form on the zero-sum subspace has absolute value at most the negative of (7), by Cauchy--Schwarz for the positive semidefinite form $-|d-e|$ on that subspace.

Equations (4)--(7) make the interior spectral heuristic concrete. They do not give a high-frequency lower bound for the complete covariance after a Sidon repair. The diagonal, the polynomial part, the long differences, boundary conditions and the unconstructed correction must all be included in such a bound.

The research evidence entry point checks every mean and covariance against a separate complete enumeration of the unrotated sample space at $(s,W)=(2,4),(3,4),(3,6),(4,6)$, in exact rational arithmetic. It also checks the two-kernel decomposition at every off-diagonal entry of these matrices. This is finite supplementary evidence; the general identities rest on the counting argument above.