For \[ R_A(t)=\mu(A\cap(A+t)), \qquad M(A)=\sup_{1/4\leq t\leq1/2}R_A(t), \] the two certified bounds are \[ \frac16\leq C=\inf_{\mu(A)=1/2}M(A)\leq\frac3{16}. \] The lower bound follows already at \(t=1/3\). The upper bound is attained by the displayed 16-cell set in `lshac-claim-sixteen-cell-upper-bound`. Their gap is \[ \frac3{16}-\frac16=\frac1{48}. \] This falls short of the candidate's requested \(10^{-8}\) gap. The exact value remains unresolved in this entry.

Let \[ A_0=A,\qquad A_1=A+\frac13,\qquad A_2=A+\frac23 \] and define the integer-valued multiplicity \[ N(x)=\mathbf1_{A_0}(x)+\mathbf1_{A_1}(x)+\mathbf1_{A_2}(x). \] For \(N\in\{0,1,2,3\}\), the pointwise inequality \[ \binom N2\geq N-1 \] holds. Integration gives \[ \sum_{0\leq i<j\leq2}\mu(A_i\cap A_j) =\int_{\mathbb T}\binom{N(x)}2\,dx \geq\int_{\mathbb T}(N(x)-1)\,dx =\frac12. \] Translation invariance and the evenness \(R_A(t)=R_A(-t)\) make all three intersections on the left equal to \(R_A(1/3)\). Therefore \[ 3R_A(1/3)\geq\frac12, \qquad R_A(1/3)\geq\frac16. \] Since \(1/3\in[1/4,1/2]\), every admissible set satisfies \(M(A)\geq1/6\).

Partition \(\mathbb T\) into the half-open cells \[ I_i=[i/16,(i+1)/16),\qquad i\in\mathbb Z/16\mathbb Z, \] and take \[ A=I_0\cup I_1\cup I_2\cup I_3\cup I_4\cup I_5\cup I_7\cup I_{10}. \] Then \(\mu(A)=8/16=1/2\). At the grid shifts \(j/16\), for \(j=4,5,6,7,8\), exact cyclic intersection counts are \[ 3,3,3,3,2. \] Thus the corresponding overlaps are those counts divided by 16.

For a general shift \(t=(j+u)/16\), where \(j\) is an integer and \(0\leq u\leq1\), each shifted cell overlaps cells at the two neighboring grid positions. Consequently \[ R_A(t)=\frac{(1-u)c_j+u c_{j+1}}{16}, \] where \(c_j\) is the cyclic intersection count at shift \(j\). The autocorrelation is linear between consecutive grid shifts. Its maximum on \([1/4,1/2]\) occurs at a grid endpoint and equals \(3/16\). Hence \(C\leq3/16\).

The function \(R_A(t)=\mu(A\cap(A+t))\) is the covariogram, also called set covariance or indicator autocorrelation. The covariogram literature located in this audit concentrates on reconstructing convex bodies from the full function and on its geometric regularity. Averkov and Bianchi define the same translate-intersection functional and solve Matheron's planar convex-body reconstruction conjecture. Their hypotheses and conclusion do not provide an extremal bound for arbitrary measurable subsets of the circle.

Popular-difference theorems seek shifts with large intersections or richer additive patterns, commonly allowing the shift to come from a neighborhood or a structured set. The present problem restricts the shift to the closed arc \([1/4,1/2]\), which stays away from zero. The targeted search found no source determining this constant or closing the interval \([1/6,3/16]\).

This audit covered covariogram, set covariance, autocorrelation of indicators on compact groups, popular differences, and prescribed difference intervals. It records search coverage rather than a claim of novelty.

The program represents a cyclic half-set by a bit mask. For every grid shift \(j\), the intersection count is the population count of the mask intersected with its cyclic rotation by \(j\). It verifies the five construction counts and uses the linear interpolation formula to certify the continuous supremum.

It also enumerates every half-subset for grid orders 4, 8, 12, 16, and 20. The exact discrete optima are respectively \[ 1/4,\quad2/8,\quad3/12,\quad3/16,\quad4/20. \] These discrete lower bounds apply only to unions of cells on the named grid. The enumeration gives no rounding theorem for arbitrary measurable sets.

For each order, the artifact sorts all optimizing index tuples and hashes their newline-delimited representation. The eight-line standard output has SHA-256 digest `2dceff6f0feed5733f8c45f87f2a24cc126a203d79abc1373a797ff3667fe868`.