Let \[ I_n(z)=\sum_{k=0}^{2n}i_{n,k}z^k \] be the independence polynomial of \(C_4\mathbin{\square}P_n\). Exact integer computation gives \[ i_{n,k}^2\geq i_{n,k-1}i_{n,k+1} \] for every \(1\leq n\leq3000\) and \(1\leq k\leq2n-1\). The sweep checks exactly \[ \sum_{n=1}^{3000}(2n-1)=9{,}000{,}000 \] inequalities. All are strict.

The independent masks in one four-cycle column are \[ 0,1,2,4,5,8,10. \] Cycle symmetry reduces their terminal polynomials to \(a_n\) for the empty mask, \(b_n\) for each singleton, and \(c_n\) for each opposite pair. Starting with \[ (a_1,b_1,c_1)=(1,z,z^2), \] the exact transfer is \[ \begin{aligned} a_{n+1}&=a_n+4b_n+2c_n,\\ b_{n+1}&=z(a_n+3b_n+c_n),\\ c_{n+1}&=z^2(a_n+2b_n+c_n), \end{aligned} \qquad I_n=a_n+4b_n+2c_n. \] The program checks this reduction against the full seven-state transfer through \(n=20\).

At \(n=1\), the least margin is 14 at \(k=1\). For every \(2\leq n\leq3000\), the least margin occurs at \(k=2n-1\), where the adjacent triple is \[ \bigl(4(n^2+n-2),4n,2\bigr) \] and the margin is \(8n^2-8n+16\). The requested lengths \(3001\) through \(10000\) remain unchecked.

Calkin and Wilf build independent sets in a rectangular grid by adjoining compatible column masks. Their Section 1 defines the mask set and the zero-intersection compatibility matrix. The transfer in this record adds a fugacity \(z^{|m|}\) for each mask and uses cyclic adjacency inside the four-vertex column. Their paper studies total counts and the hard-square constant, rather than coefficient log-concavity for the cylindrical width-four strip.

Chudnovsky and Seymour prove that every claw-free graph has a real-rooted independence polynomial, which implies log-concavity by Newton's inequalities. For \(n\geq2\), \(C_4\mathbin{\square}P_n\) contains an induced claw centered at a vertex with two cycle neighbors and one path neighbor. The theorem therefore supplies no certificate here. The obstruction is visible at \(n=2\): \[ I_2(z)=1+8z+16z^2+8z^3+2z^4 \] has discriminant \(-26624\), hence one conjugate pair of nonreal roots.

Focused searches for the exact Cartesian product, cylindrical grid independence polynomials, and grid-strip coefficient log-concavity found the two methodological sources listed here and no published result covering the stated finite range. Publication novelty remains unverified.

The program implements the three-state transfer displayed in the claim. A separate seven-state calculation uses the masks \([0,1,2,4,5,8,10]\), admits consecutive masks exactly when their bitwise intersection is zero, and matches every reduced-transfer row through \(n=20\).

For each length, the program computes every integer margin before advancing the transfer. It records the least margin and verifies the observed closed form for that minimum. The SHA-256 summary covers one line per length containing \(n\), the minimizing index, the margin, and its adjacent coefficient triple. Eight complete coefficient rows are serialized with fixed-width length prefixes and hashed separately.

The reproduced run took 131.73 seconds on the entry-research host. Raising `LAST_N` continues the same exact calculation. Runtime grows steeply because both the number and bit length of the coefficients increase. A run through 10000 should use checkpointing or a faster big-integer backend.