Write \[ v_j=(s_j,s_{j+1},\ldots,s_{j+423})\in\mathbf F_2^{424}. \] Since the endpoint coefficients are fixed at one, an internal tap set \(T\subseteq\{1,\ldots,599\}\) is feasible exactly when \[ \bigoplus_{j\in T}v_j=v_0\oplus v_{600}. \] The exact verifier forms all 599 internal columns and all 179,101 pairwise xor values. It tests internal subsets of sizes zero through four with singleton and pair tables. For size five, it checks all 35,641,099 triples against the pair table. If the two sides of one of these table identities share an index, cancellation gives a smaller solution already excluded. The sweep finds no solution with at most five internal taps. Every admissible vector has two endpoint taps, so its total weight is at least 8.

For the upper bound, interpret the hexadecimal integer ``` 104020000042401000000000000100000010004020001702093a81120945a122179a133c52f6f5d7860361026a5b18b00f8b704122171b46d22ccedce3191b108a19415c43e921a850ae209 ``` with coefficient \(c_j\) in bit \(j\), starting at the least significant bit. It has 601 bits, both endpoint bits are one, and its Hamming weight is 186. Exact substitution gives zero violations among all 424 recurrence equations.

Row reduction of the 424 recurrence rows together with the two endpoint rows gives rank 426 and affine dimension 175. The exact minimum inside the certified interval \[ 8\leq w_{\min}\leq186 \] remains open. A two-minute pseudo-Boolean solver probe at threshold 150 ended without a model or an unsatisfiability proof and contributes no further bound.

Massey's shift-register synthesis paper gives an iterative solution for the shortest linear feedback shift register generating a prescribed finite sequence. That is the standard finite linear-complexity problem. Its objective is register length. Here the degree is fixed at 600 and the number of nonzero connection coefficients is minimized.

After the two endpoint coefficients are fixed, the present equations ask for a minimum-weight vector with a prescribed binary syndrome. Berlekamp, McEliece, and van Tilborg proved the general decoding and code-weight problems NP-complete. Their result explains why Gaussian elimination settles feasibility and dimension while leaving the minimum-weight search hard. It gives no instance-specific lower bound.

Winterhof surveys linear complexity for several number-theoretic binary sequences, including Legendre, two-prime, Thue-Morse, and prime-omega sequences. The survey gives useful neighboring theory but does not treat the characteristic function of ordinary primes used here. A targeted title, abstract, and citation search on 2026-07-25 found no published computation for this exact 1024-bit prime-indicator prefix, fixed degree, endpoint convention, or tap-weight objective. The novelty claim therefore remains unverified.

The verifier sieves primality through 1025, builds the 424 by 601 binary recurrence matrix, and appends the two endpoint equations. Exact bitwise Gaussian elimination returns rank 426.

For the lower bound, the program stores all singleton and pair xors among the 599 internal columns. It exhausts 35,641,099 triples to finish the size-five test. The 179,101 pair xors are all distinct. No target representation occurs with zero through five internal columns.

The same run decodes the incumbent, checks its endpoints and weight, and substitutes it in all 424 equations. The recurrence row matrix, complete column list, and sorted pair-xor table have separate SHA-256 digests. The canonical report has SHA-256 digest `82e26220eae169ac7f57db344802bc55f25db5e9a2d728e06725e0dc586cd74b`.