Let \(M\) be the requested maximum. The support \[ \{0,45,49,53,94\} \] has the inverse displayed in `wfci127-claim-weight-85-witness`, so \(M\geq85\). The argument in `wfci127-claim-complement-upper-bound` proves \(M\leq101\). Therefore \[ 85\leq M\leq101. \] Both endpoints are odd, as every inverse of an odd-weight polynomial evaluates to 1 at \(x=1\). The exact maximum remains open in this fixture. Its complete normalized search space has \[ \binom{126}{4}=10{,}009{,}125 \] supports.

Set \[ f=1+x^{45}+x^{49}+x^{53}+x^{94}. \] In hexadecimal coefficient encoding, with the coefficient of \(x^i\) in bit \(i\), ``` f = 0x400000000022200000000001 f^{-1} = 0x3bf17593bf175d3fb175d3fb377f3fb3 ``` The inverse has 85 set bits. Exact carryless cyclic multiplication gives \[ f\,f^{-1}\equiv1\pmod{x^{127}+1}. \] The executable record repeats this check using integer bit operations. It also verifies the equivalent complement identity.

Write \[ J=1+x+\cdots+x^{126}. \] For every weight-five \(f\), cyclic multiplication gives \(fJ=J\), because each coefficient of \(fJ\) is the parity of five. Suppose \(fg=1\), and set \(h=J+g\). Then \[ fh=fJ+fg=J+1, \] whose weight is 126.

If \(t=\operatorname{wt}(h)\), the product \(fh\) is the xor of \(t\) cyclic shifts of the five-term polynomial \(f\). Hence \[ 126=\operatorname{wt}(fh)\leq5t, \] so \(t\geq26\). Also \(f(1)=1\) and \(fg=1\) imply \(g(1)=1\), making \(\operatorname{wt}(g)\) odd and \(t=127-\operatorname{wt}(g)\) even. It follows that \[ \operatorname{wt}(g)=127-t\leq101. \] For the weight-85 witness, \(h\) has weight 42 and the verifier checks \(fh=J+1\) exactly.

Misoczki, Tillich, Sendrier, and Barreto formulate QC-MDPC McEliece systems with binary circulant blocks and identify the circulant algebra with a quotient polynomial ring. Their work explains why sparse binary cyclic polynomials and invertibility arise together in code-based cryptography. Guo, Johansson, and Stankovski state the rank and coprimality criterion for a binary circulant block in their reaction-attack analysis.

Those sources address code construction, rank, and security. The audited passages do not give the maximum inverse weight for normalized weight-five units at length 127, nor an exhaustive table that contains it. A targeted exact computation is still needed to close the interval in this fixture.

The program represents a residue by a 127-bit integer and multiplication by \(x^k\) as a cyclic rotation of its coefficient word. It verifies the displayed inverse directly. Setting \(h=J+g\), it also checks \(fh=J+1\), \(\operatorname{wt}(g)=85\), and \(\operatorname{wt}(h)=42\).

The eight-line standard output has SHA-256 digest `19774a855b737562d11bd1d30f217e6f0edee0756b2d4668c37a001a0d57161c`.

The program below is a replayable exhaustive algorithm. It enumerates the 10,009,125 four-subsets of \(\{1,\ldots,126\}\). For each support it applies polynomial extended Euclid against \(x^{127}+1\), reduces a unit's Bezout coefficient cyclically, and records its inverse weight. A SHA-256 stream covers one canonical line per support, including nonunits.

This full sweep was not executed for this fixture. Its output and checksum are therefore deliberately unspecified. The completed certificate is the one-support verifier above, and the exact extremum remains between 85 and 101.