Write \[ \rho(2,8)=\max_{F:\mathbb F_2^8\to\mathbb F_2}\min_{Q\in RM(2,8)}\operatorname{wt}(F+Q). \] The cubic in rm28-claim-cubic-witness-distance-88 has distance 88 from \(RM(2,8)\), so \(\rho(2,8)\geq 88\). Wang proves \(\rho(2,7)=40\) in Theorem 11 of the cited paper. The recursive inequality \[ \rho(k,m)\leq \rho(k,m-1)+\rho(k-1,m-1) \] and the known value \(\rho(1,7)=56\) give \[ \rho(2,8)\leq 40+56=96. \] Gillot and Langevin's current specialist page records the same interval and labels the second-order case in eight variables as open. A complete answer still needs either a global upper bound of 88 or an exhaustive classification that identifies a larger value in the interval.

Brier and Langevin record the following cubic; this packet checks its distance independently. Use variables \(a,b,c,d,e,f,g,h\), and set \[ F=abc+adg+afh+agh+bdh+beg+ceh+def. \] Split on \(a\). With \(y=(b,c,d,e,f,g,h)\), write \(F=G(y)+aP(y)\), where \[ G=bdh+beg+ceh+def,\qquad P=bc+dg+fh+gh. \] Every quadratic correction has the form \(H(y)+A_0(y)+aA_1(y)\), where \(H\) is homogeneous quadratic and \(A_0,A_1\) are affine. The two affine corrections can therefore be optimized independently on the slices. If \(H\) ranges over the 21-dimensional space of homogeneous quadratics in seven variables, this gives \[ d_2(F)=\min_H\bigl(d_1(G+H)+d_1(G+P+H)\bigr). \] Shifting \(H\) by \(P\) swaps the two summands. Translating a coordinate of \(y\) shifts \(G\) by its coordinate derivative and shifts each quadratic by an affine function, which leaves \(d_1\) unchanged. Thus the pair score is invariant under \(P\) and the seven coordinate derivatives of \(G\). These eight quadratics are independent, leaving a quotient of dimension 13. Exhaustive Walsh-transform evaluation of its 8,192 representatives gives the score histogram \[ \begin{array}{c|rrrrrrr} \text{score}&88&92&96&100&104&108&112\\ \text{count}&28&1016&2968&3024&1092&56&8. \end{array} \] The minimum is 88. The quadratic \(Q=ad+bc+bd+bg\) gives \(\operatorname{wt}(F+Q)=88\) directly, with slice weights 40 and 48. The replay also fixes the truth-table ordering and checks both table digests.

Khoruzhii, Gelß, and Pokutta define \[ \rho_{2,3}(m)=\max_{F\in RM(3,m)}d_2(F) \] and record \(\rho_{2,3}(8)=88\), based on Hou's complete classification of cubic forms through eight variables. Since \(RM(3,8)\) is a subclass of all 8-variable Boolean functions, this value supplies the lower bound \(\rho(2,8)\geq88\). Global equality remains the canonical target. The distinction between the relative maximum and the full maximum prevents the cubic classification from being presented as a complete solution.

The audit checked the IEEE proceedings paper of Brier and Langevin, Wang's final 2019 paper, Gillot and Langevin's 2023 paper and specialist project page, Gao's January 2026 paper, and arXiv:2607.02365v1. Queries included `RM(2,8) covering radius`, `rho(2,8)`, and `second-order nonlinearity 8 variables`, with date filters and targeted searches of arXiv, DOI records, and the Toulon project pages. Brier and Langevin give the displayed distance-88 cubic and the split-by-one-variable quotient method. Wang supplies \(\rho(2,7)=40\), hence the current recursive upper bound 96. The 2023 paper and February 2024 project page display 88 through 96 and call the full case open. Gao's 2026 introduction still cites 88 as a lower bound obtained from the cubic classification. Khoruzhii, Gelß, and Pokutta distinguish the relative cubic value from the full radius in July 2026. The audit located no source that closes the full interval. This is a dated search report rather than an absence theorem.

The model used 37 Boolean coefficients for an arbitrary degree-at-most-two correction and 256 XOR truth coordinates. Coefficients were ordered as the constant, eight linear terms, and 28 quadratic monomials in lexicographic pair order. For each input \(x\), its error bit was the XOR of \(F(x)\) with every selected correction monomial that evaluates to one, and the objective was the integer sum of those 256 error bits. Z3 Optimize minimized this distance under a 120,000 ms timeout. CPython 3.9.6 with z3 5.0.0 returned `unknown` after 120.31 seconds, with reason `sat.canceled`. Its transient lower and upper values, 18 and 256, are solver-state bounds and supply no mathematical distance certificate.

The reusable residue is this 256-row encoding and the observed failure boundary for a direct optimization model. The successful replay exploits slice decomposition, derivative symmetry, and a 13-dimensional quotient. Another solver run is justified after adding comparable symmetry reduction or after replacing Optimize with proof-producing decision instances for successive distance thresholds.

Here \(B(3,4,7)\) means the 7-variable Boolean functions whose algebraic normal forms have no terms below degree three and have degree at most four, viewed modulo \(RM(2,7)\) and under affine equivalence. Reproduce a canonical representative list and certify completeness by orbit-stabilizer counts totaling \(2^{70}\). For every representative, compute second-order nonlinearity exactly, retain full certificates for the high-nonlinearity strata, and compare the resulting cover set with Gillot and Langevin's published project files. The pinned repository has no detected license, so use a clean-room implementation and compare factual outputs until reuse terms are available. The output should include canonical ANFs, stabilizer orders, orbit sizes, exact distances, nearest quadratics, and stable digests.

Stop only after every representative has an exact distance and the orbit mass closes. Then feed the certified high-distance classes into the one-variable split \(F(y,a)=H(y)+aG(y)\) for eight variables. A retry is justified if the affine canonicalization or exact decoder first receives an independent small-case oracle and a proof-producing completeness check. This classification is a bounded input to the global problem, so any resulting upper bound still needs a complete reduction covering every remaining degree stratum.

Join source_lines with LF, append a terminal LF, save the result as `rm28_exact.py` in a disposable directory, and run the recorded command. The program derives the eight-dimensional invariance span, enumerates the 13-dimensional quotient, evaluates both affine distances by exact integer Walsh transforms, and asserts the full histogram. It then evaluates the displayed 8-variable witness and nearest quadratic directly. Truth tables use integer inputs 0 through 255, with bit i assigned to variable i. The independent bitset implementation in rm28-artifact-bitset-crosscheck tests the same result through exhaustive affine masks.

Join source_lines with LF, append a terminal LF, and save the result as `rm28_bitset.py` in a disposable directory. This program builds truth tables as Python integers, obtains affine distance by checking all 256 affine truth masks, and uses a separate echelon-basis implementation for the eight-dimensional invariance span. For each of the 8,192 quotient representatives it also retests the score after every one of the eight span-generator shifts, for 65,536 invariance checks. The score histogram, minimum, direct correction weight, and both truth-table digests match rm28-artifact-exact-cubic-distance.

Join source_lines with LF, append a terminal LF, and save the result as `rm28_full.c` in a disposable directory. The recorded command compiles and runs a clean C11 implementation. It visits all \(2^{21}\) homogeneous quadratics by Gray code and compares each of the two 7-variable slices with every affine function. This route uses no quotient reduction, row reduction, Walsh transform, or Python code. Its full-space histogram is exactly 256 times the quotient histogram in rm28-artifact-exact-cubic-distance, which independently checks the eight-dimensional invariance multiplicity as well as the minimum 88.