Let \(M\) be the maximum in the candidate question. The point set certified in this record has 44 graph edges and four triangles. Its one-skeleton is connected. Exact elimination over \(\mathbb F_2\) gives \[ \operatorname{rank}\partial_1=19,\qquad \operatorname{rank}\partial_2=4, \] and hence \[ \beta_1=44-19-4=21. \] This raises the checked lower bound from 6 to 21.

Every clique complex is a flag complex. Theorem 10 and Corollary 11 of Beers and Botnan show that the largest first Betti number of any flag complex on 20 vertices is attained by the balanced Turán graph \(K_{10,10}\). Its clique complex is the graph itself and has \[ \beta_1=(10-1)(10-1)=81. \] The Hamming Rips complexes in this problem form a subclass, so \(M\leq81\). Combining both certificates gives \[ \boxed{21\leq M\leq81}. \] The exact value remains open. The witness is a strict optimum under replacement of any one selected word, but that neighborhood certificate does not exclude a better set reached through two or more simultaneous replacements.

Two fixed-seed simulated-annealing runs proposed 2,600,000 one-point moves in total. They began with the candidate's beta-one 6 set and later restarted from improved incumbents. The best verified value rose successively through 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, and 21. The second run proposed 2,100,000 moves without improving 21. These heuristic totals do not count distinct 20-subsets and carry no global exclusion force.

The executable artifact replaces the heuristic's final neighborhood check with an exact one. Every changed one-point replacement is inspected, and each connected neighbor has beta one at most 20.

Beers and Botnan prove the sharp extremal result for all flag complexes on 20 vertices, giving 81. Their theorem ignores Hamming realizability. No primary source located in this audit treats the exact induced 20-point problem in \(\{0,1\}^8\). Closing the remaining gap calls for a global branch-and-bound, an integer or semidefinite certificate that incorporates boundary rank, or a Hamming-specific homological inequality.

Use Hamming distance at most two on the following lexicographically ordered words: ``` 00000011 00000111 00001101 00010010 00010110 00011100 00100011 00101001 00110010 00111000 11000011 11000111 11001101 11010010 11010110 11011100 11100011 11101001 11110010 11111000 ``` The newline-separated point list has SHA-256 digest `a883d10312be85b66176b6c5f7063d7da5ca5074e31ab61adc83ad0b179ac7bf`. There are eight distance-one edges and 36 distance-two edges. The canonical 44-edge list has digest `d253d61bcc4dcdb3e0ffbf26613eda3d5c75e74e9d3aea36fde64cb8cf485596`.

The only triangles, in zero-based point indices, are \[ (0,1,6),\ (3,4,8),\ (10,11,16),\ (13,14,18). \] Their edge-boundary columns are `45`, `9800`, `128000000`, and `5800000000` in hexadecimal when edges are ordered lexicographically. Each has a different leading bit, so the four columns are independent. The verifier also constructs the vertex-edge boundary matrix and row reduces both matrices over \(\mathbb F_2\). The complete canonical certificate has SHA-256 digest `680e1588932fa8c474296aa5aa86b524cbbd22e5058213fa26d3237d34797a33`.

For the local optimization audit, remove each selected word in turn and insert each of the 236 unselected words. This gives \(20\cdot236=4{,}720\) changed sets. Exactly 3,520 are connected. Their beta-one histogram is 96 at 16, 1,392 at 17, 1,664 at 18, 356 at 19, and 12 at 20. None reaches 21, so the witness is a strict one-point-replacement optimum. The neighborhood report digest is `ea087a3b6c2151c3a982db846b3839254a1127cd6c55ea99a3e5c049b80f4963`.