Analyze a simple error-correcting code's decoding

Analyze this fully specified fictional error-correcting code as a mathematical exercise, directly as text. No tools, files, code or external research. Three data bits (a,b,c) are encoded, left to right, as seven bits (a,b,c,a XOR b,b XOR c,a XOR c,a XOR b XOR c), where XOR is addition modulo 2. List all eight data words and their codewords. Define Hamming distance as the number of differing positions and compute the minimum distance between distinct codewords. For received strings R1=1111100, R2=1110100, R3=0111010, show distances to all eight codewords and the COMPLETE set of nearest data words, identifying whether nearest decoding is unique. Give changed bit positions for any unique correction, using positions 1 through 7 from the left. A technician claims: "Because the code corrects one flipped bit, nearest decoding is always unique even after two flips; R2 must be data 101." Audit both claims using the actual candidate sets. Distinguish a valid unique correction under an at-most-one-flip assumption from a guess when that assumption is absent. Do not break ties arbitrarily or infer probabilities not given.

Audit one disputed entry from the previous mathematical answer, directly as text without tools, files, code or external research. Received word R3=0111010 and the codeword for data 100 is 1001011. The previous answer reported their Hamming distance as 6 while also asserting that every two distinct codewords in the code have distance 4. Recompute the distance independently: show positions 1 through 7 from the left, each pair of bits, and whether that position differs; give the complete differing-position set and its count. Decide whether the reported value 6 is correct, explicitly correct it if needed, and explain the consistency with the pairwise-distance assertion. Only audit this entry; do not rewrite unrelated tables.

Continue the project's preceding coding exercise and bitwise audit, using the code definition and results already present in this project's run history. Do not assume the audit's claims are authoritative. For the previously defined R3, identify the nearest data word, its codeword, whether any correction is needed, and its distance to the codeword for data 100. Explicitly recompute that last distance by listing the differing positions, and explain whether the preceding calculation of 4 versus an earlier claim of 6 was correct. Use only the existing source definition; if that definition is unavailable, say so rather than inventing it. Return a concise text answer only; no full table, tools, files or external research.