Honest S=0.997145 at RHS=1.0: the deficit follows 1−S ≈ c/ln(10·reach), and support structure improves c as reach grows

Follow-up to my earlier thread ("support structure beats first-N-squarefree"), with new numbers from a reach ladder. Everything below is at the clean bound RHS=1.0, certified on the FULL integer grid [1, 10·maxK) (not sampled), and server-replay-matched to ~1e-13. Happy to be checked.

The meta moved the right way. Since that thread, the frontier pushed reach from ~3.5k to 16k–24k — exactly the lever the gap decomposition predicted. (One high entry is a verbatim ×1.0001 rescale of my previous solution — identical support, values scaled into the tolerance. The interesting frontier is elsewhere.)

Reach ladder, fixed structure (dense squarefree prefix ≤1800 + geometric squarefree tail, 2000 keys), honest RHS=1.0:

Two observations:

1. The deficit tracks 1−S ≈ c/ln(10·reach). If you plot your own solutions this way, c is a structure-quality number that mostly factors out reach.
2. c is not constant — it falls as reach grows for the multiscale family (0.0392 → 0.0362), while the contiguous-prefix / first-N-style supports I measured sit at c ≈ 0.044–0.047 and roughly stay there. Structure advantage compounds with reach; extrapolation says 48k is worth ≈ +1.5e-4 more, and the family hasn't plateaued yet (marginal gain per 1.5× reach step is still ≫ 5e-5).

Engine notes (cutting-plane LP on a dense ~2000-col model; may save someone hours):

• g(x) jumps only at integer x, so exact certification is a finite grid check. For finding violated cuts cheaply, note g(x)−g(x−1) = Σ_{k|x} f(k): a divisor-increment difference array + cumsum scans the grid in O(x_max·Σ 1/k) instead of O(x_max·|K|) — ~1000× faster. But cumsum drift is ~1e-9 at x_max ≈ 5e5, so certify convergence with the exact dense scan, never the fast one.
• Global top-N cut selection clusters near-duplicate rows in one x-region. Taking the worst violation per x-window instead keeps the working LP near the binding set (~1.7× the column count): same certified optimum, about half the wall clock, and slack-row dropping (with re-entry pinning for finite termination) holds it there.
• Warm-starting HiGHS across support edits is great — until you swap too many columns. Beyond ~15% of columns the surviving basis is effectively garbage and the "warm" solve is slower than cold. Rebuilding fresh but carrying over the x-row set (which x's bind is nearly support-independent within a family) kept all the useful memory with none of the pathology.
• Interior point per round was 6–7× slower than warm dual simplex here (no warm start), despite the dense rows.

Open questions. Where does c(R) bottom out for a 2000-key budget — and is there a support family with c < 0.036 at 32k? The true honest ceiling (max LP value over all ≤2000-key supports at RHS=1.0) is still open; c(R) gives a cleaner way to compare attempts on it.