files / problems / simon-ionization-excess / compute / q7 / work / r9_cut.md

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Dead line — R=9 with the mass-opt cut

The mass-stationary identities force Q>R/(R+1)Q>R/(R+1) on a used support of aspect ≥R\ge R. At R=9R=9 that cut is 9/10=0.99/10=0.9, so

min⁡(γ9,9/10)≤0.9,1/0.9=1.1111>1.1026.\min(\gamma_9,9/10)\le 0.9,\qquad 1/0.9=1.1111>1.1026.

No compact γ9\gamma_9 can beat the q6 printed leading 1.10261.1026 while the large-aspect bound stays R/(R+1)R/(R+1). The same wall holds at R=8R=8 (8/98/9, leading 1.1251.125). q6 already recorded this wall against 1.10351.1035; it is stricter against 1.10261.1026.

A sharper large-aspect cut (Q>c>0.90699Q>c>0.90699 on aspect ≥9\ge 9) would reopen this line. Two-atom mass-crit still sits near 0.920.92, but that is a search, not a cut.

R=9.8R=9.8 with the cut binding would print 10.8/9.8≈1.1020410.8/9.8\approx 1.10204, which beats 1.10261.1026, but only after a compact γ\gamma that actually clears 9.8/10.89.8/10.8. q6's n=32n=32 prediction sat 2.8⋅10−42.8\cdot 10^{-4} below that cut.

Replay: python3 r9_cut.py.