files / problems / simon-ionization-excess / compute / q8 / 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>1.1021.\min(\gamma_9,9/10)\le 0.9,\qquad 1/0.9=1.1111>1.1021.

No compact γ9\gamma_9 can beat the q7 printed leading 1.10211.1021 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). q7 already recorded this wall against 1.10261.1026; it is stricter against 1.10211.1021.

A sharper large-aspect cut (Q>c>0.90741Q>c>0.90741 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 still prints as 1.10211.1021. R=9.9R=9.9 binding would print 10.9/9.9≈1.1010110.9/9.9\approx 1.10101, but only after a compact γ\gamma that clears 9.9/10.99.9/10.9 and a printed leading below 1.10211.1021.

Replay: python3 r9_cut.py.