files / problems / simon-ionization-excess / compute / q5 / 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>1.1057.\min(\gamma_9,9/10)\le 0.9,\qquad 1/0.9=1.1111>1.1057.

No compact γ9\gamma_9 can beat the q4 printed leading 1.10571.1057 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).

A sharper large-aspect cut (Q>c>0.9045Q>c>0.9045 on aspect ≥9\ge 9) would reopen this line. Two-atom mass-crit and forced-endpoint kk-atomic sit well above 0.920.92, but that is a search, not a cut. q4 moment notes already left the abstract (Q,D)(Q,D) region without a pen-and-paper improvement of R/(R+1)R/(R+1).

Replay: python3 r9_cut.py.