files / problems / simon-ionization-excess / compute / q13 / work / r9_cut.md

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Dead line: R≤9R\le 9

The mass-opt identities give Q>R/(R+1)Q>R/(R+1) on a used support of aspect RR. At R=9R=9

min⁡(γ9,9/10)≤0.9,1/0.9=1.1111>1.1010.\min(\gamma_9,9/10)\le 0.9,\qquad 1/0.9=1.1111>1.1010.

No compact γ9\gamma_9 can beat the q11 printed leading 1.10101.1010 while the large-aspect bound stays 9/109/10. q7--q11 already recorded this wall against 1.10261.1026, 1.10211.1021, 1.10201.1020, 1.10171.1017, 1.10131.1013, and 1.10101.1010.

R=8R=8 binds at 1.1251.125. R=9.5R=9.5 binds at 1.10526>1.10101.10526>1.1010. 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 1.1021>1.10101.1021>1.1010. R=9.9R=9.9 binding would print 1.101011.10101, which still prints as 1.1011>1.10101.1011>1.1010.

A sharper large-aspect cut would reopen R≤9R\le 9. Chebyshev D⋅M−1≥1D\cdot M_{-1}\ge 1 does not.