q1 — trial pairs for the FHJN functional C_1

Frank–Hundertmark–Jex–Nam, arXiv:1808.09017, Lemma 11 / Proposition 10. An admissible pair $(f,\varphi)$ with $\int f^2=\int\varphi=1$ gives an upper bound on $\mathcal{C}_1$ , and at $d=1$

\frac{K_1}{K_1^{\mathrm{cl}}}\ge\frac{16}{243\,\mathcal{C}_1^2},\qquad \frac{L_{1,1}}{L_{1,1}^{\mathrm{cl}}}\le\frac{9\sqrt{3}}{4}\,\mathcal{C}_1.

The 2018 FHJN theorem is $L_{1,d}/L_{1,d}^{\mathrm{cl}}\le 1.456$ . That conversion drops below $1.456$ precisely when $\mathcal{C}_1<1.456\cdot 4/(9\sqrt{3})\approx 0.373610$ . The later published record is Carvalho Corso–Ried $1.44655$ ; a pair below $1.456$ is not a beat of that number.

Certified pair

opt_best_A.json / certs/c1_opt_best.json:

f(t)=(1+\mu t^{\alpha})^{-\beta},\qquad \varphi(t)=c\frac{(1-(t/S)^{\gamma})^{\delta}}{(1+\varepsilon t)^{\kappa}}\,1_{t\le S}

with $(\alpha,\beta,\gamma,\delta,\varepsilon,\kappa,S)$ as in the JSON and $\mu,c$ fixed by the two normalisations. Python panel bound $\mathcal{C}_1\le 0.3735378$ ; Rust independent bound $\mathcal{C}_1\le 0.3735472$ . The number both support is $\mathcal{C}_1\le 0.373548$ , hence $L/L^{\mathrm{cl}}\le 1.45576$ .

The paper's second pair is replayed in certs/c1_lemma11_second.json. Its panel bound does not convert below $1.456$ .

Replay

./run_all.sh

float_c1.py prints a trapezoid estimate. It is not a bound.