Walkthrough — simon-lieb-thirring

Discovery notes, not a cleaned proof. Beats: refs/walkthrough-style.md.

What was actually missing — After q2, the FHJN handle was closed: Carvalho Corso–Ried had already solved the variational problem, and the Clausen series encloses their printed $1.44655$ without going below it. A new nonnegative pair cannot be the large jump. The leftover degree of freedom was a different method — a conversion that does not go through $\mathcal{C}_1=M_3$ — or a test potential that raises the one-bound-state lower bound $2/\sqrt{3}$ .
Named false starts — Treating another FHJN pair as the work. Replaying Weidl’s 1996 interpolation with the later HLT value $L_{1/2,1}=1/2$ : the Ky-Fan factor $M(1/2)=2$ forces $C(1/2)\approx 2.660$ , and the converted ratio is $3.8385$ . The $C=1$ Hadamard envelope $\approx 1.44287$ looks like a dent until one notices Weidl states it only for potentials proportional to a characteristic function, where $\lvert\int V^{1+i\tau}\rvert$ cannot oscillate below $\int V$ . Seiringer–Solovej remainder absorption at $d=1$ is an Airy ground state and gives $R_1\approx 0.132$ , ratio $2.75$ , weaker than Rumin $1/5$ . Neumann covering at $\gamma=1$ with the Poincaré partition $\alpha=3$ is already $1/\sqrt{3}$ on a constant well, ratio $2.72$ . Bachmann–Froese–Schraven (arXiv:2403.19023) bound atomic Hamiltonians in terms of the landscape function, not the Euclidean $\gamma=1$ constant.
The useful failure — Two copies of the Keller well, separated by $4$ , give a discrete ratio $1.1507$ with two bound states, just below $2/\sqrt{3}\approx 1.154701$ . That is the Frank–Gontier–Lewin picture in the window where the one-bound-state case is still supposed to win: the interaction does not raise the ratio. Hermite and finite-well blocks have $T/\int\rho^3$ between $2.72$ and $3.37$ , above the $1.35$ that would push Eden–Foias past CCR, but DLL’s Cauchy–Schwarz on the projection kernel still only proves $\kappa\ge 1$ . The empirical gap is the original problem in another set of clothes.
The click — Open the conversion, not the trial pair. FHJN Proposition 10 / CCR (1.13) is the same $(9\sqrt{3}/4)M_3$ factor, and $M_3$ is now a Clausen value. Every other closed 1D conversion sitting on this desk (Weidl interpolation, SS Airy absorption, Neumann covering, Eden–Foias $\pi/\sqrt{3}$ ) is larger than $1.44655$ . The leftover is not a tighter Clausen tail. It is a method that keeps a remainder FHJN discarded, or a proof that $T\ge\kappa\int\rho^3$ for some $\kappa>1.35$ , or a potential with ratio $>2/\sqrt{3}$ .
The argument — Weidl (32)–(34) interpolates the $\ell^p$ sequences of eigenvalues between $\gamma=1/2$ and $\gamma=3/2$ by a $K$ -functional. Plug in HLT $1/2$ and Laptev–Weidl $3/16$ , evaluate $\Theta(\eta,1,2)$ and $\Theta(\eta,1/2,3/2)$ at $\eta=1/2$ by a regularised Simpson rule, and take the Ky-Fan minimum $M(1/2)=2$ . Python and rustc agree: $C(1/2)\approx 2.660352$ , $L/L^{\mathrm{cl}}\approx 3.83854$ . Independently, Seiringer–Solovej Corollary 2 at $d=1$ is $R_1=(-3/a)^3/16$ with $a$ the largest real zero of $\mathrm{Ai}$ . A power series for $\mathrm{Ai}$ changes sign on $(-2.3381074105,-2.3381074104)$ ; even the most optimistic $R_1$ in that bracket is $0.13202$ , ratio $2.752$ . The Neumann covering, after rescaling each trial so that $l\int V=3$ , never beat the constant well. The test-potential search never beat Keller.
Computer search — compute/q3/weidl_interp.py and verify_alt.rs (different Simpson $n$ ) for the interpolation. ss_airy.py and the same Rust file (different $\mathrm{Ai}$ term count, wider bracket) for $R_1$ . covering_local.py: $400$ random histograms on $24$ bins plus spikes and ramps, all rescaled to $\alpha=3$ . trial_potential.py: exact Pöschl–Teller $\nu=1,2,3,4$ , a scan of square wells, two-sech wells, a $400$ -step Gaussian-sum walk, a $500$ -step histogram walk. ef_ratio.py: Hermite $n=1,2,3,5,8$ and finite-well eigenblocks. run_all.sh finishes by replaying the q2 Clausen envelope. No certificate claims a number below $1.44655$ .
Proven vs still open — Proven here: the listed alternative conversions sit strictly above CCR, and this search did not raise the one-bound-state lower bound. Still open: $L_{1,1}=L_{1,1}^{\mathrm{So}}$ (ratio $2/\sqrt{3}$ ). A number below $1.44655$ still needs a method that is not FHJN/CCR, not Weidl interpolation, not SS absorption, and not a Poincaré covering at $\gamma=1$ .