Retracted attack result

The proposed Choi--Lam obstruction cancels exactly

The previous weighted-symbol argument omitted the admissible multipliers $O_i=x_i p_i p_4^2$ for $i=1,2,3$ . Their cyclic sum survives the specialization and contributes $R_\perp=t^2(4F-x_4\partial_4F)$ .

Together with $E_4=t^2(F+t^2/2)$ and $R_4=t^2(3x_4partial_4F-t^2)$ , the omitted direction gives

$-\frac{12}{7}E_4+\frac{3}{7}R_\perp+\frac{1}{7}R_4=-t^4.$

The proposed leading obstruction is therefore itself a legal constraint combination. The theorem claim is withdrawn in full. The Choi--Lam case remains open, and the failed numerical Slater point is not evidence.

Attack result

One energy moment cannot control the next kinetic moment

For every $m\ge3$ , take $H=p^2/2+x^{2m}$ . If $\lambda_n\to\infty$ are eigenvalues, fix $E>\lambda_0$ and form the stationary mixtures

$\rho_n=(1-\alpha_n)|\psi_0\rangle\langle\psi_0| +\alpha_n|\psi_n\rangle\langle\psi_n|, \qquad \alpha_n=\frac{E-\lambda_0}{\lambda_n-\lambda_0}.$

At $K=m$ , the only surviving energy-shell row is $L(H)=E$ , so every $\rho_n$ is feasible. The virial identity and Cauchy--Schwarz give

$\operatorname{tr}(\rho_np^4) \ge \alpha_n\left(\frac{2m}{m+1}\right)^2\lambda_n^2 \longrightarrow\infty.$

Since $2\le K-1$ , this contradicts the literal range in Conjecture 1. The degree budget instead forces $k\le K-\deg(V)/2+1$ . For quartics this is exactly $k\le K-1$ , which explains why the paper's proved control and numerical evidence remain untouched. Keren already flags the same low-budget phenomenon for quartics at K=2; the new point is that an analytic family lies inside the broader literal quantifiers.

Exact target

Bound every degree-admissible momentum moment

For a polynomial Schrödinger Hamiltonian $H=\frac12\sum_jp_j^2+V(q)$ that is bounded below with discrete spectrum, let $\mathcal F_K^{\mathrm{SE}}(E)$ be the level- $K$ moment functionals satisfying positivity, stationarity, and every degree-admissible eigenstate relation $L(O(H-E))=0$ .

$\text{Is }\sup_{L\in\mathcal F_K^{\mathrm{SE}}(E)} L(p_j^{2k})<\infty \quad\text{for }k\le K-\frac{\deg V}{2}+1\,?$

Keren formulates the broader literal conjecture after proving that stationarity alone admits an exact spurious moment point in two dimensions. His larger conjecture asks whether sufficiently high eigenstate truncations also force the positional moments into the genuine moment cone.

Known frontier

The energy shell changes the recession geometry

In Keren's two-dimensional quartic control, $\langle p_j^4\rangle$ is unbounded under stationarity alone and bounded after the eigenstate rows are added. The next moment $\langle p_j^6\rangle$ remains unbounded at the same truncation, matching the predicted degree budget.

A proof is known when the shifted quartic potential is a sum of squares. It uses the exact identity

$\frac12\{p_j^2,g^2\} =(gp_j)^*(gp_j)-\frac12\partial_j^2(g^2).$

The conjecture starts where nonnegative polynomials cease to be sums of squares. Schmüdgen's strict Weyl Positivstellensatz does not directly cover this boundary, degree-truncated setting. The bosonic SDP paradox shows why density of Weyl sums of squares cannot by itself make a fixed level exact, while the standard noncommutative convergence theorem assumes the bounded/Archimedean structure absent here.

Intrinsic object

The filtered energy-shell ordered quotient

Let $\mathcal W_d$ be the filtered Weyl *-algebra. Quotient its Hermitian degree- $2K$ piece by the real span $\mathcal R_{K,E}$ of all admissible stationarity and energy-shell relations:

$\mathcal A_{K,E} =\mathcal W_{d,\le2K}^{\mathrm h}/\mathcal R_{K,E}.$

Give it the closed cone generated by the images of $A^*A$ with $\deg A\le K$ . Bootstrap points are normalized positive functionals on this single ordered space. A certificate

$C[1]-[p_j^{2k}]\in\mathcal Q_{K,E}$

bounds the moment for every feasible presentation at once. This remains the right intrinsic formulation, but failure need not be represented by a single linear recession ray: a normalized spectrahedral slice can escape along a nonlinear curve.

Stopping boundary

The intrinsic leading-symbol route is exhausted

The exact cancellation removes the only structural obstruction produced by the weighted filtration. The remaining options are a lower-order moment-matrix analysis or a numerical search for asymptotic feasible points. Neither presently supplies a universal object or exact translation, so the program pauses rather than cataloguing moment configurations.

Hunt calibration

Why this target was selected

The hunt also examined the radial derivative on the graded Möbius algebra, Gromov's open scalar-curvature construction direction, varying-boundary log canonical models, and K-theory of quotient ring spectra. The first and third naturally return to finite classifications if their structural routes fail; the scalar-curvature target lacks a comparably decisive first gate; the K-theory paper substantially answers its own named questions.

Here one universal object retains the information flattened by the usual moment presentation: the energy-shell order itself. Success or failure can be expressed by one exact cone certificate, without a combinatorial residual space.

Research record

The energy-shell quantum bootstrap