Retraction: the proposed Choi–Lam dual obstruction is false

Retraction

The theorem claim previously recorded in this file is withdrawn. The weighted-top-symbol calculation omitted three admissible stationarity directions. Once they are restored, the proposed contradiction disappears by an exact identity.

No dual-certificate obstruction, primal unboundedness result, or counterexample to Daniel Keren’s quartic conjecture follows from the earlier argument.

The omitted directions

Keep the notation

$$F=x_1^2x_2^2+x_2^2x_3^2+x_3^2x_1^2+x_4^4 -4x_1x_2x_3x_4$$

and specialize the weighted top symbol at

$$p_1=p_2=p_3=0,\qquad p_4=t.$$

The earlier audit retained the energy multiplier $p_4^2$ and the stationarity multiplier $x_4p_4^3$. It incorrectly asserted that these were the only surviving directions.

For each $i=1,2,3$, the degree-four multiplier

$$O_i=x_i p_i p_4^2$$

is invariant under the even sign-change group and is allowed at $K=3$. Its top Poisson symbol survives because differentiating $O_i$ with respect to $p_i$ removes the momentum that is set to zero:

$$\{H,O_i\}_{\mathrm{top}}\big|_{p=(0,0,0,t)} =x_i\partial_iF\,t^2.$$

Cyclic averaging therefore supplies the additional stationarity direction

$$R_\perp=t^2\sum_{i=1}^3x_i\partial_iF =t^2(4F-x_4\partial_4F),$$

where Euler’s identity for the quartic $F$ was used.

The two directions already retained were

$$E_4=t^2\left(F+\frac12t^2\right)$$

from the energy row, and

$$R_4=t^2(3x_4\partial_4F-t^2)$$

from the $x_4p_4^3$ stationarity row.

Exact cancellation

These three legal constraint directions satisfy

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

Indeed, the $t^2F$ coefficient is

$$-\frac{12}{7}+4\frac{3}{7}=0,$$

the $t^2x_4\partial_4F$ coefficient is

$$-\frac{3}{7}+3\frac{1}{7}=0,$$

and the $t^4$ coefficient is

$$-\frac{12}{7}\frac12-\frac17=-1.$$

Thus the entire proposed leading obstruction lies in the span of admissible constraints. The five-monomial SOS argument never becomes applicable.

Consequence for the research route

The Choi–Lam quartic remains a natural non-SOS test, but the weighted leading filtration does not decide it. The attempted numerical Slater point was also invalid: an independent reconstruction found equality residuals above 5,000.

Continuing from here would require a lower-filtration SDP analysis or an explicit asymptotic moment construction. No intrinsic reduction is currently known, and that work would naturally become the moment-configuration search the program was designed to avoid. The route should therefore pause and return to HUNT.

Sources and status

• Daniel Keren, Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone, arXiv:2608.07047v2.
• M.-D. Choi and T.-Y. Lam, Extremal positive semidefinite forms, Math. Ann. 231 (1977), 1–18.

The retraction identity is exact and independently reproduced by the accompanying rational-arithmetic script. It corrects an FPRD research note that was live briefly and was never a Publication.