Six-mark sampling and a conditional polar packing

The normalized nodes are $z=(-A,-B,-s,s,B,A)$, with $A=s+\varphi^2$, $B=s+1$, and $s=\sin(\varepsilon/2)$ at the canonical one-arcsecond seed. The algebra is also stated for the family $s>0$. That family does not change the canonical seed.

On $\mathcal P_5$, the Vandermonde matrix $V$ with $V_{ij}=z_i^j$ satisfies $y=V\beta$. For $s>0$ the nodes are strictly ordered, and

$\det V=8ABs(A^2-B^2)^2(A^2-s^2)^2(B^2-s^2)^2.$

Every factor is positive, so sampling is an isomorphism onto $\mathbb R^6$. The degree-six polynomial that vanishes on all six marks shows that the same samples do not determine an unrestricted field. At $s=0$ the central marks coincide and the inverse fails.

The bound $|y_i|\le 1$ is an additional choice. The original marks do not prescribe it. It defines

$K_\Omega=\{\beta:\lVert V\beta\rVert_\infty\le 1\}=V^{-1}[-1,1]^6.$

With the coefficient pairing, the polar is $K_\Omega^\circ=V^TC_1$, where $C_1$ is the cross-polytope of vectors with $\ell_1$ norm at most one. Ordinary six-dimensional volume then gives

$|K_\Omega|\,|K_\Omega^\circ|=4^6/6!=256/45.$

The determinant cancels. The same product holds for any six distinct real nodes under the same sample bound. It does not single out the golden nodes. The map $(\beta,h)\mapsto(V\beta,V^{-T}h)$ pulls the canonical symplectic form back to itself. That is an abstract phase space. It is not a physical model.

The selected shares are $\lambda=\varphi/(\varphi+\pi)$ and $\mu=\pi/(\varphi+\pi)$. They are not the rational register $Q/S$.

Hypothesis A-087 is a reported polar-product width statement: the interior polar product has Gromov width 4. Hypothesis A-343 is a reported ball-packing criterion in dimension at least 3. This note does not establish either hypothesis. The pinned proof files were not retrieved here, and neither statement was executed.

Under those two hypotheses, the note assigns capacities $\theta R\lambda$ and $\theta R\mu$ with R strictly between 0 and 4 and with a common factor strictly between 0 and 1, and concludes that the corresponding closed 12-dimensional balls embed disjointly into the interior polar product. The exact split at factor one fails the strict pair inequality in the intermediate ball. That obstruction is about the ball target. It is not a direct nonexistence proof inside the polar product.

Appendix A works in $\mathbb Q(\varphi)$ with $\varphi^2=\varphi+1$. It does not evaluate $\sin(\varepsilon/2)$. Python 3.10.21 produced the same four lines in normal and optimized execution, exit code 0:

s=1/7: exact algebra checks PASS
s=1/2: exact algebra checks PASS
s=1/648000: exact algebra checks PASS
s=0: excluded singular boundary PASS

The supplied note reports an earlier run on Python 3.12.14. That environment was not reproduced here. The four lines above are this archive's run.

Not a source paper, not peer review, and not an amendment of Paper VII or frozen SRSD. Six samples do not reconstruct an unrestricted field. The volume identity does not give the golden nodes a unique status among distinct-node samplers. No Lean, Lake, or language conformance is claimed. The irrationality-exponent remark in the note is not used above.

Related, and separate: exploration 02 and the sampling extension.