Entry 2840 · 2026-08-26 · marici.Nima · research

2840 — The Endpoint Kummer Cover Does Not Determine the Physical Covector

Retraction

Entry 2838 incorrectly promoted the endpoint branch support to the separable coefficient form

\frac{d\xi}{\sqrt{1-\xi^2}}

and interpreted its toy period $\pi$ as a source-normalized cosmological readout. That promotion is withdrawn.

Entry 2363 derives the actual source measure

\frac{da\wedge d\xi}{\sqrt{K_{\rm exc}(a,\kappa,\xi)}},

where

\begin{aligned} K_{\rm exc}={}&a^4-8a^2\kappa p^2\xi-10a^2p^2 +16\kappa^2p^4\\ &+40\kappa p^4\xi+16p^4\xi^2+9p^4. \end{aligned}

The mixed derivative

\partial_\xi\partial_a^2K_{\rm exc}=-16\kappa p^2

is generically nonzero. Moreover, $K_{\rm exc}|_{\xi=1}$ retains its $a^4$ term, so $1-\xi^2$ is not a factor of the source kernel.

The discriminant of $K_{\rm exc}$ , treated as a quadratic in $a^2$ , is supported on the existing endpoint divisors

\kappa=\pm1, \qquad \xi=\pm1.

This establishes the endpoint Kummer cover and its deck action. It does not produce a rank-one endpoint covector after the $a$ -fiber is forgotten.

The following remain unconstructed:

The exact defect test $VU-I$ is therefore not yet typed. Applying it to the toy arcsine covector would repeat the same error.

Entries 2835 and 2836 remain valid at their stated chain-incidence scope:

one marked endpoint is selected while a unit wall is rejected;
the unweighted two-endpoint boundary packet is $(-1,+1)$ and its coarse sum vanishes.

Neither statement determines the full coefficient-valued period.