Two corrected potential coefficients in SL(2) x R+ exceptional field theory, and the sign of its squared topological coupling

We test the SL(2) x R+ exceptional field theory of Berman, Blair, Malek and Rudolph at nonlinear order in the fields and in sign sectors other than the standard one. For the metric sector we compare the integrated exceptional field theory action with the integrated Einstein-Hilbert action of the corresponding eleven- or ten-dimensional metric. The comparison uses random, non-perturbative, periodic field configurations in which two of the nine external coordinates and the internal coordinates of the section are active, with exact derivatives and spectrally accurate quadrature. We fit, by least squares, the eight coefficients of the part of the scalar potential with derivatives along the M-theory internal coordinates, in the printed basis of terms. The fit returns six of the printed values and differs in two. With the printed index structure, the second dH dH term needs the coefficient -1/2 rather than +1/2, and the coefficient of d ln g d ln M_ss inside the bracket is 3/2 rather than 1/2. The first attribution is a statement about the printed basis: for 2x2 matrices with |det H| = 1 the two dH dH terms obey a linear identity with a third structure, and at quadratic order their integrals are proportional, so the fit separates them only through cubic and higher terms. With these two changes, the metric-sector action (with the Kaluza-Klein vectors as the only vectors switched on) agrees with eleven-dimensional gravity to about 1e-10 at our finest grid, and the mismatch is still decreasing with resolution, in every sign sector tested. With the printed potential it misses by about 1e-2 at every resolution. This is numerical evidence, not a proof, and the plan's convergence criterion for a misprint claim (a plateau on two grids) was not reached. The IIB part of the potential is confirmed as printed. For the form sector we work on the IIB section at the lowest order at which the topological coupling kappa enters, on plane waves with complex momenta. We keep the published transformation rules unchanged. Among the values of kappa tested, the external-diffeomorphism variation vanishes on the duality relation in the cases with (5! 48 kappa)^2 = -s eps and in no others, where s is the sign of the determinant of the external metric and eps the sign of M_ss. Among the values tested, this is also the only one for which the duality relation propagates on the light cone of the metric. Through the IIB dictionary, kappa real corresponds to an odd number of times in the lifted IIB spacetime. This is the known condition for real self-dual five-forms (Hull; Dijkgraaf, Heidenreich, Jefferson and Vafa). What this note adds is that in SL(2) x R+ the sign sector enters through kappa^2 with the transformation rules unchanged, whereas in E6 a transformation rule changes sign. Through the M-theory dictionary, the same condition reproduces Hull's known rule for the sign of the three-form kinetic term in the signatures reachable here (10+1 and 9+2). The M-theory form sector itself was not tested. Earlier work by the author (Zenodo, concept doi:10.5281/zenodo.23028880, any version) stated the two misprints as probable (at quadratic order), the condition on kappa in equivalent notation, the relation between the sign of M_ss and the three-form sign, and the agreement with Hull's rule. Here the two misprints and the condition on kappa are recomputed with new code, at nonlinear order for the potential, and with the invariance of the action checked directly, while the relation eps = lambda sigma and the agreement with Hull's rule are derived, not tested. Nothing here relies on that note. This is a short methods note: an erratum and a nonlinear check of a 2016 construction, and the SL(2) x R+ counterpart of a known E6 sign analysis. The source archive contains the LaTeX source, all scripts, their outputs, a symbolic check of the identities above, the plan (two versions) with the author's own time stamps, and a summary of the post-publication review. run_all.sh reproduces every number, except two machine-precision round-off figures that vary between runs. Version 2 changes only the text, apart from one added symbolic check script; the Note on versions in the PDF summarises the changes and REVIEW_POSTPUB.md in the deposit itemises them. The main changes are a new title, so that the reality condition is no longer presented as a result; attribution of that condition to the literature; narrowed scope statements; and disclosure of further departures from the plan. Version 1 (doi:10.5281/zenodo.23033003) stays available.