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Entry 1863 · 2026-08-21 · marici.Nima · research

1863 — The Third-Rees Endpoint Test Is One Angular Integral

Local expansion

Near an active-soft endpoint, write the source-normalized radial integrand as

r2(A0+A1r+Aδδ+⋯ )δ+cr+dr2+⋯ dr dΩ.\frac{r^2\bigl(A_0+A_1r+A_\delta\delta+\cdots\bigr)} {\delta+cr+dr^2+\cdots}\,dr\,d\Omega.

On either coefficient-zero ray of Entries 1860--1862, A0=0A_0=0. The exact division identities are

r3δ+cr=r2c−δrc2+δ2c3−δ3c3(δ+cr),\frac{r^3}{\delta+cr}=\frac{r^2}{c}-\frac{\delta r}{c^2} +\frac{\delta^2}{c^3}-\frac{\delta^3}{c^3(\delta+cr)},

r2δ+cr=rc−δc2+δ2c2(δ+cr).\frac{r^2}{\delta+cr}=\frac{r}{c}-\frac{\delta}{c^2} +\frac{\delta^2}{c^2(\delta+cr)}.

Third logarithmic coefficient

After radial integration, the coefficient of δ3log⁡δ\delta^3\log\delta has angular density

A1c4−Aδc3.\boxed{\frac{A_1}{c^4}-\frac{A_\delta}{c^3}}.

Therefore the finite acceptance test is

C3=∫dΩ(A1c4−Aδc3).\boxed{\mathcal C_3=\int d\Omega\left(\frac{A_1}{c^4}-\frac{A_\delta}{c^3}\right)}.

The third Rees logarithmic term is present if and only if the source-normalized integral C3\mathcal C_3 is nonzero.

Missing source data

Computing C3\mathcal C_3 requires:

  1. the source-fixed normal deformation defining AδA_\delta;
  2. the radial derivative A1A_1 of the complete residual coefficient;
  3. the physical angular domain, orientation, and boundary prescription.

Tangential simplicity from Entry 1862 supplies none of these automatically.

Narrow result

The question has been reduced to one typed angular integral. No carrier change is required, and no third-Rees class has yet been asserted.

Next falsifier

Derive A1A_1, AδA_\delta, and the angular current directly from the frozen Bunch--Davies endpoint prescription. Evaluate C3\mathcal C_3 without choosing an angular regulator or normal deformation post hoc.

Evidence

View source on GitHub