# Author-contact note DRAFT — DO NOT SEND

**Status:** internal draft only.  
**Flag:** `POTENTIAL_PUBLISHED_MTAU_SIGN_ISSUE=EXTERNALLY_REVIEWABLE` (still **not CONFIRMED**).  
**Do not send** until an explicit human decision. No public accusation. No social-media claim.

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Dear authors,

We have been carefully re-deriving the mixed-time contribution that appears in the Euclidean extended-coordinate expressions for the renormalized stress-energy tensor, starting from the definitions in:

- Taylor, Breen & Ottewill, arXiv:2201.05174v3 (Schwarzschild), and  
- Arrechea, Breen, Ottewill & Taylor, arXiv:2307.10307 (Reissner–Nordström).

We may simply be misunderstanding a convention, and we would be grateful for clarification.

## 1. Green-function definition (as we read it)

Both papers write (up to κ vs κ₊)

\[
G(x,x')=\frac{1}{8\pi^{2}}\sum_{l}(2l+1)P_{l}(\cos\gamma)\sum_{n}e^{in\kappa\Delta\tau}\,g_{nl},
\]

with \(\Delta x\equiv x'-x\) (so \(\Delta\tau=\tau'-\tau\)), Euclidean metric
\(ds^{2}=f\,d\tau^{2}+dr^{2}/f+r^{2}d\Omega^{2}\), \(W=G-K\), and
\(\mathsf{g}_{nl}=g_{nl}-k_{nl}\).

## 2. Δτ convention

We take \(\Delta\tau=\tau'-\tau\) exactly as implied by \(\Delta x\equiv x'-x\).

## 3. Three-line mixed derivative (independent check)

1. \(\partial_{\tau}\partial_{\tau'}\exp[in\kappa(\tau'-\tau)]=+n^{2}\kappa^{2}\exp(\cdots)\).  
2. Euclidean coincidence: \(g^{\tau\tau'}\to +1/f\).  
3. Folding \(\pm n\) into \(n\ge 1\) with prefactor \(1/(8\pi^{2})\) yields

\[
[g^{\tau\tau'}W_{,\tau\tau'}]_{\mathrm{analytic}}
=+\frac{1}{4\pi^{2}}\sum_{l}(2l+1)\sum_{n\ge 1}\frac{n^{2}\kappa^{2}}{f}\mathsf{g}_{nl}
+\frac{1}{4\pi^{2}}\{\mathcal{T}_{10}^{(p)}+\mathcal{D}_{22}^{(-)}+\tfrac{1}{f}(\mathcal{T}_{11}^{(p)}+\mathcal{D}_{21}^{(-)})\}.
\]

## 4. Printed formula

The displayed mixed-time formulae (2022 Schw; 2023 RN with κ→κ₊) read with an overall minus on both the mode sum and the geometric brace:

\[
[g^{\tau\tau'}W_{,\tau\tau'}]
=-\frac{1}{4\pi^{2}}\sum_{l}(2l+1)\sum_{n\ge 1}\frac{n^{2}\kappa^{2}}{f}\mathsf{g}_{nl}
-\frac{1}{4\pi^{2}}\{\cdots\}.
\]

Under the definitions above, that printed RHS equals **minus** the analytic \([g^{\tau\tau'}W_{,\tau\tau'}]\).

## 5. Independent finite-separation check (mode-term sign)

A minimal finite-Δτ evaluation of
\(\lim(1/f)(-\mathrm{d}^{2}G_{\mathrm{mode}}/\mathrm{d}(\Delta\tau)^{2})\)
on synthetic and frozen radial-Green samples **independently reproduces the mode-term sign** (analytic PLUS, opposite the printed overall-minus mode piece), to high relative accuracy. Separately, a source-audited comparison of the local geometric brace finds the same overall-minus relative to the analytic brace under \(W=G-K\); that brace result is not claimed as an independent numerical derivation. The mode-term check uses only the published Green mode expansion and Euclidean raising — not a private renormalization pipeline.

## 6. Ward diagnostic (internal only)

As an internal consistency diagnostic (not a claim about your published tables), flipping only the mixed-time overall sign to the analytic PLUS structure improved a Ward residual and a Howard scoreboard comparison in our assembly. We do **not** treat that as proof of what numerical sign your code used. A displayed-formula issue does not prove that a private numerical implementation used that displayed sign.

## 7. Request

Could you confirm whether the overall minus in the mixed-time formula is intentional under an alternate index / Synge / Wick convention that we have missed, or whether the RHS should carry the opposite overall sign in order to represent \([g^{\tau\tau'}W_{,\tau\tau'}]\) with \(W=G-K\) under the Green, Δτ, and Euclidean metric definitions printed in the papers?

Thank you very much for your time and for these careful papers.

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*Draft only. Filename MUST remain AUTHOR_CONTACT_NOTE_DRAFT_DO_NOT_SEND.md — **DO NOT SEND**.*  
*Hardened 2026-09-15 for pack-hardening-before-author-contact (mode-term vs brace wording).*