# External Review Email Draft

> **SUPERSEDED (2026-05-30) — do not send as-is.** The three questions this email
> asks have been resolved in-house: Q1 by a sine-kernel/GUE control (observed `D`
> is a z=−0.85 draw), Q2 by proof (an identity), Q3 by a window/unfolding sweep
> (null persists and deepens to 100k), and the residual representation-sector
> question by the Path-(ii) triple desk pre-check. See
> [`EXTERNAL_REVIEW_PACKET.md`](EXTERNAL_REVIEW_PACKET.md) ("Review Questions —
> Resolved In-House") and the closure receipts. This draft is retained as a record
> of the original ask; the framing no longer depends on review, so any future
> outreach would be a finished-result share, not a request for adjudication.

## Short Version

**Subject:** Quick sanity check request: bounded-null read on Riemann zero statistics

Hi [Name],

Could I ask for a small sanity check on a Riemann-zero-statistics note? This is
not a proof attempt and not an RH claim. The point is the opposite: we ran a
Sundog-style structural-zero apparatus against three Riemann-adjacent substrates
and now think the correct conclusion is a bounded null.

The claim I want checked is:

> Across three lanes -- functional-equation Z2 reflection, the linear explicit
> formula, and a nonlinear consecutive-gap S2 reversibility test -- we found no
> structural-zero edge. Each null has a distinct substrate-level reason: Z2
> reflection is identity-zero on the registered features, the explicit formula
> is linear in the test function, and the nonlinear S2 statistic is real but
> pinned to zero in expectation by the GUE / sine-kernel baseline.

The most useful review would be a short "yes, that framing is conservative" or
"no, this overstates X." In particular, I would love your view on three
questions:

1. Does GUE / sine-kernel symmetry already predict reversibility for consecutive
   unfolded gap pairs, making our Probe 05 result (`D=-0.0064`, inside floor
   `0.0424`) just a standard null?
2. Is the C1 explicit-formula diagnosis right: odd-test-function cancellation
   on a symmetric zero multiset is a linearity/identity fact, not a
   structural-zero receipt?
3. Are any of the three nulls better explained as window/unfolding artifacts
   than as substrate-level limitations?

Packet: [link or attachment]

No endorsement requested, and a one-paragraph reply is genuinely enough. A
negative answer is useful here; the goal is to prevent overclaiming before this
goes anywhere public.

Thanks,
[Your name]

## Slightly Warmer Version

**Subject:** Small Riemann-zero sanity check, mostly to prevent overclaiming

Hi [Name],

I have a small, bounded sanity-check request if you have the bandwidth. We have
been testing a Sundog "structural-zero" apparatus against a few Riemann-adjacent
objects. The result is not a positive claim. It is a bounded null: after three
lanes, the conservative conclusion appears to be that this apparatus has no
structural-zero edge on the low-lying zero statistics we examined.

The three lanes are:

- functional-equation Z2 reflection: null by identity-zero on the registered
  gap / absolute-center features;
- explicit-formula parity: null because the explicit formula is linear in the
  test function, so odd cancellation on a symmetric zero multiset is free;
- nonlinear consecutive-gap S2 swap: a real reversibility statistic, but GUE /
  sine-kernel symmetry already predicts zero mean, and the observed value is
  tiny (`D=-0.0064`, floor `0.0424`).

What I want to know is whether that "three lanes, three substrate causes, no
structural-zero edge" framing is mathematically honest. If it is too strong, or
if one of the nulls is better called an implementation/window artifact, I want
to quarantine it before any public-facing page exists.

The review packet is here: [link or attachment]

The packet is written so a 10-minute skim should be enough to answer the main
question. No endorsement requested; even "this is standard, cite X, and don't
call it Y" would be extremely helpful.

Thanks,
[Your name]

## Follow-Up If They Say Yes

Thank you. The most useful path is:

1. Read the load-bearing statement and three-lane table in
   `RIEMANN_BOUNDED_NULL_SYNTHESIS.md`.
2. Check the three questions in `EXTERNAL_REVIEW_PACKET.md`.
3. Reply with any of:
   - "framing seems conservative";
   - "quarantine this";
   - "rename / rephrase this";
   - "cite this standard reference";
   - "I only checked Probe 05 / C1."

I am not asking for a full referee report.

## Follow-Up If They Decline

No worries at all, and thank you for considering it. If there is someone who
would be a better fit for a quick point-process / explicit-formula sanity check,
I would be grateful for a pointer.