Route–T Toolkit Note (v0.3): TR1–TR3 Extraction Module and Referee FAQ for the SAPZ Navier–Stokes Package (v5.6r1)

# Overview

This record releases **Route–T Toolkit Note (v0.3)**, a referee-facing “interface document” that abstracts the
**Route–T transport-bypass** mechanism from the SAPZ Navier–Stokes package (v5.6r1).

It is designed to be **auditable and reusable**:
- TR1–TR3 are stated as a three-step extraction module,
- an explicit **object dictionary** prevents symbol-role confusion (envelope residual vs extraction residual),
- and an **Approximation Bridge (AB)** gives a checklist contract for passing identities from smooth approximants to weak limits.

This note introduces **no new analytic input** beyond the validated statements of the main package.

# Related records

- Main package (v5.6r1): **10.5281/zenodo.18835992**
- Supplementary notes (corollary ledger + suitability/LEI contract, v0.4): **10.5281/zenodo.18843294**

# What is included (v0.3)

Files:
- `SAPZ_RouteT_Toolkit_v5.6r1_v0.3.tex` (and PDF if included)

Contents:
1) **TR1 (Representation)**: transport residual is rewritten as a localized commutator pairing plus named error budgets.  
2) **TR2 (Positivity)**: symmetrization + kernel positivity yield a lower bound by a scale-matched defect  
   \[
   \mathsf{T}_\varepsilon(t;\chi)=\varepsilon^{-5}\int_{B_\varepsilon(0)}\int_{\Omega}\chi(x)\,|u(x,t)-u(x-h,t)|^2\,dx\,dh .
   \]
3) **TR3 (Local-to-sup)**: near-maximizer localization promotes local averages to a sup-type defect.
4) **Template extraction theorem**:
   \[
   (r^{\mathrm{tr}}_\varepsilon(t))_+\ \ge\ c_{\mathrm{tr}}\mathsf{T}_\varepsilon(t)\ -\ \bigl(\mathrm{Err}^{\mathrm{tr}}_\varepsilon(t)+\mathrm{Err}^{\mathrm{pos}}_\varepsilon(t)\bigr).
   \]
5) **Approximation Bridge (AB)**: checklist stability of TR1–TR3 under \(L^2\) strong / \(H^1\) weak convergence.
6) **One-page instantiation snapshot** mapping toolkit labels (A, P, T3, B, LEI) to the v5.6r1 package.
7) **New in v0.3:** a **one-page “Failure modes / Referee FAQ”** plus a **quick audit checklist**.

# What is new in v0.3

- Added a **Failure modes / Referee FAQ (one page)** documenting common objections:
  - symbol-role confusion (envelope residual vs extraction residual),
  - scaling misunderstandings (spurious \(\varepsilon^3\) factors),
  - concerns about hidden regularity/pressure cancellations in positivity,
  - measurability of near-maximizers,
  - approximation-bridge assumptions (no hidden derivatives),
  - explicit error-budget naming and absorption rules.
- Added a **quick audit checklist**: a 5-item referee checklist to mechanically verify TR1–TR3 + AB.

# Label-first convention

References are **label-first**: components are cited by their names (Gate A, CT3 persistence, Route–T extraction, Gate B),
with theorem numbers only as secondary parenthetical metadata. Numbers are **secondary**.

# Scope & non-claims

- This note is not a turbulence phenomenology paper.
- It does not claim Route–T applies verbatim to every transport PDE; it provides an auditable interface designed for adaptation.
- It does not replace the main package; it documents a reusable module extracted from it.

# Author

Lee Byoungwoo