External review, round 3 · reviews v5 · verdict: minor issues

# External verification: 06-mark-inertia

- **Subproject:** 08-ilang-propagation
- **Package:** 06-mark-inertia
- **Verified version:** v5
- **External round:** 3 of 3
- **Date:** 2026-10-25T11:28:44+01:00
- **Focus points:** none

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VERDICT: minor issues

## Summary

The second-order dressed-level formula, the inverse-curvature expansion, and the weight formulas are consistent with the stated contract and perturbative counting. The supplied diagonalisation code provides an independent check of the required second-order coefficients, and its outputs support the reported agreement. The remaining problems concern an overstatement of finite-size uniformity and two subleading ring asymptotics; they do not invalidate the leading large-volume common-speed result.

## Issues

### I1. The heavy-body odd-part estimate is not uniform in finite volume
- **Location:** Step 8, Eq. (6.10), first relation and its stated finite-size range
- **Severity:** minor
- **Problem:** The first relation is claimed for every \(L\) and \(\ell\), but its derivation uses an estimate that fails when the uniform mode dominates at fixed \(L\). For example, take the linear grip, fixed \(L\ge32\), \(r\le1\), and fixed heavy \(\varepsilon_k\), then let \(m_h\to0\). The zero mode gives
  \[
  \langle\!\langle1/E\rangle\!\rangle\sim\frac{F_1^2}{2Lm_h^2},
  \qquad R=O(m_h^2).
  \]
  Its lower-branch curvature ratio is \(4\varepsilon_k/(m_h+2\varepsilon_k)\), while its formal upper-branch ratio is \(4\varepsilon_k/(2\varepsilon_k-m_h)\). Consequently,
  \[
  \rho_{k,-}-\rho_k^{\rm e}
  =-\frac{m_h}{\varepsilon_k}+O(m_h^2),
  \]
  rather than the asserted \(2\varepsilon_kR[1+O(\varepsilon_k^{-2})]=O(m_h^2)\). The neglected other-branch contribution therefore cannot be bounded uniformly as claimed.
- **Suggested fix:** Restrict this estimate to the large-volume regime \(L\gg\ell\), with the other stated finite-size conditions, or retain the uniform-mode contribution explicitly. Keep the exact finite-\(L\) formulas (6.3) and (6.6) separate from this asymptotic estimate.

### I2. The first rapidity identity has an overly strong error estimate
- **Location:** Step 6, Eq. (6.7), first relation
- **Severity:** minor
- **Problem:** For the two-quantum contribution, convergence of the limiting rapidity integrals does not imply the stated relative \(O(\ell^{-1})\) error. Configurations with one soft momentum and the other of lattice order produce logarithmic corrections. For example, using centered momenta and \(c_0=1\), the coefficient relevant to this soft–hard contribution contains the nonzero finite part
  \[
  \operatorname{FP}\int_{-\pi}^{\pi}
  \left[
  \frac{q^2}{(2|\sin(q/2)|)^4}
  -\frac{1}{2(2|\sin(q/2)|)^2}
  \right]\,dq
  =\frac{\pi}{6}(4\ln2-1).
  \]
  Integration over the soft partner supplies \(\ln\ell\). Thus the discrepancy can be \(O(\ln\ell)\) against a leading sum of order \(\ell\), rather than having the claimed relative \(O(\ell^{-1})\) bound.
- **Suggested fix:** Replace the relative error in this relation by \(O(\ln\ell/\ell)\), or supply an additional cancellation establishing the stronger estimate. The weaker bound is already absorbed by the later \(O(\delta_\ell)\) remainder, so the leading conclusions need not change.

### I3. The ring heavy-body scale depends on the grip
- **Location:** Step 8, Size of R paragraph
- **Severity:** minor
- **Problem:** The statement that the ring heavy regime “requires \(\varepsilon_k\gg c_0\ell/\ln\ell\)” applies to the one-quantum, linear-grip contribution, not to every grip in (G2). The two-quantum numerator has a soft-momentum contribution of order \(\ln^2\ell\), so
  \[
  R_1=\Theta\!\left(\frac{\ln\ell}{c_0\ell}\right),
  \qquad
  R_2=\Theta\!\left(\frac{\ln^2\ell}{c_0\ell}\right).
  \]
  The supplied Part B output also exhibits this distinction. With a fixed nonzero two-quantum contribution, the condition \(\varepsilon_kR\gg1+r^2\) can therefore be reached on the smaller scale \(c_0\ell/\ln^2\ell\), up to grip-dependent constants.
- **Suggested fix:** State the heavy condition primarily as \(\varepsilon_kR\gg1+r^2\). Identify \(c_0\ell/\ln\ell\) as the linear-grip scale, or as a stronger sufficient condition for the general case, rather than a necessary scale for every grip.

## Focus points

None given.

Author: Ivan Honis · ivan.honis@ndot.io · License: CC BY 4.0 · Human–AI note