External review, round 2 · reviews v3 · verdict: major errors

# External verification: 08-tilt

- **Subproject:** 08-ilang-propagation
- **Package:** 08-tilt
- **Verified version:** v3
- **External round:** 2 of 3
- **Date:** 2026-10-26T17:30:59+01:00
- **Focus points:** none

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VERDICT: major errors

## Summary

The exact symmetry statements, finite-cluster diagonalization, and static response to transverse sources are largely consistent with the stated operators. However, the claimed reduction to two independent anchored oscillator fields for arbitrary push strength is not controlled: eliminating the mixed coordinate–momentum term produces a static correction that does not vanish at small wave number. There are also unresolved convention and scaling qualifications in the R2 coefficients and the fixed-volume expansion.

## Issues

### I1. The mixed push term cannot be discarded solely by derivative counting
- **Location:** Step 6, long-wavelength identification; Result (8.10)
- **Severity:** major
- **Problem:** The estimate \(p\sim\nu q\) used to suppress the mixed term neglects the momentum shift generated by that same term. In the displayed long-wavelength Hamiltonian, the local momentum-dependent contribution, including the bulk kinetic term, is
  \[
  \frac{1+h_p/T_0}{2}(p_1^2+p_2^2)
  -h_p\epsilon_h(q_1p_2-q_2p_1)
  +\frac{h_pT_0}{2}(q_1^2+q_2^2).
  \]
  Completing the momentum square gives a static coordinate coefficient
  \[
  \kappa_{\mathrm{eff}}
  =h_pT_0-\frac{h_p^2}{1+h_p/T_0}
  =\frac{h_pT_0}{1+h_p/T_0},
  \]
  rather than \(h_pT_0\). Thus the omitted term generates a zero-derivative correction of relative order \(h_p/T_0\), not merely a correction of order \(k\). Since the package explicitly does not assume \(\eta=h_p/(JS)\ll1\), its stated independent-copy dictionary and range of validity do not follow. Proper low-mode projection can require further local matching, but does not justify replacing this calculation by \(p\sim\nu q\).
- **Suggested fix:** Eliminate the momenta consistently and match the resulting local coefficients before asserting the oscillator-field identification. Alternatively, impose a weak-push condition \(h_p/T_0\ll1\), and explicitly include the additional error of order \(h_p/T_0\); small wave number alone is insufficient.

### I2. The fixed-volume expansion needs a specified scaling of the push
- **Location:** Step 4, first paragraph and zero-mode discussion; Result (8.7)
- **Severity:** minor
- **Problem:** The stated \(m=S-\Delta S+O(1/S)\) expansion at fixed \(L\) implicitly treats \(\eta=h_p/(JS)\) as fixed. At fixed physical \(h_p>0\), however, the package’s own zero-mode estimate gives
  \[
  \Delta S_{\mathrm{zero}}\simeq
  \frac12\sqrt{\frac{2zJS}{Nh_p}},
  \]
  which grows as \(\sqrt S\). The collective-mode expansion then has a different power counting, and small occupation relative to \(S\) does not by itself establish the advertised \(O(1/S)\) remainder. This does not invalidate the bulk value \(0.0784\) after the thermodynamic limit, but it leaves the claimed fixed-volume expansion and its uniformity insufficiently specified.
- **Suggested fix:** State explicitly whether \(\eta\), rather than \(h_p\), is held fixed in the fixed-volume \(S\to\infty\) expansion. If \(h_p\) is fixed, separate the collective-mode power counting from the bulk expansion and qualify the remainder and order of limits accordingly.

### I3. The closed R2 coefficient table depends on an unstated endpoint convention
- **Location:** Step 11, Eq. (8.17); Result (8.17)
- **Severity:** minor
- **Problem:** A changed link is undirected, but the table uses \(\epsilon_i\), defined as the class of its selected endpoint \(x\). Interchanging \(x\) and \(y\) leaves the contract unchanged while sending both \(\epsilon_i\to-\epsilon_i\) and \(\mathbf e_i\to-\mathbf e_i\). The integral formula is invariant under this replacement, whereas the first two rows of the closed table, written only with \(\epsilon_1\epsilon_2\), are not. Those rows implicitly assume consistently directed parallel link vectors; this convention is not stated as a restriction of the formulas.
- **Suggested fix:** Specify a common positive-axis orientation convention for every link used in the table, or retain the directed-vector factors. For parallel link vectors, the parity-dependent factors should involve \(\epsilon_1\epsilon_2(\mathbf e_1\cdot\mathbf e_2)\), which is invariant under endpoint interchange.

## Focus points

None given.

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