External review, round 1 · reviews v2 · verdict: major errors

# External verification: 10-field-own-contract

- **Subproject:** 05-ilang-free-fall
- **Package:** 10-field-own-contract
- **Verified version:** v2
- **External round:** 1 of 2
- **Date:** 2026-10-11T14:17:49+02:00
- **Focus points:** none

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VERDICT: major errors

## Summary

The compression, leading second-order on-site correction, and first hopping correction are largely consistent with the stated Schrieffer–Wolff construction. However, the corrected body contract is not necessarily in the positive-mass class claimed, even within the stated fast-field regime. The independent-cell fall calculation also covers only an additional linear-profile subclass, not the arbitrary cell data requested. Some accuracy and validity statements need narrower qualifications.

## Issues

### I1. Corrected on-site masses need not be positive
- **Location:** Step 2(b), Eq. (10.11), and Step 3(c)
- **Severity:** major
- **Problem:** Membership in \(\mathcal C_{\rm lev}\) requires \(M_{x,k}>0\), but R1 does not ensure
  \[
  N(x)M_k+\gamma_x^{(3)}M_k^3>0.
  \]
  For example, take
  \[
  C_f=\operatorname{diag}(0,\Delta),\qquad
  \varphi_0=|0\rangle,\qquad
  \Phi_x=\begin{pmatrix}n&v\\v&n+a\end{pmatrix},
  \quad a<0.
  \]
  Then \(N(x)=n>0\), \(\gamma_x^{(3)}=av^2/\Delta^2<0\), and all three perturbative parameters can be small. Nevertheless, choosing \(n<|a|v^2M_0^2/\Delta^2\) makes the retained corrected mass negative on every level. Thus the claimed classification with first corrections is false for allowed members, and \(V_{x,k}+M_{x,k}\) is then not the upper on-site eigenvalue.
- **Suggested fix:** Retain the unconditional classification only at leading order in \(\epsilon_2,\epsilon_3\). For the corrected classification and “upper eigenvalue” interpretation, impose and state a positive-mass margin relative to the corrections and remainder; otherwise explicitly identify failure of the class condition.

### I2. Arbitrary cell data are replaced by linear cell profiles
- **Location:** Step 4(c), Eq. (10.20), and its control discussion
- **Severity:** major
- **Problem:** The question allows arbitrary positive \(\Delta_x\) and arbitrary nonnegative \(\varepsilon_x\). Step 4(c) instead assumes
  \[
  \gamma_x=\gamma_0+\gamma_1x,\qquad \rho_{xx}=\rho_0+\rho_1x,
  \]
  an additional restriction not implied by \(n_x=1+gx\). Without it, the effective scalar potential and hopping generally do not have the linear profiles required by (4.16), and the finite difference called \(\gamma_1\) need not be constant. For instance, constant \(\Delta_x\) and \(\varepsilon_x^2/\Delta_x=\gamma_0+\delta(-1)^x>0\) are admissible but have alternating gradients. The assertion that any position dependence of \(\gamma_x\) gives a nonzero single \(\gamma_1\) and the stated drift acceleration is therefore unjustified.
- **Suggested fix:** State explicitly that (10.20) and the associated independence conditions apply only to linear, weakly varying cell profiles. Address the remaining cell descriptions separately, or explain why the quoted linear-profile drift result cannot settle their fall; do not present the linear calculation as covering every cell description in scope.

### I3. A sufficient gap criterion is presented as necessary
- **Location:** Setup and assumptions, A third number of the class; Regimes; final Open issues bullet
- **Severity:** minor
- **Problem:** Small \(\epsilon_3\) and the R1 inequality are sufficient norm conditions for the stated perturbative construction, not necessary conditions for spectral separation or an effective description. Consequently, “without it the two diagonal blocks … are not separated” and “where … ≥1 … the expansion fails” are too strong. A sharp field remains exactly reducible even with arbitrarily large \(\nu\); large excited-sector terms can also increase rather than close the relevant gap. Additionally, the stated limit \(\|\hat C-e_0\|\to\tau+N_{\max}M_{\max}\) is an upper bound, not generally an equality.
- **Suggested fix:** Replace these claims by “separation is not guaranteed” and “the displayed norm-based expansion is not controlled.” Write the body norm estimate with ≤, and preserve the explicitly identified sharp-field exception.

### I4. Clock accuracy does not include the retained cubic correction
- **Location:** Step 3(c), Eqs. (10.15)–(10.16), and the clock row of Step 5
- **Severity:** minor
- **Problem:** Equation (10.15) retains the first \(\epsilon_3\) correction, but the clock accuracy range in (10.16) controls only the quadratic term \(-\gamma_xM_k^2\). It therefore supplies a leading-order-in-\(\epsilon_3\) range, not a range for all terms retained in Step 3. This matters especially for an interior field eigenvalue: \(\gamma_x\) can cancel to zero while \(\gamma_x^{(3)}\neq0\), so the “all λ” case based on \(\gamma_x=0\) does not survive the cubic correction for three or more levels. The independence of neighboring spacings is likewise assessed only after dropping that correction.
- **Suggested fix:** Qualify (10.16) and its summaries explicitly as leading order in \(\epsilon_2,\epsilon_3\), with the quoted local-clock approximations. Add an order estimate for the cubic spectral-curvature error and state the spacing-independence conditions at the retained order, including cancellation cases.

## Focus points

None given.

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