External review, round 1 · reviews v1 · verdict: major errors

# External verification: 11-bound-pair

- **Subproject:** 05-ilang-free-fall
- **Package:** 11-bound-pair
- **Verified version:** v1
- **External round:** 1 of 2
- **Date:** 2026-10-11T14:04:36+02:00
- **Focus points:** none

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VERDICT: major errors

## Summary

The fibre contract, the simple-eigenvalue derivative identities, and the leading upper-band bound-state formulas are largely correct, and the supplied numerical checks support those formulas in the tested regimes. However, the claims of contract-split invariance are inconsistent with M6, the treatment of degenerate bound levels does not cover the stated scope, and the clock calculation does not attain the first-order rate accuracy of the cited moving-clock formula. There are also two smaller errors concerning a subleading binding contribution and an exceptional total label.

## Issues

### I1. Incorrect claims of contract-split invariance
- **Location:** Step 2, Eqs. (11.3)–(11.4), and Result item 1
- **Severity:** major
- **Problem:** Even under the constant redistributions retained in Step 2, the table is inconsistent: \(M_i\), defined as half a gap, remains unchanged while \(E_b\) shifts by \(-(c_1+c_2)\). Therefore the literal expression \(E_b(P)-(M_1+M_2)\) is not invariant. The distance from the continuum edge is invariant under this redistribution, but that edge is no longer numerically \(M_1+M_2\).

  More generally, M6 requires invariance under every allowed redistribution leaving total \(C\) unchanged, not just those preserving the displayed record formula. For example, moving \(h\sigma^{(1)}_3\) from \(C^{\rm D}_1\) into the body–medium pair term preserves \(C\), leaves \(U\) finite-range, and does not change the centred record vectors. Nevertheless, the listed \(C_{\rm pair}\), its rest coefficient, its bound eigenvectors, and generally its dispersion change. Consequently, the claimed invariance of quantities extracted from the listed \(C_{\rm pair}\) has not been established.
- **Suggested fix:** Define the effective body operator invariantly from the total contract and the stated weak-record limit, accounting for body-only terms wherever they are listed. Define the binding-cost difference using a separately identified continuum-edge eigenvalue, rather than treating the invariant half-gaps as shifting reference costs. Reassess the well-definedness table under all M6 redistributions.

### I2. Degenerate bound levels are not covered
- **Location:** Setup and assumptions, Simple level; Steps 3–5, especially Eq. (11.9); Result scope
- **Severity:** major
- **Problem:** The question includes every bound level, whereas the derivation assumes simplicity and then claims that Eqs. (11.5)–(11.9) extend to each analytic branch of a degenerate level. In Eq. (11.9), removing only one eigenvector leaves the other vectors of the degenerate eigenspace in \(Q_P\), so the stated inverse remains singular. Also, eigenvectors appropriate to branches under different parameter variations need not coincide; a single arbitrary vector in the degenerate eigenspace cannot generally supply all the asserted derivatives.
- **Suggested fix:** Treat degenerate levels using the full eigenspace projector, the reduced resolvent on its complement, and the appropriate effective perturbation matrices for each parameter direction. State the branch-selection conditions and distinguish the identities valid for arbitrary stationary eigenvectors from branch-dependent derivative formulas.

### I3. Clock formula omits the first-order correction to the rate
- **Location:** Step 10(a), Eq. (11.19), and Result item 5
- **Severity:** major
- **Problem:** Equation (11.19) retains only the energy term linear in \(d_k\), giving a rate factor evaluated at \(M_i=m_i\). This is a zeroth-order rate approximation, not the first-order rate accuracy exhibited by input (5.21). Generically, the quadratic dependence of the bound energy on \(M_i\) produces the correction
  \[
  \Gamma_i
  =\partial_{M_i}E_b(m_i)
   +\bar d_i\,\partial_{M_i}^{\,2}E_b(m_i)
   +O(d_{\max}^{\,2}),
  \qquad
  \bar d_i=\frac{\pi(N_i-1)}{N_i\lambda_{0,i}},
  \]
  with the other data held fixed. This correction is generally nonzero for a bound pair.

  The stated remainder does not resolve this omission: at clock-reading intervals \(\lambda\sim\lambda_{0,i}\), its term \(\delta_i^2g_0\lambda\) is of first order in \(\delta_i\). Thus it permits an error precisely as large as the first-order clock-rate correction requested.
- **Suggested fix:** Include the mean-level correction, equivalently evaluating the energy derivative at the mean clock mass to the appropriate order, and justify the reading-distribution remainder on a stated range of \(\lambda/\lambda_{0,i}\). Distinguish linear-order energy perturbation from first-order accuracy of the clock rate.

### I4. Wrong order for the contribution of w₂₂
- **Location:** Step 7, Other numbers
- **Severity:** minor
- **Problem:** The statement that \(w_{22}\) enters this upper-band level at order \(\epsilon^4\) is incorrect. The \(\lvert22\rangle\) component first appears at order \(\epsilon^2\), since reaching it requires two branch flips. Hence
  \[
  \frac{\partial E_b}{\partial w_{22}}
  =|\psi_{22}(0)|^2=O(\epsilon^4).
  \]
  Since \(w_{22}=O(\epsilon^2)\), its first possible contribution to the energy is generically \(O(\epsilon^6)\), not \(O(\epsilon^4)\).
- **Suggested fix:** Correct the stated order to \(\epsilon^6\). Keep the overall \(O(\epsilon^4)\) energy remainder, which receives other contributions.

### I5. Eigenvector formula is undefined at j = 0
- **Location:** Step 7, Eqs. (11.11)–(11.13), and Existence and number
- **Severity:** minor
- **Problem:** The derivation extends its result to every \(P\in(-\pi,\pi]\), but when \(a_1=a_2\) and \(P=\pm\pi/2\), \(j(P)=0\). Then \(\alpha(P)\) is undefined and the displayed expression for \(\rho(P)\) divides by zero. These are admissible exceptional members expressly requiring treatment under the question’s assumptions.
- **Suggested fix:** State the limiting eigenvector at \(j=0\), which at leading order is supported at \(r=0\) in branch \(\lvert11\rangle\), or restrict the displayed eigenvector formula to the neighbourhood of \(P=0\) actually requested.

## Focus points

None given.

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