03-ilang-space / 10-loop-data
10loop dataverified
Summary The loop data of the witness geometry in a recording medium, the phases of cyclic products, and how they relate to the metric.
# External verification: 10-loop-data
- **Subproject:** 03-ilang-space
- **Package:** 10-loop-data
- **Verified version:** v1
- **External round:** 1 of 2
- **Date:** 2026-10-08T16:56:20+02:00
- **Focus points:** none
---
VERDICT: minor issues
## Summary
The overlap expansion, loop-phase signs, metric bound, vanishing cases, and examples are correctly derived. The derivation answers all requested items and carefully controls the small-\(\lambda\) phase branch. Two minor qualifications are needed: the geometric interpretation of saturation excludes a degenerate case, and one consistency check presents a leading-order equality as exact.
## Issues
### I1. Saturation reformulation needs a nondegeneracy qualification
- **Location:** Step 5(iii), discussion following Eq. (10.14).
- **Severity:** minor
- **Problem:** Equation (10.14) correctly includes \(v=u_{h_2}-u_{h_1}=0\). However, its subsequent reformulation as the two conditions \(g(v,w)=0\) and \(w\in\mathbb Cv\) is equivalent only when \(v\neq0\). If \(v=0\) and \(w\neq0\), the leading-order bound is trivially saturated, but \(w\notin\mathbb Cv=\{0\}\). Such coincident record vectors are permitted by the setting.
- **Suggested fix:** Explicitly restrict the two-condition interpretation to \(v\neq0\), retaining \(v=0\) as the separate trivial saturation case already included in (10.14).
### I2. Leading-order saturation is written as an exact phase equality
- **Location:** Consistency check 3, final sentence.
- **Severity:** minor
- **Problem:** The statement that the unit triangle “gives \(\lvert\Phi\rvert=\lambda^2\)” is not exact. The preceding consistency check establishes
\[
|\Phi|=\arctan(\tan^2\lambda)
=\lambda^2+\frac23\lambda^4+O(\lambda^6)
\]
near zero. Saturation concerns only the leading \(\lambda^2\) coefficient.
- **Suggested fix:** Write \(\lvert\Phi\rvert=\lambda^2+O(\lambda^4)\), or explicitly state that the leading coefficient saturates the bound.
## Focus points
None given.