03-ilang-space / 15-bodies
15bodiesverified
Summary What a body, its position and the distance between two bodies are in the witness geometry, and whether the state of the medium determines them.
# External verification: 15-bodies
- **Subproject:** 03-ilang-space
- **Package:** 15-bodies
- **Verified version:** v2
- **External round:** 2 of 2
- **Date:** 2026-10-08T16:08:54+02:00
- **Focus points:** none
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VERDICT: minor issues
## Summary
The exact evolution, partial traces, leading-order overlap expansion, and cyclic-subspace criterion are correct. The split and label invariance arguments, configuration-distinguishability condition, and both examples are also sound. Two isolated statements overstate what the medium determines; they conflict with qualifications and counterexamples correctly given elsewhere in the derivation.
## Issues
### I1. Instantaneous recoverability claims overlook constant-distance cases
- **Location:** Step 7, “Leading order, orbit and instantaneous state,” item 3; Result, “Line” bullet.
- **Severity:** minor
- **Problem:** Step 7 states without qualification that the configuration and \(D\) are not functions of the instantaneous medium state at every \(\lambda\). The Result’s line summary similarly says “but not at every \(\lambda\)” for all its positive cases, including \(c_T=-1,\ n=2\). In that case there is only one configuration, so both the configuration and \(D\) are functions of the medium state at every \(\lambda\), as Step 8 correctly acknowledges. More generally, failure to recover the configuration does not imply failure to recover \(D\): in the fermionic window example with \(m=1\), \(D=\sqrt2\) is constant across all configurations and remains recoverable even at complete revivals.
- **Suggested fix:** Use “not necessarily functions … at every \(\lambda\)” and distinguish configuration recovery from distance recovery. Preserve explicitly the singleton-configuration and constant-\(D\) exceptions already identified in Steps 8 and 9.
### I2. The full-view discussion still implies that the medium generally encodes \(D\)
- **Location:** Step 5, final sentence: “The medium carries \(D\) in a different form, through the pair vector \(s\).”
- **Severity:** minor
- **Problem:** This unqualified statement is false in general. The medium’s leading-order state records the pair-vector sum, which need not determine \(D\). For example, the symmetric line configurations \(\{h_1,h_3\}\) and \(\{h_2,h_2\}\) have the same pair vector and identical medium states for every \(\lambda\), but distances \(2\sigma_X\) and \(0\). Steps 7 and 8 correctly establish this limitation, so the sentence creates an internal inconsistency rather than a computational error.
- **Suggested fix:** State that the medium records the pair vector \(s\), and that this determines \(D\) only when \(D\) is constant on each realizable fibre of the pair-sum map, as explained in Step 7.
## Focus points
None given.