05-ilang-free-fall / 15-curved-profile-fall
15curved profile fallverified
Determines how a body at rest accelerates when its level profiles are curved, how far the local slope describes the fall, and whether the fall stays the same for all clock levels.
# External verification: 15-curved-profile-fall
- **Subproject:** 05-ilang-free-fall
- **Package:** 15-curved-profile-fall
- **Verified version:** v2
- **External round:** 1 of 3
- **Date:** 2026-10-18T12:34:24+02:00
- **Focus points:** none
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VERDICT: minor issues
## Summary
The first-order acceleration formula, its separation into oscillating and non-oscillating terms, and the window-averaging bounds are consistent with the stated perturbative regime. The factor-class counterexamples correctly show that exact level independence fails for finite-width starts, while the stated local-slope limit restores it. Two minor issues remain: an auxiliary envelope moment is misidentified as a physical position moment, and the pure-curvature case of the control family needs a short additional argument.
## Issues
### I1. Auxiliary fourth moment is not the body’s position fourth moment
- **Location:** Step 3, Eq. (15.16) and following paragraph
- **Severity:** minor
- **Problem:** The definition \(W_4^4=\langle g|\eta^4|g\rangle\) is a fourth moment of the scalar envelope, not the physical position distribution of \(B_{k,+}[g]\). Since \(\hat y=\eta+\alpha\rho_2\), the actual fourth position moment is
\[
\langle\psi_0|\hat y^4|\psi_0\rangle
=\|(\eta^2+\alpha^2)g\|^2+\|\{\alpha,\eta\}g\|^2,
\]
which generally differs from \(\|\eta^2g\|^2\). The bounds themselves consistently use the explicitly defined envelope norm, so their algebra is unaffected.
- **Suggested fix:** Identify \(W_4\) as an auxiliary envelope fourth-moment width throughout. If a condition in terms of the body’s physical fourth position moment is intended, derive the required comparison separately.
### I2. Pure-curvature control is not explicitly established
- **Location:** Step 8(b) and Result entry (15.14)
- **Severity:** minor
- **Problem:** The explicit control argument covers \(g_n\ne0\), but the question also includes \(g_n=0\), \(h_n\ne0\). The final observation that \(b_k(0)\) depends on the level is not by itself sufficient: the constant part of \(b_k\) cancels from \(c_g\) because \(\int j=0\), and the secular coefficient involves \(b_k\omega_k'\), not \(b_k(0)\) alone. Thus the pure-curvature conclusion is asserted without the short argument requested for this case.
- **Suggested fix:** Apply the chirped, even-density starts used in Step 8(a) to the control coefficients. For sufficiently small momentum support, the unequal \(p^2\) coefficients of \(b_k\) and \(b_l\) give \(c_{g,k}\ne c_{g,l}\), while the secular terms vanish by parity. State the flat-profile exception separately.
## Focus points
None given.