# Question: 15-curved-profile-fall

- **Subproject:** 05-ilang-free-fall
- **Package:** 15-curved-profile-fall
- **Equation tags:** (15.k)
- **Created:** 2026-10-18

## Goal

For a body whose clock level is conserved, as in 04-branch-fall, but whose level profiles are not linear in the position label, determine how a start at rest in the upper branch accelerates, and how far the result of the linear profiles, taken with the local slope at the start, describes it. No field object; the position dependence is written into the body's own contract (A10), which is an assumption of this package (M8).

**Setting.**
- **Body and records.** A body $b$, the only instance of its type, and a recording medium $c$, with the records $K_x=\kappa\,x\,X$, places $(x,s,a)$ as for Model D, $x$ on a chain $\lvert x\rvert\le L$ and $x\in\mathbb Z$ as the limit of long chains (A1). As in 04-branch-fall, the term $\kappa\,x\,\langle X\rangle$ of the records is counted in every $V_{x,k}$; results in the weak-record regime ($\kappa\to0$ at fixed $\lambda$, chain and start). $\bar x_{\rm lab}(\lambda)$ is the mean position label.
- **Level blocks (the class of 04-branch-fall with $\zeta_{x,k}=1$).** On the clock level $k$ (conserved):
  $$C^{(k)}=\frac{\tau_k}2\sum_x\Bigl(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert\otimes(i\sigma_1)\Bigr)+\sum_x\lvert x\rangle\langle x\rvert\otimes\bigl(M_{x,k}\,\sigma_3+V_{x,k}\,\mathbb 1_2\bigr),$$
  with $\tau_k>0$, $M_{x,k}>0$ on the chain and real $V_{x,k}$.
- **Quadratic level profiles about a point $x_0$.** With $y:=x-x_0$, for every level $k$:
  $$M_{x,k}=M_k+G_k\,y+\tfrac12H_k\,y^2,\qquad V_{x,k}=-\mathcal F_k\,y-\tfrac12\mathcal H_k\,y^2 .$$
  The **slopes at $x_0$** are $G_k$, $\mathcal F_k$; the **curvatures** are $H_k$, $\mathcal H_k$. With $H_k=\mathcal H_k=0$ these are the linear level profiles of 04-branch-fall, with the origin moved to $x_0$.
- **Starts.** The branch start $B_{k,+}[g]$ of 04-branch-fall, built from the upper branch of the level block with all slopes and curvatures set to zero ($M_{x,k}\to M_k$, $V_{x,k}\to0$), with $g$ smooth and $2\pi$-periodic; it is **centred at $x_0$** if its mean position label is $x_0$, and **at rest** if its weight in $p$ has the mean $0$ on $(-\pi,\pi]$. $\delta p^2$ is the variance of that weight and $W^2$ the variance of the position label of the start. The start is not adapted to the profiles; this is part of the assumption (M9).
- **Order.** First order jointly in $G_k,\mathcal F_k,H_k,\mathcal H_k$ at fixed $\lambda$, fixed start and fixed chain, followed by the limit of a long chain, as in 04-branch-fall.

1. **Acceleration in a quadratic profile.** One level $k$, a start $B_{k,+}[g]$ at rest centred at $x_0$.
   - (a) Determine the first-order acceleration $\mathrm d^2\bar x_{\rm lab}/\mathrm d\lambda^2$ as a function of $\lambda$: separate the oscillating terms from the rest, and give the rest exactly as a function of $\lambda$, with its coefficients as averages over the start.
   - (b) Determine the part of (a) that the curvatures contribute, and how it depends on $\lambda$ and on the start (through $W^2$, $\delta p^2$ or other averages); state whether it grows with $\lambda$, with the power.
2. **Local slope.** Determine the conditions on $H_k$, $\mathcal H_k$, $W$, $\delta p$ and $\lambda$ under which the non-oscillating part of (a), averaged over a window, equals the drift acceleration at rest of the linear profiles with the slopes at $x_0$, $\frac{\tau_k^2}{M_k}(\mathcal F_k-G_k)$ of (4.16), up to a stated relative error. State whether a limit of the start exists in which this holds exactly at first order, and the order of the limits.
3. **Level independence.** Bodies with $N\ge2$ levels.
   - (a) The factor class: $M_{x,k}=(m+d_k)\,n_x$, $V_{x,k}=0$, $\tau_k=\tau$, with $n_x=1+g\,y+\tfrac12h\,y^2$ and $d_k:=\frac{2\pi k}{N\lambda_0}$. Determine whether the non-oscillating first-order acceleration of item 1 is the same for all levels at every $\lambda$, for every start of the class of item 1.
   - (b) Control: $M_{x,k}=m\,n_x+d_k$ (the factor on the rest term only), otherwise as in (a). The same question.

## Inputs

From 04-branch-fall@v3 (this subproject). Setting there: the class $\mathcal C_{\rm lev}$ of level blocks $C^{(k)}=\frac{\tau_k}2\sum_x\zeta_{x,k}\bigl(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert\otimes(i\sigma_1)\bigr)+\sum_x\lvert x\rangle\langle x\rvert\otimes(M_{x,k}\sigma_3+V_{x,k}\mathbb 1_2)$ with the linear level profiles $M_{x,k}=M_k+G_kx$, $V_{x,k}=-\mathcal F_kx$, $\zeta_{x,k}=1+g_{\zeta,k}(x+\frac12)$; $\omega_k(p):=\sqrt{M_k^2+\tau_k^2\sin^2p}$; the branch start $B_{k,\sigma}[g]=\int\frac{\mathrm dp}{2\pi}g(p)\lvert p\rangle\otimes\beta_{k,\sigma}(p)\otimes\lvert f_k\rangle$ with $\beta_{k,\sigma}$ the eigenvectors of the block with all gradients zero; $\langle F\rangle_g:=\int\frac{\mathrm dp}{2\pi}\lvert g\rvert^2F$; first order in the gradients at fixed $\lambda$, start and chain, then a long chain; weak-record regime; $\mathsf b_k$ a bound linear in the gradients.

Eq. (4.7) of 04-branch-fall, the coefficient of $\lambda$ in the first-order velocity of $B_{k,\sigma}[g]$ is $\langle a_{k,\sigma}\rangle_g$, with

$$
\boxed{\;a_{k,\sigma}(p)=\frac{g_{\zeta,k}\tau_k^4\sin^2p-G_kM_k\tau_k^2\cos2p}{\omega_k^2}+\sigma\,\mathcal F_k\,\frac{\tau_k^2\bigl(M_k^2\cos2p-\tau_k^2\sin^4p\bigr)}{\omega_k^3}\;},\qquad\lvert a_{k,\sigma}(p)\rvert\le\frac{\tau_k^2}{M_k}\,\mathsf b_k .
$$

Eq. (4.11) of 04-branch-fall, for a start concentrated at $p_0$ with spread $\delta p$, after the long-window limit of the window average $\overline{a^{(1)}}$ of the first-order acceleration:

$$
\bigl\lvert\langle a_{k,\sigma}\rangle_g-a_{k,\sigma}(p_0)\bigr\rvert\le\tfrac12\sup_p\lvert a_{k,\sigma}''\rvert\,\delta p^2\ \ (\text{exact}),\qquad\text{drift acceleration}=\lim_{\delta p\to0}\lim_{\Lambda\to\infty}\overline{a^{(1)}}=a_{k,\sigma}(p_0).
$$

Eq. (4.16) of 04-branch-fall:

$$
\boxed{\;a_{k,\sigma}(0)=\frac{\tau_k^2}{M_k}\,\bigl(\sigma\,\mathcal F_k-G_k\bigr)\;}
$$

Eq. (4.19) of 04-branch-fall, for every member of $\mathcal C_{\rm lev}$ with linear level profiles and $N\ge2$:

$$
\begin{aligned}
a_{k,+}(0)=a_+\ \forall k&\iff\frac{\tau_k^2}{M_k}(\mathcal F_k-G_k)=a_+\ \forall k,\qquad
a_{k,-}(0)=a_-\ \forall k\iff\frac{\tau_k^2}{M_k}(\mathcal F_k+G_k)=-a_-\ \forall k,\\
\text{both, with }a_+=a_-=a&\iff\mathcal F_k=0\ \text{ and }\ \frac{\tau_k^2G_k}{M_k}=-a\quad\forall k .
\end{aligned}
$$

## Assumptions

- The setting under "Goal": the body, the quadratic level profiles, the factor class, the control and the branch starts are assumptions of this package (M8, M9).
- Standard first-order perturbation theory in a term of the contract, plane waves, and the averaging of oscillating terms may be used, with their conditions stated.

## Scope

- **Class.** Items 1 and 2: every level block of the setting with quadratic level profiles, every start $B_{k,+}[g]$ at rest centred at $x_0$ with finite $W$. Item 3: the factor class and its control, $N\ge2$.
- **Control.** Item 3(b).
- Out of scope: the lower branch; starts in motion; profiles of the hopping ($\zeta_{x,k}\neq1$); terms of the profiles beyond second order in $y$; second order in the slopes and curvatures; long falls; clocks and their rates; fields that are not sharp; two bodies.

## Depth

- Item 1: derive (needed by item 2 and by criterion 10, which compares fields that are not linear).
- Item 2: short argument (needed by criterion 10: when a fall at a point is defined).
- Item 3: short argument (needed by criterion 10: whether the level-independent fall of the factor class survives a curved profile).

## Expected result

- Item 1: a function of $\lambda$ with explicit coefficients; a term with its dependence on $\lambda$ and on the start.
- Item 2: conditions as inequalities, and a limit with its order.
- Item 3: two yes/no answers, each with its reason and, where no, the leading difference between two levels.

Give every main result a tag $(15.k)$.

Consistency checks, at most three: for example zero curvatures against (4.7) and (4.16), and $\tau_k\to0$.

## Code

None.

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