# Question: 09-unsharp-field

- **Subproject:** 05-ilang-free-fall
- **Package:** 09-unsharp-field
- **Equation tags:** (9.k)
- **Created:** 2026-10-11

## Goal

Determine what the results on the fall of a body, on its independence of the clock level, on the clock rate and on the relation (4.24) become when the field that acts on the body is static and is not in a sharp state. This is the first part of criterion 8.

**Setting.**
- **Objects.** A body $b$, a field $f$ and a recording medium $c$, of pairwise different types, one instance each.
- **Contract (the class $\mathcal S$ of static fields).** In the order $b$, $f$, $c$:
  $$C=\sum_{\nu\in\mathcal N}C^{(\nu)}_b\otimes P_\nu\otimes\mathbb 1_c+\mathbb 1_b\otimes C_f\otimes\mathbb 1_c+\sum_x\lvert x\rangle\langle x\rvert\otimes\mathbb 1_f\otimes\kappa\,x\,X .$$
  Here $\mathcal N$ is a finite set of **sectors**; the $P_\nu$ are orthogonal projectors on $\mathcal H_f$ with $P_\nu P_{\nu'}=\delta_{\nu\nu'}P_\nu$ and $\sum_\nu P_\nu=\mathbb 1_f$; $C_f$ is self-adjoint on $\mathcal H_f$ with $[C_f,P_\nu]=0$ for all $\nu$; every **sector contract** $C^{(\nu)}_b$ is a self-adjoint operator on $\mathcal H_b$ of the class $\mathcal C_{\rm lev}$ with linear level profiles (next item); the last term is the record term of Model D, with $\kappa>0$ and $X$ self-adjoint on $\mathcal H_c$. Two different sectors have sector contracts that differ by more than a real multiple of $\mathbb 1_b$. The first sum is one pair term between $b$ and $f$ (A5). $C$ is the contract (2.1) with $r=\{c\}$, the field operators $\Phi_\nu=P_\nu$ and $B_\nu=C^{(\nu)}_b$. These field operators commute with each other and with $C_f$ (a **static** field, (2.15)), and the sector generator of the sector $\nu$ is $C^{(\nu)}=C^{(\nu)}_b\otimes\mathbb 1_c+\sum_x\lvert x\rangle\langle x\rvert\otimes\kappa\,x\,X$.
- **Sector contracts (the class $\mathcal C_{\rm lev}$).** A member of $\mathcal C_{\rm lev}$ has the places $(x,s,a)$ of Model D, with $x$ on a chain $\lvert x\rvert\le L$ and $x\in\mathbb Z$ as the limit of long chains (A1), and, with the vectors $\lvert f_k\rangle$ of Model D, the contract
  $$C_b=\sum_{k=0}^{N-1}C^{(k)}\otimes\lvert f_k\rangle\langle f_k\rvert,\qquad C^{(k)}=\frac{\tau_k}2\sum_x\zeta_{x,k}\Bigl(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert\otimes(i\sigma_1)\Bigr)+\sum_x\lvert x\rangle\langle x\rvert\otimes\bigl(M_{x,k}\,\sigma_3+V_{x,k}\,\mathbb 1_2\bigr),$$
  with real numbers $\tau_k>0$, $\zeta_{x,k}>0$, $M_{x,k}>0$ and $V_{x,k}$. $C^{(k)}$ is the **level block** of the clock level $k$; the level is conserved. **Linear level profiles:** $M_{x,k}=M_k+G_kx$, $V_{x,k}=-\mathcal F_kx$, $\zeta_{x,k}=1+g_{\zeta,k}(x+\tfrac12)$, with $M_k>0$, $\lvert G_k\rvert(L+1)<M_k$ and $\lvert g_{\zeta,k}\rvert(L+1)<1$. The **level data** are $\tau_k$ and $M_k$; the **gradients** are $G_k$, $\mathcal F_k$ and $g_{\zeta,k}$. The class $\mathcal C$ of 03-clock-and-fall with its linear profiles is the subclass $\tau_k=\tau$, $M_k=m+d_k$, $G_k=m\,g_m+d_k\,g_\eta$, $\mathcal F_k=\mathcal F$, $g_{\zeta,k}=g_\zeta$, with $d_k:=\frac{2\pi k}{N\lambda_0}$; Model D is its member with all gradients zero. All sector contracts have the same places (one $N$). The level data and the gradients of the sector $\nu$ are written $\tau^\nu_k$, $M^\nu_k$, $G^\nu_k$, $\mathcal F^\nu_k$, $g^\nu_{\zeta,k}$.
- **Records and regime.** As in 04-branch-fall, applied in every sector: the level profiles of the sector $\nu$ are those of its effective body contract, (8.19) of 04-ilang-time/08-maximal-speed for the sector generator $C^{(\nu)}$, so the term $\kappa\,x\,\langle X\rangle$ of the records is counted in every $V^\nu_{x,k}$. The mean position label is $\bar x_{\rm lab}(\lambda):=\sum_xx\,p_x(\lambda)$, with $p_x$ the place weights of the view of the body. Unless an item says otherwise, results are asked in the weak-record regime, $\kappa\to0$ at fixed $\lambda$, chain and start.
- **Start, weights and sector means.** The start is $\Psi_0=\phi\otimes\varphi\otimes\chi$, with $\phi$ the start of the body named in each item, $\varphi$ any unit vector of $\mathcal H_f$ and $\chi$ the state of the medium, $\sigma_X>0$. $w_\nu:=\lVert P_\nu\varphi\rVert^2$ and $S:=\{\nu:w_\nu>0\}$. The field is **sharp** if $S$ has one element. For numbers $F^\nu$, $F'^\nu$ attached to the sectors: $\langle F\rangle_w:=\sum_\nu w_\nu F^\nu$, ${\rm Cov}_w(F,F'):=\langle FF'\rangle_w-\langle F\rangle_w\langle F'\rangle_w$ and ${\rm Var}_w(F):={\rm Cov}_w(F,F)$.
- **Branches and branch starts of a sector.** With all gradients of the sector $\nu$ set to zero, the level block of the level $k$ is $-\tau^\nu_k\sin\hat p\,\sigma_1+M^\nu_k\sigma_3$, the block (5.5) of 04-ilang-time/05-moving-clock with $\tau\to\tau^\nu_k$ and $\mu_k\to M^\nu_k$. Its eigenvalues are $\pm\omega^\nu_k(p)$, $\omega^\nu_k(p):=\sqrt{(M^\nu_k)^2+(\tau^\nu_k)^2\sin^2p}$, with the eigenvectors $\beta^\nu_{k,\pm}(p)$ of (5.6) of 04-ilang-time/05-moving-clock (same substitution). For a level $k$, a branch $\sigma\in\{+,-\}$ and a smooth $2\pi$-periodic function $g$ with $\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\lvert g\rvert^2=1$, the **branch start** $B^\nu_{k,\sigma}[g]$ of the sector $\nu$ is
  $$\lvert\phi\rangle=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,g(p)\,\lvert p\rangle\otimes\beta^\nu_{k,\sigma}(p)\otimes\lvert f_k\rangle$$
  on the infinite chain. $\langle F\rangle_g:=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\lvert g(p)\rvert^2F(p)$. The start is **concentrated at $p_0$** with the spread $\delta p$ if, on the arc $(p_0-\pi,p_0+\pi]$, the weight $\lvert g\rvert^2\frac{\mathrm dp}{2\pi}$ has the mean $p_0$ and the variance $\delta p^2$; it is **at rest** if $p_0=0$. A branch start is not adapted to the gradients (M9).
- **Orders and sector quantities.** "First order in the gradients" means: first order jointly in the gradients of all sectors of $S$ and of all levels involved, at fixed $\lambda$, fixed start and fixed chain, followed by the limit of a long chain. For a member of $\mathcal C_{\rm lev}$, the **drift acceleration** $a_{k,\sigma}(p_0)$ of 04-branch-fall is the first-order acceleration of $\bar x_{\rm lab}$ for a branch start, averaged over a long window, followed by the limit of a sharp label, (4.10), (4.11). The **sector drift acceleration** $a^\nu_{k,\sigma}(p_0)$ is this quantity for the sector contract $C^{(\nu)}_b$. The **sector displacement** $\Delta x^\nu_{k,\sigma}(\lambda)$ is the first-order term of $\bar x_{\rm lab}(\lambda)-\bar x_{\rm lab}(0)$ for the evolution with the sector generator $C^{(\nu)}$ from the start $B^\nu_{k,\sigma}[g]\otimes\chi$.
- **Subclasses.**
  - $\mathcal S_{\rm grad}$ (only the gradients depend on the sector): $\tau^\nu_k=\tau_k$ and $M^\nu_k=M_k$ for all $\nu$. The branches and the branch starts are then the same in all sectors and are written $B_{k,\sigma}[g]$.
  - $\mathcal S^{\mathcal C}_{\rm grad}$: the members of $\mathcal S_{\rm grad}$ whose sector contracts are of the class $\mathcal C$ with one set $\tau$, $m$, $N$, $\lambda_0$ and with the linear profiles of 03-clock-and-fall, with the gradients $g^\nu_m$, $g^\nu_\eta$, $g^\nu_\zeta$, $\mathcal F^\nu$.
  - $\mathcal S_{\rm a}$ (a factor on the rest and clock term): $C^{(\nu)}_b=A+\sum_x(n^\nu+\gamma^\nu x)\,\lvert x\rangle\langle x\rvert\otimes\sigma_3\otimes(m\mathbb 1_N+D)$, with $A$ the motion term of Model D and real numbers $n^\nu>0$, $\gamma^\nu$ with $\lvert\gamma^\nu\rvert(L+1)<n^\nu$. This is the lapse-type pair term of the base problem with $G=\sigma_3\otimes(m\mathbb 1_N+D)$ and $\Phi_x=\sum_\nu(n^\nu+\gamma^\nu x)P_\nu$.
  - $\mathcal S_{\rm c}$ (a factor on the whole body contract): $C^{(\nu)}_b=\tfrac12\bigl\{n^\nu\mathbb 1+\gamma^\nu\hat x,\ C_{\rm D}\bigr\}$ (anticommutator), with $C_{\rm D}$ the contract of Model D and the same conditions on $n^\nu$, $\gamma^\nu$.

  In $\mathcal S_{\rm a}$ and $\mathcal S_{\rm c}$ the value $n^\nu$ of the factor at $x=0$ depends on the sector, and with it the level data. For these two subclasses $\bar n:=\langle n\rangle_w$ and $\bar\gamma:=\langle\gamma\rangle_w$.

1. **The mixture.** For every member of $\mathcal S$ and every start $\phi\otimes\varphi\otimes\chi$:
   - (a) Express the place weights $p_x(\lambda)$ of the view of the body, the mean position label, and the joint statistics of the position label and the clock label at one $\lambda$ through the quantities of the sectors. State what is exact and what it rests on.
   - (b) For a reading on the body at $\lambda_1$ with the outcome $r$, determine the weights with which the sectors enter the statistics of all later readings on the body and the medium. State the condition under which a given reading leaves the weights unchanged.
   - (c) Determine which of the following are well defined (M6) under the replacements (2.20) and under $C^{(\nu)}_b\to C^{(\nu)}_b+c_\nu\mathbb 1_b$, $C_f\to C_f-\sum_\nu c_\nu P_\nu$ with real $c_\nu$: the weights $w_\nu$; a sector contract; the differences $C^{(\nu)}_b-C^{(\nu')}_b$; the level data and the gradients of a sector.

2. **Fall when only the gradients depend on the sector.** Members of $\mathcal S_{\rm grad}$, starts $B_{k,\sigma}[g]\otimes\varphi\otimes\chi$.
   - (a) Determine, to first order in the gradients, through which functions of the weights and of the sector gradients the place weights $p_x(\lambda)$ depend on the state of the field. Determine the drift acceleration of the mean position label, and state in which form (4.16), (4.19), (4.22) and (4.24) hold for it.
   - (b) For a start at rest: determine the variance over the sectors of the sector displacements, ${\rm Var}_w\bigl(\Delta x_{k,\sigma}(\lambda)\bigr)$, in its part that grows fastest with $\lambda$, with its coefficient, and state the order of the limits in which it is obtained. Determine the necessary and sufficient condition on the sectors of $S$ under which this part vanishes for the level $k$ and the branch $\sigma$, and the condition under which it vanishes for all levels.
   - (c) Determine the necessary and sufficient condition under which the sector drift accelerations from rest are the same for all levels in every sector of $S$: in the upper branch, and in both branches. Determine whether a drift acceleration of the mean position label that is the same for all levels implies this condition, and give a control. State what distinguishes the two cases in readings of the position.

3. **Clock when only the gradients depend on the sector.** Members of $\mathcal S^{\mathcal C}_{\rm grad}$, starts $S_{\rm clock}(x_0)\otimes\varphi\otimes\chi$, and the rate (3.23) in its wide-start form for every sector.
   - (a) Determine the clock statistics at $x_0$ in terms of the sectors, the mean over the sectors of the advance and its rate, and the variance over the sectors of the advance, as functions of $\lambda$ and $x_0$. Determine the range of $\lambda$ in which the standard deviation of the advance over the sectors stays below a given fraction $\epsilon$ of one tick.
   - (b) Determine the ratio of the rates at two places $x_1$, $x_2$, with the rates defined from the mean advances of two runs compared at equal $\lambda$ (A8), and the first order in the gradients of $\bigl(\bar r(x+1)-\bar r(x)\bigr)/\bar r(x)$ for these rates $\bar r$.
   - (c) Determine the covariance over the sectors of the advance at $x_0$ and of the sector displacement of item 2(b) for the level $k$, in its part that grows fastest with $\lambda$. State the condition on the sectors under which the two are functions of each other.
   - (d) Determine the condition on the sectors and the weights under which the relation (4.24) holds between the drift acceleration of the mean position label and the rates $\bar r$ of (b). State separately what holds sector by sector.

4. **A factor whose value is not sharp.** Members of $\mathcal S_{\rm a}$ and of $\mathcal S_{\rm c}$.
   - (a) For each sector give the level data and the gradients, the rate at $x$ in the wide-start form (state the data of the class $\mathcal C$ to which (3.23) is applied), the bound (5.11), and the sector drift accelerations from rest in both branches. State whether these are the same for all levels within a sector, and whether (4.24) holds within a sector.
   - (b) Determine the components of the branch start $B^{\nu_0}_{k,\sigma}[g]$ of a reference sector $\nu_0$ in the two branches of another sector $\nu$, and the weight of the other branch for a start at rest in terms of $\delta p$. State for which starts and in which limit (4.16) gives the sector drift acceleration in every sector of $S$, for $\mathcal S_{\rm a}$ and for $\mathcal S_{\rm c}$.
   - (c) For the start $S_{\rm clock}(0)$: determine the mean over the sectors of the advance and its variance. Determine the mean $\bar a$ over the sectors of the sector drift accelerations from rest, and the number $\bar\varrho$, the first order in the gradients of $\bigl(\bar r(x+1)-\bar r(x)\bigr)/\bar r(x)$ at $x=0$ for the rates $\bar r$ defined from the mean advances. Express $\bar a/\bar\varrho$ through the moments of $n$ and $\gamma$ under $w$. Determine the necessary and sufficient condition on $w$ under which it equals the value that (4.24) gives for a sharp field with the factor $\bar n+\bar\gamma x$, and its leading deviation from that value for a narrow distribution of $n$, in terms of ${\rm Var}_w(n)$ and ${\rm Cov}_w(n,\gamma)$.

5. **Summary.** Give a table with one row for each of: the rate of the clock at a place; the ratio of the rates at two places; the drift acceleration from rest; its independence of the clock level; the relation (4.24). For each row state what holds within every sector, what holds for the means over the sectors in $\mathcal S_{\rm grad}$, in $\mathcal S_{\rm a}$ and in $\mathcal S_{\rm c}$, and to which order, with the tags of the results used.

## Inputs

From 02-field-object@v2 (this subproject). Setting there: a body $b$, a field $f$ and a set $r$ of further objects (possibly empty), of pairwise different types, $\mathcal H=\mathcal H_{br}\otimes\mathcal H_f$ with $\mathcal H_{br}:=\mathcal H_b\otimes\mathcal H_r$; $C_{br}$ is a sum of pair terms among $b$ and the objects of $r$, $C_f$ is self-adjoint on $\mathcal H_f$, and $B_j$ (on $\mathcal H_b$), $\Phi_j$ (on $\mathcal H_f$), $j\in J$, are finite families of self-adjoint operators; $B_j$ also denotes $B_j\otimes\mathbb 1_r$. The start is $\Psi_0=\Psi_{br}\otimes\varphi$ and $\lambda\ge0$; $V(\lambda)$ is the view of $\{b\}\cup r$. A field state $\varphi$ is **sharp** if $\Phi_j\varphi=n_j\varphi$ for all $j$, with real numbers $n_j$. A **schedule** consists of readings with projectors $\Pi_i\otimes\mathbb 1_f$ at $0\le\lambda_1\le\dots\le\lambda_k$, where $\Pi_i$ is diagonal in the joint places of $b$ and $r$; $p(r_1,\dots,r_k)$ is the joint distribution of its outcomes.

Eq. (2.1) of 02-field-object, the contract:

$$
C=C_{br}\otimes\mathbb 1_f+\sum_{j\in J}B_j\otimes\mathbb 1_r\otimes\Phi_j+\mathbb 1_{br}\otimes C_f
$$

Eq. (2.15) of 02-field-object. Let the $\Phi_j$ commute with each other and with $C_f$, with the joint spectral projectors $P_\nu$ labelled by the joint eigenvalues $\nu=(\nu_j)_{j\in J}\in\mathcal N$:

$$
\Phi_j=\sum_{\nu\in\mathcal N}\nu_j\,P_\nu,\qquad C=\sum_{\nu\in\mathcal N}C^{(\nu)}\otimes P_\nu+\mathbb 1_{br}\otimes C_f,\qquad C^{(\nu)}:=C_{br}+\sum_{j\in J}\nu_j\,B_j\otimes\mathbb 1_r .
$$

Eq. (2.16) of 02-field-object, with $w_\nu:=\lVert P_\nu\varphi\rVert^2$, $S:=\{\nu:w_\nu>0\}$, $\varphi_\nu:=P_\nu\varphi/\sqrt{w_\nu}$ and $\Psi^{(\nu)}(\lambda):=e^{-iC^{(\nu)}\lambda}\Psi_{br}$; exact, for every $\Psi_{br}$, every $\varphi$ and every $\lambda$:

$$
\Psi(\lambda)=\sum_{\nu\in S}\sqrt{w_\nu}\;\Psi^{(\nu)}(\lambda)\otimes e^{-iC_f\lambda}\varphi_\nu,\qquad V(\lambda)=\sum_{\nu\in S}w_\nu\,\lvert\Psi^{(\nu)}(\lambda)\rangle\langle\Psi^{(\nu)}(\lambda)\rvert .
$$

Eq. (2.17) of 02-field-object, for every schedule, with $p^{(\nu)}$ the statistics of the schedule computed with $C^{(\nu)}$ and the start $\Psi_{br}$:

$$
p(r_1,\dots,r_k)=\sum_{\nu\in S}w_\nu\,p^{(\nu)}(r_1,\dots,r_k) .
$$

Eq. (2.19) of 02-field-object. For such a field the view $V(\lambda)$ is pure for every $\Psi_{br}$ and every $\lambda\ge0$ if and only if

$$
\sum_{j\in J}(\nu_j-\nu'_j)\,B_j\in\mathbb R\,\mathbb 1_b\qquad\text{for all }\nu,\nu'\in S .
$$

Eq. (2.20) of 02-field-object, the replacements of $C_{br}$, the $\Phi_j$ and $C_f$ at fixed $B_j$ that leave $C$ unchanged (exactly these, if $\mathbb 1_b$ and the $B_j$ are linearly independent):

$$
\Phi_j\to\Phi_j+a_j\,\mathbb 1_f,\qquad C_f\to C_f+c\,\mathbb 1_f,\qquad C_{br}\to C_{br}-\sum_{j\in J}a_j\,B_j\otimes\mathbb 1_r-c\,\mathbb 1_{br},\qquad a_j,c\in\mathbb R .
$$

From 05-moving-clock@v3 of 04-ilang-time. Setting there: one Model D body with the contract $C_b=C_{\rm D}$ of the base problem, the records $K_x=\kappa xX$ and the start $\lvert\phi\rangle\otimes\lvert\chi\rangle$; $d_k:=\frac{2\pi k}{N\lambda_0}$; the plane waves $\lvert p\rangle=\sum_xe^{ipx}\lvert x\rangle$, $p\in(-\pi,\pi]$.

Eq. (5.5) of 04-ilang-time/05-moving-clock:

$$
C_b\bigl(\lvert p\rangle\otimes\beta\otimes\lvert f_k\rangle\bigr)=\lvert p\rangle\otimes C_k(p)\beta\otimes\lvert f_k\rangle,\qquad C_k(p):=-\tau\sin p\,\sigma_1+\mu_k\sigma_3,\qquad\mu_k:=m+d_k .
$$

Eq. (5.6) of 04-ilang-time/05-moving-clock, the eigenvalues $\pm\omega_k(p)$ and the real eigenvectors $\lvert p\rangle\otimes\beta_{k,\pm}(p)\otimes\lvert f_k\rangle$ of $C_b$:

$$
\omega_k(p)=\sqrt{\mu_k^2+\tau^2\sin^2p},\quad
\beta_{k,+}=\begin{pmatrix}\cos\frac{\theta_k}2\\ \sin\frac{\theta_k}2\end{pmatrix},\quad
\beta_{k,-}=\begin{pmatrix}-\sin\frac{\theta_k}2\\ \cos\frac{\theta_k}2\end{pmatrix},\quad
\tan\theta_k(p)=-\frac{\tau\sin p}{\mu_k},\ \ \lvert\theta_k\rvert<\frac\pi2 .
$$

Eq. (5.11) of 04-ilang-time/05-moving-clock, the bound (17.7) of 03-ilang-space/17-motion for this $C_b$ (the maximal speed):

$$
v_{\max}=\tau\,\kappa\sigma_X=\max_h\sum_{h'}\lvert t_{hh'}\rvert\,d_0(h,h') .
$$

From 08-maximal-speed@v3 of 04-ilang-time. Setting there: two bodies $a$, $b$ of different types and a medium $c$, with body contracts $C_A$, $C_B$ (diagonal entries $t^A_{hh}$, $t^B_{kk}$) and records $K^A_h$, $K^B_k$; the start is a product with the medium in $\chi$.

Eq. (8.1) of 04-ilang-time/08-maximal-speed, all replacements of the diagonal entries of the body contracts and of the records that leave the total contract unchanged, with arbitrary real $\alpha^A_h$, $\alpha^B_k$ and an arbitrary self-adjoint $Z$ on $\mathcal H_c$:

$$
t'^A_{hh}=t^A_{hh}+\alpha^A_h,\quad K'^A_h=K^A_h+Z-\alpha^A_h\mathbb 1_c,\qquad
t'^B_{kk}=t^B_{kk}+\alpha^B_k,\quad K'^B_k=K^B_k-Z-\alpha^B_k\mathbb 1_c ,
$$

Eq. (8.19) of 04-ilang-time/08-maximal-speed, the effective body contracts:

$$
C^{\rm eff}_A\otimes\mathbb 1_b+\mathbb 1_a\otimes C^{\rm eff}_B:=(\mathbb 1_{ab}\otimes\langle\chi\rvert)\,C\,(\mathbb 1_{ab}\otimes\lvert\chi\rangle),\qquad
C^{\rm eff}_A=C_A+\sum_h\langle K^A_h\rangle\lvert h\rangle\langle h\rvert,\quad
C^{\rm eff}_B=C_B+\sum_k\langle K^B_k\rangle\lvert k\rangle\langle k\rvert .
$$

Eq. (8.20) of 04-ilang-time/08-maximal-speed:

$$
\text{(8.1):}\quad C^{\rm eff}_A\mapsto C^{\rm eff}_A+\langle Z\rangle\,\mathbb 1_a,\quad C^{\rm eff}_B\mapsto C^{\rm eff}_B-\langle Z\rangle\,\mathbb 1_b;\qquad \chi\mapsto e^{i\theta}\chi:\quad C^{\rm eff}_A,\ C^{\rm eff}_B\ \text{unchanged}.
$$

From 03-ilang-space/17-motion@v2. Setting there: $C=C_b\otimes\mathbb 1_c+\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$; the start is $\lvert\phi\rangle\otimes\lvert\chi\rangle$; $\langle X\rangle:=\langle\chi\vert X\vert\chi\rangle$, $\lvert u_h\rangle:=(K_h-\langle K_h\rangle)\lvert\chi\rangle$, $d_0(h,h')=\lVert u_h-u_{h'}\rVert$; with $p_h(\lambda)$ the place weights of the view of $b$, $\bar x(\lambda):=\sum_hp_h(\lambda)\,u_h$.

Eq. (17.15) of 03-ilang-space/17-motion, for the chain $h_m$, $m\in\mathbb Z$, with $K_{h_m}=m\,X$, $X$ self-adjoint on $\mathcal H_c$, $\sigma_X:=(\langle X^2\rangle-\langle X\rangle^2)^{1/2}>0$ and $e:=(X-\langle X\rangle)\chi$, so that $\lVert e\rVert=\sigma_X$ and $u_{h_m}=m\,e$:

$$
d_0(h_m,h_n)=\lvert m-n\rvert\,\sigma_X,\qquad \bar x=\bar m\,e,\quad \bar m:=\sum_mm\,p_{h_m},\qquad s^2=\sigma_X^2\sum_m(m-\bar m)^2p_{h_m} .
$$

From 03-clock-and-fall@v3 (this subproject). Setting there: one body $b$ of the class $\mathcal C$ and a medium $c$ with the records $K_x=\kappa\,x\,X$. The class $\mathcal C$: places $(x,s,a)$ as for Model D and, with real numbers $m_x>0$, $\eta_x>0$, $\zeta_x>0$ and $V_x$,
$$C_b=A_\zeta+\sum_x\lvert x\rangle\langle x\rvert\otimes\Bigl[\sigma_3\otimes(m_x\mathbb 1_N+\eta_xD)+V_x\,\mathbb 1_2\otimes\mathbb 1_N\Bigr],\qquad A_\zeta=\frac\tau2\sum_x\zeta_x\Bigl(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert\otimes(i\sigma_1)\Bigr)\otimes\mathbb 1_N ,$$
with the linear profiles $m_x=m\,(1+g_mx)$, $\eta_x=1+g_\eta x$, $\zeta_x=1+g_\zeta(x+\tfrac12)$, $V_x=-\mathcal F\,x$ on a chain $\lvert x\rvert\le L$. Notation fixed there:
- $P_x:=\lvert x\rangle\langle x\rvert$, $\hat x:=\sum_xx\,P_x$, $d_k:=\frac{2\pi k}{N\lambda_0}$; $C_{\rm eff}$ is the effective body contract (8.19) of 04-ilang-time/08-maximal-speed, and all profiles are those of $C_{\rm eff}$;
- on the clock level $k$ (clock label in $\lvert f_k\rangle$; the level is conserved): $M_{x,k}:=m_x+\eta_xd_k$, the on-site block $M_{x,k}\sigma_3+V_x\mathbb 1_2$ and the hopping block $T_x=-\tfrac{i\tau}2\zeta_x\sigma_1$ from $x$ to $x+1$; for the linear profiles $M_{x,k}=M_k+G_kx$ with $M_k:=m+d_k$ and $G_k:=m\,g_m+d_k\,g_\eta$;
- the start $S_{\rm clock}(x_0)$: $\phi=\phi_X\otimes\lvert1\rangle\otimes\lvert0\rangle$ with $\phi_X(x)=c_W\,w\bigl((x-x_0)/W\bigr)$, $w$ real and smooth with support in $[-1,1]$ (a bare branch label $s=1$ and the clock label in the place $\lvert0\rangle$);
- $\tilde\phi_X(p):=\sum_xe^{-ipx}\phi_X(x)$, $\langle f\rangle_W:=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\lvert\tilde\phi_X(p)\rvert^2f(p)$, and $\delta_W^2:=\sum_x(\phi_X(x+1)-\phi_X(x))^2=O(W^{-2})$.

The variants named there, each with one number $g$: (a) $g_m=g_\eta=g$, $g_\zeta=0$, $\mathcal F=0$; (b1) $g_m=g$, the others zero; (b2) $g_\eta=g$, the others zero; (c) $g_m=g_\eta=g_\zeta=g$, $\mathcal F=0$; (d) $\mathcal F\neq0$, the others zero. The variant (f) of 04-branch-fall, a constant under the factor: the on-site block of the level $k$ is $(1+gx)\bigl[(m+d_k)\,\sigma_3+c_{\rm f}\,\mathbb 1_2\bigr]$ with a real number $c_{\rm f}$, that is $g_m=g_\eta=g$, $g_\zeta=0$, $\mathcal F=-c_{\rm f}\,g$.

Eq. (3.36) of 03-clock-and-fall, for the start $S_{\rm clock}(x_0)$ at zeroth order in the gradients and to first order in the level spacing, with the local parameters $\tau^0:=\tau(1+g_\zeta x_0)$, $m^0:=m_{x_0}$, $\eta^0:=\eta_{x_0}$, $\bar\mu:=m^0+\eta^0\frac{\pi(N-1)}{N\lambda_0}$ and $\Gamma(p):=\bar\mu/\sqrt{\bar\mu^2+\tau^{0\,2}\sin^2p}$. The clock statistics is, to this order, a mixture of the distributions of a ring clock with the advances $\pm\Gamma(p)\eta^0\lambda/\lambda_0$ and the weights $\tfrac12(1\pm\Gamma(p))\lvert\tilde\phi_X(p)\rvert^2\frac{\mathrm dp}{2\pi}$; $\vartheta_W$ is defined as the mean advance of this mixture, and $r_W$ is the rate at finite $W$:

$$
\vartheta_W(\lambda):=\Bigl\langle\tfrac{1+\Gamma}2\Gamma-\tfrac{1-\Gamma}2\Gamma\Bigr\rangle_W\frac{\eta^0\lambda}{\lambda_0}=(1-\varsigma_W)\frac{\eta^0\lambda}{\lambda_0},\qquad r_W(x_0):=\frac{\mathrm d\vartheta_W}{\mathrm d\lambda}=\frac{\eta^0}{\lambda_0}\,(1-\varsigma_W),\qquad\varsigma_W(x_0):=\Bigl\langle\frac{\tau^{0\,2}\sin^2p}{\bar\mu^2+\tau^{0\,2}\sin^2p}\Bigr\rangle_W,
$$

Eq. (3.23) of 03-clock-and-fall, the rate at $x_0$ (its wide-start form and its form at finite $W$), with $E^{\pm}_{x,k}:=V_x\pm M_{x,k}$:

$$
\text{rate at }x_0:=\lim_{W\to\infty}r_W(x_0)=\frac{\eta_{x_0}}{\lambda_0}=\frac{N}{2\pi}\bigl(E^+_{x_0,k+1}-E^+_{x_0,k}\bigr),\ \ \text{tick }\frac{\lambda_0}{\eta_{x_0}};\qquad\text{at finite }W:\ \ r_W=\frac{\eta_{x_0}}{\lambda_0}(1-\varsigma_W),\ \ \text{tick }\frac{\lambda_0}{\eta_{x_0}(1-\varsigma_W)},\ \ \Bigl\lvert r_W-\frac{\eta_{x_0}}{\lambda_0}\Bigr\rvert\le\frac{\eta_{x_0}}{\lambda_0}\,\delta_W^2\frac{\tau^{0\,2}}{m_{x_0}^2}.
$$

From 04-branch-fall@v3 (this subproject). Setting there: a body of the class $\mathcal C_{\rm lev}$ with linear level profiles (as under "Goal", one contract in the place of a sector contract), a medium with the records $K_x=\kappa\,x\,X$, branch starts $B_{k,\sigma}[g]$ as under "Goal"; first order in the gradients of the level $k$; records at leading order in $\kappa$. Inside one level the index $k$ is dropped: $\tau,M,G,\mathcal F,g_\zeta,\omega$; a prime is $\partial_p$; $\mathsf b:=\lvert G\rvert+M\lvert g_\zeta\rvert+\lvert\mathcal F\rvert$; $X_g:=\bigl(\int\frac{\mathrm dp}{2\pi}\lvert g'\rvert^2\bigr)^{1/2}$.

Eq. (4.7) of 04-branch-fall. The first-order velocity of $B_{k,\sigma}[g]$ is $\lambda\,\langle a_{k,\sigma}\rangle_g$ plus a rest that is bounded for all $\lambda\ge0$, with the coefficient

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

Eq. (4.10) of 04-branch-fall, with $a^{(1)}(\lambda)$ the first-order acceleration of $\bar x_{\rm lab}$ and $\overline{a^{(1)}}$ its average over a window $[\lambda_1,\lambda_1+\Lambda]$, for all $\lambda_1\ge0$, $\Lambda>0$ and every smooth $g$:

$$
\Bigl\lvert\overline{a^{(1)}}-\langle a_{k,\sigma}\rangle_g\Bigr\rvert\le\frac{2\tau}{M\Lambda}\,\mathsf b\,\Bigl(X_g+\frac{\tau}{2M}\Bigr)\ \ (\text{exact}),\qquad\lim_{\Lambda\to\infty}\overline{a^{(1)}}=\langle a_{k,\sigma}\rangle_g .
$$

Eq. (4.11) of 04-branch-fall, for every sequence of branch starts concentrated at $p_0$ with $\delta p\to0$:

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

Eq. (4.16) of 04-branch-fall, the drift acceleration at rest:

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

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

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

Eq. (4.22) of 04-branch-fall, for the class $\mathcal C$ and every $N\ge2$:

$$
a_{k,+}(0)\ \text{level-independent}\iff\mathcal F=m(g_m-g_\eta);\qquad a_{k,-}(0)\ \text{level-independent}\iff\mathcal F=-m(g_m-g_\eta);\qquad\text{value }-\tau^2g_\eta\text{ in each case};
$$

Eq. (4.24) of 04-branch-fall, for the class $\mathcal C$ under the hypothesis $a_{k,+}(0)=a\neq0$ for all $k$, at first order in the gradients and at leading order in $\kappa$. Here $r(x)=\eta_x/\lambda_0$ is the rate (3.23) in its wide-start form, $d_0:=\kappa\sigma_X$ is the distance of neighbouring positions in the background of (17.15), $\bar x=d_0\,\bar x_{\rm lab}\,\hat e$ with $\hat e:=e/\sigma_X$, and $v_{\max}=\tau d_0$ is the bound (5.11) of 04-ilang-time/05-moving-clock:

$$
\boxed{\;\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\Big\vert_{\rm drift}=d_0\,a_{k,+}(0)\,\hat e=-\,\frac{v_{\max}^2}{d_0}\;\frac{r(x+1)-r(x)}{r(x)}\;\hat e\;}
$$

## Assumptions

- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5), and $\lambda\ge0$.
- The three types are pairwise different and each has one instance, so no swap condition applies.
- The class $\mathcal S$ and its subclasses, the linear level profiles, the product start, the branch starts and the starts $S_{\rm clock}(x_0)$ are assumptions of this package (M8, M9). No state of the field is preferred (A11): every result is asked for every $\varphi$, through the weights $w_\nu$.
- The quoted results of 03-clock-and-fall and 04-branch-fall are used for each sector contract, with their starts, their orders and their limits; the package says where a statement needs a limit to be uniform over the sectors of $S$.
- A comparison of two runs at equal $\lambda$ uses the order of $\lambda$ (A8); the package says where it relies on it.
- Every limit is stated with its order: the limit of a sharp label, an average over a window, the growth of $\lambda$ and the expansion parameters. Where two of them do not commute, both orders are given, each with the quantity it defines. A statement about a rate of change (a velocity, an acceleration, the rate of a clock) is derived for that rate; it is not inferred from a bound or an estimate of the quantity whose derivative it is.
- Every statement with "all", "iff" or "well defined" is stated with its exceptional members (a sector of weight zero, sectors with equal data, a gradient that vanishes); a condition that is only sufficient is called sufficient.

## Scope

- **Class.** Item 1: every member of $\mathcal S$, every start. Item 2: every member of $\mathcal S_{\rm grad}$, both branches, every level, every $\varphi$. Item 3: every member of $\mathcal S^{\mathcal C}_{\rm grad}$, every $x_0$ on the chain, every $\varphi$. Item 4: every member of $\mathcal S_{\rm a}$ and of $\mathcal S_{\rm c}$, every $\varphi$. Item 5: as the results state.
- **Controls.** A sharp field ($S$ with one element) against a field that is not sharp, in every item; in item 2(c) a member whose mean drift acceleration is the same for all levels while its sector drift accelerations are not; in item 4(c) a distribution with a sharp $n$ against one in which $n$ is not sharp.
- In scope:
  - the statistics of the body and of its clock as a mixture over the sectors;
  - the fall, its independence of the level, the clock rate and the relation (4.24): sector by sector, and for the means over the sectors, at first order in the gradients;
  - the spread over the sectors of the displacement and of the advance of the clock, and their covariance.
- Out of scope:
  - a field whose operators do not commute with each other or with its own contract (10-field-own-contract and later packages);
  - the second order in the gradients within a sector, the fall in motion (the order $v^2$), lattice terms, and records beyond leading order in $\kappa$;
  - readings on the field, two bodies in one field, and instances of one type;
  - any physical meaning beyond the definitions (M1–M4); no theory of gravity and no theory that contains time may be used as an input.

## Depth

- Item 1: short argument.
- Item 2: (a) short argument; (b), (c) derive.
- Item 3: short argument.
- Item 4: (a), (b) short argument; (c) derive.
- Item 5: standard (a table of results with their tags).

## Expected result

- Item 1: relations with a statement on exactness; a formula for the weights with a condition; four yes/no answers.
- Item 2: a statement on the dependence with a closed form; a leading term with two equivalences; an equivalence, a yes/no answer and a control.
- Item 3: a mixture with a mean, a variance and a range; a closed form and its first order; a leading term with a condition; a condition and a statement.
- Item 4: lists of data with yes/no answers; components with a weight and a statement; two means, a variance, a closed form, an equivalence and a leading term.
- Item 5: a table.

Give every main result a tag $(9.k)$.

Consistency checks, at most three: for example a sharp field, two sectors with equal weights, and all gradients zero.

## Code

None.

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