Version 1 · earlier version; the current one is v3 · Review of v1

# A body in a static field that is not sharp: mixture over sectors, fall, clock rate and the relation (4.24)

- **Subproject:** 05-ilang-free-fall
- **Package:** 09-unsharp-field
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-11

## Setup and assumptions

**Setting (assumption of the package, M8).** The class $\mathcal S$ of the question: a body $b$, a field $f$, a medium $c$, with $C=\sum_\nu C^{(\nu)}_b\otimes P_\nu\otimes\mathbb 1_c+\mathbb 1_b\otimes C_f\otimes\mathbb 1_c+\sum_x\lvert x\rangle\langle x\rvert\otimes\mathbb 1_f\otimes\kappa xX$, $[C_f,P_\nu]=0$, sector contracts in $\mathcal C_{\rm lev}$ with linear level profiles, sector generators $C^{(\nu)}=C^{(\nu)}_b\otimes\mathbb 1_c+\sum_x\lvert x\rangle\langle x\rvert\otimes\kappa xX$, start $\phi\otimes\varphi\otimes\chi$, $w_\nu=\lVert P_\nu\varphi\rVert^2$, $S=\{\nu:w_\nu>0\}$, and the subclasses $\mathcal S_{\rm grad}$, $\mathcal S^{\mathcal C}_{\rm grad}$, $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$.

**Inputs (as quoted in the question).** (2.1), (2.15)–(2.17), (2.19), (2.20) of 02-field-object; (3.23), (3.36) of 03-clock-and-fall; (4.7), (4.10), (4.11), (4.16), (4.19), (4.22), (4.24) of 04-branch-fall; (5.5), (5.6), (5.11) of 04-ilang-time/05-moving-clock; (8.1), (8.19), (8.20) of 04-ilang-time/08-maximal-speed; (17.15) of 03-ilang-space/17-motion.

**Conventions of this file.**
- A superscript $(\nu)$ or $\nu$ marks a quantity computed with the sector generator $C^{(\nu)}$ from the start $\phi\otimes\chi$. $\mu_k:=m+d_k$, $\sigma_w(F):={\rm Var}_w(F)^{1/2}$, $\mathbf g^\nu_k:=(G^\nu_k,\mathcal F^\nu_k,g^\nu_{\zeta,k})$.
- Positions are in label units; $\bar x=d_0\,\bar x_{\rm lab}\,\hat e$ with $d_0=\kappa\sigma_X$ by (17.15) of 03-ilang-space/17-motion, the same in all sectors because the record term does not depend on the sector.
- Level profiles are those of the effective body contract (8.19) of 04-ilang-time/08-maximal-speed. Results are at leading order in $\kappa$ (weak records), first order jointly in the gradients at fixed $\lambda$, start and chain, then the long chain.
- Status words: **exact** (an identity or a bound with all constants), **order estimate** (leading behaviour with its calculation). Every result carries its class, its order of limits and its status in brackets.

## Derivation

### Step 1. The mixture (item 1)

**(a)** Apply (2.16) with $r=\{c\}$ and $\Psi_{br}=\phi\otimes\chi$; the partial trace over $c$ and every expectation are linear in the view:

$$
p_x(\lambda)=\sum_{\nu\in S}w_\nu\,p^{(\nu)}_x(\lambda),\qquad \bar x_{\rm lab}(\lambda)=\sum_{\nu\in S}w_\nu\,\bar x^{(\nu)}_{\rm lab}(\lambda),\qquad p(x,a;\lambda)=\sum_{\nu\in S}w_\nu\,p^{(\nu)}(x,a;\lambda).
\tag{9.1}
$$

[Class: every member of $\mathcal S$, every $\phi,\varphi,\chi$, every $\lambda\ge0$, every $\kappa$ and chain. Status: exact.] It rests on three facts only: the start is a product of $\phi\otimes\chi$ and $\varphi$; the $P_\nu$ are orthogonal projectors that commute with $C_f$, so the companion states $e^{-iC_f\lambda}\varphi_\nu$ stay orthonormal and every cross term between two sectors vanishes in the view (1.7); no reading acts on the field. Functionals that are not linear in the view do not average; for the position label

$$
{\rm Var}(x)(\lambda)=\bigl\langle{\rm Var}^{(\nu)}(x)\bigr\rangle_w+{\rm Var}_w\bigl(\bar x^{(\nu)}_{\rm lab}(\lambda)\bigr)\qquad(\text{exact, from (9.1)}).
\tag{9.2}
$$

**(b)** Let the reading at $\lambda_1$ have the projector $\Pi_r\otimes\mathbb 1_c\otimes\mathbb 1_f$, $p^{(\nu)}(r):=\lVert(\Pi_r\otimes\mathbb 1_c)\Psi^{(\nu)}(\lambda_1)\rVert^2$ and $p(r)=\sum_\nu w_\nu p^{(\nu)}(r)>0$. Writing (2.17) for the schedule $(r,r_2,\dots,r_k)$ and dividing by $p(r)$:

$$
p(r_2,\dots,r_k\mid r)=\sum_{\nu\in S}w_\nu(r)\,p^{(\nu)}(r_2,\dots,r_k\mid r),\qquad w_\nu(r)=\frac{w_\nu\,p^{(\nu)}(r)}{p(r)};\qquad w_\nu(r)=w_\nu\ \forall\nu\in S\iff p^{(\nu)}(r)=p(r)\ \forall\nu\in S .
\tag{9.3}
$$

[Class as (9.1), every schedule on body and medium. Status: exact.] A sector with $p^{(\nu)}(r)=0$ leaves $S$; a sector of weight zero never enters. The reading leaves the weights unchanged for all its outcomes iff its distribution is the same in all sectors of $S$. Example: a reading of the clock level (conserved by every sector generator) leaves them unchanged; a position reading at $\lambda_1>0$ changes them in general.

**(c)** For a traceless self-adjoint $T$ on $\mathcal H_b\otimes\mathcal H_c$, ${\rm Tr}_{bc}[(T\otimes\mathbb 1_f)C]=\sum_\nu{\rm Tr}(TC^{(\nu)})\,P_\nu$. Different sectors differ by more than a multiple of $\mathbb 1$, so a generic $T$ gives pairwise different coefficients, and the $P_\nu$ are the spectral projectors of an operator fixed by $C$. Given the $P_\nu$, $(\mathbb 1\otimes P_\nu)C(\mathbb 1\otimes P_\nu)=C^{(\nu)}\otimes P_\nu+\mathbb 1\otimes C_fP_\nu$ fixes $C^{(\nu)}$ up to $C^{(\nu)}\to C^{(\nu)}+c_\nu\mathbb 1$, $C_fP_\nu\to(C_f-c_\nu)P_\nu$. The replacement (2.20) with $\Phi_\nu=P_\nu$ keeps the spectral projectors and gives $C^{(\mu)}\to C^{(\mu)}-c\,\mathbb 1$ for all $\mu$ (the terms $\sum_\nu a_\nu B_\nu$ cancel). Hence:

$$
\begin{aligned}
&w_\nu:\ \text{yes (fixed by $C$ and $\varphi$, up to the naming of the sectors)};\qquad C^{(\nu)}_b:\ \text{no (only up to }c_\nu\mathbb 1_b);\\
&C^{(\nu)}_b-C^{(\nu')}_b:\ \text{yes under (2.20), no under the shifts }c_\nu\ \text{(only modulo }\mathbb R\,\mathbb 1_b);\qquad \tau^\nu_k,M^\nu_k,G^\nu_k,\mathcal F^\nu_k,g^\nu_{\zeta,k}:\ \text{yes}.
\end{aligned}
\tag{9.4}
$$

The shift $c_\nu$ changes only $V^\nu_{x,k}\to V^\nu_{x,k}+c_\nu$, and $\mathcal F^\nu_k=V^\nu_{x,k}-V^\nu_{x+1,k}$ is unchanged; the split between body and records is fixed by the effective body contract ((8.19), (8.20) of 04-ilang-time/08-maximal-speed). Without the condition that sectors differ by more than a multiple of $\mathbb 1_b$, only the sum of the weights of such sectors is well defined. [Class: every member of $\mathcal S$. Status: exact.]

### Step 2. Fall in $\mathcal S_{\rm grad}$: first order (item 2a)

In $\mathcal S_{\rm grad}$ the level block of the level $k$ in the sector $\nu$ is $C_0+G^\nu_k\,\mathcal W_G+\mathcal F^\nu_k\,\mathcal W_{\mathcal F}+g^\nu_{\zeta,k}\,\mathcal W_\zeta$ with operators $C_0,\mathcal W_i$ that do not depend on the sector, and the start $B_{k,\sigma}[g]\otimes\chi$ is the same in all sectors and stays in the level $k$. On a finite chain $e^{-iC^{(\nu)}\lambda}$ is entire in the gradients, so $p^{(\nu)}_x=p^0_x+\mathbf g^\nu_k\cdot\mathbf q_x+O(\lvert\mathbf g^\nu_k\rvert^2)$ with sector-independent $p^0_x,\mathbf q_x$, and by (9.1)

$$
p_x(\lambda)=p^0_x(\lambda)+\langle\mathbf g_k\rangle_w\cdot\mathbf q_x(\lambda)+O\bigl(\langle\lvert\mathbf g_k\rvert^2\rangle_w\bigr).
\tag{9.5}
$$

To first order the place weights depend on the state of the field only through the three numbers $\langle G_k\rangle_w$, $\langle\mathcal F_k\rangle_w$, $\langle g_{\zeta,k}\rangle_w$: they are those of the sharp field with the mean gradients. By (2.17) the same holds for every schedule. The second order contains $\langle g_ig_j\rangle_w=\langle g_i\rangle_w\langle g_j\rangle_w+{\rm Cov}_w(g_i,g_j)$, so the field state first enters beyond its means through the covariances of the gradients. [Class: $\mathcal S_{\rm grad}$, every level, branch, $g$, $\varphi$; fixed $\lambda$ and chain. Status: exact first-order statement.]

The first-order acceleration, its window average and $a_{k,\sigma}(p)$ of (4.7) are linear in $\mathbf g^\nu_k$ with sector-independent coefficients, and $S$ is finite. Averaging (4.10), (4.11) with the weights $w_\nu$:

$$
\bar a_{k,\sigma}(p_0):=\lim_{\delta p\to0}\lim_{\Lambda\to\infty}\overline{a^{(1)}}=\langle a_{k,\sigma}(p_0)\rangle_w=a_{k,\sigma}(p_0)\Big\vert_{\mathbf g_k\to\langle\mathbf g_k\rangle_w},\qquad \bar a_{k,\sigma}(0)=\frac{\tau_k^2}{M_k}\bigl(\sigma\langle\mathcal F_k\rangle_w-\langle G_k\rangle_w\bigr),
\tag{9.6}
$$

with $\bigl\lvert\overline{a^{(1)}}-\langle\langle a_{k,\sigma}\rangle_g\rangle_w\bigr\rvert\le\frac{2\tau_k}{M_k\Lambda}\langle\mathsf b_k\rangle_w\bigl(X_g+\frac{\tau_k}{2M_k}\bigr)$ and $\bigl\lvert\langle\langle a_{k,\sigma}\rangle_g\rangle_w-\bar a_{k,\sigma}(p_0)\bigr\rvert\le\frac12\langle\sup_p\lvert a''_{k,\sigma}\rvert\rangle_w\,\delta p^2$. [Class: as (9.5). Limits: first order in the gradients, long chain, leading $\kappa$, then $\Lambda\to\infty$, then $\delta p\to0$. Status: exact bounds, exact limit.] Forms of the earlier results for the mean position label:
- (4.16) holds as (9.6), and (4.19) holds with $G_k,\mathcal F_k$ replaced by $\langle G_k\rangle_w,\langle\mathcal F_k\rangle_w$; in particular both branches with one value $a$ iff $\langle\mathcal F_k\rangle_w=0$ and $\tau_k^2\langle G_k\rangle_w/M_k=-a$ for all $k$.
- (4.22) holds in $\mathcal S^{\mathcal C}_{\rm grad}$ as: $\bar a_{k,\pm}(0)$ is level-independent iff $\langle\mathcal F\rangle_w=\pm m\bigl(\langle g_m\rangle_w-\langle g_\eta\rangle_w\bigr)$, with the value $-\tau^2\langle g_\eta\rangle_w$.
- (4.24) holds for the mean rates $\bar r$ under the condition (9.15) of Step 5.

### Step 3. Spread of the displacement over the sectors (item 2b)

By (4.7) the first-order velocity in the sector $\nu$ is $\lambda\langle a^\nu_{k,\sigma}\rangle_g+\rho^\nu(\lambda)$. Both terms are linear in $\mathbf g^\nu_k$, so $\rho^\nu=\mathbf g^\nu_k\cdot\boldsymbol\rho$ with three sector-independent functions bounded for all $\lambda\ge0$: $R_i(g):=\sup_\lambda\lvert\rho_i\rvert<\infty$ and $R^\nu:=\sum_i\lvert g^\nu_{i,k}\rvert R_i(g)$. Integrating from $0$: $\Delta x^\nu_{k,\sigma}(\lambda)=\frac12\lambda^2\langle a^\nu_{k,\sigma}\rangle_g+E^\nu(\lambda)$ with $\lvert E^\nu\rvert\le\lambda R^\nu$. With the Cauchy–Schwarz inequality for ${\rm Cov}_w$:

$$
{\rm Var}_w\bigl(\Delta x_{k,\sigma}(\lambda)\bigr)=\frac{\lambda^4}4\,{\rm Var}_w\bigl(\langle a_{k,\sigma}\rangle_g\bigr)+{\rm rem},\qquad\lvert{\rm rem}\rvert\le\lambda^3\,\sigma_w\bigl(\langle a_{k,\sigma}\rangle_g\bigr)\langle R^2\rangle_w^{1/2}+\lambda^2\langle R^2\rangle_w .
\tag{9.7}
$$

[Class: $\mathcal S_{\rm grad}$, every $k,\sigma,g,\varphi$, all $\lambda\ge0$. Status: exact.] For a start at rest, (4.11) gives $\lvert\langle a^\nu_{k,\sigma}\rangle_g-a^\nu_{k,\sigma}(0)\rvert\le\frac12\sup\lvert a^{\nu\prime\prime}_{k,\sigma}\rvert\delta p^2$, and with (4.16)

$$
\lim_{\delta p\to0}\ \lim_{\lambda\to\infty}\frac{{\rm Var}_w\bigl(\Delta x_{k,\sigma}(\lambda)\bigr)}{\lambda^4}=\frac14{\rm Var}_w\bigl(a_{k,\sigma}(0)\bigr)=\frac{\tau_k^4}{4M_k^2}\Bigl[{\rm Var}_w(\mathcal F_k)+{\rm Var}_w(G_k)-2\sigma\,{\rm Cov}_w(\mathcal F_k,G_k)\Bigr].
\tag{9.8}
$$

[Limits, in this order: first-order term of each sector at fixed $\lambda$, $g$, chain; long chain; leading $\kappa$; $\lambda\to\infty$ at fixed $g$; $\delta p\to0$. Status: exact limit.] The last two limits commute only along sequences with $R_i(g)/\lambda\to0$; for $\delta p\to0$ at fixed $\lambda$ the bound in (9.7) diverges with the position spread $X_g$ of the start, and the inputs define no quantity in that order. Since the zeroth-order displacement is the same in all sectors of $\mathcal S_{\rm grad}$, (9.7) is also the leading (second) order in the gradients of the term ${\rm Var}_w(\bar x^{(\nu)}_{\rm lab})$ of (9.2). Because $\tau_k^2/M_k>0$:

$$
\begin{aligned}
(9.8)=0\ \text{for }(k,\sigma)&\iff\sigma\mathcal F^\nu_k-G^\nu_k\ \text{is the same for all }\nu\in S;\qquad\text{for }k\text{ and both }\sigma\iff\mathcal F^\nu_k\text{ and }G^\nu_k\text{ are each the same on }S;\\
\text{for all levels}&\iff\text{the respective condition holds for every }k=0,\dots,N-1 .
\end{aligned}
\tag{9.9}
$$

[Status: exact.] Exceptional members: a sharp field satisfies (9.9) trivially; sectors of weight zero do not enter; sectors of $S$ that differ only in $g_{\zeta,k}$ or only in other levels satisfy (9.9) although the field is not sharp. At finite $\delta p$ the coefficient $\frac14{\rm Var}_w(\langle a_{k,\sigma}\rangle_g)$ vanishes iff $\langle a^\nu_{k,\sigma}\rangle_g$ is the same on $S$, and differs from (9.8) by $O(\delta p^2)$. When the coefficient vanishes, $\lvert{\rm rem}\rvert\le\lambda^2\langle R^2\rangle_w$.

### Step 4. Independence of the level: sectors against the mean (item 2c)

(4.19) applied to each sector contract gives, for $N\ge2$:

$$
\begin{aligned}
a^\nu_{k,+}(0)\ \text{level-independent }\forall\nu\in S&\iff\forall\nu\in S\ \exists a^\nu_+:\ \tfrac{\tau_k^2}{M_k}(\mathcal F^\nu_k-G^\nu_k)=a^\nu_+\ \forall k;\\
\text{both branches}&\iff\forall\nu\in S:\ \tfrac{\tau_k^2\mathcal F^\nu_k}{M_k}=\tfrac{a^\nu_+-a^\nu_-}2\ \text{ and }\ \tfrac{\tau_k^2G^\nu_k}{M_k}=-\tfrac{a^\nu_++a^\nu_-}2\ \ \forall k
\end{aligned}
\tag{9.10}
$$

(with one value $a^\nu$ in both branches iff $\mathcal F^\nu_k=0$ and $\tau_k^2G^\nu_k/M_k=-a^\nu$). The values $a^\nu_\pm$ may differ between sectors. [Class: $\mathcal S_{\rm grad}$. Limits as (4.11). Status: exact.] A level-independent $\bar a_{k,\sigma}(0)$ is by (9.6) one condition on the means and does **not** imply (9.10). Control, a member of $\mathcal S^{\mathcal C}_{\rm grad}$ with two sectors, $\varepsilon\ne0$:

$$
w_1=w_2=\tfrac12,\quad g^{1,2}_\eta=g,\quad g^{1,2}_m=g\pm\varepsilon,\quad g^\nu_\zeta=\mathcal F^\nu=0:\qquad a^{1,2}_{k,\sigma}(0)=-\tau^2g\mp\frac{\tau^2\varepsilon m}{\mu_k},\qquad\bar a_{k,\sigma}(0)=-\tau^2g\ \ \forall k,\sigma .
\tag{9.11}
$$

The mean drift is the same for all levels in both branches, the sector drifts are not. In readings of the position, the mean and by (9.5) every first-order statistic are the same in the two cases. A start in one level determines, through (9.1) and (9.2), the distribution over the sectors of the sector drifts (weights $w_\nu$ at the values $a^\nu_{k,\sigma}(0)$), whose first moment is $\bar a_{k,\sigma}(0)$ and whose variance is (9.8). Under (9.10) this distribution is the same for all levels; in the control ${\rm Var}_w(a_{k,\sigma}(0))=\tau^4\varepsilon^2m^2/\mu_k^2$ depends on $k$. The distinction is of second order in the gradients. A single body cannot compare two levels inside one sector, so a member that violates (9.10) with a level-independent distribution is not distinguished by such readings.

### Step 5. Clock in $\mathcal S^{\mathcal C}_{\rm grad}$ (item 3)

**(a)** By (9.1) the statistics of the clock label is the mixture with the weights $w_\nu$ of the sector statistics (3.36), with the local parameters $\tau^{0,\nu},m^{0,\nu},\eta^{0,\nu}$ of the sector; in the wide-start form each sector is a ring clock with the advance $\vartheta^\nu=r^\nu(x_0)\lambda$, $r^\nu(x_0)=\eta^\nu_{x_0}/\lambda_0$ by (3.23). With $\eta^\nu_{x_0}=1+g^\nu_\eta x_0$:

$$
\bar\vartheta(\lambda;x_0)=\bigl(1+\langle g_\eta\rangle_wx_0\bigr)\frac\lambda{\lambda_0},\qquad\bar r(x_0):=\frac{\mathrm d\bar\vartheta}{\mathrm d\lambda}=\frac{1+\langle g_\eta\rangle_wx_0}{\lambda_0},\qquad{\rm Var}_w\bigl(\vartheta(\lambda;x_0)\bigr)={\rm Var}_w(g_\eta)\,x_0^2\,\frac{\lambda^2}{\lambda_0^2} .
\tag{9.12}
$$

The standard deviation stays below $\epsilon$ ticks (advance $\epsilon$) iff $\lambda<\lambda_\epsilon:=\epsilon\lambda_0/\bigl(\lvert x_0\rvert\sigma_w(g_\eta)\bigr)$, that is for fewer than $\epsilon(1+\langle g_\eta\rangle_wx_0)/(\lvert x_0\rvert\sigma_w(g_\eta))$ mean ticks; for all $\lambda$ if $x_0\,\sigma_w(g_\eta)=0$. At finite $W$: $\bar r_W=\langle\frac{\eta_{x_0}}{\lambda_0}(1-\varsigma_W)\rangle_w$ and $\lvert\bar r_W-\bar r\rvert\le\langle\frac{\eta_{x_0}}{\lambda_0}\delta_W^2\frac{\tau^{0\,2}}{m_{x_0}^2}\rangle_w$. [Class: $\mathcal S^{\mathcal C}_{\rm grad}$, every $x_0$, every $\varphi$. Orders: those of (3.36) in every sector (zeroth order in the gradients at the local parameters, first order in the level spacing), then $W\to\infty$, uniform because $S$ is finite. Status: the mixture is exact; (9.12) is exact given (3.23).] The place $x=0$ is the place where the factors $\eta^\nu_x$ of all sectors agree.

**(b)** Two runs with the starts $S_{\rm clock}(x_{1,2})\otimes\varphi\otimes\chi$ (the same $\varphi$), their mean advances compared at equal $\lambda$ (this uses the order of $\lambda$, A8):

$$
\frac{\bar r(x_1)}{\bar r(x_2)}=\frac{\bar\vartheta(\lambda;x_1)}{\bar\vartheta(\lambda;x_2)}=\frac{1+\langle g_\eta\rangle_wx_1}{1+\langle g_\eta\rangle_wx_2},\qquad\frac{\bar r(x+1)-\bar r(x)}{\bar r(x)}=\frac{\langle g_\eta\rangle_w}{1+\langle g_\eta\rangle_wx}=\langle g_\eta\rangle_w+O(g^2).
\tag{9.13}
$$

[Status: exact given (9.12); independent of $\lambda$ and of the unit of $\lambda$.] It is the ratio of the means, not the mean of the sector ratios $\eta^\nu_{x_1}/\eta^\nu_{x_2}$: two runs do not share a sector.

**(c)** With $\vartheta^\nu=(1+g^\nu_\eta x_0)\lambda/\lambda_0$ and $\Delta x^\nu_{k,\sigma}$ of Step 3 at the same $\lambda$, and $a^\nu_{k,\sigma}(0)=\tau^2(\sigma\mathcal F^\nu-mg^\nu_m-d_kg^\nu_\eta)/\mu_k$ from (4.16):

$$
{\rm Cov}_w\bigl(\vartheta,\Delta x_{k,\sigma}\bigr)=\frac{x_0\lambda^3}{2\lambda_0}{\rm Cov}_w\bigl(g_\eta,\langle a_{k,\sigma}\rangle_g\bigr)+{\rm rem}',\qquad\lim_{\delta p\to0}\lim_{\lambda\to\infty}\frac{{\rm Cov}_w}{\lambda^3}=\frac{x_0\tau^2}{2\lambda_0\mu_k}\Bigl[\sigma{\rm Cov}_w(g_\eta,\mathcal F)-m\,{\rm Cov}_w(g_\eta,g_m)-d_k{\rm Var}_w(g_\eta)\Bigr],
\tag{9.14}
$$

with $\lvert{\rm rem}'\rvert\le\lvert x_0\rvert\lambda^2\sigma_w(g_\eta)\langle R^2\rangle_w^{1/2}/\lambda_0$. [Class: $\mathcal S^{\mathcal C}_{\rm grad}$. Limits as (9.8). Status: exact.] For $x_0\ne0$ the advance and the leading displacement are functions of each other on $S$ iff $g^\nu_\eta=g^{\nu'}_\eta\Leftrightarrow a^\nu_{k,\sigma}(0)=a^{\nu'}_{k,\sigma}(0)$ for all $\nu,\nu'\in S$. A sufficient condition, which gives the same affine relation for all levels, is $\mathcal F^\nu=\sigma m(g^\nu_m-g^\nu_\eta)$ in every sector of $S$: then $a^\nu_{k,\sigma}(0)=-\tau^2g^\nu_\eta$, the limit in (9.14) is $-x_0\tau^2{\rm Var}_w(g_\eta)/(2\lambda_0)$, and the correlation is $-{\rm sgn}(x_0)$. At $x_0=0$ the advance is the same in all sectors and the covariance vanishes.

**(d)** By (9.6), $\bar a_{k,+}(0)=\tau^2\bigl(\langle\mathcal F\rangle_w-m\langle g_m\rangle_w-d_k\langle g_\eta\rangle_w\bigr)/\mu_k$, and by (9.13) with $v_{\max}=\tau d_0$ the right side of (4.24) for the rates $\bar r$ is $-d_0\tau^2\langle g_\eta\rangle_w\hat e$. Equating:

$$
d_0\,\bar a_{k,+}(0)\,\hat e=-\frac{v_{\max}^2}{d_0}\,\frac{\bar r(x+1)-\bar r(x)}{\bar r(x)}\,\hat e\quad\iff\quad\bigl\langle\mathcal F-m(g_m-g_\eta)\bigr\rangle_w=0 .
\tag{9.15}
$$

[Class: $\mathcal S^{\mathcal C}_{\rm grad}$, upper branch; first order in the gradients, leading $\kappa$, limits of (9.6). Status: exact.] The condition does not contain $k$: it holds for one level iff for all, and then $\bar a_{k,+}(0)=-\tau^2\langle g_\eta\rangle_w$ (the lower branch has $-m$ in place of $m$). Sector by sector, (4.24) holds in the sector $\nu$ iff $\mathcal F^\nu=m(g^\nu_m-g^\nu_\eta)$ (by (4.22)). This for all $\nu\in S$ is sufficient for (9.15) for every $w$ on $S$, and not necessary: in the control (9.11) the relation (9.15) holds and (4.24) holds in no sector, and for $w_1\ne w_2$ it fails, $\langle\mathcal F-m(g_m-g_\eta)\rangle_w=-m\varepsilon(w_1-w_2)$. The relation for the means is a property of the contract together with the state of the field.

### Step 6. Sector data of $\mathcal S_{\rm a}$ and $\mathcal S_{\rm c}$ (item 4a)

In $\mathcal S_{\rm a}$ the on-site block of the level $k$ is $(n^\nu+\gamma^\nu x)\mu_k\sigma_3$ and the motion term is $A$. In $\mathcal S_{\rm c}$, $\frac12\{\hat x,\lvert x+1\rangle\langle x\rvert\}=(x+\frac12)\lvert x+1\rangle\langle x\rvert$ gives the hopping factor $n^\nu+\gamma^\nu(x+\frac12)$, and the on-site block is the same. The effective contract adds $\mathcal F^\nu_k=-\kappa\langle X\rangle$, the same for all sectors and levels; it vanishes at the leading order in $\kappa$ used here. Each sector is the member of the class $\mathcal C$ with the data $(\tau^\nu,\,n^\nu m,\,N,\,\lambda_0/n^\nu)$ (since $n^\nu D$ is $D$ with $\lambda_0\to\lambda_0/n^\nu$) and $g^\nu:=\gamma^\nu/n^\nu$:

$$
\begin{array}{l|c|c}
 & \mathcal S_{\rm a} & \mathcal S_{\rm c}\\\hline
\tau^\nu_k,\ M^\nu_k,\ G^\nu_k,\ g^\nu_{\zeta,k},\ \mathcal F^\nu_k & \tau,\ n^\nu\mu_k,\ \gamma^\nu\mu_k,\ 0,\ 0 & n^\nu\tau,\ n^\nu\mu_k,\ \gamma^\nu\mu_k,\ \gamma^\nu/n^\nu,\ 0\\
\text{class }\mathcal C & \text{variant (a)},\ g=g^\nu,\ \tau^\nu=\tau & \text{variant (c)},\ g=g^\nu,\ \tau^\nu=n^\nu\tau\\
r^\nu(x)\ \text{(3.23), wide start} & (n^\nu+\gamma^\nu x)/\lambda_0 & (n^\nu+\gamma^\nu x)/\lambda_0\\
v^\nu_{\max}\ \text{(5.11)} & \tau d_0 & n^\nu\tau d_0\\
a^\nu_{k,\pm}(0)\ \text{(4.16)} & -\tau^2\gamma^\nu/n^\nu & -\tau^2n^\nu\gamma^\nu
\end{array}
\tag{9.16}
$$

Within a sector the drift acceleration from rest is the same for all levels and both branches (yes, both subclasses), and (4.24) holds with the sector's own $v^\nu_{\max}$ (yes, both): $-(v^{\nu}_{\max})^2\gamma^\nu/(d_0n^\nu)=d_0a^\nu_{k,\pm}(0)$ at first order. [Class: every member of $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$. Orders: those of (3.23), (4.16), (4.24). Status: exact given the inputs.]

### Step 7. Branch starts in different sectors (item 4b)

By (5.6) of 04-ilang-time/05-moving-clock, $\tan\theta^\nu_k=-\tau^\nu_k\sin p/M^\nu_k$. In $\mathcal S_{\rm c}$ this is $-\tau\sin p/\mu_k$ for every $\nu$: the branches and branch starts are the same in all sectors. In $\mathcal S_{\rm a}$, with $\Delta^\nu_k(p):=\theta^{\nu_0}_k-\theta^\nu_k=\arctan\frac{\tau\sin p}{n^\nu\mu_k}-\arctan\frac{\tau\sin p}{n^{\nu_0}\mu_k}$:

$$
\beta^{\nu_0}_{k,\sigma}=\cos\tfrac{\Delta^\nu_k}2\,\beta^\nu_{k,\sigma}+\sigma\sin\tfrac{\Delta^\nu_k}2\,\beta^\nu_{k,-\sigma},\qquad{\sf W}^\nu_k[g]:=\bigl\langle\sin^2\tfrac{\Delta^\nu_k}2\bigr\rangle_g\le\frac{\tau^2}{4\mu_k^2}\Bigl(\frac1{n^\nu}-\frac1{n^{\nu_0}}\Bigr)^2\delta p^2\ \ (\text{start at rest}),
\tag{9.17}
$$

so $B^{\nu_0}_{k,\sigma}[g]$ has in the sector $\nu$ the amplitudes $g\cos\frac{\Delta}2$ in the branch $\sigma$ and $\sigma g\sin\frac\Delta2$ in the branch $-\sigma$. The bound uses $\sin^2\frac\Delta2\le\frac{\Delta^2}4$, $\lvert\arctan u-\arctan u'\rvert\le\lvert u-u'\rvert$ and $\langle\sin^2p\rangle_g\le\delta p^2$; it is the leading term when $\langle p^4\rangle_g=o(\delta p^2)$. [Status: exact identity and bound.] For smooth $g$ the start is a branch start of the sector $\nu$ iff $n^\nu=n^{\nu_0}$.

For $n^\nu\ne n^{\nu_0}$ write the start as $\phi_\sigma+\phi_{-\sigma}$ in the branches of $\nu$. The first-order velocity is a quadratic form, $v^{(1)}[\phi]=v^{(1)}[\phi_\sigma]+v^{(1)}[\phi_{-\sigma}]+\mathsf X(\lambda)$. By (4.7) for each component, and since $a^\nu_{k,+}(p)=a^\nu_{k,-}(p)=:a^\nu_k(p)$ for $\mathcal F=0$, the first two terms are $\lambda\langle a^\nu_k\rangle_g$ plus a bounded rest. For the cross part $\mathsf X$: on momentum amplitudes $\hat x=i\partial_p$, the level block is $H_0(p)+G\hat x\sigma_3$ with $H_0=\omega(P_+-P_-)$, the velocity is $H_0'(p)=-\tau\cos p\,\sigma_1$, and its first-order Heisenberg term is $\mathcal V_1(\lambda)=i\int_0^\lambda\mathrm ds\,[W_I(s),v_I(\lambda)]$ with $W=G\hat x\sigma_3$. In the blocks $P_\sigma\mathcal V_1P_{-\sigma}$ every term carries $e^{\pm2i\omega\lambda}$ or is bounded; the growing ones are $\lambda\,c(p)e^{\pm2i\omega\lambda}$ (branch-diagonal part of $W_I$ against the off-diagonal part of $v_I$) and $\lambda^2\omega'(p)c(p)e^{\pm2i\omega\lambda}$ (factors $s$, $\lambda$ arise only from $\partial_pe^{\pm i\omega s}$, always with $\omega'$). One integration by parts ($\omega'e^{2i\omega\lambda}=\partial_pe^{2i\omega\lambda}/2i\lambda$) reduces the second kind to the first, and stationary phase at the non-degenerate stationary points $p\in\{0,\pi,\pm\frac\pi2\}$ of $\omega$ gives $\int\mathrm dp\,A(p)e^{\pm2i\omega\lambda}=O(\lambda^{-1/2})$ for smooth $A$. Hence $\mathsf X(\lambda)=O(\lambda^{1/2})$, and

$$
\mathcal S_{\rm a}:\quad\overline{a^{(1),\nu}}=\langle a^\nu_k\rangle_g+O(\Lambda^{-1/2})\ \ (\Lambda\to\infty,\ \text{fixed }\lambda_1,g),\qquad\lim_{\delta p\to0}\lim_{\Lambda\to\infty}\overline{a^{(1),\nu}}=a^\nu_k(0)=-\tau^2\frac{\gamma^\nu}{n^\nu}\quad\forall\nu\in S .
\tag{9.18}
$$

[Status: order estimate for the cross part, with the calculation above; exact for the two branch parts.] Statement: in $\mathcal S_{\rm c}$, (4.16) gives the sector drift acceleration in every sector of $S$ for every branch start, with the limits and the exact bounds of (4.10), (4.11), uniform on the finite $S$. In $\mathcal S_{\rm a}$ this holds at finite $\delta p$ and $\Lambda$ only in the sectors with $n^\nu=n^{\nu_0}$; in the others it holds for starts at rest in the limit $\Lambda\to\infty$ followed by $\delta p\to0$, in which ${\sf W}^\nu_k\to0$ by (9.17), without the bound (4.10).

### Step 8. Means over the sectors in $\mathcal S_{\rm a}$ and $\mathcal S_{\rm c}$ (item 4c)

From (9.16) and (9.1), for the start $S_{\rm clock}(0)$ (at $x_0$: $n\to n+\gamma x_0$), in both subclasses:

$$
\bar\vartheta(\lambda)=\bar n\,\frac\lambda{\lambda_0},\qquad{\rm Var}_w(\vartheta)={\rm Var}_w(n)\frac{\lambda^2}{\lambda_0^2},\qquad\bar r(x)=\frac{\bar n+\bar\gamma x}{\lambda_0},\qquad\bar\varrho=\frac{\bar r(1)-\bar r(0)}{\bar r(0)}=\frac{\bar\gamma}{\bar n},\qquad\bar a=\begin{cases}-\tau^2\langle\gamma/n\rangle_w&\mathcal S_{\rm a}\\-\tau^2\langle n\gamma\rangle_w&\mathcal S_{\rm c}\end{cases}
\tag{9.19}
$$

Here the advance is not sharp already at $x_0=0$: its standard deviation is below $\epsilon$ ticks iff $\lambda<\epsilon\lambda_0/\sigma_w(n)$. By (9.18) and Step 7, $\bar a$ is also the drift acceleration from rest of the mean position label. For $\bar\gamma\ne0$, using $\langle\gamma\rangle_w=\langle n\rangle_w\langle\gamma/n\rangle_w+{\rm Cov}_w(n,\gamma/n)$ and $\langle n\gamma\rangle_w=\bar n\bar\gamma+{\rm Cov}_w(n,\gamma)$:

$$
\frac{\bar a}{\bar\varrho}=-\tau^2\,\frac{\bar n\,\langle\gamma/n\rangle_w}{\bar\gamma}\ \ (\mathcal S_{\rm a}),\qquad\frac{\bar a}{\bar\varrho}=-\tau^2\bar n^2\Bigl(1+\frac{{\rm Cov}_w(n,\gamma)}{\bar n\bar\gamma}\Bigr)\ \ (\mathcal S_{\rm c});\qquad\text{sharp value: }-\tau^2\ (\mathcal S_{\rm a}),\ \ -\tau^2\bar n^2\ (\mathcal S_{\rm c}),
\tag{9.20}
$$

where the sharp value is $-v_{\max}^2/d_0^2$ of (4.24) for the factor $\bar n+\bar\gamma x$, by (9.16). Equality holds iff ${\rm Cov}_w(n,\gamma/n)=0$ in $\mathcal S_{\rm a}$, and iff ${\rm Cov}_w(n,\gamma)=0$ in $\mathcal S_{\rm c}$. A sharp $n$ (${\rm Var}_w(n)=0$) is sufficient in both and necessary in neither. Controls with $n$ not sharp: $\gamma^\nu=g\,n^\nu$ gives equality in $\mathcal S_{\rm a}$ and $\bar a/\bar\varrho=-\tau^2\langle n^2\rangle_w$ in $\mathcal S_{\rm c}$; a sector-independent $\gamma$ gives equality in $\mathcal S_{\rm c}$ and $-\tau^2\bar n\langle1/n\rangle_w$, of modulus above $\tau^2$, in $\mathcal S_{\rm a}$. For a narrow distribution, $u:=(n-\bar n)/\bar n$ with $\lvert u\rvert\le\epsilon<1$ on $S$ and $\frac1{1+u}=1-u+u^2-\frac{u^3}{1+u}$:

$$
\mathcal S_{\rm a}:\quad\frac{\bar a}{\bar\varrho}=-\tau^2\Bigl[1+\frac{{\rm Var}_w(n)}{\bar n^2}-\frac{{\rm Cov}_w(n,\gamma)}{\bar n\bar\gamma}+\rho_3\Bigr],\qquad\rho_3=\frac{\langle(\gamma-\bar\gamma)u^2\rangle_w-\langle\gamma u^3/(1+u)\rangle_w}{\bar\gamma},
\tag{9.21}
$$

with $\lvert\rho_3\rvert\le\epsilon^2\bigl[\max_S\lvert\gamma-\bar\gamma\rvert+\epsilon\max_S\lvert\gamma\rvert/(1-\epsilon)\bigr]/\lvert\bar\gamma\rvert$, of third order in the joint spread of $n$ and $\gamma$; in $\mathcal S_{\rm c}$ the deviation ${\rm Cov}_w(n,\gamma)/(\bar n\bar\gamma)$ of (9.20) is exact. [Class: every member of $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$, every $\varphi$ with $\bar\gamma\ne0$. Orders: those of (9.16), (9.18). Status: (9.19)–(9.21) exact given (9.16); $\bar a$ as a drift of the mean label in $\mathcal S_{\rm a}$ has the status of (9.18).]

## Result

Summary (item 5). "Sector" is every sector of $S$; orders are first order in the gradients and leading order in $\kappa$, with the limits of the cited tags.

| Quantity | Within every sector | Means in $\mathcal S_{\rm grad}$ | Means in $\mathcal S_{\rm a}$ | Means in $\mathcal S_{\rm c}$ |
| --- | --- | --- | --- | --- |
| Rate of the clock at $x$ | $\eta^\nu_x/\lambda_0$, (3.23) | $(1+\langle g_\eta\rangle_wx)/\lambda_0$; spread of the advance $\lvert x\rvert\sigma_w(g_\eta)\lambda/\lambda_0$, (9.12) | $(\bar n+\bar\gamma x)/\lambda_0$; spread $\sigma_w(n+\gamma x)\lambda/\lambda_0$, (9.19) | as $\mathcal S_{\rm a}$, (9.19) |
| Ratio of the rates at $x_1,x_2$ | $\eta^\nu_{x_1}/\eta^\nu_{x_2}$ | ratio of the means, (9.13) | $(\bar n+\bar\gamma x_1)/(\bar n+\bar\gamma x_2)$ | as $\mathcal S_{\rm a}$ |
| Drift acceleration from rest | (4.16); (9.16) | (4.16) at the mean gradients, (9.6); spread (9.8) | $-\tau^2\langle\gamma/n\rangle_w$, (9.19), limit (9.18) | $-\tau^2\langle n\gamma\rangle_w$, (9.19) |
| Its independence of the level | (4.19), i.e. (9.10) | (4.19) at the mean gradients; weaker than (9.10), control (9.11) | yes, both branches, (9.16) | yes, both branches, (9.16) |
| Relation (4.24) | iff $\mathcal F^\nu=m(g^\nu_m-g^\nu_\eta)$; always in $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$ with $v^\nu_{\max}$ | iff $\langle\mathcal F-m(g_m-g_\eta)\rangle_w=0$, (9.15) | iff ${\rm Cov}_w(n,\gamma/n)=0$, (9.20), (9.21) | iff ${\rm Cov}_w(n,\gamma)=0$, with $v_{\max}=\bar n\tau d_0$, (9.20) |

Tagged results: (9.1) mixture; (9.2) total variance; (9.3) weights after a reading; (9.4) well-defined quantities; (9.5) first-order place weights; (9.6) mean drift acceleration; (9.7), (9.8) spread of the displacement; (9.9) its vanishing; (9.10) level independence in every sector; (9.11) control; (9.12) clock mixture; (9.13) ratio of mean rates; (9.14) covariance; (9.15) relation (4.24) for the means; (9.16) sector data of $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$; (9.17) branch components; (9.18) drift in every sector of $\mathcal S_{\rm a}$; (9.19) means; (9.20) ratio $\bar a/\bar\varrho$; (9.21) leading deviation.

## Consistency checks

1. **Dimensions.** Labels are pure numbers; $\tau,M,G,\mathcal F,1/\lambda_0$ are costs ($1/\lambda$), $g_\zeta,g_m,g_\eta,n,\gamma/n$ are pure numbers. (9.6): $\tau^2G/M\sim\lambda^{-2}$. (9.8): $\lambda^4\cdot\lambda^{-4}$, a pure number. (9.14): $\lambda^3\tau^2/\lambda_0\sim1$. (9.20): $\bar a/\bar\varrho\sim\tau^2=v_{\max}^2/d_0^2$. Passed.
2. **Sharp field.** For $S=\{\nu\}$ all ${\rm Var}_w$ and ${\rm Cov}_w$ vanish: (9.3) gives $w_\nu(r)=1$, (9.6) is (4.16), (9.8) and (9.14) vanish, (9.13) is $\eta_{x_1}/\eta_{x_2}$, (9.15) is the condition of (4.22), (9.20) gives the sharp values, and $\rho_3=0$. Passed.
3. **Two sectors with equal weights, control (9.11).** Direct evaluation: $\frac12(a^1+a^2)=-\tau^2g$ and ${\rm Var}_w=\frac14(a^1-a^2)^2=\tau^4\varepsilon^2m^2/\mu_k^2$, equal to (9.8) with ${\rm Var}_w(G_k)=m^2\varepsilon^2$, $M_k=\mu_k$; (9.15) reads $\frac12(-m\varepsilon)+\frac12(m\varepsilon)=0$. Passed.

## Open issues

- The cross part of the first-order velocity in $\mathcal S_{\rm a}$, (9.18), is an order estimate $O(\lambda^{1/2})$ by stationary phase; a bound with explicit constants, of the kind of (4.10), was not derived.
- The constants $R_i(g)$ of (9.7) are taken from the boundedness stated in (4.7); their explicit form in terms of $X_g$ is not quoted, so the joint limit $\lambda\to\infty$, $\delta p\to0$ of (9.8) is stated only in the order given.
- The term $\langle{\rm Var}^{(\nu)}(x)\rangle_w$ of (9.2) at second order in the gradients is outside the scope; only the term over the sectors is determined.

## Methods used

- Block decomposition of a contract over commuting projectors; mixture (incoherent sum) over sectors
- Conditional weights after a reading (Bayes rule from joint statistics)
- First-order perturbation theory (Dyson expansion) in the gradients; linearity in the sector gradients
- Law of total variance; Cauchy–Schwarz inequality for covariances
- Branch (eigenvector) decomposition of two-band blocks; stationary phase and integration by parts for oscillatory integrals
- Moment expansion of a ratio of means

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