Version 3 · current

# 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:** v3
- **Mode:** external regeneration
- **Date:** 2026-10-11

## Changes from previous version

External regeneration after `verification.md` (external round 1; I1–I6, all minor). Everything that the six issues do not touch is the text of v2, word for word, with the same tags and the same order of the steps.

- **I1 (the maximal speed in $\mathcal S_{\rm c}$).** Step 6: the row $v^\nu_{\max}$ of (9.16) is **restated** as the reference speed $v^\nu_{\max}:=\tau^\nu d_0$, the bound (5.11) of 04-ilang-time/05-moving-clock at zero gradients; its values are unchanged. **New (9.25)**: the bound (5.11) of 04-ilang-time/05-moving-clock for the full sector contract, with the row sums and the ends of the chain. The words "reference speed" are added where $v^\nu_{\max}$ or $v_{\max}$ is used: after (9.23), after (9.20), in the Result table and in the JSON twin.
- **I2 (narrow distribution).** Step 8: the display (9.21) and the bound on $\rho_3$ are unchanged. The clause "of third order in the joint spread of $n$ and $\gamma$" is replaced by **new (9.26)** (the split of $\rho_3$ into the mixed moment $\mathsf m_3$ and a rest $\rho_4$, with bounds, and the form with three moments) and by the statement of the assumption on the joint distribution under which the two moments give the leading deviation, of the leading deviation without it, and a control.
- **I3 (degenerate cases).** Step 5(c): the qualifications $x_0\ne0$, $\lambda>0$, ${\rm Var}_w(g_\eta)>0$, and the condition at $x_0=0$. Step 8 after (9.19): the case ${\rm Var}_w(n)=0$.
- **I4 (scope of the clock rows).** Result: the cells of the rows "rate", "ratio" and "relation (4.24)" in the columns "Within every sector" and "Means in $\mathcal S_{\rm grad}$" are restricted to sector contracts of the class $\mathcal C$ and to $\mathcal S^{\mathcal C}_{\rm grad}$; the sentence above the table and the last bullet of Step 2 say the same.
- **I5 (zero gradients of a level).** Step 3 after (9.8): the growth of the bound is stated for $\langle\mathsf b_k^2\rangle_w>0$ and $\lambda>0$, and the case $\langle\mathsf b_k^2\rangle_w=0$ is stated. The same qualification is added in the bracket of (9.14) and in the second open issue.
- **I6 (vanishing mean gradient).** Step 8: **new (9.27)**, the comparison with the sharp field without division, for every state of the field, and the statement for $\bar\gamma=0$ with a control. The class bracket of Step 8, the Result table and the JSON twin name the range $\bar\gamma\ne0$ of (9.20), (9.21), (9.24), (9.26).
- **Tags.** Displays unchanged: (9.1)–(9.15), (9.17)–(9.24). Restated: (9.16) (the label of the row $v^\nu_{\max}$). New: (9.25), (9.26), (9.27). Withdrawn: none.
- Consistency checks 1 and 2 are extended to the new displays; one keyword is added under "Methods used".

## Response to verification

- **I1: Accepted and fixed in Step 6** (row $v^\nu_{\max}$ of (9.16), new (9.25) with the paragraph around it, sentence after (9.23)), in Step 8 (sentence after (9.20)) and in the Result table. $n^\nu\tau d_0$ is now named the reference speed at zero gradients; the bound of the full sector contract is given with its dependence on the place and with the ends of the chain.
- **I2: Accepted and fixed in Step 8** (new (9.26) and the paragraph "Which narrowness is needed"). The two-moment form holds iff the mixed moment $\mathsf m_3$ is small against $\bar\gamma\,{\rm Var}_w(n)$; joint narrowness is sufficient; for narrow $n$ alone three moments are needed. The example of the verifier is used as the control.
- **I3: Accepted and fixed in Step 5(c)** (paragraph after (9.14)) **and in Step 8** (sentences after (9.19)).
- **I4: Accepted and fixed in the Result section** (sentence above the table and the three clock rows) and in the last bullet of Step 2.
- **I5: Accepted and fixed in Step 3** (paragraph after (9.8)), in the bracket of (9.14) in Step 5(c) and in the second open issue.
- **I6: Accepted and fixed in Step 8** (new (9.27) and the paragraph "Every state of the field; the case $\bar\gamma=0$", class bracket) and in the Result table.

## 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. As in the question and in 04-branch-fall, "leading order in $\kappa$" means that the body evolves by the effective body contract. The plain gradient $\mathcal F_X:=-\kappa\langle X\rangle$ of the record term $\kappa\,x\,\langle X\rangle$, which is counted in every $V^\nu_{x,k}$, is one of the first-order gradients and is kept; no order is assumed between $\mathcal F_X$ and the gradients produced by the field.
- 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: for a start whose clock label is in one vector $\lvert f_k\rangle$, the reading of the clock label $a$ (the partition of the places $(x,s,a)$ by $a$) leaves them unchanged at every $\lambda_1$: every sector generator commutes with $\lvert f_k\rangle\langle f_k\rvert$, so the state of every sector stays a product with $\lvert f_k\rangle$ and $p^{(\nu)}(a)=\lvert\langle a\vert f_k\rangle\rvert^2=1/N$ for all $\nu$. 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. This is a statement for $\mathcal S^{\mathcal C}_{\rm grad}$, the subclass in which the rates are defined; for a member of $\mathcal S_{\rm grad}$ outside it the quoted inputs define no rate, and no form of (4.24) is stated.

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

By (4.7) the first-order velocity in the sector $\nu$ is $\lambda\langle a^\nu_{k,\sigma}\rangle_g+\rho^\nu(\lambda)$. The rest $\rho^\nu$ is bounded with explicit constants by (4.10), which holds for every window, also for the window $[0,\lambda]$ that starts at $\lambda=0$. For this window $\overline{a^{(1)}}=\bigl(v^{(1),\nu}(\lambda)-v^{(1),\nu}(0)\bigr)/\lambda$, with $v^{(1),\nu}$ the first-order velocity, so $\lvert\rho^\nu(\lambda)-\rho^\nu(0)\rvert\le\frac{2\tau_k}{M_k}\mathsf b^\nu_k\bigl(X_g+\frac{\tau_k}{2M_k}\bigr)$ and $\rho^\nu(0)=v^{(1),\nu}(0)$. The velocity at $\lambda=0$: $\mathrm d\bar x_{\rm lab}/\mathrm d\lambda=\langle i[C,\hat x]\rangle$ (exact); the record term and the on-site terms commute with $\hat x$, and on the level $k$ the hopping term gives $i[C,\hat x]=\hat v_0+g^\nu_{\zeta,k}\,\tfrac12\{\hat x,\hat v_0\}$ with $\hat v_0=-\tau_k\cos\hat p\,\sigma_1$. The start does not depend on the gradients, hence $v^{(1),\nu}(0)=g^\nu_{\zeta,k}\,{\rm Re}\langle\hat x\phi\vert\hat v_0\phi\rangle$. Here $\lVert\hat v_0\rVert=\tau_k$; $\hat x=i\partial_p$ acts on the amplitude $g\,\beta_{k,\sigma}$, so $\lVert\hat x\phi\rVert^2=X_g^2+\frac14\langle\theta_k'^2\rangle_g$, with $\lvert\theta_k'\rvert=\tau_kM_k\lvert\cos p\rvert/\omega_k^2\le\tau_k/M_k$ from (5.6) of 04-ilang-time/05-moving-clock; and $M_k\lvert g^\nu_{\zeta,k}\rvert\le\mathsf b^\nu_k$. Together:

$$
\lvert v^{(1),\nu}(0)\rvert\le\tau_k\lvert g^\nu_{\zeta,k}\rvert\Bigl(X_g+\frac{\tau_k}{2M_k}\Bigr),\qquad\lvert\rho^\nu(\lambda)\rvert\le R^\nu:=\frac{3\tau_k}{M_k}\,\mathsf b^\nu_k\,\Bigl(X_g+\frac{\tau_k}{2M_k}\Bigr)\ \ \forall\lambda\ge0,\qquad\mathsf b^\nu_k=\lvert G^\nu_k\rvert+M_k\lvert g^\nu_{\zeta,k}\rvert+\lvert\mathcal F^\nu_k\rvert .
\tag{9.22}
$$

[Class: $\mathcal S_{\rm grad}$, every $k$, $\sigma$, smooth $g$, all $\lambda\ge0$; first order in the gradients, long chain, leading $\kappa$. Status: exact bound.] 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 can be combined. By (9.7), (9.22) and (4.11), ${\rm Var}_w\bigl(\Delta x_{k,\sigma}(\lambda)\bigr)/\lambda^4\to\frac14{\rm Var}_w\bigl(a_{k,\sigma}(0)\bigr)$ along every sequence of starts at rest and of values of $\lambda$ with $\delta p\to0$, $\lambda\to\infty$ and $X_g/\lambda\to0$, because then $\langle R^2\rangle_w^{1/2}/\lambda=\frac{3\tau_k}{M_k}\langle\mathsf b_k^2\rangle_w^{1/2}\bigl(X_g+\frac{\tau_k}{2M_k}\bigr)/\lambda\to0$. This condition is sufficient; whether it is necessary is not shown. For the other order, $\delta p\to0$ at fixed $\lambda$ and then $\lambda\to\infty$, the following is shown: the coefficient $\frac14{\rm Var}_w(\langle a_{k,\sigma}\rangle_g)$ of (9.7) tends to $\frac14{\rm Var}_w(a_{k,\sigma}(0))$, and, for $\langle\mathsf b_k^2\rangle_w>0$ and at fixed $\lambda>0$, the bound on ${\rm rem}$ grows without limit along every such sequence on which $\lvert g(\pi)\rvert^2$ stays away from $1$, since $X_g\ge\bigl\lvert1-\lvert g(\pi)\rvert^2\bigr\rvert/(2\,\delta p)$ (from $\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,p\,\partial_p\lvert g\rvert^2=\lvert g(\pi)\rvert^2-1$ and the Cauchy–Schwarz inequality). Not shown, for $\langle\mathsf b_k^2\rangle_w>0$: whether ${\rm Var}_w\bigl(\Delta x_{k,\sigma}(\lambda)\bigr)$ has a limit for $\delta p\to0$ at fixed $\lambda$, and its value; no quantity is defined in that order. Exceptional case $\langle\mathsf b_k^2\rangle_w=0$: all gradients of the level $k$ vanish in every sector of $S$ (this is admissible also for a field that is not sharp, whose sectors differ in other levels). Then $R^\nu=0$ and $\langle a^\nu_{k,\sigma}\rangle_g=0$, so $\Delta x^\nu_{k,\sigma}(\lambda)=0$ for every $\nu\in S$, every $\lambda$ and every $g$; the bound on ${\rm rem}$ is zero, ${\rm Var}_w\bigl(\Delta x_{k,\sigma}(\lambda)\bigr)=0$, and its limit is zero in either 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 $\lvert{\rm rem}'\rvert\le\lvert x_0\rvert\lambda^2\sigma_w(g_\eta)\langle R^2\rangle_w^{1/2}/\lambda_0$ and $R^\nu$ of (9.22) at $\tau_k=\tau$, $M_k=\mu_k$. [Class: $\mathcal S^{\mathcal C}_{\rm grad}$. Limits as (9.8), with the same sufficient condition $X_g/\lambda\to0$ for the combined limit and the same statement on the other order; there the bound on ${\rm rem}'$ grows only at fixed $\lambda>0$ and for $x_0\,\sigma_w(g_\eta)\,\langle\mathsf b_k^2\rangle_w\ne0$, and if this product vanishes the covariance is zero for every $\lambda$ and $g$. Status: exact.] For $x_0\ne0$ and $\lambda>0$ the advance and the leading displacement $\frac12\lambda^2a^\nu_{k,\sigma}(0)$ 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 over the sectors of the advance and the leading displacement is $-{\rm sgn}(x_0)$ for $x_0\ne0$, $\lambda>0$ and ${\rm Var}_w(g_\eta)>0$; for ${\rm Var}_w(g_\eta)=0$ both are the same in all sectors of $S$ and no correlation coefficient is defined. At $x_0=0$ the advance is the same in all sectors and the covariance vanishes; for $\lambda>0$ the two are then functions of each other iff the leading displacement is the same in all sectors of $S$ as well, that is iff $a^\nu_{k,\sigma}(0)$ is the same for all $\nu\in S$, the condition (9.9) for $(k,\sigma)$. At $\lambda=0$ both vanish in every sector.

**(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. By the convention of this file it is one of the first-order gradients, $\mathcal F^\nu_k=\mathcal F_X$, and it is kept; it vanishes iff $\langle X\rangle=0$ ($\kappa>0$). 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 $\lambda_0\to\lambda_0/n^\nu$), $g^\nu:=\gamma^\nu/n^\nu$ and the plain gradient $\mathcal F=\mathcal F_X$:

$$
\begin{array}{l|c|c}
 & \mathcal S_{\rm a} & \mathcal S_{\rm c}\\\hline
\tau^\nu_k,\ M^\nu_k,\ G^\nu_k,\ g^\nu_{\zeta,k},\ \mathcal F^\nu_k & \tau,\ n^\nu\mu_k,\ \gamma^\nu\mu_k,\ 0,\ \mathcal F_X & n^\nu\tau,\ n^\nu\mu_k,\ \gamma^\nu\mu_k,\ \gamma^\nu/n^\nu,\ \mathcal F_X\\
\text{class }\mathcal C & g_m=g_\eta=g^\nu,\ g_\zeta=0,\ \mathcal F=\mathcal F_X,\ \tau^\nu=\tau & g_m=g_\eta=g_\zeta=g^\nu,\ \mathcal F=\mathcal F_X,\ \tau^\nu=n^\nu\tau\\
\text{variant, iff }\langle X\rangle=0 & \text{(a)},\ g=g^\nu & \text{(c)},\ g=g^\nu\\
r^\nu(x)\ \text{(3.23), wide start} & (n^\nu+\gamma^\nu x)/\lambda_0 & (n^\nu+\gamma^\nu x)/\lambda_0\\
v^\nu_{\max}:=\tau^\nu d_0\ \text{(reference speed: (5.11) of 04-ilang-time/05-moving-clock at zero gradients)} & \tau d_0 & n^\nu\tau d_0\\
a^\nu_{k,\sigma}(0)\ \text{(4.16)} & -\tau^2\dfrac{\gamma^\nu}{n^\nu}+\sigma\dfrac{\tau^2\mathcal F_X}{n^\nu\mu_k} & -\tau^2n^\nu\gamma^\nu+\sigma\dfrac{\tau^2n^\nu\mathcal F_X}{\mu_k}
\end{array}
\tag{9.16}
$$

The level data, the rate $r^\nu(x)$ and $v^\nu_{\max}$ hold for every $\langle X\rangle$: the plain term is the same for all levels and cancels in $E^+_{x,k+1}-E^+_{x,k}$ of (3.23), and (5.11) of 04-ilang-time/05-moving-clock contains only off-diagonal entries.

The row $v^\nu_{\max}$ of (9.16) is the **reference speed** of the sector, $v^\nu_{\max}:=\tau^\nu d_0$: the bound (5.11) of 04-ilang-time/05-moving-clock for the sector contract with its gradients set to zero. It is the speed that enters (4.24), whose $v_{\max}=\tau d_0$ is defined in the same way. In $\mathcal S_{\rm c}$ it is not the bound (5.11) of 04-ilang-time/05-moving-clock for the full sector contract, because the gradient $\gamma^\nu$ changes the hopping entries. With $d_0(h,h')=\lvert x-x'\rvert\,d_0$ by (17.15) of 03-ilang-space/17-motion, only the hopping entries enter the bound. In the place basis there is one entry for each neighbour; between $x$ and $x+1$ its modulus is $\frac\tau2$ in $\mathcal S_{\rm a}$ and $\frac\tau2\bigl(n^\nu+\gamma^\nu(x+\frac12)\bigr)>0$ in $\mathcal S_{\rm c}$. In $\mathcal S_{\rm c}$ the row sum at a place with the position label $x$ is therefore $\tau d_0\,(n^\nu+\gamma^\nu x)$ for $\lvert x\rvert<L$ and $\frac{\tau d_0}2\bigl(n^\nu\pm\gamma^\nu(L-\frac12)\bigr)$ at the ends $x=\pm L$; in $\mathcal S_{\rm a}$ it is $\tau d_0$ and $\frac{\tau d_0}2$. With $t^\nu_{hh'}$ the entries of $C^{(\nu)}_b$ in the place basis, for a chain with $L\ge1$:

$$
\max_h\sum_{h'}\lvert t^\nu_{hh'}\rvert\,d_0(h,h')=\begin{cases}\tau\,d_0&\mathcal S_{\rm a}\\[0.5ex]\tau\,d_0\,\bigl(n^\nu+\lvert\gamma^\nu\rvert(L-1)\bigr)&\mathcal S_{\rm c}\end{cases}
\tag{9.25}
$$

[Class: every member of $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$, every $\langle X\rangle$, chain $\lvert x\rvert\le L$ with $L\ge1$. Status: exact.] In $\mathcal S_{\rm c}$ the maximum is taken at the interior place $x={\rm sgn}(\gamma^\nu)(L-1)$; the ends give less because $\lvert\gamma^\nu\rvert(L+1)<n^\nu$. This bound depends on the gradient and on the length of the chain, and by the same condition it lies in $[\,n^\nu\tau d_0,\,2n^\nu\tau d_0)$. It equals $v^\nu_{\max}\bigl(1+\lvert g^\nu\rvert(L-1)\bigr)$, so at a fixed chain it differs from the reference speed at first order in the gradient. In (4.24) the speed multiplies the first-order quantity $(r(x+1)-r(x))/r(x)$, so replacing $v^\nu_{\max}$ by (9.25) changes that relation at second order only, which is outside the order of this file. Every statement below that contains $v^\nu_{\max}$ or $v_{\max}$ refers to the reference speed.

By (4.16) the drift acceleration is the sum of the part produced by the field and of the part of the records, with $\tau^\nu=\tau$ in $\mathcal S_{\rm a}$ and $\tau^\nu=n^\nu\tau$ in $\mathcal S_{\rm c}$:

$$
\begin{aligned}
&a^\nu_{k,\sigma}(0)=a^\nu_{\rm f}+\sigma\,\delta a^\nu_k,\qquad a^\nu_{\rm f}:=-\frac{(\tau^\nu)^2\gamma^\nu}{n^\nu},\qquad\delta a^\nu_k:=\frac{(\tau^\nu)^2\mathcal F_X}{n^\nu\mu_k}=-\frac{(\tau^\nu)^2\kappa\langle X\rangle}{n^\nu\mu_k},\qquad\frac{\delta a^\nu_k}{a^\nu_{\rm f}}=\frac{\kappa\langle X\rangle}{\gamma^\nu\mu_k}\ \ (\gamma^\nu\neq0);\\
&a^\nu_{k,\sigma}(0)\ \text{the same for all }k\iff a^\nu_{k,+}(0)=a^\nu_{k,-}(0)\iff\text{(4.24) holds in the sector }\nu\text{ with }v^\nu_{\max}\iff\langle X\rangle=0 .
\end{aligned}
\tag{9.23}
$$

The equivalences use $N\ge2$, the pairwise different $\mu_k=m+d_k$ and $\tau^\nu,n^\nu,\mu_k,\kappa>0$. Within a sector the drift acceleration from rest is the same for all levels iff $\langle X\rangle=0$, and the same in both branches iff $\langle X\rangle=0$ (both subclasses). (4.24) holds with the sector's own reference speed $v^\nu_{\max}$ iff $\langle X\rangle=0$ (both): $-(v^{\nu}_{\max})^2\gamma^\nu/(d_0n^\nu)=d_0a^\nu_{\rm f}$ at first order, so $d_0a^\nu_{k,+}(0)$ differs from the right side of (4.24) by $d_0\,\delta a^\nu_k$ (for $\gamma^\nu=0$ and $\langle X\rangle=0$ the relation reads $0=0$). For every $\langle X\rangle$ the part $a^\nu_{\rm f}=\frac12\bigl(a^\nu_{k,+}(0)+a^\nu_{k,-}(0)\bigr)$ is the same for all levels and satisfies this relation. For $\langle X\rangle\ne0$ the part $\sigma\,\delta a^\nu_k$ depends on the level through $1/\mu_k$ and is odd in the branch; against the part produced by the field it has the size $\kappa\lvert\langle X\rangle\rvert/(\lvert\gamma^\nu\rvert\mu_k)\le\kappa\lvert\langle X\rangle\rvert/(\lvert\gamma^\nu\rvert m)$, the same in both subclasses. The condition $\langle X\rangle=0$ refers to the listed form of the subclasses; in invariant terms it is $\mathcal F^\nu_k=0$ for the effective body contract, which is well defined by (9.4). [Class: every member of $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$, every $\langle X\rangle$. 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 (the amplitudes $g\cos\frac\Delta2$ and $\sigma g\sin\frac\Delta2$ are smooth, and (4.7) is quadratic in the amplitude), the first two terms are $\lambda\,\bigl\langle\cos^2\frac{\Delta^\nu_k}2\,a^\nu_{k,\sigma}+\sin^2\frac{\Delta^\nu_k}2\,a^\nu_{k,-\sigma}\bigr\rangle_g$ plus a bounded rest. The equality $a^\nu_{k,+}(p)=a^\nu_{k,-}(p)$ is not used: by (4.7), $a^\nu_{k,\sigma}(p)-a^\nu_{k,-\sigma}(p)=2\sigma\mathcal F_X\,\tau^2\bigl((M^\nu_k)^2\cos2p-\tau^2\sin^4p\bigr)/(\omega^\nu_k)^3$, which vanishes identically iff $\langle X\rangle=0$ and has the modulus at most $2\tau^2\lvert\mathcal F_X\rvert/M^\nu_k$ (from $\lvert M^2\cos2p-\tau^2\sin^4p\rvert\le\omega^2$ and $\omega\ge M$). The coefficient therefore differs from $\langle a^\nu_{k,\sigma}\rangle_g$ by at most $\frac{2\tau^2\kappa\lvert\langle X\rangle\rvert}{n^\nu\mu_k}\,{\sf W}^\nu_k[g]$. For the cross part $\mathsf X$: on momentum amplitudes $\hat x=i\partial_p$, the level block is $H_0(p)+\hat x\,(G\sigma_3-\mathcal F_X\mathbb 1_2)$ 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 $W=\hat x\,(G\sigma_3-\mathcal F_X\mathbb 1_2)$. 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 $\omega'$); the constant matrix that multiplies $\hat x$ in $W$ does not enter this counting, so it holds for both parts of $W$. 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

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

The limit $\delta p\to0$ uses ${\sf W}^\nu_k\to0$ by (9.17) and (4.11) for $\langle a^\nu_{k,\sigma}\rangle_g$. [Class: $\mathcal S_{\rm a}$, starts $B^{\nu_0}_{k,\sigma}[g]$ at rest, every $\langle X\rangle$. Limits: first order in the gradients, long chain, leading $\kappa$, $\Lambda\to\infty$ at fixed $\lambda_1$ and $g$, then $\delta p\to0$. 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). Both statements hold for every $\langle X\rangle$, with $\mathcal F^\nu_k=\mathcal F_X$ in (4.16); for $\langle X\rangle\ne0$ the value depends on the level and the branch, (9.23).

### 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:

$$
\begin{aligned}
&\bar\vartheta(\lambda)=\bar n\,\frac\lambda{\lambda_0},\qquad{\rm Var}_w(\vartheta)={\rm Var}_w(n)\frac{\lambda^2}{\lambda_0^2},\qquad\bar r(x)=\frac{\bar n+\bar\gamma x}{\lambda_0},\qquad\bar\varrho=\frac{\bar r(1)-\bar r(0)}{\bar r(0)}=\frac{\bar\gamma}{\bar n},\\
&\bar a_{k,\sigma}(0):=\langle a_{k,\sigma}(0)\rangle_w=\bar a+\sigma\,\delta\bar a_k,\qquad\bar a=\begin{cases}-\tau^2\langle\gamma/n\rangle_w&\mathcal S_{\rm a}\\-\tau^2\langle n\gamma\rangle_w&\mathcal S_{\rm c}\end{cases}\qquad\delta\bar a_k=\frac{\tau^2\mathcal F_X}{\mu_k}\times\begin{cases}\langle1/n\rangle_w&\mathcal S_{\rm a}\\\bar n&\mathcal S_{\rm c}\end{cases}
\end{aligned}
\tag{9.19}
$$

Here the advance is not sharp already at $x_0=0$ when ${\rm Var}_w(n)>0$: its standard deviation $\sigma_w(n)\lambda/\lambda_0$ is then below $\epsilon$ ticks iff $\lambda<\epsilon\lambda_0/\sigma_w(n)$. For ${\rm Var}_w(n)=0$ the advance is the same in all sectors of $S$ and its spread over the sectors is zero for all $\lambda$, within the orders of (3.36) and (3.23). By (9.18) and Step 7, $\bar a_{k,\sigma}(0)$ is also the drift acceleration from rest of the mean position label for the level $k$ and the branch $\sigma$. The clock quantities of (9.19) hold for every $\langle X\rangle$. The mean of the sector drift accelerations equals $\bar a$, the part produced by the field, iff $\langle X\rangle=0$ (because $\langle1/n\rangle_w>0$, $\bar n>0$); for $\langle X\rangle\ne0$ the part $\sigma\,\delta\bar a_k$ depends on the level and is odd in the branch, with $\delta\bar a_k/\bar a=\kappa\langle X\rangle\langle1/n\rangle_w/(\mu_k\langle\gamma/n\rangle_w)$ in $\mathcal S_{\rm a}$ and $\kappa\langle X\rangle\,\bar n/(\mu_k\langle n\gamma\rangle_w)$ in $\mathcal S_{\rm c}$ (for $\bar a\ne0$). 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 $\bar n+\bar\gamma x$, with the reference speed $v_{\max}$ of (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 $\mathcal S_{\rm a}$. These statements concern the part $\bar a$ produced by the field and hold for every $\langle X\rangle$; for the mean $\bar a_{k,\sigma}(0)$ of the sector drift accelerations they hold as stated iff $\langle X\rangle=0$, and in general:

$$
\begin{aligned}
&\frac{\bar a_{k,\sigma}(0)}{\bar\varrho}=\frac{\bar a}{\bar\varrho}+\sigma\,\frac{\tau^2\mathcal F_X}{\mu_k\,\bar\gamma}\times\begin{cases}\bar n\,\langle1/n\rangle_w&\mathcal S_{\rm a}\\ \bar n^2&\mathcal S_{\rm c}\end{cases}\\
&\frac{\bar a_{k,\sigma}(0)}{\bar\varrho}=\text{sharp value for one }(k,\sigma)\iff\begin{cases}{\rm Cov}_w(n,\gamma/n)=-\sigma\,\dfrac{\mathcal F_X}{\mu_k}\,\bar n\,\langle1/n\rangle_w&\mathcal S_{\rm a}\\[1ex]{\rm Cov}_w(n,\gamma)=\sigma\,\dfrac{\mathcal F_X}{\mu_k}\,\bar n&\mathcal S_{\rm c}\end{cases}\\
&\text{for all levels, or for one level in both branches}\iff\langle X\rangle=0\ \text{ and }\ {\rm Cov}_w(n,\gamma/n)=0\ (\mathcal S_{\rm a}),\ \ {\rm Cov}_w(n,\gamma)=0\ (\mathcal S_{\rm c}).
\end{aligned}
\tag{9.24}
$$

The last line follows because the right sides of the second line depend on $k$ through $1/\mu_k$ and are odd in $\sigma$, while the left sides do not depend on $k$, $\sigma$ ($N\ge2$). For $\langle X\rangle\ne0$ a sharp $n$ is not sufficient: the relation (4.24) then fails for the sharp field itself, by (9.23), and equality for one $(k,\sigma)$ needs a covariance that compensates the part of the records for that level and branch only. 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 $\lvert\rho_3\rvert\le\epsilon^2\bigl[\max_S\lvert\gamma-\bar\gamma\rvert+\epsilon\max_S\lvert\gamma\rvert/(1-\epsilon)\bigr]/\lvert\bar\gamma\rvert$, which is of third order only when the spread of $\gamma$ is small as well, see (9.26); in $\mathcal S_{\rm c}$ the deviation ${\rm Cov}_w(n,\gamma)/(\bar n\bar\gamma)$ of (9.20) is exact. In the part of the records of (9.24), the same expansion gives $\bar n\langle1/n\rangle_w=1+{\rm Var}_w(n)/\bar n^2-\langle u^3/(1+u)\rangle_w$ in $\mathcal S_{\rm a}$, with $\lvert\langle u^3/(1+u)\rangle_w\rvert\le\epsilon^3/(1-\epsilon)$; in $\mathcal S_{\rm c}$ that part is exact.

**Which narrowness is needed ($\mathcal S_{\rm a}$).** The identity (9.21) and the bound on $\rho_3$ are exact, and the error of the two-moment form ${\rm Var}_w(n)/\bar n^2-{\rm Cov}_w(n,\gamma)/(\bar n\bar\gamma)$ is exactly $\rho_3$. Let $\epsilon_\gamma:=\max_S\lvert\gamma-\bar\gamma\rvert/\lvert\bar\gamma\rvert$ be the relative spread of $\gamma$, so that $\max_S\lvert\gamma\rvert\le(1+\epsilon_\gamma)\lvert\bar\gamma\rvert$. With $\langle u^2\rangle_w={\rm Var}_w(n)/\bar n^2$ and $\lvert u^3/(1+u)\rvert\le u^2\epsilon/(1-\epsilon)$:

$$
\begin{aligned}
&\rho_3=\frac{\mathsf m_3}{\bar n^2\bar\gamma}+\rho_4,\qquad\mathsf m_3:=\bigl\langle(\gamma-\bar\gamma)(n-\bar n)^2\bigr\rangle_w,\qquad\rho_4:=-\frac{\langle\gamma\,u^3/(1+u)\rangle_w}{\bar\gamma},\qquad\Bigl\lvert\frac{\mathsf m_3}{\bar n^2\bar\gamma}\Bigr\rvert\le\epsilon_\gamma\,\frac{{\rm Var}_w(n)}{\bar n^2},\qquad\lvert\rho_4\rvert\le\frac{\epsilon\,(1+\epsilon_\gamma)}{1-\epsilon}\,\frac{{\rm Var}_w(n)}{\bar n^2};\\
&\mathcal S_{\rm a}:\quad\frac{\bar a}{\bar\varrho}=-\tau^2\Bigl[1-\frac{{\rm Cov}_w(n,\gamma)}{\bar n\bar\gamma}+\frac{{\rm Var}_w(n)}{\bar n^2}+\frac{\mathsf m_3}{\bar n^2\bar\gamma}+\rho_4\Bigr].
\end{aligned}
\tag{9.26}
$$

[Status: exact identity and exact bounds.] The mixed moment is $\mathsf m_3={\rm Cov}_w\bigl(\gamma,(n-\bar n)^2\bigr)$. Consider a family of members and field states with $\epsilon\to0$ at bounded $\epsilon_\gamma$ and ${\rm Var}_w(n)>0$. By (9.26), $\rho_4$ is always small against ${\rm Var}_w(n)/\bar n^2$, hence:
- **Assumption on the joint distribution.** The error $\rho_3$ of the two-moment form is small against ${\rm Var}_w(n)/\bar n^2$ iff $\mathsf m_3/\bigl(\bar\gamma\,{\rm Var}_w(n)\bigr)\to0$: the mixed moment is small against $\bar\gamma\,{\rm Var}_w(n)$. This condition is necessary and sufficient. Joint narrowness, $\epsilon_\gamma\to0$ together with $\epsilon\to0$, is sufficient for it, and then $\lvert\rho_3\rvert\le\bigl[\epsilon_\gamma+\epsilon(1+\epsilon_\gamma)/(1-\epsilon)\bigr]{\rm Var}_w(n)/\bar n^2$, of third order in $(\epsilon,\epsilon_\gamma)$, while the two terms of the two-moment form are of second order (${\rm Var}_w(n)/\bar n^2\le\epsilon^2$, $\lvert{\rm Cov}_w(n,\gamma)\rvert/(\bar n\lvert\bar\gamma\rvert)\le\epsilon\,\epsilon_\gamma$). Under the assumption the leading deviation of $\bar a/\bar\varrho$ from $-\tau^2$ is $-\tau^2\bigl[{\rm Var}_w(n)/\bar n^2-{\rm Cov}_w(n,\gamma)/(\bar n\bar\gamma)\bigr]$. Exceptional members: if the two terms cancel to this accuracy, the two moments do not give the leading deviation (for $\gamma^\nu=g\,n^\nu$ they cancel exactly and the deviation is zero).
- **Without the assumption** (narrow $n$ alone). The deviation is given by the second line of (9.26). The term $-{\rm Cov}_w(n,\gamma)/(\bar n\bar\gamma)$ is of first order in $\epsilon$ (modulus at most $\epsilon\,\epsilon_\gamma$), and it alone is the leading deviation when it is not small against $\epsilon$. The second order in $\epsilon$ is ${\rm Var}_w(n)/\bar n^2+\mathsf m_3/(\bar n^2\bar\gamma)=\langle\gamma\,(n-\bar n)^2\rangle_w/(\bar n^2\bar\gamma)$, and it is the leading deviation when ${\rm Cov}_w(n,\gamma)=0$ and it does not vanish itself. The term with ${\rm Var}_w(n)$ thus enters only together with the mixed moment, which is of the same order and is not a function of ${\rm Var}_w(n)$ and ${\rm Cov}_w(n,\gamma)$: three moments are needed.
- **The form asked for in item 4(c),** the leading deviation for a narrow distribution of $n$ in terms of ${\rm Var}_w(n)$ and ${\rm Cov}_w(n,\gamma)$ alone, holds only under the assumption above; narrowness of $n$ alone does not give it. Control: three sectors with $w=(\frac14,\frac12,\frac14)$, $n=(1-\epsilon,1,1+\epsilon)$, $\gamma=(2h,h,2h)$, $h\ne0$ small enough to be admissible. Then $\bar n=1$, $\bar\gamma=\frac32h$, ${\rm Cov}_w(n,\gamma)=0$, ${\rm Var}_w(n)=\frac12\epsilon^2$, $\epsilon_\gamma=\frac13$, $\mathsf m_3/(\bar n^2\bar\gamma)=\frac16\epsilon^2$, and directly $\bar n\langle\gamma/n\rangle_w/\bar\gamma=1+\frac{2\epsilon^2}{3(1-\epsilon^2)}$: the deviation is $\frac23\epsilon^2+O(\epsilon^4)$, as (9.26) gives, and not the $\frac12\epsilon^2$ of the two-moment form.

**Every state of the field; the case $\bar\gamma=0$.** (9.20), (9.21), (9.24) and (9.26) divide by $\bar\gamma$ and hold for $\bar\gamma\ne0$. For $\bar\gamma=0$ the mean rates $\bar r$ are the same at all places, $\bar\varrho=0$, and neither $\bar a/\bar\varrho$ nor $\bar a_{k,\sigma}(0)/\bar\varrho$ is defined, also when the numerator vanishes. The comparison with the sharp field with the factor $\bar n+\bar\gamma x$ is made without division. Let $a^{\rm sh}$ be the drift acceleration from rest that (4.24) gives for that sharp field, with its reference speed $v_{\max}$ of (9.16). From (9.19) and the two covariance identities used for (9.20), for every $\varphi$:

$$
\bar a_{k,\sigma}(0)-a^{\rm sh}=\bigl(\bar a-a^{\rm sh}\bigr)+\sigma\,\delta\bar a_k,\qquad\bar a-a^{\rm sh}=\begin{cases}\dfrac{\tau^2}{\bar n}\,{\rm Cov}_w(n,\gamma/n)&\mathcal S_{\rm a}\\[1ex]-\tau^2\,{\rm Cov}_w(n,\gamma)&\mathcal S_{\rm c}\end{cases}\qquad a^{\rm sh}:=-\frac{v_{\max}^2}{d_0^2}\,\bar\varrho=\begin{cases}-\tau^2\,\bar\gamma/\bar n&\mathcal S_{\rm a}\\-\tau^2\,\bar n\,\bar\gamma&\mathcal S_{\rm c}\end{cases}
\tag{9.27}
$$

Hence $\bar a=a^{\rm sh}$ iff ${\rm Cov}_w(n,\gamma/n)=0$ in $\mathcal S_{\rm a}$ and iff ${\rm Cov}_w(n,\gamma)=0$ in $\mathcal S_{\rm c}$, and with $\delta\bar a_k$ of (9.19) the conditions of the second and third lines of (9.24) are the conditions for $\bar a_{k,\sigma}(0)=a^{\rm sh}$. In this form they hold for every $\varphi$, $\bar\gamma=0$ included; for $\bar\gamma\ne0$ they are the statements of (9.20) and (9.24). For $\bar\gamma=0$: $a^{\rm sh}=0$, and $\bar a=-\tau^2\langle\gamma/n\rangle_w=\tau^2{\rm Cov}_w(n,\gamma/n)/\bar n$ in $\mathcal S_{\rm a}$, $\bar a=-\tau^2\langle n\gamma\rangle_w=-\tau^2{\rm Cov}_w(n,\gamma)$ in $\mathcal S_{\rm c}$. This need not vanish: the mean position label can have a drift acceleration while the mean rates do not depend on the place. Control: two sectors with $w_1=w_2=\frac12$, $\gamma^{1,2}=\pm\gamma_0\ne0$ and $n^1\ne n^2$ give $\bar\gamma=0$ and $\bar a=-\frac12\tau^2\gamma_0\,(1/n^1-1/n^2)\ne0$ in $\mathcal S_{\rm a}$, $\bar a=-\frac12\tau^2\gamma_0\,(n^1-n^2)\ne0$ in $\mathcal S_{\rm c}$; with a sharp $n$ and $\bar\gamma=0$, $\bar a=a^{\rm sh}=0$. For narrow $n$ the form of (9.21), (9.26) without division is ${\rm Cov}_w(n,\gamma/n)={\rm Cov}_w(n,\gamma)/\bar n-\bigl(\bar\gamma\,{\rm Var}_w(n)+\mathsf m_3\bigr)/\bar n^2+\langle\gamma\,u^3/(1+u)\rangle_w$, with $\lvert\langle\gamma\,u^3/(1+u)\rangle_w\rvert\le\frac{\epsilon}{1-\epsilon}\max_S\lvert\gamma\rvert\,{\rm Var}_w(n)/\bar n^2$, for every $\bar\gamma$; at $\bar\gamma=0$ the term with ${\rm Var}_w(n)$ is absent and the second order is the mixed moment alone. [Class: every member of $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$, every $\langle X\rangle$; (9.19) and (9.27) for every $\varphi$; (9.20), (9.21), (9.24), (9.26) for every $\varphi$ with $\bar\gamma\ne0$. Orders: those of (9.16), (9.18). Status: (9.19)–(9.21), (9.24), (9.26), (9.27) exact given (9.16); the statements on the leading deviation hold under the assumptions named with them; $\bar a_{k,\sigma}(0)$ 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. $\mathcal F_X=-\kappa\langle X\rangle$; in the columns of $\mathcal S_{\rm a}$ and $\mathcal S_{\rm c}$ every entry carries its condition on $\langle X\rangle$. The clock rows (rate, ratio, relation (4.24)) are stated for sector contracts of the class $\mathcal C$ and, in the column of $\mathcal S_{\rm grad}$, for its subclass $\mathcal S^{\mathcal C}_{\rm grad}$: for a general member of $\mathcal C_{\rm lev}$ the numbers $\eta^\nu_x$, $g^\nu_\eta$, $g^\nu_m$ and the rate (3.23) are not defined, and only the exact mixture (9.1) of the clock statistics holds. $v^\nu_{\max}$ and $v_{\max}$ are the reference speeds of (9.16); the bound (5.11) of 04-ilang-time/05-moving-clock for the full sector contract is (9.25).

| 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 $x$ | sector contract of the class $\mathcal C$: $\eta^\nu_x/\lambda_0$, (3.23); other members of $\mathcal C_{\rm lev}$: no rate defined | in $\mathcal S^{\mathcal C}_{\rm grad}$: $(1+\langle g_\eta\rangle_wx)/\lambda_0$; spread of the advance $\lvert x\rvert\sigma_w(g_\eta)\lambda/\lambda_0$, (9.12); outside $\mathcal S^{\mathcal C}_{\rm grad}$: only the mixture (9.1) | $(\bar n+\bar\gamma x)/\lambda_0$; spread $\sigma_w(n+\gamma x)\lambda/\lambda_0$, (9.19); for every $\langle X\rangle$ | as $\mathcal S_{\rm a}$, (9.19) |
| Ratio of the rates at $x_1,x_2$ | class $\mathcal C$: $\eta^\nu_{x_1}/\eta^\nu_{x_2}$ | in $\mathcal S^{\mathcal C}_{\rm grad}$: ratio of the means, (9.13) | $(\bar n+\bar\gamma x_1)/(\bar n+\bar\gamma x_2)$; for every $\langle X\rangle$ | as $\mathcal S_{\rm a}$ |
| Drift acceleration from rest | (4.16); (9.16), (9.23) | (4.16) at the mean gradients, (9.6); spread (9.8) | $-\tau^2\langle\gamma/n\rangle_w+\sigma\tau^2\mathcal F_X\langle1/n\rangle_w/\mu_k$, (9.19), limit (9.18); the second term vanishes iff $\langle X\rangle=0$ | $-\tau^2\langle n\gamma\rangle_w+\sigma\tau^2\mathcal F_X\bar n/\mu_k$, (9.19); the second term vanishes iff $\langle X\rangle=0$ |
| Its independence of the level | (4.19), i.e. (9.10); in $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$ iff $\langle X\rangle=0$, (9.23) | (4.19) at the mean gradients; weaker than (9.10), control (9.11) | iff $\langle X\rangle=0$, then in both branches, (9.19) | iff $\langle X\rangle=0$, then in both branches, (9.19) |
| Relation (4.24) | class $\mathcal C$: iff $\mathcal F^\nu=m(g^\nu_m-g^\nu_\eta)$; in $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$ iff $\langle X\rangle=0$, with the reference speed $v^\nu_{\max}$, (9.23) | in $\mathcal S^{\mathcal C}_{\rm grad}$: iff $\langle\mathcal F-m(g_m-g_\eta)\rangle_w=0$, (9.15) | for all levels iff $\langle X\rangle=0$ and ${\rm Cov}_w(n,\gamma/n)=0$, for every $\varphi$ in the form (9.27); ratio (9.20), (9.24) for $\bar\gamma\ne0$ only; deviation (9.21), by two moments only under the assumption of (9.26) | for all levels iff $\langle X\rangle=0$ and ${\rm Cov}_w(n,\gamma)=0$, with the reference speed $v_{\max}=\bar n\tau d_0$, for every $\varphi$ in the form (9.27); ratio (9.20), (9.24) for $\bar\gamma\ne0$ only |

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 $\mathcal S_{\rm a}$, $\mathcal S_{\rm c}$ with the plain gradient of the records and the reference speed; (9.17) branch components; (9.18) drift in every sector of $\mathcal S_{\rm a}$; (9.19) means; (9.20) ratio $\bar a/\bar\varrho$ for the part produced by the field, for $\bar\gamma\ne0$; (9.21) deviation in two moments with its exact error $\rho_3$; (9.22) explicit bound on the rest of the first-order velocity; (9.23) split of the sector drift acceleration and the condition $\langle X\rangle=0$; (9.24) ratio for the mean with the part of the records, for $\bar\gamma\ne0$; (9.25) the bound (5.11) of 04-ilang-time/05-moving-clock for the full sector contract; (9.26) split of $\rho_3$, the assumption for the two-moment form and the form with three moments; (9.27) comparison with the sharp field without division, for every state of the field, with the case $\bar\gamma=0$.

## 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): $\bar a/\bar\varrho\sim\tau^2=v_{\max}^2/d_0^2$. $\mathcal F_X=-\kappa\langle X\rangle$ is a cost: (9.22) gives $R^\nu\sim\lambda^{-1}$, a velocity in labels; (9.23), (9.19): $\tau^2\mathcal F_X/(n\mu_k)\sim\lambda^{-2}$ and $\kappa\langle X\rangle/(\gamma\mu_k)$ is a pure number; (9.24): $\tau^2\mathcal F_X/(\mu_k\bar\gamma)\sim\tau^2$. (9.25): $\tau d_0$, a cost times a distance, as (5.11) of 04-ilang-time/05-moving-clock. (9.26): $\mathsf m_3/(\bar n^2\bar\gamma)$ and $\rho_4$ are pure numbers. (9.27): $\tau^2{\rm Cov}_w(n,\gamma/n)/\bar n\sim\lambda^{-2}$. 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 $\rho_3=0$. With $\bar n\langle1/n\rangle_w=1$, (9.19) and (9.24) reduce to the sector values of (9.23): $\bar a_{k,\sigma}(0)/\bar\varrho=(a_{\rm f}+\sigma\,\delta a_k)\,n/\gamma$. For $\langle X\rangle=0$ the parts of the records in (9.16), (9.18), (9.19), (9.23), (9.24) vanish. In (9.26), $\mathsf m_3=0$ and $\rho_4=0$; in (9.27), $a^{\rm sh}=a_{\rm f}$ of (9.23), $\bar a=a^{\rm sh}$ and $\bar a_{k,\sigma}(0)-a^{\rm sh}=\sigma\,\delta a_k$. 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. This includes the part with the plain gradient $\mathcal F_X$.
- For (9.8) and (9.14) the order "$\delta p\to0$ at fixed $\lambda$, then $\lambda\to\infty$" is not determined when $\langle\mathsf b_k^2\rangle_w>0$: the bound (9.22) contains $X_g$. The combined limit is shown under the sufficient condition $X_g/\lambda\to0$; its necessity is not shown. For $\langle\mathsf b_k^2\rangle_w=0$ the spread and the covariance vanish identically, and the limit is zero in either order.
- 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
- Heisenberg equation for the velocity operator; window bound at $\lambda_1=0$; uncertainty inequality on the circle
- 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
- Row sums of the hopping entries for a position-dependent hopping factor

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