Version 2 · earlier version; the current one is v3 · External review, round 2: minor issues

# Common maximal speed: two body types recorded by one medium

- **Subproject:** 04-ilang-time
- **Package:** 08-maximal-speed
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-10

## Changes from previous version

- **Step 1 (I2).** The claimed redundancy of the parameters of (8.1) is removed. The parametrization is one-to-one; the data in question give a different replacement, which moves a constant between $C_A$ and $C_B$.
- **Step 7 (I1, I3), rewritten.** Under (8.1) the diagonal of a displayed body contract is a convention. The step now builds the convention-independent effective body contracts (8.19), (8.20), proves an exact state-dependent bound (8.21) and its Model D form (8.22), (8.23), which is uniform in the chain length, states the order of the limits, and lists which quantities of 05 and 06 are well defined. The v1 sentence "$M^T_k$ is well defined as half the gap of the displayed $C_T$" is withdrawn.
- **Step 9 (I1).** (8.16)–(8.18) are kept, with their reading made exact: $M^T_k$ is the positive eigenvalue of the body contract only in the displayed Model D description. The convention-independent relations (8.24)–(8.26) are new.
- **Own corrections.** (8.8) for equal finite lengths needs $L\ge2$, and the general finite condition is added. The branch of (8.14) and the accuracy in $\epsilon_T$ of the condition in (8.15) are stated. The scaling argument of Step 5 uses off-diagonal entries only. The dimension check uses that record vectors are costs.
- **Tags.** (8.1)–(8.18) keep their meaning; (8.19)–(8.26) are new; no tag is retired.

## Response to verification

- **I1: Accepted and fixed in Step 7 (parts 1, 2, 4) and Step 9.** The half-gap of a displayed $C_T$ is not invariant under place-dependent $\alpha^T$. The rest gap is now taken from the effective body contract (8.19), which keeps the scalar parts $\langle K^T_h\rangle$ of the record terms at zeroth order in $\kappa_T$ and is invariant under the whole family (8.1) up to $\pm\langle Z\rangle\mathbb 1$ (8.20). (8.16) is stated as an identity of the displayed description; the convention-independent relations are (8.25), (8.26). The rebooking of the verification is consistency check 2.
- **I2: Accepted and fixed in Step 1**, paragraph after (8.1).
- **I3: Accepted and fixed in Step 7, parts 1–3.** The state-dependent Duhamel bounds (8.21), (8.23) are uniform in the chain length for starts with bounded $\lVert\hat x_T\phi_T\rVert$; the order of the limits is stated under "Setup and assumptions".

## Setup and assumptions

**Setting (question.md).** Objects $a$ (type $A$, places $h\in H_A$), $b$ (type $B$, places $k\in H_B$) and the medium $c$; the three types are pairwise different and each has one instance, so (1.11) imposes nothing. The contract is

$$
C=C_A\otimes\mathbb 1_b\otimes\mathbb 1_c+\mathbb 1_a\otimes C_B\otimes\mathbb 1_c+\sum_h\lvert h\rangle\langle h\rvert_a\otimes\mathbb 1_b\otimes K^A_h+\sum_k\mathbb 1_a\otimes\lvert k\rangle\langle k\rvert_b\otimes K^B_k ,
$$

with $C_A=\sum_{h,h'}t^A_{hh'}\lvert h\rangle\langle h'\rvert$, $C_B=\sum_{k,k'}t^B_{kk'}\lvert k\rangle\langle k'\rvert$, $K^A_h$, $K^B_k$ self-adjoint, no contract between $a$ and $b$, and the start $\lvert\phi\rangle_a\otimes\lvert\vartheta\rangle_b\otimes\lvert\chi\rangle_c$. All contracts are independent of $\lambda$ (A5), and $\lambda\ge0$.

**Quoted definitions.** $\langle X\rangle:=\langle\chi\vert X\vert\chi\rangle$; $u^A_h:=(K^A_h-\langle K^A_h\rangle)\chi$, $u^B_k:=(K^B_k-\langle K^B_k\rangle)\chi$, all in $\mathcal N=\{v:\langle\chi\vert v\rangle=0\}$; $g(v,w)=\operatorname{Re}\langle v\vert w\rangle$; $d^a_0(h,h')=\lVert u^A_h-u^A_{h'}\rVert$, $d^b_0(k,k')=\lVert u^B_k-u^B_{k'}\rVert$ (16.8); $\bar x_a=\sum_hp^a_hu^A_h$, $v_a=\mathrm d\bar x_a/\mathrm d\lambda$, and likewise for $b$; $a^A_{hh'}=\lvert t^A_{hh'}\rvert d^a_0(h,h')$, $v^A_{\max}=\varrho(a^A)$, and likewise for $B$.

**Inputs.** (16.3), (16.4), (16.8), (16.11), (16.13) of 03-ilang-space/16-common-space; (17.4), (17.7), (17.15), (17.20) of 03-ilang-space/17-motion; (5.11), (5.22), (5.25), (5.28), (5.29) of 05-moving-clock; (6.7), (6.15), (6.16), (6.20) of 06-inertia-internal-energy.

**Notation of this package.** $T\in\{A,B\}$ labels the type; $\phi_A:=\phi$, $\phi_B:=\vartheta$. In items 2 and 3, $e_T:=(X_T-\langle X_T\rangle)\chi$ with $\lVert e_T\rVert=\sigma_{X_T}>0$. The *record step* $\ell_T$ and the *hop modulus* $\eta_T$ are: for a chain (17.20), $\ell_T:=e_T$ and $\eta_T:=t_T$; for Model D, $\ell_T:=\kappa_Te_T$ and $\eta_T:=\tau_T/2$. $d_1^T:=\lVert\ell_T\rVert$ is the distance of neighbouring points ($d_1$ of (6.20) for Model D). The *displayed Model D description* is the one of the question: $C_T=C^{\rm D}_T:=A_T+B_T$ and $K^T_x=\kappa_T\,x\,X_T$; $\hat x_T$ is its position label, as an operator on $\mathcal H_T$. In item 3: $\rho_T:=\tau_T/\bar\mu_T$, $\nu$ is the common drift speed and $\beta_T:=\nu/v^T_{\max}$.

**Exact statements, statements to an order, and order of the limits.**
- Exact, for every finite system (A1): Steps 1–6; in Step 7, (8.19)–(8.23); in Step 9, the identities (8.16), (8.24), (8.25) and the equivalences (8.17), (8.18), (8.26), as statements about the quantities defined there. Chains on $\mathbb Z$ are the limits $L_T\to\infty$ of open chains with $L_T$ positions; item 3 uses them, with $v^T_{\max}=\tau_Td_1^T$ (8.7).
- To an order: (8.11) (leading order in $\kappa_A$, $\kappa_B$, with the remainder (8.23)); Step 8 (in addition first order in $\epsilon_T$, leading order in the spread $\delta p_T$, expansion in small labels $p_{0T}$); in Step 9, only the statement that $I^T_k$ is the inertia (first order in the force, average over long $\lambda$-intervals).
- Order of the limits in item 3: (i) $\kappa_A,\kappa_B\to0$ at fixed $\lambda$ and fixed starts with finite $\lVert\hat x_T\phi_T\rVert$; (ii) $L_T\to\infty$, which by (8.23) commutes with (i) for such starts; (iii) the orders of 05 and 06 in $\epsilon_T$, $\delta p_T$ and the force; (iv) last, small $p_{0T}$.

## Derivation

### Step 1. Replacements that leave $C$ unchanged, and their action (item 1a)

Write $t'^A_{hh}=t^A_{hh}+\alpha^A_h$, $t'^B_{kk}=t^B_{kk}+\alpha^B_k$, $K'^A_h=K^A_h+\Delta^A_h$, $K'^B_k=K^B_k+\Delta^B_k$, with $\alpha^A_h,\alpha^B_k$ real and $\Delta^A_h,\Delta^B_k$ self-adjoint (the new operators must be self-adjoint). With $\mathbb 1_a=\sum_h\lvert h\rangle\langle h\rvert$ and $\mathbb 1_b=\sum_k\lvert k\rangle\langle k\rvert$,

$$
C'-C=\sum_{h,k}\lvert h\rangle\langle h\rvert_a\otimes\lvert k\rangle\langle k\rvert_b\otimes\bigl[(\alpha^A_h+\alpha^B_k)\mathbb 1_c+\Delta^A_h+\Delta^B_k\bigr].
$$

The projectors $\lvert h\rangle\langle h\rvert\otimes\lvert k\rangle\langle k\rvert$ are linearly independent, so $C'=C$ iff every bracket vanishes: $\Delta^A_h+\alpha^A_h\mathbb 1_c=-(\Delta^B_k+\alpha^B_k\mathbb 1_c)$ for all $h,k$. The left side does not depend on $k$ and the right side not on $h$, so both equal one self-adjoint operator $Z$. Conversely every such choice gives $C'=C$. Hence all replacements are

$$
t'^A_{hh}=t^A_{hh}+\alpha^A_h,\quad K'^A_h=K^A_h+Z-\alpha^A_h\mathbb 1_c,\qquad
t'^B_{kk}=t^B_{kk}+\alpha^B_k,\quad K'^B_k=K^B_k-Z-\alpha^B_k\mathbb 1_c ,
\tag{8.1}
$$

with $\alpha^A_h,\alpha^B_k\in\mathbb R$ and $Z=Z^\dagger$ on $\mathcal H_c$ arbitrary. The parametrization is one-to-one: $\alpha^A_h=t'^A_{hh}-t^A_{hh}$, $\alpha^B_k=t'^B_{kk}-t^B_{kk}$ and $Z=K'^A_h-K^A_h+\alpha^A_h\mathbb 1_c$ are read off the replacement. For $\alpha^A=\alpha^B=0$, (8.1) is (16.3). The data $\alpha^A_h=c$, $\alpha^B_k=-c$, $Z=c\,\mathbb 1_c$ ($c\in\mathbb R$) leave every $K$ unchanged and give $C_A\mapsto C_A+c\,\mathbb 1_a$, $C_B\mapsto C_B-c\,\mathbb 1_b$: a constant moved between the two body contracts.

These are conventions of the description (A5, M6): $Z$ moves a single-object term of $c$ between the pair terms $a$–$c$ and $b$–$c$; $\alpha^A_h$ and $\alpha^B_k$ move a single-object term of a body between the two parts in which its pair term with $c$ is written. In particular the diagonal of a displayed body contract is fixed only up to an arbitrary real function of the place, so its spectrum is not well defined (used in Step 7).

*Action.* Multiples of $\mathbb 1_c$ cancel in $K-\langle K\rangle$, so $u^A_h\mapsto u^A_h+z$ and $u^B_k\mapsto u^B_k-z$ with $z=(Z-\langle Z\rangle)\chi$; every $z\in\mathcal N$ occurs (16.4). $C$ and the start are unchanged, hence so are $\Psi(\lambda)$, the views and the place weights $p^a_h$, $p^b_k$. With $\sum_hp^a_h=\sum_kp^b_k=1$:

$$
\begin{aligned}
\text{replacement (8.1):}\quad&\bar x_a\mapsto\bar x_a+z,\quad\bar x_b\mapsto\bar x_b-z,\quad v_a\mapsto v_a,\quad v_b\mapsto v_b,\quad v^A_{\max}\mapsto v^A_{\max},\quad v^B_{\max}\mapsto v^B_{\max};\\
\chi\mapsto e^{i\theta}\chi:\quad&(\bar x_a,\bar x_b,v_a,v_b)\mapsto e^{i\theta}(\bar x_a,\bar x_b,v_a,v_b),\quad v^A_{\max}\mapsto v^A_{\max},\quad v^B_{\max}\mapsto v^B_{\max}.
\end{aligned}
\tag{8.2}
$$

Reasons. First line: $z$ does not depend on $\lambda$; $d^a_0$, $d^b_0$ contain only differences of record vectors of one type; the off-diagonal $t$ are not replaced, and $a^T_{hh}=0$ for any $t^T_{hh}$ because $d_0(h,h)=0$. Second line: $\langle X\rangle$ is unchanged, so every record vector is multiplied by $e^{i\theta}$ (16.4); the state changes by a common phase (1.2), so the views are unchanged; norms of differences, hence $d_0$ and $a^T$, are unchanged. Phases of $\phi$ and $\vartheta$ change nothing.

### Step 2. Well-defined quantities (item 1b)

A quantity is well defined (M6) iff it is invariant under both lines of (8.2), for all $\alpha^A$, $\alpha^B$, $Z$, $\theta$. Since $\langle e^{i\theta}v\vert e^{i\theta}w\rangle=\langle v\vert w\rangle$:

$$
\begin{aligned}
&\text{well defined:}&&\lVert v_a\rVert,\quad\lVert v_b\rVert,\quad g(v_a,v_b),\quad\lVert v_a-v_b\rVert,\quad v^A_{\max},\quad v^B_{\max},\quad v^A_{\max}/v^B_{\max}\ \ (\text{if }v^B_{\max}>0);\\
&\text{not well defined:}&&\bar x_a-\bar x_b\ \ (\text{neither the vector nor its norm}).
\end{aligned}
\tag{8.3}
$$

Indeed $\bar x_a-\bar x_b\mapsto\bar x_a-\bar x_b+2z$ with $z\in\mathcal N$ arbitrary, and $\bar x_a-\bar x_b\in\mathcal N$: at a given $\lambda$ it can be given any value in $\mathcal N$, for example $0$. (For $\dim\mathcal H_c=1$ all record vectors vanish and the statement is empty.) In addition, the norm of its change between two values of $\lambda$ is well defined, being the norm of an integral of $v_a-v_b$.

### Step 3. The bound for one body in the presence of the other (item 1c)

Let $\lvert\psi_h\rangle:=(\langle h\rvert_a\otimes\mathbb 1_{bc})\lvert\Psi\rangle\in\mathcal H_b\otimes\mathcal H_c$, so that $p^a_h=\lVert\psi_h\rVert^2$ and $(V_a)_{h'h}=\langle\psi_h\vert\psi_{h'}\rangle$ by (1.5). From (1.9), $i\,\mathrm d\psi_h/\mathrm d\lambda=\sum_{h'}t^A_{hh'}\psi_{h'}+Q_h\psi_h$ with $Q_h:=C_B\otimes\mathbb 1_c+\mathbb 1_b\otimes K^A_h+\sum_k\lvert k\rangle\langle k\rvert_b\otimes K^B_k$ self-adjoint. Then $\mathrm dp^a_h/\mathrm d\lambda=2\operatorname{Re}\langle\psi_h\vert\dot\psi_h\rangle$, and $\langle\psi_h\vert Q_h\vert\psi_h\rangle$ is real, so

$$
\frac{\mathrm dp^a_h}{\mathrm d\lambda}=\sum_{h'}J^a_{h'\to h},\qquad J^a_{h'\to h}:=2\operatorname{Im}\bigl[t^A_{hh'}\,(V_a)_{h'h}\bigr]=-J^a_{h\to h'} .
\tag{8.4}
$$

This is (17.4) with $V_b$ replaced by $V_a$: the body $b$, its contract and its records enter only through the view $V_a$. By the antisymmetry, $v_a=\sum_h\dot p^a_hu^A_h=\tfrac12\sum_{h,h'}J^a_{h'\to h}(u^A_h-u^A_{h'})$. With $\lvert J^a_{h'\to h}\rvert\le2\lvert t^A_{hh'}\rvert\,\lvert(V_a)_{h'h}\rvert$, the Cauchy–Schwarz inequality $\lvert(V_a)_{h'h}\rvert\le q_hq_{h'}$, $q_h:=\sqrt{p^a_h}$, and the Rayleigh bound for the real symmetric matrix $a^A$ and the unit vector $q$:

$$
\lVert v_a(\lambda)\rVert\le\sum_{h,h'}a^A_{hh'}\,q_h(\lambda)\,q_{h'}(\lambda)\le\varrho(a^A)=v^A_{\max}\qquad\text{for every start }\phi\otimes\vartheta\otimes\chi\text{ and every }\lambda\in\mathbb R .
\tag{8.5}
$$

So the answer is **yes**, exactly, in particular for every $\lambda\ge0$; the same holds for $b$. The product form of the start is used only to define $\chi$, hence the record vectors. Both sides of (8.5) are well defined by (8.3). The entries of $a^A$ are

$$
a^A_{hh'}=\lvert t^A_{hh'}\rvert\;\bigl\lVert\bigl(K^A_h-K^A_{h'}-\langle K^A_h-K^A_{h'}\rangle\bigr)\chi\bigr\rVert ,
\tag{8.6}
$$

so $v^A_{\max}$ is a function of the off-diagonal $t^A_{hh'}$, the differences $K^A_h-K^A_{h'}$ and $\chi$ only. It does **not** depend on $C_B$, on the $K^B_k$ or on $\vartheta$ (nor on $\phi$ or the $t^A_{hh}$). The only datum shared by $v^A_{\max}$ and $v^B_{\max}$ is the start $\chi$ of the medium. The velocity $v_a(\lambda)$ itself does depend on them, through $V_a$ in (8.4).

### Step 4. Maximal speeds of chain types and of Model D types (items 2a, 2b)

*Chain.* $t^T_{h_mh_n}=t_T\,\delta_{\lvert m-n\rvert,1}$ and $d_0=\lvert m-n\rvert\sigma_{X_T}$ (17.15), so $a^T=t_T\sigma_{X_T}$ times the adjacency matrix of a path with $L_T$ vertices, whose largest eigenvalue is $2\cos\frac{\pi}{L_T+1}$ (standard).
*Model D.* The record vector of the place $(x,s,a)$ is $x\,\ell_T$ up to a common shift. The off-diagonal entries of $C_T$ come from $A_T$, which joins $(x,s,a)$ to $(x\pm1,s',a)$, $s'\neq s$, with modulus $\tau_T/2$ and $d_0=\kappa_T\sigma_{X_T}$, and from $B_T$, which joins places with the same $x$, where $d_0=0$. So $a^T=\frac{\tau_T}2\kappa_T\sigma_{X_T}$ times the adjacency matrix of $2N_T$ disjoint paths with $L_T$ vertices; $m_T$, $N_T$, $\lambda_{0T}$ do not enter. In both families, exactly,

$$
v^T_{\max}=2\,\eta_T\,d_1^T\cos\frac{\pi}{L_T+1}\ \xrightarrow[L_T\to\infty]{}\ 2\,\eta_T\,d_1^T=
\begin{cases}2\,t_T\,\sigma_{X_T}&\text{chain, (17.20)},\\ \tau_T\,\kappa_T\,\sigma_{X_T}&\text{Model D, (5.11)}.\end{cases}
\tag{8.7}
$$

Hence, for chains on $\mathbb Z$ and for finite chains with $L_A=L_B\ge2$, exactly,

$$
\text{(a) chains: } v^A_{\max}=v^B_{\max}\iff t_A\,\sigma_{X_A}=t_B\,\sigma_{X_B};\qquad
\text{(b) Model D: } v^A_{\max}=v^B_{\max}\iff\tau_A\,\kappa_A\,\sigma_{X_A}=\tau_B\,\kappa_B\,\sigma_{X_B}.
\tag{8.8}
$$

For finite chains in general the exact condition is $\eta_Ad_1^A\cos\frac{\pi}{L_A+1}=\eta_Bd_1^B\cos\frac{\pi}{L_B+1}$; for $L_A=L_B=1$ both maximal speeds vanish, whatever the parameters.

### Step 5. The language does not imply equal maximal speeds (item 2c)

**No.** Take two open chains with the same $L\ge2$, $X_A=X_B=X$ with $\sigma_X>0$, and $t_A=r\,t_B$, $r>0$. This is an admissible description: finitely many objects and places (A1); $C$ self-adjoint and independent of $\lambda$, a sum of the pair terms $a$–$c$ and $b$–$c$ (Laws 3, 4, A5); a unit product start (A3); no identical instances, so Law 5 and A6 impose nothing; Laws 1, 2 and A2, A4, A7–A9 do not restrict the contract. By (8.7),

$$
\frac{v^A_{\max}}{v^B_{\max}}=\frac{t_A\,\sigma_{X_A}}{t_B\,\sigma_{X_B}}=r\qquad\text{takes every value in }(0,\infty).
\tag{8.9}
$$

In general, multiplying the off-diagonal entries $t^A_{hh'}$, $h\neq h'$, by $s>0$ maps admissible descriptions to admissible descriptions, multiplies $v^A_{\max}$ by $s$ and leaves $v^B_{\max}$ unchanged, by (8.6). Equality of the maximal speeds is a condition on the description.

### Step 6. Common points (item 2d)

Let both chains have $L$ positions, with labels $j_T,\dots,j_T+L-1$. Positions are the points: places at one position have $d_0=0$, different positions have $d_0=\lvert x-x'\rvert d_1^T>0$. So $\lvert X_A\rvert=\lvert X_B\rvert=L$ and $U_T-\bar u^T=\{(j-\tfrac{L-1}2)\ell_T:\ j=0,\dots,L-1\}$. By (16.13), a correspondence with $D_A=X_A$, $D_B=X_B$ exists iff $L\ge2$ and these two centred sets are equal. Their elements of largest norm are $\pm\tfrac{L-1}2\ell_T$, so equality forces $\ell_B=\pm\ell_A$; conversely $\ell_B=\pm\ell_A$ gives equal sets. Therefore, exactly, for every $L\ge2$,

$$
\text{full correspondence}\iff\ell_B=\pm\ell_A\ \Longrightarrow\ d_1^A=d_1^B;\qquad\text{then}\quad v^A_{\max}=v^B_{\max}\iff\eta_A=\eta_B .
\tag{8.10}
$$

- Records: the two types have the same record vectors up to one translation $t_\pi\in\mathcal N$ (16.11), which the replacement (8.1) with $z=t_\pi/2$ removes; neighbouring points are equally far apart for both types. For chains: $(X_B-\langle X_B\rangle)\chi=\pm(X_A-\langle X_A\rangle)\chi$, hence $\sigma_{X_A}=\sigma_{X_B}$. For Model D: $\kappa_B(X_B-\langle X_B\rangle)\chi=\pm\kappa_A(X_A-\langle X_A\rangle)\chi$, hence $\kappa_A\sigma_{X_A}=\kappa_B\sigma_{X_B}$. Nothing follows for $X_A$, $X_B$ as operators, nor for $t_T$, $\tau_T$, $m_T$, $N_T$, $\lambda_{0T}$.
- Condition: by (8.7) with equal $L$ and equal $d_1^T$, $v^A_{\max}=v^B_{\max}$ iff $t_A=t_B$ (chains), iff $\tau_A=\tau_B$ (Model D). Common points remove the records from the condition (8.8) and leave the hopping.

### Step 7. Effective body contracts; why the one-body results apply (item 3a)

**1. A convention-independent split (exact; setting of item 1).** By (8.1), a displayed body contract is fixed only up to $\operatorname{diag}(\alpha^T)$ and the record operators only up to scalars. A statement about the spectrum of a displayed $C_T$, or about record terms "of first order in $\kappa$", therefore holds in one description only. Take instead the partial expectation of $C$ in the medium start. Because there is no contract between $a$ and $b$, it is a sum of two one-body operators, the **effective body contracts**:

$$
C^{\rm eff}_A\otimes\mathbb 1_b+\mathbb 1_a\otimes C^{\rm eff}_B:=(\mathbb 1_{ab}\otimes\langle\chi\rvert)\,C\,(\mathbb 1_{ab}\otimes\lvert\chi\rangle),\qquad
C^{\rm eff}_A=C_A+\sum_h\langle K^A_h\rangle\lvert h\rangle\langle h\rvert,\quad
C^{\rm eff}_B=C_B+\sum_k\langle K^B_k\rangle\lvert k\rangle\langle k\rvert .
\tag{8.19}
$$

The left side is built from $C$ and $\chi$ alone, and its split into two one-body operators is unique up to a constant. Indeed, by (8.1) the $\alpha$'s cancel in $t'_{hh}+\langle K'_h\rangle$, and

$$
\text{(8.1):}\quad C^{\rm eff}_A\mapsto C^{\rm eff}_A+\langle Z\rangle\,\mathbb 1_a,\quad C^{\rm eff}_B\mapsto C^{\rm eff}_B-\langle Z\rangle\,\mathbb 1_b;\qquad \chi\mapsto e^{i\theta}\chi:\quad C^{\rm eff}_A,\ C^{\rm eff}_B\ \text{unchanged}.
\tag{8.20}
$$

Hence, for one type, the eigenvectors of $C^{\rm eff}_T$ and the differences of its eigenvalues are well defined (M6), and so is the sum of an eigenvalue of $C^{\rm eff}_A$ and one of $C^{\rm eff}_B$. A single eigenvalue of $C^{\rm eff}_T$ is not, and nothing about the spectrum of a displayed $C_T$ is. By (1.8) the cost of the start is $L=\langle\phi\vert C^{\rm eff}_A\vert\phi\rangle+\langle\vartheta\vert C^{\rm eff}_B\vert\vartheta\rangle$ exactly: $C^{\rm eff}_T$ gives body $T$ its share of the cost, fixed up to the constant.

The rest, $\hat C_{\rm rec}:=C-C^{\rm eff}_A-C^{\rm eff}_B$ (identities suppressed), is invariant under (8.1). It has $K-\langle K\rangle$ in place of the $K$'s, so $\hat C_{\rm rec}(\phi'\otimes\vartheta'\otimes\chi)=\sum_{h,k}\phi'_h\vartheta'_k\lvert hk\rangle\otimes(u^A_h+u^B_k)$. Let $\phi(s):=e^{-iC^{\rm eff}_As}\phi$ and $\vartheta(s):=e^{-iC^{\rm eff}_Bs}\vartheta$; the two effective contracts act on different factors and commute. Duhamel's formula $e^{-iC\lambda}=e^{-iC_0\lambda}-i\int_0^\lambda e^{-iC(\lambda-s)}\hat C_{\rm rec}\,e^{-iC_0s}\,\mathrm ds$ with $C_0:=C-\hat C_{\rm rec}$, the unitarity of $e^{-iC(\lambda-s)}$ and the orthonormality of the $\lvert hk\rangle$ give, for $\lambda\ge0$,

$$
\bigl\lVert\Psi(\lambda)-\phi(\lambda)\otimes\vartheta(\lambda)\otimes\chi\bigr\rVert\le\varsigma(\lambda):=\int_0^\lambda\Bigl(\sum_{h,k}\lvert\phi_h(s)\rvert^2\,\lvert\vartheta_k(s)\rvert^2\,\lVert u^A_h+u^B_k\rVert^2\Bigr)^{1/2}\mathrm ds,
\qquad
\bigl\lVert V_a(\lambda)-\lvert\phi(\lambda)\rangle\langle\phi(\lambda)\rvert\bigr\rVert_1\le2\,\varsigma(\lambda),
\tag{8.21}
$$

and likewise for $V_b$. The second bound holds because the partial trace does not increase the trace norm and $\lVert\,\lvert\psi\rangle\langle\psi\rvert-\lvert\psi'\rangle\langle\psi'\rvert\,\rVert_1\le2\lVert\psi-\psi'\rVert$. (8.21) is exact, for every description of item 1, every product start and every finite size. $\varsigma$ is invariant under (8.1), because $z$ cancels in $u^A_h+u^B_k$ and (8.20) changes $\phi(s)$, $\vartheta(s)$ by phases, and under the phase of $\chi$. The convention-independent form of the weak-record regime is $\varsigma(\lambda)\ll1$: the view of each body is then the one generated by its own effective body contract.

**2. Model D types.** In the displayed description, (8.19) gives $C^{\rm eff}_T=C^{\rm D}_T+\kappa_T\langle X_T\rangle\hat x_T$. By (8.20), in every description of the family (8.1),

$$
C^{\rm eff}_T=C^{\rm D}_T+\gamma_T\,\hat x_T+c_T\,\mathbb 1,\qquad\gamma_T:=\kappa_T\langle X_T\rangle,\qquad c_A=-c_B=\langle Z\rangle .
\tag{8.22}
$$

$C^{\rm D}_T+c_T\mathbb 1$ is the part of $C^{\rm eff}_T$ that commutes with the translations of the position label, and $\gamma_T\hat x_T$ is a uniform cost gradient of first order in $\kappa_T$, of the kind treated in 06. The displayed body contract of a rebooked description is $C^{\rm D}_T+\operatorname{diag}(\alpha^T)$ instead, and its records contain the scalars $-\alpha^T_h$ at zeroth order in $\kappa_T$; (8.19) puts them back. A cost gradient with a force $F$, as in 06, is a further diagonal term of $C^{\rm eff}_T$ and is included in $C^{\rm D}_T$ below.

The record vectors are $u^T_{(x,s,a)}=x\,\ell_T$ in the displayed description, so $\lVert u^A_h+u^B_k\rVert\le d_1^A\lvert x_a\rvert+d_1^B\lvert x_b\rvert$. All diagonal terms commute with $\hat x_T$, so $[C^{\rm eff}_T,\hat x_T]=[A_T,\hat x_T]$, of norm $\tau_T\cos\frac{\pi}{L_T+1}\le\tau_T$; hence $\lVert\hat x_T\phi_T(s)\rVert\le\lVert\hat x_T\phi_T\rVert+\tau_Ts$. With the Minkowski inequality in (8.21), and with Duhamel's formula once more for $C^{\rm eff}_T$ against $C^{\rm D}_T+c_T\mathbb 1$ (the phases cancel, $c_A+c_B=0$):

$$
\varsigma(\lambda)\le\sum_{T}d_1^T\,\Lambda_T(\lambda),\qquad
\Bigl\lVert\Psi(\lambda)-e^{-iC^{\rm D}_A\lambda}\phi\otimes e^{-iC^{\rm D}_B\lambda}\vartheta\otimes\chi\Bigr\rVert\le\sum_{T}\bigl(d_1^T+\lvert\gamma_T\rvert\bigr)\Lambda_T(\lambda),\qquad
\Lambda_T(\lambda):=\lambda\,\lVert\hat x_T\phi_T\rVert+\tfrac12\tau_T\lambda^2 .
\tag{8.23}
$$

Both bounds are exact. Since $d_1^T+\lvert\gamma_T\rvert=\kappa_T(\sigma_{X_T}+\lvert\langle X_T\rangle\rvert)$, they give

$$
\lvert\Psi(\lambda)\rangle=e^{-iC^{\rm D}_A\lambda}\lvert\phi\rangle\otimes e^{-iC^{\rm D}_B\lambda}\lvert\vartheta\rangle\otimes\lvert\chi\rangle+O(\kappa_A)+O(\kappa_B),\qquad
V_a(\lambda)=e^{-iC^{\rm D}_A\lambda}\lvert\phi\rangle\langle\phi\rvert e^{iC^{\rm D}_A\lambda}+O(\kappa_A)+O(\kappa_B),
\tag{8.11}
$$

and likewise for $V_b$, with the remainders bounded by (8.23) and twice (8.23). In the same way $[C,\hat x_T]=[A_T,\hat x_T]$ gives $\lVert\hat x_T\Psi(\lambda)\rVert\le\lVert\hat x_T\phi_T\rVert+\tau_T\lambda$, so the mean position label differs from its leading-order value by at most $2(\lVert\hat x_T\phi_T\rVert+\tau_T\lambda)$ times the right side of (8.23).

**3. Uniformity and order of the limits.** The right sides of (8.23) do not contain $L_A$, $L_B$. The weak-record regime for both bodies is $(d_1^T+\lvert\gamma_T\rvert)\Lambda_T(\lambda)\ll1$ for $T=A,B$. For starts with $\lVert\hat x_T\phi_T\rVert$ bounded uniformly in $L_T$, the limits $\kappa_T\to0$ and $L_T\to\infty$ therefore commute. The starts of 05 and 06, of spread $\delta p_T>0$ and centred at the label $x^0_T$, are of this kind: $\lVert\hat x_T\phi_T\rVert$ is of order $\lvert x^0_T\rvert+1/\delta p_T$. Because this grows as $\delta p_T\to0$ and $\Lambda_T$ grows with $\lambda$, the limit $\kappa_T\to0$ is taken first, at fixed $\delta p_T$ and fixed $\lambda$ (a tick, or an averaging interval). For a start without a bounded position moment, (8.21) still holds at every finite $L_T$, but its bound grows like $\kappa_TL_T$; then $\kappa_T\to0$ must be taken at fixed finite $L_T$, before $L_T\to\infty$.

**4. Application to each body, and well-defined quantities.**
- By (8.11) the leading-order view of each body is the one generated by its own contract $C^{\rm D}_T$ and its own start. The other body, its contract, its records and its start do not enter. The one-body setting of 05 and 06 is the same argument with the other body removed, and it has the same effective contract (8.22).
- By (8.6) the differences $x\,\ell_T$ of the record vectors of a body, hence the background (17.15) in which its position is measured, depend only on its own records and on $\chi$.
- Every quoted result of 05 and 06 is a statement about these two things only: the place weights of the view of the body (position, branch and clock labels) and the differences of its record vectors.

Hence (5.11)–(5.29) and (6.7)–(6.20) hold for $a$ with the parameters of type $A$ and for $b$ with those of type $B$, with "$O(\kappa)$" bounded by (8.23).

Under the whole family (8.1) and the phase of $\chi$, the following are invariant: the starts and the views, hence the ticks and $R_T$ (functions of the clock-label weights of the view); $\lVert\bar v_T\rVert$, by (8.2); $v^T_{\max}$; the parameters $\tau_T$, $N_T$, $\lambda_{0T}$ (off-diagonal entries of $C_T$), $\bar\mu_T$ and $m_T$ (half the difference of the diagonal entries of $C^{\rm eff}_T$ at $s=1$ and $s=2$, at equal $x$ and $a$, is $\bar\mu_T$), $d_1^T$ and $\gamma_T$; hence $\rho_T$ and $\epsilon_T$. "The branch of positive eigenvalue" of 05 and 06 is the upper branch of $C^{\rm D}_T+c_T\mathbb 1$. Not invariant are the diagonal and the spectrum of a displayed $C_T$, and the eigenvalues of $C^{\rm eff}_T$ (8.20). None of the quoted formulas may be evaluated with the displayed body contract of a rebooked description.

### Step 8. Moving clocks (item 3b)

Let $\nu:=\lVert\bar v_a\rVert=\lVert\bar v_b\rVert$ for the drift velocities (5.25) of the two clock-carrying starts. By Step 7, (5.28) holds for each type with its own $v^T_{\max}$ of (8.7): $R_T=1+\nu^2/\bigl(2(v^T_{\max})^2\bigr)+O(p_{0T}^4)$. By (5.22) and (5.25), $\beta_T^2=\rho_T^2p_{0T}^2+O(p_{0T}^4)$, so the two labels are small together. Subtracting,

$$
R_A-R_B=\frac{\nu^2}2\Bigl[\frac1{(v^A_{\max})^2}-\frac1{(v^B_{\max})^2}\Bigr]+O(p_{0A}^4)+O(p_{0B}^4)
=\frac{\nu^2}2\Bigl[\frac1{\tau_A^2\kappa_A^2\sigma_{X_A}^2}-\frac1{\tau_B^2\kappa_B^2\sigma_{X_B}^2}\Bigr]+\dots
\tag{8.12}
$$

To this order (second order in $p_{0T}$, first order in $\epsilon_T$, leading order in $\delta p_T$, $\kappa_A$, $\kappa_B$, in the order of limits of the setup) the tick ratio of type $T$ is the function $\nu\mapsto1+\nu^2/\bigl(2(v^T_{\max})^2\bigr)$ of the drift speed. The two functions are equal iff their coefficients are:

$$
R_A(\nu)=R_B(\nu)\ \text{to this order}\iff v^A_{\max}=v^B_{\max}\iff\tau_A\,\kappa_A\,\sigma_{X_A}=\tau_B\,\kappa_B\,\sigma_{X_B} .
\tag{8.13}
$$

No condition arises on $m_T$, $N_T$, $\lambda_{0T}$, nor on $\tau_T$ and $\kappa_T\sigma_{X_T}$ separately. Both sides of (8.12) and the condition (8.13) contain only quantities that are invariant by Step 7, part 4.

*The relations (5.29).* By (5.22), $\sin^2p_{0T}=(R_T^2-1)/\rho_T^2$. Inserting this in (5.29) eliminates the label and gives the relation between tick ratio and drift speed of type $T$, at the orders of (5.29); on the branch with $R_T\to1$ for $\beta_T\to0$ (small labels) it is solved by the series on the right:

$$
R_T^2\bigl(1-\beta_T^2\bigr)=1+\frac{(R_T^2-1)^2}{\rho_T^2}
\quad\Longrightarrow\quad
R_T=1+\frac{\beta_T^2}2+\Bigl(\frac38+\frac1{2\rho_T^2}\Bigr)\beta_T^4+O(\beta_T^6).
\tag{8.14}
$$

(With $r:=R_T^2-1$ the relation reads $r=\beta_T^2+\beta_T^2r+r^2/\rho_T^2$; iterating, $r=\beta_T^2+(1+\rho_T^{-2})\beta_T^4+O(\beta_T^6)$, and $R_T=1+\tfrac r2-\tfrac{r^2}8+\dots$) Under (8.13), $\beta_A=\beta_B=:\beta$ and

$$
R_A-R_B=\frac{\beta^4}2\Bigl(\frac{\bar\mu_A^2}{\tau_A^2}-\frac{\bar\mu_B^2}{\tau_B^2}\Bigr)+O(\beta^6);\qquad
\text{(5.29) agree}\iff\text{(8.13) and }\ \frac{\bar\mu_A}{\tau_A}=\frac{\bar\mu_B}{\tau_B},\quad\bar\mu_T=m_T+\frac{\pi(N_T-1)}{N_T\lambda_{0T}} .
\tag{8.15}
$$

"Agree" means that the relation (8.14) between $R$ and $\nu$ is the same for both types. The equivalence follows from the coefficients of $\nu^2$ and $\nu^4$ in (8.14), since $\rho_T>0$. So the answer is **no**: under (8.13) the left sides of (5.29) have the same form, but the right sides agree only under the additional condition in (8.15). It is independent of (8.13), because $v^T_{\max}$ does not contain $m_T$, $N_T$, $\lambda_{0T}$. The two relations agree at order $\beta^2$ and differ at order $\beta^4$. Since (5.22) holds to first order in $\epsilon_T$, the additional condition is fixed up to relative terms $O((N_T\epsilon_T)^2)$; at zeroth order it reads $m_A/\tau_A=m_B/\tau_B$. The same condition makes $v_{\rm top}$ of (5.28) equal, since $v^T_{\rm top}=v^T_{\max}\bigl(\sqrt{1+\rho_T^{-2}}-\rho_T^{-1}\bigr)$. By (8.14), the leading order (8.12) is accurate for $\beta_T^2\ll\min(1,\rho_T^2)$.

### Step 9. Inertia, and the relation of the conditions (items 3c, 3d)

*Which rest cost is well defined.* By (8.22), on the plane wave with label $0$ and clock level $k$, the translation-invariant part $C^{\rm D}_T+c_T\mathbb 1$ of the effective body contract has the two eigenvalues

$$
E^T_{k,\pm}=c_T\pm M^T_k,\qquad M^T_k:=m_T+\frac{2\pi k}{N_T\lambda_{0T}},\qquad\Delta^T_k:=E^T_{k,+}-E^T_{k,-}=2M^T_k ,
\tag{8.24}
$$

with $c_A=-c_B=\langle Z\rangle$ an arbitrary real number. By (8.20), well defined are the branch gap $\Delta^T_k$, the level differences $E^T_{k,+}-E^T_{k',+}=M^T_k-M^T_{k'}$, and the sum $E^A_{k,+}+E^B_{k',+}$. The rest cost $E^T_{k,+}$ itself is not. So the rest cost of (6.7), "the positive eigenvalue of the body contract", is the number $M^T_k$ only in the displayed description: for a displayed $C_T$ this needs $\alpha^T=0$, and for $C^{\rm eff}_T$ it holds iff $\langle Z\rangle=0$. What is well defined is $M^T_k$ as half the branch gap. That is the quantity 06 uses: by (6.7) and (6.15), $\mu^T_{\rm av}(k)=\omega_k''(0)\,\ell_T\,g(\ell_T,\cdot\,)$ with $\omega_k''(0)=\tau_T^2/M^T_k=2\tau_T^2/\Delta^T_k$, the curvature of the upper branch, which does not change with $c_T$. Hence $I^T_k$ is well defined, and by Step 7 (6.20) holds for each body.

*The relation.* With $(v^T_{\max})^2=\tau_T^2\kappa_T^2\sigma_{X_T}^2$ from (8.7), for every clock level $k=0,\dots,N_T-1$,

$$
I^T_k=\frac{M^T_k}{(v^T_{\max})^2},\qquad\text{that is}\qquad M^T_k=I^T_k\,(v^T_{\max})^2,\qquad M^T_k=m_T+\frac{2\pi k}{N_T\lambda_{0T}} ,
\tag{8.16}
$$

equivalently, by (6.16), $M^T_k\,\mu^T_{\rm av}(k)=(v^T_{\max})^2\,n_T\,g(n_T,\cdot\,)$ with $n_T=e_T/\sigma_{X_T}$. (8.16) is an exact identity between the quantities defined in (6.7), (6.20) and (5.11). Read with $M^T_k$ as the rest cost, the positive eigenvalue of the body contract, it holds **in the displayed Model D description only**. Read with $M^T_k=\Delta^T_k/2$, it holds in every description of the family (8.1). That $I^T_k$ is the inertia of the body holds at the orders of (6.20). The convention-independent relations between background inertia and rest cost are, exactly,

$$
\Delta^T_k=2\,I^T_k\,(v^T_{\max})^2,\qquad E^T_{k,+}-E^T_{k',+}=\bigl(I^T_k-I^T_{k'}\bigr)(v^T_{\max})^2;\qquad\text{whereas}\quad E^T_{k,+}=c_T+I^T_k\,(v^T_{\max})^2 .
\tag{8.25}
$$

The first two contain only well-defined quantities. The third shows that the relation between the rest cost and the background inertia is affine, with the well-defined slope $(v^T_{\max})^2$ and the conventional intercept $c_T$. A proportionality of rest cost and inertia is a property of the bookings with $\langle Z\rangle=0$, not of the system.

*The condition.* The factor $M^T_k/I^T_k=(v^T_{\max})^2$ is the same for all levels of one type. Hence, with $M^T_k=\Delta^T_k/2$,

$$
\frac{M^A_k}{I^A_k}=\frac{M^B_{k'}}{I^B_{k'}}\ \ \text{for all levels }k,k'\ (\text{equivalently, for one pair})\iff v^A_{\max}=v^B_{\max}\iff\tau_A\,\kappa_A\,\sigma_{X_A}=\tau_B\,\kappa_B\,\sigma_{X_B} .
\tag{8.17}
$$

In convention-independent form, for two Model D types in every description of the family (8.1), and with the second statement for $N_A,N_B\ge2$:

$$
\frac{\Delta^A_k}{I^A_k}=\frac{\Delta^B_{k'}}{I^B_{k'}}\ \ \forall k,k'
\iff
\frac{E^A_{k,+}-E^A_{l,+}}{I^A_k-I^A_l}=\frac{E^B_{k',+}-E^B_{l',+}}{I^B_{k'}-I^B_{l'}}\ \ \forall k\neq l,\ k'\neq l'
\iff v^A_{\max}=v^B_{\max} .
\tag{8.26}
$$

The affine relations $E_+=c_T+I\,(v^T_{\max})^2$ of the two types coincide iff $v^A_{\max}=v^B_{\max}$ and $c_A=c_B$, that is $\langle Z\rangle=0$. The second requirement is a convention and not a condition on the types. In addition, for the label inertia $I^{\rm lab}_k=M_k/\tau^2$ the relation to $M_k=\Delta_k/2$ is the same for both types iff $\tau_A=\tau_B$; this coincides with (8.17) iff $d_1^A=d_1^B$, for example under (8.10).

*Relation of the conditions (item 3d).* Both conditions are the equality of the maximal speeds:

$$
\text{(8.13)}\iff v^A_{\max}=v^B_{\max}\iff\text{(8.17)}\iff\text{(8.26)} .
\tag{8.18}
$$

A common maximal speed is therefore necessary and sufficient for a common leading-order law $R=1+\nu^2/(2v_{\max}^2)$ of moving clocks (to the order stated at (8.13)), and for a common relation $\Delta_k=2I_k\,v_{\max}^2$ between branch gap and background inertia, equivalently a common slope $v_{\max}^2$ of rest cost against inertia (exact in the sense stated at (8.25)). The reason is that the same combination $\tau_T\kappa_T\sigma_{X_T}$ of (5.11) appears in (5.28) and in (6.20). A common maximal speed does not imply that (5.29) agree beyond the leading order (8.15). With common points (8.10), all these conditions become $\tau_A=\tau_B$.

## Result

- **Item 1.** (a) All replacements: (8.1), parametrized one-to-one; action on the six quantities and under the phase of $\chi$: (8.2). (b) Well-defined quantities: (8.3); $\bar x_a-\bar x_b$ is not well defined. (c) Yes: the bound (8.5) holds exactly for every product start and every $\lambda$; $v^A_{\max}$ does not depend on $C_B$, $K^B_k$, $\vartheta$ (8.6).
- **Item 2.** (a), (b) Maximal speeds (8.7) and the conditions (8.8). (c) No (8.9). (d) A full correspondence holds iff $\ell_B=\pm\ell_A$, which gives equal distances of neighbouring points; then the maximal speeds are equal iff $t_A=t_B$, respectively $\tau_A=\tau_B$ (8.10).
- **Item 3.** (a) Effective body contracts (8.19) and their transformation (8.20); the exact bound (8.21); for Model D the form (8.22), the bounds (8.23) uniform in the chain length, and the factorization (8.11). (b) $R_A-R_B$ at leading order (8.12); the condition (8.13); the relation (8.14) and the answer no (8.15). (c) In the displayed description $I^T_k=M^T_k/(v^T_{\max})^2$ (8.16) with the condition (8.17); the rest eigenvalues and the branch gap (8.24); the convention-independent relations (8.25) and condition (8.26). (d) All conditions are $v^A_{\max}=v^B_{\max}$ (8.18).

## Consistency checks

1. **Dimensions and rescaling.** Let $\mathsf c=[\lambda]^{-1}$, the dimension of a cost. Then $t$, $\tau$, $m$, $M$, $E$, $\Delta$, $\gamma$, $K$ and, because $u=(K-\langle K\rangle)\chi$, the record vectors and $d_1$ have dimension $\mathsf c$. So $[v_{\max}]=[\eta\,d_1]=\mathsf c^2=[\bar x/\lambda]$; $\varsigma$, $d_1\Lambda$, $\beta$, $\rho$, $R$ are dimensionless; $[I_k]=[M/v_{\max}^2]=\mathsf c^{-3}$, the inverse of $[a/F]=\mathsf c^3$ ($g(u,F)$ is a cost, so $F$ is dimensionless); in (8.25), $\mathsf c=\mathsf c^{-3}\mathsf c^{4}$. The rescaling $\kappa_T\mapsto s\kappa_T$, $X_T\mapsto X_T/s$ leaves the records unchanged and gives $\sigma_{X_T}\mapsto\sigma_{X_T}/s$, $\langle X_T\rangle\mapsto\langle X_T\rangle/s$: $d_1^T$, $\gamma_T$, $v^T_{\max}$, $I^T_k$, the bounds (8.23) and the conditions (8.8), (8.10), (8.13), (8.15), (8.17), (8.26) are unchanged. Passed.
2. **The rebookings of the verification.** (i) $\alpha^A_{(x,s,a)}=\delta\,(\sigma_3)_{ss}$, $\alpha^B=0$, $Z=0$: the displayed contract becomes $C^{\rm D}_A+\delta\,\sigma_3\otimes\mathbb 1_N$, with half-gap $M^A_k+\delta$ at label $0$, and $\langle K'^A_h\rangle=\gamma_Ax-\delta(\sigma_3)_{ss}$. (8.19) gives $C^{\rm eff}_A=C^{\rm D}_A+\gamma_A\hat x_A$, unchanged, as (8.20) states for $\langle Z\rangle=0$; so $\Delta^A_k$, $I^A_k$ and (8.25) are unchanged, while the half-gap of the displayed contract is not. (ii) $\alpha^A=c$, $\alpha^B=-c$, $Z=c\,\mathbb 1_c$: $E^A_{k,+}=M^A_k+c$ and $E^B_{k',+}=M^B_{k'}-c$; gaps, level differences and the sum are unchanged, and the proportionality $E_+=I\,v_{\max}^2$ is lost. Under $a\leftrightarrow b$: $Z\mapsto-Z$, $c_A\leftrightarrow c_B$, and all conditions are symmetric. Passed.
3. **Equal parameters, and a special case of (8.14).** For equal parameters, (8.8), (8.13), (8.17), (8.26) hold and (8.12), (8.15) vanish. For $\rho_T=1$, $\sin^2p_0=0.01$: (5.22) gives $R=\sqrt{1.01}=1.0049876$, and (5.22), (5.25) give $\beta^2=0.01\cdot0.99/1.01=0.0098020$. The expansion (8.14) gives $1+0.0049010+0.875\cdot0.00009608=1.0049851$. The difference $2.5\cdot10^{-6}$ is of order $\beta^6=9.4\cdot10^{-7}$. Passed.

## Open issues

- Equality of the maximal speeds is a condition on the description (8.9); nothing in this package selects it (out of scope).
- The rest cost as a cost is not well defined: only the branch gap, cost differences within a type and the total of the two bodies are (8.24). A later package that needs $E=I\,v_{\max}^2$ as a proportionality must state the booking $\langle Z\rangle=0$ as a convention; nothing in this setting singles it out.
- The gradient $\gamma_T=\kappa_T\langle X_T\rangle$ of (8.22) is convention-independent and of the same order in $\kappa_T$ as the force on the position label $\mathcal F=\kappa\,g(e,F)$ of 06. It does not enter $\mu_{\rm av}$, which is the coefficient of $F$, and its effect on the views is bounded by (8.23). For $\langle X_T\rangle\neq0$ it should add an $F$-independent mean acceleration of order $\kappa_T^2$, which the quoted results do not describe; whether 06 accounts for it cannot be checked from the quoted inputs.
- Item 3 inherits the orders of 05 and 06. Beyond leading order in $\kappa$ (out of scope), the influence of $b$ on the view of $a$ vanishes if $\kappa_A=0$ or $\kappa_B=0$, so it is of order $\kappa_A\kappa_B$.
- $v^T_{\max}$ is an upper bound. The drift speeds (5.25) of Model D do not exceed $v^T_{\rm top}<v^T_{\max}$, which depends on $\rho_T$ as well; equal $v_{\max}$ does not give equal $v_{\rm top}$ (8.15).
- Only chain types and Model D types are compared, as the question asks; other types are covered only by (8.5), (8.6), (8.9) and (8.19)–(8.21).

## Methods used

- Linear independence of place projectors; rebooking of single-object terms (A5)
- Invariance analysis under description conventions (M6)
- Partial expectation in the medium start (effective one-body contracts)
- Continuity equation for place weights, Cauchy–Schwarz inequality, Rayleigh bound for a symmetric matrix
- Spectrum of the path-graph adjacency matrix
- Duhamel formula with state-dependent bounds; commutator bound for the position moment; Minkowski inequality; contractivity of the partial trace
- Elimination of a parameter and power-series inversion

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