04-ilang-time / 08-maximal-speed
08maximal speedverified
Determines whether bodies of different types recorded by one medium have the same maximal speed, the conditions for it, and what it means for moving clocks and inertia.
# Common maximal speed: two body types recorded by one medium
- **Subproject:** 04-ilang-time
- **Package:** 08-maximal-speed
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-10
## 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).
**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. 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). 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 and statements to an order.**
- Steps 1–6 (items 1 and 2) are exact, for finite systems (A1); the chains on $\mathbb Z$ of (17.20) and (5.11) are the limits $L\to\infty$ of open chains with $L$ positions.
- Steps 7–9 (item 3) use the weak-record regime for both bodies: leading order in $\kappa_A$ and $\kappa_B$ at fixed $\lambda$ and fixed start. Step 8 uses in addition the orders of (5.28), (5.29): first order in $\epsilon_T$, leading order in the spread $\delta p_T$, and small labels $p_{0T}^2\ll1$. Step 9 uses the orders of (6.20): first order in the force, average over long $\lambda$-intervals.
## 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 data $(\alpha^A,\alpha^B,Z)$ and $(\alpha^A+c,\alpha^B-c,Z+c\mathbb 1_c)$, $c\in\mathbb R$, give the same replacement. For $\alpha^A=\alpha^B=0$, (8.1) is (16.3). 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.
*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). 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 $\kappa_Txe_T$. The off-diagonal entries of $C_T$ come from $A$, 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 $B$, 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
$$
v^T_{\max}=2\,\eta_T\,d_1^T\cos\frac{\pi}{L_T+1}\ \xrightarrow[L_T\to\infty]{}\ 2\,\eta_T\,d_1^T=
\begin{cases}2\,t_T\,\sigma_{X_T}&\text{chain, (17.20)},\\ \tau_T\,\kappa_T\,\sigma_{X_T}&\text{Model D, (5.11)}.\end{cases}
\tag{8.7}
$$
Hence, for chains on $\mathbb Z$ (and for finite chains with $L_A=L_B$), 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 with $L_A\neq L_B$ each side carries its factor $\cos\frac{\pi}{L_T+1}$.
### 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, $C_A\mapsto s\,C_A$ ($s>0$) 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 $\lvert X_A\rvert=\lvert X_B\rvert=L$, $U_T=\{j\,\ell_T\}$ 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
$$
\text{full correspondence}\iff\ell_B=\pm\ell_A\ \Longrightarrow\ d_1^A=d_1^B;\qquad\text{then}\quad v^A_{\max}=v^B_{\max}\iff\eta_A=\eta_B .
\tag{8.10}
$$
- Records: the two types have the same record vectors up to one translation $t_\pi\in\mathcal N$ (16.11), which a replacement (8.1) 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 $d_1^T$, exactly, $v^A_{\max}=v^B_{\max}$ 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. Why the one-body results apply to each body (item 3a)
Split $C=C_0+C_{\rm rec}$ with $C_0:=C_A\otimes\mathbb 1\otimes\mathbb 1+\mathbb 1\otimes C_B\otimes\mathbb 1$ and $C_{\rm rec}$ the record terms, of first order in $\kappa_A$, $\kappa_B$. A cost gradient in the diagonal entries of a body, as in 06-inertia-internal-energy, is part of its $C_T$. There is no contract between $a$ and $b$, so the two terms of $C_0$ act on different factors and commute. Duhamel's formula $e^{-iC\lambda}=e^{-iC_0\lambda}-i\int_0^\lambda e^{-iC(\lambda-s)}C_{\rm rec}\,e^{-iC_0s}\,\mathrm ds$ then gives, at fixed $\lambda$ and fixed start,
$$
\lvert\Psi(\lambda)\rangle=e^{-iC_A\lambda}\lvert\phi\rangle\otimes e^{-iC_B\lambda}\lvert\vartheta\rangle\otimes\lvert\chi\rangle+O(\kappa_A)+O(\kappa_B),\qquad
V_a(\lambda)=e^{-iC_A\lambda}\lvert\phi\rangle\langle\phi\rvert e^{iC_A\lambda}+O(\kappa_A)+O(\kappa_B),
\tag{8.11}
$$
and likewise for $V_b$. Three facts follow.
1. At leading order the view of each body is that of its own body contract alone, which is the defining property of the weak-record regime, and it equals the leading-order view of the one-body setting with the same $C_T$ and the same start.
2. By (8.6) the record vectors $\kappa_Txe_T$ of a body, hence the background (17.15) in which its position is measured, are those of the one-body setting with the medium start $\chi$.
3. Every quoted result of 05-moving-clock and 06-inertia-internal-energy is a statement about these two things only: the place weights of the body (position, branch and clock labels) and 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 "$O(\kappa)$" read as "$O(\kappa_A)+O(\kappa_B)$". The quantities used below are well defined: $\lVert\bar v_T\rVert$ by (8.2), because $\bar x(\lambda)-\bar x(0)$ is invariant, and $M^T_k$ as half the gap of (6.7), which a uniform $\alpha^T$ in (8.1) does not change.
### 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 $\delta p_T$, $\kappa_A$, $\kappa_B$) 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.
*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):
$$
R_T^2\bigl(1-\beta_T^2\bigr)=1+\frac{(R_T^2-1)^2}{\rho_T^2}
\quad\Longrightarrow\quad
R_T=1+\frac{\beta_T^2}2+\Bigl(\frac38+\frac1{2\rho_T^2}\Bigr)\beta_T^4+O(\beta_T^6).
\tag{8.14}
$$
(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
$$
R_A-R_B=\frac{\beta^4}2\Bigl(\frac{\bar\mu_A^2}{\tau_A^2}-\frac{\bar\mu_B^2}{\tau_B^2}\Bigr)+O(\beta^6);\qquad
\text{(5.29) agree}\iff\text{(8.13) and }\ \frac{\bar\mu_A}{\tau_A}=\frac{\bar\mu_B}{\tau_B},\quad\bar\mu_T=m_T+\frac{\pi(N_T-1)}{N_T\lambda_{0T}} .
\tag{8.15}
$$
So the answer is **no**: under (8.13) the left sides of (5.29) have the same form, but the right sides, as functions of $R$ or of $\nu$, 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 (5.29) agree at order $\beta^2$ and differ at order $\beta^4$. The same additional 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)
By Step 7, (6.20) holds for each body. 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 $n_T=e_T/\sigma_{X_T}$. As an identity between the quantities defined in (6.7), (6.20) and (5.11), (8.16) is exact; that $I^T_k$ is the inertia of the body holds at the orders of (6.20). The relation between background inertia and rest cost is thus a proportionality with the factor $M^T_k/I^T_k=(v^T_{\max})^2$, the same for all levels of one type. Hence
$$
\frac{M^A_k}{I^A_k}=\frac{M^B_{k'}}{I^B_{k'}}\ \ \text{for all levels }k,k'\ (\text{equivalently, for one pair})\iff v^A_{\max}=v^B_{\max}\iff\tau_A\,\kappa_A\,\sigma_{X_A}=\tau_B\,\kappa_B\,\sigma_{X_B} .
\tag{8.17}
$$
In addition, for the label inertia $I^{\rm lab}_k=M_k/\tau^2$ the relation to the rest cost 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:
$$
\text{(8.13)}\iff v^A_{\max}=v^B_{\max}\iff\text{(8.17)} .
\tag{8.18}
$$
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 $M_k=I_k\,v_{\max}^2$ between rest cost and background inertia (exact in the sense stated at (8.16)). 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 three conditions become $\tau_A=\tau_B$.
## Result
- **Item 1.** (a) All replacements: (8.1); 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) 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) $I^T_k=M^T_k/(v^T_{\max})^2$ (8.16) and the condition (8.17). (d) Both conditions are $v^A_{\max}=v^B_{\max}$ (8.18).
## Consistency checks
1. **Dimensions and rescaling.** Let $\mathsf c=[\lambda]^{-1}$ (cost) and $\mathsf L=[u]$. Then $[v_{\max}]=[\eta\,d_1]=\mathsf{cL}=[\bar x/\lambda]$; $\beta$, $\rho$, $R$ are dimensionless; $[I_k]=[M/v_{\max}^2]=\mathsf c^{-1}\mathsf L^{-2}$, which is the inverse of $[a/F]=\mathsf{Lc}^2/(\mathsf c/\mathsf L)$. The rescaling $\kappa_T\mapsto s\kappa_T$, $X_T\mapsto X_T/s$ leaves the records unchanged and gives $\sigma_{X_T}\mapsto\sigma_{X_T}/s$: $d_1^T$, $v^T_{\max}$, $I^T_k$ and the conditions (8.8), (8.10), (8.13), (8.15), (8.17) are unchanged. Passed.
2. **Equal parameters and exchange of $a$ and $b$.** For equal parameters, (8.8), (8.13), (8.17) hold and (8.12), (8.15) vanish. Under $a\leftrightarrow b$, (8.1) goes into itself with $Z\mapsto-Z$, $\alpha^A\leftrightarrow\alpha^B$, (8.2) with $z\mapsto-z$; (8.12) and (8.15) change sign; (8.3) and all conditions are symmetric. Passed.
3. **Special case of (8.14).** $\rho_T=1$, $\sin^2p_0=0.01$: (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).
- Item 3 inherits the orders of 05-moving-clock and 06-inertia-internal-energy. 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 $\kappa_A\kappa_B$; and the mean $\langle X_T\rangle$ can be rebooked by (8.1), with $\alpha^T_x\propto x$, as a uniform cost gradient on body $T$ of first order in $\kappa_T$.
- $v^T_{\max}$ 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) and (8.9).
## Methods used
- Linear independence of place projectors; rebooking of single-object terms (A5)
- Invariance analysis under description conventions (M6)
- Continuity equation for place weights, Cauchy–Schwarz inequality, Rayleigh bound for a symmetric matrix
- Spectrum of the path-graph adjacency matrix
- Duhamel (first-order perturbation) expansion in the record strength
- Elimination of a parameter and power-series inversion