Version 1 · earlier version; the current one is v5 · Review of v1

# Two types of bodies in one background: equal fall and equal clock response

- **Subproject:** 05-ilang-free-fall
- **Package:** 08-two-types
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-11

## Setup and assumptions

**Setting (assumptions of this package, M8).** Objects $b_1$, $b_2$, $c$ (and $f$ in Step 5) of pairwise different types, one instance each; no swap condition applies and every unit vector is an admissible start. The contract is $C=F_1+F_2$ with $F_i:=C_i+\kappa_i\,\hat x_i\otimes X$, $\kappa_i>0$, and $C_i$ of the class $\mathcal C$ of 03-clock-and-fall: data $\tau_i>0$, $m_i>0$, $N_i\ge2$, $\lambda_{0,i}>0$, linear profiles with the gradients $g_{m,i}$, $g_{\eta,i}$, $g_{\zeta,i}$, $\mathcal F_i$ on a chain $\lvert x\rvert\le L$, $L\ge1$. The start is a product with the medium in $\chi$, $\sigma_X>0$. All profiles are those of $C^{\rm eff}_i=C_i+\kappa_i\langle X\rangle\hat x_i$, (8.19) of 04-ilang-time/08-maximal-speed; so $\mathcal F_i=\mathcal F^{\rm own}_i-\kappa_i\langle X\rangle$, with $\mathcal F^{\rm own}_i$ the gradient written in $C_i$.

**Quoted inputs and the quantities built from them.**
- (17.15) of 03-ilang-space/17-motion: $u^{(i)}_x=\kappa_i\,x\,e$, $d_{0,i}=\kappa_i\sigma_X$, $\hat e=e/\sigma_X$. (5.11) of 04-ilang-time/05-moving-clock at zero gradients: $v_i=\tau_i\,d_{0,i}$.
- (3.23): $r_i(x)=\eta^{(i)}_x/\lambda_{0,i}$ with $\eta^{(i)}_x=1+g_{\eta,i}x$ (wide start $S_{\rm clock}(x)$).
- (4.16): $a^{(i)}_{k,\sigma}(0)=\frac{\tau_i^2}{M^{(i)}_k}\bigl(\sigma\mathcal F_i-G^{(i)}_k\bigr)$, $M^{(i)}_k=m_i+d^{(i)}_k$, $G^{(i)}_k=m_ig_{m,i}+d^{(i)}_kg_{\eta,i}$, $d^{(i)}_k=\frac{2\pi k}{N_i\lambda_{0,i}}$; $A^{(i)}_{k,\sigma}:=d_{0,i}\,a^{(i)}_{k,\sigma}(0)$. Also (4.22), (4.24).
- (5.19), (5.20), (5.25) of 05-rate-comparison; (2.3), (2.20) of 02-field-object; (8.1), (8.20) of 04-ilang-time/08-maximal-speed.

**Regime and orders (A13).** Branch starts at rest; first order in the gradients ($\lvert g\rvert L\ll1$); the drift acceleration of (4.16) (window average, then a sharp label); records at leading order in $\kappa_i$; rates in the wide-start form; $\lambda\ge0$.

**Names.** $(\mathrm L_i)$: $\mathcal F_i=m_i(g_{m,i}-g_{\eta,i})$. FA: the two types fall alike from rest. RL, RD: the clocks respond alike per label, per distance. A tag $(8.k)$ without a package name is a tag of this package; the three quoted tags of 04-ilang-time/08-maximal-speed always carry that name.

## Derivation

### Step 1. One type in the background (item 1)

**(a)** By (3.23), $\bigl(r_i(x+1)-r_i(x)\bigr)/r_i(x)=g_{\eta,i}/(1+g_{\eta,i}x)$, whose first order in the gradients does not depend on $x$. With (4.16):

$$
\varrho_i=g_{\eta,i},\qquad A^{(i)}_{k,\sigma}=\frac{d_{0,i}\,\tau_i^2}{M^{(i)}_k}\Bigl(\sigma\,\mathcal F_i-m_ig_{m,i}-d^{(i)}_kg_{\eta,i}\Bigr).
\tag{8.1}
$$

Since $d_{0,i}>0$, $A^{(i)}_{k,+}$ is level-independent exactly when $a^{(i)}_{k,+}(0)$ is. By (4.22) and $v_i=\tau_id_{0,i}$:

$$
A^{(i)}_{k,+}\ \text{the same for all }k\iff(\mathrm L_i):\ \mathcal F_i=m_i\,(g_{m,i}-g_{\eta,i});\qquad\text{value}\quad A^{(i)}=-d_{0,i}\,\tau_i^2\,g_{\eta,i}=-\frac{v_i^2}{d_{0,i}}\,\varrho_i .
\tag{8.2}
$$

**(b)** The three operations act as follows.
- *(8.1) of 04-ilang-time/08-maximal-speed.* By (8.20) of 04-ilang-time/08-maximal-speed, $C^{\rm eff}_i\to C^{\rm eff}_i\pm\langle Z\rangle\mathbb 1$: $V^{(i)}_x$ gets a constant, and no gradient and no off-diagonal entry changes. The record vectors become $u^{(i)}_x\pm(Z-\langle Z\rangle)\chi$, the same shift for all positions of a type, so all differences $u^{(i)}_x-u^{(i)}_{x'}$ and $d_{0,i}$ are unchanged. Hence none of the eight quantities changes; $\kappa_i$ is defined only together with $X$.
- *Unit of $\lambda$.* Every entry of $C$ is divided by $s$: $\tau_i$, $m_i$, $d^{(i)}_k$, $\mathcal F_i$, $\kappa_iX$ are divided by $s$ ($\lambda_{0,i}\to s\lambda_{0,i}$), while $g_{m,i}$, $g_{\eta,i}$, $g_{\zeta,i}$ are unchanged, because the profiles are normalised.
- *$X\to tX$, $\kappa_i\to\kappa_i/t$.* $\sigma_X\to t\sigma_X$, so $d_{0,i}$ and the effective contracts are unchanged.

With the factor $1/s$ of $\kappa_iX$ put into $\kappa_i$, and with (8.1):

$$
\Bigl(\tau_i,\ \kappa_i,\ d_{0,i},\ v_i,\ \varrho_i,\ \tfrac{\varrho_i}{d_{0,i}},\ A^{(i)}_{k,+},\ \tfrac{A^{(i)}_{k,+}}{v_i^2}\Bigr)\longmapsto\Bigl(\tfrac{\tau_i}{s},\ \tfrac{\kappa_i}{st},\ \tfrac{d_{0,i}}{s},\ \tfrac{v_i}{s^2},\ \varrho_i,\ s\,\tfrac{\varrho_i}{d_{0,i}},\ \tfrac{A^{(i)}_{k,+}}{s^3},\ s\,\tfrac{A^{(i)}_{k,+}}{v_i^2}\Bigr).
\tag{8.3}
$$

| | $\tau_i$ | $\kappa_i$ | $d_{0,i}$ | $v_i$ | $\varrho_i$ | $\varrho_i/d_{0,i}$ | $A^{(i)}_{k,+}$ | $A^{(i)}_{k,+}/v_i^2$ |
|---|---|---|---|---|---|---|---|---|
| well defined | no | no | no | no | yes | no | no | no |
| ratio of the two types well defined | yes | yes | yes | yes | yes | yes | yes | yes |

All ratios are well defined because $s$ and $t$ are common to the two types: there is one $\lambda$ and one $X$. The product $A^{(i)}_{k,+}\,d_{0,i}/v_i^2$ is well defined as well. Step 7 tests the ratios against a wider class of changes of description.

### Step 2. Fall alike (item 2)

**(a)** *Necessity.* If $A^{(i)}_{k,+}=A\neq0$ for all $k$ and $i$, then for each $i$ the acceleration is level-independent, so $(\mathrm L_i)$ holds and $A=-d_{0,i}\tau_i^2g_{\eta,i}$ by (8.2), for both $i$. *Sufficiency.* $(\mathrm L_1)$, $(\mathrm L_2)$ give the values (8.2), which are then equal and non-zero. With $d_{0,i}=\kappa_i\sigma_X$ and one $\sigma_X>0$:

$$
\text{FA}\iff(\mathrm L_1),\ (\mathrm L_2),\ \ \kappa_1\tau_1^2\,g_{\eta,1}=\kappa_2\tau_2^2\,g_{\eta,2}\neq0\iff(\mathrm L_1),\ (\mathrm L_2),\ \ v_1^2\,\frac{\varrho_1}{d_{0,1}}=v_2^2\,\frac{\varrho_2}{d_{0,2}}\neq0;\qquad A=-v_i^2\,\frac{\varrho_i}{d_{0,i}} .
\tag{8.4}
$$

**(b)** By (4.22), $a^{(i)}_{k,-}(0)$ is level-independent iff $\mathcal F_i=-m_i(g_{m,i}-g_{\eta,i})$. Together with $(\mathrm L_i)$ and $m_i>0$ this is $\mathcal F_i=0$, $g_{m,i}=g_{\eta,i}$, and the value is $-\tau_i^2g_{\eta,i}$ in both branches. Hence, by the argument of (a):

$$
A^{(i)}_{k,\sigma}=A\neq0\ \ \text{for all }k,\ \sigma=\pm,\ i=1,2\iff\mathcal F_i=0,\ \ g_{m,i}=g_{\eta,i}\ \ (i=1,2)\ \ \text{and}\ \ v_1^2\,\frac{\varrho_1}{d_{0,1}}=v_2^2\,\frac{\varrho_2}{d_{0,2}}\neq0 .
\tag{8.5}
$$

Here $\mathcal F_i=0$ is a condition on the effective profile: $\mathcal F^{\rm own}_i=\kappa_i\langle X\rangle$. $g_{\zeta,i}$ is free in (8.4) and (8.5).

**(c)** The same argument without the factors $d_{0,i}$:

$$
a^{(1)}_{k,+}(0)=a^{(2)}_{k',+}(0)\ \ \text{for all }k,k'\iff(\mathrm L_1),\ (\mathrm L_2)\ \ \text{and}\ \ \tau_1^2\,g_{\eta,1}=\tau_2^2\,g_{\eta,2} .
\tag{8.6}
$$

This statement is well defined: both sides scale by $1/s^2$ under (8.3), the labels belong to the types (A2), and their orientation is fixed by $\kappa_i>0$. It is not a statement about the space of 03-ilang-space (M2): it counts the own positions of each type. By (8.4) it coincides with FA iff $d_{0,1}=d_{0,2}$, that is $\kappa_1=\kappa_2$.

### Step 3. Clocks respond alike (item 3)

From (8.1) and $d_{0,i}=\kappa_i\sigma_X$:

$$
\text{RL}:\ \ \varrho_1=\varrho_2\iff g_{\eta,1}=g_{\eta,2};\qquad\text{RD}:\ \ \frac{\varrho_1}{d_{0,1}}=\frac{\varrho_2}{d_{0,2}}\iff\frac{g_{\eta,1}}{\kappa_1}=\frac{g_{\eta,2}}{\kappa_2} .
\tag{8.7}
$$

- *Same distance.* Positions $x$ and $x+n$ of the type $i$ are the distance $\ell=n\,d_{0,i}$ apart by (17.15) of 03-ilang-space/17-motion, and at first order $\bigl(r_i(x+n)-r_i(x)\bigr)/r_i(x)=n\varrho_i=\ell\,\varrho_i/d_{0,i}$. For two intervals of the same length $\ell$, one in the positions of each type, the relative rate differences agree iff RD. RL compares the clocks over $d_{0,1}$ for one type and over $d_{0,2}$ for the other.
- *What they rely on (A8).* Each $\varrho_i$ compares the rates of one type for two starts at equal $\lambda$: it uses the order of $\lambda$ within the type, and the unit of $\lambda$ cancels. RL then compares two numbers and uses no equal $\lambda$ between the types. RD uses in addition $d_{0,1}/d_{0,2}=\kappa_1/\kappa_2$, a ratio of entries of one contract $C$ that belong to different types; this ratio presupposes one $\lambda$ for both types (Step 7).

### Step 4. Relation (item 4)

**(a)** Assume FA. By (8.4) both $\varrho_i/d_{0,i}$ are non-zero and $v_1^2\varrho_1/d_{0,1}=v_2^2\varrho_2/d_{0,2}$, so RD holds iff $v_1^2=v_2^2$, that is iff $v_1=v_2$ ($v_i>0$). Assume RD. If $(\mathrm L_1)$, $(\mathrm L_2)$, $\varrho_1\neq0$ and $v_1=v_2$, then (8.2) gives $A^{(1)}=A^{(2)}\neq0$, which is FA; conversely FA gives $(\mathrm L_i)$ and $\varrho_i\neq0$ by (8.4) and $v_1=v_2$ by the first part. Hence neither statement implies the other, and

$$
\text{FA}\ \Longrightarrow\ \bigl(\text{RD}\iff v_1=v_2\bigr);\qquad\text{RD}\ \Longrightarrow\ \bigl(\text{FA}\iff(\mathrm L_1),\ (\mathrm L_2),\ \varrho_1\neq0,\ v_1=v_2\bigr).
\tag{8.8}
$$

**(b)** Controls, both of the variant (a) of 03-clock-and-fall for each type ($g_{m,i}=g_{\eta,i}=g_i$, $g_{\zeta,i}=0$, $\mathcal F_i=0$, so $(\mathrm L_i)$ holds), with $\kappa_1=\kappa_2$, $\tau_2=2\tau_1$ and $m_i$, $N_i$, $\lambda_{0,i}$ arbitrary:

$$
\text{K1}:\ g_1=4g_2\neq0\ \Rightarrow\ A^{(1)}=A^{(2)}=-4d_0\tau_1^2g_2,\ \ \frac{\varrho_1}{d_{0,1}}=4\,\frac{\varrho_2}{d_{0,2}};\qquad\text{K2}:\ g_1=g_2\neq0\ \Rightarrow\ \varrho_1=\varrho_2,\ \ \frac{\varrho_1}{d_{0,1}}=\frac{\varrho_2}{d_{0,2}},\ \ A^{(2)}=4A^{(1)} .
\tag{8.9}
$$

K1 falls alike while the clocks respond differently; in K2 the clocks respond alike while the types fall differently. In both $v_2=2v_1$, as (8.8) requires.

**(c)** With $v_1=v_2=:v$, FA implies RD with no further condition. RD implies FA only together with $(\mathrm L_1)$, $(\mathrm L_2)$ and $\varrho_1\neq0$, which are conditions inside each type. For two types that satisfy these, any two of FA, RD, $v_1=v_2$ imply the third, and (8.2) reads $A=-v^2\varrho_i/d_{0,i}$ with one $v$. The condition $v_1=v_2$ is $\tau_1\kappa_1=\tau_2\kappa_2$, not $\tau_1=\tau_2$.

### Step 5. A common field (item 5)

**(a)** (2.1) of 02-field-object couples the field to one body, so the argument of (2.3) is repeated for two. For $\Psi\in\mathcal H_{b_1}\otimes\mathcal H_{b_2}\otimes\mathcal H_c$: $(B_i\otimes\Phi)(\Psi\otimes\varphi)=\gamma\,(B_i\Psi)\otimes\varphi$ and $C_f\varphi=e_f\varphi$, hence

$$
C\,(\Psi\otimes\varphi)=\bigl((C_{\rm red}+e_f)\Psi\bigr)\otimes\varphi,\qquad C_{\rm red}=\sum_{i=1,2}\Bigl[A_i+\sum_xP^{(i)}_x\otimes(1+\gamma\theta_ix)\,\sigma_3\otimes\bigl(m_i\mathbb 1_{N_i}+D_i\bigr)+\kappa_i\,\hat x_i\otimes X\Bigr],
\tag{8.10}
$$

with $A_i$ the motion term of $C^{\rm D}_i$. So $e^{-iC\lambda}(\Psi_0\otimes\varphi)=e^{-ie_f\lambda}\bigl(e^{-iC_{\rm red}\lambda}\Psi_0\bigr)\otimes\varphi$: the field stays a factor, $e_f$ is a common phase (1.2), and the bodies and the medium evolve by $C_{\rm red}$, which has the form of the setting. Reading off the effective profiles ($V^{(i)}_x=\kappa_i\langle X\rangle x$):

$$
g_{m,i}=g_{\eta,i}=\gamma\theta_i,\qquad g_{\zeta,i}=0,\qquad\mathcal F_i=-\kappa_i\langle X\rangle;\qquad\text{class }\mathcal C\iff\lvert\gamma\rvert\,L\,\max(\lvert\theta_1\rvert,\lvert\theta_2\rvert)<1 .
\tag{8.11}
$$

The range is the positivity of $1+\gamma\theta_ix$ on $\lvert x\rvert\le L$; the first-order results need $\lvert\gamma\theta_i\rvert L\ll1$.

**(b)** Insert (8.11) into (8.1)–(8.8); $\varrho_i=\gamma\theta_i$.
- *$(\mathrm L_i)$.* It reads $\kappa_i\langle X\rangle=0$, that is $\langle X\rangle=0$, and contains neither $\theta_i$ nor $\gamma$. For $\langle X\rangle\neq0$, (8.1) gives $A^{(i)}_{k,+}=-d_{0,i}\tau_i^2\bigl(\gamma\theta_i+\kappa_i\langle X\rangle/M^{(i)}_k\bigr)$: the level-dependent part is of the order $\kappa_i^2$ and the level-independent part of the order $\kappa_i$, so at the leading order in $\kappa_i$ of (4.24) $(\mathrm L_i)$ holds for every $\langle X\rangle$.
- *The other conditions*, with $(\mathrm L_i)$ in this sense and $\gamma\neq0$:

$$
\begin{aligned}
&A^{(i)}=-\sigma_X\,\kappa_i\tau_i^2\,\theta_i\,\gamma;\qquad\text{FA and (8.5)}\iff\kappa_1\tau_1^2\theta_1=\kappa_2\tau_2^2\theta_2\neq0;\qquad(8.6)\iff\tau_1^2\theta_1=\tau_2^2\theta_2;\\
&\text{RL}\iff\theta_1=\theta_2;\qquad\text{RD}\iff\frac{\theta_1}{\kappa_1}=\frac{\theta_2}{\kappa_2};\qquad\text{FA and RD}\iff\frac{\theta_1}{\kappa_1}=\frac{\theta_2}{\kappa_2}\neq0,\ \ \tau_1\kappa_1=\tau_2\kappa_2 .
\end{aligned}
\tag{8.12}
$$

(8.5) adds nothing to FA because $\mathcal F_i=0$ and $g_{m,i}=g_{\eta,i}$ hold already. No condition of (8.12) contains $\gamma$: each holds for every sharp value $\gamma\neq0$ of the range (8.11) once it holds for one. Only the common value $A$ is proportional to $\gamma$. At $\gamma=0$, RL and RD hold trivially and FA fails ($A=0$).

**(c)** The replacements are: (2.20) for two bodies, $\Phi\to\Phi+a\mathbb 1_f$, $C_f\to C_f+c\,\mathbb 1_f$, $C^{\rm D}_i\to C^{\rm D}_i-aB_i$ (and $-c$), which leaves $C$ unchanged by direct substitution, keeps $\theta_i$ and gives $\gamma\to\gamma+a$; $\Phi\to t\Phi$, $\theta_i\to\theta_i/t$, $\gamma\to t\gamma$; and those of (8.3), which leave $\theta_i$ unchanged. There is one $\Phi$, so $a$ and $t$ are common to the two types.

$$
\text{well defined: }\ \theta_1=\theta_2,\quad\text{the value of }\theta_1/\theta_2,\quad\frac{\theta_1}{d_{0,1}}=\frac{\theta_2}{d_{0,2}};\qquad\text{not well defined: the value of }\theta_i\gamma\ \ (\theta_i\gamma\to\theta_i\gamma+a\theta_i).
\tag{8.13}
$$

After (2.20) the own contracts are of the class $\mathcal C$ with the own gradients $-a\theta_i$, and the well-defined number is the gradient of $C_{\rm red}$, $g_{\eta,i}=-a\theta_i+\theta_i(\gamma+a)=\varrho_i$. It equals $\theta_i\gamma$ only in the descriptions whose body contracts are Model D contracts, a convention that fixes the zero of $\Phi$. Differences $\theta_i(\gamma-\gamma')$ of two sharp values are well defined.

**(d)** By (8.12), with $\Theta_i:=v_i^2\theta_i/d_{0,i}=\sigma_X\kappa_i\tau_i^2\theta_i$:

$$
\text{common coupling for the fall:}\ \ \Theta_1=\Theta_2\ \ (\text{then }A=-\Theta\gamma);\qquad\text{common coupling for the clocks:}\ \ \frac{\theta_1}{d_{0,1}}=\frac{\theta_2}{d_{0,2}} .
\tag{8.14}
$$

Both are well defined by (8.13) and (8.3). They coincide iff $v_1=v_2$. Equal couplings per label, $\theta_1=\theta_2$, are RL only: they give RD iff $\kappa_1=\kappa_2$ and FA iff $\kappa_1\tau_1^2=\kappa_2\tau_2^2$.

### Step 6. Equivalences of the setting (items 6a, 6b)

**(a)** For the class $\mathcal C$ with $\tau_i>0$, $C_i$ has the entries $\mp\tfrac{i}{2}\tau_i\zeta_x\neq0$ between $x$ and $x+1$, so $J_1\neq0$ and $J_2\neq0$. With the medium read, (5.19) leaves two rows: $X$ diagonal, with $a_1$, $a_2$ free; $X$ not diagonal, with $a_1=a_2=a_c$. With the medium not read, (5.20) has all $a_1,a_2>0$ for every $X$. Hence

$$
\text{the relative factor of }b_1,b_2\text{ is detectable}\iff\text{the medium is read and }X\text{ is not diagonal in the places of the medium}.
\tag{8.15}
$$

**(b)** In (5.20), $C_i/a_i$ is of the class $\mathcal C$ with every entry divided by $a_i$, and $X$, $\chi$ are unchanged. With (8.1), (4.16) and $v_i=\tau_id_{0,i}$:

$$
\begin{aligned}
&\tau_i,\ m_i,\ d^{(i)}_k,\ \mathcal F_i,\ \kappa_i,\ d_{0,i}\ \to\ \tfrac1{a_i}\,(\,\cdot\,);\qquad\lambda_{0,i}\to a_i\lambda_{0,i};\qquad g_{m,i},\ g_{\eta,i},\ g_{\zeta,i},\ \varrho_i\ \ \text{unchanged};\\
&v_i\to\frac{v_i}{a_i^2},\qquad\frac{\varrho_i}{d_{0,i}}\to a_i\,\frac{\varrho_i}{d_{0,i}},\qquad a^{(i)}_{k,\sigma}(0)\to\frac{a^{(i)}_{k,\sigma}(0)}{a_i^2},\qquad A^{(i)}_{k,\sigma}\to\frac{A^{(i)}_{k,\sigma}}{a_i^3},\qquad\frac{A^{(i)}_{k,+}}{v_i^2}\to a_i\,\frac{A^{(i)}_{k,+}}{v_i^2} .
\end{aligned}
\tag{8.16}
$$

$C^{\rm eff}_i/a_i$ has the eigenvectors of $C^{\rm eff}_i$, so the levels, the branches, the branch starts and $S_{\rm clock}$ are the same, and $(\mathrm L_i)$ is unchanged. The rates obey $r'_i=r_i/a_i$, as in (5.25).

### Step 7. All equivalent descriptions of the setting

The statements of item 6(c) are defined for descriptions of the form of the setting. The following lemma lists all of them.

**Lemma.** Let $(a_1,a_2;C')$ be an equivalence of $C$, the medium read or not, with $C'=F'_1+F'_2$ again of the form of the setting: $F'_i=C'_i+\kappa'_i\hat x_i\otimes X'$, $C'_i$ of the class $\mathcal C$ on the same places with linear profiles, $\kappa'_i>0$. Then

$$
F'_i=\frac{F_i}{a_i}+c_i\,\mathbb 1,\qquad\text{hence}\qquad C'^{\rm eff}_i=\frac{C^{\rm eff}_i}{a_i}+c_i\,\mathbb 1,\qquad\kappa'_i\,(X'-\langle X'\rangle)=\frac{\kappa_i}{a_i}\,(X-\langle X\rangle),
\tag{8.17}
$$

and the rules (8.16) hold for it. Conversely $F_1/a_1+F_2/a_2+c\,\mathbb 1$ is an equivalence for all $a_1,a_2>0$ if the medium is not read (5.20), and if it is read and $X$ is diagonal (5.19); for $X$ not diagonal and the medium read, only for $a_1=a_2$ (5.19).

*Proof.*
1. Take the schedule with one event: $b_1$, the reading into single places $h$ (projectors $P_h$), at $\lambda$. With $U=e^{-iC\lambda}$ and $U'=e^{-iC'a_1\lambda}$, the equality of the statistics for every start gives $U'^{\dagger}P_hU'=U^{\dagger}P_hU$ (a self-adjoint operator is fixed by its expectation values). So $W(\lambda):=U'U^{\dagger}$ commutes with every $P_h$, for all $\lambda\ge0$.
2. The derivatives of $W$ at $\lambda=0$ commute with every $P_h$ as well: $i\dot W(0)=a_1C'-C=:R$ and $-\ddot W(0)=R^2+[C,R]$. Hence $R=\sum_hP_h\otimes R_h$, with $R_h$ on $\mathcal H_{b_2}\otimes\mathcal H_c$, and $[C,R]$ is block diagonal in the places of $b_1$. Only the off-diagonal part of $C_1$ is not block diagonal, so $(C_1)_{hh'}(R_{h'}-R_h)=0$ for $h\neq h'$.
3. $R$ block diagonal means that $a_1C'_1-C_1$ is diagonal in the places of $b_1$. The hopping entries give $a_1\tau'_1\zeta'_x=\tau_1\zeta_x$ on all bonds. The clock entries give $a_1\eta'_x/\lambda'_{0,1}=\eta_x/\lambda_{0,1}$, because $D'_1=(\lambda_{0,1}/\lambda'_{0,1})D_1$ and $D_1$ is not diagonal in the clock label ($\langle a\vert D\vert a'\rangle=\frac{2\pi}{N\lambda_0}\bigl(e^{2\pi i(a-a')/N}-1\bigr)^{-1}\neq0$ for $a\neq a'$). With the normalised linear profiles: $\tau'_1=\tau_1/a_1$, $\lambda'_{0,1}=a_1\lambda_{0,1}$, $g'_{\zeta,1}=g_{\zeta,1}$, $g'_{\eta,1}=g_{\eta,1}$. What remains is $R_h=s\,\mu(x)+\nu(x)+x\,Y+R_0$ for $h=(x,s,a)$, $s=\pm1$, with $\mu(x):=a_1m'_x-m_x$, $\nu(x):=a_1V'_x-V_x$ (the own profiles), $Y:=a_1\kappa'_1X'-\kappa_1X$ and $R_0$ independent of $h$.
4. The hopping connects $(x,s,a)$ and $(x+1,-s,a)$, so by item 2 $s\,[\mu(x)+\mu(x+1)]\,\mathbb 1=\nu(x+1)-\nu(x)+Y$ for $s=\pm1$. Hence $\mu(x)+\mu(x+1)=0$ on at least two bonds, so the linear $\mu$ vanishes ($m'_1=m_1/a_1$, $g'_{m,1}=g_{m,1}$), and $Y=-\nu'\mathbb 1_c$ with $\nu(x)=\nu_0+\nu'x$. Then $a_1F'_1=C_1+\nu_0+\nu'\hat x_1+\hat x_1\otimes(\kappa_1X-\nu')=F_1+\nu_0$. The same holds for $b_2$ with $a_2$.
5. Taking $\langle\chi\vert\cdot\vert\chi\rangle$ of $F'_i$ gives $C'^{\rm eff}_i$, and $F'_i-C'^{\rm eff}_i$ gives the record part. The split of $F'_i$ into $C'_i$ and the record is free up to the replacements of Step 1(b), which change none of the quantities in (8.16). $\blacksquare$

### Step 8. What is the same in every equivalent description (items 6c, 6d, 6e)

**(c)** By (8.17) every equivalent description of the setting acts by (8.16) with its factors. So in it: $(\mathrm L_i)$ is unchanged; FA reads $(\mathrm L_i)$ and $A^{(1)}/a_1^3=A^{(2)}/a_2^3\neq0$; $v'_1=v'_2$ reads $v_1/a_1^2=v_2/a_2^2$; RL is unchanged; RD reads $a_1\varrho_1/d_{0,1}=a_2\varrho_2/d_{0,2}$; and for (4.24) the hypothesis is unchanged ($a'^{(i)}_{k,+}=a^{(i)}_{k,+}/a_i^2$) and $A'^{(i)}=A^{(i)}/a_i^3=-v_i^2\varrho_i/(a_i^3d_{0,i})=-v'^2_i\varrho'_i/d'_{0,i}$. In the first column below $a_1=a_2$ by (8.15). In the other two all $a_1,a_2>0$ occur, and the equivalence $a_1=2$, $a_2=1$ makes a true FA, a true $v_1=v_2$ and a true RD with $\varrho_i\neq0$ false ($A'^{(1)}=A/8$, $v'_1=v/4$, $\varrho'_1/d'_{0,1}=2\varrho_1/d_{0,1}$).

$$
\begin{array}{l|c|c|c}
\text{statement} & \text{read, }X\text{ not diagonal} & \text{read, }X\text{ diagonal} & \text{not read}\\\hline
\text{FA} & \text{same} & \text{not the same} & \text{not the same}\\
v_1=v_2 & \text{same} & \text{not the same} & \text{not the same}\\
\text{RL} & \text{same} & \text{same} & \text{same}\\
\text{RD} & \text{same} & \text{not the same} & \text{not the same}\\
\text{(4.24) for each type} & \text{same} & \text{same} & \text{same}
\end{array}
\tag{8.18}
$$

**(d)** Medium not read. By (8.16), $v'_1=v'_2$ iff $a_1/a_2=\sqrt{v_1/v_2}$, which has a solution for every pair; the common factor left free is the unit of $\lambda$. In this description, with $v'$ the common speed,

$$
\frac{A'^{(1)}_{k,+}}{A'^{(2)}_{k',+}}=\Bigl(\frac{v_2}{v_1}\Bigr)^{3/2}\frac{A^{(1)}_{k,+}}{A^{(2)}_{k',+}},\qquad\varrho'_i=\varrho_i,\qquad\frac{\varrho'_1/d'_{0,1}}{\varrho'_2/d'_{0,2}}=\sqrt{\frac{v_1}{v_2}}\;\frac{\varrho_1/d_{0,1}}{\varrho_2/d_{0,2}},\qquad A'^{(i)}=-v'^2\,\frac{\varrho_i}{d'_{0,i}}\ \ \text{under }(\mathrm L_i).
\tag{8.19}
$$

A description in which the types fall alike needs $(\mathrm L_i)$ and $(a_1/a_2)^3=A^{(1)}/A^{(2)}>0$. By (8.17) there are no other equivalent descriptions of the setting, and $\operatorname{sign}A^{(i)}=-\operatorname{sign}g_{\eta,i}$ by (8.2), so

$$
\text{an equivalent description with FA exists}\iff(\mathrm L_1),\ (\mathrm L_2)\ \ \text{and}\ \ g_{\eta,1}\,g_{\eta,2}>0 ;\qquad\text{in it}\quad\frac{v'_1}{v'_2}=\frac{v_1}{v_2}\Bigl(\frac{A^{(2)}}{A^{(1)}}\Bigr)^{2/3} .
\tag{8.20}
$$

**(e)** *Relative factor detectable.* All equivalent descriptions of the setting are $C/\alpha+c$: a change of the unit of $\lambda$, (8.3) with $s=\alpha$. FA, $v_1=v_2$, RL, RD and every ratio of (8.3) mean the same in all of them, as in Steps 2–4.

*Relative factor not detectable.* By (8.16) the following are the same in all equivalent descriptions: $(\mathrm L_i)$, $\operatorname{sign}A^{(i)}$ (the direction of the fall along $\hat e$), $\varrho_i$, the relation (4.24) of each type, RL, and the numbers $A^{(i)}/v_i^{3/2}$. An equality of the magnitudes of the falls, or of the responses per distance, has a meaning only relative to a convention that fixes $a_1/a_2$. With the convention of equal maximal speeds, $v'_1=v'_2$, FA and RD become by (8.8) one statement that no equivalence changes:

$$
(\mathrm L_1),\ (\mathrm L_2)\quad\text{and}\quad\frac{A^{(1)}}{v_1^{3/2}}=\frac{A^{(2)}}{v_2^{3/2}}\neq0\iff(\mathrm L_1),\ (\mathrm L_2)\quad\text{and}\quad\varrho_1\sqrt{\frac{\tau_1}{\kappa_1}}=\varrho_2\sqrt{\frac{\tau_2}{\kappa_2}}\neq0 .
\tag{8.21}
$$

The second form uses $A^{(i)}/v_i^{3/2}=-\varrho_i\sqrt{\tau_i/d_{0,i}}$ from (8.2). Without a convention, "fall alike" can only mean (8.20): each type falls level-independently, and both in the same direction. The equal maximal speed is then a convention, not a condition (M7).

## Result

Class: all pairs of bodies of the class $\mathcal C$ with linear profiles, all $\kappa_i>0$, every $X$, $\chi$ with $\sigma_X>0$; branch starts at rest, first order in the gradients, leading order in $\kappa_i$. Item 5: all pairs of Model D bodies, all $\theta_i$, every sharp stationary field state in the range (8.11).

1. **One type.** (8.1): $\varrho_i=g_{\eta,i}$. (8.2): level-independent fall iff $(\mathrm L_i)$, value $-v_i^2\varrho_i/d_{0,i}$. (8.3) with its table: of the eight quantities only $\varrho_i$ is well defined; all ratios between the types are.
2. **Fall alike.** (8.4) in the data and through $v_i$ and the responses; (8.5) for both branches; (8.6) for the labels, well defined but not a statement about distances.
3. **Clocks.** (8.7); RD compares over the same distance; RL uses no common $\lambda$ of the two types, RD does.
4. **Relation.** (8.8): neither implies the other; controls (8.9); with $(\mathrm L_i)$ and $\varrho_i\neq0$, any two of FA, RD, $v_1=v_2$ imply the third.
5. **Common field.** Generator (8.10), gradients and range (8.11), conditions (8.12), all independent of $\gamma$; (8.13); common couplings (8.14).
6. **Equivalent descriptions.** Detectability (8.15); rules (8.16); all equivalent descriptions of the setting (8.17); table (8.18); the description of equal maximal speeds (8.19) and the existence condition (8.20); the convention-free and the convention-relative meaning (8.21).

Controls (M9): K1, K2 of (8.9); the columns of (8.18) (medium read against not read, $X$ diagonal against not diagonal).

## Consistency checks

1. **Dimensions and $a_1=a_2$.** With costs in units of $1/\lambda$: $d_{0,i}\sim\lambda^{-1}$, $v_i\sim\lambda^{-2}$, $A\sim\lambda^{-3}$, and (8.2) gives $\lambda^{-4}/\lambda^{-1}$. The rules (8.16) with $a_1=a_2=s$ are (8.3) with $t=1$, and then every row of (8.18) is "same", as in its first column. Passed.
2. **Identical data.** For $\tau_1=\tau_2$, $\kappa_1=\kappa_2$ and equal gradients: $v_1=v_2$, RL and RD hold, and (8.4) reduces to $(\mathrm L)$ and $g_\eta\neq0$, as (8.8) states. With the medium not read, (5.20) with $a_1\neq a_2$ still gives a description without FA, as (8.18) states. Passed.
3. **One gradient zero.** For $g_{\eta,2}=0$ and $(\mathrm L_2)$: $A^{(2)}=0$, so FA fails by (8.4), RD holds iff $g_{\eta,1}=0$ by (8.7), and no equivalent description with FA exists by (8.20), in agreement with $A'^{(2)}=A^{(2)}/a_2^3=0$. Passed.

## Open issues

- Item 5: with Model D body contracts the effective gradient is $\mathcal F_i=-\kappa_i\langle X\rangle$, so the level-independence is exact only for $\langle X\rangle=0$; for $\langle X\rangle\neq0$ it holds at the leading order in $\kappa_i$. The conditions on the six numbers in (8.12) are stated with this qualification.
- (8.17) lists the equivalent descriptions that are again of the form of the setting (class $\mathcal C$, linear profiles, one operator of the medium, at least three positions). For an equivalent contract outside this form, for example with a term $\Delta$ of (5.19) that is not a constant, the statements of (8.18) are not defined here.
- All results are of first order in the gradients, from rest and at leading order in the records; the fall in motion, the lattice terms and a field that is not sharp are out of scope.
- The convention $v'_1=v'_2$ in (8.21) is a choice. Whether anything in the language selects it belongs to criterion 10.

## Methods used

- Level-independence conditions of the drift acceleration (linear algebra in the level index)
- Scaling analysis under changes of the unit of the evolution parameter and of operator normalisations
- Invariance under replacements of contract terms at fixed total contract
- Reduction of a contract on an eigenstate of the field operators
- Operator identities from equal reading statistics (polarisation), expansion of unitary groups to second order
- Connectivity argument for operators commuting with a hopping contract

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