Version 2 · earlier version; the current one is v5 · External review, round 1: major errors

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

- **Subproject:** 05-ilang-free-fall
- **Package:** 08-two-types
- **Version:** v2
- **Mode:** internal regeneration
- **Date:** 2026-10-11

## Changes from previous version

Local regeneration after `review.v1.md` (I1, I2). All tags (8.1)–(8.21) are kept. The displays (8.12) and (8.14) are restated under their old tags; no other display changed. Changed passages, in the order of the text:

1. **Header.** Version and mode.
2. **Step 2(c), the last sentence after (8.6)** (I2). v1: "By (8.4) it coincides with FA iff $d_{0,1}=d_{0,2}$, that is $\kappa_1=\kappa_2$." This fails for a zero common value, where (8.6) holds and FA does not. The sentence is replaced by the comparison in both directions, with the zero value treated. The display (8.6) is unchanged.
3. **Step 5(b)** (I1), four passages:
   - the opening line now also gives $A^{(i)}_{k,+}$ from (8.1) and (8.11);
   - the bullet "$(\mathrm L_i)$" is rewritten. The claim of v1 that $(\mathrm L_i)$ holds "at the leading order in $\kappa_i$" for every $\langle X\rangle$ is withdrawn: the level-dependent term is of the first order in $\kappa_i$ in $a^{(i)}_{k,+}(0)$ and is a gradient that (4.16) keeps;
   - the lead-in of the bullet "The other conditions" and the display (8.12): every condition now stands with the condition on $\langle X\rangle$ under which it holds, and for $\langle X\rangle\neq0$ the display says what fails and gives the level-dependent part with its bound;
   - the paragraph after (8.12), rewritten accordingly (range of the approximation, dependence on $\gamma$).
4. **Step 5(d)** (I1): the lead-in sentence; (8.14) restated with the same condition and with the statement that holds for every $\langle X\rangle$; one sentence added after "They coincide iff $v_1=v_2$."; the last sentence now carries "$\neq0$" and "for $\langle X\rangle=0$" (in v1 it gave "RD iff $\kappa_1=\kappa_2$" also for $\theta_1=\theta_2=0$, where RD holds for all $\kappa_i$).
5. **Result, item 5**, and **Open issues, first item**: brought in line with (8.12) and (8.14).
6. **JSON twin:** key results 8.6, 8.12, 8.14; one definition added ($\epsilon^{(i)}_k$); the first open issue.

The Setup, Steps 1, 2(a), 2(b), 3, 4, 5(a), 5(c), 6, 7, 8, the other items of the Result and of the Open issues, the consistency checks and the methods are word for word those of v1.

## 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. Its relation to FA follows from (8.4) and (8.6), which differ by the factors $\kappa_i$ and by the condition $\neq0$: FA $\Longrightarrow$ ((8.6) $\iff\kappa_1=\kappa_2$), and (8.6) $\Longrightarrow$ (FA $\iff\kappa_1=\kappa_2$ and $g_{\eta,1}\neq0$). In the first direction the common value is non-zero, so $\kappa_1\tau_1^2g_{\eta,1}=\kappa_2\tau_2^2g_{\eta,2}$ and $\tau_1^2g_{\eta,1}=\tau_2^2g_{\eta,2}$ hold together iff $\kappa_1=\kappa_2$, that is $d_{0,1}=d_{0,2}$. In the second direction, if the common value $-\tau_i^2g_{\eta,i}$ of (8.6) is non-zero, FA holds iff $\kappa_1=\kappa_2$; if it is zero ($g_{\eta,1}=g_{\eta,2}=0$), (8.6) holds and FA fails for all $\kappa_1$, $\kappa_2$, because FA requires $A\neq0$.

### 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); $\varrho_i=\gamma\theta_i$, and (8.1) reads $A^{(i)}_{k,+}=-d_{0,i}\tau_i^2\bigl(\gamma\theta_i+\kappa_i\langle X\rangle/M^{(i)}_k\bigr)$.
- *$(\mathrm L_i)$.* It reads $\kappa_i\langle X\rangle=0$, that is $\langle X\rangle=0$ ($\kappa_i>0$). It contains none of the six numbers and not $\gamma$: it is a condition on the start of the medium, the same for both types, and its well-defined form is $\mathcal F_1=\mathcal F_2=0$ for the effective contracts. For $\langle X\rangle\neq0$ it fails: $M^{(i)}_0\neq M^{(i)}_1$ ($N_i\ge2$), so $A^{(i)}_{k,+}$ depends on the level for every $\theta_i$ and $\gamma$. Then FA, (8.5) and (8.6), which all require a level-independent fall of each type, fail for all $\theta_1$, $\theta_2$, $\gamma$. The level-dependent term $-\tau_i^2\kappa_i\langle X\rangle/M^{(i)}_k$ of $a^{(i)}_{k,+}(0)$ is of the first order in $\kappa_i$ and of the first order in the gradients: it is the gradient $\mathcal F_i$, which the setting counts in $V^{(i)}_x$ and which (4.16) keeps. The leading order in the records does not remove it.
- *The other conditions*, for $\gamma\neq0$, each with the condition on $\langle X\rangle$ under which it holds. RL and RD contain only $\varrho_i$ and $d_{0,i}$ and do not need $(\mathrm L_i)$; neither does $v_1=v_2$, which is $\tau_1\kappa_1=\tau_2\kappa_2$.

$$
\begin{aligned}
&\text{every }\langle X\rangle,\ \gamma\neq0:\quad\text{RL}\iff\theta_1=\theta_2;\qquad\text{RD}\iff\frac{\theta_1}{\kappa_1}=\frac{\theta_2}{\kappa_2};\\
&\langle X\rangle=0,\ \gamma\neq0:\quad A^{(i)}=-\sigma_X\,\kappa_i\tau_i^2\,\theta_i\,\gamma;\qquad\text{FA and (8.5)}\iff\kappa_1\tau_1^2\theta_1=\kappa_2\tau_2^2\theta_2\neq0;\qquad(8.6)\iff\tau_1^2\theta_1=\tau_2^2\theta_2;\\
&\qquad\qquad\text{FA and RD}\iff\frac{\theta_1}{\kappa_1}=\frac{\theta_2}{\kappa_2}\neq0,\ \ \tau_1\kappa_1=\tau_2\kappa_2;\\
&\langle X\rangle\neq0:\quad(\mathrm L_i),\ \text{FA},\ (8.5),\ (8.6)\ \text{fail for all }\theta_1,\theta_2,\gamma;\qquad A^{(i)}_{k,+}=-\sigma_X\,\kappa_i\tau_i^2\,\theta_i\,\gamma\,\bigl(1+\epsilon^{(i)}_k\bigr),\\
&\qquad\qquad\epsilon^{(i)}_k:=\frac{\kappa_i\,\langle X\rangle}{M^{(i)}_k\,\gamma\theta_i},\qquad\lvert\epsilon^{(i)}_k\rvert\le\epsilon_i:=\frac{\kappa_i\,\lvert\langle X\rangle\rvert}{m_i\,\lvert\gamma\theta_i\rvert}\qquad(\gamma\theta_i\neq0).
\end{aligned}
\tag{8.12}
$$

For $\langle X\rangle=0$, (8.5) adds nothing to FA because $\mathcal F_i=0$ and $g_{m,i}=g_{\eta,i}$ hold already. For $\langle X\rangle\neq0$, $\epsilon^{(i)}_k$ is the level-dependent part of $A^{(i)}_{k,+}$ relative to the level-independent part; the bound uses $M^{(i)}_k\ge m_i$, and $\epsilon^{(i)}_0\neq\epsilon^{(i)}_1$. Exactly, $A^{(1)}_{k,+}/A^{(2)}_{k',+}=\frac{\kappa_1\tau_1^2\theta_1}{\kappa_2\tau_2^2\theta_2}\cdot\frac{1+\epsilon^{(1)}_k}{1+\epsilon^{(2)}_{k'}}$. So the condition $\kappa_1\tau_1^2\theta_1=\kappa_2\tau_2^2\theta_2\neq0$ then gives a fall that is alike only up to the relative deviations $\epsilon^{(i)}_k$. They are small on the range $\kappa_i\lvert\langle X\rangle\rvert\ll m_i\lvert\gamma\theta_i\rvert$ (with $\lvert\gamma\theta_i\rvert L\ll1$ for the first order); at fixed $\langle X\rangle$ and $\gamma\theta_i\neq0$ the bound $\epsilon_i$ is linear in $\kappa_i$, and it vanishes for no $\kappa_i>0$.

*Dependence on $\gamma$.* No condition of the first three lines of (8.12) contains $\gamma$: each holds for every sharp value $\gamma\neq0$ of the range (8.11) once it holds for one, RL and RD for every $\langle X\rangle$ and the others for $\langle X\rangle=0$. Only the common value $A$ is proportional to $\gamma$. For $\langle X\rangle\neq0$ the failures of the fourth line hold for every $\gamma$, and the accuracy of the approximate statement depends on $\gamma$ through $\epsilon_i\propto1/\lvert\gamma\rvert$. At $\gamma=0$, RL and RD hold trivially and FA fails ($A=0$ for $\langle X\rangle=0$, a level-dependent fall otherwise).

**(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 $\Theta_i:=v_i^2\theta_i/d_{0,i}=\sigma_X\kappa_i\tau_i^2\theta_i$: by (8.1) and (8.11), $A^{(i)}_{k,+}=-\Theta_i\gamma-d_{0,i}\tau_i^2\kappa_i\langle X\rangle/M^{(i)}_k$, and the second term does not contain the field. With $\gamma$, $\gamma'$ two sharp values of the range (8.11):

$$
\begin{aligned}
&\text{common coupling for the fall:}\ \ \Theta_1=\Theta_2=:\Theta;\qquad\text{then, for every }\langle X\rangle:\ \ A^{(i)}_{k,+}(\gamma)-A^{(i)}_{k,+}(\gamma')=-\Theta\,(\gamma-\gamma')\ \ \text{for all }k,\ i;\\
&\qquad\langle X\rangle=0:\ \ A^{(i)}_{k,+}=-\Theta\gamma\ \ \text{for all }k,\ i,\ \ \text{and FA}\iff\Theta_1=\Theta_2\neq0\ \ (\gamma\neq0);\qquad\langle X\rangle\neq0:\ \ A^{(i)}_{k,+}=-\Theta\gamma\,\bigl(1+\epsilon^{(i)}_k\bigr)\ \ (\Theta\gamma\neq0),\ \ \text{FA fails};\\
&\text{common coupling for the clocks:}\ \ \frac{\theta_1}{d_{0,1}}=\frac{\theta_2}{d_{0,2}}\qquad(\text{RD for }\gamma\neq0,\ \text{for every }\langle X\rangle).
\end{aligned}
\tag{8.14}
$$

Both are well defined by (8.13) and (8.3). They coincide iff $v_1=v_2$. The name for the fall rests on the first line, which holds for every $\langle X\rangle$: the part of the fall that the field produces is the same for all levels, and it is the same for the two types iff $\Theta_1=\Theta_2$; that the whole fall is alike needs $\langle X\rangle=0$ and $\Theta\gamma\neq0$ in addition. Equal couplings per label, $\theta_1=\theta_2\neq0$, are RL only: they give RD iff $\kappa_1=\kappa_2$ and, for $\langle X\rangle=0$, 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): RL and RD are conditions on the six numbers for every $\langle X\rangle$; $(\mathrm L_i)$ is $\langle X\rangle=0$, a condition on the start of the medium; FA, (8.5) and (8.6) are conditions on the six numbers exactly for $\langle X\rangle=0$, and none of these conditions contains $\gamma$. For $\langle X\rangle\neq0$, FA, (8.5) and (8.6) fail for all couplings, and the fall is alike only up to the relative deviations $\epsilon^{(i)}_k$, $\lvert\epsilon^{(i)}_k\rvert\le\kappa_i\lvert\langle X\rangle\rvert/(m_i\lvert\gamma\theta_i\rvert)$. (8.13). Common couplings (8.14): for the fall $\Theta_1=\Theta_2$, which makes the part of the fall produced by the field equal for all levels and both types for every $\langle X\rangle$, and the whole fall for $\langle X\rangle=0$; for the clocks $\theta_1/d_{0,1}=\theta_2/d_{0,2}$, for every $\langle X\rangle$.
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 conditions for the fall in (8.12) and (8.14) are exact only for $\langle X\rangle=0$, a condition on the start of the medium and not on the six numbers. For $\langle X\rangle\neq0$ the fall of each type depends on the level and FA fails for all couplings; (8.12) gives the size of the level-dependent part and the range on which it is small.
- (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