Version 1 · earlier version; the current one is v3 · Review of v1

# Two clocks of one body: gap responses, common response and fall

- **Subproject:** 05-ilang-free-fall
- **Package:** 06-two-clocks
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-11

## Setup and assumptions

- **Setting.** One body $b$ and a medium $c$ with the records $K_x=\kappa\,x\,X$; no field object: the position dependence sits in the body contract (A10; an assumption, M8). Local member, gaps, labels, responses, named gaps, branch starts and drift acceleration are those of `question.md`. All profiles are those of $C_{\rm eff}$, (8.19) of 04-ilang-time/08-maximal-speed. Nothing below says that the language selects such descriptions (M10).
- **Classes.** **[L]**: every member of $\mathcal C_{\rm lev}$ with linear level profiles and $N\ge2$ levels, all level offsets, both branches, all labels. **[C]**: the class $\mathcal C$ with the linear profiles of 03-clock-and-fall ($\tau_k=\tau$, $M_k=m+d_k$, $G_k=m\,g_m+d_kg_\eta$, $\mathcal F_k=\mathcal F$, $g_{\zeta,k}=g_\zeta$, $V_k=0$), every $N\ge2$.
- **Inputs, as quoted in the question.** (2.5) of 04-ilang-time/02-clock; (5.5), (5.6), (5.11) of 04-ilang-time/05-moving-clock; (8.1), (8.19), (8.20) of 04-ilang-time/08-maximal-speed; (17.15) of 03-ilang-space/17-motion; (3.2), (3.23), (3.36) of 03-clock-and-fall; (4.7), (4.10), (4.11), (4.16)–(4.19), (4.22), (4.24), (4.25) of 04-branch-fall.
- **Approximations.** Responses: first order in the gradients; the second order is not estimated. Fall: the drift acceleration of 04-branch-fall (first order in the gradients, long window, sharp label), at leading order in $\kappa$ (weak-record regime).
- **Notation.**

$$
e_{k,\sigma}(p):=V_k+\sigma\,\omega_k(p),\qquad \Omega^0_\gamma:=e_{k,\sigma}(p)-e_{k',\sigma'}(p'),\qquad u_k:=\frac{G_k}{M_k},\qquad W_k:=\sqrt{M_k^2+\tau_k^2}=\omega_k(\tfrac\pi2).
\tag{6.1}
$$

  A label $\gamma$ is *admissible* if $\Omega^0_\gamma\neq0$; it is a *rest label* if $\sin p=\sin p'=0$, and otherwise it *has a moving label*. The *plain part* of the level $k$ is $V_{x,k}\mathbb 1_2$. Statements about level gaps assume $\Omega^0_{\gamma_{\rm lev}(j)}=M_{j+1}-M_j+V_{j+1}-V_j\neq0$ for all $j\le N-2$ (true in [C]).
- **Level offsets and the quoted results of 04-branch-fall.** The level is conserved, so a start in the level $k$ stays in (level $k$)$\,\otimes\,\mathcal H_c$. On this subspace the total contract with the offset is the total contract without it plus $V_k\mathbb 1$. The state changes by the common phase $e^{-iV_k\lambda}$, which is not data (1.2): no view, no weight $p_x$ and no $\bar x_{\rm lab}$ changes, and the vectors $\beta_{k,\sigma}$ are the same. Hence (4.7)–(4.19) hold as quoted for every $V_k$.

## Derivation

### Step 1. Response of a general gap (item 1(a))

At $x_0=0$ the local member has $M_{0,k}=M_k$, $V_{0,k}=V_k$, $\zeta^0_k(0)=1$, so $\varepsilon_{k,\sigma}(p;0)=e_{k,\sigma}(p)$ and $\Omega_\gamma(0)=\Omega^0_\gamma$ exactly. At $x_0=1$: $M_{1,k}=M_k+G_k$, $V_{1,k}=V_k-\mathcal F_k$, $\zeta^0_k(1)=1+g_{\zeta,k}$. The Taylor expansion of the square root ($\omega_k\ge M_k>0$) gives

$$
\varepsilon_{k,\sigma}(p;1)-\varepsilon_{k,\sigma}(p;0)=-\mathcal F_k+\sigma\,\delta\omega_k(p)+O(\text{gradients}^2),\qquad
\delta\omega_k(p):=\frac{M_kG_k+\tau_k^2g_{\zeta,k}\sin^2p}{\omega_k(p)}=g_{\zeta,k}\,\omega_k(p)+(u_k-g_{\zeta,k})\frac{M_k^2}{\omega_k(p)} .
\tag{6.2}
$$

The difference of two such shifts, divided by $\Omega^0_\gamma$, is the response. **[L]**, every admissible label $\gamma=(k,\sigma,p;k',\sigma',p')$:

$$
\boxed{\;\rho_\gamma=\frac{\sigma\,\delta\omega_k(p)-\sigma'\,\delta\omega_{k'}(p')-(\mathcal F_k-\mathcal F_{k'})}{V_k-V_{k'}+\sigma\,\omega_k(p)-\sigma'\,\omega_{k'}(p')}\;}
\tag{6.3}
$$

It is a linear form in the gradients: the coefficient of $G_k$ is $\sigma M_k/(\omega_k(p)\,\Omega^0_\gamma)$, of $g_{\zeta,k}$ it is $\sigma\tau_k^2\sin^2p/(\omega_k(p)\,\Omega^0_\gamma)$, of $\mathcal F_k$ it is $-1/\Omega^0_\gamma$; the gradients of the second level enter with $(k,\sigma,p)\to(k',\sigma',p')$ and the opposite sign. Inside one level ($k=k'$) the terms $\mathcal F_k$ and $V_k$ cancel. With $\sigma/\omega-\sigma'/\omega'=-\sigma\sigma'(\sigma\omega-\sigma'\omega')/(\omega\omega')$ and the second form of (6.2):

$$
\rho_\gamma=g_{\zeta,k}-\sigma\sigma'\,(u_k-g_{\zeta,k})\,\frac{M_k^2}{\omega_k(p)\,\omega_k(p')}\qquad(k=k').
\tag{6.4}
$$

So the gaps inside one level contain only $G_k$ and $g_{\zeta,k}$. The gradient $\mathcal F_k$ of the plain part enters only the gaps between two levels, and only as $\mathcal F_k-\mathcal F_{k'}$, together with the difference $V_k-V_{k'}$ of the level offsets in the denominator.

### Step 2. The named gaps (item 1(b))

**[L]**. From (6.3) with $\omega_k(0)=M_k$, $\delta\omega_k(0)=G_k$; from (6.4) with $\sigma\sigma'=-1$, $p=p'=0$; and from (6.4) with $\sigma\sigma'=+1$, $\omega_k(\tfrac\pi2)=W_k$, $\omega_k(0)=M_k$:

$$
\rho_{\rm lev}(k)=\frac{(G_{k+1}-\mathcal F_{k+1})-(G_k-\mathcal F_k)}{(M_{k+1}+V_{k+1})-(M_k+V_k)},\qquad
\rho_{\rm br}(k)=\frac{G_k}{M_k}=u_k,\qquad
\rho_{\rm band}(k)=g_{\zeta,k}-\frac{M_k}{W_k}\,(u_k-g_{\zeta,k}) .
\tag{6.5}
$$

**[C]**. Here $\varepsilon_{k,+}(0;x_0)=E^+_{x_0,k}$, so by (3.23), exactly in the gradients and for every $k\le N-2$,

$$
\Omega_{\gamma_{\rm lev}(k)}(x_0)=E^+_{x_0,k+1}-E^+_{x_0,k}=\frac{2\pi}{N}\,r(x_0),\qquad \rho_{\rm lev}(k)=g_\eta=\frac{r(1)-r(0)}{r(0)} .
\tag{6.6}
$$

The level gap is $2\pi/N$ times the rate in its wide-start form. The rate $r_W$ at finite $W$ carries the further factor $1-\varsigma_W$, which depends on the place only through $\tau^0/\bar\mu$.

### Step 3. Starts and readings (item 1(c))

The body evolves by $C_{\rm loc}(x_0)$ alone ($\kappa=0$), so (2.5) of 04-ilang-time/02-clock applies; $\beta_{k,\sigma}$, $\theta_k$ are those of the local member. $g_0$ is a label weight concentrated at $0$ with the spread $\delta p$, and the limit $\delta p\to0$ is taken at fixed $\lambda$. Each eigenvector evolves by its phase, and (1.3) gives, with $\langle a\vert f_k\rangle=N^{-1/2}e^{2\pi ika/N}$, $\langle\beta_{k,+}(0)\vert\beta_{k+1,+}(0)\rangle=1$ and $\langle\beta_{k,+}(0)\vert\beta_{k,+}(\tfrac\pi2)\rangle=\cos\tfrac{\theta_k(\pi/2)}2\neq0$:

$$
\begin{array}{l|l|l|l}
\text{gap}&\text{start: label weight, internal factor at }p&\text{reading}&\text{statistics for }\delta p\to0\\\hline
\gamma_{\rm lev}(k)&g_0(p),\ \ \tfrac1{\sqrt2}\bigl[\beta_{k,+}(p)\otimes\lvert f_k\rangle+\beta_{k+1,+}(p)\otimes\lvert f_{k+1}\rangle\bigr]&\text{clock label }a&\tfrac1N\bigl[1+\cos\bigl(\Omega_{\gamma_{\rm lev}(k)}(x_0)\,\lambda-\tfrac{2\pi a}N\bigr)\bigr]\\
\gamma_{\rm band}(k)&\tfrac1{\sqrt2}\bigl[g_0(p)+g_0(p-\tfrac\pi2)\bigr],\ \ \beta_{k,+}(p)\otimes\lvert f_k\rangle&r=x\bmod4&\tfrac14\bigl[1+\cos\tfrac{\theta_k(\pi/2)}2\,\cos\bigl(\Omega_{\gamma_{\rm band}(k)}(x_0)\,\lambda-\tfrac{\pi r}2\bigr)\bigr]
\end{array}
\tag{6.7}
$$

Both limits are periodic in $\lambda$, with the period $2\pi/\lvert\Omega_\gamma(x_0)\rvert$. At a finite spread neither is: for the first row the exact statistics is $\tfrac1N\bigl[1+\bigl\langle\cos\tfrac{\theta_{k+1}-\theta_k}2\cos\bigl((\varepsilon_{k+1,+}(p;x_0)-\varepsilon_{k,+}(p;x_0))\lambda-\tfrac{2\pi a}N\bigr)\bigr\rangle_{g_0}\bigr]$, a continuous superposition of gaps.

**The branch gap has no such example.** Since $\theta_k(0)=\theta_k(\pi)=0$, the eigenvectors with the labels $(k,+,0)$ and $(k,-,0)$ are $\lvert p\rangle\otimes\lvert s{=}1\rangle\otimes\lvert f_k\rangle$ and $\lvert p\rangle\otimes\lvert s{=}2\rangle\otimes\lvert f_k\rangle$ with $p\in\{0,\pi\}$. Every reading projector $\Pi_r$ is diagonal in the places (A4), hence in the branch label $s$, so the coefficient of this term of (2.5) vanishes:

$$
\bigl(\langle p\rvert\otimes\langle s{=}2\rvert\otimes\langle f_k\rvert\bigr)\,\Pi_r\,\bigl(\lvert p'\rangle\otimes\lvert s{=}1\rangle\otimes\lvert f_k\rangle\bigr)=0\quad\text{for every reading and }p,p'\in\{0,\pi\}:\qquad\gamma_{\rm br}(k)\ \text{is in no statistics of the body}.
\tag{6.8}
$$

The nearest term: start with the weight $g_0(p-p_0)$, $\sin p_0\neq0$, and the internal factor $\lvert s{=}1\rangle\otimes\lvert f_k\rangle=\cos\tfrac{\theta_k}2\beta_{k,+}-\sin\tfrac{\theta_k}2\beta_{k,-}$; reading of the branch label. Then $p(s{=}1;\lambda)\to1-\tfrac12\sin^2\theta_k(p_0)\,\bigl[1-\cos\bigl(2\omega_k(p_0;x_0)\lambda\bigr)\bigr]$, periodic, with the gap $(k,+,p_0;k,-,p_0)$. By (6.4) its response is $g_{\zeta,k}+(u_k-g_{\zeta,k})M_k^2/\omega_k(p_0)^2$. For $p_0\to0$ the response tends to $\rho_{\rm br}(k)$ and the coefficient $\tfrac12\sin^2\theta_k(p_0)$ tends to zero: the branch gap is the lower edge of the band $[2M_k,2W_k]$ of these frequencies.

### Step 4. Well-definedness (item 1(d))

With one body, (8.1) of 04-ilang-time/08-maximal-speed changes $C_{\rm eff}$ by $\langle Z\rangle\mathbb 1$ at most, (8.20) there. So $V_{x,k}\mapsto V_{x,k}+z$ for all $x$, $k$ with one $z$ (only the offsets change, $V_k\mapsto V_k+z$), and $C_{\rm loc}(x_0)\mapsto C_{\rm loc}(x_0)+z\mathbb 1$. A change of the unit, $\lambda'=\lambda/s$ with $s>0$, replaces $C$ by $sC$ in (1.9). **[L]**:

$$
\text{(8.1)}:\ \ \Omega_\gamma\mapsto\Omega_\gamma,\ \ \rho_\gamma\mapsto\rho_\gamma;\qquad\qquad \lambda\mapsto\lambda/s:\ \ \Omega_\gamma\mapsto s\,\Omega_\gamma,\ \ \rho_\gamma\mapsto\rho_\gamma .
\tag{6.9}
$$

Gaps are well defined under (8.1), and under a change of the unit only up to one common positive factor (their ratios are well defined). Responses are well defined under both: (6.3) is homogeneous of degree zero in $C$.

### Step 5. Common response inside one level (item 2(a))

**[L]**, one level $k$:

$$
\boxed{\;\text{all admissible gaps inside the level }k\text{ have a common response}\iff G_k=M_k\,g_{\zeta,k};\qquad\text{its value is }g_{\zeta,k}=u_k\;}
\tag{6.10}
$$

- ($\Leftarrow$) If $u_k=g_{\zeta,k}$, (6.4) gives $\rho_\gamma=g_{\zeta,k}$ for both branches and all $p$, $p'$.
- ($\Rightarrow$) The branch gap and the band gap are admissible ($2M_k>0$, $W_k-M_k>0$ since $\tau_k>0$). By (6.5), $\rho_{\rm br}(k)-\rho_{\rm band}(k)=(u_k-g_{\zeta,k})(1+M_k/W_k)$, which vanishes only if $u_k=g_{\zeta,k}$.

The condition does not contain $\tau_k$ (beyond $\tau_k>0$). Control: the variant (a), where $u_k=g\neq0=g_\zeta$.

### Step 6. Common response of all gaps (item 2(b))

**[L]**, with one real number $c$:

$$
\boxed{\;\text{all admissible gaps have the common response }g_*\iff G_k=g_*M_k,\quad g_{\zeta,k}=g_*,\quad \mathcal F_k+g_*V_k=c\qquad\forall k\;}
\tag{6.11}
$$

- ($\Leftarrow$) Then $\delta\omega_k=g_*\omega_k$ by (6.2), and the numerator of (6.3) is $g_*(\sigma\omega_k-\sigma'\omega_{k'})+g_*(V_k-V_{k'})=g_*\Omega^0_\gamma$.
- ($\Rightarrow$) The gaps inside each level have the common response $g_*$, so by (6.10) $u_k=g_{\zeta,k}=g_*$. For $k\neq k'$ the label $(k,+,p;k',-,p')$ has $\Omega^0_\gamma=V_k-V_{k'}+\omega_k(p)+\omega_{k'}(p')$, which is not constant in $p$ ($\tau_k>0$), so it is admissible for some $p$. For it, (6.3) equals $g_*$ only if $-(\mathcal F_k-\mathcal F_{k'})=g_*(V_k-V_{k'})$.

The general form of the profiles follows by inserting (6.11) into the linear profiles; the anticommutator reproduces $\zeta_{x,k}=1+g_*(x+\tfrac12)$ on the bond $(x,x+1)$. With $C_b^0$ the member with all gradients zero (level offsets included):

$$
M_{x,k}=(1+g_*x)\,M_k,\qquad \zeta_{x,k}=1+g_*(x+\tfrac12),\qquad V_{x,k}=(1+g_*x)\,V_k-c\,x;\qquad C_b=\tfrac12\bigl\{\mathbb 1+g_*\hat x,\ C_b^0\bigr\}-c\,\hat x .
\tag{6.12}
$$

One position factor multiplies the whole body contract, level offsets included, up to a term $-c\,\hat x$ common to all levels. For $g_*\neq0$ put $V_\circ:=c/g_*$; then $\mathcal F_k=-g_*(V_k-V_\circ)$. Under $V_k\mapsto V_k+z$ of Step 4, $c\mapsto c+g_*z$ and $V_\circ\mapsto V_\circ+z$: the condition and $V_k-V_\circ$ are well defined. Controls: the variant (a) (fails $g_{\zeta,k}=g_*$); a member with the first two conditions and $\mathcal F_0+g_*V_0\neq\mathcal F_1+g_*V_1$ (fails for the gaps between the levels $0$ and $1$).

### Step 7. The class $\mathcal C$ (item 2(c))

**[C]**. Insert $G_k=m\,g_m+d_kg_\eta$, $M_k=m+d_k$ into (6.10), (6.11) and use $d_0=0$, $d_1\neq0$, $m>0$; the third condition of (6.11) holds identically, since $\mathcal F_k=\mathcal F$ and $V_k=0$:

$$
\begin{aligned}
\text{level }k\text{ alone has a common response}&\iff m\,(g_m-g_\zeta)+d_k\,(g_\eta-g_\zeta)=0;\\
\text{every level has one}\iff\text{all admissible gaps have a common response}&\iff g_m=g_\eta=g_\zeta\ (=g_*),\quad\mathcal F\ \text{arbitrary}.
\end{aligned}
\tag{6.13}
$$

By (6.5), $\rho_{\rm lev}=g_\eta$, $\rho_{\rm br}(k)=(m\,g_m+d_kg_\eta)/M_k$, $\rho_{\rm band}(k)=g_\zeta-\tfrac{M_k}{W_k}(\rho_{\rm br}(k)-g_\zeta)$, with $W_k=\sqrt{M_k^2+\tau^2}$. For $g\neq0$ (in (d): $\mathcal F\neq0$):

$$
\begin{array}{l|c|c|c|l}
\text{variant}&\rho_{\rm lev}&\rho_{\rm br}(k)&\rho_{\rm band}(k)&\text{equal responses}\\\hline
\text{(a)}&g&g&-\frac{M_k}{W_k}g&\text{level}=\text{branch}\neq\text{band}\\
\text{(b1)}&0&\frac{m\,g}{M_k}&-\frac{m\,g}{W_k}&\text{none}\\
\text{(b2)}&g&\frac{d_kg}{M_k}&-\frac{d_kg}{W_k}&\text{none for }k\ge1;\ \ k=0:\ \text{branch}=\text{band}=0\neq\text{level}\\
\text{(c)}&g&g&g&\text{all three}\\
\text{(d)}&0&0&0&\text{all three (zero)}\\
\text{(f)}&g&g&-\frac{M_k}{W_k}g&\text{level}=\text{branch}\neq\text{band}
\end{array}
\tag{6.14}
$$

### Step 8. The exact form (item 2(d))

**[L]** (only the values of the profiles at $x_0$ and $x_1$ enter; no expansion):

$$
\exists\,\nu:\ \Omega_\gamma(x_1)=\nu\,\Omega_\gamma(x_0)\ \ \forall\gamma\iff\exists\,\nu>0,\ c'\in\mathbb R:\ \ M_{x_1,k}=\nu M_{x_0,k},\ \ \zeta^0_k(x_1)=\nu\,\zeta^0_k(x_0),\ \ V_{x_1,k}=\nu V_{x_0,k}+c'\ \ \forall k\iff C_{\rm loc}(x_1)=\nu\,C_{\rm loc}(x_0)+c'\,\mathbb 1 .
\tag{6.15}
$$

- ($\Leftarrow$) Then $\omega_k(p;x_1)=\nu\,\omega_k(p;x_0)$ and $\varepsilon_{k,\sigma}(p;x_1)=\nu\,\varepsilon_{k,\sigma}(p;x_0)+c'$; every gap is multiplied by $\nu$.
- ($\Rightarrow$) The labels $(k,+,p;k,-,p)$ give $\omega_k(p;x_1)=\nu\,\omega_k(p;x_0)$ for all $p$, so $\nu>0$; at $p=0$ and $p=\tfrac\pi2$ this gives the first two conditions (all quantities are positive). A label between the levels $k\neq k'$ then gives $V_{x_1,k}-V_{x_1,k'}=\nu(V_{x_0,k}-V_{x_0,k'})$.
- The last equivalence holds because the on-site block of the level $k$ of $C_{\rm loc}(x_0)$ is $M_{x_0,k}\sigma_3+V_{x_0,k}\mathbb 1_2$ and its hopping factor is $\tau_k\zeta^0_k(x_0)$.

The linear profiles (6.12) satisfy (6.15) exactly for every pair, with $\nu=(1+g_*x_1)/(1+g_*x_0)$.

### Step 9. Common response and fall (item 3(a))

**[L]** with (6.11) and $g_*\neq0$. Inserting $G_k=g_*M_k$ and $\mathcal F_k=-g_*(V_k-V_\circ)$ into (4.16):

$$
\boxed{\;a_{k,\sigma}(0)=-\tau_k^2\Bigl(g_*-\sigma\frac{\mathcal F_k}{M_k}\Bigr)=-g_*\,\tau_k^2\,\frac{M_k+\sigma\,(V_k-V_\circ)}{M_k}\;}
\tag{6.16}
$$

$$
\begin{aligned}
a_{k,+}(0)=a_+\ \forall k&\iff\tau_k^2\,\frac{M_k+V_k-V_\circ}{M_k}=-\frac{a_+}{g_*}\ \ \forall k,\qquad
a_{k,-}(0)=a_-\ \forall k\iff\tau_k^2\,\frac{M_k-V_k+V_\circ}{M_k}=-\frac{a_-}{g_*}\ \ \forall k,\\
\text{both, with }a_+=a_-=a&\iff V_k=V_\circ\ \text{ and }\ \tau_k=\tau\quad\forall k;\qquad\text{then }a=-g_*\tau^2 .
\end{aligned}
\tag{6.17}
$$

- The first two lines are (6.16) read in both directions (they are the first line of (4.19) with (6.11) inserted).
- Third line: by (4.19), both with one value $\iff\mathcal F_k=0$ and $\tau_k^2u_k=-a$ for all $k$. With $u_k=g_*\neq0$: $\mathcal F_k=0\iff V_k=V_\circ$, and $\tau_k^2=-a/g_*$ for all $k$ $\iff$ the positive numbers $\tau_k$ are equal. Conversely (6.16) gives $-g_*\tau^2$.
- The condition $V_k=V_\circ$ concerns the plain parts: every plain part is $V_\circ\mathbb 1_2$ at every position, a multiple of $\mathbb 1$. The condition $\tau_k=\tau$ concerns the numbers $\tau_k$. For one branch alone the two do not separate: one equation per level ties $V_k-V_\circ$ to $\tau_k$.
- Controls. [C] with $g_m=g_\eta=g_\zeta=g\neq0$ and $\mathcal F\neq0$ (the variant (c) with a constant under the factor, $V_\circ=\mathcal F/g$): common response $g$ by (6.13), but $a_{k,\sigma}(0)=-\tau^2g+\sigma\tau^2\mathcal F/M_k$ depends on the level. A member with $V_k=V_\circ$ and $\tau_0\neq\tau_1$: $a_{k,\sigma}(0)=-g_*\tau_k^2$. The variant (d) (common response $0$): $a_{k,\sigma}(0)=\sigma\tau^2\mathcal F/M_k\neq0$, a fall without any response.

### Step 10. The converse (item 3(b))

**[L]**. By (4.19) and (6.5), for $a\neq0$:

$$
a_{k,+}(0)=a_{k,-}(0)=a\ \ \forall k\iff\mathcal F_k=0\ \text{ and }\ \rho_{\rm br}(k)=-\frac{a}{\tau_k^2}\quad\forall k .
\tag{6.18}
$$

Under this hypothesis, (6.3) with $\delta\omega_k(0)=\delta\omega_k(\pi)=G_k=-aM_k/\tau_k^2$ gives for the rest labels

$$
\rho_\gamma=-a\,\frac{\sigma M_k/\tau_k^2-\sigma'M_{k'}/\tau_{k'}^2}{V_k-V_{k'}+\sigma M_k-\sigma'M_{k'}};\qquad\text{all admissible rest gaps have a common response}\iff\tau_k=\tau\ \text{and}\ V_k\ \text{level-independent};\quad\text{value }-\frac a{\tau^2}.
\tag{6.19}
$$

- ($\Leftarrow$) is read off. ($\Rightarrow$) The branch gaps give $-a/\tau_k^2$, so $\tau_k=\tau$. For $k\neq k'$ one of the labels $(k,+,0;k',-,0)$, $(k',+,0;k,-,0)$ is admissible ($\Omega^0=\pm(V_k-V_{k'})+M_k+M_{k'}$), and its response is $-a/\tau^2$ only if $V_k=V_{k'}$.
- What follows from the hypothesis alone: the branch gap of each level responds with $-a/\tau_k^2$. In the subclass $\tau_k=\tau$ with equal level offsets, which contains [C], all rest gaps (branch gaps and level gaps) have the common response $-a/\tau^2$.
- What need not: every gap with a moving label, since by (6.4) it needs $g_{\zeta,k}=u_k$ and $g_{\zeta,k}$ does not enter (4.16); the branch gaps of different levels when the $\tau_k$ differ; the gaps between levels when the $V_k$ differ.
- Control: the variant (a). $a_{k,\pm}(0)=-\tau^2g$ for all $k$ by (4.25), and $\rho_{\rm band}(k)=-\tfrac{M_k}{W_k}g\neq g=\rho_{\rm br}(k)$ by (6.14). Further controls: $\mathcal F_k=0$, $G_k=-aM_k/\tau_k^2$ with $\tau_0\neq\tau_1$, or with $V_0\neq V_1$.

### Step 11. The order $v^2$ (item 3(c))

**[L]** with (6.11), $g_*\neq0$. (4.17) gives $c_{2,k,\sigma}=2g_*\bigl(1+\tfrac{M_k^2}{\tau_k^2}\bigr)-\sigma\tfrac{\mathcal F_k}{M_k}\bigl(\tfrac32+\tfrac{2M_k^2}{\tau_k^2}\bigr)$. With the further conditions of the third line of (6.17):

$$
c_{2,k,\sigma}=2g_*\Bigl(1+\frac{M_k^2}{\tau^2}\Bigr),\qquad a_{k,\sigma}=-g_*\tau^2+2g_*\Bigl(1+\frac{M_k^2}{\tau^2}\Bigr)v^2+O(v^4)\qquad\text{(both branches)} .
\tag{6.20}
$$

Through the order $v^2$ the fall is the same for all levels exactly $\iff$ the numbers $M_k$ are equal; so exactly no as soon as two levels differ in $M_k$ (always in [C]). To leading order in $M_k/\tau$ yes: $c_{2,k,\sigma}=2g_*\,[1+O(M_k^2/\tau^2)]$.

### Step 12. Answer to criterion 4 (item 3(d))

**[L]**. From (6.11), (6.17), (6.18): for $g_*\neq0$, $a\neq0$ and $a=-g_*\tau^2$,

$$
\Bigl[\text{(6.11)}\Bigr]\wedge\Bigl[V_k\ \text{level-independent},\ \tau_k=\tau\Bigr]\iff\Bigl[a_{k,\pm}(0)=a\ \forall k\Bigr]\wedge\Bigl[\tau_k=\tau,\ \ g_{\zeta,k}=-\tfrac a{\tau^2}\ \forall k,\ \ V_k\ \text{level-independent}\Bigr].
\tag{6.21}
$$

Both sides are the members with $\tau_k=\tau$, $G_k=g_*M_k$, $g_{\zeta,k}=g_*$, $\mathcal F_k=0$ and equal level offsets (left: (6.17); right: (6.18) gives $\mathcal F_k=0$, $u_k=-a/\tau^2$, and then (6.11)).
- A common response of all periodic processes does not imply a fall that is the same for all levels. It does exactly when, in addition, the plain part is a multiple of $\mathbb 1$ and the $\tau_k$ are equal (controls of Step 9).
- A fall that is the same for all levels and both branches does not imply a common response. It does exactly when, in addition, the $\tau_k$ are equal, $g_{\zeta,k}=-a/\tau^2$ in every level and the level offsets are equal (controls of Step 10). Each further condition is necessary, by (6.11).

### Step 13. The relation behind (4.24) (item 4(a))

**[L]**. In (4.18) write $\delta E^\sigma_k=\tfrac\sigma2\delta\Delta_k+\tfrac12(\delta E^+_k+\delta E^-_k)$, and use $\delta\Delta_k/\Delta_k=G_k/M_k=\rho_{\rm br}(k)$, $\delta E^+_k+\delta E^-_k=-2\mathcal F_k$, $2\bar t_k=\tau_k$:

$$
\boxed{\;a_{k,\sigma}(0)=-(2\bar t_k)^2\Bigl[\frac{\delta\Delta_k}{\Delta_k}+\sigma\,\frac{\delta E^+_k+\delta E^-_k}{\Delta_k}\Bigr]=-\tau_k^2\Bigl[\rho_{\rm br}(k)-\sigma\,\frac{\mathcal F_k}{M_k}\Bigr],\qquad\tfrac12\bigl[a_{k,+}(0)+a_{k,-}(0)\bigr]=-\tau_k^2\,\rho_{\rm br}(k)\;}
\tag{6.22}
$$

Both equations are identities for every member of [L]: no condition on the gradients, the offsets or the $\tau_k$. The mean of the falls of the two branches is $-\tau_k^2$ times the response of the branch gap of the same level. The term that is odd in the branch vanishes if and only if $\delta E^+_k+\delta E^-_k=0$, that is $\mathcal F_k=0$: no gradient of the plain part.

### Step 14. The relation with a given gap (item 4(b))

**[L]**. For a branch $\sigma$ and a named gap, the relation (4.24) with the response of that gap in the place of $\bigl(r(x+1)-r(x)\bigr)/r(x)$ is

$$
R^\sigma_\gamma:\qquad a_{k,\sigma}(0)=-\tau_k^2\,\rho_\gamma\ \ \text{for all levels }k,\qquad\text{in space:}\ \ \frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\Big\vert_{\rm drift}=-\frac{v_k^2}{d_0}\,\rho_\gamma\,\hat e,\quad v_k:=\tau_kd_0 ,
\tag{6.23}
$$

with $\gamma=\gamma_{\rm br}(k)$ or $\gamma_{\rm band}(k)$ of the same level, or with every level gap $\gamma_{\rm lev}(j)$, $j\le N-2$. The necessary and sufficient conditions:

$$
\begin{array}{l|l|l}
\text{gap}&\text{one branch }\sigma&\text{both branches}\\\hline
\gamma_{\rm br}(k)&\mathcal F_k=0\ \ \forall k&\mathcal F_k=0\ \ \forall k\\
\gamma_{\rm band}(k)&\sigma\,\mathcal F_k=(G_k-M_kg_{\zeta,k})\bigl(1+\tfrac{M_k}{W_k}\bigr)\ \ \forall k&\mathcal F_k=0,\ \ G_k=M_kg_{\zeta,k}\ \ \forall k\\
\gamma_{\rm lev}(j)&\exists\,g_\ell:\ G_k-\sigma\mathcal F_k=g_\ell M_k\ \forall k,\ \ (1-\sigma)\mathcal F_k+g_\ell V_k\ \text{level-independent}&\exists\,g_\ell:\ \mathcal F_k=0,\ G_k=g_\ell M_k\ \forall k,\ \ g_\ell V_k\ \text{level-independent}
\end{array}
\tag{6.24}
$$

- Branch gap: by (6.22), $a_{k,\sigma}(0)+\tau_k^2\rho_{\rm br}(k)=\sigma\tau_k^2\mathcal F_k/M_k$, zero if and only if $\mathcal F_k=0$.
- Band gap: by (6.22), $R^\sigma_{\rm band}\iff\rho_{\rm br}(k)-\rho_{\rm band}(k)=\sigma\mathcal F_k/M_k$, and $\rho_{\rm br}(k)-\rho_{\rm band}(k)=(u_k-g_{\zeta,k})(1+M_k/W_k)$ by (6.5). For both signs: $\mathcal F_k=0$, and then $u_k=g_{\zeta,k}$ since $1+M_k/W_k>0$.
- Level gap, ($\Rightarrow$): put $g_\ell:=\rho_{\rm lev}(0)$; then $a_{k,\sigma}(0)=-\tau_k^2g_\ell$ is $G_k-\sigma\mathcal F_k=g_\ell M_k$ by (4.16). Inserting $G_k=g_\ell M_k+\sigma\mathcal F_k$ into (6.5), $\rho_{\rm lev}(j)=g_\ell$ becomes $(\sigma-1)(\mathcal F_{j+1}-\mathcal F_j)=g_\ell(V_{j+1}-V_j)$ for all $j$. ($\Leftarrow$): the same two steps backwards. Both branches: add and subtract the conditions of $\sigma=\pm$.
- In [C] the row $\gamma_{\rm lev}$, $\sigma=+$ reads $g_\ell=g_\eta$, $\mathcal F=m(g_m-g_\eta)$: the hypothesis of (4.24), by (4.22).

With the response of every admissible gap, for one branch or both:

$$
\boxed{\;R^\sigma_\gamma\ \text{for every admissible }\gamma\iff G_k=g_*M_k,\ \ g_{\zeta,k}=g_*,\ \ \mathcal F_k=0\ \ \forall k,\ \ g_*V_k\ \text{level-independent}\;}\qquad(g_*\neq0:)\quad C_b-V_\circ\mathbb 1=\tfrac12\bigl\{\mathbb 1+g_*\hat x,\ C_b^0-V_\circ\mathbb 1\bigr\} .
\tag{6.25}
$$

- ($\Rightarrow$) The left side of $R^\sigma_\gamma$ does not depend on $\gamma$, so all admissible gaps have one response $g_*$: (6.11). Then (6.22) with $\rho_{\rm br}(k)=g_*$ gives $\mathcal F_k=0$, and the third condition of (6.11) becomes $g_*V_k=c$. ($\Leftarrow$) (6.11) holds with $c=g_*V_k$, and (6.16) gives $a_{k,\sigma}(0)=-\tau_k^2g_*$.
- For $g_*\neq0$ the level offsets are equal, $V_k=V_\circ$, and the level blocks of $C_b^0-V_\circ\mathbb 1$ have the spectrum $\pm\omega_k(p)$: the factor multiplies the body contract measured from the common symmetry point of the spectra of its level blocks. The numbers $\tau_k$ may differ.
- So (4.24) rests on two facts for both branches: the identity (6.22), which ties the fall to the branch gap of the same level when $\mathcal F_k=0$; and the equality of the response of the level gap with that of the branch gap, which is a condition on the contract (the row $\gamma_{\rm lev}$).

### Step 15. Speeds and orders (items 4(c), 4(d))

**Speeds.** [L], at zero gradients (the order needed in a first-order relation). The hopping part of $C_b$ has the matrix elements $\mp\tfrac i2(\sigma_1)_{s's}T_{a'a}$ between $(x,s,a)$ and $(x\pm1,s',a')$, with $T:=\sum_k\tau_k\lvert f_k\rangle\langle f_k\rvert$ and $T_{a'a}=\tfrac1N\sum_k\tau_ke^{2\pi ik(a'-a)/N}$. Places with equal $x$ have the distance $0$, neighbours $d_0=\kappa\sigma_X$ by (17.15) of 03-ilang-space/17-motion. The formula (5.11) of 04-ilang-time/05-moving-clock gives

$$
v_{\max}=d_0\,\tau_{*},\qquad\tau_{*}:=\sum_{n=0}^{N-1}\Bigl\lvert\frac1N\sum_{k=0}^{N-1}\tau_k\,e^{2\pi ikn/N}\Bigr\rvert\ \ge\ \max_k\tau_k,\qquad v_k=\tau_kd_0\le v_{\max} .
\tag{6.26}
$$

The inequality follows from $\tau_k=\sum_nT_{n0}\,e^{-2\pi ikn/N}$. The relation of the level $k$ contains the level speed $v_k$ (the value of the same formula for the level block $C^{(k)}_0$ on the places $(x,s)$), not the bound $v_{\max}$ of the body. All levels contain $v_{\max}$ if and only if $\tau_k=\tau$ for all $k$; then $\tau_{*}=\tau$.

**Orders.** For each relation of (6.24), (6.25):
- Gradients: first order. $a_{k,\sigma}(0)$ and $\rho_\gamma$ are first-order coefficients and the relation between them is exact; the second order is not determined here.
- Drift velocity: zeroth order in $v$. By (4.17) the correction is $c_{2,k,\sigma}v^2+O(v^4)$; under (6.25)

$$
a_{k,\sigma}=-\tau_k^2\,g_*\Bigl[1-\frac{2v^2}{\tau_k^2}\Bigl(1+\frac{M_k^2}{\tau_k^2}\Bigr)\Bigr]+O(v^4),
\tag{6.27}
$$

  with a level-dependent part of relative order $v^2M_k^2/\tau_k^4$. The window and the spread add the errors of (4.10), (4.11).
- $M_k/\tau_k$: no expansion; at rest the relations are exact in the level data. Only the level-independence of the $v^2$ term holds to leading order in $M_k/\tau_k$.
- $\kappa$: leading order (weak-record regime). In the space form both sides are of the order $\kappa^1$ through $d_0$; the next order is not determined here.

### Step 16. Control: where the factor sits (item 5)

**[C]**, $g\neq0$, in (f) $c_{\rm f}\neq0$; $R^\pm$ means the relation (6.23) for both branches and all levels. From (6.14), (4.25), (6.24):

$$
\begin{array}{l|c|c|c|c|c|c|c|c}
\text{variant}&\rho_{\rm lev}&\rho_{\rm br}(k)&\rho_{\rm band}(k)&a_{k,+}(0)&a_{k,-}(0)&R^\pm_{\rm lev}&R^\pm_{\rm br}&R^\pm_{\rm band}\\\hline
\text{(a)}&g&g&-\frac{M_k}{W_k}g&-\tau^2g&-\tau^2g&\text{yes}&\text{yes}&\text{no}\\
\text{(c)}&g&g&g&-\tau^2g&-\tau^2g&\text{yes}&\text{yes}&\text{yes}\\
\text{(f)}&g&g&-\frac{M_k}{W_k}g&-\tau^2g\bigl(1+\frac{c_{\rm f}}{M_k}\bigr)&-\tau^2g\bigl(1-\frac{c_{\rm f}}{M_k}\bigr)&\text{no}&\text{no}&\text{no}
\end{array}
\tag{6.28}
$$

In (f) each relation fails for a single branch as well: $\mathcal F=-c_{\rm f}g\neq0$ violates the rows of (6.24) for all levels ($M_0\neq M_1$). The variants (a) and (c) agree in the level gap, the branch gap and the fall from rest of both branches. They differ in every gap with a moving label. In one body the ratio of the band gap to the level gap changes from place to place at the relative rate $\rho_{\rm band}-\rho_{\rm lev}=-g\,(1+M_k/W_k)$ in (a) and $0$ in (c); the two readings of (6.7) show it. They also differ in the $v^2$ coefficient of the fall, $g\,\ell_k$ against $g\,(1+\ell_k)$ in (4.25).

## Result

All results are for the class [L], except (6.6), (6.13), (6.14), (6.28), which are for [C].
- **Item 1.** Response of a general gap (6.3), inside one level (6.4); named gaps (6.5); in [C] the level gap is $2\pi/N$ times the rate (6.6). Readings that contain the level gap and the band gap (6.7); none contains the branch gap (6.8). Gaps are defined up to the unit of $\lambda$, responses are well defined (6.9).
- **Item 2.** One level: $G_k=M_kg_{\zeta,k}$ (6.10). All gaps: (6.11), with the profiles (6.12). Class $\mathcal C$: $g_m=g_\eta=g_\zeta$, $\mathcal F$ arbitrary (6.13), variants (6.14). Exact form: $C_{\rm loc}(x_1)=\nu\,C_{\rm loc}(x_0)+c'\mathbb 1$ (6.15).
- **Item 3.** Fall under a common response (6.16), three equivalences (6.17). Converse (6.18), (6.19), with the variant (a) as control. Order $v^2$ (6.20). Criterion 4: neither property implies the other; the exact further conditions are in (6.21).
- **Item 4.** Identity (6.22). The relation (6.23) holds with a named gap under (6.24), and with every gap under (6.25). Bound and level speed (6.26); orders, with (6.27).
- **Item 5.** Table (6.28).

## Consistency checks

1. **Units and zero gradients.** Under $C\mapsto sC$: $\rho_\gamma$ has degree $0$, $a_{k,\sigma}(0)=\tfrac{\tau_k^2}{M_k}(\sigma\mathcal F_k-G_k)$ and $\tau_k^2\rho_\gamma$ have degree $2$ (position labels per $\lambda^2$), so (6.22), (6.23) are homogeneous. With all gradients zero, (6.3) and (4.16) give $\rho_\gamma=0$ and $a_{k,\sigma}(0)=0$. Passed.
2. **Variant (c) against an exact rescaling.** There $C_{\rm loc}(x_0)=(1+gx_0)\,C_{\rm loc}(0)$ exactly, so every gap at $x_0=1$ is $(1+g)$ times the gap at $0$ and $\rho_\gamma=g$ for every label. (6.3) with $\delta\omega_k=g\,\omega_k$, $\mathcal F=0$, $V_k=0$ gives the same, and (6.15) holds with $\nu=1+gx_0$. Passed.
3. **$\tau_k\to0$.** The band gap tends to $\tau_k^2\zeta^0_k(x_0)^2/(2M_{x_0,k})$, whose relative first-order change between $x_0=0$ and $1$ is $2g_{\zeta,k}-u_k$; (6.5) with $W_k\to M_k$ gives $g_{\zeta,k}-(u_k-g_{\zeta,k})$, the same. A body that cannot hop does not fall, $a_{k,\sigma}(0)\to0$ in (6.22), while $\rho_{\rm br}(k)$ stays finite, in agreement with $v_k\to0$ in (6.23). Passed.

## Open issues

- Item 1(c) asked for a start and a reading for each named gap. For the branch gap none exists, (6.8): the term with the label $\gamma_{\rm br}(k)$ has the coefficient zero in every reading of a body that evolves by the local member. The gap is reached only as the limit $p_0\to0$ of the frequencies $2\omega_k(p_0)$, with a vanishing coefficient. The identity (6.22) therefore ties the fall to a gap that is not a process in the sense of the definition of this package. Whether a reading in a larger description (a record object, A8) shows it is not examined.
- The statistics (6.7) are limits of a sharp label at fixed $\lambda$; the rate of convergence is not estimated.
- Readings of the question fixed here: "the relation holds for all levels with the level gap" is taken for every pair of a level $k$ and a level gap $\gamma_{\rm lev}(j)$; the relation of the level $k$ is written with $v_k=\tau_kd_0$; $v_{\max}$ in (6.26) is evaluated at zero gradients.
- Statements about level gaps assume that the level gaps are not zero at zero gradients.
- Not treated (out of scope): the second order in the gradients, starts in a superposition of levels, records beyond the leading order in $\kappa$, a field object.
- M10: the results state what members of $\mathcal C_{\rm lev}$ give; they do not show that the language selects the members of (6.25).

## Methods used

- First-order perturbation of the eigenvalues of a $2\times2$ block family
- Spectral decomposition of reading statistics (gaps of a self-driven part)
- Linear algebra of the conditions (both directions of each equivalence)
- Invariance analysis under contract replacements and rescaling of $\lambda$
- Discrete Fourier transform on $\mathbb Z_N$ (circulant hopping matrix)
- Limiting cases and homogeneity checks

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