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

# Typical or forced: how special the universal couplings of one body are

- **Subproject:** 05-ilang-free-fall
- **Package:** 17-typical-or-forced
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-18

## Changes from previous version

- Step 6.3, new (17.14): R2 is analysed as written (a factor $c_k$ on each level, not necessarily common) and, separately, in the common-factor form R2$_{\rm c}$, each with its M6 answer, necessity, sufficiency and control (I1).
- Step 6.2, (17.10): the equivalence with R1 is stated for R2$_{\rm c}$, not for R2 (I1); the sufficiency for $\mathcal U_5$ is restated for R1 with $\alpha\neq0$.
- Step 6.1, (17.9): relistings (M4) (a constant into the field pair term) and (M5) (an on-site $\sigma_1$ term moved from the record term to the own term) added (I1, I2).
- Step 6.3, item 1: the invariance of R2 is restricted to relistings that leave the coupling unchanged; (M4) changes the verdict (I2).
- Step 4, new (17.13): the first equality of (4.24) is derived for $\mathcal C^+$ by reduction, level by level, to a member of $\mathcal C$; the Setup assumption and the Open issue on it are replaced (I3).
- Step 3 and "Regime and orders": the rate is defined with frozen coefficients, freezing before $W\to\infty$; the commuting factorization is stated as exact for the frozen contract only; the hierarchy for reading it as an approximation is stated, its error listed as open (I4).
- Step 7, Result (17.7), (17.9), (17.10), Open issues, Methods: adjusted to the changes above.

## Response to verification

- I1: Accepted and fixed in Step 6.3 ((17.14) for R2 as written; R2$_{\rm c}$ for one common factor), Step 6.2 ((17.10)) and Step 7.
- I2: Accepted and fixed in Step 6.1 ((M4)) and Step 6.3, item 1.
- I3: Accepted and fixed in Step 4 ((17.13)).
- I4: Accepted and fixed in Step 3 (convention, order of limits, hierarchy) and in "Regime and orders".

## Setup and assumptions

**Description studied (assumption, M8, M9).** One body of Model D type, the only instance of its type ($\tau>0$, $m>0$, $N\ge2$, $d_k=\frac{2\pi k}{N\lambda_0}$, $M_k=m+d_k$), recorded through $K_x=\kappa xX$ in the weak-record regime, with $\kappa x\langle X\rangle$ counted in the on-site potential. A field $f$ in a sharp state $\varphi$ with linear profile $N(x)=n_1x$, $n_1\neq0$ (origin at the zero of the profile), coupled by the lapse-type pair term $\sum_x\lvert x\rangle\langle x\rvert\otimes G\otimes\Phi_x$ with
$G=\sigma_3\otimes(\alpha_0\mathbb 1_N+\alpha_1D)+\mathbb 1_2\otimes(\beta_0\mathbb 1_N+\beta_1D)$, $\alpha_i,\beta_i\in\mathbb R$, and optionally by a factor on the motion term whose sharp profile gives $\zeta_x=1+g_\zeta(x+\frac12)$. $G$ and Model D are diagonal in $\lvert f_k\rangle$, so the levels decouple, and on level $k$ the body sees the level block of 04-branch-fall with $\tau_k=\tau$ and
$$
M_{x,k}=M_k+G_kx,\ \ G_k=mg_m+d_kg_\eta,\qquad V_{x,k}=-\mathcal F_kx,\ \ \mathcal F_k=\mathcal F+d_k\mathcal F_1,\qquad
mg_m=n_1\alpha_0,\ \ g_\eta=n_1\alpha_1,\ \ \mathcal F=\mathcal F_{\rm rec}-n_1\beta_0,\ \ \mathcal F_1=-n_1\beta_1,
\tag{17.1}
$$
with $\mathcal F_{\rm rec}:=-\kappa\langle X\rangle$ the record part. For $n_1\neq0$, fixed $\mathcal F_{\rm rec}$ and with the motion factor present, $(\alpha_0,\alpha_1,\beta_0,\beta_1,g_\zeta)\mapsto\boldsymbol g=(g_m,g_\eta,\mathcal F,\mathcal F_1,g_\zeta)$ is an affine bijection onto $\mathbb R^5$: every $\boldsymbol g$ is an admissible member of $\mathcal C^+$. The *setting's listing* is: own term Model D, coupling $G\otimes\Phi_x$, record term $\sum_x\lvert x\rangle\langle x\rvert\otimes K_x$.

**Regime and orders.** First order in the gradients ($\lVert\boldsymbol g\rVert\,\lvert x\rvert\ll1$ on the region considered); branch starts at rest, drift velocity $v$. "Exactly" means with the full $c_{2,k,\sigma}$ of (4.17); "leading order in $M_k/\tau$" means dropping in $c_{2,k,\sigma}$ the terms of relative order $M_k^2/\tau^2$. Dimensions refer to $\boldsymbol g\in\mathbb R^5$ at fixed $(\tau,m,N,\lambda_0,\mathcal F_{\rm rec})$. Norm: Euclidean norm of the scaled vector $\hat{\boldsymbol g}:=(g_m,g_\eta,\mathcal F/m,\mathcal F_1,g_\zeta)$, whose components are all per label step. The rate at a place is that of the frozen contract, with freezing before $W\to\infty$ (Step 3).

**Inputs** (quoted in question.md):
- (4.16) of 04-branch-fall: $a_{k,\sigma}(0)=\frac{\tau_k^2}{M_k}(\sigma\mathcal F_k-G_k)$, level by level, for $\mathcal C_{\rm lev}$ with linear level profiles.
- (4.17) of 04-branch-fall: $c_{2,k,\sigma}=g_\zeta+\frac{G_k}{M_k}\bigl(1+\frac{2M_k^2}{\tau^2}\bigr)-\sigma\frac{\mathcal F_k}{M_k}\bigl(\frac32+\frac{2M_k^2}{\tau^2}\bigr)$; (4.20): the upper branch falls alike on all levels through $v^2$ iff $\frac{\tau^2}{M_k}(\mathcal F_k-G_k)=a_0$ for all $k$ and $c_{2,k,+}=g_\zeta-\frac{\mathcal F_k}{2M_k}-\frac{a_0}{\tau^2}\bigl(1+\frac{2M_k^2}{\tau^2}\bigr)$ is independent of $k$.
- (4.24) of 04-branch-fall, for $\mathcal C$ with $a_{k,+}(0)=a\neq0$: $\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\big\vert_{\rm drift}=d_0a_{k,+}(0)\hat e=-\frac{v_{\max}^2}{d_0}\frac{r(x+1)-r(x)}{r(x)}\hat e$, $v_{\max}=\tau d_0$.
- (3.23) of 03-clock-and-fall: rate at $x_0$ $:=\lim_{W\to\infty}r_W(x_0)=\frac{\eta_{x_0}}{\lambda_0}=\frac N{2\pi}\bigl(E^+_{x_0,k+1}-E^+_{x_0,k}\bigr)$, $E^\pm_{x,k}=V_{x,k}\pm M_{x,k}$.
- (4.22) of 04-branch-fall: used in Step 4 and in the checks; (4.19): used in the checks.

**Use of (4.24) in $\mathcal C^+$.** Since $v_{\max}=\tau d_0$, given its first equality (derived for $\mathcal C^+$ in Step 4, (17.13)), (4.24) is equivalent to its label form $a_{k,+}(0)=-\tau^2\frac{r(x+1)-r(x)}{r(x)}$. "(4.24) holds on level $k$" means that the label form holds on level $k$.

## Derivation

### Step 1. Fall from rest (item 1a, 1c)

By (4.16) and (17.1), with $d_k=M_k-m$: $\sigma\mathcal F_k-G_k=\sigma\mathcal F-mg_m+d_k(\sigma\mathcal F_1-g_\eta)=m\ell_\sigma+M_k(\sigma\mathcal F_1-g_\eta)$, hence
$$
a_{k,\sigma}(0)=\tau^2(\sigma\mathcal F_1-g_\eta)+\tau^2\,\frac{m}{M_k}\,\ell_\sigma,
\qquad
\ell_\sigma:=\sigma\frac{\mathcal F}{m}-g_m+g_\eta-\sigma\mathcal F_1 .
\tag{17.2}
$$
For $N\ge2$, $k\mapsto m/M_k$ is strictly decreasing, so $a_{k,\sigma}(0)$ is the same on all levels iff $\ell_\sigma=0$, with value $a_\sigma=\tau^2(\sigma\mathcal F_1-g_\eta)$. Since $\ell_++\ell_-=2(g_\eta-g_m)$, $\ell_+-\ell_-=2(\mathcal F/m-\mathcal F_1)$ and, on $\mathcal U_4^{\pm}$, $a_+-a_-=2\tau^2\mathcal F_1$:
$$
\begin{aligned}
\mathcal U_4^{+}&=\{\mathcal F=m(g_m-g_\eta+\mathcal F_1)\},\ \ a_+=\tau^2(\mathcal F_1-g_\eta);\qquad
\mathcal U_4^{-}=\{\mathcal F=m(\mathcal F_1+g_\eta-g_m)\},\ \ a_-=-\tau^2(\mathcal F_1+g_\eta);\\
\mathcal U_4^{\pm}&=\mathcal U_4^+\cap\mathcal U_4^-=\{g_m=g_\eta,\ \mathcal F=m\mathcal F_1\};\qquad
\mathcal U_4^{=}=\mathcal U_4^{\pm}\cap\{a_+=a_-\}=\{g_m=g_\eta,\ \mathcal F=\mathcal F_1=0\},\ \ a=-\tau^2g_\eta .
\end{aligned}
\tag{17.3}
$$
In $\hat{\boldsymbol g}$, $\ell_\pm=\boldsymbol n_\pm\cdot\hat{\boldsymbol g}$ with $\boldsymbol n_\pm=(-1,1,\pm1,\mp1,0)$, $\lvert\boldsymbol n_\pm\rvert=2$, $\boldsymbol n_+\cdot\boldsymbol n_-=0$, and $\mathcal U_4^=$ adds the independent equation $\mathcal F_1=0$. Dimensions $4,4,3,2$, codimensions $1,1,2,3$, the same for $N=2$ and $N\ge3$.

### Step 2. Through the order $v^2$ (item 1a, 1c)

*Upper branch.* On $\mathcal U_4^+$, $\mathcal F_k=\Delta+M_k\mathcal F_1$ with $\Delta:=m(g_m-g_\eta)$, so (4.20)(ii) reads
$$
c_{2,k,+}=g_\zeta-\frac{\mathcal F_1}2-\frac{a_+}{\tau^2}-h(M_k),\qquad h(u):=\frac{\Delta}{2u}+\frac{2a_+}{\tau^4}\,u^2 ,
\tag{17.4}
$$
and the condition is that $h(M_k)$ is independent of $k$.
1. *Exactly, $N=2$:* $h(M_0)=h(M_1)$ $\iff$ $\frac{\Delta}{2}\frac{M_1-M_0}{M_0M_1}=\frac{2a_+}{\tau^4}(M_1-M_0)(M_1+M_0)$ $\iff$ $\Delta=\frac{4a_+}{\tau^4}M_0M_1(M_0+M_1)$. This equation does not contain $\mathcal F$, while $\ell_+$ does, so the two are independent.
2. *Exactly, $N\ge3$:* $h(M_k)=h_0$ means that $p(u)=\frac{2a_+}{\tau^4}u^3-h_0u+\frac\Delta2$ vanishes at $N\ge3$ distinct positive points. If $a_+\neq0$, $p$ is a cubic without $u^2$ term, so its three roots sum to zero, impossible for three positive roots. Hence $a_+=0$, then $p$ is affine with three roots, so $h_0=\Delta=0$.
3. *Leading order in $M_k/\tau$:* the $u^2$ term of $h$ is of relative order $M_k^2/\tau^2$ and is dropped; $\Delta/(2M_k)$ is $k$-independent iff $\Delta=0$, for every $N\ge2$. On the resulting set the residual is $c_{2,k,+}-c_{2,0,+}=-\frac{2a_+}{\tau^4}(M_k^2-m^2)$.

*Lower branch.* In (4.16)–(4.17), $\mathcal F_k$ enters only as $\sigma\mathcal F_k$, so the lower-branch conditions are those of the upper branch under the reflection $R:(\mathcal F,\mathcal F_1)\mapsto(-\mathcal F,-\mathcal F_1)$, which maps $a_+$ to $a_-$. Hence:
$$
\begin{aligned}
&\mathcal U^+_{4,v}=\Bigl\{\ell_+=0,\ m(g_m-g_\eta)=\tfrac{4M_0M_1(M_0+M_1)}{\tau^2}(\mathcal F_1-g_\eta)\Bigr\}\ (N=2,\ \dim3);\qquad
\mathcal U^+_{4,v}=\{g_m=g_\eta=\mathcal F_1,\ \mathcal F=mg_\eta\}\ (N\ge3,\ \dim2,\ a_+=0);\\
&\text{leading order in }M_k/\tau,\ N\ge2:\ \ \mathcal U^+_{4,v,0}=\{g_m=g_\eta,\ \mathcal F=m\mathcal F_1\}=\mathcal U_4^{\pm}\ (\dim3);\qquad
\mathcal U^-_{4,v}=R\,\mathcal U^+_{4,v},\ \ \mathcal U^-_{4,v,0}=R\,\mathcal U^+_{4,v,0};\\
&\text{both branches: exactly }\mathcal U^+_{4,v}\cap\mathcal U^-_{4,v}=\{g_m=g_\eta=\mathcal F=\mathcal F_1=0\}\ (\dim1,\ a_\pm=0);\ \ \text{at leading order }\mathcal U_4^{\pm}\ (\dim3).
\end{aligned}
\tag{17.5}
$$
For both branches exactly with $N=2$: $\ell_\pm=0$ gives $\Delta=0$, $\mathcal F=m\mathcal F_1$, then item 1 gives $a_+=a_-=0$, so $\mathcal F_1=g_\eta=0$; for $N\ge3$ intersect the two sets of item 2. $\mathcal U_4^\pm$ is $R$-invariant.

### Step 3. Rate at a place (item 1b)

*Convention.* As in (3.23), and as the question asks (zeroth order in the gradients, wide-start form), the rate at $x_0$ is the rate of the frozen contract $C_{x_0}$: the translation-invariant contract with the coefficients of $C$ at $x_0$. Freezing comes first; $W\to\infty$ is then taken for $C_{x_0}$, not for the actual inhomogeneous evolution. Read as an approximation to the clock of an actual start of width $W$ at $x_0$, observed over an interval $\Lambda$, it needs $W\gg1$ (so that $\varsigma_W$ of (3.23) is small) and $(W+\tau\Lambda)\lVert\hat{\boldsymbol g}\rVert\ll1$ (the profiles vary little over the labels the start covers; $v_{\max}/d_0=\tau$ bounds the label speed). Its error in this regime is not derived here.

$C_{x_0}$ has the on-site part $\sigma_3\otimes(m_{x_0}\mathbb 1_N+\eta_{x_0}D)+\mathbb 1_2\otimes(V_{x_0}\mathbb 1_N-\mathcal F_1x_0D)$, where $m_{x_0}=m(1+g_mx_0)$, $\eta_{x_0}=1+g_\eta x_0$, $V_{x_0}=-\mathcal Fx_0$, and hopping $\tau\zeta_{x_0}$. Write it as $C^{\mathcal C}_{x_0}-\mathcal F_1x_0\,\mathbb 1\otimes\mathbb 1_2\otimes D$, with $C^{\mathcal C}_{x_0}$ the frozen member of $\mathcal C$ with the same $(g_m,g_\eta,\mathcal F,g_\zeta)$. The added term is the identity on position and branch, and $D$ commutes with the clock factors of $C^{\mathcal C}_{x_0}$ ($\mathbb 1_N$ and $m_{x_0}\mathbb 1_N+\eta_{x_0}D$). So the two terms commute and $e^{-iC_{x_0}\lambda}=e^{-iC^{\mathcal C}_{x_0}\lambda}e^{i\mathcal F_1x_0D\lambda}$, exactly for the frozen contract. (On the full chain $-\mathcal F_1xD$ does not commute with the hopping; its effect there is the level-dependent force $d_k\mathcal F_1$, contained in (17.2).) By (3.23), for $W\to\infty$ the first factor acts on the clock of the upper-branch start as $e^{-i\eta_{x_0}D\lambda}$ up to a common phase: the level phases are $E^+_{x_0,k}$, affine in $d_k$ with slope $\eta_{x_0}$. The product acts as $e^{-i(\eta_{x_0}-\mathcal F_1x_0)D\lambda}$, which is the clock evolution of $\mathcal C$ with $\eta_{x_0}\to\eta_{x_0}-\mathcal F_1x_0$. By (3.23),
$$
r(x_0)=\frac{1+(g_\eta-\mathcal F_1)\,x_0}{\lambda_0}=\frac N{2\pi}\bigl(E^+_{x_0,k+1}-E^+_{x_0,k}\bigr),
\qquad E^+_{x,k}=m(1+g_mx)-\mathcal Fx+d_k\bigl(1+(g_\eta-\mathcal F_1)x\bigr),
\tag{17.6}
$$
the same for every $k$, because $E^+_{x,k}$ is affine in $d_k$.

### Step 4. Relation (4.24) and $\mathcal U_5$ (item 1b, 1c)

*First equality of (4.24).* Fix a level $k$ with $a_{k,+}(0)\neq0$. Model D, the coupling and the record term commute with the projector $\lvert f_k\rangle\langle f_k\rvert$ on the clock label, and the factor $\varphi$ of the sharp field splits off. So a branch start on level $k$ evolves, together with the medium, under the level-$k$ block and the record term alone. The member of $\mathcal C$ with $g'_\eta=(G_k-\mathcal F_k)/M_k$, $g'_m=g'_\eta+\mathcal F_k/m$, $\mathcal F'=\mathcal F_k$, $g'_\zeta=g_\zeta$ has the same level-$k$ block ($mg'_m+d_kg'_\eta=M_kg'_\eta+\mathcal F_k=G_k$), the same records and the same branch start. By (4.22) its fall from rest is level-independent, with value $-\tau^2g'_\eta=\frac{\tau^2}{M_k}(\mathcal F_k-G_k)=a_{k,+}(0)\neq0$ by (4.16). So (4.24) applies to it. Its first equality, applied to the start on level $k$, whose evolution is the same in both descriptions, gives for the member of $\mathcal C^+$
$$
\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\Big\vert_{\rm drift}=d_0\,a_{k,+}(0)\,\hat e\qquad(\text{start on level }k,\ \ a_{k,+}(0)\neq0),
\tag{17.13}
$$
at the order of (4.24): first order in the gradients, weak records. What (4.24) excludes from the drift is excluded in the same way, since the two evolutions coincide. On $\mathcal U_5$ (below) every level has $a_{k,+}(0)\neq0$.

*Label form.* By (17.6), $\frac{r(x+1)-r(x)}{r(x)}=\frac{g_\eta-\mathcal F_1}{1+(g_\eta-\mathcal F_1)x}=g_\eta-\mathcal F_1+O(\lVert\boldsymbol g\rVert^2)$. With (17.2), the defect of the label form of (4.24) on level $k$ is
$$
\mathcal D_k:=a_{k,+}(0)+\tau^2\frac{r(x+1)-r(x)}{r(x)}=\tau^2\frac{m}{M_k}\,\ell_+ +O(\lVert\boldsymbol g\rVert^2),
\qquad
\mathcal U_5=\{\ell_+=0,\ \mathcal F_1\neq g_\eta\}=\mathcal U_4^+\setminus\{a_+=0\}.
\tag{17.7}
$$
Since $m/M_k\neq0$, (4.24) holds on one level iff it holds on every level, iff $\ell_+=0$. The common value is then $a_+=\tau^2(\mathcal F_1-g_\eta)$, nonzero iff $\mathcal F_1\neq g_\eta$. $\{\ell_+=0,\mathcal F_1=g_\eta\}$ has codimension 2, so $\mathcal U_5$ is an open dense part of the hyperplane $\mathcal U_4^+$: dimension 4, codimension 1, for every $N\ge2$.

### Step 5. Stability (item 2)

The distance of $\hat{\boldsymbol g}$ from $\mathcal U_4^+$ is $\delta=\lvert\ell_+\rvert/2$. By (17.2) and (17.7), $a_{k,+}(0)$ and $\mathcal D_k$ are monotone in $k$, with extreme levels $k=0$ ($M_0=m$) and $k=N-1$:
$$
\max_{k,k'}\bigl\lvert a_{k,+}(0)-a_{k',+}(0)\bigr\rvert=2\tau^2\delta\,\frac{d_{N-1}}{m+d_{N-1}},
\qquad
\max_k\lvert\mathcal D_k\rvert=\lvert\mathcal D_0\rvert=2\tau^2\delta .
\tag{17.8}
$$
For the distance $\delta_\pm$ from $\mathcal U_4^\pm$, the orthogonality of $\boldsymbol n_\pm$ gives $\delta_\pm^2=(\ell_+^2+\ell_-^2)/4$. On branch $\sigma$ the largest level difference is therefore at most $2\tau^2\delta_\pm d_{N-1}/M_{N-1}$, with equality when $\ell_{-\sigma}=0$. Both quantities are of first order in $\delta$, with the exact coefficients above at first order in the gradients.

### Step 6. Requirements (item 3)

**6.1 Relistings.** For any self-adjoint $Q$ on the body and any pair term $C_\gamma$ of the body,
$$
C_{\rm own}\to C_{\rm own}-Q,\qquad C_\gamma\to C_\gamma+Q\otimes\mathbb 1
\tag{17.9}
$$
leaves $C$ unchanged, so by A5 and M6 the two listings are the same description. Instances:
- (M1) $Q=c\,\mathbb 1$ into the record term: the zero of the body's own cost.
- (M2) $Q=\epsilon\sum_x\lvert x\rangle\langle x\rvert\otimes\sigma_3\otimes\mathbb 1_N$, $0<\lvert\epsilon\rvert<m$, into the record pair term: the own term stays of Model D form, with $m\to m-\epsilon$.
- (M3) $Q=-c\sum_x\lvert x\rangle\langle x\rvert\otimes G$ into the field pair term: the coupling stays $G\otimes(\Phi_x-c)$ and the own on-site term becomes $\sigma_3\otimes(m+D)+cG$.
- (M4) $Q=c\,\mathbb 1$, $c\neq0$, into the field pair term: the pair term becomes $\sum_x\lvert x\rangle\langle x\rvert\otimes G\otimes\Phi_x+c\,\mathbb 1$, with the part $c\,\mathbb 1$ proportional to $\mathbb 1_2$ on the branch label.
- (M5) $Q=-\epsilon\sum_x\lvert x\rangle\langle x\rvert\otimes\sigma_1\otimes\mathbb 1_N$, $\epsilon\neq0$, into the record pair term: the own on-site block of level $k$ becomes $M_k\sigma_3+\epsilon\sigma_1$.

The effective level blocks in $\varphi$ and in the medium background are computed from the total $C$. Hence $M_{x,k}$, $\zeta_{x,k}$, and $V_{x,k}$ up to one common constant are invariant under (17.9) and under the common phase. So are $\boldsymbol g$, $M_k$, (17.2), (17.6) and the sets (17.3), (17.5), (17.7).

**6.2 R1.** In the setting's listing, with (17.1), $n_1\neq0$ and R2$_{\rm c}$ the common-factor form of R2 (6.3),
$$
\text{R1}\iff\text{R2}_{\rm c}\iff G=\alpha\,\sigma_3\otimes(m\mathbb 1_N+D)\iff g_m=g_\eta,\ \ \mathcal F_1=0,\ \ \mathcal F=\mathcal F_{\rm rec};\qquad\text{then }\ \ell_\pm=\pm\mathcal F_{\rm rec}/m .
\tag{17.10}
$$
1. *Well defined?* No. After (M2), the own on-site term is $\sigma_3\otimes((m-\epsilon)\mathbb 1_N+D)$. Then $G=\alpha\sigma_3\otimes(m+D)$ with $\alpha\neq0$ fails R1 (its $\mathbb 1_N$ and $D$ coefficients have ratio $m\neq m-\epsilon$), while $G'=\alpha\sigma_3\otimes((m-\epsilon)\mathbb 1_N+D)$ satisfies it. For one and the same $C$, the verdict of R1 depends on the listing. (M1) changes it as well.
2. *Relation (setting's listing).* If $\mathcal F_{\rm rec}=0$, R1 is equivalent to $\mathcal U_4^=$ by (17.3). It is therefore sufficient for $\mathcal U_4^\pm$, and for $\mathcal U_5$ when $\alpha\neq0$ ($a=-\tau^2n_1\alpha$). It is not necessary for either: take $G=(\alpha\sigma_3+\beta\mathbb 1_2)\otimes(m\mathbb 1_N+D)$ with $\beta\neq0$; by (17.1), $g_m=g_\eta=n_1\alpha$ and $\mathcal F=m\mathcal F_1=-mn_1\beta$, so $G\in\mathcal U_4^\pm$, and $G\in\mathcal U_5$ if $\beta\neq-\alpha$, yet $G$ fails R1. If $\mathcal F_{\rm rec}\neq0$, then $\ell_\pm\neq0$, so R1 excludes $\mathcal U_4^\pm$ and $\mathcal U_5$. So R1 is not necessary for $\mathcal U_4^\pm$ or $\mathcal U_5$ for any $\mathcal F_{\rm rec}$. It is sufficient for $\mathcal U_4^\pm$ iff $\mathcal F_{\rm rec}=0$, and R1 with $\alpha\neq0$ is sufficient for $\mathcal U_5$ iff $\mathcal F_{\rm rec}=0$.
3. *Controls.* (a) Setting's listing with $\mathcal F_{\rm rec}\neq0$: every R1 coupling fails item 4 on both branches. (b) $\mathcal F_{\rm rec}=0$: $G'$ of item 1 satisfies R1 in the listing (M2), and by (17.1) $\ell_\pm=n_1\alpha\epsilon/m\neq0$, so item 4 fails on both branches.

**6.3 R2.** Two readings are distinguished:
- (R2) as written: on every level $k$ the level block of $G$ is $c_k$ times the traceless part, on the branch label, of the own on-site block of level $k$, with real factors $c_k$ that may depend on $k$; and the coupling has no part proportional to $\mathbb 1_2$;
- (R2$_{\rm c}$) the same with one common factor, $c_k=\alpha$ for all $k$.

In the setting's listing the own block $M_k\sigma_3$ is traceless and nonzero, and $G$ has the level block $(\alpha_0+\alpha_1d_k)\sigma_3+(\beta_0+\beta_1d_k)\mathbb 1_2$. Its $\sigma_3$ part is always a multiple of $M_k\sigma_3$, with $c_k=(\alpha_0+\alpha_1d_k)/M_k$, and with two distinct $d_k$ its $\mathbb 1_2$ part vanishes on all levels iff $\beta_0=\beta_1=0$. By (17.1):
$$
\text{R2}\iff G=\sigma_3\otimes(\alpha_0\mathbb 1_N+\alpha_1D)\iff\mathcal F_1=0,\ \ \mathcal F=\mathcal F_{\rm rec};\qquad\text{then }\ \ell_\pm=\pm\frac{\mathcal F_{\rm rec}}m+g_\eta-g_m ,
\tag{17.14}
$$
with $g_m,g_\eta,g_\zeta$ free (codimension 2). R2$_{\rm c}$ adds $\alpha_0=\alpha m$, $\alpha_1=\alpha$, i.e. $g_m=g_\eta$: it is the set (17.10).
1. *Well defined?* Neither form. Both are invariant under (M1), which leaves the coupling unchanged and adds to the own block only a multiple of $\mathbb 1_2$, which the traceless part does not see. This does not extend to relistings that change the coupling. After (M4), every coupling that satisfies R2 or R2$_{\rm c}$ in the setting's listing has the $\mathbb 1_2$ part $c\,\mathbb 1$. (M5) changes the verdict even if "the coupling" is restricted to the part of the pair term that acts on the field: the own block becomes $M_k\sigma_3+\epsilon\sigma_1$, of which $(\alpha_0+\alpha_1d_k)\sigma_3$ is a multiple only where it vanishes, so every such coupling with $(\alpha_0,\alpha_1)\neq(0,0)$ fails both forms after (M5). Under (M2), R2 is invariant (the own block $(M_k-\epsilon)\sigma_3$ stays nonzero, $c_k\to(\alpha_0+\alpha_1d_k)/(M_k-\epsilon)$), but R2$_{\rm c}$ is not: $M_k/(M_k-\epsilon)$ depends on $k$, so $G$ and $G'$ of 6.2 exchange their verdicts. R2$_{\rm c}$ is invariant under (M3) when $1+c\alpha\neq0$ (the own block becomes $(1+c\alpha)M_k\sigma_3$).
2. *Relation (setting's listing).* R2$_{\rm c}$: as R1 in 6.2. R2: by (17.14) and (17.3), the intersection of R2 with $\mathcal U_4^\pm$ is $\mathcal U_4^=$ if $\mathcal F_{\rm rec}=0$ and empty otherwise; its intersection with $\mathcal U_4^+$ is $\{\mathcal F_1=0,\ \mathcal F=\mathcal F_{\rm rec},\ m(g_m-g_\eta)=\mathcal F_{\rm rec}\}$, non-empty for every $\mathcal F_{\rm rec}$, and meets $\mathcal U_5$ where $g_\eta\neq0$. R2 is not sufficient for $\mathcal U_4^+$, $\mathcal U_4^\pm$ or $\mathcal U_5$ (control in item 3). It is not necessary for any of them: $G_\beta=\alpha\sigma_3\otimes(m\mathbb 1_N+D)+\mathbb 1_2\otimes(\beta_0\mathbb 1_N+\beta D)$ with $\beta\neq0$ and $n_1\beta_0=\mathcal F_{\rm rec}+mn_1\beta$ has, by (17.1), $g_m=g_\eta=n_1\alpha$, $\mathcal F_1=-n_1\beta$, $\mathcal F=m\mathcal F_1$. So it lies in $\mathcal U_4^\pm$, and in $\mathcal U_5$ if $\beta\neq-\alpha$, yet fails R2. R2 is thus unrelated to $\mathcal U_4^\pm$ and $\mathcal U_5$, for every $\mathcal F_{\rm rec}$.
3. *Controls.* R2: $G=\epsilon\,\sigma_3\otimes D$ with $\epsilon\neq0$ and $n_1\epsilon\neq\pm\mathcal F_{\rm rec}/m$ satisfies R2 ($c_k=\epsilon d_k/M_k$) and has $\ell_\pm\neq0$ by (17.14), so item 4 fails on both branches; for $\mathcal F_{\rm rec}=0$, (17.2) gives $a_{k,\sigma}(0)=-\tau^2n_1\epsilon\,d_k/M_k$. R2$_{\rm c}$: the controls (a), (b) of 6.2, with (M2) applied to the traceless block.

**6.4 R3.** Let $B_{x,k}:=M_{x,k}\sigma_3+V_{x,k}\mathbb 1_2$ be the effective on-site block of level $k$ (in $\varphi$ and the medium background), and $M_k=\frac12(E^+_{0,k}-E^-_{0,k})$ its half-gap between branches. The requirements are:
- (R3) there is a level-independent operator $\Gamma$ on the branch label with $B_{x+1,k}-B_{x,k}=M_k\Gamma$ for every $k$;
- (R3$^+$) $E^+_{x+1,k}-E^+_{x,k}=\gamma_+M_k$ for every $k$, with one real $\gamma_+$.

Both are taken at first order in the gradients.
1. *Well defined?* Yes. Only differences of effective costs at neighbouring places (where the common constant cancels) and branch gaps enter, and both are computed from the total $C$. So R3 and R3$^+$ are invariant under (17.9) and under the common phase.
2. *Relation.* In $\mathcal C^+$, $B_{x+1,k}-B_{x,k}=G_k\sigma_3-\mathcal F_k\mathbb 1_2$, so $\Gamma=\gamma\sigma_3-\phi\mathbb 1_2$. With two distinct $M_k$:
$$
\text{R3}\iff G_k=\gamma M_k,\ \mathcal F_k=\phi M_k\ \forall k\iff g_m=g_\eta=\gamma,\ \mathcal F_1=\phi,\ \mathcal F=m\phi\iff\boldsymbol g\in\mathcal U_4^{\pm};\qquad
\text{R3}^+\iff\ell_+=0\iff\boldsymbol g\in\mathcal U_4^{+},
\tag{17.11}
$$
where R3$^+$ uses $\mathcal F_k-G_k=m\ell_++M_k(\mathcal F_1-g_\eta)$ from Step 1. Hence: $\Gamma$ traceless ($\phi=0$) $\iff\mathcal U_4^=$; R3$^+$ with $\gamma_+=g_\eta-\mathcal F_1\neq0$ $\iff\mathcal U_5$; and $a_+=-\tau^2\gamma_+$. R3 is necessary and sufficient for $\mathcal U_4^\pm$, so by (17.2) no member of $\mathcal C^+$ satisfies R3 and fails item 4 from rest. Through $v^2$ exactly it is not sufficient. Control: R3 with $\phi\neq\gamma$, i.e. $a_+\neq0$, has $\Delta=0$ and so fails (17.5) for $N=2$ and for $N\ge3$.
3. In the setting's listing, $B_{x+1,k}-B_{x,k}=n_1G^{(k)}-\mathcal F_{\rm rec}\mathbb 1_2$, with $G^{(k)}$ the level block of $G$. R3 therefore reads $n_1G^{(k)}=M_k\Gamma+\mathcal F_{\rm rec}\mathbb 1_2$ for all $k$: it ties the coupling to the body's half-gaps and to the record force.

### Step 7. Verdict (item 4)

At first order in the gradients, every set of (17.3), (17.5) and (17.7) has as closure a proper linear subspace of $\mathbb R^5$:
$$
\operatorname{int}\,\mathcal U=\emptyset\quad\text{and}\quad\mathbb R^5\setminus\overline{\mathcal U}\ \text{is open and dense, for every set }\mathcal U\text{ of (17.3), (17.5), (17.7)} .
\tag{17.12}
$$
The couplings that give items 4 and 5 are therefore exceptional (codimension 1 to 4), not typical. By (17.8) they are broken linearly in the distance. They are selected by the well-defined requirement R3 (R3$^+$ for $\mathcal U_4^+$ and $\mathcal U_5$), not by R1 or R2 (as written or as R2$_{\rm c}$), which are not well defined; R2 as written is moreover unrelated to them. By (17.1) every $\boldsymbol g\in\mathbb R^5$ is an admissible description, so within $\mathcal C^+$ the language does not enforce R3. This follows from admissibility (A10, M8), not from items 1–3 (M10).

## Result

Class throughout: $\mathcal C^+$ as in the Setup (one Model D body, the only one of its type; sharp field with linear profile, $n_1\neq0$; weak records), $N\ge2$, first order in the gradients, branch starts at rest, unless stated.

- **(17.1)** (definition) Parametrization of $\mathcal C^+$ by $\boldsymbol g$, with $\mathcal F=\mathcal F_{\rm rec}-n_1\beta_0$ and $\mathcal F_1=-n_1\beta_1$. It is onto $\mathbb R^5$ at fixed $\mathcal F_{\rm rec}$.
- **(17.2)** (formula) $a_{k,\sigma}(0)=\tau^2(\sigma\mathcal F_1-g_\eta)+\tau^2\frac m{M_k}\ell_\sigma$.
- **(17.3)** (criterion) For $N=2$ and $N\ge3$ alike:
  - the fall from rest is level-independent on branch $\sigma$ iff $\ell_\sigma=0$;
  - on both branches iff $g_m=g_\eta$ and $\mathcal F=m\mathcal F_1$;
  - on both branches with $a_+=a_-$ iff in addition $\mathcal F_1=0$.

  Codimensions $1,1,2,3$.
- **(17.4)–(17.5)** (criterion) Upper branch through $v^2$, exactly:
  - $N=2$: $\ell_+=0$ and $m(g_m-g_\eta)=\frac{4M_0M_1(M_0+M_1)}{\tau^2}(\mathcal F_1-g_\eta)$, codimension 2;
  - $N\ge3$: $g_m=g_\eta=\mathcal F_1$ and $\mathcal F=mg_\eta$, codimension 3, with $a_+=0$.

  At leading order in $M_k/\tau$ (residual of relative order $M_k^2/\tau^2$) the set is $\mathcal U_4^\pm$, codimension 2, for every $N\ge2$. The lower branch gives the images under $(\mathcal F,\mathcal F_1)\mapsto(-\mathcal F,-\mathcal F_1)$. Both branches: exactly, only $g_\zeta$ may be nonzero (codimension 4, zero fall), for $N=2$ and $N\ge3$; at leading order, $\mathcal U_4^\pm$.
- **(17.6)** (formula) Wide-start rate at $x_0$, at zeroth order in the gradients: $r(x_0)=[1+(g_\eta-\mathcal F_1)x_0]/\lambda_0=\frac N{2\pi}(E^+_{x_0,k+1}-E^+_{x_0,k})$ for every $k$.
- **(17.13)** (formula) First equality of (4.24) in $\mathcal C^+$: for a branch start on level $k$ with $a_{k,+}(0)\neq0$, $\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\big\vert_{\rm drift}=d_0a_{k,+}(0)\hat e$, with $a_{k,+}(0)$ of (17.2), at first order in the gradients and with weak records.
- **(17.7)** (criterion) The defect of the label form of (4.24) on level $k$ is $\mathcal D_k=\tau^2\frac m{M_k}\ell_+$ up to $O(\lVert\boldsymbol g\rVert^2)$. (4.24) holds on one level iff it holds on all levels, iff $\ell_+=0$. $\mathcal U_5=\mathcal U_4^+\setminus\{\mathcal F_1=g_\eta\}$ has dimension 4 and codimension 1, for $N=2$ and $N\ge3$. On $\mathcal U_5$ the label form is equivalent to (4.24) by (17.13).
- **(17.8)** (formula) At distance $\delta$ from $\mathcal U_4^+$ in the norm of $\hat{\boldsymbol g}$: the largest level difference of $a_{k,+}(0)$ is $2\tau^2\delta\,d_{N-1}/(m+d_{N-1})$, and the largest defect of (4.24) is $2\tau^2\delta$. Both are of order $\delta^1$. At distance $\delta_\pm$ from $\mathcal U_4^\pm$, the largest level difference on each branch is at most $2\tau^2\delta_\pm d_{N-1}/M_{N-1}$.
- **(17.9)** (definition) Relistings (M1)–(M5). The effective level blocks, up to one common constant, are invariant, and so are $\boldsymbol g$ and all sets above.
- **(17.10)** (negative) R1 and R2$_{\rm c}$ (R2 with one common factor on all levels) are not well defined: (M2) changes their verdict for a fixed $C$, (M1) does so for R1, and (M4) and (M5) do so for R2$_{\rm c}$. In the setting's listing they define the same set, $g_m=g_\eta$, $\mathcal F_1=0$, $\mathcal F=\mathcal F_{\rm rec}$. This set equals $\mathcal U_4^=$ if $\mathcal F_{\rm rec}=0$ and is disjoint from $\mathcal U_4^\pm$ and $\mathcal U_5$ if $\mathcal F_{\rm rec}\neq0$. In the setting's listing, R1 and R2$_{\rm c}$ are not necessary for $\mathcal U_4^\pm$ or $\mathcal U_5$; they are sufficient for $\mathcal U_4^\pm$ iff $\mathcal F_{\rm rec}=0$; with $\alpha\neq0$ they are sufficient for $\mathcal U_5$ iff $\mathcal F_{\rm rec}=0$. Controls that satisfy them and fail item 4 on both branches: any R1 coupling with $\mathcal F_{\rm rec}\neq0$; or, for $\mathcal F_{\rm rec}=0$, $G'=\alpha\sigma_3\otimes((m-\epsilon)\mathbb 1_N+D)$ with $\alpha\neq0$ in the listing (M2).
- **(17.14)** (negative) R2 as written (a factor $c_k$ on every level, not necessarily common) is not well defined: (M4) and (M5) change its verdict for a fixed $C$, although it is invariant under (M1) and (M2). In the setting's listing it is equivalent to $\mathcal F_1=0$, $\mathcal F=\mathcal F_{\rm rec}$ (codimension 2; $g_m,g_\eta,g_\zeta$ free). For every $\mathcal F_{\rm rec}$ it is neither necessary nor sufficient for $\mathcal U_4^\pm$, $\mathcal U_4^+$ or $\mathcal U_5$; its intersection with $\mathcal U_4^\pm$ is $\mathcal U_4^=$ if $\mathcal F_{\rm rec}=0$ and empty otherwise. Control: $G=\epsilon\,\sigma_3\otimes D$ with $\epsilon\neq0$ and $n_1\epsilon\neq\pm\mathcal F_{\rm rec}/m$ satisfies R2 and fails item 4 on both branches.
- **(17.11)** (theorem) R3 (block gradient $=M_k\Gamma$ with a level-independent $\Gamma$) is well defined and is equivalent to $\mathcal U_4^\pm$. With $\Gamma$ traceless it is equivalent to $\mathcal U_4^=$. R3$^+$ is equivalent to $\mathcal U_4^+$, and R3$^+$ with $\gamma_+\neq0$ to $\mathcal U_5$. There is no control in $\mathcal C^+$ from rest. Through $v^2$ exactly, R3 with $a_+\neq0$ fails.
- **(17.12)** (theorem) Verdict: the sets are exceptional, with empty interior and positive codimension; they are broken at first order in $\delta$; they are selected by R3; and within $\mathcal C^+$ they are not enforced by the language, since every $\boldsymbol g$ is admissible.

## Consistency checks

1. **Dimensions.** Costs $\tau,m,d_k,\mathcal F$ are per unit of $\lambda$, and $g_m,g_\eta,g_\zeta,\mathcal F_1$ are per label step. Then $a_{k,\sigma}(0)$ and $\tau^2(\mathcal F_1-g_\eta)$ have dimension $[\lambda]^{-2}$; $\ell_\sigma$, $h(u)$ and $\delta$ are dimensionless; $r$ is per unit of $\lambda$; (17.8) has dimension $[\lambda]^{-2}$. Passed.
2. **Subclass $\mathcal C$ ($\mathcal F_1=0$).** (17.3) gives $\mathcal F=\pm m(g_m-g_\eta)$ with value $-\tau^2g_\eta$, which is (4.22). $\mathcal U_4^=$ gives $\mathcal F_k=0$ and $\tau^2G_k/M_k=\tau^2g_\eta=-a$, which is (4.19). (17.6) reduces to $\eta_{x_0}/\lambda_0$, which is (3.23). (17.7) gives (4.24) on $\mathcal U_4^+$ with $a\neq0$. Passed.
3. **Model D without field and records ($\boldsymbol g=0$).** $a_{k,\sigma}(0)=0$; $\boldsymbol g=0$ lies in every set of (17.3) and (17.5); $c_{2,k,+}=0$ by (17.4); $r=1/\lambda_0$. Passed.

## Open issues

- (17.13) uses that the first equality of (4.24), quoted per level $k$, depends only on the block of the start's level; this is read from the quoted form, not re-checked in 04-branch-fall.
- (17.6) is the frozen-coefficient, $W\to\infty$ form only. The finite-$W$ rate, its bound in (3.23), and the error of the frozen rate as an approximation to readings of the actual inhomogeneous evolution (hierarchy of Step 3) are not derived for $\mathcal F_1\neq0$.
- The lower branch through $v^2$ uses (4.17) at $\sigma=-1$ level by level, in the form analogous to (4.20), which is quoted for the upper branch only.
- $\mathcal F_{\rm rec}=-\kappa\langle X\rangle$ depends on the medium background. Whether the background of 03-ilang-space fixes $\langle X\rangle=0$ is not quoted.
- R3 is equivalent to the property it selects. Whether a requirement outside $\mathcal C^+$ selects it (couplings that mix branches or levels, non-linear profiles, fields that are not sharp, several bodies) is out of scope.

## Methods used

- affine parametrization of a coupling class
- partial fractions in the level label and injectivity of Möbius maps
- root counting for a depressed cubic
- frozen-coefficient definition of a local rate; commuting factorization of the frozen evolution
- reduction, level by level, to a member of the subclass with the same level block
- relisting argument for M6 invariance, with explicit controls
- distance to hyperplanes, linear stability

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