Version 6 · current

# Fall of a bound pair of two bodies in a sharp field

- **Subproject:** 05-ilang-free-fall
- **Package:** 13-pair-fall
- **Version:** v6
- **Mode:** external regeneration
- **Date:** 2026-10-11

## Changes from previous version

Trigger: `verification.r3.md` (external round 3 on v5: I1, I2 and I3, all minor) and the notes of the main model with it: the three issues are instances of two kinds of defect, and every instance of each kind is to be removed, the further places that the main model found in v5 included. No display (13.1)–(13.35) is changed in its formula. No tag is renumbered or withdrawn. New tag: (13.36). The question is unchanged since v5. Every sentence of v5 that is not listed here is kept word for word.

- **Header.** Version; this section and the section "Response to verification" replace those of v5 (which answered `verification.r2.md`).
- **Kind 1 (I1): a necessary condition read off an upper bound.**
  - Step 7.1, bullet "Rate parameter and reading": restated from its fifth sentence on (its first four sentences are those of v5). The two bounds (the bound $2\pi\lvert\delta r\rvert\lambda$ on the effect of a term on $P_{N_i}$, and the $O$-remainder of (11.19)) are named as upper bounds. (13.35) is the hierarchy under which they allow a term to be separated from the remainder; it is called neither necessary nor sufficient for a reading to resolve the term. What a negative statement would need (a lower bound on the remainder) and what a positive one would need (the sensitivity of the reading distribution at the chosen advance) is said. The last sentence of the bullet ("Without (13.35) …") is restated.
  - Step 7.2, bullet "What $\varrho_i$ is a statement on": the sentence before its sub-bullets and the sub-bullet "Two pairs without gradients" ("can be resolved only under", "requires").
  - Result, item 4(a); table, third row; Open issues, fifth item; Setup, bullet "Two notions of a rate"; Methods used, last keyword; twin: (13.17), (13.35), the definition entry of (13.35), the fifth open issue.
  - Every other sentence that states a necessary condition was read for the argument that carries it.
    - Changed: Step 1.2 ("for this start of the field only" claimed more than is derived); Step 1.4 (see I2); Step 6.5, bullet "Comparison" ("would need a binding that depends on the levels": the argument, the strict monotonicity of $u$, is named, and "a different value for each pair of levels" is corrected to pairs that differ in the level of one part); Step 7.1, third bullet (the lower bound $s\gtrsim1/(2\delta P)$: its argument is named); Result, item 2(b) ("the vanishing case needs $g_1=g_2$": $g_1=g_2$ is part of a sufficient condition).
    - Read and left unchanged, because the argument stands in the step: Setup (the two sets of data coincide iff $M^{(i)}_{k_i}=\bar M_i$: algebra); Step 2.3 ($\mathcal D_\eta$ independent of $\eta$ iff $\mathcal K=0$: linear independence); Step 5.3 (i) (the phase construction), (N) (that construction, analyticity in $\lambda$, the vanishing mean), (13.32) (disjoint carriers), (13.33) (the recursion); Step 6.5, claim (mixed second difference); Step 7.3 (the member $\varrho_i=0$: linearity). In the Result and in the twin the argument of (N) is now named.
- **Kind 2 (I2, I3): a summary sentence that has lost a condition, an order or an exceptional member of its step.**
  - I2: Step 1.4 is restated with the criterion in both directions, the scalar and the reducible case, and the statement that a non-zero binding of $\mathcal W$ is not reducible. Result, item 1(a); twin (13.1); Methods used, one keyword added.
  - I3: Step 7.3, bullet "Equal to $\tau_i^2$": "at leading order in $\mathsf R$" is stated, and the new display (13.36) separates the matching of the fall from the matching of the clock relation beyond leading order. Result, item 4(b); table, third row; twin (13.19), (13.23) and the new entry (13.36).
  - Places named by the main model.
    - The member (13.23) now carries "at leading order in $\mathsf R$" and its exact form: Result, item 3(b); table, first row.
    - "(13.19), of order $\epsilon^2$" now carries the condition $g_1M_1+g_2M_2\neq0$ and the order $\epsilon^4$ on that member: Result, item 4(b); table, third row.
    - The relation (13.20) names the member $g_{\rm c}=0$: Result, item 4(c); Step 7.4, one insertion.
    - The statement on the clock levels is a statement on the leading coefficient, with its consequence for the falls: Result, item 3(c); table, second row; Step 6.5, bullets "Hence" and "Comparison".
- **Read-through of the Result, the table and the twin.** Every sentence was read against the step and the display it cites; each now carries its class, its order and its exceptional members. Changed in this pass, besides the places above:
  - Result, head: "Every statement is at first order …" (item 1 is exact).
  - Item 1: (a) "exactly, for all $\lambda$", "for this start", A10 for the position-dependent term; (b) "exactly", "on the levels", "as the coefficient read from $C_{\rm bod}$", what the argument uses.
  - Item 2: each of (a)–(d) names the class and the order. (a) the statement on the rests. (b) the conditions of the vanishing case and "to all orders in the gradients". (c) the class of sequences, "of both parts", the bounded rest, "a fixed $\mathcal P_0$" and "for both parts" in the second determination, the arguments of (i) and of (N), the exceptional members of the last sub-bullet. (d) "do not enter $\mathcal M_P$".
  - Item 3: the paragraph of v5 is split into "At rest" (new in the Result; Step 6.1) and (a)–(d). The three cases of (b), the leading dependence of (c) and item (d) are now stated in the Result (in v5 in Step 6 only).
  - Item 4: split into (a)–(c). (a) the orders of (13.17), (13.18), (13.21). (b) each of the three answers with its order and its members. (c) the class and both signs.
  - Table: all cells of the columns "Bound pair" and "Pair agrees with its parts" (order in $\mathsf R$, a fixed $u$ or a fixed binding, exact forms, exceptional members). Second row, last cell: "at no fixed $u$" is corrected to "at no fixed binding" (Step 6.5 fixes the binding, not $u$). First column: "exact in the data at first order in the gradients".
  - Twin: every key result names its class and its order. Changed in content: (13.1), (13.11), (13.13), (13.15), (13.16), (13.17), (13.19), (13.20), (13.23), (13.25), (13.29), (13.35); new: (13.36).
  - Open issues, fourth item: (13.36) is added to the list.
- **Code.** `code/v6/pair_fall_check.py` is the script of v5 with the paths changed; no table is added, because no statement that is new in v6 rests on a number. All numbers of v5 are reproduced. Changed in the section "Code": the bullets "File" and "Run"; the path of the script in Steps 4.4 and 5.3.

## Response to verification

- **I1: Accepted and fixed in Steps 7.1 and 7.2** (with Result, item 4(a); table, third row; Open issues, fifth item).
  - What the bounds are. $\lvert\partial_\vartheta P_N\rvert<2\pi$ bounds the effect of a term $\delta r$ of the rate parameter on the distribution from above, and the $O$-terms of (11.19) bound the remainder from above. Neither bounds anything from below.
  - What (13.35) is now. The hierarchy under which the two bounds allow a term to be separated from the remainder (the bound of its effect exceeds the bound of the remainder): a conservative hierarchy for the control of the remainder, as the issue proposes. It is no longer called necessary; "only under", "only if" and "resolved … only" are removed in Steps 7.1 and 7.2, the Result, the Open issues and the twin.
  - Both directions of the issue are stated. Where (13.35) fails, it does not follow that a reading cannot resolve the term: the remainder can be smaller than its bound, and a negative statement would need a lower bound on it. Where (13.35) holds, a positive statement needs in addition the sensitivity of the reading distribution at the chosen advance: a lower bound $\varkappa$ on $\lvert\partial_\vartheta P_{N_i}\rvert$ between the two advances, with which the term is separated as soon as $\varkappa\,\lvert\delta r\rvert\lambda$ exceeds the bound of the remainder; at an integer advance the sensitivity vanishes. No positive and no negative statement on resolution is made.
  - The distinction between the rate parameter and a rate inferred from readings is kept.
- **I2: Accepted and fixed in Step 1.4** (with Result, item 1(a)). The obstruction is stated with its conditions, in both directions: a term $U'\otimes\Phi'$ is not a sum of pair terms iff $\Phi'$ is not a multiple of $\mathbb 1_f$ and $U'$ is not of the form $u_1\otimes\mathbb 1+\mathbb 1\otimes u_2$. The two cases of the issue are named: the scalar case ($\Phi'=c\,\mathbb 1_f$: the pair term $c\,U'$) and the reducible case (two pair terms of a body and the field). Added: a non-zero binding of the class $\mathcal W$ is not reducible on a chain with $L>R$, so for the bindings of the question the obstruction depends on the field factor alone.
- **I3: Accepted and fixed in Step 7.3** (with Result, item 4(b), and the table, third row). "$\mathsf w_i^2$ equals $\tau_i^2$ only on the member (13.23)" is now a statement at leading order in $\mathsf R$, in the step, the Result, the table and the twin. Beyond leading order the two conditions are kept apart in the new display (13.36): $\mathsf w_i^2=\tau_i^2$ is $a_{\rm pair}=-\tau_i^2\varrho_i$ with $\varrho_i$ of (13.21), as the issue proposes, and the matching of the fall is $a_{\rm pair}=-\tau_i^2g_i$. Each holds on one ratio $\mathsf z_i(u)\,(1+O(\epsilon^2))$; the two ratios are the same iff $\varrho_i=g_i$ there, which fails for small $\epsilon$ wherever the coefficient of order $\epsilon^2$ of (13.18) does not vanish on the member (an example is given). The exact member of Step 6.4 is no longer identified with the member of the clock relation.

## Setup and assumptions

- **Setting.** Objects, contract $C$, class $\mathcal W$, records, field, levels, bound level, packets, orders and the regime $\mathsf R$ are those of the question, used literally. Assumptions of the package (M8, A11): two Model D bodies, one medium with one operator and one strength, a binding of the class $\mathcal W$ without a factor of the field, the field coupled to the rest and clock terms only, the field start $\varphi$ sharp and stationary, and a packet of a bound level as the start (M9). Equal parameters of the two types are never assumed.
- **Inputs, as quoted in the question:** (11.1), (11.2), (11.5)–(11.9), (11.11)–(11.14), (11.16), (11.19)–(11.22) of 11-bound-pair; (8.10), (8.11) of 08-two-types; (2.1), (2.3) of 02-field-object; (4.7), (4.10), (4.11), (4.16), (4.24) of 04-branch-fall; (5.1), (5.5), (5.6), (5.11) of 04-ilang-time/05-moving-clock; (17.15) of 03-ilang-space/17-motion; (16.11) of 03-ilang-space/16-common-space. With (11.14) the question quotes the statement of Step 8 of 11-bound-pair on the differentiation of (11.12) in the data; with (11.11) the statement of its Step 7 (the reduction to $H_{\rm eff}$ with a remainder in norm); and three statements of its Step 10 on the dependence of the level on the rest costs. The statement of Step 7 and the statements of Step 10 are used in Step 7.2; the statement of Step 8 is used in Step 5.3.
- **Notation added here.**
  - The packet weight is written $g(P)$ and enters only through $g$, $g'$ and $\langle F\rangle_g:=\int\mathrm dP\,\lvert g(P)\rvert^2F(P)$. The common gradient ($g$ in the question) is written $g_{\rm c}:=g_1=g_2$.
  - $\mathfrak h:=\ell^2(\mathbb Z)\otimes\mathbb C^2\otimes\mathbb C^2$ is the fibre space; $\lvert P;\psi\rangle(x_1,x_2)=(2\pi)^{-1/2}e^{iP(\eta x_1+(1-\eta)x_2)}\psi(x_1-x_2)$, so that $\langle P;\psi\vert P';\psi'\rangle=\delta(P-P')\langle\psi\vert\psi'\rangle$ and $\lVert\Phi_g\rVert=1$.
  - $\Gamma(P)>0$ is the distance of $E_b(P)$ from the rest of the spectrum of $H(P)$; $R_P:=Q_P(H(P)-E_b(P))^{-1}Q_P$ with $\lVert R_P\rVert=1/\Gamma(P)$; $\mathcal U_P(\lambda):=\mathbb 1-e^{-i(H(P)-E_b(P))\lambda}$; $I_g$ as in (11.8); $\Gamma_g:=\min_{I_g}\Gamma>0$.
  - $\mathsf b:=\lvert g_1\rvert M_1+\lvert g_2\rvert M_2+2\kappa\lvert\langle X\rangle\rvert$, $\mathsf d_\eta:=\lvert1-\eta\rvert\lvert g_1\rvert M_1+\lvert\eta\rvert\lvert g_2\rvert M_2+\lvert1-2\eta\rvert\kappa\lvert\langle X\rangle\rvert$, $\tau_{\min}:=\min(\tau_1,\tau_2)$.
  - Two sets of data (Step 7). The **level data**: both parts on their levels, $(M_1,M_2)=(M^{(1)}_{k_1},M^{(2)}_{k_2})$; $a_i$, $a$, $u$, $\Lambda$, $\mathfrak m_i$, $\tau_{\rm p}$ without a label are taken with them, as in (13.13)–(13.16), (13.19), (13.20), (13.23), (13.34). The **clock data of the part $i$**: the mean rest cost $\bar M_i$ for the part $i$, whose clock is read, and the level $M_j=M^{(j)}_{k_j}$ for the other part $j\neq i$; quantities taken with them carry the label $[i]$: $a_{i[i]}:=\tau_i^2/(4\bar M_i)$, $a_{[i]}:=a_{i[i]}+a_j$, $u_{[i]}:=\lvert w\rvert/a_{[i]}$, $\Lambda_{[i]}:=1+\varsigma/\sqrt{1+u_{[i]}^2}$. In (11.19), (11.20), (13.17), (13.18), (13.21), (13.22) the symbols $M_i$, $a_i$, $a$, $u$, $\Lambda$ are those of the clock data of the part $i$, as (11.20) states: there $u=u_{[i]}$ and $\Lambda=\Lambda_{[i]}$. The two sets coincide iff $M^{(i)}_{k_i}=\bar M_i$, which is possible only for odd $N_i$ and $k_i=(N_i-1)/2$. Both $u$ and $u_{[i]}$ are fixed in $\mathsf R$, and $u_{[i]}/u=a/a_{[i]}$ does not depend on the binding, so an ordered limit in $u$ is one in $u_{[i]}$ as well.
  - Two notions of a rate (Step 7.1): the **rate parameter** of the level, the number that multiplies $\lambda$ in the advance of (11.19), and the rate that a **reading** of the clock label gives, which (11.19) fixes within its remainder only.
- **Standard facts used.**
  - F1 (first order in a term of the contract): $e^{-i(H_0+\Delta)\lambda}=e^{-iH_0\lambda}-i\int_0^\lambda\mathrm ds\,e^{-iH_0(\lambda-s)}\Delta\,e^{-iH_0s}+O(\Delta^2)$ at fixed $\lambda$.
  - F2 (isolated simple eigenvalue of an analytic family): $E_b$ and $\psi_P$ are analytic in $P$ and in the data, $Q_P\psi_P'=-R_PV_\eta\psi_P$, and $\langle\psi_P\vert\psi_P'\rangle\in i\mathbb R$.
  - F3: an eigenvector of $H(P)$ for an eigenvalue outside the bands (11.2) decays exponentially in $\lvert r\rvert$ (finite range of the hopping and of $W$), locally uniformly in $P$; hence $\hat r\psi_P\in\mathfrak h$.
  - F4 (long chain): at fixed $\lambda$ the first-order coefficients on the chain $\lvert x_i\rvert\le L$ converge for $L\to\infty$ to those of the infinite chain, because the hopping has range one and $\Phi_g$, $\hat x_i\Phi_g$ are square summable.
  - F5 (spectral measures): for a bounded self-adjoint $H$ and vectors $\phi$, $\chi$, the map $S\mapsto\langle\phi\vert\mathbb 1_S(H)\vert\chi\rangle$ on the Borel sets is a finite complex measure carried by the spectrum of $H$; two finite complex measures on $\mathbb R$ with the same Fourier transform are equal; the spectrum of a norm-continuous family of bounded self-adjoint operators moves continuously.
- **Approximations and their range.** First order in $(g_1,g_2,\kappa\langle X\rangle)$ at fixed $\lambda$ (the size of the second order is not estimated). Records at leading order in $\kappa$ with the convention of the question: the bodies evolve by $C_{\rm bod}$. Items 3 and 4 at leading order in $\mathsf R$ at fixed $u$; their remainders are not uniform in $u$. Trivial member: for $g_1=g_2=0$ and $\kappa\langle X\rangle=0$ every first-order quantity below is zero; a statement with "iff", "only", "no" or "not at all" names this member where it would be false on it.

## Derivation

### Step 1. Reduced contract (item 1a)

1. Every term of $C$ is the identity on $\mathcal H_f$, or $\theta_i\,\hat x_i\otimes G_i\otimes\Phi$, or $C_f$. With $\Phi\varphi=\gamma\varphi$ and $C_f\varphi=e_f\varphi$, for every $\Psi\in\mathcal H_{b_1}\otimes\mathcal H_{b_2}\otimes\mathcal H_c$ (the computation of (2.3) and (8.10); $U$ is carried along unchanged):

$$
C\,(\Psi\otimes\varphi)=\bigl((C_{\rm red}+e_f)\Psi\bigr)\otimes\varphi,\qquad C_{\rm red}=\sum_{i=1,2}\Bigl[A_i+\sum_xP^{(i)}_x\otimes(1+g_ix)\,G_i+\kappa\,\hat x_i\otimes X\Bigr]+U .
\tag{13.1}
$$

2. Hence $e^{-iC\lambda}(\Psi\otimes\varphi)=e^{-ie_f\lambda}\,(e^{-iC_{\rm red}\lambda}\Psi)\otimes\varphi$ for all $\lambda$: the view of $\{b_1,b_2,c\}$ is pure and evolves by $C_{\rm red}$ (exact; derived for this start of the field, and not claimed for another one).
3. $C_{\rm red}$ is the contract of an admissible description of $b_1,b_2,c$. It is self-adjoint and independent of $\lambda$; $U$ is a pair term of $b_1,b_2$; $\kappa\,\hat x_i\otimes X$ are pair terms of $b_i,c$; $A_i+\sum_xP^{(i)}_x\otimes(1+g_ix)G_i$ acts on one object and is included in a pair term (A5). Its position-dependent rest and clock term is then a property of the type of $b_i$ (A10).
4. A binding with a factor of the field is a term $U'\otimes\Phi'$ with $U'$ self-adjoint on $\mathcal H_{b_1}\otimes\mathcal H_{b_2}$ and $\Phi'$ self-adjoint on $\mathcal H_f$, or a sum of such terms, for example $\sum_{x_1,x_2}\lvert x_1\rangle\langle x_1\rvert\otimes\lvert x_2\rangle\langle x_2\rvert\otimes W(x_1-x_2)\otimes\Phi_{x_1x_2}$. Decompose the operators of each object into $\mathbb C\mathbb 1$ and the traceless ones. A sum of pair terms has no component that is traceless on the three objects $b_1$, $b_2$, $f$ at once (a pair term is the identity on all objects but two), and an operator on $\mathcal H_{b_1}\otimes\mathcal H_{b_2}\otimes\mathcal H_f$ without such a component is a sum of pair terms (each of its other components is the identity on at least one of the three). The component of $U'\otimes\Phi'$ that is traceless on all three is the product of the part of $U'$ that is traceless on both bodies and the traceless part of $\Phi'$. Hence $U'\otimes\Phi'$ is not a sum of pair terms, and so not of the form (1.10), iff $U'\neq u_1\otimes\mathbb 1+\mathbb 1\otimes u_2$ and $\Phi'\notin\mathbb C\mathbb 1$. Under these two conditions, and under them only, a binding with a factor of the field needs a term on three objects, which A5 does not contain. The two cases without the obstruction:
   - Scalar case, $\Phi'=c\,\mathbb 1_f$: the term is $c\,U'$, an admissible pair term of $b_1$, $b_2$. The binding then carries a number, not an operator of the field.
   - Reducible case, $U'=u_1\otimes\mathbb 1+\mathbb 1\otimes u_2$: the term is $u_1\otimes\Phi'+u_2\otimes\Phi'$, two admissible pair terms of a body and the field. It couples each body to the field, as the lapse-type terms of $C$ do, and binds nothing. A non-zero binding of the class $\mathcal W$ is not of this form on a chain with $L>R$: its blocks would be $W(x_1-x_2)=\mu_1(x_1)\otimes\mathbb 1+\mathbb 1\otimes\mu_2(x_2)$; for $\lvert x_1-x_2\rvert>R$ the block is zero, which gives $\mu_1(x_1)=c\,\mathbb 1$ and $\mu_2(x_2)=-c\,\mathbb 1$, with one constant $c$ for all positions (every $x_1$ is farther than $R$ from $x_2=L$ or from $x_2=-L$, and $x_1=0$ from both; likewise for $x_2$); hence $W\equiv0$.

### Step 2. The operator on the two bodies (item 1b)

1. The partial expectation of $C-e_f$ in $\chi\otimes\varphi$ is, by (13.1),

$$
C_{\rm bod}=C^{\rm D}_1+C^{\rm D}_2+U+\sum_{i=1,2}g_i\,\hat x_i\otimes G_i+\kappa\langle X\rangle\,(\hat x_1+\hat x_2).
\tag{13.2}
$$

2. $[C_{\rm bod},D_i]=0$, so the levels are conserved, and $G_i=M_i\sigma^{(i)}_3$ on the levels $k_1,k_2$. With $\hat x_1=\hat X_\eta+(1-\eta)\hat r$ and $\hat x_2=\hat X_\eta-\eta\hat r$ the gradient terms $\Delta:=C_{\rm bod}-C_{\rm pair}$ are

$$
\begin{aligned}
\Delta&=\hat X_\eta\,\mathcal K+\hat r\,\mathcal D_\eta,\qquad \mathcal K:=g_1M_1\sigma^{(1)}_3+g_2M_2\sigma^{(2)}_3+2\kappa\langle X\rangle,\\
\mathcal D_\eta&:=(1-\eta)\,g_1M_1\sigma^{(1)}_3-\eta\,g_2M_2\sigma^{(2)}_3+(1-2\eta)\,\kappa\langle X\rangle .
\end{aligned}
\tag{13.3}
$$

3. Well-definedness (M6). A replacement of the listed terms that leaves $C$ unchanged leaves $\langle\chi\otimes\varphi\vert C\vert\chi\otimes\varphi\rangle$ unchanged. It can change the listed $C_f$ by a multiple of $\mathbb 1$, hence $e_f$: $C_{\rm bod}$ is fixed by $C$, $\chi$, $\varphi$ up to an additive real constant, which drops out of everything below. Let $b(x_1,x_2)$ be the block of $C_{\rm bod}$ that is diagonal in both positions, an operator on the internal labels. For $\lvert x_1-x_2\rvert>R$ it is $(1+g_1x_1)G_1+(1+g_2x_2)G_2+\kappa\langle X\rangle(x_1+x_2)$ up to the constant. $G_1$ and $G_2$ are traceless and act on the labels of one body each, so the three parts are linearly independent: $\kappa\langle X\rangle$ is the normalized trace of $b(x_1+1,x_2)-b(x_1,x_2)$ (both arguments in that region), $g_1G_1$ is its traceless part, and $G_1$ is the traceless part of $b(0,x_2)$ on the labels of $b_1$; likewise for $b_2$.
   - $g_1$, $g_2$, $\kappa\langle X\rangle$: fixed by $C_{\rm bod}$, well defined, independent of $\eta$. They are the coefficients read from $C_{\rm bod}$. The listed numbers $\gamma\theta_i$ and $\kappa\langle\chi\vert X\vert\chi\rangle$ can change under a relisting such as $X\to X-c$, $C^{\rm D}_i\to C^{\rm D}_i+\kappa c\,\hat x_i$. Used: the label $0$ of the two types (A2) and the finite range of $W$.
   - The operator that multiplies the centre label, $\mathcal K$ (on all levels $g_1G_1+g_2G_2+2\kappa\langle X\rangle$): fixed by $C_{\rm bod}$, independent of $\eta$.
   - The operator that multiplies $\hat r$: fixed by $C_{\rm bod}$ once $\eta$ is given. It depends on the convention, $\mathcal D_\eta=\mathcal D_0-\eta\,\mathcal K$, and is independent of $\eta$ iff $\mathcal K=0$, i.e. (as $M_i>0$) iff all three gradients vanish.

### Step 3. First-order velocities from a bound level (item 2a)

Class: every $W\in\mathcal W$, all data, every bound level, every $g\in\mathcal G(\mathcal P_0)$, all gradients. On the levels $k_1,k_2$ write $H_0:=C_{\rm pair}$ and $\Phi_g(\lambda):=e^{-iH_0\lambda}\Phi_g$.

1. $\Delta$ is diagonal in the positions, so $i[C_{\rm bod},\hat x_i]=V_i$ and $\mathrm d\bar m_i/\mathrm d\lambda=\langle V_i\rangle$ in the evolved state, exactly. The first-order velocity $v^{(1)}_i$ is the part of this expectation that is linear in the gradients. By F1,

$$
v^{(1)}_i(\lambda)=2\,\mathrm{Re}\,\bigl\langle\Phi_g(\lambda)\big\vert V_i\big\vert\Phi^{(1)}(\lambda)\bigr\rangle,\qquad
\Phi^{(1)}(\lambda):=-i\int_0^\lambda\mathrm ds\;e^{-iH_0(\lambda-s)}\,\Delta\,\Phi_g(s),
\tag{13.4}
$$

and the same formula with $V_i\to\hat r$ gives the first-order change of $\bar m_1-\bar m_2$. On a finite chain all operators are bounded; the long-chain limit is F4, with (13.5).
2. In the fibres $\Phi_g(s)$ is $g\,e^{-iE_bs}\psi_P$, and $\hat X_\eta$ acts as $i\partial_P$ ($g$ has compact support in $\mathcal P_0$), so

$$
\Delta\,\Phi_g(s)\ \longleftrightarrow\ e^{-iE_b(P)s}\bigl[s\,E_b'\,g\,\mathcal K\psi_P+\xi_P\bigr],\qquad
\xi_P:=i\,g'\,\mathcal K\psi_P+i\,g\,\mathcal K\psi_P'+g\,\hat r\,\mathcal D_\eta\psi_P ,
\tag{13.5}
$$

a vector of $\mathfrak h$ with $\int\mathrm dP\,\lVert\xi_P\rVert^2<\infty$ (F2, F3, $g\in C^1$ with compact support).
3. Split (13.5) along $\psi_P$ and in $Q_P\mathfrak h$. On $\psi_P$ the propagator is $e^{-iE_b(\lambda-s)}$. On $Q_P\mathfrak h$, $\int_0^\lambda\mathrm ds\,e^{-i(H-E_b)(\lambda-s)}=-iR_P\,\mathcal U_P(\lambda)$ and $\int_0^\lambda\mathrm ds\,s\,e^{-i(H-E_b)(\lambda-s)}=-i\lambda R_P+R_P^2\,\mathcal U_P(\lambda)$.
4. Part along $\psi_P$. $\langle V_i\rangle_P=E_b'$ by (11.7), and $\langle\mathcal K\rangle_P$, $\langle\hat r\mathcal D_\eta\rangle_P$ are real. The terms with $\lambda^2$ and with $\mathcal D_\eta$ are imaginary and drop out. With $2\,\mathrm{Re}\langle\psi_P\vert\mathcal K\vert\psi_P'\rangle=\partial_P\langle\mathcal K\rangle_P$ the rest is $\lambda\int\mathrm dP\,E_b'\,\partial_P\bigl(\lvert g\rvert^2\langle\mathcal K\rangle_P\bigr)=-\lambda\,\langle E_b''\langle\mathcal K\rangle\rangle_g$ (partial integration).
5. Part in $Q_P\mathfrak h$. The term $-i\lambda R_P$ gives $-\lambda\int\mathrm dP\,\lvert g\rvert^2E_b'\,2\,\mathrm{Re}\langle V_i\psi_P\vert R_P\vert\mathcal K\psi_P\rangle$. Since $V_1-V_2=i[H(P),\hat r]$ and $(H-E_b)\psi_P=0$,

$$
\begin{aligned}
R_PV_1\psi_P&=R_PV_\eta\psi_P+i(1-\eta)\,Q_P\hat r\psi_P,\qquad R_PV_2\psi_P=R_PV_\eta\psi_P-i\eta\,Q_P\hat r\psi_P,\\
2\,\mathrm{Re}\,\langle V_i\psi_P\vert R_P\vert\mathcal K\psi_P\rangle&=-\partial_P\langle\mathcal K\rangle_P\qquad(i=1,2).
\end{aligned}
\tag{13.6}
$$

The second line holds because $\partial_P\langle\mathcal K\rangle_P=-2\,\mathrm{Re}\langle\mathcal K\psi_P\vert R_P\vert V_\eta\psi_P\rangle$ (F2) and $\langle Q_P\hat r\psi_P\vert\mathcal K\psi_P\rangle=\langle\hat r\mathcal K\rangle_P-\langle\hat r\rangle_P\langle\mathcal K\rangle_P$ is real ($[\hat r,\mathcal K]=0$).
6. Result, for $i=1,2$ and every $\lambda$:

$$
\boxed{\;v^{(1)}_i(\lambda)=\lambda\,\langle\alpha\rangle_g+\rho_i(\lambda)\;},\qquad \alpha(P):=-E_b''(P)\,\langle\mathcal K\rangle_P+E_b'(P)\,\partial_P\langle\mathcal K\rangle_P ,
\tag{13.7}
$$

$$
\alpha=g_1\alpha_1+g_2\alpha_2+\kappa\langle X\rangle\,\alpha_X,\qquad
\alpha_i(P)=-M_i\bigl[E_b''\,\langle\sigma^{(i)}_3\rangle_P-E_b'\,\partial_P\langle\sigma^{(i)}_3\rangle_P\bigr],\qquad \alpha_X(P)=-2E_b''(P),
\tag{13.8}
$$

$$
\begin{aligned}
\rho_i(\lambda)&=-2\,\mathrm{Re}\int\mathrm dP\,\Bigl[i\lvert g\rvert^2E_b'\,\langle V_i\psi_P\vert R_P^2\,\mathcal U_P(\lambda)\vert\mathcal K\psi_P\rangle+g^{*}\langle V_i\psi_P\vert R_P\,\mathcal U_P(\lambda)\vert\xi_P\rangle\Bigr],\\
\lvert\rho_i(\lambda)\rvert&\le\tau_i\,\mathsf B_g,\qquad \mathsf B_g:=\frac4{\Gamma_g}\Bigl[\mathsf b\Bigl(\frac{\tau_{\min}}{\Gamma_g}+\lVert g'\rVert_2+\max_{I_g}\lVert\psi_P'\rVert\Bigr)+\mathsf d_\eta\max_{I_g}\lVert\hat r\psi_P\rVert\Bigr].
\end{aligned}
\tag{13.9}
$$

The bound uses $\lVert V_i\rVert\le\tau_i$, $\lvert E_b'\rvert\le\tau_{\min}$, $\lVert\mathcal U_P\rVert\le2$, $\lVert\mathcal K\rVert\le\mathsf b$, $\lVert\mathcal D_\eta\rVert\le\mathsf d_\eta$ and $\int\lvert g\rvert\lvert g'\rvert\le\lVert g'\rVert_2$.
7. Status. Both parts have the same coefficient $\langle\alpha\rangle_g$ of the growing part. The rests of the two parts are not equal on the class, and $\rho_i(0)=0$: they differ in the runs of the numerical sets S2–S5 and of S1 with $g_1$ only (section "Code"; at values of $\lambda$ within the range of each run, where the first-order separation changes), and they coincide under the conditions of Step 4.3, where $\bar m_1-\bar m_2=0$ (set S1 with the common or the plain gradient). The rest is bounded for all $\lambda$, in particular for all $\lambda\ge0$, under the conditions of the class alone: the level is isolated on the compact $I_g\subset\mathcal P_0$ ($\Gamma_g>0$) and $g\in\mathcal G(\mathcal P_0)$ ($\lVert g'\rVert_2<\infty$). $\rho_i$ does not depend on $\eta$ or on the phase of $\psi_P$; the single terms of the bound do. By (11.5) and F2, $\langle\sigma^{(i)}_3\rangle_P=\partial E_b/\partial M_i$ and $\partial_P\langle\sigma^{(i)}_3\rangle_P=\partial E_b'/\partial M_i$. Numerical check: section "Code".

### Step 4. Mean label separation (item 2b)

1. The computation of Step 3 with $V_i\to\hat r$ ($\langle\hat r\rangle_P$ is real; (13.6) is not used) gives, for every member of the class of item 2,

$$
\begin{aligned}
(\bar m_1-\bar m_2)^{(1)}(\lambda)&=\lambda\,\langle\beta\rangle_g+\sigma(\lambda),\qquad \lvert\sigma(\lambda)\rvert\le\max_{I_g}\lVert\hat r\psi_P\rVert\;\mathsf B_g,\\
\beta(P)&:=-\langle\mathcal K\rangle_P\,\partial_P\langle\hat r\rangle_P-2E_b'(P)\,\mathrm{Re}\,\langle\hat r\psi_P\vert R_P\vert\mathcal K\psi_P\rangle ,
\end{aligned}
\tag{13.10}
$$

where $\sigma$ is the first line of (13.9) with $V_i\psi_P\to\hat r\psi_P$. There is no term $\propto\lambda^2$: the first-order relative velocity $\rho_1-\rho_2=\langle\beta\rangle_g+\dot\sigma$ has no growing part. This holds alike for the common gradient and for $g_1\neq g_2$.
2. Bounded case (sufficient condition). Let $W(-r)^{*}=W(r)$ for all $r$ (complex conjugate of the matrix in the branch basis); every diagonal on-site binding has this property. The antiunitary $(\Theta\psi)(r):=\psi(-r)^{*}$ then commutes with $H(P)$ for every $P$ and $\eta$ ($\Theta S\Theta^{-1}=S^\dagger$ in (11.1)) and with $\mathcal K$, and $\Theta\hat r\Theta^{-1}=-\hat r$. For a simple level $\Theta\psi_P=e^{i\phi}\psi_P$, hence $\langle\hat r\rangle_P=0$ for all $P$ and $\langle\hat r\psi_P\vert R_P\vert\mathcal K\psi_P\rangle\in i\mathbb R$. So $\beta\equiv0$, and the first-order change of the separation is $\sigma(\lambda)$: bounded for all $\lambda\ge0$, for all values of the gradients.
3. Vanishing case (sufficient condition). Let $\tau_1=\tau_2$, $M_1=M_2$, $W(-r)=\mathsf S\,W(r)\,\mathsf S$ with $\mathsf S$ the exchange of the two branch labels, and $g_1=g_2$ (any $\kappa\langle X\rangle$). The exchange $\Sigma$ of the labels of the two bodies then commutes with $C_{\rm bod}$ and with the joint shift, $\Sigma\Phi_g=\pm\Phi_g$ (simple level) and $\Sigma\hat r\Sigma=-\hat r$. Hence $\bar m_1-\bar m_2=0$ for all $\lambda$, to all orders in the gradients.
4. Otherwise. For a binding without the property of 2, $\beta$ need not vanish: for the numerical control (`code/v6/pair_fall_check.py`, set S5) $\langle\beta\rangle_g\neq0$ for each of the three gradients (from the numerical level), so by (13.10) the first-order separation grows linearly in $\lambda$; the run of that set shows the slope over its range $\lambda\le400$ ($\lambda\le600$ for $g_1$ only): the internal structure of the level follows the first-order change $-\lambda\langle\mathcal K\rangle_P$ of the total label. This rests on that numerical control, not on an analytic example. With the property of 2 and without the conditions of 3 the change is bounded for all $\lambda$ (item 2 of this step) and need not vanish: in the runs of the sets S1–S4 (same file; $\lambda\le3000$ for S1–S3, $\lambda\le200$ for S4), except S1 with the common or the plain gradient, it is not zero and stays bounded over the range of the run. Whether it converges for $\lambda\to\infty$ is not derived, and the runs do not show it (section "Code", item 2b).

### Step 5. Drift acceleration (items 2c, 2d)

1. The first-order acceleration is $a^{(1)}_i=\mathrm dv^{(1)}_i/\mathrm d\lambda=\langle\alpha\rangle_g+\dot\rho_i(\lambda)$. Its rest is the derivative of (13.9) under the integral ($H(P)$ is bounded, $R_P\,\partial_\lambda\mathcal U_P=i\,Q_Pe^{-i(H-E_b)\lambda}$, and $\int\mathrm dP\,\lvert g\rvert\,\lVert\xi_P\rVert<\infty$):

$$
\dot\rho_i(\lambda)=2\,\mathrm{Re}\int\mathrm dP\,\Bigl[\lvert g\rvert^2E_b'\,\langle V_i\psi_P\vert R_P\,e^{-i(H(P)-E_b)\lambda}\vert\mathcal K\psi_P\rangle-i\,g^{*}\langle V_i\psi_P\vert Q_P\,e^{-i(H(P)-E_b)\lambda}\vert\xi_P\rangle\Bigr],\qquad
\lvert\dot\rho_i(\lambda)\rvert\le\tfrac12\,\tau_i\,\Gamma_g\,\mathsf B_g .
\tag{13.27}
$$

The bound uses the estimates listed under (13.9), $\lVert e^{-i(H-E_b)\lambda}\rVert=1$ and $\int\lvert g\rvert\,\lVert\xi_P\rVert\le\mathsf b\,(\lVert g'\rVert_2+\max_{I_g}\lVert\psi_P'\rVert)+\mathsf d_\eta\max_{I_g}\lVert\hat r\psi_P\rVert$; it holds for all $\lambda$. So $\dot\rho_i$ is bounded for every packet of the class. It is not shown to be small, and its decay for $\lambda\to\infty$ is neither derived nor excluded; in the numerical sets it is not negligible over the range of $\lambda$ used (section "Code": on half of the range the mean rates deviate from $\langle\alpha\rangle_g$ by up to $1.2\cdot10^{-2}$). Its window average is a difference quotient of the velocity, exactly: $\overline{a^{(1)}_i}=\langle\alpha\rangle_g+[\rho_i(\lambda_1+\Lambda)-\rho_i(\lambda_1)]/\Lambda$, so $\lvert\overline{a^{(1)}_i}-\langle\alpha\rangle_g\rvert\le2\tau_i\mathsf B_g/\Lambda$ for all $\lambda_1\ge0$, $\Lambda>0$.
2. Sharp total label. Class of sequences $\mathcal S(P_0,I)$: packets of $\mathcal G(\mathcal P_0)$ with supports in one compact interval $I\subset\mathcal P_0$, mean $P_0$ and $\delta P\to0$. Taylor's formula with $\int(P-P_0)\lvert g\rvert^2=0$ gives $\lvert\langle\alpha\rangle_g-\alpha(P_0)\rvert\le\frac12\max_I\lvert\alpha''\rvert\,\delta P^2$. Hence, with the window first and the sharp label second, for both parts:

$$
\boxed{\;a^{\rm drift}(P_0):=\lim_{\delta P\to0}\,\lim_{\Lambda\to\infty}\overline{a^{(1)}_i}=g_1\,\alpha_1(P_0)+g_2\,\alpha_2(P_0)+\kappa\langle X\rangle\,\alpha_X(P_0)\;},\qquad \alpha_i=-M_i\Bigl[E_b''\,\frac{\partial E_b}{\partial M_i}-E_b'\,\frac{\partial E_b'}{\partial M_i}\Bigr],\quad\alpha_X=-2E_b'' .
\tag{13.11}
$$

Here $E_b'=\langle V_i\rangle_P$ (11.7), $\partial E_b/\partial M_i=\langle\sigma^{(i)}_3\rangle_P$ (11.5), and $E_b''=-\eta^2\langle A_1\rangle_P-(1-\eta)^2\langle A_2\rangle_P-2\mathcal S_\eta(P)$ (11.9).
3. The other order (item 2(c) as restricted on 2026-10-11). Fix a window $[\lambda_1,\lambda_1+\Lambda]$, $\lambda_1\ge0$, $\Lambda>0$. The class of sequences is $\mathcal S(P_0,I)$ of 2, unchanged: it restricts $\lvert g\rvert^2$ (support in $I$, mean $P_0$, $\delta P\to0$) and leaves the phase of $g$ free. A class that restricts the phase is another class; no statement is made for it. All statements are for one part $i$. The item asks a determination (the other order exists and agrees with the defining order, or it does not exist) for two groups, the members with $g_1=g_2=0$ and the levels of items 3 and 4 at $P_0=0$; for the other members it asks three things, marked (i), (ii), (iii) below.
   - The fixed-window quantity. In (13.9) the packet enters through $\lvert g\rvert^2$ and through $g^{*}g'=\tfrac12(\lvert g\rvert^2)'+i\,\mathsf j_g$ with $\mathsf j_g:=\mathrm{Im}(g^{*}g')$, a continuous function with support in $I$. Put $F_{i,\lambda}(P):=\langle V_i\psi_P\vert R_P\,\mathcal U_P(\lambda)\vert\mathcal K\psi_P\rangle$ and $\mathsf A_{i,\lambda}(P):=-2\,\mathrm{Re}\bigl[iE_b'\langle V_i\psi_P\vert R_P^2\,\mathcal U_P(\lambda)\vert\mathcal K\psi_P\rangle+\langle V_i\psi_P\vert R_P\,\mathcal U_P(\lambda)\vert\,i\mathcal K\psi_P'+\hat r\mathcal D_\eta\psi_P\rangle\bigr]$, both analytic in $P\in\mathcal P_0$ (F2, F3) and independent of $g$. With $\xi_P$ of (13.5) and a partial integration of the term with $(\lvert g\rvert^2)'$, (13.9) is $\rho_i(\lambda)=\langle\mathsf A_{i,\lambda}-\partial_P\mathrm{Im}F_{i,\lambda}\rangle_g+2\int\mathrm dP\,\mathsf j_g\,\mathrm{Re}F_{i,\lambda}$. With the difference quotient of 1, exactly:

$$
\overline{a^{(1)}_i}=\langle\alpha\rangle_g+\frac1\Lambda\Bigl[\langle\mathsf G_i\rangle_g+2\int\mathrm dP\;\mathsf j_g(P)\,\mathsf f_i(P)\Bigr],\qquad
\mathsf f_i(P):=\mathrm{Re}\bigl[F_{i,\lambda_1+\Lambda}(P)-F_{i,\lambda_1}(P)\bigr],\quad
\mathsf G_i(P):=\Bigl[\mathsf A_{i,\lambda}(P)-\partial_P\mathrm{Im}F_{i,\lambda}(P)\Bigr]_{\lambda=\lambda_1}^{\lambda=\lambda_1+\Lambda} .
\tag{13.28}
$$

   $\mathsf f_i$ depends on the level, the gradients and the window; it does not depend on $\eta$ or on the phase of $\psi_P$ ($\mathsf G_i$ and $\mathsf j_g$ do; the bracket of (13.28) does not). The joint shift by $n$ labels, $g\to g\,e^{-inP}$, is the special case $\mathsf j_g\to\mathsf j_g-n\lvert g\rvert^2$.
   - (i) Existence at a fixed window (necessary and sufficient condition, for every member of the class of item 2). For every sequence of the class $\langle\alpha\rangle_g\to\alpha(P_0)$ and $\langle\mathsf G_i\rangle_g\to\mathsf G_i(P_0)$ (continuous functions on $I$; Chebyshev's inequality). The last term of (13.28) is not controlled by $\lvert g\rvert^2$. For a real number $c$ put $\tilde g:=g\,e^{ic\phi}$ with $\phi(P):=\int_{P_0}^P\mathsf f_i$. Then $\lvert\tilde g\rvert^2=\lvert g\rvert^2$, so $\tilde g$ belongs to a sequence of the class whenever $g$ does, and $\mathsf j_{\tilde g}=\mathsf j_g+c\,\mathsf f_i\lvert g\rvert^2$ changes the window average by $\frac{2c}\Lambda\langle\mathsf f_i^2\rangle_g$. $\mathsf f_i$ is analytic on the interval $\mathcal P_0$. If it does not vanish identically, its zeros are isolated and $\langle\mathsf f_i^2\rangle_g>0$ for every packet; with $c=c_n$ chosen along a given sequence, the window averages then diverge, or converge to any prescribed real number. This holds also if $\mathsf f_i(P_0)=0$; then $c_n\to\infty$ (the phase of the packet varies faster and faster). If $\mathsf f_i\equiv0$ on $\mathcal P_0$, the limit is $\alpha(P_0)+\mathsf G_i(P_0)/\Lambda$ for every sequence. Hence: at a fixed window the limit $\delta P\to0$ exists for every sequence of $\mathcal S(P_0,I)$ iff $\mathsf f_i\equiv0$ on $\mathcal P_0$ (equivalently on $I$).
   - (ii) A necessary condition (N) for the other order. "The other order exists on the class" means: for one $\lambda_1$ and all $\Lambda$ beyond some $\Lambda_0$ the limit $\delta P\to0$ exists for every sequence of the class, and the limit $\Lambda\to\infty$ of the result exists. Define $c_i$ by (13.29); it is analytic in $\lambda$ ($H(P)$ is bounded) and in $P$, and $\mathsf f_i(P)=c_i(\lambda_1;P)-c_i(\lambda_1+\Lambda;P)$. If the other order exists, $c_i(\lambda;P)=c_i(\lambda_1;P)$ for all $\lambda\ge\lambda_1+\Lambda_0$, hence for all $\lambda$ (analyticity). The mean of $c_i(\lambda;P)$ over $\lambda\in[0,T]$ tends to zero for $T\to\infty$ (mean ergodic theorem: $H(P)-E_b(P)$ has no kernel on $Q_P\mathfrak h$, simple level), so this constant is zero. So the other order exists on $\mathcal S(P_0,I)$ only if (N): $c_i\equiv0$ on $\mathbb R\times\mathcal P_0$, where

$$
c_i(\lambda;P):=\mathrm{Re}\,\bigl\langle V_i\psi_P\big\vert R_P\,e^{-i(H(P)-E_b(P))\lambda}\big\vert\mathcal K\psi_P\bigr\rangle,\qquad
c_i(0;P)=-\tfrac12\,\partial_P\langle\mathcal K\rangle_P,\qquad
\partial_\lambda^2c_i(0;P)=-\tfrac12\,\bigl\langle\,[V_i,[H(P),\mathcal K]]\,\bigr\rangle_P .
\tag{13.29}
$$

   The value at $\lambda=0$ is (13.6); the second derivative is $-\mathrm{Re}\langle V_i\psi_P\vert(H-E_b)\vert\mathcal K\psi_P\rangle$ with $(H-E_b)\mathcal K\psi_P=[H,\mathcal K]\psi_P$. Conversely, if $c_i\equiv0$ then $\mathsf f_i\equiv0$ for every window, and the limit $\delta P\to0$ exists for every window and every sequence. For a member with $c_i\not\equiv0$ there is a $P_1\in\mathcal P_0$ with $c_i(\cdot\,;P_1)\not\equiv0$; this function of $\lambda$ is analytic and, by its vanishing mean, not constant. So for every $\lambda_1$ the limit $\delta P\to0$ fails to exist on the class for all $\Lambda$ outside a set without accumulation point.
   - (ii) A sufficient condition (S), and the spectral form of (N) and (S). For $P\in\mathcal P_0$ let $\Omega_P:=\operatorname{spec}H(P)\setminus\{E_b(P)\}$, a compact set at the distance $\Gamma(P)$ from $E_b(P)$, and let $\nu_{i,P}$ be the complex measure of (13.31) on the Borel sets $S\subseteq\mathbb R$ (F5; $\mathbb 1_S$ is the indicator function of $S$). It is finite and carried by $\Omega_P-E_b(P)$, hence by $\lvert\omega\rvert\ge\Gamma(P)$, and with $R_P=Q_P(H(P)-E_b)^{-1}Q_P$:

$$
\nu_{i,P}(S):=\bigl\langle V_i\psi_P\big\vert\,Q_P\,\mathbb 1_S\bigl(H(P)-E_b(P)\bigr)\,\big\vert\mathcal K\psi_P\bigr\rangle,\qquad
c_i(\lambda;P)=\mathrm{Re}\int\frac{e^{-i\omega\lambda}}{\omega}\,\mathrm d\nu_{i,P}(\omega),\qquad
F_{i,\lambda}(P)=\int\frac{1-e^{-i\omega\lambda}}{\omega}\,\mathrm d\nu_{i,P}(\omega).
\tag{13.31}
$$

     - (N) at $P$, that is $c_i(\cdot\,;P)\equiv0$, holds iff $\nu_{i,P}(S)=\overline{\nu_{i,P}(-S)}$ for every $S$. Reason: $2c_i(\lambda;P)$ is the Fourier transform of the finite measure $\omega^{-1}\,\mathrm d\bigl[\nu_{i,P}(\omega)-\overline{\nu_{i,P}(-\omega)}\bigr]$ (F5). Equivalently: $\mathrm{Re}\,F_{i,\lambda}(P)=0$ for all $\lambda$, because $\mathrm{Re}\,F_{i,\lambda}=c_i(0;\cdot)-c_i(\lambda;\cdot)$ and the mean of $c_i$ over $\lambda$ vanishes.
     - (S) at $P$ is: $F_{i,\lambda}(P)=0$ for all $\lambda$. It holds iff $\nu_{i,P}=0$: the Fourier transform of $\omega^{-1}\nu_{i,P}$ is then constant, so this measure is a multiple of the point mass at $\omega=0$, where it has no mass.
     - (S) on $\mathcal P_0$ is sufficient for the other order, which then agrees with the defining order. Proof: $\mathsf f_i\equiv0$ for every window, $\mathsf G_i=\mathsf A_{i,\lambda_1+\Lambda}-\mathsf A_{i,\lambda_1}$, and $\lvert\mathsf A_{i,\lambda}(P)\rvert\le\frac{4\tau_i}{\Gamma(P)}\bigl[\mathsf b\,\bigl(\frac{\tau_{\min}}{\Gamma(P)}+\lVert\psi_P'\rVert\bigr)+\mathsf d_\eta\lVert\hat r\psi_P\rVert\bigr]$ for all $\lambda$ (the estimates listed under (13.9)). By (13.28) the limit $\delta P\to0$ is $\alpha(P_0)+[\mathsf A_{i,\lambda_1+\Lambda}(P_0)-\mathsf A_{i,\lambda_1}(P_0)]/\Lambda$ for every sequence, and the limit $\Lambda\to\infty$ of that is $\alpha(P_0)$: the two limits commute.
     - (S) implies (N). A member with (N) and without (S) has $\mathrm{Re}\,F_{i,\lambda}\equiv0$ for all $\lambda$ and $\mathrm{Im}\,F_{i,\lambda}\not\equiv0$ for some $\lambda$.
   - Where (N) and (S) coincide. The level has **no mirror pair at $P$** if no two points $E$, $E'$ of $\Omega_P$ satisfy $E+E'=2E_b(P)$, that is, if the compact sets $\Omega_P-E_b(P)$ and $E_b(P)-\Omega_P$ are disjoint. Then (N) at $P$ implies $\nu_{i,P}=0$: for a Borel set $S\subseteq\Omega_P-E_b(P)$ the set $-S$ does not meet the carrier of $\nu_{i,P}$, so $\nu_{i,P}(S)=\overline{\nu_{i,P}(-S)}=0$. If the level has no mirror pair at one $P_1\in\mathcal P_0$, it has none on a neighbourhood of $P_1$ (the two sets have a positive distance, and $E_b(P)$ and the spectrum of $H(P)$ move continuously, F5). For a member with (N), $F_{i,\lambda}(\cdot)$ then vanishes on that neighbourhood for every $\lambda$, hence on $\mathcal P_0$ (it is analytic in $P$): (S) holds. With the two bullets before:

$$
\text{no mirror pair at one }P_1\in\mathcal P_0:\qquad
\text{the other order exists on }\mathcal S(P_0,I)\iff c_i\equiv0\ \text{on}\ \mathbb R\times\mathcal P_0\iff F_{i,\lambda}\equiv0\ \text{on}\ \mathcal P_0\ \text{for all}\ \lambda ,
\tag{13.32}
$$

   and the other order then agrees with the defining order. For these members the criterion is complete. Among them is every member whose level lies below, or above, the whole rest of the spectrum of $H(P_1)$ at one $P_1\in\mathcal P_0$.
   - The eigenvector form of (S). If $\psi_P$ is an eigenvector of $\mathcal K$ ($Q_P\mathcal K\psi_P=0$), then $\nu_{i,P}=0$. On $\mathcal P_0$ this form holds iff $g_1=g_2=0$: for $(g_1,g_2)\neq(0,0)$, every binding of $\mathcal W$ and all data, $\psi_P$ is an eigenvector of $\mathcal K$ at no $P$ with $0<\lvert P\rvert<\pi$. Proof. $\mathcal K$ is diagonal in the branch states $\lvert s_1s_2\rangle$, and two of them that differ in the label $s_i$ only have eigenvalues that differ by $2g_iM_i$. So an eigenspace of $\mathcal K$ lies in an eigenspace of $\Sigma:=\sigma^{(1)}_3$ if $g_2=0$, of $\Sigma:=\sigma^{(2)}_3$ if $g_1=0$, and of $\Sigma:=\sigma^{(1)}_3\sigma^{(2)}_3$ if $g_1g_2\neq0$; hence $\Sigma\psi_P=s\,\psi_P$ with $s=\pm1$. Let $A_\Sigma$ be $A_1$, $A_2$, $A_1+A_2$ in the three cases: it anticommutes with $\Sigma$, and $H(P)-A_\Sigma-W(\hat r)$ commutes with $\Sigma$. The part of $(H(P)-E_b)\psi_P=0$ in the eigenspace $-s$ of $\Sigma$ is $A_\Sigma\psi_P+\frac12(1-s\Sigma)W(\hat r)\psi_P=0$, that is, by (11.1),

$$
\tfrac i2\bigl[B\,\psi_P(r+1)-B^{\dagger}\psi_P(r-1)\bigr]=-\tfrac12\,(1-s\Sigma)\,W(r)\,\psi_P(r)\quad(r\in\mathbb Z),\qquad
B:=\tau_1e^{i\eta P}\sigma^{(1)}_1,\quad -\tau_2e^{-i(1-\eta)P}\sigma^{(2)}_1,\quad\text{or their sum}
\tag{13.33}
$$

   in the three cases. $B$ is normal, so $\lVert B\phi\rVert=\lVert B^{\dagger}\phi\rVert$, and it is invertible for $0<\lvert P\rvert<\pi$ (the eigenvalues $\pm\tau_1e^{i\eta P}\pm'\tau_2e^{-i(1-\eta)P}$ of the sum vanish only for $\tau_1=\tau_2$ and $e^{iP}=\pm1$). For $\lvert r\rvert>R$ the right side is zero, so $\lVert B\psi_P(r+1)\rVert=\lVert B\psi_P(r-1)\rVert$, and square summability gives $\psi_P(r)=0$ for $\lvert r\rvert\ge R$. Where $\psi_P(r)=0$ the right side is zero as well, so $\psi_P(r)=\psi_P(r+1)=0$ implies $\psi_P(r-1)=0$. Starting from $r=R$ this gives $\psi_P\equiv0$, a contradiction.
   - Bindings that commute with $\mathcal K$. Let $[W(r),\mathcal K]=0$ for all $r$ (every diagonal binding). Then $[H,\mathcal K]=g_1M_1[A_1,\sigma^{(1)}_3]+g_2M_2[A_2,\sigma^{(2)}_3]$. With $\mathsf T_1:=e^{i\eta P}S^\dagger$, $\mathsf T_2:=e^{i(1-\eta)P}S$, so that $A_i=\frac{i\tau_i}2\sigma^{(i)}_1(\mathsf T_i-\mathsf T_i^\dagger)$ and $V_i=-\frac{\tau_i}2\sigma^{(i)}_1(\mathsf T_i+\mathsf T_i^\dagger)$: $[A_i,\sigma^{(i)}_3]=\tau_i\sigma^{(i)}_2(\mathsf T_i-\mathsf T_i^\dagger)$, $[V_i,[A_i,\sigma^{(i)}_3]]=-i\tau_i^2\sigma^{(i)}_3(\mathsf T_i^2-\mathsf T_i^{\dagger2})$ and $[V_i,[A_j,\sigma^{(j)}_3]]=0$ for $j\neq i$. Only the gradient of the part itself enters:

$$
\partial_\lambda^2c_i(0;P)=-\,g_i\,M_i\,\tau_i^2\;\mathrm{Im}\,\bigl\langle\sigma^{(i)}_3\,\mathsf T_i^{\,2}\bigr\rangle_P\quad\bigl([W(r),\mathcal K]=0\bigr),\qquad
\mathrm{Im}\,\bigl\langle\sigma^{(i)}_3\,\mathsf T_i^{\,2}\bigr\rangle_P=-\,\varsigma\,\frac{a_j\,\sin2P}{\sqrt{w^2+j(P)^2}}+O(\epsilon)\quad(\text{level (11.12)},\ j\neq i).
\tag{13.30}
$$

   The second relation uses (11.13) at a fixed $P$ with $j(P)\neq0$: $\langle\mathsf T_1^2\rangle_P=e^{2i\eta P}\sum_r\psi_P(r)^{*}\psi_P(r+2)=e^{i(2\eta P-\alpha)}\,\mathcal N_P^2\,\frac{2\rho}{1-\rho^2}=-\varsigma\,(a_1+a_2e^{2iP})/\sqrt{w^2+j^2}$, likewise $\langle\mathsf T_2^2\rangle_P=-\varsigma\,(a_2+a_1e^{2iP})/\sqrt{w^2+j^2}$, and $\sigma^{(i)}_3=1$ on $\lvert11\rangle$; the remainder is that of (11.13).
   - Determination for the members with $g_1=g_2=0$ (first group of the item): the other order exists on the class and agrees with the defining order. Here $\mathcal K=2\kappa\langle X\rangle\mathbb 1$, so $\psi_P$ is an eigenvector of $\mathcal K$ for every $P\in\mathcal P_0$ and (S) holds. This group contains:
     - the member with vanishing gradients: both orders give zero;
     - every member with $g_1=g_2=0$ and a plain gradient: both orders give $-2\kappa\langle X\rangle E_b''(P_0)$, for every binding of $\mathcal W$, every bound level and every $P_0$, the levels of items 3 and 4 at $P_0=0$ included.
   - Determination for the levels of items 3 and 4 at $P_0=0$ (second group of the item; for small $\epsilon$ at fixed hatted data and a fixed $\mathcal P_0$): the other order exists and agrees iff $g_1=g_2=0$ (previous bullet). For $(g_1,g_2)\neq(0,0)$ and any $\kappa\langle X\rangle$ it does not exist, for both parts: (N) fails, so the defining order exists, the other one does not, and the two limits do not commute. For the part $i$ with $g_i\neq0$: by (13.30), $\partial_\lambda^2c_i(0;P)\neq0$ at a fixed $P\in\mathcal P_0$ with $0<\lvert P\rvert<\pi/2$ (its leading term is of order $\epsilon^0$ in units of $\lvert g_i\rvert M_i\tau_i^2$, its remainder of order $\epsilon$). For the part $i$ with $g_i=0\neq g_j$: $c_i(0;P)=-\frac12g_jM_j\,\partial_P\langle\sigma^{(j)}_3\rangle_P$, and (11.5) with (11.12), differentiated once in $M_j$ at fixed $P$ (the statement quoted with (11.14); also the estimate of Step 7.2), gives $\langle\sigma^{(j)}_3\rangle_P=1-\frac{a_j}{M_j}\bigl[1+\varsigma\,\frac{a_j+a_i\cos2P}{\sqrt{w^2+j(P)^2}}\bigr]+O(\epsilon^4)$. The fraction is strictly increasing in $\cos2P$ for $\lvert P\rvert\le\pi/4$ (its derivative in $\cos2P$ is $a_i(w^2+a_i^2+a_ia_j\cos2P)/(w^2+j^2)^{3/2}>0$). So at a fixed $P\in\mathcal P_0$ with $0<\lvert P\rvert\le\pi/4$ the difference $\langle\sigma^{(j)}_3\rangle_P-\langle\sigma^{(j)}_3\rangle_0$ is of order $\epsilon^2$ and not zero, against a remainder $O(\epsilon^4)$: $\langle\sigma^{(j)}_3\rangle_P$ is not constant on $\mathcal P_0$. Hence, for these levels, the other order exists on the class iff $g_1=g_2=0$. At the point $P_0=0$ itself $\mathsf f_i(0)=0$ for every window ($\mathcal R$ of Step 6.1 commutes with $H(0)$ and with $\mathcal K$, $\mathcal R\psi_0=\pm\psi_0$ and $\mathcal RV_i(0)\mathcal R^{-1}=-V_i(0)$ give $F_{i,\lambda}(0)=0$); this does not help, because $\mathsf f_i\not\equiv0$ on $\mathcal P_0$.
   - The other members ($(g_1,g_2)\neq(0,0)$; levels other than those of items 3 and 4). The answers to (i) and (ii) are the conditions above: $\mathsf f_i\equiv0$ on $\mathcal P_0$ (13.28); (N) with (13.29), named necessary; (S) with (13.31), named sufficient. By (13.33) the eigenvector form of (S) does not hold for any of them. Every member with $c_i\not\equiv0$ violates (N): the defining order exists, the other one does not, and the two limits do not commute. In particular every member for which $\langle\mathcal K\rangle_P$ is not constant on $\mathcal P_0$. Among them:
     - Numerical members, from the numerical bound level of the script (tables "other order" and "second derivative" of `code/v6/pair_fall_check.py`, section "Code"), for the common gradient and for $g_1$ only, the configurations of the tables. Here $\partial_P\langle\mathcal K\rangle_P=g_1M_1\,\partial_P\langle\sigma^{(1)}_3\rangle_P+g_2M_2\,\partial_P\langle\sigma^{(2)}_3\rangle_P$. The sets S1–S3 are levels of the kind of items 3 and 4 at finite $\epsilon$: their numbers illustrate the derived statement of the bullet before and do not replace it.
       - The control set S5 (binding of range 1 without symmetry) at $P=0.2$, the total label of its evolution: $\partial_P\langle\sigma^{(1)}_3\rangle_P=-3.28\cdot10^{-2}$ and $\partial_P\langle\sigma^{(2)}_3\rangle_P=-3.89\cdot10^{-2}$, so $\partial_P\langle\mathcal K\rangle_P=-7.94\cdot10^{-2}\,g_{\rm c}$ for the common gradient and $-3.28\cdot10^{-2}\,g_1$ for $g_1$ only. Directly, for the window $[0,20]$ and the common gradient: $\mathsf f_1(0.2)=1.20\cdot10^{-2}\,g_{\rm c}$ and $\mathsf f_2(0.2)=-6.28\cdot10^{-2}\,g_{\rm c}$.
       - The sets S1–S4 (diagonal on-site bindings; packets at $P_0=0$ with supports $\lvert P\rvert\le0.5$) at $P=0.2$: $\partial_P\langle\sigma^{(1)}_3\rangle_P=-2.10\cdot10^{-3}$, $-1.49\cdot10^{-3}$, $+1.66\cdot10^{-3}$, $-2.40\cdot10^{-2}$, and $\mathsf f_1(0.2)\neq0$, $\mathsf f_2(0.2)\neq0$ for the three windows of the table, while $\lvert\mathsf f_i(0)\rvert\le1.5\cdot10^{-15}$. The set S4 is outside the regime $\mathsf R$; for it the statement is numerical only: $\partial_\lambda^2c_1(0;0.2)=-5.22\cdot10^{-2}\,g_1$.
       - The numbers change by less than $3\cdot10^{-15}$ with the range of the fibre and agree with a symmetric difference in $P$ to $2\cdot10^{-10}$. For S1–S4 the first relation of (13.30) holds to $2\cdot10^{-16}$, and for S1–S3 the second one to a relative $1.9\cdot10^{-3}$ (S1, S2) and $1.8\cdot10^{-2}$ (S3).
   - (iii) A member with (N) and without (S). By (13.32) its level has a mirror pair at every $P\in\mathcal P_0$. The answer to the third question of the item has two parts.
     - Excluded: in the part of the class in which the level has no mirror pair at one $P_1\in\mathcal P_0$. There (N) implies (S), by (13.32).
     - Neither shown to exist nor excluded: in the rest of the class, the levels with a mirror pair at every $P\in\mathcal P_0$. Reason: (N) is then the reflection property $\nu_{i,P}(S)=\overline{\nu_{i,P}(-S)}$ of a non-zero complex spectral measure of $H(P)$, for all $P$ of an interval. It is not reduced here to a condition on the data, and no short argument that decides it was found. The conditions that are reduced to expectation values, $\langle\mathcal K\rangle_P$ constant on $\mathcal P_0$ and a vanishing double commutator (13.29), are two of infinitely many (one for each derivative of $c_i$ at $\lambda=0$); every member examined here violates at least one of them, and they are not shown to exclude every member.
     - No member of the class with $(g_1,g_2)\neq(0,0)$ that satisfies (N) is exhibited, with or without (S); whether the class contains one is not decided.
     - For a member with (N) and without (S) the limit $\delta P\to0$ exists at every window and is $\alpha(P_0)+\mathsf G_i(P_0)/\Lambda$; whether it tends to $\alpha(P_0)$ for $\Lambda\to\infty$ is not derived, because $\partial_P\mathrm{Im}F_{i,\lambda}(P_0)$ contains $\partial_Pe^{-i(H(P)-E_b)\lambda}$, which is bounded here only by a multiple of $\lambda$. A classification of these members is not asked.
   - The evaluation of the other order is not asked (item 2(c) of the question) and is not given.
4. What multiplies the gradients. $E_b''(P_0)$ multiplies all three, but it is the whole coefficient only for $\kappa\langle X\rangle$: $\alpha_X=-2E_b''$, exactly, on the whole class. For $g_i$ it is weighted by $M_i\,\partial E_b/\partial M_i=M_i\langle\sigma^{(i)}_3\rangle_{P_0}$, and for $E_b'(P_0)\neq0$ the term $M_iE_b'\,\partial E_b'/\partial M_i$ enters in addition. Exceptional members:
   - $E_b''(P_0)=0$: $\alpha_X=0$, $\alpha_i=M_iE_b'\,\partial E_b'/\partial M_i$; at $P_0=0$ the inertia $\mathfrak m_b$ is not defined.
   - $E_b'(P_0)=0$: $\alpha_i=-M_iE_b''\langle\sigma^{(i)}_3\rangle_{P_0}$; at $P_0=0$ then, for $E_b''(0)\neq0$, $a^{\rm drift}(0)=-\langle\mathcal K\rangle_0/\mathfrak m_b$. The class does not imply $E_b'(0)=0$: the numerical level of the set S5 has $E_b'(0)=-1.17\cdot10^{-4}$ (section "Code"; a numerical member, not an analytic one). A sufficient condition for $E_b'(0)=0$ is $W(-r)=\sigma^{(1)}_3\sigma^{(2)}_3W(r)\sigma^{(1)}_3\sigma^{(2)}_3$ together with a level that the map $P\to-P$ takes to itself.
   - $\partial E_b'/\partial M_i=\partial_P\langle\sigma^{(i)}_3\rangle_{P_0}=0$: the additional term of $\alpha_i$ vanishes also for $E_b'(P_0)\neq0$.
   - $\langle\sigma^{(i)}_3\rangle_{P_0}=0$: the first term of $\alpha_i$ vanishes.
5. Common gradient (item 2d). With (11.6),

$$
\alpha_1+\alpha_2=-E_b''\,\mathcal M_P+E_b'\,\mathcal M_P',\qquad
\mathcal M_P:=M_1\langle\sigma^{(1)}_3\rangle_P+M_2\langle\sigma^{(2)}_3\rangle_P=E_b(P)-\langle A_1\rangle_P-\langle A_2\rangle_P-\langle U\rangle_P .
\tag{13.12}
$$

Of the five terms of the scaling relation only the two rest terms enter the coefficient of $g_{\rm c}$, through $\mathcal M_P$ and its derivative, next to the factors $E_b''$ and $E_b'$. The motion terms $\langle A_i\rangle_P$ and the binding term $\langle U\rangle_P$, which the field does not multiply, do not enter $\mathcal M_P$; $\mathcal M_P=E_b(P)$ iff $\langle A_1\rangle_P+\langle A_2\rangle_P+\langle U\rangle_P=0$.

### Step 6. Diagonal on-site binding in the regime $\mathsf R$ (item 3)

Class: every diagonal on-site binding with $\hat w_{11}\neq0$, the level (11.12), fixed hatted data, $P_0=0$.

1. At rest. $(\mathcal R\psi)(r):=\sigma^{(1)}_3\sigma^{(2)}_3\psi(-r)$ gives $\mathcal RH(P)\mathcal R^{-1}=H(-P)$ for a diagonal on-site binding, and by (11.12) the level near $\mathcal B_{++}$ is the only one for small $\epsilon$. So $E_b(-P)=E_b(P)$ and $E_b'(0)=0$, exactly. By (11.8), without gradients $\mathrm d\bar m_i/\mathrm d\lambda=\langle E_b'\rangle_g=O(\delta P^2)$ for all $\lambda$ (zero for even $\lvert g\rvert^2$): the pair is at rest in the sharp-total-label limit.
2. Closed form. (13.11) at $P_0=0$ is $a_{\rm pair}:=a^{\rm drift}(0)=-E_b''(0)\bigl[g_1M_1\langle\sigma^{(1)}_3\rangle_0+g_2M_2\langle\sigma^{(2)}_3\rangle_0+2\kappa\langle X\rangle\bigr]$, exactly. With (11.14) and (11.16):

$$
\boxed{\;a_{\rm pair}=\frac{\varsigma}{\sqrt{1+u^2}}\;\frac{g_1M_1(1+r_1)+g_2M_2(1+r_2)+2\kappa\langle X\rangle(1+r_X)}{\mathfrak m_1+\mathfrak m_2}\;},\qquad r_1,r_2,r_X=O(\epsilon^2)\ \text{at fixed }u .
\tag{13.13}
$$

With the curvature inertia $\mathfrak m_b=1/E_b''(0)$, which is defined for small $\epsilon$ ($E_b''(0)\neq0$ by (11.14)), the first line of this item is the exact form $a_{\rm pair}=-\langle\mathcal K\rangle_0/\mathfrak m_b$. At leading order in $\mathsf R$ it reads $a_{\rm pair}=-(g_1M_1+g_2M_2+2\kappa\langle X\rangle)/\mathfrak m_b$: with the exact $\mathfrak m_b$ the coefficient of $\kappa\langle X\rangle$ is exact, and the coefficient of $g_i$ carries the relative remainder $\langle\sigma^{(i)}_3\rangle_0-1=-\frac{a_i}{M_i}\Lambda+O(\epsilon^4)$. Since $\mathfrak m_ia^{(i)}=-(g_iM_i+\kappa\langle X\rangle)$, at leading order $a_{\rm pair}=-\varsigma(1+u^2)^{-1/2}\,\bar a$ with the inertia-weighted mean $\bar a:=(\mathfrak m_1a^{(1)}+\mathfrak m_2a^{(2)})/(\mathfrak m_1+\mathfrak m_2)$ of the free falls. Each coefficient is of order $\epsilon^2$; the remainders are not uniform in $u$. The form with $\bar a$ is a statement on the three coefficients: where their leading terms cancel in the sum, $a_{\rm pair}$ is given by the remainders $r_1,r_2,r_X$ (Steps 6.4 and 6.5).
3. Ordered limits of the leading coefficient (first $\epsilon\to0$ at fixed $u$, then $u$; the limit in $u$ is taken through the binding $\hat w_{11}$, at fixed $\hat\tau_i$, $M_i$ and gradients):

$$
u\to0:\ \ a_{\rm pair}\to-\varsigma\,\bar a\,\bigl[1-\tfrac12u^2+O(u^4)\bigr];\qquad
u\to\infty:\ \ a_{\rm pair}\to-\varsigma\,\frac{\bar a}{u}\,\bigl[1+O(u^{-2})\bigr]\to0 .
\tag{13.14}
$$

For $w_{11}<0$ the wide pair falls with $\bar a$: the sum of the weights $g_iM_i+\kappa\langle X\rangle$ over the sum of the inertias. For $w_{11}>0$ the level lies above the band, $\mathfrak m_b<0$, and the pair accelerates opposite to $\bar a$ at every $u$; by (4.7) its wide form $-\bar a$ is, at leading order, the same mean of the drift accelerations of the free parts at $p=\pi/2$ instead of at rest. A tight pair does not fall at this order: its inertia grows like $u$, its weight does not.
4. Ratios for $\langle X\rangle=0$ (item 3b), where $a^{(i)}=-\tau_i^2g_i$ and $g_i\neq0$:

$$
\frac{a_{\rm pair}}{a^{(i)}}=-\frac{\varsigma}{\sqrt{1+u^2}}\cdot\frac{g_1M_1+g_2M_2}{g_i\,\tau_i^2\,(\mathfrak m_1+\mathfrak m_2)}\,\bigl(1+O(\epsilon^2)\bigr),\qquad
\tau_{\rm p}^2:=\frac{M_1+M_2}{M_1/\tau_1^2+M_2/\tau_2^2}.
\tag{13.15}
$$

The factor $1+O(\epsilon^2)$ holds at a fixed ratio $g_1:g_2$ with $g_1M_1+g_2M_2\neq0$. For $g_1=g_2=0$ neither the pair nor a part falls, and no ratio is defined; the three cases have at least one non-zero gradient.
   - Common gradient, $\tau_1=\tau_2$ (the free parts fall alike): both ratios are $-\varsigma/\sqrt{1+u^2}$. For $w_{11}<0$ this is $<1$ at every fixed $u$. The absolute and the relative order of the difference are to be kept apart. With $\tau=\epsilon\hat\tau$ both falls are of order $\epsilon^2\lvert g_{\rm c}\rvert$ ($a^{(i)}=-\epsilon^2\hat\tau^2g_{\rm c}$, exactly), and by (13.13), at a fixed $u$,

$$
a_{\rm pair}-a^{(i)}=-\Lambda\,a^{(i)}\,\bigl(1+O(\epsilon^2)\bigr)=\Lambda\,\tau^2g_{\rm c}\,\bigl(1+O(\epsilon^2)\bigr),\qquad
\frac{a_{\rm pair}}{a^{(i)}}-1=-\Lambda\,\bigl(1+O(\epsilon^2)\bigr),\qquad
\Lambda=1+\frac{\varsigma}{\sqrt{1+u^2}}\in(0,2).
\tag{13.34}
$$

   The absolute difference is of order $\epsilon^2\lvert g_{\rm c}\rvert$, the order of the falls themselves; the relative difference is of order $\epsilon^0$. The pair and its parts agree trivially through the orders $\epsilon^0$ and $\epsilon^1$, at which both falls vanish, and they differ at the leading non-zero order $\epsilon^2$ at every fixed $u$, because $\Lambda\neq0$. So the pair falls like its parts neither exactly nor at the leading non-zero order $\epsilon^2$ at a fixed $u$, but only in the ordered wide-pair limit, with the leading relative difference $-\tfrac12u^2$ ($\Lambda=\tfrac12u^2+O(u^4)$ for $w_{11}<0$). For $w_{11}>0$: not at all (the ratio is $-(1+u^2)^{-1/2}<0$ at every $u$; it tends to $-1$ in the wide limit, where $\Lambda\to2$).
   - Common gradient, $\tau_1\neq\tau_2$ (the free parts differ, $a^{(1)}/a^{(2)}=\tau_1^2/\tau_2^2$): the ratios are $-\varsigma(1+u^2)^{-1/2}\tau_{\rm p}^2/\tau_i^2$, with $\tau_{\rm p}^2$ strictly between $\tau_1^2$ and $\tau_2^2$. The pair falls like both of its parts at no $u$ and in no limit. In the wide limit ($w_{11}<0$) the ratio differs from $1$ by $M_j(\tau_j^2-\tau_i^2)/(M_i\tau_j^2+M_j\tau_i^2)$, $j\neq i$. Exceptional: for the part with the smaller $\tau_i$ and $w_{11}<0$ the ratio is $1$, at leading order in $\mathsf R$, at the single value $u_*=(\tau_{\rm p}^4/\tau_i^4-1)^{1/2}$; this is the member (13.23) for $g_1=g_2$.
   - $g_1\neq g_2$: at leading order in $\mathsf R$, $a_{\rm pair}=-\varsigma(1+u^2)^{-1/2}\bar a$. If $\tau_1^2g_1=\tau_2^2g_2$ the free parts fall alike and the first case applies, with the first form of (13.34). Otherwise $a^{(1)}\neq a^{(2)}$, and the pair falls like both of its parts at no $u$ and in no limit: in the wide limit ($w_{11}<0$) $a_{\rm pair}-a^{(1)}\to\mathfrak m_2(a^{(2)}-a^{(1)})/(\mathfrak m_1+\mathfrak m_2)$. Exceptional members:
     - $g_1M_1+g_2M_2=0$: no fall at this order. The leading terms of (13.13) cancel, and $a_{\rm pair}=\varsigma(1+u^2)^{-1/2}\,g_1M_1\,(r_1-r_2)/(\mathfrak m_1+\mathfrak m_2)$, of order $\epsilon^4\lvert g_1\rvert$; it need not vanish. Numerical member: the set S2 with $g_2=-0.8\,g_1$ has $a_{\rm pair}=1.77\cdot10^{-5}\,g_1$, against the single terms $\mp1.00\cdot10^{-2}\,g_1$ (from the coefficients of the numerical level at $P_0=0$, section "Code").
     - One part only. By (13.15), for $g_i\neq0$ and $j\neq i$, at leading order in $\mathsf R$,

$$
\frac{a_{\rm pair}}{a^{(i)}}=1\iff\frac{g_j}{g_i}=\mathsf z_i(u):=-\,\frac{M_i+\varsigma\sqrt{1+u^2}\;\tau_i^2\,(\mathfrak m_1+\mathfrak m_2)}{M_j},\qquad \tau_i^2\,(\mathfrak m_1+\mathfrak m_2)=M_i+M_j\,\frac{\tau_i^2}{\tau_j^2}.
\tag{13.23}
$$

   For given data and $u$ this is one value of $g_j/g_i$, for both signs of $w_{11}$. For $w_{11}<0$, $\mathsf z_i(u)>\tau_i^2/\tau_j^2$, and it tends to $\tau_i^2/\tau_j^2$, the first case, for $u\to0$. For $w_{11}>0$, $\mathsf z_i(u)<-(2M_i+M_j\tau_i^2/\tau_j^2)/M_j<0$: the two gradients have opposite signs. On this member $a_{\rm pair}=a^{(i)}\neq a^{(j)}$, because $\mathsf z_i(u)\neq\tau_i^2/\tau_j^2$ for $u>0$: it concerns one part only. The exact ratio $a_{\rm pair}/a^{(i)}$ is affine in $g_j/g_i$ with a non-zero slope, so the exact member is one value as well, $\mathsf z_i(u)\,(1+O(\epsilon^2))$.
5. Clock levels (item 3c). With $u=4\lvert w\rvert\mathfrak m_1\mathfrak m_2/(\mathfrak m_1+\mathfrak m_2)$, (13.13) reads

$$
a_{\rm pair}=\varsigma\,\frac{g_1M_1+g_2M_2+2\kappa\langle X\rangle}{\sqrt{(\mathfrak m_1+\mathfrak m_2)^2+16\,w^2\,\mathfrak m_1^2\mathfrak m_2^2}}+O(\epsilon^2)\,\frac{\lvert g_1\rvert M_1+\lvert g_2\rvert M_2+2\kappa\lvert\langle X\rangle\rvert}{\sqrt{(\mathfrak m_1+\mathfrak m_2)^2+16\,w^2\,\mathfrak m_1^2\mathfrak m_2^2}},\qquad \mathfrak m_i=\frac{M^{(i)}_{k_i}}{\tau_i^2}.
\tag{13.16}
$$

   The remainder is additive: by (13.13) each of the three terms of the numerator carries its own relative remainder $r_1,r_2,r_X=O(\epsilon^2)$. The multiplicative form, the first term times $1+O(\epsilon^2)$, holds on every set of gradients with $\lvert g_1M_1+g_2M_2+2\kappa\langle X\rangle\rvert\ge c\,(\lvert g_1\rvert M_1+\lvert g_2\rvert M_2+2\kappa\lvert\langle X\rangle\rvert)$, $c>0$ independent of $\epsilon$. It does not hold where the leading terms cancel: there $a_{\rm pair}$ is the remainder (Step 6.4, member $g_1M_1+g_2M_2=0$).
   - Claim: at a fixed binding $w_{11}\neq0$ the leading coefficient is the same for all $N_1N_2$ pairs of levels iff $g_1=g_2=0$ and $\kappa\langle X\rangle=0$. Proof: call the numerator $\mathsf N$ and the root $h$. If $\mathsf N=\varsigma c\,h$ on four pairs $\{M_1,M_1'\}\times\{M_2,M_2'\}$ ($N_i\ge2$), take the mixed second difference. It is $0$ for $\mathsf N$ and strictly positive for $h$, because $4h^3\,\partial_{\mathfrak m_1}\partial_{\mathfrak m_2}h=4q\bigl[(\mathfrak m_1+\mathfrak m_2)^2+(1+q)\mathfrak m_1\mathfrak m_2\bigr]>0$ with $q:=16w^2\mathfrak m_1\mathfrak m_2$. So $c=0$, $\mathsf N=0$ on the four pairs, and all three gradients vanish.
   - Hence, for the leading coefficient: for $\langle X\rangle=0$ iff $g_1=g_2=0$ (no fall); for $\langle X\rangle\neq0$ never.
   - Leading dependence (for $\mathsf N\neq0$): $\partial\ln a_{\rm pair}/\partial\ln M_i=g_iM_i/\mathsf N-\mathfrak m_i/(\mathfrak m_1+\mathfrak m_2)-\frac{u^2}{1+u^2}\frac{a_i}{a}$. For the common gradient with $\tau_1=\tau_2=\tau$ and $\langle X\rangle=0$ only the last term remains: $a_{\rm pair}=\varsigma g_{\rm c}\tau^2/\sqrt{1+u^2}$ with $u=\frac{4\lvert w\rvert}{\tau^2}\frac{M_1M_2}{M_1+M_2}$.
   - In the ordered wide-pair limit the leading coefficient $\varsigma\mathsf N/(\mathfrak m_1+\mathfrak m_2)$ is the same for all levels iff $\langle X\rangle=0$ and $\tau_1^2g_1=\tau_2^2g_2$ (two affine functions of $M_1,M_2$ that agree on the four pairs).
   - Comparison: a free part falls alike on all levels iff $\langle X\rangle=0$ (4.16). The pair does not, at any fixed binding $w_{11}\neq0$ and with at least one non-zero gradient, for small $\epsilon$: by the claim two pairs of levels then have different leading coefficients, of order $\epsilon^2$ at fixed hatted data, while the remainder of (13.16) is of order $\epsilon^4$. The reason is that the binding $w$ does not scale with the levels while $a$ does. What is fixed here is the binding: it is the identity on the clock labels, so all pairs of levels have the same $w$, and $u=4\lvert w\rvert\mathfrak m_1\mathfrak m_2/(\mathfrak m_1+\mathfrak m_2)$ depends on the pair of levels: it is strictly increasing in each $\mathfrak m_i$, so two pairs of levels that differ in the level of one part have different values of $u$. "At a fixed $u$" in the other statements of this step refers to one pair of levels; no statement is made for pairs of levels compared at one common value of $u$, which for all pairs of levels would need a binding that depends on the levels, by this monotonicity (out of scope). In the ordered wide-pair limit the leading coefficient is the same for all levels iff $\langle X\rangle=0$ and $\tau_1^2g_1=\tau_2^2g_2$ (previous bullet). Without a fall (all three gradients zero) the fall is the same for all levels trivially, for the pair and for its parts.
6. $\langle X\rangle\neq0$ (item 3d). It adds $-2\kappa\langle X\rangle E_b''(P_0)$ to the drift acceleration, exactly and on the whole class of item 2 (13.8). At $P_0=0$ this is $-2\kappa\langle X\rangle/\mathfrak m_b$, in $\mathsf R$ equal at leading order to $\frac{\varsigma}{\sqrt{1+u^2}}\frac{2\kappa\langle X\rangle}{\mathfrak m_1+\mathfrak m_2}$. For a free part it adds $-\kappa\langle X\rangle/\mathfrak m_i$ (4.16). In both cases the plain gradient enters as the sum of the plain gradients over the curvature inertia, without a factor $\langle\sigma_3\rangle$, and it makes the fall depend on the levels. For the pair it is in addition reduced by $(1+u^2)^{-1/2}$ against the inertia-weighted mean, and reversed for $w_{11}>0$.

### Step 7. Fall and the clocks of the parts (item 4)

Class as in Step 6, with $\langle X\rangle=0$. Two sets of data meet in this step (Setup): $a_{\rm pair}$ is taken with the level data, the rate of the clock of the part $i$ with the clock data of the part $i$.

1. Rate (item 4a). At the label $x_0$ put $n_i:=1+g_ix_0>0$. The operator $(1+g_ix_0)G_i$ is the rest and clock operator of Model D with $m_i\to n_im_i$ and $\lambda_{0,i}\to\lambda_{0,i}/n_i$, hence $M^{(i)}_k\to n_iM^{(i)}_k$ and $\bar M_i\to n_i\bar M_i$. The zeroth order in the gradients is the pair without gradients and with these data. It is invariant under the joint shift, so (11.19) applies to a packet placed at $x_0$: the reading of the clock label of the part $i$ is $P_{N_i}(a;\vartheta_i)+\mathcal R_i$ with the advance $\vartheta_i=\Gamma_i(x_0)\,n_i\lambda/\lambda_{0,i}$. The rate is $\mathrm d\vartheta_i/\mathrm d\lambda$. With (11.20):

$$
\begin{aligned}
r_i(x_0)&=\frac{1+g_ix_0}{\lambda_{0,i}}\;\Gamma_i(x_0),\qquad \Gamma_i(x_0)=1-\frac{\tau_i^2}{4\,n_i^2\bar M_i^2}\,\Lambda(x_0)+O(\epsilon^4),\\
\Lambda(x_0)&=1+\frac{\varsigma}{\sqrt{1+u(x_0)^2}},\qquad u(x_0)=\frac{\lvert w\rvert}{a_1/n_1+a_2/n_2}
\end{aligned}
\tag{13.17}
$$

Here $a_1$, $a_2$, $u$, $\Lambda$ are those of the clock data of the part $i$ ($a_i$ with $\bar M_i$ for the part that is read and $M_j$ for the other part): $u(0)=u_{[i]}$ and $\Lambda(0)=\Lambda_{[i]}$, not $u$ and $\Lambda$ of (13.13).
   - Conditions, from (11.19): the clock of the part $i$ starts in $\lvert0\rangle$; the other part is on its level; the positions and branches are in a packet of the bound level of the data at $x_0$ with mean total label $0$ and spread $\delta P$; leading order in $\kappa$.
   - Range of $\lambda$: $\mathcal R_i$ is small for $\lambda\ll\lambda_{0,i}/\delta_i$ (in $\mathsf R$ with $\delta_i\to D_i/M_i$) and $\delta P^2(1+\lambda/\lambda_{0,i})\ll1$.
   - As a statement on the pair in the field, (13.17) is the zeroth order in $(\lvert g_1\rvert+\lvert g_2\rvert)\,s$, where $s\gtrsim1/(2\delta P)$ is the width of the packet in labels (the uncertainty relation between the centre label and the total label). The first-order corrections to the reading are not derived here.
   - Rate parameter and reading. Two things are called a rate, and they are kept apart from here on. The **rate parameter** of the level is the number that multiplies $\lambda$ in the advance of (11.19): $r_i(x_0)=n_i\Gamma_i(x_0)/\lambda_{0,i}$, a function of the data at $x_0$. (13.17) is a statement on it: its first equation is this definition, and the expansion of $\Gamma_i$ is (11.20). A **reading** of the clock label gives the distribution $q_i(a;\lambda)=P_{N_i}(a;r_i\lambda)+\mathcal R_i$, hence the rate only within the remainder $\mathcal R_i$ of (11.19). What can be said on the readings of a term $\delta r$ of the rate parameter rests on two bounds, and both are upper bounds.
     - The effect of the term. Without the term the distribution $P_{N_i}$ would be $P_{N_i}(a;(r_i-\delta r)\lambda)$; put $\Xi_i(a):=P_{N_i}(a;r_i\lambda)-P_{N_i}(a;(r_i-\delta r)\lambda)$. By (5.1), $\lvert\partial_\vartheta P_N(a;\vartheta)\rvert<2\pi$, so two rate parameters that differ by $\delta r$ give distributions $P_{N_i}$ that differ by less than $2\pi\lvert\delta r\rvert\lambda$: $\lvert\Xi_i(a)\rvert<2\pi\lvert\delta r\rvert\lambda$.
     - The remainder. (11.19) gives $\lvert\mathcal R_i\rvert\le\bar{\mathcal R}_i$, where $\bar{\mathcal R}_i$ is the sum of its three $O$-terms with their constants. It gives no bound from below.
     - What the two bounds establish. The reading distribution differs from the distribution without the term by $\Xi_i(a)+\mathcal R_i$. Call the term **separated from the remainder** at $\lambda$ if $\lvert\Xi_i(a)\rvert>\bar{\mathcal R}_i$ for one $a$; the difference is then not zero and has the sign of $\Xi_i(a)$. By the first bound this can hold only where $2\pi\lvert\delta r\rvert\lambda>\bar{\mathcal R}_i$. In the form $\lvert\delta r\rvert\,\lambda\gg\bar{\mathcal R}_i$ this is the condition under which the two bounds allow the term to be separated from the remainder; for the term of order $\epsilon^2$ of (13.17) it is (13.35) below. It is a statement on the two bounds, a conservative hierarchy for the control of the remainder. It is neither a necessary nor a sufficient condition for a reading to resolve the term.
     - Not necessary. Where the condition fails, the bound $\bar{\mathcal R}_i$ does not separate the term from the remainder. That a reading cannot resolve the term does not follow: $\lvert\mathcal R_i\rvert$ can be smaller than its bound. A statement of that kind would need a lower bound on $\lvert\mathcal R_i\rvert$, which (11.19) does not contain.
     - Not sufficient. A positive statement needs in addition the sensitivity of the reading distribution to the rate at the chosen advance, that is a lower bound on $\lvert\Xi_i(a)\rvert$. If $\lvert\partial_\vartheta P_{N_i}(a;\vartheta)\rvert\ge\varkappa>0$ for one $a$ and all $\vartheta$ between $(r_i-\delta r)\lambda$ and $r_i\lambda$, then $\lvert\Xi_i(a)\rvert\ge\varkappa\,\lvert\delta r\rvert\lambda$ (the derivative keeps its sign; mean value theorem), and the term is separated from the remainder as soon as $\varkappa\,\lvert\delta r\rvert\lambda>\bar{\mathcal R}_i$. Where the sensitivity vanishes this fails: at an integer advance $P_N(a;\vartheta)\in\{0,1\}$ for every $a$ by (5.1), all $\partial_\vartheta P_N(a;\vartheta)$ vanish there, and $\Xi_i(a)$ is of second order in $\lvert\delta r\rvert\lambda$ when one of the two advances is an integer. The constants of $\bar{\mathcal R}_i$ and the sensitivity $\varkappa$ at a given advance are not computed here; no positive and no negative statement on the resolution of a term by a reading is made.

   For the term of order $\epsilon^2$ of (13.17), $\delta r=-\frac{\tau_i^2}{4n_i\bar M_i^2}\Lambda(x_0)/\lambda_{0,i}$, the condition $\lvert\delta r\rvert\,\lambda\gg\bar{\mathcal R}_i$ is the hierarchy

$$
\epsilon^2\,\ell\ \gg\ \delta_i^2\,(1+\ell)^2+\delta P^2\,(1+\ell)+\kappa,\qquad \ell:=\frac{\lambda}{\lambda_{0,i}} .
\tag{13.35}
$$

   The right side is $\bar{\mathcal R}_i$, that is $\mathcal R_i$ as (11.19) writes it (each term with the constant of its $O$; the last one is its record term $O(\kappa)$), and $\epsilon^2$ stands for $\tau_i^2\Lambda/(4\bar M_i^2)$ at a fixed $u$. (13.35) amounts to: $\delta_i\ll\epsilon$ (level spread; in $\mathsf R$, $\delta_i=D_i/M_i$), $\delta P\ll\epsilon$ (packet width), and $\ell$ in the range $\max(\kappa,\delta_i^2,\delta P^2)/\epsilon^2\ll\ell\ll\epsilon^2/\delta_i^2$ (record strength and range of $\lambda$), which is not empty iff in addition $\kappa\,\delta_i^2\ll\epsilon^4$; it is to be taken together with the range of $\lambda$ above, in which $\mathcal R_i$ is small. The conditions of the two bullets "Conditions" and "Range of $\lambda$" ($\mathcal R_i$ small) do not imply (13.35). Where (13.35) fails, the term of order $\epsilon^2$ of (13.17) is established here as a statement on the rate parameter, and nothing is claimed on its readings. Where (13.35) holds, the two bounds allow its separation from the remainder, and the separation follows under the condition on the sensitivity stated above.
2. Relative change per label, to first order in the gradients ($\varrho_i=\partial_{x_0}\ln r_i$ at $x_0=0$):

$$
\varrho_i=g_i+\frac{\tau_i^2}{4\bar M_i^2}\Bigl[2\Lambda\,g_i+\frac{\varsigma\,u^2}{(1+u^2)^{3/2}}\,\frac{g_1a_1+g_2a_2}{a}\Bigr]+O(\epsilon^4)\,(\lvert g_1\rvert+\lvert g_2\rvert) .
\tag{13.18}
$$

In (13.18), as in (13.17), $a_1$, $a_2$, $a$, $u$, $\Lambda$ are those of the clock data of the part $i$ and carry the label $[i]$ wherever they meet the level data. At leading order $\varrho_i=g_i$, as for the free part. The term of order $\epsilon^2$ contains the gradient of the other part. Both are derived as follows.
   - Derivative used. $\Gamma_i=\partial E_b(0)/\partial M_i$, exactly, as an identity for the number $\Gamma_i$ of (11.19), by (11.5); and at the label $x_0$ the data are $n_1M_1$, $n_2M_2$ with $\tau_1$, $\tau_2$ and $W$ unchanged (item 1). The rate parameter is $r_i=n_i\Gamma_i/\lambda_{0,i}$, so its change with $x_0$ is the derivative of a known function of the data, not of an estimate. To first order in the gradients, without the expansion in $\mathsf R$ (with $M_i=\bar M_i$ for the part that is read, as in (13.17)):

$$
\varrho_i=g_i+\frac1{\Gamma_i}\sum_{j=1,2}g_j\,M_j\,\frac{\partial^2E_b(0)}{\partial M_j\,\partial M_i} .
\tag{13.21}
$$

   This uses the second derivatives of the level at $P=0$ in the rest costs $M_1$, $M_2$: one derivative in the data more than (11.14) and (11.20). A derivative of the remainder of (11.20) is thereby reduced to (11.12): that remainder is $\partial/\partial M_i$ of the remainder of (11.12).
   - What $\varrho_i$ is a statement on. $\varrho_i=\partial_{x_0}\ln r_i$ at $x_0=0$ is the relative change per label of the rate parameter of (13.17), which the question defines at zeroth order in the gradients, with the data at $x_0$. (13.21) is exact in the data in this sense only: it is an identity between derivatives of the level at $P=0$ (no expansion in $\mathsf R$; first order in the gradients). It is not an identity for a rate inferred from readings, which (11.19) fixes within $\mathcal R_i$ only. (13.18), (13.22) and (13.24)–(13.26) are statements on the rate parameter as well. What the bounds of item 1 allow for the readings of them:
     - Two pairs without gradients, with the data at the labels $0$ and $\Delta x$ of the chain: (11.19) applies to each, and a term $\delta\varrho$ of $\varrho_i$ changes the difference of the two advances by $\delta\varrho\,\Delta x\,\ell$, to first order in the gradients. By item 1 the bounds allow the term to be separated from the remainders of the two readings where (13.35) holds with $\epsilon^2\to\lvert\delta\varrho\rvert\,\Delta x$: that is $\lvert g_i\rvert\Delta x$ for the leading term $g_i$, and $\epsilon^2(\lvert g_1\rvert+\lvert g_2\rvert)\Delta x$ for the terms of order $\epsilon^2$ of (13.18), for $\nu_i$ of (13.24), for (13.25) and for the terms of order $\epsilon^2$ of (13.26). Since $(\lvert g_1\rvert+\lvert g_2\rvert)\Delta x<4$ on the chain, the second hierarchy implies (13.35) up to that factor. As in item 1, these hierarchies are statements on upper bounds: where one fails, it does not follow that the readings cannot resolve the term, and where it holds, a positive statement needs in addition the sensitivity of the two reading distributions at the chosen advance.
     - One pair in the field, read at two places: the corrections to the reading of first order in the gradients (the gradient across the packet, of relative order $(\lvert g_1\rvert+\lvert g_2\rvert)s$, item 1 of this step) enter at the order of $\varrho_i$ itself and are not derived here. No statement on such readings is made: for them (13.18) and (13.24)–(13.26), their terms of order $\epsilon^2$ included, are statements on the rate parameter only.
   - Regime $\mathsf R$. (11.12) at $P=0$ is $E_b(0)=M_1+M_2+\mathsf e(a)+\mathcal E$ with $\mathsf e(a):=a+\varsigma\sqrt{w^2+a^2}$ and the remainder $\mathcal E=O(\epsilon^4)$. With $\partial a/\partial M_j=-a_j/M_j$, $\mathsf e'=\Lambda$ and $\mathsf e''=\varsigma u^2/\bigl(a(1+u^2)^{3/2}\bigr)$, for $i,j=1,2$:

$$
\frac{\partial^2E_b(0)}{\partial M_j\,\partial M_i}=\delta_{ij}\,\frac{2a_i}{M_i^2}\,\Lambda+\frac{a_i\,a_j}{M_iM_j}\cdot\frac{\varsigma\,u^2}{a\,(1+u^2)^{3/2}}+\frac{\partial^2\mathcal E}{\partial M_j\,\partial M_i},\qquad \frac{\partial^2\mathcal E}{\partial M_j\,\partial M_i}=O(\epsilon^4)\ \text{at fixed }u .
\tag{13.22}
$$

   The two explicit terms are of order $\epsilon^2$.
   - Remainder of (13.22): the uniform estimate. Conditions: fixed hatted data with $\hat w_{11}\neq0$ (a fixed $u\in(0,\infty)$) and $\epsilon$ below a bound that depends on them. Let $M=(M_1,M_2)$, let $M^0$ be the real rest costs, $m_0:=\min(M^0_1,M^0_2)$ and $D_\delta:=\{M\in\mathbb C^2:\lvert M_i-M^0_i\rvert\le\delta\}$ with $\delta\le m_0/4$.
     - (i) In (11.11), $H(P)=H_0+\epsilon H_1+\epsilon^2H_2$ is linear in $M$, and $H_1$, $H_2$, $\Pi$ do not depend on $M$. For $M\in D_\delta$, $H_0$ is diagonal in the branches and its eigenvalue $M_1+M_2$ keeps the distance $\ge\frac32m_0$ from the other three. The reduction of Step 7 of 11-bound-pair (degenerate perturbation theory: the Riesz projector of $H(P)$ on the circle of radius $m_0$ around $M^0_1+M^0_2$, and the transformation back to $\Pi\mathfrak h$) is an expansion in $\epsilon$ that converges in norm for $\epsilon\lVert H_1\rVert+\epsilon^2\lVert H_2\rVert<m_0/4$. Its terms are built from $H_1$, $H_2$, $\Pi$ and the resolvent of $H_0$ on that circle, whose norm is $\le2/m_0$ on $D_\delta$, and its bounds use only these norms. So it holds on $D_\delta$ as for real $M$ (as a similarity; it is unitary for real $M$): $H_{\rm eff}$ is analytic in $M\in D_\delta$, its terms up to order $\epsilon^2$ are those of (11.11) (rational in $M$), its term of order $\epsilon^3$ vanishes for real $M$ by (11.11) and hence on $D_\delta$ (identity theorem), and the rest is analytic with norm $\le c_4\epsilon^4$ on $D_\delta$.
     - (ii) At $P=0$ the term of order $\epsilon^2$ of (11.11) is $\epsilon^2\hat h(M)$ with $\hat h$ independent of $\epsilon$: on the coefficient of $\lvert11\rangle$ it is $(a+w\,\delta_{r,0})\psi(r)-\frac a2[\psi(r+2)+\psi(r-2)]$. For real $M$ its eigenvalue $\mathsf e(a)$ is simple, at the distance $a(\sqrt{1+u^2}-1)$ from the rest $[0,2a]$ of its spectrum. Hence, for $\delta$ small (depending on the hatted data and on $u$, not on $\epsilon$), $\mathsf e(a)$ is a simple eigenvalue of $\epsilon^2\hat h(M)$, analytic on $D_\delta$, and the resolvent of $\epsilon^2\hat h(M)$ on a circle of radius $\gamma\epsilon^2$ around it is bounded by $C/\epsilon^2$ on $D_\delta$ (continuity on the compact $D_\delta$, and scaling).
     - (iii) Standard perturbation theory for an isolated simple eigenvalue of an analytic family, applied to the perturbation of norm $\le c_4\epsilon^4$: for $c_4C\epsilon^2<\frac12$ the level $E_b(0)$ is analytic in $M\in D_\delta$ and $\lvert\mathcal E\rvert\le c\,\epsilon^4$ there, with $c$ independent of $\epsilon$. Cauchy's estimate on the polydisc $D_\delta$ gives $\lvert\partial_{M_j}\partial_{M_i}\mathcal E\rvert\le2c\,\epsilon^4/\delta^2$ at $M^0$.
     - Range. $\delta$, $\gamma$, $C$, $c$ depend on the hatted data and on $u$; they are not bounded here for $u\to0$ (where $\gamma\to0$) or $u\to\infty$. This is the range that the third quoted statement of Step 10 of 11-bound-pair gives for its bounds. At a fixed $P$ with $j(P)\neq0$ the three steps hold with $J$, $J^{*}$ of (11.11) in the place of $a$.
   - Result. Insert (13.22) in (13.21), with $1/\Gamma_i=1+O(\epsilon^2)$ by (11.20) and $a_i/M_i=\tau_i^2/(4\bar M_i^2)$ for the part that is read: this is (13.18). Its leading order, $\varrho_i=g_i+O(\epsilon^2)(\lvert g_1\rvert+\lvert g_2\rvert)$, uses only that the second derivatives (13.22) are of order $\epsilon^2$; its term of order $\epsilon^2$ uses their values, with the remainder $O(\epsilon^4)$. What each display rests on:
     - (13.21): the definition of $\Gamma_i$ in (11.19), with (11.5); exact in the data as an identity for the rate parameter (bullet "What $\varrho_i$ is a statement on"), not for a rate inferred from readings.
     - (13.22), values: (11.12) differentiated twice in $M_1$, $M_2$. For $i=j$ the value is the first quoted statement of Step 10 of 11-bound-pair; the mixed value is obtained here in the same way.
     - (13.22), remainder: the uniform estimate above, that is (11.11) with the statement of its Step 7, extended to complex rest costs by standard analytic perturbation theory. It gives the dependence on $M_i$ on the scale $M_i$ that the second quoted statement of Step 10 states. The statement of Step 8, quoted with (11.14), is not used for (13.22).
     - (13.18): (13.21), (13.22) and (11.20). (13.24), (13.25): the mixed value of (13.22) with its remainder. (13.26): all values of (13.22) with their remainders.
     - Numerical check of (13.22) and of the term of order $\epsilon^2$: section "Code".
3. The number $\mathsf w_i^2$ (item 4b). It is formed with $\varrho_i$, the relative change of the rate parameter; (13.19), (13.20) and (13.24)–(13.26) are statements on it in the sense of item 2 of this step. In this item $u$, $\mathfrak m_i$, $\tau_{\rm p}$ are those of the level data and come from $a_{\rm pair}$ (13.13); $u_{[i]}$, $a_{[i]}$ are those of the clock data of the part $i$ and come from $\varrho_i$ (13.18); $j\neq i$ is the other part. By (13.21), (13.22), $\varrho_i$ is linear in the two gradients:

$$
\varrho_i=g_i\,\bigl(1+O(\epsilon^2)\bigr)+\nu_i\,g_j\,\bigl(1+O(\epsilon^2)\bigr),\qquad \nu_i:=\frac{\tau_i^2}{4\bar M_i^2}\cdot\frac{\varsigma\,u_{[i]}^2}{(1+u_{[i]}^2)^{3/2}}\cdot\frac{a_j}{a_{[i]}} .
\tag{13.24}
$$

$\nu_i$ is the coefficient of the gradient of the other part in (13.18): of order $\epsilon^2$, non-zero, with the sign of $w_{11}$. The two factors $1+O(\epsilon^2)$ depend on the data, not on the gradients. $\mathsf w_i^2$ is defined where $\varrho_i\neq0$:
   - at a fixed ratio $g_1:g_2$ with $g_i\neq0$, for small $\epsilon$: there $\varrho_i=g_i(1+O(\epsilon^2))\neq0$, and (13.19) holds;
   - for $g_i=0\neq g_j$: there $\varrho_i=\nu_i\,g_j\,(1+O(\epsilon^2))\neq0$, and (13.25) holds;
   - not on the member $\varrho_i=0$ with $g_j\neq0$. By (13.24) it is the single ratio $g_i/g_j=-\nu_i\,(1+O(\epsilon^2))$: of order $\epsilon^2$, with the sign opposite to that of $w_{11}$. There $a_{\rm pair}\neq0$ by (13.13), and $\lvert\mathsf w_i^2\rvert\to\infty$ as the ratio is approached;
   - not for $g_1=g_2=0$, where $\varrho_i=0$ and $a_{\rm pair}=0$.

   For $g_i\neq0$ at a fixed ratio $g_1:g_2$:

$$
\mathsf w_i^2:=-\frac{a_{\rm pair}}{\varrho_i}=-\frac{\varsigma}{\sqrt{1+u^2}}\cdot\frac{g_1M_1+g_2M_2}{g_i\,(\mathfrak m_1+\mathfrak m_2)}\,\bigl(1+O(\epsilon^2)\bigr)\qquad\Bigl[\,=-\frac{\varsigma\,\tau_{\rm p}^2}{\sqrt{1+u^2}}\,\bigl(1+O(\epsilon^2)\bigr)\ \text{for } g_1=g_2\Bigr].
\tag{13.19}
$$

   - Range of (13.19). The factor $1+O(\epsilon^2)$ holds uniformly on every set of ratios with $\lvert g_i\rvert\ge c\,\lvert g_j\rvert$ and $\lvert g_1M_1+g_2M_2\rvert\ge c\,(\lvert g_1\rvert M_1+\lvert g_2\rvert M_2)$, with $c>0$ independent of $\epsilon$ ($g_j=0$ included). For smaller $\lvert g_i/g_j\rvert$ the right-hand side of (13.19) is to be multiplied by $(1+\nu_ig_j/g_i)^{-1}$, by (13.24); the factor $1+O(\epsilon^2)$ is then uniform on $\lvert g_i+\nu_ig_j\rvert\ge c\,(\lvert g_i\rvert+\lvert\nu_ig_j\rvert)$. The additional factor is $1+O(\epsilon^2g_j/g_i)$ for $\lvert g_i/g_j\rvert\gg\epsilon^2$, of order one for $g_i/g_j$ of order $\epsilon^2$, and it has its pole on the member $\varrho_i=0$.
   - Equal to $\tau_i^2$: no, except on one member; this is a statement at leading order in $\mathsf R$. At leading order $\varrho_i=g_i$ and $a^{(i)}=-\tau_i^2g_i$, so $\mathsf w_i^2=\tau_i^2$ iff $a_{\rm pair}=a^{(i)}$, i.e. iff $g_j/g_i=\mathsf z_i(u)$ of (13.23): one value of the ratio of the gradients at given data and $u$, for both signs of $w_{11}$. For the common gradient, at leading order, $\mathsf w_i^2/\tau_i^2=-\varsigma(1+u^2)^{-1/2}\tau_{\rm p}^2/\tau_i^2$, which is $\neq1$ except at $u_*$ of Step 6 for one part; for $w_{11}>0$ it is negative. Beyond leading order the matching of the clock relation and the matching of the fall are two conditions (first order in the gradients, no expansion in $\mathsf R$; $\varrho_i$ of (13.21)):

$$
\mathsf w_i^2=\tau_i^2\iff a_{\rm pair}=-\tau_i^2\,\varrho_i\quad(\varrho_i\neq0),\qquad\qquad
a_{\rm pair}=a^{(i)}\iff a_{\rm pair}=-\tau_i^2\,g_i .
\tag{13.36}
$$

   Both conditions are linear in $(g_1,g_2)$. In the second the coefficient of $g_j$ is $-E_b''(0)\,M_j\langle\sigma^{(j)}_3\rangle_0$, of order $\epsilon^2$ and not zero by (11.14); in the first, $\tau_i^2\,\Gamma_i^{-1}M_j\,\partial^2E_b(0)/\partial M_j\partial M_i$ is added to it, of order $\epsilon^4$ by (13.22). So for small $\epsilon$ at fixed hatted data each condition holds on one ratio $g_j/g_i=\mathsf z_i(u)\,(1+O(\epsilon^2))$ (the second is the exact member of Step 6.4), and (13.23) is the leading order of both. The two ratios are the same iff $\varrho_i=g_i$ on that ratio. By (13.18), with its remainder $O(\epsilon^4)$, this fails for small $\epsilon$ whenever $2\Lambda_{[i]}+\varsigma\,u_{[i]}^2(1+u_{[i]}^2)^{-3/2}\,\bigl(a_{i[i]}+\mathsf z_i(u)\,a_j\bigr)/a_{[i]}\neq0$, and (13.23) does not make this number vanish: for $w_{11}<0$ and coinciding level and clock data it is $u^2\,\frac{a_j}{a}\,\bigl(1-\tau_i^2/\tau_j^2\bigr)+O(u^4)$, not zero for $\tau_1\neq\tau_2$ and small $u$. Hence the member on which the pair falls like the part $i$ is not in general the member on which $\mathsf w_i^2=\tau_i^2$: the two agree at leading order in $\mathsf R$ and can differ at relative order $\epsilon^2$.
   - The same for $i=1,2$: where both numbers are defined, $\mathsf w_1^2/\mathsf w_2^2=\varrho_2/\varrho_1$ (exact; both contain the same $a_{\rm pair}$), which is $g_2/g_1$ at leading order at a fixed ratio with $g_1,g_2\neq0$. So: the same at leading order iff $g_1=g_2$; exceptional is the member $a_{\rm pair}=0$ (at leading order $g_1M_1+g_2M_2=0$), where both numbers vanish. For $g_1=g_2\neq0$, (13.18) gives at relative order $\epsilon^2$

$$
\frac{\mathsf w_1^2}{\mathsf w_2^2}=\frac{\varrho_2}{\varrho_1}=1+\frac{\tau_2^2}{4\bar M_2^2}\,\mathsf y(u_{[2]})-\frac{\tau_1^2}{4\bar M_1^2}\,\mathsf y(u_{[1]})+O(\epsilon^4),\qquad \mathsf y(u):=2+\varsigma\,\frac{2+3u^2}{(1+u^2)^{3/2}} .
\tag{13.26}
$$

   $\mathsf y(u)=2\Lambda+\varsigma u^2(1+u^2)^{-3/2}$ is the bracket of (13.18) for $g_1=g_2$, and $\mathsf y(u)>0$ for $u>0$ and both signs, because $(2+3u^2)^2<4(1+u^2)^3$. So the two numbers are the same at relative order $\epsilon^2$ iff $\tau_1^2\,\mathsf y(u_{[1]})/\bar M_1^2=\tau_2^2\,\mathsf y(u_{[2]})/\bar M_2^2$. This reduces to $\tau_1/\bar M_1=\tau_2/\bar M_2$ for the members with $u_{[1]}=u_{[2]}$, i.e. $\tau_1^2\,(1/\bar M_1-1/M_1)=\tau_2^2\,(1/\bar M_2-1/M_2)$, in particular when each part is on the level with $M^{(i)}_{k_i}=\bar M_i$; and in the ordered limits $u\to\infty$ (both signs, $\mathsf y\to2$) and $u\to0$ for $w_{11}>0$ ($\mathsf y\to4$). For $w_{11}<0$ and $u\to0$, $\mathsf y(u)=\tfrac34u^4+O(u^6)$: both terms of order $\epsilon^2$ vanish in the limit.
   - Data of the pair: yes. It contains the binding ($\varsigma$, $u$) and the clock levels (through $u$, and through $\tau_{\rm p}$ if $\tau_1\neq\tau_2$).
   - The case $g_i=0\neq g_j$. The free part $i$ then does not fall and its clock rate does not change, so (4.24) is empty for it. In the pair, (13.13) divided by (13.24) gives

$$
\mathsf w_i^2=-\frac{\varsigma}{\sqrt{1+u^2}}\cdot\frac{M_j}{(\mathfrak m_1+\mathfrak m_2)\,\nu_i}\,\bigl(1+O(\epsilon^2)\bigr)=-\,4\,\frac{\bar M_i^2\,M_j}{M_i}\cdot\frac{u\,(1+u_{[i]}^2)^{3/2}}{u_{[i]}^3\,\sqrt{1+u^2}}\,\bigl(1+O(\epsilon^2)\bigr)\qquad(g_i=0\neq g_j),
\tag{13.25}
$$

   with $M_i=M^{(i)}_{k_i}$, $M_j=M^{(j)}_{k_j}$; for $M^{(i)}_{k_i}=\bar M_i$ it is $-4M_iM_j(1+u^2)/u^2$. The second form uses $M_j/(\mathfrak m_1+\mathfrak m_2)=4a_1a_2M_j/a$ and $a_{[i]}/a=u/u_{[i]}$. It is of order $\epsilon^0$, while (13.19) is of order $\epsilon^2$; its remainder is a relative $O(\epsilon^2)$ at fixed $u$, not uniform in $u$. The three answers for (13.25):
   - $g_i=0\neq g_j$, equal to $\tau_i^2$: no, at every fixed $u$ and in both ordered limits. It is negative for both signs of $w_{11}$, and $\tau_i^2>0$ is of order $\epsilon^2$. For $u\to0$ it grows like $-u^{-2}$; for $u\to\infty$ it tends to $-4\bar M_i^2M_j/M_i$.
   - $g_i=0\neq g_j$, the same for $i=1,2$: no. The other part has $g_j\neq0$ and the number $\mathsf w_j^2$ of (13.19), of order $\epsilon^2$: $\mathsf w_i^2/\mathsf w_j^2=\varrho_j/\varrho_i=\nu_i^{-1}(1+O(\epsilon^2))$, of order $\epsilon^{-2}$, with the sign of $w_{11}$.
   - $g_i=0\neq g_j$, data of the pair: yes. The binding (through $u$, $u_{[i]}$; not its sign at this order), the levels of both parts ($M_i$, $M_j$), and the mean rest cost $\bar M_i$ of the part that is read.
   - Derivatives used: (13.19) is (13.13) divided by (13.18). (13.13) uses (11.14) and (11.16): the first derivatives of (11.12) in $M_i$, and $E_b''(0)$. Of (13.18) only the leading order $\varrho_i=g_i+O(\epsilon^2)(\lvert g_1\rvert+\lvert g_2\rvert)$ enters, hence the second derivatives (13.22) only through their order $\epsilon^2$, not through their values. The factor $1+O(\epsilon^2)$ holds at a fixed ratio $g_1:g_2$ with $g_i\neq0$ and $g_1M_1+g_2M_2\neq0$. For $g_1M_1+g_2M_2=0$ the leading term vanishes (no fall at this order, Step 6.4); then $\mathsf w_i^2=O(\epsilon^4)$ by (13.13), and its value is not determined here. (13.24), (13.25), (13.26) and the place of the member $\varrho_i=0$ use the values of the second derivatives (13.22), with their remainder $O(\epsilon^4)$: the mixed one in (13.24), (13.25), all of them in (13.26).
4. Free parts alike with one speed (item 4c): $g_1=g_2=g_{\rm c}$ and $\tau_1=\tau_2=\tau$, so that $a^{(i)}=-\tau^2g_{\rm c}$ is (4.24) with $v_{\max}=\tau d_0$. Both parts of the pair have the drift acceleration $a_{\rm pair}$, and $\bar x=d_0\,\bar m\,\hat e$ for both (17.15). Hence

$$
\frac{\mathrm d^2\bar x}{\mathrm d\lambda^2}\Big\vert_{\rm drift}=-\,\frac{\mathsf w^2}{\tau^2}\;\frac{v_{\max}^2}{d_0}\;\varrho_i\;\hat e,\qquad \frac{\mathsf w^2}{\tau^2}=-\frac{\varsigma}{\sqrt{1+u^2}}\,\bigl(1+O(\epsilon^2)\bigr).
\tag{13.20}
$$

The pair obeys a relation of the form (4.24) in which $v_{\max}^2$ is replaced by $-\varsigma\,v_{\max}^2/\sqrt{1+u^2}$. With the speed $\tau d_0$ itself it holds, for $g_{\rm c}\neq0$, neither exactly nor at the leading non-zero order at a fixed $u$. Both sides of (4.24) are of order $\epsilon^2\lvert g_{\rm c}\rvert\,d_0$ and agree trivially through the orders $\epsilon^0$ and $\epsilon^1$, at which both vanish. At the order $\epsilon^2$ they differ, by (13.34) and $\varrho_i=g_{\rm c}(1+O(\epsilon^2))$, by $\Lambda\,\tau^2g_{\rm c}\,d_0\,\hat e\,(1+O(\epsilon^2))$: an absolute deviation of order $\epsilon^2$, and a relative one, $\mathsf w^2/\tau^2-1=-\Lambda\,(1+O(\epsilon^2))$, of order $\epsilon^0$ ($\Lambda$ of the level data). For $w_{11}<0$ it holds only in the ordered wide-pair limit, with the leading deviation $\mathsf w^2/\tau^2-1=-\tfrac12u^2+O(u^4)$; for $w_{11}>0$ not at all ($\mathsf w^2/\tau^2=-(1+u^2)^{-1/2}<0$ at every $u$; it tends to $-1$ in the wide limit). (13.20) is (13.19) for $g_1=g_2=g_{\rm c}\neq0$ and $\tau_1=\tau_2$, so it uses the same derivatives: (11.14), (11.16), and of (13.22) only the order $\epsilon^2$. In (13.20) $u$ is that of the level data; the clock data of the part $i$ enter through $\varrho_i$ (13.18) and change only the factor $1+O(\epsilon^2)$. For $g_{\rm c}=0$ both sides of (13.20) vanish, and the relation holds trivially with any speed.

## Result

Items 2–4 are statements at first order in $(g_1,g_2,\kappa\langle X\rangle)$ at fixed $\lambda$, followed by the long chain, with the records at leading order in $\kappa$ (the bodies evolve by $C_{\rm bod}$); item 1 is exact. Each sentence below names its class, its order and its exceptional members again. "Derived" means derived here from the quoted inputs and standard facts; "numerical" means exhibited with the numerical level or the numerical evolution of the script (section "Code"). For vanishing gradients every first-order quantity is zero; a statement with "iff", "only" or "no" names this member where it would be false on it.

- **Item 1** (the contract $C$ of the question; exact, no expansion; derived).
  - (a) Generator. For a field that starts in $\varphi$ with $\Phi\varphi=\gamma\varphi$ and $C_f\varphi=e_f\varphi$, exactly and for all $\lambda$: the two bodies and the medium evolve by $C_{\rm red}$ (13.1), and their view is pure. This is derived for this start of the field and not claimed for another one. $C_{\rm red}$ is the contract of an admissible description of $b_1,b_2,c$ (A5): yes; its position-dependent rest and clock term $(1+g_ix)\,G_i$ is then a property of the type of $b_i$ (A10).
  - (a) Binding with a factor of the field. A term $U'\otimes\Phi'$, with $U'$ on the two bodies and $\Phi'$ on $\mathcal H_f$, is not a sum of pair terms, so that it would need a term on three objects that (1.10) does not contain, iff $\Phi'$ is not a multiple of $\mathbb 1_f$ and $U'$ is not of the form $u_1\otimes\mathbb 1+\mathbb 1\otimes u_2$ (Step 1.4). In the scalar case $\Phi'=c\,\mathbb 1_f$ it is the admissible pair term $c\,U'$ of the two bodies. In the reducible case $U'=u_1\otimes\mathbb 1+\mathbb 1\otimes u_2$ it is the sum of the admissible pair terms $u_i\otimes\Phi'$ of a body and the field. A non-zero binding of the class $\mathcal W$ on a chain with $L>R$ is not reducible; for it the obstruction holds iff the field factor is not a multiple of $\mathbb 1_f$.
  - (b) Operator. With the medium in $\chi$ and the field in $\varphi$, exactly: $C_{\rm bod}$ is (13.2), fixed by $C,\chi,\varphi$ up to an additive constant. On the levels $k_1$, $k_2$ its gradient terms are $\hat X_\eta\,\mathcal K+\hat r\,\mathcal D_\eta$ (13.3).
  - (b) Fixed by $C_{\rm bod}$ and well defined under every replacement of the listed terms that leaves $C$ unchanged: $g_1$ yes; $g_2$ yes; $\kappa\langle X\rangle$ yes, each as the coefficient read from $C_{\rm bod}$ (the listed numbers $\gamma\theta_i$ and $\kappa\langle\chi\vert X\vert\chi\rangle$ can change under a relisting; the argument uses the label $0$ of the two types and the finite range of $W$); the operator $\mathcal K$ at the centre label yes; the operator $\mathcal D_\eta$ at $\hat r$ yes, for given $\eta$. Of the five only $\mathcal D_\eta=\mathcal D_0-\eta\,\mathcal K$ depends on $\eta$, and it does iff at least one gradient is non-zero (for vanishing gradients $\mathcal K=\mathcal D_\eta=0$). The centre label $\hat X_\eta$ itself depends on $\eta$ in every case.
- **Item 2** (class of item 2: every binding $W\in\mathcal W$, all data of the two bodies, every bound level in the sense of the setting, every packet $g\in\mathcal G(\mathcal P_0)$, all values of the three gradients; exact relations at first order in the gradients; derived, except the members named "numerical").
  - (a) For every binding of $\mathcal W$, every bound level and every packet of $\mathcal G(\mathcal P_0)$, at first order in the gradients and for every $\lambda$: $v^{(1)}_i=\lambda\langle\alpha\rangle_g+\rho_i(\lambda)$ (13.7), with $\alpha$ of (13.8) and the same coefficient $\langle\alpha\rangle_g$ for both parts, and $\lvert\rho_i\rvert\le\tau_i\mathsf B_g$ for all $\lambda$ (13.9), under the conditions of the class alone. The rests $\rho_1$, $\rho_2$ are not equal on the class (numerical: the runs of S2–S5 and of S1 with $g_1$ only); they coincide under the sufficient condition of Step 4.3.
  - (b) For every binding of $\mathcal W$, every bound level and every packet of $\mathcal G(\mathcal P_0)$, at first order in the gradients: $(\bar m_1-\bar m_2)^{(1)}=\lambda\langle\beta\rangle_g+\sigma(\lambda)$ with $\sigma$ bounded for all $\lambda$ (13.10); no member has a term $\propto\lambda^2$. The first-order separation stays bounded for all $\lambda\ge0$ and all gradients if $W(-r)^{*}=W(r)$ (a sufficient condition; it holds for all diagonal on-site bindings). The separation is zero for all $\lambda$, to all orders in the gradients, under the sufficient condition of Step 4.3: $\tau_1=\tau_2$, $M_1=M_2$, $W(-r)=\mathsf S\,W(r)\,\mathsf S$ and $g_1=g_2$, with any $\kappa\langle X\rangle$. For a binding without the property $W(-r)^{*}=W(r)$, $\langle\beta\rangle_g\neq0$ for the control set S5 (numerical level; no analytic example), so that the separation grows linearly by (13.10); the run of S5 shows the slope for $\lambda\le400$. The answers are the same for the common gradient and for $g_1\neq g_2$, except that the sufficient condition of the vanishing case contains $g_1=g_2$; that $g_1=g_2$ is necessary for it is not derived (numerical: the set S1, which meets the other conditions, has a non-zero separation with $g_1$ only).
  - (c) For every binding of $\mathcal W$ and every bound level, at first order in the gradients, and for every sequence of the class $\mathcal S(P_0,I)$ (packets of $\mathcal G(\mathcal P_0)$ with supports in one compact $I\subset\mathcal P_0$, mean $P_0$ and $\delta P\to0$; the phase of the packets is free): with the window average first and $\delta P\to0$ second, both limits exist, and the drift acceleration of both parts is $a^{\rm drift}(P_0)=g_1\alpha_1(P_0)+g_2\alpha_2(P_0)+\kappa\langle X\rangle\alpha_X(P_0)$ (13.11). The rest of the first-order acceleration is bounded for all $\lambda$ (13.27); its decay for $\lambda\to\infty$ is neither derived nor excluded. The other order (the sharp total label first), on the same class of sequences and for one part $i$, as the item reads since 2026-10-11:
    - Determination, members with $g_1=g_2=0$ (every binding of $\mathcal W$, every bound level and every $P_0$; any $\kappa\langle X\rangle$): the other order exists and agrees with the defining order (the limits commute); both give $-2\kappa\langle X\rangle E_b''(P_0)$ (derived).
    - Determination, the levels of items 3 and 4 at $P_0=0$ (diagonal on-site binding, level (11.12); small $\epsilon$ at fixed hatted data and a fixed $\mathcal P_0$; any $\kappa\langle X\rangle$): the other order exists and agrees iff $g_1=g_2=0$; for $(g_1,g_2)\neq(0,0)$ it does not exist, for both parts, and the limits do not commute (derived; (13.29), (13.30)).
    - Other members, (i): at a fixed window the limit $\delta P\to0$ exists for every sequence of $\mathcal S(P_0,I)$ iff $\mathsf f_i\equiv0$ on $\mathcal P_0$ (13.28); necessary and sufficient, for every member of the class of item 2. The necessity is carried by a construction: a phase factor on the packets, which the class allows, shifts the window average by any amount unless $\mathsf f_i\equiv0$.
    - Other members, (ii): necessary for the other order is (N), $c_i\equiv0$ on $\mathbb R\times\mathcal P_0$ (13.29), equivalently $\mathrm{Re}\,F_{i,\lambda}\equiv0$ for all $\lambda$; the argument is (i) for all large windows, the analyticity of $c_i$ in $\lambda$ and its vanishing mean. Sufficient is (S), $F_{i,\lambda}\equiv0$ on $\mathcal P_0$ for all $\lambda$ (13.31); the other order then agrees with the defining order. The eigenvector form of (S), $Q_P\mathcal K\psi_P=0$ on $\mathcal P_0$, holds iff $g_1=g_2=0$ (13.33). Every member with $c_i\not\equiv0$ has no other order, in particular whenever $\langle\mathcal K\rangle_P$ is not constant on $\mathcal P_0$: the levels of the sets S1–S5 with the common gradient or with $g_1$ only (numerical).
    - Other members, (iii): a member with (N) and without (S) is excluded wherever the level has no mirror pair at one $P_1\in\mathcal P_0$; there (N), (S) and the existence of the other order are equivalent (13.32). Where the level has a mirror pair at every $P\in\mathcal P_0$, such a member is neither shown to exist nor excluded (Step 5.3, with the reason). No member with $(g_1,g_2)\neq(0,0)$ that satisfies (N) is exhibited. The evaluation of that order is not asked.
    - What multiplies the gradients, for every member of the class of item 2: $E_b''(P_0)$ multiplies all three, and it is the whole coefficient of $\kappa\langle X\rangle$ ($\alpha_X=-2E_b''$, exactly). It is not the only property of the level: $\partial E_b/\partial M_i=\langle\sigma^{(i)}_3\rangle_{P_0}$ weights it in the coefficient of $g_i$, and for $E_b'(P_0)\neq0$ the term $M_iE_b'\,\partial E_b'/\partial M_i$ enters in addition. Exceptional members (Step 5.4): $E_b'(P_0)=0$ or $\partial_P\langle\sigma^{(i)}_3\rangle_{P_0}=0$ (no additional term); $\langle\sigma^{(i)}_3\rangle_{P_0}=0$ (no first term); $E_b''(P_0)=0$ ($\alpha_X=0$, and $\mathfrak m_b$ is not defined at $P_0=0$).
  - (d) For every binding of $\mathcal W$ and every bound level, with the common gradient $g_1=g_2=g_{\rm c}$, exactly at first order: the coefficient of $g_{\rm c}$ is $-E_b''\,\mathcal M_P+E_b'\,\mathcal M_P'$ (13.12). Of the five terms of (11.6) only the two rest terms enter it, through $\mathcal M_P$ and $\mathcal M_P'$, next to the factors $E_b''$ and $E_b'$; the motion terms and the binding term do not enter $\mathcal M_P$.
- **Item 3** (class of item 3: every diagonal on-site binding with $\hat w_{11}\neq0$, both signs of $w_{11}$, the bound level (11.12), packets concentrated at $P_0=0$; first order in the gradients; leading order in $\mathsf R$ at fixed hatted data, hence at a fixed $u\in(0,\infty)$, with remainders that are not uniform in $u$; a limit in $u$ is an ordered limit, first $\epsilon\to0$ at fixed $u$; derived from (11.14), (11.16)).
  - At rest. For a diagonal on-site binding with $\hat w_{11}\neq0$ and the level (11.12), for small $\epsilon$: $E_b(-P)=E_b(P)$ and $E_b'(0)=0$, exactly. Without gradients the mean velocity of both parts is $\langle E_b'\rangle_g=O(\delta P^2)$ for all $\lambda$ (11.8), so the pair is at rest in the sharp-total-label limit (Step 6.1).
  - (a) For this class, exactly in the data: $a_{\rm pair}=-\langle\mathcal K\rangle_0/\mathfrak m_b$. At leading order in $\mathsf R$ at a fixed $u$ it is the closed form (13.13), equal to $-\varsigma(1+u^2)^{-1/2}\,\bar a$, with a relative remainder $O(\epsilon^2)$ on each of its three coefficients. Where the leading terms of the three cancel in the sum, $a_{\rm pair}$ is given by these remainders and is of order $\epsilon^4$ times the gradients. Ordered limits of the leading coefficient (13.14): $-\varsigma\,\bar a\,[1-\tfrac12u^2+O(u^4)]$ for $u\to0$, and $-\varsigma(\bar a/u)[1+O(u^{-2})]\to0$ for $u\to\infty$.
  - (b) For this class with $\langle X\rangle=0$, at leading order in $\mathsf R$ at a fixed $u$ and at a fixed ratio $g_1:g_2$ with $g_i\neq0$ and $g_1M_1+g_2M_2\neq0$: the ratio $a_{\rm pair}/a^{(i)}$ is (13.15), up to a factor $1+O(\epsilon^2)$. The three cases have at least one non-zero gradient; for $g_1=g_2=0$ nothing falls and no ratio is defined.
    - Common gradient $g_{\rm c}\neq0$ and $\tau_1=\tau_2$ (the free parts fall alike): both ratios are $-\varsigma(1+u^2)^{-1/2}$ at leading order in $\mathsf R$. The pair falls like its parts neither exactly nor at the leading non-zero order $\epsilon^2$ at a fixed $u$: $a_{\rm pair}-a^{(i)}=-\Lambda\,a^{(i)}(1+O(\epsilon^2))$ (13.34), of order $\epsilon^2$ like the falls themselves, a relative difference $-\Lambda$ of order $\epsilon^0$; through the orders $\epsilon^0$ and $\epsilon^1$ both falls vanish. For $w_{11}<0$ it does so in the ordered wide-pair limit, with the leading relative difference $-\tfrac12u^2$; for $w_{11}>0$ not at all (the ratio is negative at every $u$).
    - Common gradient $g_{\rm c}\neq0$ and $\tau_1\neq\tau_2$ (the free parts differ): the ratios are $-\varsigma(1+u^2)^{-1/2}\tau_{\rm p}^2/\tau_i^2$ at leading order in $\mathsf R$. The pair falls like both of its parts at no $u$ and in no limit; this is exact, because $a^{(1)}\neq a^{(2)}$. In the ordered wide-pair limit for $w_{11}<0$ the ratio differs from $1$ by $M_j(\tau_j^2-\tau_i^2)/(M_i\tau_j^2+M_j\tau_i^2)$. Exceptional, at leading order in $\mathsf R$: for $w_{11}<0$ and the part with the smaller $\tau_i$ the ratio is $1$ at the single value $u=u_*$.
    - $g_1\neq g_2$: at leading order in $\mathsf R$ at a fixed $u$, $a_{\rm pair}=-\varsigma(1+u^2)^{-1/2}\,\bar a$. If $\tau_1^2g_1=\tau_2^2g_2$ the free parts fall alike, and the first case applies with the first form of (13.34). Otherwise the pair falls like both of its parts at no $u$ and in no limit (exact, because $a^{(1)}\neq a^{(2)}$); in the ordered wide-pair limit for $w_{11}<0$, $a_{\rm pair}-a^{(1)}\to\mathfrak m_2(a^{(2)}-a^{(1)})/(\mathfrak m_1+\mathfrak m_2)$.
    - One part. For $g_i\neq0$, at leading order in $\mathsf R$ at a fixed $u$, the pair falls like the part $i$ iff $g_j/g_i=\mathsf z_i(u)$ (13.23), for both signs of $w_{11}$; for the common gradient this is $u=u_*$. Without the expansion in $\mathsf R$ the member is one ratio as well, $\mathsf z_i(u)\,(1+O(\epsilon^2))$. On it $a^{(j)}\neq a^{(i)}$: it concerns one part only.
    - Exceptional member $g_1M_1+g_2M_2=0$ with $g_1\neq0$: the pair has no fall at the leading order; $a_{\rm pair}$ is of order $\epsilon^4\lvert g_1\rvert$ and need not vanish (numerical: S2 with $g_2=-0.8\,g_1$).
  - (c) For this class at a fixed binding $w_{11}\neq0$: the leading coefficient of $a_{\rm pair}$ in $\mathsf R$, the first term of (13.16), is the same for all pairs of clock levels iff all three gradients vanish (no fall): for $\langle X\rangle=0$ iff $g_1=g_2=0$, for $\langle X\rangle\neq0$ never. Hence, with at least one non-zero gradient, the falls of two pairs of levels differ for small $\epsilon$, at the order $\epsilon^2$ of the falls. Leading dependence, where the numerator $\mathsf N=g_1M_1+g_2M_2+2\kappa\langle X\rangle$ does not vanish: $\partial\ln a_{\rm pair}/\partial\ln M_i=g_iM_i/\mathsf N-\mathfrak m_i/(\mathfrak m_1+\mathfrak m_2)-\frac{u^2}{1+u^2}\frac{a_i}{a}$. In the ordered wide-pair limit the leading coefficient is the same for all pairs of levels iff $\langle X\rangle=0$ and $\tau_1^2g_1=\tau_2^2g_2$. A free part falls alike on all its levels iff $\langle X\rangle=0$ (4.16), exactly at first order in the gradients. No statement is made for pairs of levels compared at one common value of $u$.
  - (d) For every binding of $\mathcal W$ and every bound level, exactly at first order in the gradients: $\langle X\rangle\neq0$ adds $-2\kappa\langle X\rangle E_b''(P_0)$ to the drift acceleration (13.8). For the class of item 3 this is $-2\kappa\langle X\rangle/\mathfrak m_b$, at leading order in $\mathsf R$ at a fixed $u$ equal to $\frac{\varsigma}{\sqrt{1+u^2}}\frac{2\kappa\langle X\rangle}{\mathfrak m_1+\mathfrak m_2}$. For a free part it adds $-\kappa\langle X\rangle/\mathfrak m_i$ (4.16). In both cases the plain gradient enters as the sum of the plain gradients over the curvature inertia, without a factor $\langle\sigma_3\rangle$, and it makes the fall depend on the levels; for the pair it is in addition reduced by $(1+u^2)^{-1/2}$ against the inertia-weighted mean, and reversed for $w_{11}>0$.
- **Item 4** (class of item 3 with $\langle X\rangle=0$; derived from (11.19), (11.20), from (11.12), and from (11.11) with the statement of its Step 7).
  - (a) Rate. For a diagonal on-site binding with $\hat w_{11}\neq0$, the level (11.12) and $\langle X\rangle=0$, at zeroth order in the gradients (with the data of the two parts at $x_0$), at leading order in $\kappa$, under the conditions and in the range of $\lambda$ of (11.19) (Step 7.1): the rate parameter of the clock of the part $i$, the number that multiplies $\lambda$ in the advance of (11.19), is (13.17), with $\Gamma_i$ up to $O(\epsilon^4)$ at a fixed $u$. It is taken with the clock data of the part $i$ ($u_{[i]}$, $\Lambda_{[i]}$), not with the level data of (13.13). It is a statement on the rate parameter, not on a rate inferred from readings, which (11.19) fixes within its remainder.
  - (a) Relative change. For the same class, to first order in the gradients and without the expansion in $\mathsf R$: $\varrho_i$ is (13.21), exact in the data as an identity for the rate parameter, not for a rate inferred from readings. In $\mathsf R$ at a fixed $u$ it is (13.18), with the remainder $O(\epsilon^4)(\lvert g_1\rvert+\lvert g_2\rvert)$, from (13.22); at leading order $\varrho_i=g_i$, and the term of order $\epsilon^2$ contains the gradient of the other part.
  - (a) Readings. (13.35) is the hierarchy under which two upper bounds, the bound $2\pi\lvert\delta r\rvert\lambda$ on the effect of a term $\delta r$ of the rate parameter on the distribution $P_{N_i}$ and the $O$-remainder of (11.19), allow the term of order $\epsilon^2$ of (13.17) to be separated from the remainder: $\delta_i\ll\epsilon$, $\delta P\ll\epsilon$ and $\max(\kappa,\delta_i^2,\delta P^2)/\epsilon^2\ll\lambda/\lambda_{0,i}\ll\epsilon^2/\delta_i^2$. It is neither necessary nor sufficient for a reading to resolve the term. Where it fails, it does not follow that a reading cannot resolve the term (that would need a lower bound on the remainder). Where it holds, a positive statement needs in addition the sensitivity of the reading distribution to the rate at the chosen advance, a lower bound on the change of $P_{N_i}$; the sensitivity vanishes at an integer advance. No positive and no negative statement on resolution is made. For the terms of order $\epsilon^2$ of $\varrho_i$ and the readings of two pairs without gradients at labels $\Delta x$ apart, the same holds with $\epsilon^2\to\epsilon^2(\lvert g_1\rvert+\lvert g_2\rvert)\Delta x$. For the readings of one pair in the field no statement is made: (13.18) and (13.24)–(13.26) are statements on the rate parameter (Step 7.2).
  - (b) The number $\mathsf w_i^2=-a_{\rm pair}/\varrho_i$, for the same class. It is formed with the relative change $\varrho_i$ of the rate parameter, at first order in the gradients; $a_{\rm pair}$ is taken with the level data, $\varrho_i$ with the clock data of the part $i$.
    - Value, fixed ratio: at a fixed ratio $g_1:g_2$ with $g_i\neq0$ and $g_1M_1+g_2M_2\neq0$, at leading order in $\mathsf R$ at a fixed $u$ (relative remainder $O(\epsilon^2)$), it is (13.19), of order $\epsilon^2$; for $g_1=g_2$ it is $-\varsigma\,\tau_{\rm p}^2/\sqrt{1+u^2}$. On the member $g_1M_1+g_2M_2=0$ with $g_i\neq0$ its leading term vanishes; it is then of order $\epsilon^4$ and not determined here. For $\lvert g_i/g_j\rvert$ of order $\epsilon^2$ the factor $(1+\nu_ig_j/g_i)^{-1}$ of Step 7.3 enters.
    - Value, $g_i=0\neq g_j$: at leading order in $\mathsf R$ at a fixed $u$ it is (13.25), negative for both signs of $w_{11}$ and of order $\epsilon^0$.
    - Not defined: on the member $\varrho_i=0$, the ratio $g_i/g_j=-\nu_i\,(1+O(\epsilon^2))$ of (13.24), and for $g_1=g_2=0$.
    - Equal to $\tau_i^2$: for $g_i\neq0$, at leading order in $\mathsf R$ at a fixed $u$, iff $g_j/g_i=\mathsf z_i(u)$ (13.23), the ratio on which the pair falls like the part $i$ at that order; for the common gradient this is $u=u_*$ ($\tau_1\neq\tau_2$, $w_{11}<0$, the part with the smaller $\tau_i$). Beyond leading order the condition is $a_{\rm pair}=-\tau_i^2\varrho_i$ with $\varrho_i$ of (13.21), which is not the matching of the fall, $a_{\rm pair}=-\tau_i^2g_i$ (13.36): each holds on one ratio $\mathsf z_i(u)\,(1+O(\epsilon^2))$, and the two ratios are the same iff $\varrho_i=g_i$ there, which fails for small $\epsilon$ wherever the coefficient of order $\epsilon^2$ of (13.18) does not vanish on the member (Step 7.3, with an example). For $g_i=0\neq g_j$ it does not equal $\tau_i^2$ at any fixed $u$ or in either ordered limit.
    - The same for $i=1,2$: where both numbers are defined, $\mathsf w_1^2/\mathsf w_2^2=\varrho_2/\varrho_1$, exactly. At leading order in $\mathsf R$, at a fixed ratio with $g_1,g_2\neq0$, they are the same iff $g_1=g_2$ (on the member $a_{\rm pair}=0$ both vanish). For $g_1=g_2\neq0$ they are the same at relative order $\epsilon^2$ iff the condition of (13.26) holds. For $g_i=0\neq g_j$ they differ by the factor $\nu_i^{-1}$, of order $\epsilon^{-2}$.
    - Data of the pair: yes. (13.19) contains the binding ($\varsigma$, $u$) and the clock levels (through $u$, and through $\tau_{\rm p}$ if $\tau_1\neq\tau_2$); (13.25) contains the binding, the levels of both parts and the mean rest cost $\bar M_i$.
  - (c) Relation (13.20). For the same class with the common gradient $g_{\rm c}\neq0$ and $\tau_1=\tau_2=\tau$, at a fixed $u$ of the level data: the pair obeys the relation of the form (4.24) with $v_{\max}^2$ replaced by $-\varsigma\,v_{\max}^2/\sqrt{1+u^2}$, up to a relative $O(\epsilon^2)$, with the relative change $\varrho_i$ of the rate parameter of either part. With the speed $\tau d_0$ itself the relation fails at the leading non-zero order $\epsilon^2$ at every fixed $u$: the absolute deviation is of order $\epsilon^2$, the relative deviation is $\mathsf w^2/\tau^2-1=-\Lambda\,(1+O(\epsilon^2))$, of order $\epsilon^0$ (13.34); through the orders $\epsilon^0$ and $\epsilon^1$ both sides vanish. For $w_{11}<0$ it holds in the ordered wide-pair limit, with the leading deviation $-\tfrac12u^2$; for $w_{11}>0$ it holds in no limit (the factor is negative at every $u$). For $g_{\rm c}=0$ both sides vanish, and the relation holds trivially with any speed.
  - What (13.18) rests on: the second derivatives of (11.12) in $M_1,M_2$ at $P=0$, with remainders $O(\epsilon^4)$ at fixed $u$ by the uniform estimate of Step 7.2, which extends (11.11) with the statement of its Step 7 to complex rest costs by standard analytic perturbation theory (constants not uniform in $u$). Its term of order $\epsilon^2$, and with it (13.24)–(13.26), uses their values (13.22); its leading order $\varrho_i=g_i$, and with it the factors $1+O(\epsilon^2)$ of (13.19) and (13.20), uses only that they are of order $\epsilon^2$.
- **Item 5.** In the table $u$ and $\Lambda$ are those of the level data, and $u_{[i]}$ that of the clock data of the part $i$. For the pair, "clock rate" is the rate parameter of (13.17). All cells are at first order in the gradients; the cells of the pair are for a diagonal on-site binding with $\hat w_{11}\neq0$, the level (11.12) and $P_0=0$, and each names its order in $\mathsf R$.

| | Free part $i$ (upper branch, from rest) | Bound pair (level (11.12), $P_0=0$) | Pair agrees with its parts |
|---|---|---|---|
| Drift acceleration from rest | $a^{(i)}=-(g_iM_i+\kappa\langle X\rangle)/\mathfrak m_i$, exact in the data at first order in the gradients; (4.16), (8.11) | exact in the data: $a_{\rm pair}=-\langle\mathcal K\rangle_0/\mathfrak m_b$; at leading order in $\mathsf R$ at a fixed $u$: $a_{\rm pair}=-\varsigma(1+u^2)^{-1/2}\,\bar a$, with a relative $O(\epsilon^2)$ on each of the three coefficients, not uniform in $u$; where their leading terms cancel, of order $\epsilon^4$ times the gradients; (13.11), (13.13) | for $\langle X\rangle=0$ and $(g_1,g_2)\neq(0,0)$, at leading order in $\mathsf R$: like both parts at no fixed $u$ (for $a^{(1)}=a^{(2)}$ the difference is $-\Lambda a^{(i)}$: absolute order $\epsilon^2$, relative order $\epsilon^0$); in the ordered limit $u\to0$ iff $a^{(1)}=a^{(2)}$ and $w_{11}<0$; like one part $i$ ($g_i\neq0$), at leading order in $\mathsf R$ at a fixed $u$, iff $g_j/g_i=\mathsf z_i(u)$, for both signs of $w_{11}$ (the exact member is one ratio $\mathsf z_i(u)(1+O(\epsilon^2))$), which for the common gradient is $u=u_*$ ($\tau_1\neq\tau_2$, $w_{11}<0$, the part with the smaller $\tau_i$); for vanishing gradients: trivially; (13.14), (13.15), (13.23), (13.34) |
| Independent of the clock levels | iff $\kappa\langle X\rangle=0$, exact at first order in the gradients; (4.16) | at a fixed binding $w_{11}\neq0$, the leading coefficient in $\mathsf R$ (first term of (13.16)) is the same for all pairs of levels iff $g_1=g_2=0$ and $\langle X\rangle=0$ (no fall); with a non-zero gradient the falls of two pairs of levels differ for small $\epsilon$; (13.16) | with a non-zero gradient: at no fixed binding $w_{11}\neq0$, for small $\epsilon$; in the ordered limit $u\to0$ the leading coefficient is the same for all pairs of levels iff $\langle X\rangle=0$ and $\tau_1^2g_1=\tau_2^2g_2$; for vanishing gradients: trivially; (13.16) |
| Number that ties it to the relative change of a clock rate ($\langle X\rangle=0$) | $\tau_i^2$, defined for $g_i\neq0$, from $a^{(i)}=-\tau_i^2g_i$ with the relative change $g_i$ of its clock rate; (4.24) | formed with the rate parameter of (13.17), not with a rate inferred from readings ((13.35) is a hierarchy on upper bounds); at leading order in $\mathsf R$ at a fixed $u$: for a fixed ratio with $g_i\neq0$ and $g_1M_1+g_2M_2\neq0$, $\mathsf w_i^2=-\varsigma(1+u^2)^{-1/2}(g_1M_1+g_2M_2)/(g_i(\mathfrak m_1+\mathfrak m_2))$, of order $\epsilon^2$ (of order $\epsilon^4$ and not determined where $g_1M_1+g_2M_2=0$); for $g_i=0\neq g_j$, $\mathsf w_i^2=-4\bar M_i^2M_j\,u\,(1+u_{[i]}^2)^{3/2}/(M_i\,u_{[i]}^3\sqrt{1+u^2})$, negative, of order $\epsilon^0$; not defined at $g_i/g_j=-\nu_i(1+O(\epsilon^2))$ and for $g_1=g_2=0$; (13.18), (13.19), (13.24), (13.25), (13.35) | for $g_1=g_2\neq0$ and $\tau_1=\tau_2=\tau$, at a fixed $u$: $\mathsf w^2/\tau^2=-\varsigma(1+u^2)^{-1/2}(1+O(\epsilon^2))$, with the relative deviation $-\Lambda$ of order $\epsilon^0$ and the absolute deviation $-\Lambda\tau^2$ of order $\epsilon^2$; equal to $1$ only for $w_{11}<0$ in the ordered limit $u\to0$; in general, for $g_i\neq0$ and at leading order in $\mathsf R$ at a fixed $u$, $\mathsf w_i^2=\tau_i^2$ iff $g_j/g_i=\mathsf z_i(u)$ (one part; for the common gradient $u=u_*$); beyond leading order the condition is $a_{\rm pair}=-\tau_i^2\varrho_i$, which is not the matching of the fall; for $g_i=0\neq g_j$ at no fixed $u$ and in neither ordered limit; (13.19), (13.20), (13.23), (13.25), (13.34), (13.36) |

## Consistency checks

1. **One body, and no binding.** (13.8) applied to one free body on a level ($E=\sigma\omega$, $\langle\sigma_3\rangle=\partial E/\partial M=\sigma M/\omega$, $\mathcal K=G\sigma_3-\mathcal F$) gives $\alpha=-GM(\omega\omega')'/\omega^2+\sigma\mathcal F\omega''=-GM\tau^2\cos2p/\omega^2+\sigma\mathcal F\tau^2(M^2\cos2p-\tau^2\sin^4p)/\omega^3$. This is (4.7) with $g_\zeta=0$, for all $p$ and both branches. In the pair, the ordered limit of no binding ($u\to0$, $w_{11}<0$) of (13.13) is $\bar a$, equal to $a^{(1)}=a^{(2)}$ when the free parts fall alike. Passed.
2. **Common gradient against (11.6).** With (11.14) at $P=0$ (each value up to $O(\epsilon^4)$, the remainder of (11.14)): $\mathcal M_0=M_1+M_2-a\Lambda$ from $\langle\sigma^{(i)}_3\rangle_0$, and $E_b-\langle A_1\rangle_0-\langle A_2\rangle_0-\langle U\rangle_0=M_1+M_2+a+\varsigma\sqrt{w^2+a^2}-2a\Lambda-\varsigma w^2/\sqrt{w^2+a^2}=M_1+M_2-a\Lambda$. Passed, in agreement with (13.12).
3. **Dimensions.** Labels and $P$ are pure numbers and costs are $1/\lambda$. $g_i$ is per label, $\kappa\langle X\rangle$ a cost per label, $E_b''$ and $1/\mathfrak m$ are costs. (13.8), (13.13): cost$^2$, i.e. labels$/\lambda^2$. (13.10): $\beta$ is a cost, i.e. labels$/\lambda$. (13.17): $1/\lambda$. (13.19): $\mathsf w_i^2$ is a cost$^2$, like $\tau_i^2$. Passed.

## Open issues

- The second order in the gradients, and the range of $\lambda$ over which the first order describes the motion, are not estimated (out of scope); (13.9) and (13.10) bound first-order terms only.
- Item 2(c), the other order of the limits, for the members with $(g_1,g_2)\neq(0,0)$ other than the levels of items 3 and 4. A member that satisfies the necessary condition (N) without the sufficient condition (S) is excluded where the level has no mirror pair at one $P_1\in\mathcal P_0$ (13.32); where the level has a mirror pair at every $P\in\mathcal P_0$ it is neither shown to exist nor excluded (Step 5.3 (iii), with the reason). No member with $(g_1,g_2)\neq(0,0)$ that satisfies (N) is exhibited. The class $\mathcal S(P_0,I)$ leaves the phase of the packets free; no statement is made for a class that restricts it. Outside the scope of the question: a classification of the members between (N) and (S), and the evaluation of the limits in the other order.
- A binding of $\mathcal W$ with $\beta\not\equiv0$ is exhibited numerically (set S5), not analytically; the same holds for a level with $E_b'(0)\neq0$ (Step 5.4).
- The remainders of (13.13)–(13.20), (13.23)–(13.26), (13.34) and (13.36) (a relative $O(\epsilon^2)$, or $O(\epsilon^4)$ against terms of order $\epsilon^2$ in (13.17), (13.18) and (13.26)) are not determined (for the sets S1–S3, with $\tau_i/M_i\le0.2$: numerically up to 4 % for (13.13) and up to 7 % for (13.22)) and are not uniform in $u$.
- The first-order corrections to the clock reading of a pair in the field (the gradient across the packet) are not derived. For this reason (13.18) and (13.24)–(13.26) are statements on the rate parameter only, as far as the readings of one pair in the field are concerned. Whether a reading resolves a term of order $\epsilon^2$ of the rate parameter is not decided: (13.35) is a hierarchy on two upper bounds; its constants (those of the remainder of (11.19)), a lower bound on that remainder and the sensitivity of the reading distribution at a given advance are not computed, so neither a positive nor a negative statement on resolution is made.
- The results hold for a binding and for motion terms without a factor of the field, the class of the question. Whether another admissible coupling makes the pair fall like its parts is outside this package.
- The remainder $O(\epsilon^4)$ of (13.22) rests on the uniform estimate of Step 7.2: the reduction (11.11) of 11-bound-pair, extended to complex rest costs by standard analytic perturbation theory. It is an argument at the physicist-formal standard; its constants $\delta$, $c$ are not computed and are not bounded for $u\to0$ or $u\to\infty$. The numerical check finds the scaling $\epsilon^4$ for the sets S1–S3 at three scales (section "Code", item 4). (13.24)–(13.26) and the place of the member $\varrho_i=0$ use the values (13.22) and share this estimate.
- Whether the rest $\dot\rho_i$ of the first-order acceleration decays for $\lambda\to\infty$ at a fixed packet is not derived (Step 5.1); the drift acceleration does not need it. Likewise, whether the bounded first-order separation of Step 4.2 converges for $\lambda\to\infty$ is not derived.

## Code

- **File:** `code/v6/pair_fall_check.py`, role: check. Outputs: `code/v6/output/summary.txt` (all numbers below), `code/v6/output/results.json`, `code/v6/output/first_order_motion.png`. It is the script of v5 with the paths changed; the evolution, its parameters, its tables and its numbers are those of v5. No table is added: no statement that is new in v6 (the scalar and the reducible case of Step 1.4, the bounds of Step 7.1, (13.36)) rests on a number.
- **What it does.**
  - It builds the fibre contract (11.1) on $\lvert r\rvert\le48$, finds the bound level attached to $\mathcal B_{++}$ on a grid of 161 values of $P$, and computes $\alpha$ (13.8) and $\beta$ (13.10) from it ($E_b''$ by (11.9); the reduced resolvent by full diagonalization).
  - It builds $\Phi_g$ with $g\propto\cos^4\bigl(\pi(P-P_0)/(2P_c)\bigr)$ on the chain $\lvert x_i\rvert\le60$ (121 sites, dimension 58564), evolves it by $C_{\rm bod}$ (13.2) with small gradients of both signs (Chebyshev expansion), and takes the symmetric difference.
  - The slope of a first-order quantity is its mean rate over $[0,\lambda_{\max}]$ with the weight $\propto\sin^4(\pi\lambda/\lambda_{\max})$. The sampling step is chosen so that no multiple of $2\pi/\text{step}$ lies near a frequency $\lvert E_n(P)-E_b(P)\rvert$ of the rest.
  - From the numerical level at $P=0$ (fibre $\lvert r\rvert\le64$) it computes the clock ratio $\Gamma_i=\langle\sigma^{(i)}_3\rangle_0$ and its derivative in the data, $T_{ij}:=M_j\,\partial\Gamma_i/\partial M_j=M_j\,\partial^2E_b(0)/\partial M_j\partial M_i=-2M_j\,\mathrm{Re}\langle\sigma^{(i)}_3\psi_0\vert R_0\vert\sigma^{(j)}_3\psi_0\rangle$, and compares them with (11.20), with (13.22) and, through (13.21), with (13.18).
  - From the numerical level at $P=0$ and $P=0.2$ (fibre $\lvert r\rvert\le96$) it computes $\partial_P\langle\sigma^{(i)}_3\rangle_P=-2\,\mathrm{Re}\langle\sigma^{(i)}_3\psi_P\vert R_P\vert V_\eta\psi_P\rangle$, $\partial_P\langle\mathcal K\rangle_P$ and $\mathsf f_i(P)=\mathrm{Re}\,F_{i,\Lambda}(P)$ of Step 5.3 for the windows $[0,\Lambda]$, $\Lambda=5,10,20$, by the sum over the levels of the fibre, per unit gradient (common gradient; $g_1$ only), and checks the identity (13.6).
  - From the numerical level at $P=0.1$ and $P=0.2$ (fibre $\lvert r\rvert\le96$) it computes $\partial_\lambda^2c_i(0;P)$ of (13.29) in four ways: by the sum over the levels of the fibre, by the double commutator of (13.29), by the first relation of (13.30) and, for S1–S3, by its second relation; per unit gradient of one part.
- **Sets.** Each with the common gradient, with $g_1$ only ($g_2=0$), and with a plain gradient in the place of $\kappa\langle X\rangle$.
  - S1: $\tau_i=0.2$, $M_i=1$, $w_{11}=-0.02$ ($u=1.00$). S2: $\tau=(0.20,0.18)$, $M=(1,1.25)$, $w_{11}=-0.02$ ($u=1.21$). S3: $\tau=(0.20,0.17)$, $M=(1,0.9)$, $w_{11}=+0.022$ ($u=1.22$). S4: $\tau=(1,0.8)$, $M=(1,1.2)$, $w_{11}=-0.9$. In all four the other $w_{s_1s_2}$ are non-zero, $P_0=0$ and $P_c=0.5$ ($\delta P=0.106$).
  - S5 (control): the data of S4 with a binding of range 1 that is neither diagonal nor symmetric, $P_0=0.2$, $P_c=0.4$ ($\delta P=0.085$).
  - Range: $\lambda\le3000$ (S1–S3), $200$ (S4), $400$ (S5). Gradients: $5\cdot10^{-7}$ (S1–S3), $2\cdot10^{-6}$ (S4, S5).
- **Item 2a.** The slopes of $v^{(1)}_1$ and $v^{(1)}_2$ agree with $\langle\alpha\rangle_g$ of (13.7), in all three configurations, to a relative $2\cdot10^{-5}$ (S1), $1.3\cdot10^{-5}$ (S2), $6\cdot10^{-6}$ (S3), $4\cdot10^{-7}$ (S4), $3\cdot10^{-5}$ (S5). Over the range of each run the rest $\lvert v^{(1)}_i-\lambda\langle\alpha\rangle_g\rvert$ is at most 3 % of the linear part at $\lambda_{\max}$.
- **Item 2b.** All numbers per unit gradient, over the range of each run.
  - S1–S4: $\lvert\langle\beta\rangle_g\rvert\le4\cdot10^{-15}$ (numerical level), in all configurations.
  - The configurations with a non-zero first-order separation are S1 with $g_1$ only and the three configurations of S2, S3 and S4. In them the separation stays bounded over the range of the run, as (13.10) with $\beta\equiv0$ states for all $\lambda$. Its largest value, for the common gradient, $g_1$ only and the plain gradient: $270$ (S1, $g_1$ only); $39.0$, $201$, $71.1$ (S2); $43.5$, $157$, $31.8$ (S3); $1.21$, $2.41$, $0.96$ (S4). In these ten configurations its weighted mean rate over the full range, times $\lambda_{\max}$, is below 0.5 % of the largest value of the same configuration: 0.43 % (S1 with $g_1$ only), 0.23 % (S2, all three), 0.17 % (S3, all three), below 0.002 % (S4). A convergence to a constant is not shown: over the two halves of the range the weighted mean rates times the half range are 22 % and 11 % of the largest value for S1 with $g_1$ only, and 14 % and 4.5 % for S2 with $g_1$ only.
  - For S1 with the common and the plain gradient the separation is zero within the noise of the difference ($3\cdot10^{-8}$ against $270$ for $g_1$ only), as in Step 4.3; a ratio of its mean rate to its largest value is a ratio of noise and is not stated.
  - S5: over $\lambda\le400$ the slopes are $-0.08662$, $-0.02819$, $-0.07457$ against $\langle\beta\rangle_g=-0.08651$, $-0.02813$, $-0.07444$; with $\lambda\le600$ the second slope is $-0.028139$.
- **Item 2c (members for the other order of the limits, Step 5.3).** Table "other order": the numerical level of the five sets at $P=0$ and $P=0.2$; all numbers per unit gradient.
  - $\partial_P\langle\sigma^{(i)}_3\rangle_P$ at $P=0.2$, $i=1,2$: S1 $-2.102\cdot10^{-3}$ (both); S2 $-1.490\cdot10^{-3}$, $-1.333\cdot10^{-3}$; S3 $+1.657\cdot10^{-3}$, $+1.946\cdot10^{-3}$; S4 $-2.398\cdot10^{-2}$, $-2.300\cdot10^{-2}$; S5 $-3.277\cdot10^{-2}$, $-3.888\cdot10^{-2}$. Hence $\partial_P\langle\mathcal K\rangle_P$ for the common gradient: $-4.20\cdot10^{-3}$, $-3.16\cdot10^{-3}$, $+3.41\cdot10^{-3}$, $-5.16\cdot10^{-2}$, $-7.94\cdot10^{-2}$; for $g_1$ only it is $M_1=1$ times the first number of each set.
  - $\mathsf f_i(P)=\mathrm{Re}\,F_{i,\Lambda}(P)$ at $P=0.2$ for $\Lambda=5,10,20$: non-zero for all sets, both parts, both configurations. S5 with the common gradient: $\mathsf f_1=7.07\cdot10^{-2}$, $2.34\cdot10^{-2}$, $1.20\cdot10^{-2}$ and $\mathsf f_2=2.83\cdot10^{-2}$, $1.11\cdot10^{-1}$, $-6.28\cdot10^{-2}$. The smallest value of the table is $3.7\cdot10^{-6}$ ($\mathsf f_2$ of S2 with $g_1$ only, $\Lambda=5$).
  - At $P=0$: for S1–S4 (symmetry $\mathcal R$ of Step 6.1) $\lvert\partial_P\langle\sigma^{(i)}_3\rangle_0\rvert\le2.4\cdot10^{-17}$ and $\lvert\mathsf f_i(0)\rvert\le1.5\cdot10^{-15}$. For S5: $E_b'(0)=-1.17\cdot10^{-4}$, $\partial_P\langle\mathcal K\rangle_0=3.36\cdot10^{-4}$ for the common gradient, and $\mathsf f_i(0)\neq0$.
  - Checks. The identity (13.6), $2\,\mathrm{Re}\langle V_i\psi_P\vert R_P\vert\mathcal K\psi_P\rangle=-\partial_P\langle\mathcal K\rangle_P$ for $i=1,2$, holds to $3\cdot10^{-16}$. A symmetric difference of $\langle\sigma^{(i)}_3\rangle_P$ in $P$ (step $10^{-4}$) agrees with the resolvent form to $2\cdot10^{-10}$. Fibre range $64$ against $96$: $3\cdot10^{-15}$. At these $P$ the level is outside the bands, with a distance to the rest of the spectrum of at least $8\cdot10^{-3}$.
  - Comparison for S1–S3 at $P=0.2$ (a check of the table; the formula is the derivative in $P$ of $\langle\sigma^{(i)}_3\rangle_P$ of Step 5.3): the leading order obtained from (11.12) by one derivative in $M_i$ and one in $P$, $\varsigma\,\frac{a_i}{M_i}\,\frac{2a_j\sin2P\,(w^2+a_j^2+a_ia_j\cos2P)}{(w^2+j(P)^2)^{3/2}}$, deviates from the numbers above by $8\cdot10^{-4}$ (S1), up to $3\cdot10^{-3}$ (S2) and $-4.3\cdot10^{-2}$ (S3).
  - Table "second derivative" (added in v4; (13.29), (13.30)), at $P=0.1$ and $P=0.2$, per unit gradient of one part. The sum over the levels of the fibre agrees with the double commutator of (13.29) to $4\cdot10^{-16}$ for all five sets. For S1–S4 it agrees with the first relation of (13.30) to $2\cdot10^{-16}$, and the gradient of the other part contributes less than $4\cdot10^{-16}$. Values at $P=0.2$ for the parts $1$, $2$: S1 $-5.552\cdot10^{-3}$ (both); S2 $-3.919\cdot10^{-3}$, $-6.128\cdot10^{-3}$; S3 $+4.354\cdot10^{-3}$, $+3.529\cdot10^{-3}$; S4 $-5.216\cdot10^{-2}$, $-7.683\cdot10^{-2}$; at $P=0.1$ they are about half of these. The second relation of (13.30) deviates by a relative $-1.7\cdot10^{-3}$ to $-1.9\cdot10^{-3}$ (S1), $-0.6\cdot10^{-3}$ to $-1.6\cdot10^{-3}$ (S2) and $-1.7\cdot10^{-2}$ (S3). For S5, whose binding does not commute with $\mathcal K$, the first relation of (13.30) does not apply (it deviates by up to $2\cdot10^{-2}$) and the gradient of the other part contributes (up to $1.3\cdot10^{-3}$), as (13.29) allows. Fibre range $64$ against $96$: $6\cdot10^{-16}$.
- **Item 3.** The coefficients of (13.13) deviate from those of the numerical level at $P_0=0$ by $-5\cdot10^{-4}$ to $+2.5\cdot10^{-3}$ (S1), up to $2.9\cdot10^{-3}$ (S2), and $-2.2\cdot10^{-2}$ to $-3.8\cdot10^{-2}$ (S3), against $(\tau_i/M_i)^2\le0.04$. With $\tau\to s\tau$, $w\to s^2w$ and $s=1,\tfrac12,\tfrac14$ the deviations fall by factors $3.8$ to $4.5$ per step. The slopes of the evolution agree, to the same deviations, with (13.13) averaged over the packet through (11.12). The member of Step 6.4 with cancelling leading terms is formed from these coefficients: for S2 they are $-9.9939\cdot10^{-3}$ for $g_1$ and $-1.25145\cdot10^{-2}$ for $g_2$.
- **Item 4 (the derivative of the clock ratio in the data).** Sets S1–S3 and their scalings $\tau\to s\tau$, $w\to s^2w$, $s=1,\tfrac12,\tfrac14$. In this table the rest costs $M_i$ of the set are used for the part that is read as well ($\bar M_i=M_i$), so the clock data and the level data coincide.
  - The four numbers $T_{ij}$ deviate from (13.22) by a relative $5.4\cdot10^{-3}$ to $1.1\cdot10^{-2}$ (S1, S2) and $-2.7\cdot10^{-2}$ to $-7.0\cdot10^{-2}$ (S3) at $s=1$. Per step in $s$ the relative deviations fall by factors $3.8$ to $4.0$ and the absolute ones by $15$ to $16$: for these three sets and three scales the deviation from (13.22) scales like $\epsilon^4$, in agreement with Step 7.2.
  - $\varrho_i-g_i$ from (13.21), for the common gradient, $g_1$ only and $g_2$ only, deviates from the term of order $\epsilon^2$ of (13.18) by up to $1.6\cdot10^{-2}$ (S1, S2) and $5.5\cdot10^{-2}$ (S3) at $s=1$, with factors $3.9$ to $4.0$ per step. The column of the common gradient is the term $\frac{\tau_i^2}{4\bar M_i^2}\mathsf y$ of (13.26); the columns of the gradient of the other part are $\nu_i$ of (13.24).
  - $1-\Gamma_i$ deviates from (11.20) by $7\cdot10^{-3}$ to $9\cdot10^{-3}$ (S1, S2) and $-4\cdot10^{-2}$ (S3) at $s=1$, with factors $3.9$ to $4.0$ per step.
  - Checks: a symmetric difference of $\Gamma_i$ in $M_j$ (relative step $10^{-4}$) agrees with the resolvent form to $7\cdot10^{-8}$; fibre range $48$ against $64$: $5\cdot10^{-12}$.
- **Cross-checks of (13.23) and (13.25).** They are formed here from numbers that the script prints; the script does not print them, and no statement rests on them.
  - (13.25), part $2$ with $g_2=0\neq g_1$ and $\bar M_i=M_i$: minus the coefficient of $g_1$ of the level at $P_0=0$, divided by $T_{21}/\Gamma_2$, is $-7.927$ (S1), $-8.314$ (S2), $-5.868$ (S3), against $-4M_1M_2(1+u^2)/u^2=-8$, $-8.395$, $-6.017$: deviations $-0.9$ %, $-1.0$ %, $-2.5$ %, and a negative number for both signs of $w_{11}$.
  - (13.23), S2: with the coefficients $-9.9939\cdot10^{-3}$ of $g_1$ and $-1.25145\cdot10^{-2}$ of $g_2$ of the level at $P_0=0$ and $a^{(1)}=-0.04\,g_1$, the pair falls like the part $1$ at $g_2/g_1=2.3977$, against $\mathsf z_1(u)=2.3994$.
- **Convergence** (configuration $g_1$ only; relative change of the slopes).
  - Chain: $45$ against $60$: $3\cdot10^{-7}$ (S1), $5\cdot10^{-8}$ (S4); $75$ against $60$: $1.5\cdot10^{-6}$ (S5).
  - Gradient doubled: $2\cdot10^{-6}$. Other sampling step: below $10^{-6}$. Fibre range $48$ against $64$: $10^{-13}$ in $\alpha$.
  - Narrower packet ($P_c=0.35$): the same agreement with its own $\langle\alpha\rangle_g$.
  - Range of $\lambda$: on half of the range the slopes deviate by up to $1.2\cdot10^{-2}$; the full range reduces this to the values of item 2a.
- **Run.** Once, in the foreground: 533 s (v5: 341 s, for the same computation). The summary reproduces the numbers of v5 on every line; only the run times differ. No package was installed.

## Methods used

- Reduction of a contract on an eigenstate of the field operators; partial expectation
- Decomposition of operators into the identity and traceless parts (pair terms against a three-object term)
- Fibre decomposition under the joint shift of two bodies
- First-order perturbation theory in a term of the contract (Duhamel formula)
- Perturbation theory of an isolated simple eigenvalue; reduced resolvent
- Commutator identities for velocity operators; Hellmann–Feynman relations
- Antiunitary and exchange symmetries of the fibre contract
- Window averaging of bounded oscillating terms; packets with a sharp total label
- Phase-modulated sequences of packets; isolated zeros of real-analytic functions
- Spectral measures; uniqueness of the Fourier transform of a finite complex measure
- Two-step recursion for an eigenvector in a symmetry sector of the fibre contract
- Leading-order expansion in a scaling regime with ordered limits; absolute and relative orders
- Mixed second differences (strict supermodularity) for a uniqueness argument
- Numerical evolution by a Chebyshev expansion with a symmetric difference in the gradients
- Second derivatives of an eigenvalue in the data; Cauchy estimate for the derivatives of an analytic remainder
- Analytic perturbation theory for an isolated part of the spectrum (Riesz projector); identity theorem
- Mean ergodic theorem for the time average of a unitary group
- Separation of a term of a rate parameter from the remainder of a reading distribution: upper bounds, and the sensitivity of the distribution (mean value theorem)

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