# Question: 12-changing-field

- **Subproject:** 05-ilang-free-fall
- **Package:** 12-changing-field
- **Equation tags:** (12.k)
- **Created:** 2026-10-11

## Goal

Determine what a body sees when the field is not stationary, so that the factor it produces changes with $\lambda$: through which quantities of the field the body depends on it at first order, how the body falls in a gradient that changes with $\lambda$, and under which condition this fall is the same for all clock levels. This is the third part of criterion 8.

**Setting.**
- **Objects.** A body $b$, a field $f$ and a recording medium $c$, of pairwise different types, one instance each.
- **Contract.** In the order $b$, $f$, $c$:
  $$C=C_{\rm D}\otimes\mathbb 1_f\otimes\mathbb 1_c+\bigl(\hat x\otimes G\bigr)\otimes Y\otimes\mathbb 1_c+\mathbb 1_b\otimes C_f\otimes\mathbb 1_c+\sum_x\lvert x\rangle\langle x\rvert\otimes\mathbb 1_f\otimes\kappa\,x\,X,\qquad G:=\sigma_3\otimes(m\mathbb 1_N+D),$$
  with $C_{\rm D}$ the contract of Model D on the chain $\lvert x\rvert\le L$ ($x\in\mathbb Z$ as the limit of long chains, A1), $\hat x:=\sum_xx\,\lvert x\rangle\langle x\rvert$, and self-adjoint operators $Y$ (the **gradient operator**) and $C_f$ on a finite-dimensional $\mathcal H_f$, $\lVert Y\rVert(L+1)<1$. This is the lapse-type pair term of the base problem with the factor $\Phi_x=\mathbb 1_f+x\,Y$ on the rest and clock term $\sum_x\lvert x\rangle\langle x\rvert\otimes G$; the motion term carries no factor. $C$ is the contract (2.1) with $r=\{c\}$. $Y$ need not commute with $C_f$. The form of the contract is an assumption of this package (M8).
- **Start of the field.** Any unit vector $\varphi$ of $\mathcal H_f$; the start is $\Psi_0=\phi\otimes\varphi\otimes\chi$, with $\phi$ a start of the body and $\chi$ the state of the medium, $\sigma_X>0$. Named cases: *sharp and stationary*, $Y\varphi=\gamma\varphi$ and $C_f\varphi=e_f\varphi$; *sharp and not stationary*, $Y\varphi=\gamma\varphi$ while (2.11) fails; *not sharp*. No state of the field is preferred (A11).
- **The gradient seen at $\lambda$.** $\gamma(\lambda):=\langle\varphi\vert e^{iC_f\lambda}\,Y\,e^{-iC_f\lambda}\vert\varphi\rangle$. The **field frequencies** are the differences $e-e'$ of eigenvalues of $C_f$.
- **Levels, branches and branch starts.** $d_k:=\frac{2\pi k}{N\lambda_0}$, $M_k:=m+d_k$. On the clock level $k$ the contract of Model D acts as the block (5.5) of 04-ilang-time/05-moving-clock, with the eigenvalues $\pm\omega_k(p)$ and the eigenvectors $\beta_{k,\pm}(p)$ of (5.6) of 04-ilang-time/05-moving-clock. For a level $k$, a branch $\sigma\in\{+,-\}$ and a smooth $2\pi$-periodic function $g$ with $\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\lvert g\rvert^2=1$, the **branch start** $B_{k,\sigma}[g]$ is
  $$\lvert\phi\rangle=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\,g(p)\,\lvert p\rangle\otimes\beta_{k,\sigma}(p)\otimes\lvert f_k\rangle$$
  on the infinite chain. $\langle F\rangle_g:=\int_{-\pi}^{\pi}\frac{\mathrm dp}{2\pi}\lvert g(p)\rvert^2F(p)$. The start is **concentrated at $p_0$** with the spread $\delta p$ if, on the arc $(p_0-\pi,p_0+\pi]$, the weight $\lvert g\rvert^2\frac{\mathrm dp}{2\pi}$ has the mean $p_0$ and the variance $\delta p^2$; it is **at rest** if $p_0=0$. A **sharp-label limit** is a limit along a sequence of such starts with $\delta p\to0$; the package names the class of sequences for which each limit is taken.
- **Orders.** "First order in the field" means first order in $Y$, at fixed $\lambda$, fixed start and fixed chain, followed by the limit of a long chain. The mean position label is $\bar x_{\rm lab}(\lambda):=\sum_xx\,p_x(\lambda)$, with $p_x$ the place weights of the view of the body. Records: the weak-record regime, $\kappa\to0$ at fixed $\lambda$, chain and start, with the term $\kappa\,x\,\langle X\rangle$ counted as a plain gradient of the effective body contract, as in 04-branch-fall; in items 2–4 set $\langle X\rangle=0$.

1. **First order in the field.** For every $C_f$, $Y$, $\varphi$ and every start $\phi\otimes\varphi\otimes\chi$:
   - (a) Determine through which functions of $(\varphi,C_f,Y)$ the first-order term of the view of $\{b,c\}$, and of the statistics of every schedule of readings on $b$ and $c$, depends on the field.
   - (b) Give the generator on $\mathcal H_b\otimes\mathcal H_c$, depending on $\lambda$, that reproduces this first-order term, and state whether it is the contract of an admissible description (A5). State what (a) becomes for a sharp and stationary field, by (2.3), and for a field with (2.11), by (2.12).
   - (c) State, from (2.14) and from the first-order result, at which order in $Y$ and in $\lambda$ the view of $\{b,c\}$ first differs from the one produced by the generator of (b), for a sharp start that is not stationary.

2. **Fall in a gradient that changes with $\lambda$.** For a branch start $B_{k,\sigma}[g]$ of Model D:
   - (a) Determine the first-order velocity of the mean position label as a functional of $\gamma(\cdot)$ on $[0,\lambda]$, with its kernel given as an average $\langle\cdot\rangle_g$ of an explicit function of $p$. Check it against (4.7) for a constant $\gamma$ (there $G_k=\gamma M_k$, $\mathcal F_k=0$, $g_{\zeta,k}=0$).
   - (b) For starts at rest, in a sharp-label limit: determine the kernel in closed form, and the first-order velocity and acceleration for $\gamma(\lambda)=\gamma_0+\gamma_1\cos(\Omega\lambda+\vartheta)$ with real $\gamma_0$, $\gamma_1$, $\vartheta$ and $\Omega>0$. Separate the part that oscillates with the frequency $\Omega$ of the field, the parts that oscillate with other frequencies, and any part that grows with $\lambda$. State the order of the limits used.

3. **Independence of the clock level.**
   - (a) For $\Omega\neq2M_k$: determine the complex number $\mathcal A_{k,\sigma}(\Omega)$ for which the part of the first-order acceleration of item 2(b) that oscillates with the frequency $\Omega$ is $\operatorname{Re}\bigl[\mathcal A_{k,\sigma}(\Omega)\,\gamma_1e^{i(\Omega\lambda+\vartheta)}\bigr]$. Determine its value for $\Omega\to0$ and compare it with (4.16); determine the necessary and sufficient condition under which $\mathcal A_{k,\sigma}(\Omega)$ is the same for all levels, and its leading dependence on the level for $\Omega\ll2M_k$.
   - (b) For $\Omega=2M_k$: determine how the first-order velocity grows with $\lambda$, and what distinguishes this level from the others.
   - (c) State for which field frequencies, relative to the level data, the fall in the field is the same for all levels to a given relative accuracy $\delta$, and which statement replaces "the fall is the same for all levels" for a general $\gamma(\lambda)$.

4. **Example and controls.** A field with two places $\lvert0\rangle$, $\lvert1\rangle$, $Y=y\,\lvert1\rangle\langle1\rvert$ with a real number $y$, $C_f=\gamma_f\bigl(\lvert0\rangle\langle1\rvert+\lvert1\rangle\langle0\rvert\bigr)$ with $\gamma_f\neq0$, and $\varphi=\lvert0\rangle$: the field (2.13) coupled as a gradient.
   - (a) Determine $\gamma(\lambda)$ and evaluate items 2(b) and 3 for it.
   - (b) Controls: a sharp and stationary field (state what items 2 and 3 reduce to); a field with $[Y,C_f]=0$ that is not sharp (state $\gamma(\lambda)$ and which of the results of this package describe it, and which do not).

5. **Summary.** State in a table, for each of: the factor that the body sees; the drift acceleration from rest; its independence of the clock level: what a field that is not stationary changes, at which order in $Y$, and for which field frequencies, with the tags of the results used.

## Inputs

From 02-field-object@v2 (this subproject). Setting there: a body $b$, a field $f$ and a set $r$ of further objects (possibly empty), of pairwise different types, $\mathcal H=\mathcal H_{br}\otimes\mathcal H_f$ with $\mathcal H_{br}:=\mathcal H_b\otimes\mathcal H_r$; $C_{br}$ is a sum of pair terms among $b$ and the objects of $r$, $C_f$ is self-adjoint on $\mathcal H_f$, and $B_j$ (on $\mathcal H_b$), $\Phi_j$ (on $\mathcal H_f$), $j\in J$, are finite families of self-adjoint operators; $B_j$ also denotes $B_j\otimes\mathbb 1_r$. The start is $\Psi_0=\Psi_{br}\otimes\varphi$ and $\lambda\ge0$; $V(\lambda)$ is the view of $\{b\}\cup r$. A field state $\varphi$ is **sharp** if $\Phi_j\varphi=n_j\varphi$ for all $j$, with real numbers $n_j$. A **schedule** consists of readings with projectors $\Pi_i\otimes\mathbb 1_f$ at $0\le\lambda_1\le\dots\le\lambda_k$, where $\Pi_i$ is diagonal in the joint places of $b$ and $r$; $p(r_1,\dots,r_k)$ is the joint distribution of its outcomes.

Eq. (2.1) of 02-field-object, the contract:

$$
C=C_{br}\otimes\mathbb 1_f+\sum_{j\in J}B_j\otimes\mathbb 1_r\otimes\Phi_j+\mathbb 1_{br}\otimes C_f
$$

Eq. (2.3) of 02-field-object, for a field state with $\Phi_j\varphi=n_j\varphi$ and $C_f\varphi=e_f\varphi$ (sharp and stationary), and every $\Psi\in\mathcal H_{br}$:

$$
C\,(\Psi\otimes\varphi)=\bigl((C_{\rm red}+e_f)\Psi\bigr)\otimes\varphi,\qquad C_{\rm red}:=C_{br}+\sum_{j\in J}n_j\,B_j\otimes\mathbb 1_r .
$$

Eq. (2.11) of 02-field-object. Let $\varphi$ be sharp with the numbers $n_j$, and let $\mathbb 1_b$ and the $B_j$ be linearly independent. The view $V(\lambda)$ is pure for every $\Psi_{br}$ and every $\lambda\ge0$ if and only if

$$
\Phi_j\,C_f^{\,k}\,\varphi=n_j\,C_f^{\,k}\,\varphi\qquad\text{for all } j\in J \text{ and all } k\ge0 .
$$

Eq. (2.12) of 02-field-object. If (2.11) holds (linear independence is not needed), then with $C_{\rm red}$ of (2.3), $\Psi_{\rm red}(\lambda):=e^{-iC_{\rm red}\lambda}\Psi_{br}$ and $V_f$ the view of $\{f\}$:

$$
\Psi(\lambda)=\Psi_{\rm red}(\lambda)\otimes e^{-iC_f\lambda}\varphi,\qquad V(\lambda)=\lvert\Psi_{\rm red}(\lambda)\rangle\langle\Psi_{\rm red}(\lambda)\rvert,\qquad V_f(\lambda)=\lvert e^{-iC_f\lambda}\varphi\rangle\langle e^{-iC_f\lambda}\varphi\rvert .
$$

Eq. (2.13) of 02-field-object, an example: a field with two places $\lvert0\rangle$, $\lvert1\rangle$, one index, and $\gamma\neq0$; $\varphi$ is sharp and (2.11) fails at $k=1$:

$$
\Phi_1=\lvert1\rangle\langle1\rvert,\qquad C_f=\gamma\bigl(\lvert0\rangle\langle1\rvert+\lvert1\rangle\langle0\rvert\bigr),\qquad \varphi=\lvert0\rangle:\qquad n_1=0,\qquad \omega_1^{(1)}=\Phi_1C_f\varphi=\gamma\lvert1\rangle\neq0 .
$$

Eq. (2.14) of 02-field-object, for every sharp $\varphi$, every contract of the form (2.1) and every start $\Psi_{br}$, as an expansion for $\lambda\to0$ at fixed operators and start:

$$
1-\operatorname{Tr}V(\lambda)^2=\frac{\lambda^4}{2}\sum_{j,j'\in J}\Gamma_{jj'}\,\bigl\langle C_f\varphi\bigm\vert(\Phi_j-n_j)(\Phi_{j'}-n_{j'})\bigm\vert C_f\varphi\bigr\rangle+O(\lambda^5),\qquad \Gamma_{jj'}=\langle B_jB_{j'}\rangle_{\Psi_{br}}-\langle B_j\rangle_{\Psi_{br}}\langle B_{j'}\rangle_{\Psi_{br}} .
$$

From 05-moving-clock@v3 of 04-ilang-time. Setting there: one Model D body with the contract $C_b=C_{\rm D}$ of the base problem, the records $K_x=\kappa xX$ and the start $\lvert\phi\rangle\otimes\lvert\chi\rangle$; $d_k:=\frac{2\pi k}{N\lambda_0}$; the plane waves $\lvert p\rangle=\sum_xe^{ipx}\lvert x\rangle$, $p\in(-\pi,\pi]$.

Eq. (5.5) of 04-ilang-time/05-moving-clock:

$$
C_b\bigl(\lvert p\rangle\otimes\beta\otimes\lvert f_k\rangle\bigr)=\lvert p\rangle\otimes C_k(p)\beta\otimes\lvert f_k\rangle,\qquad C_k(p):=-\tau\sin p\,\sigma_1+\mu_k\sigma_3,\qquad\mu_k:=m+d_k .
$$

Eq. (5.6) of 04-ilang-time/05-moving-clock, the eigenvalues $\pm\omega_k(p)$ and the real eigenvectors $\lvert p\rangle\otimes\beta_{k,\pm}(p)\otimes\lvert f_k\rangle$ of $C_b$:

$$
\omega_k(p)=\sqrt{\mu_k^2+\tau^2\sin^2p},\quad
\beta_{k,+}=\begin{pmatrix}\cos\frac{\theta_k}2\\ \sin\frac{\theta_k}2\end{pmatrix},\quad
\beta_{k,-}=\begin{pmatrix}-\sin\frac{\theta_k}2\\ \cos\frac{\theta_k}2\end{pmatrix},\quad
\tan\theta_k(p)=-\frac{\tau\sin p}{\mu_k},\ \ \lvert\theta_k\rvert<\frac\pi2 .
$$

Eq. (5.11) of 04-ilang-time/05-moving-clock, the bound (17.7) of 03-ilang-space/17-motion for this $C_b$ (the maximal speed):

$$
v_{\max}=\tau\,\kappa\sigma_X=\max_h\sum_{h'}\lvert t_{hh'}\rvert\,d_0(h,h') .
$$

From 08-maximal-speed@v3 of 04-ilang-time. Setting there: two bodies $a$, $b$ of different types and a medium $c$, with body contracts $C_A$, $C_B$ (diagonal entries $t^A_{hh}$, $t^B_{kk}$) and records $K^A_h$, $K^B_k$; the start is a product with the medium in $\chi$.

Eq. (8.1) of 04-ilang-time/08-maximal-speed, all replacements of the diagonal entries of the body contracts and of the records that leave the total contract unchanged, with arbitrary real $\alpha^A_h$, $\alpha^B_k$ and an arbitrary self-adjoint $Z$ on $\mathcal H_c$:

$$
t'^A_{hh}=t^A_{hh}+\alpha^A_h,\quad K'^A_h=K^A_h+Z-\alpha^A_h\mathbb 1_c,\qquad
t'^B_{kk}=t^B_{kk}+\alpha^B_k,\quad K'^B_k=K^B_k-Z-\alpha^B_k\mathbb 1_c ,
$$

Eq. (8.19) of 04-ilang-time/08-maximal-speed, the effective body contracts:

$$
C^{\rm eff}_A\otimes\mathbb 1_b+\mathbb 1_a\otimes C^{\rm eff}_B:=(\mathbb 1_{ab}\otimes\langle\chi\rvert)\,C\,(\mathbb 1_{ab}\otimes\lvert\chi\rangle),\qquad
C^{\rm eff}_A=C_A+\sum_h\langle K^A_h\rangle\lvert h\rangle\langle h\rvert,\quad
C^{\rm eff}_B=C_B+\sum_k\langle K^B_k\rangle\lvert k\rangle\langle k\rvert .
$$

Eq. (8.20) of 04-ilang-time/08-maximal-speed:

$$
\text{(8.1):}\quad C^{\rm eff}_A\mapsto C^{\rm eff}_A+\langle Z\rangle\,\mathbb 1_a,\quad C^{\rm eff}_B\mapsto C^{\rm eff}_B-\langle Z\rangle\,\mathbb 1_b;\qquad \chi\mapsto e^{i\theta}\chi:\quad C^{\rm eff}_A,\ C^{\rm eff}_B\ \text{unchanged}.
$$

From 04-branch-fall@v3 (this subproject). Setting there: a body with the places of Model D and a conserved clock level, whose level blocks are $C^{(k)}=\frac{\tau_k}2\sum_x\zeta_{x,k}\bigl(\lvert x+1\rangle\langle x\rvert\otimes(-i\sigma_1)+\lvert x\rangle\langle x+1\rvert\otimes(i\sigma_1)\bigr)+\sum_x\lvert x\rangle\langle x\rvert\otimes\bigl(M_{x,k}\sigma_3+V_{x,k}\mathbb 1_2\bigr)$ with the linear level profiles $M_{x,k}=M_k+G_kx$, $V_{x,k}=-\mathcal F_kx$, $\zeta_{x,k}=1+g_{\zeta,k}(x+\tfrac12)$ and contracts that do not depend on $\lambda$; a medium with the records $K_x=\kappa\,x\,X$; branch starts $B_{k,\sigma}[g]$ as under "Goal" (with $\tau\to\tau_k$); first order in the gradients $G_k$, $\mathcal F_k$, $g_{\zeta,k}$ of the level $k$, at fixed $\lambda$, start and chain, followed by the limit of a long chain; records at leading order in $\kappa$. Model D is the member $\tau_k=\tau$, $M_k=m+d_k$ with all gradients zero. Inside one level the index $k$ is dropped: $\tau,M,G,\mathcal F,g_\zeta,\omega$; a prime is $\partial_p$; $\mathsf b:=\lvert G\rvert+M\lvert g_\zeta\rvert+\lvert\mathcal F\rvert$; $X_g:=\bigl(\int\frac{\mathrm dp}{2\pi}\lvert g'\rvert^2\bigr)^{1/2}$. The **drift acceleration** $a_{k,\sigma}(p_0)$ is the first-order acceleration of $\bar x_{\rm lab}$, averaged over a long window, followed by the limit of a sharp label.

Eq. (4.2) of 04-branch-fall, the level block for the linear level profiles on the infinite chain, exactly ($\hat x=i\partial_p$ on plane-wave amplitudes):

$$
C^{(k)}=C_0+C_1,\quad C_0=-\tau\sin\hat p\,\sigma_1+M\sigma_3,\quad C_1=\tfrac12\{\hat x,B(\hat p)\},\quad B(p)=-g_\zeta\tau\sin p\,\sigma_1+G\sigma_3-\mathcal F .
$$

Eq. (4.5) of 04-branch-fall, the velocity of $B_{k,\sigma}[g]$ at zero gradients, exactly:

$$
v^{(0)}(\lambda)=\sigma\,\langle\omega_k'\rangle_g\qquad(\text{constant in }\lambda).
$$

Eq. (4.7) of 04-branch-fall. The first-order velocity of $B_{k,\sigma}[g]$ is $\lambda\,\langle a_{k,\sigma}\rangle_g$ plus a rest that is bounded for all $\lambda\ge0$, with the coefficient

$$
\boxed{\;a_{k,\sigma}(p)=\frac{g_{\zeta,k}\tau_k^4\sin^2p-G_kM_k\tau_k^2\cos2p}{\omega_k^2}+\sigma\,\mathcal F_k\,\frac{\tau_k^2\bigl(M_k^2\cos2p-\tau_k^2\sin^4p\bigr)}{\omega_k^3}\;},\qquad\lvert a_{k,\sigma}(p)\rvert\le\frac{\tau_k^2}{M_k}\,\mathsf b_k .
$$

Eq. (4.10) of 04-branch-fall, with $a^{(1)}(\lambda)$ the first-order acceleration of $\bar x_{\rm lab}$ and $\overline{a^{(1)}}$ its average over a window $[\lambda_1,\lambda_1+\Lambda]$, for all $\lambda_1\ge0$, $\Lambda>0$ and every smooth $g$:

$$
\Bigl\lvert\overline{a^{(1)}}-\langle a_{k,\sigma}\rangle_g\Bigr\rvert\le\frac{2\tau}{M\Lambda}\,\mathsf b\,\Bigl(X_g+\frac{\tau}{2M}\Bigr)\ \ (\text{exact}),\qquad\lim_{\Lambda\to\infty}\overline{a^{(1)}}=\langle a_{k,\sigma}\rangle_g .
$$

Eq. (4.11) of 04-branch-fall, for every sequence of branch starts concentrated at $p_0$ with $\delta p\to0$:

$$
\bigl\lvert\langle a_{k,\sigma}\rangle_g-a_{k,\sigma}(p_0)\bigr\rvert\le\tfrac12\sup_p\lvert a_{k,\sigma}''\rvert\,\delta p^2\ \ (\text{exact}),\qquad\text{drift acceleration}=\lim_{\delta p\to0}\lim_{\Lambda\to\infty}\overline{a^{(1)}}=a_{k,\sigma}(p_0).
$$

Eq. (4.16) of 04-branch-fall, the drift acceleration at rest:

$$
\boxed{\;a_{k,\sigma}(0)=\frac{\tau_k^2}{M_k}\,\bigl(\sigma\,\mathcal F_k-G_k\bigr)\;}
$$

## Assumptions

- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5), and $\lambda\ge0$; the dependence on $\lambda$ of what the body sees comes from the state of the field alone.
- The three types are pairwise different and each has one instance, so no swap condition applies.
- The form of the contract, the product start and the branch starts are assumptions of this package (M8, M9). The results are asked for every $\varphi$, through $\gamma(\lambda)$ (A11).
- Standard first-order perturbation theory in a term of the contract, and standard facts on plane waves and on the averaging of oscillating terms, may be used; state them with their conditions.
- Every limit is stated with its order: the limit of a sharp label, an average over a window, the growth of $\lambda$ and the expansion parameters. Where two of them do not commute, both orders are given, each with the quantity it defines. A statement about a rate of change (a velocity, an acceleration, the rate of a clock) is derived for that rate; it is not inferred from a bound or an estimate of the quantity whose derivative it is.
- Every statement with "all", "iff" or "well defined" is stated with its exceptional members (a gradient operator that is a multiple of the identity, a field frequency that vanishes or equals a gap of the body, equal data of two levels); a condition that is only sufficient is called sufficient.

## Scope

- **Class.** Item 1: every $C_f$, $Y$, $\varphi$, every start of the body and of the medium. Items 2 and 3: every level and both branches of Model D, every smooth $g$ in 2(a), starts at rest in 2(b) and 3, every $\gamma_0$, $\gamma_1$, $\vartheta$, $\Omega>0$. Item 4: the stated field.
- **Controls.** A constant $\gamma$ against a $\gamma(\lambda)$ that changes (items 2 and 3); the two controls of item 4(b).
- In scope:
  - what the body sees of a field that is not stationary, at first order in the field;
  - the first-order velocity and acceleration of a branch start in a gradient that changes with $\lambda$;
  - the conditions on the field frequencies under which the fall from rest is the same for all clock levels.
- Out of scope:
  - the second order in the field beyond the statement of item 1(c); the stationary start in an eigenvector of $C_f$ and its second-order terms (10-field-own-contract); a static field that is not sharp (09-unsharp-field);
  - a factor that changes with $\lambda$ at the reference position (a term $G\otimes N_0$ with an operator $N_0$), the clock statistics in a changing field, the fall in motion and lattice terms;
  - where the field comes from, two bodies in one field, and instances of one type;
  - any physical meaning beyond the definitions (M1–M4); no theory of gravity and no theory that contains time may be used as an input.

## Depth

- Item 1: short argument.
- Item 2: derive.
- Item 3: derive.
- Item 4: short argument.
- Item 5: standard (a table of results with their tags).

## Expected result

- Item 1: a statement with the functions named; a generator with a yes/no answer and two special cases; orders.
- Item 2: a functional with its kernel and a check; a closed form with the separated parts and the order of the limits.
- Item 3: a closed form with a limit, an equivalence and a leading term; a growth law; a condition and a statement.
- Item 4: a function and evaluations; two controls.
- Item 5: a table.

Give every main result a tag $(12.k)$.

Consistency checks, at most three: for example a constant $\gamma$, the limit $\Omega\to0$, and dimensions.

## Code

None.

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