# Question: 02-grip-weight

- **Subproject:** 08-ilang-propagation
- **Package:** 02-grip-weight
- **Equation tags:** (2.k)
- **Created:** 2026-10-24

## Goal

Determine how the weight of a held body, of its own mark and of a bond made through the field depends on the grip function, for a field of oscillator cells, and for which grips the mark and the bond are weighed like the first-order part of the cost. This is item 1 of the base problem for a field of cells. Every result is stated through quantities that the total contract and the state fix (M6).

**Setting.**
- **Field** (assumption, M8). Oscillator cells as in the base problem ("Field of cells"): one cell at each place $x$ of $X_f$, a ring ($D=1$), a square torus ($D=2$) or a cubic torus ($D=3$) of side $L$, $\lvert X_f\rvert=L^D$, each cell an object of its own type with the places $0,\dots,M-1$; the contract is the compression $C_M=\Pi H\Pi$ of
  $$H=\sum_x\tfrac12p_x^2+\frac{\kappa_h}2\sum_{\langle xy\rangle}(q_x-q_y)^2+\frac{m_h^2}2\sum_xq_x^2,\qquad\kappa_h>0,\ m_h>0,$$
  plus the terms of the bodies and the source below; $\Phi_x=q_x$. Statements about the uncompressed contract are statements about the limit $M\to\infty$, taken first; state in which sense it is taken.
- **Bodies** (assumption, M8, A15). Held Model D bodies ($\tau=0$), each at a position label identified with one cell (the separations used are cell separations; their relation to the distances of 03-ilang-space is out of scope, M2), in the eigenvector $\lvert\sigma\rangle\lvert f_k\rangle$ of $S:=\sigma_3\otimes(m\mathbb 1_N+D)$ with eigenvalue $\sigma M_k$. A body has no rest and clock term of its own; it holds on to the field by the grip of the base problem, $\sum_x\lvert x\rangle\langle x\rvert\otimes S\otimes F(q_x)$, with a real grip function $F$. Two classes of grips:
  - (G2) $F(q)=F_0+F_1q+F_2q^2$ with real $F_0,F_1,F_2$, $F_1\ne0$: the total contract stays quadratic in the $q$'s;
  - (G) a real function $F$, smooth at $q=0$, with $F'(0)\ne0$, written $F(q)=F(0)+\theta f(q)$ with $f(0)=0$, $f'(0)=1$, studied through the lowest orders in $\theta$ at which the mark and its weight appear.
- **Source** (assumption, M8, A15). One held object at $x_s$ coupled to the cell there by the pair term $g_sq_{x_s}$, $g_s$ real; first order in $g_s$.
- **State** (assumption, M9, A11). For (G2): the lowest eigenvector of the total contract, in the limit $M\to\infty$, where it exists; state the condition on the grip under which it exists. For (G): the eigenvalue of $C_M$ connected, as $\theta\to0$, to the lowest eigenvalue at $\theta=0$; its Taylor coefficients in $\theta$, each taken in the limit $M\to\infty$; state where this eigenvalue is the lowest one.
- **Definitions of this package.**
  - *Cost* of a body: the lowest eigenvalue of the description with the body (in its internal state $(k,\sigma)$) minus that without the body, both without the source.
  - *First-order part* $C^{(1)}$ and *mark* $C^{\rm mark}$: replace $F$ by $sF$ and expand the cost in $s$; $C^{(1)}$ is the term linear in $s$, at $s=1$, and $C^{\rm mark}$ is the cost minus $C^{(1)}$.
  - *Weight*: the dependence of the same eigenvalue, with the source present, on the place $x$ of the held body, $W(x)-W(x')$, as in the base problem. The weights of $C^{(1)}$ and of $C^{\rm mark}$ are defined by the same expansion in $s$, with the source present.
  - *Bond*: for two held bodies at $x_1$ and $x_2=x_1+\mathbf r$ with grips of the same class and no pair term between them, the part of their total cost that depends on $\mathbf r$, without the source; its weight is the dependence of that part on a common shift of both bodies with the source present.
  - *Weighing ratios*: $\rho_{\rm mark}$, the weight of the mark per unit of mark divided by the weight of $C^{(1)}$ per unit of $C^{(1)}$; and $\rho_{\rm bond}$, the weight of the bond per unit of bond profile divided by the same quantity for $C^{(1)}$, referred to the midpoint of the pair. Where a ratio has a vanishing denominator, say so.

**Note (2026-10-24, after external round 1).**
- (a) Class (G) is restricted to grips of a fixed shape on the scale of the field's fluctuations: $F(q)=F(0)\,\Phi(\beta q)$ with $F(0)\ne0$, $\Phi(0)=1$, $\Phi'(0)\ne0$, or $F(q)=F(0)+\theta f(q)$, where $\Phi$ and $f$ are smooth and every derivative of them is bounded by a polynomial. Results are expansions in $\beta$ (resp. $\theta$) at fixed $\Phi$ (resp. $f$). Grips outside this class are out of scope.
- (b) The reference of $\rho_{\rm bond}$ is the weight of $C^{(1)}$ per unit of $C^{(1)}$ of a single body at the midpoint of the pair with the grip of body $i$; this gives one ratio $\rho_{{\rm bond},i}$ per body, as in 07-ilang-weight/04-weight-of-bond. The two coincide for a common grip. With this reference, pairs whose first-order costs cancel need not be excluded. The two bodies may have different grips of the class.
- (c) For members whose mark weight vanishes at the lowest order (for example shapes odd about zero), state this and the lowest order at which a nonzero weight can appear; its value at that order is not asked.

1. **Cost and mark.** For (G2) exactly, and for (G) through the lowest order in $\theta$ at which $C^{\rm mark}$ appears: determine the cost, $C^{(1)}$ and $C^{\rm mark}$ of a held body as functions of $k$, $\sigma$, the grip and the field parameters ($\kappa_h$, $m_h$, $L$, $D$), and state which part is fixed by the total contract.
2. **Weight.** At first order in $g_s$: determine $W(x)-W(x')$ for both classes, split into the weights of $C^{(1)}$ and of $C^{\rm mark}$, on the three graphs, with their forms at distances from the source large against the lattice spacing, both below and above the pinning length $\sqrt{\kappa_h}/m_h$.
3. **Bond.** For two held bodies with grips of either class: determine the bond profile and its weight, referred to the midpoint, at first order in $g_s$ and through the lowest order in the grip at which the bond and its weight appear; say how the weight changes if another place of the pair is used.
4. **Which grips weigh.** Determine for which grips (and for which $\kappa_h$, $m_h$, $L$, $D$) $\rho_{\rm mark}=1$ and $\rho_{\rm bond}=1$ for every level and both branches, at which order in the grip and for which distances; where they are not 1, the first deviation, its parameter and its dependence on the distance.

## Inputs

None. The package builds only on the base problem (its notation "Model D", "Field of cells", "Grip", "Weight, inertia and cost").

## Assumptions

- The setting under "Goal". The field class, the bodies, the grips, the source and the field state are assumptions of the package (M8, M9).
- Standard mathematics applied to the stated operators may be used (quadratic forms and their lowest eigenvectors, Gaussian expectation values, perturbation theory for an isolated eigenvalue, compressions and min-max, Fourier analysis and lattice sums on the three graphs); state it with its conditions.

## Scope

- **Class.** Both grip classes, every level and both branches, the three graph families, every $L$ and $m_h>0$ in a stated regime; first order in $g_s$; the full grip dependence for (G2) where the lowest eigenvector exists.
- **Order of limits.** $M\to\infty$ first, then the size, then the distances, unless a result states another order; say how the results depend on $m_h$ and whether they have a limit as $m_h\to0$ at fixed $L$ and as $L\to\infty$.
- **Cancellations.** Name the members of the class at which a denominator of a weighing ratio vanishes (for example a body whose $C^{(1)}$ vanishes, or a pair whose first-order costs cancel) and exclude them explicitly.
- **Control.** For each answer of item 4, a description of the class, or of a stated wider class, in which it fails, or a proof that there is none. The linear grip ($F_2=0$ in (G2)) is one member to treat explicitly.
- **Other states** (A11). For one stated eigenvector of the field other than the lowest, say whether the answers of item 4 change.
- Out of scope: moving bodies and inertia; the speed with which the made field forms (04-formation); fields of identical objects (03-gas-grip); fields without pinning ($m_h=0$) and anchors (a later package).

## Depth

- Item 1: derive (needed by items 2–4).
- Item 2: derive (needed by item 4).
- Item 3: derive (needed by item 4).
- Item 4: derive (needed by item 1 of the base problem and by the later package on anchors, item 3 of the base problem).
- Other states: short argument (needed by item 1 of the base problem, A11).

## Expected result

- Item 1: expressions for the cost, $C^{(1)}$ and $C^{\rm mark}$, exact for (G2) and as an expansion for (G), with the existence condition.
- Item 2: expressions for the weights on the three graphs, with large-distance forms.
- Item 3: expressions for the bond profile and its weight.
- Item 4: conditions on the grip and the field parameters, with the order and the range of distances; the first deviation otherwise.

Give every main result a tag $(2.k)$.

Consistency checks, at most three: for example the limit of a linear grip, the sum of the weight over all places, and dimensions.

## Code

Required: check: (i) for (G2) grips, on a ring and on a cubic torus of two sizes each, $\rho_{\rm mark}$ and $\rho_{\rm bond}$ of item 4 against the exact lowest eigenvalue of the uncompressed quadratic total contract, for two levels and both branches and at least three distances; (ii) on a ring of three cells, the lowest eigenvalue of the compression $C_M$ for increasing $M$ against the uncompressed value, for one grip of each class.

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