# Question: 23-random-arrangements

- **Subproject:** 03-ilang-space
- **Package:** 23-random-arrangements
- **Equation tags:** (23.k)
- **Created:** 2026-10-09

## Goal

Determine by a precommitted numerical run, in parts T1–T4, two properties of the small-$\lambda$ chain geometry $\ell_0$ of quasi-local records in two dimensions:
- whether non-periodic (random) arrangements of places give an isotropic large-scale geometry;
- how robust the neighbour graph and $\ell_0$ are against small changes of the records.

22 showed that every fixed periodic arrangement gives a polygonal, anisotropic norm (22.29). This package tests random arrangements numerically; it addresses criterion 4. Isotropy means the same as in 22: the large-scale distance is a Euclidean norm.

**Precommitment.**
- Every model, parameter, seed, size, window, tolerance and decision rule is fixed in this question before the run. None may be changed.
- If a part cannot be run as specified, report that, and do not modify the part.
- Every reported number must be produced by the scripts and saved as raw output.
- A decision follows the stated rule only. No other criterion may be substituted after the run.
- Parts marked exploratory have no decision.

**Common setting.**
- **Medium.** An $L\times L$ torus of qubit cells $l\in(\mathbb Z/L\mathbb Z)^2$, with $\chi=\lvert0\cdots0\rangle$ (one object, as in 15 and 22). $\lvert v\rvert_T$ is the Euclidean length of the minimal image of $v$ on the torus.
- **Places and records.** Places $h$ have centres $c_h$. In T1–T3 the centres are grid points; in T4 they are not. The records are
  $$K_h=\sum_l\kappa(l-c_h)\,\sigma_x^{[l]}$$
  with one of two kernels, where $\xi=2$:
  - **separable:** $\kappa_{\rm sep}(v)=e^{-(\lvert v_1\rvert_T+\lvert v_2\rvert_T)/\xi}$, where $\lvert v_k\rvert_T$ is the minimal-image absolute value of the coordinate;
  - **isotropic:** $\kappa_{\rm iso}(v)=e^{-\lvert v\rvert_T/\xi}$.
- **Distances.** By (22.2), $d_0(h,h')^2=\sum_l(\kappa(l-c_h)-\kappa(l-c_{h'}))^2$. With the overlap $S(h,h'):=\sum_l\kappa(l-c_h)\kappa(l-c_{h'})$, this is $d_0^2=S(h,h)+S(h',h')-2S(h,h')$.
- **Neighbour graph.** $N_0$ is the graph of (4.3) applied to $d_0$: $\{x,x'\}$ is an edge iff no third place $y$ has $\max\bigl(d_0(x,y),d_0(y,x')\bigr)<d_0(x,x')$. A tie keeps the edge, as the strict inequality requires.
  - Where (N1)/(N2) of 22 decide, this is the small-$\lambda$ neighbour graph.
  - Report the number of pairs whose status rests on a tie.
- **Chain distance.** $\ell_0$ is the $d_0$-weighted shortest-path distance in $N_0$ (22.4).
- **Numerical exactness.**
  - For far pairs, $d_0$ is close to its saturation value, and the decisions rest on very small differences.
  - All comparisons that decide neighbours must be made without cancellation. For example, compare overlaps where the norms are equal, or compare $d_0^2-S(0)$-type quantities computed directly from their small terms. Never subtract nearly equal large numbers.
  - Overlaps $S$ of far pairs must be computed with relative (not only absolute) accuracy. Sums of positive terms qualify; FFT-based correlations do not, for values below $10^{-12}S(0)$.
  - Where all places have equal norms by symmetry, the code must make displacements that are equivalent under the symmetries of the torus grid give bitwise identical overlaps.
  - Report the smallest decisive margin found in each part. If it is below $100$ times the estimated rounding error, report the affected pairs.
- **Random numbers** come from `numpy.random.default_rng(seed)`, with the seeds given below.
- **Ratios and angles.** For a pair at torus displacement $\Delta$ with $r:=\lvert\Delta\rvert_T$, the ratio is $\varrho:=\ell_0/r$. The folded angle $\varphi\in[0,\pi/4]$ is the angle of $\Delta$ reduced by the symmetries of the square (reflections in the axes and diagonals).

**T1. Code check on a periodic lattice** (tests (22.25)–(22.26)).
- **Setting.** $L=32$, places at all $L^2$ grid points, separable kernel.
- **Determine** $N_0$ and $\ell_0$.
- **Decision.** Pass iff both hold:
  - $N_0$ is exactly the 4-neighbour torus grid;
  - $\max\bigl\lvert\ell_0-\rho\lVert\Delta\rVert_{1,T}\bigr\rvert\le10^{-9}\rho L$ over all pairs, where $\rho:=d_0$ of a unit step and $\lVert\Delta\rVert_{1,T}$ is the minimal-image $\ell^1$ length.

**T1b. Periodic lattice, isotropic kernel** (reference; exploratory).
- **Setting.** $L=32$, places at all grid points, isotropic kernel.
- **Determine** the neighbour offsets of $N_0$ and $B_{\rm per}$. Here $B_{\rm per}$ is the ratio of the mean $\varrho$ over pairs with $\varphi\in[3\pi/16,\pi/4]$ to the mean $\varrho$ over pairs with $\varphi\in[0,\pi/16]$, for $r\in[8,14]$.

**T2. Random arrangements: isotropy.**
- **Setting.** Isotropic kernel. The places are a random subset of the grid points: each point is included independently with probability $p=0.1$.
  - Sizes $L\in\{64,128\}$.
  - Five realizations per size: seeds $2301,\dots,2305$ for $L=64$ and $2311,\dots,2315$ for $L=128$.
- **Determine** for each realization:
  - $N_0$ and $\ell_0$;
  - the mean degree and the maximum degree of $N_0$, and its number of connected components;
  - for the pairs with $r\in[L/8,L/4]$, the mean $\varrho$ in four bins of $\varphi$ of width $\pi/16$;
  - $B:=$ (mean $\varrho$ in the bin $[3\pi/16,\pi/4]$) / (mean $\varrho$ in the bin $[0,\pi/16]$).
- **Report** for each size the mean of $B$ over the five realizations and its standard error $\mathrm{SE}$ (sample standard deviation $/\sqrt5$).
- **Decision, at $L=128$.**
  - *Isotropy supported* iff $\lvert\bar B-1\rvert<0.02$ and $\lvert\bar B-1\rvert<2\,\mathrm{SE}$.
  - *Anisotropy detected* iff $\bar B-1>0.02$ and $\bar B-1>3\,\mathrm{SE}$.
  - *Inconclusive* otherwise.
  - Report $\bar B$ at $L=64$ as well; it is not part of the decision.

**T3. Robustness against a small non-local admixture.**
- **Setting.** The first $L=128$ realization of T2 (seed 2311). Each record vector is changed to $u_h\to u_h+\varepsilon g_h$, where the $g_h$ are independent standard Gaussian vectors on all $L^2$ cells, normalized to unit length (seed 2321). In operator form, $K_h\to K_h+\varepsilon\sum_lg_{h,l}\sigma_x^{[l]}$.
  - $\varepsilon\in\{10^{-12},10^{-9},10^{-6},10^{-4},10^{-2}\}$.
  - Norms are no longer equal. Use the exactness rule of the common setting, with $d_0^2$ split into its large constant and its small terms.
- **Determine** for each $\varepsilon$:
  - (a) the number of edges of $N_0(\varepsilon)$ that are not edges of $N_0(0)$;
  - (b) the number of **long edges**, i.e. edges with $r>4/\sqrt p$;
  - (c) the maximum degree;
  - (d) the mean $\varrho$ over pairs with $r\in[16,32]$, divided by its value at $\varepsilon=0$;
  - (e) (exploratory) the length of the shortest long edge.
- **Decision.**
  - *Fragile* iff for some $\varepsilon\le10^{-4}$, (b) $>0$ and $\lvert(\mathrm d)-1\rvert>0.05$.
  - *Robust* iff for all $\varepsilon$ (including $10^{-2}$), (b) $=0$ and $\lvert(\mathrm d)-1\rvert<0.01$.
  - *Intermediate* otherwise.

**T4. Continuous centres** (exploratory, no decision).
- **Setting.** $L=128$, isotropic kernel. The centres are the T2 places of seed 2311, each shifted by an independent uniform offset in $[0,1)^2$ (seed 2331). Kernel values use the shifted centres; the norms then differ.
- **Determine** quantities (b)–(d) of T3 for this arrangement, and the spread (max − min) of the norms $S(h,h)$.

## Inputs

From 04-neighbours@v1. Eq. (4.3), neighbours, for distinct points $x,x'$:

$$
x\sim x'\;:\Longleftrightarrow\;x\asymp x'\ \text{ and there is no } y\in X_V \text{ with } \max\bigl(\alpha(x,y),\alpha(y,x')\bigr)<\alpha(x,x') .
$$

Eq. (4.6), the chain distance:

$$
\ell(x,x'):=\min\Bigl\{\sum_{j=0}^{m-1}\alpha(y_j,y_{j+1})\;:\;y_0=x,\ y_m=x',\ y_j\sim y_{j+1}\Bigr\},
\qquad \ell(x,x'):=+\infty\ \text{between components.}
$$

From 05-recording-contract@v1, in the setting of 05. Eq. (5.10):

$$
\lim_{\lambda\to0^+}\frac{\alpha(h,h';\lambda)}{\lambda}=d_0(h,h')=\bigl\lVert u_h-u_{h'}\bigr\rVert,
\qquad \lvert u_h\rangle=\bigl(K_h-\langle K_h\rangle\bigr)\lvert\chi\rangle .
$$

From 22-large-scale-geometry@v3. Qubit-cell media with $K_h=\sum_lk_{h,l}\sigma_x^{[l]}$, $k_{h,l}$ real, $\chi=\lvert0\cdots0\rangle$; $\lvert e_l\rangle$ is the place with only bit $l$ set. Eq. (22.2):

$$
\cos\alpha(h,h';\lambda)=\prod_l\bigl\lvert\cos(\lambda\Delta_l)\bigr\rvert,\qquad \Delta_l:=k_{h,l}-k_{h',l},\qquad
\lvert u_h\rangle=\sum_lk_{h,l}\lvert e_l\rangle,\qquad d_0(h,h')^2=\sum_l\Delta_l^2 .
$$

From Step 2 of 22, verbatim:
- "**(N1)** If some $y$ has $\max\bigl(d_0(x,y),d_0(y,x')\bigr)<d_0(x,x')$, then $x\not\sim x'$ for all small $\lambda$."
- "**(N2)** If every $y\notin\{x,x'\}$ has $\max\bigl(d_0(x,y),d_0(y,x')\bigr)>d_0(x,x')$, then $x\sim x'$ for all small $\lambda$."

$N_0$ is the resulting small-$\lambda$ graph. Eq. (22.4):

$$
\ell_0(x,x')=\min\Bigl\{\textstyle\sum_jd_0(y_j,y_{j+1}):\ y_0=x,\ y_m=x',\ y_j\sim y_{j+1}\text{ in }N_0\Bigr\}.
$$

From 22, item 3(a): places $h_p$, $p\in\{1,\dots,n\}^2$, on an infinite square grid of cells, with $K_{h_p}=\sum_lq^{\lvert l_1-p_1\rvert+\lvert l_2-p_2\rvert}\sigma_x^{[l]}$, $0<q<1$. Eq. (22.25):

$$
N_0=\text{square grid graph on }\{1,\dots,n\}^2 .
$$

Eq. (22.26), with $\rho$ the $d_0$ of a unit step:

$$
\ell_0(h_p,h_{p'})=\rho\bigl(\lvert\Delta_1\rvert+\lvert\Delta_2\rvert\bigr).
$$

Eq. (22.27):

$$
(X_n,\ell_0/n)\xrightarrow{\rm GH}\bigl([0,1]^2,\ \rho\lVert\cdot\rVert_1\bigr).
$$

Eq. (22.29), under the assumptions (F1)–(F3) of 22 (a fixed periodic pattern on growing convex domains whose recomputed neighbour graphs are the induced subgraphs):

$$
\text{Periodic arrangements under (F1)--(F3): the rescaled }\ell_0\text{ never converges to an isotropic limit (polygonal stable norm).}
$$

## Assumptions

- The common setting and the parts T1–T4 as specified. All contracts are independent of $\lambda$ (A5).
- Only the small-$\lambda$ quantities $d_0$, $N_0$, $\ell_0$ are computed; no evolution is needed.
- The torus replaces the infinite grid of 22. Torus effects are kept small by the distance windows ($r\le L/4$).

## Scope

- In scope: parts T1–T4, the reported numbers, the decisions, and a short discussion of what they mean for criterion 4.
- Out of scope:
  - finite $\lambda$;
  - other kernels, densities or sizes;
  - three dimensions;
  - analytic proofs of the observed behaviour;
  - any physical or spatial meaning beyond the definitions (M1, M4).

## Depth

- T1–T4: computation, with a short description of each script and its checks.
- Discussion: short argument. State what the decisions do and do not show.

## Expected result

- T1: a pass/fail decision.
- T1b: the neighbour offsets and one ratio.
- T2: per-realization and mean values with standard errors, and a decision.
- T3: a table over $\varepsilon$ and a decision.
- T4: a small table.
- Discussion: a few sentences.

Give every main result a tag $(23.k)$.

Consistency checks, at most three: for example T1 itself, the symmetry of $N_0$, and the connectivity of $N_0$.

## Code

Required: computation: parts T1–T4 exactly as specified. Scripts go in `code/v1/`, and outputs in `code/v1/output/`. Save outputs as compact summaries (JSON or CSV, each file below 100 kB): no full distance matrices, but every number used in the derivation. Record the runtime of each part.