Version 1 · current · External review, round 1: correct

# Random record operators: the generic leading-order witness geometry

- **Subproject:** 03-ilang-space
- **Package:** 03-ilang-space/07-random-records
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-08

## Setup and assumptions

**Description** (setting of 05, as quoted in question.md). Two objects $a$, $c$ of different types, each the only instance of its type; part $A=\{a\}$. Contract $C=\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$; initial state $\lvert\phi\rangle\otimes\lvert\chi\rangle$ with $\phi_h\neq0$ for every $h$, so all $n:=d_{T(a)}\ge2$ places of $a$ enter (5.10)–(5.14). $d_c:=\dim\mathcal H_c\ge2$; orthonormal basis $e_1=\lvert\chi\rangle,e_2,\dots,e_{d_c}$ of $\mathcal H_c$; $K_{ab}:=\langle e_a\vert K\vert e_b\rangle$. $E$ denotes expectation over the ensemble.

**Inputs** (05-recording-contract@v1). With $g(v,w):=\operatorname{Re}\langle v\vert w\rangle$, $(\mathcal H_c,g)$ is a real Euclidean space of dimension $2d_c$; $h\sim_0h'$ means $u_h=u_{h'}$; $\dim_0:=\dim_{\mathbb R}\operatorname{aff}_{\mathbb R}\{u_h\}$.
- (5.10): $d_0(h,h')=\lVert u_h-u_{h'}\rVert$, $\lvert u_h\rangle=\bigl(K_h-\langle K_h\rangle\bigr)\lvert\chi\rangle$, where $d_0$ is the leading-order distance, $\alpha(h,h';\lambda)=\lambda d_0(h,h')+o(\lambda)$.
- (5.11): $\bigl(H_V/{\sim_0},d_0\bigr)\cong\bigl(\{u_h\},\lVert\cdot\rVert\bigr)\subset(\mathcal H_c,g)$.
- (5.13): $\langle\chi\vert u_h\rangle=0$.
- (5.14): $\dim_0\le\min\bigl(d_{T(a)}-1,\,2d_c-2\bigr)$.

**Ensembles** (definitions of question.md). $K_1,\dots,K_n$ i.i.d. from
- $\mathrm{GUE}(s)$: $K=K^\dagger$; $K_{ab}$ ($a<b$) complex Gaussian with independent real and imaginary parts $N(0,s^2/2)$; $K_{aa}$ real $N(0,s^2)$; all independent.
- $\mathrm{GOE}(s)$: $K$ real symmetric in the basis $(e_a)$; $K_{ab}$ ($a<b$) real $N(0,s^2)$; $K_{aa}$ real $N(0,2s^2)$; all independent.

**Real coordinates.** The vectors $\{e_a,\,ie_a\}_{a\ge2}$ are $g$-orthonormal ($g(e_a,ie_b)=\operatorname{Re}(i\delta_{ab})=0$, $g(ie_a,ie_b)=\delta_{ab}$) and span $\chi^\perp=\operatorname{span}_{\mathbb C}\{e_2,\dots,e_{d_c}\}$. For $v=\sum_{a\ge2}v_ae_a$: $g(e_a,v)=\operatorname{Re}v_a$, $g(ie_a,v)=\operatorname{Im}v_a$, and $\lVert v\rVert^2=g(v,v)=\sum_{a\ge2}\bigl[(\operatorname{Re}v_a)^2+(\operatorname{Im}v_a)^2\bigr]$. So $(\chi^\perp,g)\cong\mathbb R^{2(d_c-1)}$ isometrically. Likewise $\chi^\perp_{\mathbb R}:=\operatorname{span}_{\mathbb R}\{e_2,\dots,e_{d_c}\}$ is a real subspace of $(\mathcal H_c,g)$ with $g$-orthonormal basis $e_2,\dots,e_{d_c}$, so $(\chi^\perp_{\mathbb R},g)\cong\mathbb R^{d_c-1}$.

**Assumptions.** (i) The $K_h$ are random but $\lambda$-independent (A5); every statement concerns the leading-order geometry (5.10)–(5.14), valid as $\lambda\to0^+$. (ii) "Almost surely" and "in probability" refer to the ensemble of descriptions. (iii) No meaning is attached to the configuration beyond these definitions (M1, M4).

**Standard facts used without proof.** (F1) Linear images of jointly Gaussian variables are Gaussian; functions of independent variables are independent; if $X_1,\dots,X_k$ are i.i.d. $N(0,\sigma^2)$, then $\sum_jX_j^2=\sigma^2\chi^2_k$, with density of a Gamma law of shape $k/2$ and scale $2\sigma^2$, mean $k\sigma^2$, variance $2k\sigma^4$. A complex Gaussian vector with zero mean and zero pseudo-covariance is circularly symmetric. (F2) Independent random vectors with absolutely continuous laws on $\mathbb R^m$ are almost surely in general position: any $k\le m+1$ of them are affinely independent. (F3) Chebyshev's inequality, Cauchy–Schwarz, the union bound.

## Derivation

### Step 1. The record vector is the first column of $K_h$ (both ensembles)

$\langle K_h\rangle=\langle\chi\vert K_h\vert\chi\rangle=(K_h)_{11}$ and, by completeness of $(e_a)$, $K_h\lvert\chi\rangle=\sum_a(K_h)_{a1}e_a$. Hence (5.10) gives

$$
\lvert u_h\rangle=\sum_{a=2}^{d_c}(K_h)_{a1}\,e_a\;\in\chi^\perp ,
\tag{7.1}
$$

in agreement with (5.13). Since $\chi$ and the basis are fixed, $u_h$ is a function of $K_h$ alone. The $K_h$ are i.i.d., so by (F1) the $u_h$, $h=1,\dots,n$, are **independent and identically distributed**. For given $K_h$, replacing $\chi$ by $e^{i\varphi}\chi$ (1.2) multiplies every $u_h$ by $e^{i\varphi}$, a $g$-isometry; $d_0$ and $\dim_0$ are unchanged (M6).

### Step 2. GUE: circularly symmetric Gaussian record vectors

Put $z_{h,a}:=(K_h)_{a1}=(K_h)_{1a}^{*}$ for $a=2,\dots,d_c$ (self-adjointness). The $(K_h)_{1a}$, $a\ge2$, are distinct strictly upper entries, hence independent, each with independent real and imaginary parts $N(0,s^2/2)$; complex conjugation flips the sign of the imaginary part and preserves this law. Therefore

$$
\operatorname{Re}z_{h,a},\ \operatorname{Im}z_{h,a}\quad(a=2,\dots,d_c)\quad\text{are i.i.d. } N(0,s^2/2),
\tag{7.2}
$$

so $Ez_{h,a}=0$, $E[z_{h,a}z_{h,b}^{*}]=s^2\delta_{ab}$, and $E[z_{h,a}z_{h,b}]=0$ (for $a=b$: $E(\operatorname{Re}z)^2-E(\operatorname{Im}z)^2+2iE[\operatorname{Re}z\operatorname{Im}z]=0$; for $a\neq b$: independence). For a unit vector $e=\sum_{a\ge2}c_ae_a\in\chi^\perp$, $\langle e\vert u_h\rangle=\sum_ac_a^{*}z_{h,a}$ and

$$
E\bigl\lvert\langle e\vert u_h\rangle\bigr\rvert^2=\sum_{a,b}c_a^{*}c_b\,E[z_{h,a}z_{h,b}^{*}]=s^2\sum_a\lvert c_a\rvert^2=s^2,
\qquad
E\,\langle e\vert u_h\rangle^2=0 .
\tag{7.3}
$$

Thus $u_h$ is Gaussian (F1), with mean $0$, covariance $E\lvert u_h\rangle\langle u_h\rvert=s^2\mathbb 1_{\chi^\perp}$, $\mathbb 1_{\chi^\perp}=\mathbb 1-\lvert\chi\rangle\langle\chi\rvert$, and zero pseudo-covariance: it is circularly symmetric. Its density on $\chi^\perp$, with respect to Lebesgue measure in the real coordinates, is

$$
p(u)=(\pi s^2)^{-(d_c-1)}\exp\bigl(-\lVert u\rVert^2/s^2\bigr).
\tag{7.4}
$$

It depends on $u$ only through $\lVert u\rVert$, so it does not depend on the choice of $e_2,\dots,e_{d_c}$. In the real Euclidean space $(\chi^\perp,g)$ the law is isotropic with variance $s^2/2$ per $g$-unit direction; the complex covariance $s^2$ and the real variance $s^2/2$ describe the same law.

### Step 3. GUE: distances

Let $h\neq h'$ and $w:=u_h-u_{h'}$. By Step 1 and (7.2), the $2(d_c-1)$ real coordinates of $w$ are i.i.d. $N(0,s^2)$, each being the difference of two independent $N(0,s^2/2)$ variables. With $\lVert w\rVert^2$ the sum of their squares, (F1) gives

$$
d_0(h,h')^2=\lVert u_h-u_{h'}\rVert^2\;\overset{d}{=}\;s^2\,\chi^2_{2(d_c-1)},
\qquad
p_{d_0^2}(x)=\frac{x^{d_c-2}\,e^{-x/(2s^2)}}{(d_c-2)!\,(2s^2)^{d_c-1}}\quad(x>0),
\tag{7.5}
$$

$$
E\,d_0^2=2(d_c-1)\,s^2,\qquad
\operatorname{Var}d_0^2=4(d_c-1)\,s^4,\qquad
\frac{\sqrt{\operatorname{Var}d_0^2}}{E\,d_0^2}=\frac{1}{\sqrt{d_c-1}} .
\tag{7.6}
$$

These depend only on $s$ and $d_c$: not on $h,h'$, $n$, $\phi$, $\chi$ or the basis. The mean also follows from (7.3): $E\,d_0^2=E\lVert u_h\rVert^2+E\lVert u_{h'}\rVert^2=2(d_c-1)s^2$. Since $d_0^2$ has a density, $P\bigl(d_0(h,h')=0\bigr)=0$. Almost surely $\sim_0$ is therefore trivial, and by (5.11) the leading-order space consists of $n$ distinct points $u_h$ of $(\chi^\perp,g)$.

### Step 4. GUE: dimension

By (7.1), all $u_h$ lie in $\chi^\perp\cong\mathbb R^{2(d_c-1)}$, and $\dim_0=\operatorname{rank}_{\mathbb R}\{u_h-u_1\}_{h=2}^{n}\le\min\bigl(n-1,2(d_c-1)\bigr)$. This is (5.14). The $u_h$ are independent with the density (7.4), so by (F2) they are almost surely in general position. If $n\le2d_c-1$, all $n$ points are affinely independent and $\dim_0=n-1$. Otherwise, any $2d_c-1$ of them are affinely independent and span an affine subspace of dimension $2(d_c-1)$, so $\dim_0\ge2(d_c-1)$. Hence

$$
\dim_0=\min\bigl(n-1,\;2(d_c-1)\bigr)\qquad\text{almost surely},
\tag{7.7}
$$

and the bound (5.14) is attained.

### Step 5. GOE: the real variant

Now $z_{h,a}:=(K_h)_{a1}=(K_h)_{1a}$ is real for $a\ge2$, and these are distinct strictly upper entries, i.i.d. $N(0,s^2)$. By (7.1), $u_h\in\chi^\perp_{\mathbb R}$, with i.i.d. $N(0,s^2)$ coordinates in the $g$-orthonormal basis $e_2,\dots,e_{d_c}$:

$$
u_h\sim N\bigl(0,\,s^2\mathbb 1_{\chi^\perp_{\mathbb R}}\bigr),\qquad
p(u)=(2\pi s^2)^{-(d_c-1)/2}\exp\bigl(-\lVert u\rVert^2/(2s^2)\bigr),
\tag{7.8}
$$

independent for different $h$ (Step 1). Hence $E\,g(e,u_h)^2=s^2$ for every unit $e\in\chi^\perp_{\mathbb R}$, and $E\lVert u_h\rVert^2=(d_c-1)s^2$ as for the GUE. The coordinates of $u_h-u_{h'}$ ($h\neq h'$) are i.i.d. $N(0,2s^2)$, so by (F1)

$$
d_0(h,h')^2\;\overset{d}{=}\;2s^2\,\chi^2_{d_c-1},
\qquad
p_{d_0^2}(x)=\frac{x^{(d_c-3)/2}\,e^{-x/(4s^2)}}{\Gamma\bigl(\tfrac{d_c-1}{2}\bigr)\,(4s^2)^{(d_c-1)/2}}\quad(x>0),
\tag{7.9}
$$

$$
E\,d_0^2=2(d_c-1)\,s^2,\qquad
\operatorname{Var}d_0^2=8(d_c-1)\,s^4,\qquad
\frac{\sqrt{\operatorname{Var}d_0^2}}{E\,d_0^2}=\sqrt{\frac{2}{d_c-1}} ,
\tag{7.10}
$$

which again depend only on $s$ and $d_c$. Again $\sim_0$ is almost surely trivial. The argument of Step 4 in $\chi^\perp_{\mathbb R}\cong\mathbb R^{d_c-1}$, with the density (7.8), gives

$$
\dim_0=\min\bigl(n-1,\;d_c-1\bigr)\qquad\text{almost surely}.
\tag{7.11}
$$

For $n>d_c$ this is strictly below the bound (5.14), so the GOE does not attain (5.14) there.

### Step 6. Unified form

Let $m=2(d_c-1)$ (GUE) or $m=d_c-1$ (GOE), and let $\mathcal R_m:=(\chi^\perp,g)$ or $(\chi^\perp_{\mathbb R},g)$ respectively. Let $\sigma^2:=2(d_c-1)s^2/m$, that is, $\sigma^2=s^2$ (GUE) or $2s^2$ (GOE). Steps 1–5 then read: the $u_h$ are i.i.d. isotropic Gaussian points $N\bigl(0,\tfrac{\sigma^2}{2}\mathbb 1_m\bigr)$ in the $m$-dimensional Euclidean space $\mathcal R_m$, and

$$
d_0(h,h')^2\overset{d}{=}\sigma^2\chi^2_m,\qquad
E\,d_0^2=m\sigma^2=2(d_c-1)s^2,\qquad
\frac{\sqrt{\operatorname{Var}d_0^2}}{E\,d_0^2}=\sqrt{\frac{2}{m}},\qquad
\dim_0\overset{\text{a.s.}}{=}\min(n-1,m).
\tag{7.12}
$$

Since $\mathrm{GUE}(s)$ and $\mathrm{GOE}(s)$ are the images of the $s=1$ ensembles under $K\mapsto sK$, (7.1) gives $u_h\mapsto su_h$. The law of every ratio of distances is therefore independent of $s$.

### Step 7. Regime (a): $n$ fixed, $d_c\to\infty$

Fix $h\neq h'$ and set $X:=d_0(h,h')^2/(m\sigma^2)$. By (7.12), $EX=1$ and $\operatorname{Var}X=2/m$. Let $\bar d:=E\,d_0(h,h')$; it is common to all pairs, because all pairs $(u_h,u_{h'})$ have the same law. Put $Y:=\sqrt X$ and $\mu:=EY=\bar d/(\sigma\sqrt m)$. For $x\ge0$, $\lvert\sqrt x-1\rvert=\lvert x-1\rvert/(\sqrt x+1)\le\lvert x-1\rvert$, and Cauchy–Schwarz gives

$$
\lvert\mu-1\rvert\le E\lvert Y-1\rvert\le E\lvert X-1\rvert\le\sqrt{2/m}.
\tag{7.13}
$$

Let $0<\epsilon\le1$ and $m\ge32/\epsilon^2$, so that $\sqrt{2/m}\le\epsilon/4$ and $\mu\ge3/4$. Then $\lvert Y/\mu-1\rvert\le2\bigl(\lvert Y-1\rvert+\lvert1-\mu\rvert\bigr)\le2\lvert X-1\rvert+\epsilon/2$, so $\lvert d_0/\bar d-1\rvert>\epsilon$ implies $\lvert X-1\rvert>\epsilon/4$. Chebyshev gives $P(\lvert X-1\rvert>\epsilon/4)\le32/(m\epsilon^2)$. The union bound over the $\binom n2$ pairs then gives

$$
P\Bigl(\max_{h\neq h'}\Bigl\lvert\frac{d_0(h,h')}{\bar d}-1\Bigr\rvert>\epsilon\Bigr)\le\frac{16\,n(n-1)}{m\,\epsilon^2}\xrightarrow[d_c\to\infty]{}0 ,
\tag{7.14}
$$

since $m\ge d_c-1\to\infty$ in both ensembles. By Step 6 this holds for any dependence $s=s(d_c)$. All normalized distances therefore converge to $1$ in probability, jointly. The distance matrix of $\{u_h/\bar d\}$ tends to that of the regular $(n-1)$-simplex with unit edge, and a finite subset of a Euclidean space is fixed up to isometry by its distance matrix. The normalized configuration therefore tends, up to isometry, to a regular simplex. This is consistent with (7.12): once $m\ge n-1$, $\dim_0=n-1$ almost surely. By (7.13), $\bar d=\sigma\sqrt m\,\bigl(1+O(m^{-1/2})\bigr)\simeq s\sqrt{2(d_c-1)}$ in both ensembles.

### Step 8. Regime (b): $n>m+1$

By Step 6, $u_1,\dots,u_n$ are $n$ independent Gaussian points $N\bigl(0,\tfrac{\sigma^2}{2}\mathbb 1_m\bigr)$ in the $m$-dimensional Euclidean space $\mathcal R_m$. Since $\sim_0$ is almost surely trivial, (5.11) makes the leading-order space $(H_V/{\sim_0},d_0)$ isometric to this configuration. By (7.12), for $n\ge m+1$ (in particular for $n>m+1$),

$$
\dim_0=m=\begin{cases}2(d_c-1)&\mathrm{GUE}(s)\\ d_c-1&\mathrm{GOE}(s)\end{cases}\qquad\text{almost surely},
\tag{7.15}
$$

independently of $n$. Further places only add points in the same $\mathcal R_m$ and never raise $\dim_0$. For $n>m+1$ the points are affinely dependent, while any $m+1$ of them are almost surely affinely independent. The value $m$ is fixed by the medium $c$, through $d_c$ and the complex or real class of the records, not by $n$.

## Result

1. **Record vectors (GUE):** $u_h=\sum_{a\ge2}(K_h)_{a1}e_a$ (7.1), i.i.d. in $h$, circularly symmetric complex Gaussian in $\chi^\perp$ with mean $0$ and covariance $s^2\mathbb 1_{\chi^\perp}$ (7.2)–(7.4); this is real variance $s^2/2$ per $g$-direction.
2. **Distances (GUE):** $d_0^2=s^2\chi^2_{2(d_c-1)}$ (7.5); mean $2(d_c-1)s^2$, variance $4(d_c-1)s^4$, relative standard deviation $1/\sqrt{d_c-1}$ (7.6). All depend only on $s$ and $d_c$; $\sim_0$ is trivial almost surely.
3. **Dimension (GUE):** $\dim_0=\min(n-1,2(d_c-1))$ almost surely (7.7); the bound (5.14) is attained.
4. **GOE:** $u_h\sim N(0,s^2\mathbb 1)$ in $\chi^\perp_{\mathbb R}$ (7.8); $d_0^2=2s^2\chi^2_{d_c-1}$, mean $2(d_c-1)s^2$, variance $8(d_c-1)s^4$, relative standard deviation $\sqrt{2/(d_c-1)}$ (7.9)–(7.10); $\dim_0=\min(n-1,d_c-1)$ almost surely (7.11), below (5.14) for $n>d_c$.
5. **Regimes:** unified form (7.12). (a) For fixed $n$, $\max_{h\neq h'}\lvert d_0/\bar d-1\rvert\to0$ in probability as $d_c\to\infty$, with the explicit bound (7.14): the normalized configuration tends to a regular simplex. (b) For $n\ge m+1$, $\dim_0=m$ almost surely (7.15), fixed by the medium.

All expected results of question.md are confirmed; no discrepancy in any distribution, factor or dimension. Minor sharpening: (7.15) already holds at $n=m+1$.

## Consistency checks

1. **Dimensions.** The exponent $\operatorname{Tr}K^2/s^2$ is dimensionless, so $s$ carries the unit of $K_h$, that is, of $C$. By (5.10), $d_0=\lVert u_h-u_{h'}\rVert$ carries the unit of $K_h$, since $\chi$ is a unit vector. (7.5)–(7.6) and (7.9)–(7.10) give $d_0^2\propto s^2$ and $\operatorname{Var}d_0^2\propto s^4$, while chi-square variables and relative deviations are dimensionless. The two sides agree.
2. **Qubit medium, $d_c=2$.** GUE: $u_h=z_{h,2}e_2$ lies in the complex line $\mathbb Ce_2\cong\mathbb R^2$; $m=2$ and (7.7) gives $\dim_0=2$ for $n\ge3$, since three independent points of the plane with densities are almost surely not collinear. Directly, $\lvert z_{h,2}-z_{h',2}\rvert^2$ is exponential with mean $2s^2$, which is $s^2\chi^2_2$ with relative deviation $1=1/\sqrt{d_c-1}$, as (7.5)–(7.6) state. GOE: the points lie on the line $\mathbb Re_2$, $\dim_0=1$ for every $n\ge2$, and $(x_h-x_{h'})^2$ with $x_h-x_{h'}\sim N(0,2s^2)$ is $2s^2\chi^2_1$, matching (7.9) and (7.11).
3. **$s\to0$.** $E\,d_0^2=2(d_c-1)s^2\to0$, so every $d_0\to0$ in mean square and all places merge into one point, $u_h\to0$. This is consistent with $K_h\to0$ leaving no record by (5.10). For every $s>0$, $\dim_0$ and the law of $d_0/\bar d$ do not depend on $s$ (Step 6): $s$ only sets the unit of $d_0$, and the collapse occurs only at $s=0$.

## Open issues

- $\mathrm{GOE}(s)$ is defined relative to the basis $(e_a)$ with $e_1=\chi$. It therefore uses a real structure in which $\chi$ is real, which is not part of the description data. The results (7.8)–(7.11) are invariant under real orthogonal changes of $e_2,\dots,e_{d_c}$. The GUE results do not depend on the basis at all (7.4).
- Distances that share a place, such as $d_0(1,2)$ and $d_0(1,3)$, are correlated through $u_1$. Only the union bound was needed in (7.14); the joint law is out of scope.
- All statements are leading order in $\lambda$; finite $\lambda$ and non-Gaussian ensembles are out of scope.

## Methods used

- Matrix elements in an adapted orthonormal basis
- Gaussian ensembles (GUE, GOE) and their entry distributions
- Circularly symmetric complex Gaussian vectors, real realification of $\mathbb C^{k}$
- Chi-square and Gamma distributions
- General position of absolutely continuous random vectors, affine dimension
- Chebyshev inequality, Cauchy–Schwarz, union bound, convergence in probability
- Scaling argument