Version 2 · current

# Background stability: identification of the witness geometry across $\lambda$

- **Subproject:** 03-ilang-space
- **Package:** 24-background-stability
- **Version:** v2
- **Mode:** external regeneration
- **Date:** 2026-10-09

## Changes from previous version

- **Step 8.2, "$\Lambda_N$ in general" (I1).** The candidate set is kept and named $Z_b$; it is identified with $Z_{\mathcal F}$ of (24.7) for this example, so it is discrete. New: the exact retention rule (24.30). It gives the edge indicators in terms of the values $F(k\lambda)$, and the one-sided values are obtained from the Taylor data of $F$ at the candidate. It also gives the rule for which candidates belong to $\Lambda_N$, and the qubit case $n=3$ as an illustration.
- **Step 9.** One paragraph added after (24.29): the maximum over $k$ in (24.29) is attained at $k=m$ for every $\lambda\in(0,\pi/2)$. This closes the former open issue on (c).
- **Result 3(b)** now cites (24.30). **Open issues:** the item on (b) is updated, and the item on (c) is removed.
- Nothing else is changed. Tags (24.1)–(24.29) keep their meaning; (24.30) is new.

## Response to verification

- **I1:** Accepted and fixed in Step 8.2 ("$\Lambda_N$ in general"), new (24.30), with Result 3(b) and Open issues updated. As suggested, a candidate $\lambda_0\in Z_b\setminus\Lambda_{\mathcal P}$ is retained iff the edge indicators at $\lambda_0$ and on the two adjacent candidate-free intervals are not all equal. The verifier's example ($F=\cos^2$, $n=3$, $\lambda=\pi/4$) is worked out there and is discarded.

## Setup and assumptions

- **Recording setting of 05.** Places $h\in H$ (finite), medium $\mathcal H_c$ with $d_c<\infty$ (A1), $K_h=K_h^\dagger$ independent of $\lambda$ (A5), start $\phi\otimes\chi$ with all $\phi_h\neq0$. Every place is present and $\lvert E_h(\lambda)\rangle=e^{-iK_h\lambda}\lvert\chi\rangle$ up to phase, so $W(h,h';\lambda)$ is (5.6).
- **Inputs, as quoted in question.md:** (3.1), (3.2), (3.5) (points, $W$, $\alpha=\arccos\sqrt W$); (4.1), (4.3) (relatedness, neighbours); (5.6), (5.10), (5.15), (5.16); (11.11) with its two assumptions; (15.2); (15.17); (17.1), (17.2); (22.14), (22.15). The definitions of background, identification, background interval, $\lambda_*$, trajectory and $\delta$ are those of question.md. "Discrete" = closed and locally finite in $(0,\infty)$.
- **Definition of $\Lambda_N$.** $\Lambda_N:=\{\lambda\in(0,\infty)\setminus\Lambda_{\mathcal P}:\ N_V$ is not constant on any neighbourhood of $\lambda\}$. Since $\Lambda_{\mathcal P}$ is closed (Step 3), the point set near such $\lambda$ is fixed, so the comparison is meaningful.
- **Notation.** $\lVert s\rVert_\pi:=\min_{m\in\mathbb Z}\lvert s-m\pi\rvert\in[0,\pi/2]$.
- **Neighbour rule in terms of $W$.** $\arccos\sqrt{\cdot}$ is strictly decreasing on $[0,1]$, so $\alpha(x,y)<\alpha(x,x')\iff W(x,y)>W(x,x')$, and (4.1), (4.3) read

$$
x\sim x'\iff W(x,x')>0\ \text{ and there is no } y \text{ with } \min\bigl(W(x,y),W(y,x')\bigr)>W(x,x') .
\tag{24.1}
$$

  $y\in\{x,x'\}$ never qualifies ($W(x,x)=1$ but $W(x,x')$ is not $>W(x,x')$).

## Derivation

### Step 1. $W$ is a finite exponential sum

With spectral decompositions $K_h=\sum_a\kappa^{(h)}_aQ^{(h)}_a$, $\langle\chi\vert e^{iK_{h'}\lambda}e^{-iK_h\lambda}\vert\chi\rangle=\sum_{a,b}e^{i(\kappa^{(h')}_b-\kappa^{(h)}_a)\lambda}\langle\chi\vert Q^{(h')}_bQ^{(h)}_a\vert\chi\rangle$, hence

$$
W(h,h';\lambda)=\sum_{\nu}c_\nu e^{i\nu\lambda}\qquad(\text{finitely many real }\nu),
\tag{24.2}
$$

the restriction to $\mathbb R$ of an entire function; so is every real linear combination of such $W$. **Identity theorem:** a real-analytic $f$ on $\mathbb R$ with $f\not\equiv0$ has zeros without accumulation point in $\mathbb R$ (including at $0$).

### Step 2. Item 1(a): the dichotomy

$1-W(h,h';\cdot)$ is of type (24.2) and vanishes at $\lambda=0$. Either it vanishes identically, or its zeros are isolated in $\mathbb R$; then $0$ is an isolated zero and

$$
Z(h,h')=(0,\infty)\quad\text{or}\quad Z(h,h')\ \text{discrete with}\ Z(h,h')\cap(0,\varepsilon_{hh'})=\emptyset\ \text{for some }\varepsilon_{hh'}>0 .
\tag{24.3}
$$

($Z$ may be empty or infinite.) For unit vectors, $W=1$ iff $e^{-iK_h\lambda}\chi$ and $e^{-iK_{h'}\lambda}\chi$ are the same state. By (15.2) with $Y_1=K_h$, $Y_2=K_{h'}$ (index renamed $j$):

$$
Z(h,h')=(0,\infty)\iff (K_h-K_{h'})K_h^{\,j}\lvert\chi\rangle=\mu_0K_h^{\,j}\lvert\chi\rangle\ \ (j=0,\dots,d_c-1),\quad \mu_0=\langle K_h\rangle-\langle K_{h'}\rangle ,
\tag{24.4}
$$

i.e. $K_h-K_{h'}$ acts as the number $\mu_0$ on the cyclic subspace of $K_h$ generated by $\chi$. Write $h\equiv h'$. It is an equivalence relation ($\varrho_h(\lambda)=\varrho_{h'}(\lambda)$ for all $\lambda$). The $j=0$ condition is $u_h=u_{h'}$, so $h\equiv h'\Rightarrow d_0(h,h')=0$ by (5.10); in particular $d_0>0$ forces the discrete case.

### Step 3. Item 1(b): structure of $\mathcal P(\lambda)$

Define

$$
\mathcal P_{\rm gen}:=H/{\equiv},\qquad \Lambda_{\mathcal P}:=\bigcup_{h\not\equiv h'}Z(h,h') .
\tag{24.5}
$$

A finite union of discrete sets, each avoiding $(0,\varepsilon_{hh'})$, is discrete and avoids $(0,\min\varepsilon)$. For $\lambda\notin\Lambda_{\mathcal P}$: $h\approx_\lambda h'\iff h\equiv h'$, so $\mathcal P(\lambda)=\mathcal P_{\rm gen}$. For $\lambda\in\Lambda_{\mathcal P}$: ${\equiv}\subseteq{\approx_\lambda}$, and some $h\not\equiv h'$ have $h\approx_\lambda h'$. Hence

$$
\lambda\in\Lambda_{\mathcal P}\ \Longrightarrow\ \mathcal P(\lambda)=\mathcal P_{\rm gen}/{\approx_\lambda}\ \text{ is strictly coarser than }\mathcal P_{\rm gen}:\ \text{each point of }\mathcal P(\lambda)\text{ is a union of generic points.}
\tag{24.6}
$$

So $\Lambda_{\mathcal P}$ is exactly the set where $\mathcal P(\lambda)\neq\mathcal P_{\rm gen}$. At its points generic points only **merge**: $x,x'$ merge iff $W(x,x';\lambda)=1$, an equivalence relation by (3.1). No generic point ever splits. Since $\Lambda_{\mathcal P}$ is discrete, a merged point separates again into the same generic points on both sides.

### Step 4. Item 1(c): $N_V$ changes only on a discrete set

On an open interval $U\subseteq(0,\infty)\setminus\Lambda_{\mathcal P}$ the points are $\mathcal P_{\rm gen}$. By (24.1), $N_V(\lambda)$ is a fixed Boolean function of the signs ($>0,=0,<0$) of the finite family

$$
\mathcal F:=\{W(x,x';\cdot)\}\cup\{W(x,y;\cdot)-W(x,x';\cdot)\},\qquad x,x',y\in\mathcal P_{\rm gen}\ \text{distinct}.
\tag{24.7}
$$

Each $f\in\mathcal F$ is real-analytic (Step 1). If $f\equiv0$, its sign is constant. Otherwise its zero set $Z_f$ is discrete, and by the intermediate value theorem its sign is constant on each component of the complement. With $Z_{\mathcal F}:=\bigcup_{f\not\equiv0}Z_f\cap(0,\infty)$:

$$
N_V\ \text{is constant on each component of }(0,\infty)\setminus(\Lambda_{\mathcal P}\cup Z_{\mathcal F}),\qquad \Lambda_N\subseteq Z_{\mathcal F}\setminus\Lambda_{\mathcal P}\ \text{ is discrete}.
\tag{24.8}
$$

Every $f\in\mathcal F$ is analytic at $0$, so $Z_{\mathcal F}\cap(0,\varepsilon')=\emptyset$ for some $\varepsilon'>0$. **Answer:** yes, always. No condition on the records is needed beyond the standing ones: finite $d_c$ (A1) and $\lambda$-independent $K_h$. The records only decide which $f\in\mathcal F$ vanish identically. $N_V$ can change at $\lambda_0$ only if some $f\not\equiv0$ vanishes there: a pair becomes unrelated ($W(x,x')=0$), or two witness overlaps tie ($W(x,y)=W(x,x')$).

### Step 5. Item 1(d): the stability scale

$\Lambda_{\mathcal P}\cup\Lambda_N$ is discrete: it is a closed discrete set together with a subset of the discrete set $Z_{\mathcal F}$. By Steps 3–4 it has no point in $(0,\min(\varepsilon,\varepsilon'))$. If $(0,\Lambda)$ contains $\lambda_0\in\Lambda_{\mathcal P}$, then $\mathcal P$ is not constant there (Step 3). If it contains $\lambda_0\in\Lambda_N$, then $N_V$ is not locally constant there. Otherwise $\mathcal P=\mathcal P_{\rm gen}$ and $N_V$ is locally constant on the connected set $(0,\Lambda)$, hence constant. So

$$
\lambda_*=\min\bigl(\Lambda_{\mathcal P}\cup\Lambda_N\bigr)\in(0,\infty]\qquad(\lambda_*=\infty\ \text{iff the set is empty}),\qquad \lambda_*>0\ \text{always}.
\tag{24.9}
$$

On $(0,\lambda_*)$ the background is $(\mathcal P_{\rm gen},N_V^{(0)})$, with $N_V^{(0)}$ the small-$\lambda$ graph. Under the two assumptions of 11, the $u_h$ are distinct, so $d_0>0$ and $\mathcal P_{\rm gen}$ consists of singletons (Step 2), and $N_V^{(0)}$ is (11.11). Thus (11.11) holds on all of $(0,\lambda_*)$: "sufficiently small" in 11 can be read as $\lambda<\lambda_*$. $\lambda_*$ is the first $\lambda$ at which two places merge or the graph leaves (11.11); $\lambda_*\le\min\Lambda_{\mathcal P}$, with equality iff no graph change precedes the first merger.

### Step 6. Item 2: trajectories (body of 17)

$\lvert\Psi(\lambda)\rangle=e^{-iC\lambda}(\phi\otimes\chi)$ in finite dimension, so $p_h(\lambda)=\lVert\psi_h(\lambda)\rVert^2$ from (17.1) are real exponential sums with $\sum_hp_h=1$. The background is that of the recording setting: it depends only on $K_h,\chi$ by (5.6), not on $t_{hh'}$ or $\phi$, and it exists for every $\lambda>0$.

- **Defined off $\Lambda_{\mathcal P}$: yes.** In fact $P_x(\lambda)$ is defined for every $\lambda>0$ on the points of $\mathcal P(\lambda)$. For $\lambda\notin\Lambda_{\mathcal P}$ it is a probability distribution on the fixed set $\mathcal P_{\rm gen}$, and

$$
\lambda\mapsto\bigl(P_x(\lambda)\bigr)_{x\in\mathcal P_{\rm gen}},\qquad P_x(\lambda)=\sum_{h\in x}p_h(\lambda),
\tag{24.10}
$$

  is real-analytic on all of $(0,\infty)$, since the formula makes sense at every $\lambda$.
- **Join across $\Lambda_{\mathcal P}$: yes, unconditionally.** Any $\lambda,\lambda'\notin\Lambda_{\mathcal P}$ have $\mathcal P(\lambda)=\mathcal P(\lambda')=\mathcal P_{\rm gen}$, so the identification by places maps every generic point to itself, also across $\lambda_0\in\Lambda_{\mathcal P}$. The trajectories on the two sides are restrictions of the single analytic curve (24.10). At $\lambda_0$ the trajectory lives on the coarser points $X\in\mathcal P(\lambda_0)$ and is the coarse-graining of the common limit:

$$
P_X(\lambda_0)=\sum_{x\in\mathcal P_{\rm gen},\,x\subseteq X}\ \lim_{\lambda\to\lambda_0^{\pm}}P_x(\lambda).
\tag{24.11}
$$

  A merger is a momentary loss of resolution, not a break.
- **At $\Lambda_N$:** the points, the identification and the curve (24.10) are unchanged and analytic, and $\alpha(x,x';\lambda)$ is continuous. Only the neighbour relation among the fixed points changes. So whether a transfer of weight between two points is a transfer between neighbours, and whether the hopping ($t_{hh'}\neq0$) couples only neighbours, can switch there; the trajectory itself does not.

### Step 7. Example (a): line, qubit medium

**7.1 Circle model.** By (5.16), $W(h_i,h_j;\lambda)=\cos^2((i-j)\lambda)$, so

$$
\alpha(h_i,h_j;\lambda)=\arccos\lvert\cos((i-j)\lambda)\rvert=\lVert(i-j)\lambda\rVert_\pi=\rho(z_i,z_j),\qquad z_i:=i\lambda \bmod \pi ,
\tag{24.12}
$$

where $\rho$ is the arc metric on the circle $\mathbb T:=\mathbb R/\pi\mathbb Z$. The background at $\lambda$ is isometric to $S_\lambda:=\{z_1,\dots,z_n\}\subset\mathbb T$: points correspond to distinct values of $z_i$. Since $d_0=\lvert i-j\rvert>0$ (5.15), $\mathcal P_{\rm gen}$ consists of singletons, and $h_i\approx_\lambda h_j\iff(i-j)\lambda\in\pi\mathbb Z$:

$$
\Lambda_{\mathcal P}=\{\pi p/q:\ p\ge1,\ 1\le q\le n-1,\ \gcd(p,q)=1\},\qquad \Lambda_{\mathcal P}\cap(0,\pi]=\{\pi p/q:\ 1\le p\le q\le n-1,\ \gcd(p,q)=1\}.
\tag{24.13}
$$

$W$ is even and $\pi$-periodic in $\lambda$, so

$$
\text{the background (as a labelled partition and graph) at }\lambda+\pi\text{ and at }\pi-\lambda\text{ equals that at }\lambda .
\tag{24.14}
$$

**7.2 Blocking lemma.** Let $x,x'\in\mathbb T$ with $0<d:=\rho(x,x')<\pi/2$. Put $x=0$, $x'=d$. The balls are $(-d,d)$ and $(0,2d)$ mod $\pi$. Their intersection is $(0,d)$ if $2d\le\pi-d$, and $(0,d)\cup(\pi-d,2d)$ otherwise. Hence

$$
\{y:\max(\rho(x,y),\rho(y,x'))<d\}=(\text{open short arc})\ \cup\ A(x,x'),\quad A(x,x')=\begin{cases}\emptyset,& d\le\pi/3,\\ \text{open arc of length }3d-\pi\text{ centred at }m+\pi/2,& d>\pi/3,\end{cases}
\tag{24.15}
$$

with $m$ the midpoint of the short arc. With (4.1), (4.3): for $\lvert S_\lambda\rvert\ge3$, $N_V(\lambda)$ is contained in the cycle $\mathcal C(\lambda)$ that joins cyclically adjacent points of $S_\lambda$. An edge of $\mathcal C(\lambda)$ with gap $g$ belongs to $N_V(\lambda)$ unless the gap is **bad**:

$$
\text{bad}\iff g\ge\pi/2,\ \text{ or }\ \pi/3<g<\pi/2\ \text{and}\ A\cap S_\lambda\neq\emptyset .
\tag{24.16}
$$

**7.3 At $\lambda=\pi p/q\in\Lambda_{\mathcal P}$.** $h_i\approx h_j\iff q\mid(i-j)$. Since $n>q$, $\mathcal P(\lambda)$ consists of the $q$ residue classes mod $q$, and $S_\lambda=\{\pi r/q\}$ is the regular $q$-gon. For $q\ge3$ all gaps are $\pi/q\le\pi/3$, so none is bad:

$$
q=1:\ \text{one point};\quad q=2:\ \text{two points with }W=0,\ \text{no edge};\quad q\ge3:\ N_V=C_q,\ [i]\sim[i\pm\bar p]\ (p\bar p\equiv1\bmod q).
\tag{24.17}
$$

**7.4 Generic components.** On a component $I$ of $(0,\pi)\setminus\Lambda_{\mathcal P}$ the cyclic order of the $z_i$ is fixed: a change of order needs $z_i=z_j$. So every gap is $k\lambda-m\pi$ with fixed integers $k,m$, a linear function of $\lambda$. At an endpoint $\pi p/q$ of $I$ (including $0=\pi\cdot0/1$ and $\pi=\pi\cdot1/1$), the $z_i$ tend to the $q$-gon with every vertex occupied. A vertex inside the limit of an empty gap would be the limit of points inside that gap, so every gap tends to $0$ or $\pi/q$. The endpoints are consecutive fractions of order $n-1$, so their denominators differ unless both are $1$ ($p'q-pq'=1$). If both denominators are $\ge3$, every gap stays $<\pi/3$ on $I$, and $N_V=\mathcal C(\lambda)$ there. It remains to treat the components with an endpoint in $\{0,\pi/2,\pi\}$: $I_0=(0,\pi/(n-1))$ and $J=(\pi(q_o-1)/(2q_o),\pi/2)$, with $q_o$ the largest odd number $\le n-1$, and their mirrors under (24.14). For $n=3$, $I_0=J=(0,\pi/2)$.

**7.5 $I_0$.** The $z_i=i\lambda$ lie in order on an arc of length $(n-1)\lambda<\pi$. The gaps are $\lambda$ ($n-1$ times) and $g=\pi-(n-1)\lambda$ between $z_n$ and $z_1$.
- *Small gaps.* For $n\ge4$, $\lambda<\pi/3$, so they are not bad. For $n=3$ and $\lambda>\pi/3$, the arc $A$ of the gap $(z_1,z_2)$ is $(z_1+\pi-\lambda,z_1+2\lambda)$; $z_3=z_1+2\lambda$ is an endpoint, hence not inside. The gap $(z_2,z_3)$ is symmetric. So the small gaps are never bad.
- *Big gap.* Its arc $A$ is centred at $(z_1+z_n)/2$ with half-length $(3g-\pi)/2$.
  - For $n$ odd, $z_{(n+1)/2}$ sits at the centre, so the gap is bad iff $g>\pi/3$.
  - For $n$ even, the nearest points are at distance $\lambda/2$, so the gap is bad iff $\lambda<3g-\pi$, i.e. $\lambda<2\pi/(3n-2)$. This also covers $g\ge\pi/2$, since $\pi/(2(n-1))<2\pi/(3n-2)$.

  Hence the edge $\{h_1,h_n\}$ is absent exactly for $\lambda<\lambda_*$, with ties ($\lambda=\lambda_*$) keeping it:

$$
\lambda_*(n)=\frac{\pi}{\lfloor 3n/2\rfloor-1}=\begin{cases}\dfrac{2\pi}{3(n-1)},& n\ \text{odd},\\[2mm] \dfrac{2\pi}{3n-2},& n\ \text{even},\end{cases}\qquad \frac{\pi}{2(n-1)}<\lambda_*<\frac{\pi}{n-1}=\min\Lambda_{\mathcal P}.
\tag{24.18}
$$

  $\lambda_*\notin\Lambda_{\mathcal P}$: its reduced denominator is $3(n-1)/2$ resp. $(3n-2)/2$, both $>n-1$.

**7.6 $J$ for $n\ge4$** ($q_o\ge3$). Write $\lambda=\pi/2-\varepsilon$ with $0<\varepsilon<\pi/(2q_o)$, and let $A_o=\lceil n/2\rceil\ge2$, $B_e=\lfloor n/2\rfloor\ge1$.
- *Configuration.* Even $l$ give the cluster $E=\{\pi-2j\varepsilon\}_{j\le B_e}$; odd $l$ give $O=\{\pi/2-(2j-1)\varepsilon\}_{j\le A_o}$.
- *Gaps.* The intra-cluster gaps are $2\varepsilon<\pi/q_o\le\pi/3$. The inter-cluster gaps are $G_1=\pi/2-(2A_o-3)\varepsilon$ (from $-2\varepsilon$ to $\pi/2-(2A_o-1)\varepsilon$) and $G_2=\pi/2-(2B_e-1)\varepsilon$ (from $\pi/2-\varepsilon$ to $\pi-2B_e\varepsilon$); both lie in $(0,\pi/2)$ on $J$.
- *Arcs.* By (24.15), $A(G_1)=(\pi/2+(2A_o-5)\varepsilon,\ \pi-4(A_o-1)\varepsilon)$ lies inside the closure of the empty gap $G_2$, because $2A_o-5\ge-1$ and $4(A_o-1)\ge2B_e$. Likewise $A(G_2)=(2(B_e-1)\varepsilon,\ \pi/2-(4B_e-1)\varepsilon)$ lies inside the closure of $G_1$, because $B_e\ge1$ and $A_o\le2B_e$. Open arcs inside a closed empty gap contain no point.

So there are no bad gaps on $J$, and $N_V=\mathcal C(\lambda)$ there.

**7.7 Everything on $(0,\pi]$.** Collecting 7.3–7.6 and (24.14): for $\lambda\in(0,\pi)\setminus\Lambda_{\mathcal P}$, all places are separate points and

$$
N_V(\lambda)=\begin{cases}\text{path } h_1-h_2-\dots-h_n,&\lambda\in(0,\lambda_*)\cup(\pi-\lambda_*,\pi),\\ \mathcal C(\lambda)\ (\text{cycle of cyclically adjacent }z_i=i\lambda\bmod\pi),&\lambda\in[\lambda_*,\pi-\lambda_*]\setminus\Lambda_{\mathcal P},\end{cases}
\tag{24.19}
$$

with $\mathcal C(\lambda)=h_1-\dots-h_n-h_1$ on $[\lambda_*,\pi/(n-1))$ and on its mirror. At $\Lambda_{\mathcal P}$ the background is (24.17); at $\lambda=\pi$ it is one point. $\mathcal C$ changes only at $\Lambda_{\mathcal P}$. Hence

$$
\Lambda_N\cap(0,\pi]=\{\lambda_*,\ \pi-\lambda_*\},\qquad \lambda_*(n)\ \text{from (24.18)},
\tag{24.20}
$$

and both sets are $\pi$-periodic on $(0,\infty)$.

**7.8 Relative deviation.** On $(0,\lambda_*)$, $k\lambda\le(n-1)\lambda<2\pi/3$, so $\alpha=k\lambda$ for $k\lambda\le\pi/2$ and $\alpha=\pi-k\lambda$ otherwise. The relative deviation for separation $k$ is $\max(0,2-\pi/(k\lambda))$, increasing in $k$:

$$
\delta(\lambda)=\begin{cases}0,&0<\lambda\le\frac{\pi}{2(n-1)},\\ 2-\dfrac{\pi}{(n-1)\lambda},&\frac{\pi}{2(n-1)}\le\lambda<\lambda_*,\end{cases}\qquad \sup_{(0,\lambda_*)}\delta=\begin{cases}\tfrac12,& n\ \text{odd},\\ \frac{n-2}{2(n-1)},& n\ \text{even}\end{cases}\ (\text{not attained}).
\tag{24.21}
$$

### Step 8. Example (b): line, general $X$

**8.1 Spectral condition.** Let $X=\sum_\mu\mu\,Q_\mu$, $w_\mu=\langle\chi\vert Q_\mu\vert\chi\rangle$, and $\Sigma_\chi=\{\mu:w_\mu>0\}$, which has at least two elements since $\sigma_X>0$. Then

$$
F(s)=\Bigl\lvert\sum_{\mu\in\Sigma_\chi}w_\mu e^{-is\mu}\Bigr\rvert^2,\qquad F(s)=1\iff s(\mu-\mu')\in2\pi\mathbb Z\ \ \forall\mu,\mu'\in\Sigma_\chi ,
\tag{24.22}
$$

by equality in the triangle inequality ($\sum w_\mu=1$). If all differences $\mu-\mu'$ are commensurable, they generate $g\mathbb Z$ for some $g>0$, and $\{s>0:F(s)=1\}=T\mathbb Z_{>0}$ with $T:=2\pi/g$. Otherwise this set is empty. Since $d_0=\lvert i-j\rvert\sigma_X>0$, $\mathcal P_{\rm gen}$ consists of singletons, and with (5.16):

$$
\Lambda_{\mathcal P}=\begin{cases}\{Tp/q:\ p\ge1,\ 1\le q\le n-1,\ \gcd(p,q)=1\},&\Sigma_\chi\ \text{commensurable},\\ \emptyset,&\text{otherwise},\end{cases}
\tag{24.23}
$$

with $\mathcal P(Tp/q)$ the residue classes mod $q$. Example (a): $\Sigma_\chi=\{\pm1\}$, $g=2$, $T=\pi$. $F$ is even, so in the commensurable case the background is invariant under $\lambda\to\lambda+T$ and $\lambda\to T-\lambda$.

**8.2 Bounds on $\lambda_*$.** Standard fact (Bohr): a finite exponential sum with real frequencies is almost periodic; in particular $\sup_{s\ge s_0}F(s)=F(0)=1$ for every $s_0$. Since $F(s)=1-\sigma_X^2s^2+O(s^4)$:
- $s_m:=\sup\{s>0:\ F\ \text{strictly decreasing on }[0,s]\}$ satisfies $0<s_m<\infty$.
- Let $S:=\{s>0:\ F(s)\ge F(v)\ \forall v\in[s/2,s]\}$. It is closed in $(0,\infty)$ and $S\cap(0,s_m]=\emptyset$, because there $F(s)<F(s/2)$.
- $S\neq\emptyset$. Let $\tau:=\inf\{s>s_m:F(s)\ge F(s_m/2)\}$, which is finite by almost periodicity. Then $F(\tau)=F(s_m/2)$. For $v\in[\tau/2,\tau]$: on $[s_m/2,s_m]$, $F(v)\le F(s_m/2)$ by monotonicity; on $(s_m,\tau)$, $F(v)<F(s_m/2)$ by the definition of $\tau$. So $\tau\in S$.
- Define $s_c:=\min S\in(s_m,\tau]$.

*Lower bound.* For $\lambda<s_m/(n-1)$, the values $F(k\lambda)$, $k\le n-1$, are $<1$ and strictly decreasing in $k$. So all places are separate points; $k=1$ pairs are not blocked, and each $k\ge2$ pair is blocked by $h_{i+1}$. $N_V$ is the path, as in (11.11).

*Upper bound.* Let $\lambda_1=s_c/(n-1)$. $F(s_c)>0$, since otherwise $F\equiv0$ on $[s_c/2,s_c]$, impossible for analytic $F\not\equiv0$. For each $l\in\{2,\dots,n-1\}$, the larger of $l-1$ and $n-l$ is $\ge(n-1)/2$, so its multiple of $\lambda_1$ lies in $[s_c/2,s_c)$, where $F\le F(s_c)$. So no $h_l$ blocks, and $h_1\sim h_n$ unless $\lambda_1\in\Lambda_{\mathcal P}$. Either way the background at $\lambda_1$ differs from the path background. Hence

$$
\frac{s_m}{n-1}\ \le\ \lambda_*\ \le\ \frac{s_c}{n-1}\ \le\ \frac{\tau}{n-1},\qquad\text{so }\lambda_*=\Theta(1/n)\text{ with }(n-1)\lambda_*\in[s_m,s_c].
\tag{24.24}
$$

For (a): $s_m=\pi/2$, $s_c=2\pi/3$, $\tau=3\pi/4$, consistent with (24.18).

*Where $\lambda_*$ falls.* $\lambda_*\in\Lambda_N$. If $\Sigma_\chi$ is not commensurable, $\Lambda_{\mathcal P}=\emptyset$. If it is, $s_c<T$: for small $u$, $F(T-u)=F(u)$; on $[T-s_m,T-u]$, $F(v)=F(T-v)\le F(u)$; and on $[(T-s_m)/2,T-s_m]\subset(0,T)$, $F\le M<1$; so $T-u\in S$ once $F(u)>M$. Hence $\lambda_*<T/(n-1)=\min\Lambda_{\mathcal P}$.

*$\Lambda_N$ in general.* By (24.8), $\Lambda_N\subseteq Z_b:=\{\lambda:F(k\lambda)=F(k'\lambda),\ 1\le k<k'\le n-1\}\cup\{\lambda:F(k\lambda)=0\}$, with $\Lambda_N\cap(0,\lambda_*)=\emptyset$ and $\min\Lambda_N=\lambda_*$. In the commensurable case it is $T$-periodic and symmetric under $\lambda\to T-\lambda$. $Z_b$ is the set $Z_{\mathcal F}$ of (24.7) for this example. Every $f\in\mathcal F$ is $F(k\lambda)$ or $F(k\lambda)-F(k'\lambda)$, and every such function occurs (take $h_1,h_{1+k},h_{1+k'}$). For $k\neq k'$ it is $\not\equiv0$, since it equals $\sigma_X^2(k'^2-k^2)\lambda^2+O(\lambda^4)$. So $Z_b$ is discrete.

*Exact rule.* Off $\Lambda_{\mathcal P}$ the points are the places, and $W(h_i,h_j)=F(\lvert i-j\rvert\lambda)$. By (24.1), $N_V(\lambda)$ is the set of pairs with $e_{ij}(\lambda)=1$, and

$$
\begin{aligned}
&e_{ij}(\lambda)=1\iff F(k\lambda)>0\ \text{and no }l\notin\{i,j\}\text{ has }\min\bigl(F(\lvert i-l\rvert\lambda),F(\lvert l-j\rvert\lambda)\bigr)>F(k\lambda)\qquad(k=\lvert i-j\rvert),\\
&\lambda_0\in\Lambda_N\iff\lambda_0\in Z_b\setminus\Lambda_{\mathcal P}\ \text{ and }\ e(\lambda_0^-),\ e(\lambda_0),\ e(\lambda_0^+)\ \text{are not all equal}.
\end{aligned}
\tag{24.30}
$$

Here $e=(e_{ij})$, and $e(\lambda_0^{\pm})$ is its value on $(\lambda_0,\lambda_0+\varepsilon)$ resp. $(\lambda_0-\varepsilon,\lambda_0)$. The radius $\varepsilon>0$ is chosen so that $(\lambda_0-\varepsilon,\lambda_0+\varepsilon)$ meets the discrete set $Z_b\cup\Lambda_{\mathcal P}$ only in $\lambda_0$. *Why.* $e$ depends on $\lambda$ only through the signs of $F(k\lambda)$ and $F(k\lambda)-F(k'\lambda)$. These signs are constant on each component of $(0,\infty)\setminus Z_b$ (intermediate value theorem). Hence no $\lambda\notin Z_b\cup\Lambda_{\mathcal P}$ lies in $\Lambda_N$. A candidate $\lambda_0\in Z_b\setminus\Lambda_{\mathcal P}$ lies in $\Lambda_N$ iff $N_V$ takes more than one value on $(\lambda_0-\varepsilon,\lambda_0+\varepsilon)$. On that interval $N_V$ takes only the three values above, and the same holds on every smaller interval. The one-sided values follow from $F$ at $\lambda_0$. If $f=F(k\cdot)-F(k'\cdot)$ has a zero of order $r$ at $\lambda_0$, then by Taylor's theorem its sign is $\operatorname{sgn}f^{(r)}(\lambda_0)$ on the right and $(-1)^r\operatorname{sgn}f^{(r)}(\lambda_0)$ on the left. $F(k\lambda)>0$ on both sides of each of its zeros, since $F\ge0$ and its zeros are isolated. So a candidate is retained exactly when some edge appears or disappears there. This happens at a tie $F(k\lambda_0)=F(k'\lambda_0)$ that switches a blocking, or at a zero of $F(k\lambda_0)$ for a pair that is an edge on one side. *Illustration* ((a), $n=3$): $(Z_b\setminus\Lambda_{\mathcal P})\cap(0,\pi]=\{\pi/4,\pi/3,2\pi/3,3\pi/4\}$. Rule (24.30) discards $\pi/4$ and $3\pi/4$. There only $F(2\lambda)=0$ holds, and $(h_1,h_3)$ is blocked by $h_2$ on both sides and unrelated at the candidate, so the graph is the path throughout. The rule retains $\pi/3$ and $2\pi/3$ (path $\leftrightarrow$ triangle), as in (24.20).

**8.3 Relative deviation.** With $A(s):=\arccos\sqrt{F(s)}=\sigma_Xs\,(1+O(s^2))$:

$$
\delta(\lambda)=\max_{1\le k\le n-1}\Bigl\lvert\frac{A(k\lambda)}{\sigma_Xk\lambda}-1\Bigr\rvert\ \le\ \rho_F\bigl((n-1)\lambda\bigr)\le\rho_F(s_c),\qquad \rho_F(s):=\sup_{0<u\le s}\Bigl\lvert\frac{A(u)}{\sigma_Xu}-1\Bigr\rvert=O(s^2),
\tag{24.25}
$$

a bound on $(0,\lambda_*)$ that is uniform in $n$.

### Step 9. Example (c): window medium

The $\sigma_x^{[l]}$ commute, and $\langle0\vert e^{\mp i\lambda\sigma_x}\vert0\rangle=\cos\lambda$. In $e^{iK_{h_j}\lambda}e^{-iK_{h_i}\lambda}$ only the sites $l\in w_i\triangle w_j$ survive, and $\lvert w_i\triangle w_j\rvert=2k$ by (15.17). Hence, for all $\lambda$,

$$
W(h_i,h_j;\lambda)=(\cos\lambda)^{4k},\qquad\sqrt W=c^{\,k},\quad c:=\cos^2\lambda,\quad k=\min(\lvert i-j\rvert,m).
\tag{24.26}
$$

- $c=1$ ($\lambda\in\pi\mathbb Z$): all places form one point.
- $c=0$ ($\lambda\in\pi/2+\pi\mathbb Z$): all $W=0$, so the points are singletons and no pair is related.
- $c\in(0,1)$: the points are singletons and $\alpha=\arccos(c^k)$ with $0<\alpha_1<\dots<\alpha_m<\pi/2$. This is the same order pattern as (22.14). (4.3) depends on $\alpha$ only through comparisons among the $\alpha$'s and with $\pi/2$, so $N_V$ is (22.15).

$$
\begin{aligned}
&\lambda\in\pi\mathbb Z_{>0}:&&\mathcal P=\{H\},\ N_V\ \text{one vertex};\\
&\lambda\in\tfrac\pi2+\pi\mathbb Z_{\ge0}:&&\text{singletons},\ N_V=\emptyset\ (\text{no edges});\\
&\text{otherwise}:&&\text{singletons},\ h_i\sim h_j\iff\lvert i-j\rvert=1\ \text{or}\ \lvert i-j\rvert\ge2m-1 .
\end{aligned}
\tag{24.27}
$$

$$
\Lambda_{\mathcal P}=\pi\mathbb Z_{>0},\qquad \Lambda_N=\tfrac\pi2+\pi\mathbb Z_{\ge0},\qquad \lambda_*=\tfrac\pi2\ \ \text{for all }n\ge2m,\ m\ge2\ (\text{independent of }n).
\tag{24.28}
$$

$\pi/2\in\Lambda_N$ because the graph (22.15), which has edges, becomes edgeless there.

*Relative deviation.* For $\lvert u\rvert<\pi/2$, $\ln\cos u=-\sum_{j\ge1}a_ju^{2j}$ with $a_j>0$, $a_1=\tfrac12$, $a_2=\tfrac1{12}$. With $s=\sqrt{2k}\lambda<\pi/2$:

$$
2k\ln\cos\frac{s}{\sqrt{2k}}=-\sum_ja_js^{2j}(2k)^{1-j}>\ln\cos s ,
$$

so $\alpha_k<\sqrt{2k}\lambda$. For $s\ge\pi/2$ this holds trivially, since $\alpha_k<\pi/2$. With $d_0=\sqrt{2k}$ from (15.17):

$$
\delta(\lambda)=\max_{1\le k\le m}\Bigl[1-\frac{\arccos(\cos^{2k}\lambda)}{\sqrt{2k}\,\lambda}\Bigr]\in(0,1),\qquad \delta=\tfrac{(2m-1)\lambda^2}{12}+O(\lambda^4)\ (\lambda\to0),\qquad \delta\to1-\tfrac1{\sqrt{2m}}\ (\lambda\to\tfrac\pi2).
\tag{24.29}
$$

The small-$\lambda$ form follows from $\alpha_k/(\sqrt{2k}\lambda)=1-(2k-1)\lambda^2/12+O(\lambda^4)$, which is largest in deviation at $k=m$.

*Maximum at $k=m$ for every $\lambda\in(0,\pi/2)$.* Put $t:=-\ln c>0$. Then $-\ln\cos\alpha_k=kt$, so $\alpha_k^2/k=t/G(\alpha_k)$ with $G(\theta):=-\ln\cos\theta/\theta^2=\sum_ja_j\theta^{2j-2}$. $G$ is strictly increasing on $(0,\pi/2)$, since all $a_j>0$. Since $\alpha_k$ increases with $k$, $\alpha_k/(\sqrt{2k}\lambda)=\sqrt{t/(2G(\alpha_k))}/\lambda$ strictly decreases in $k$. So the maximum in (24.29) is attained at $k=m$: $\delta(\lambda)=1-\arccos(\cos^{2m}\lambda)/(\sqrt{2m}\,\lambda)$.

## Result

1. **(a)** $Z(h,h')$ is either $(0,\infty)$ or discrete and bounded away from $0$ (24.3). The first case holds iff (24.4), which implies $d_0=0$. **(b)** $\mathcal P(\lambda)=\mathcal P_{\rm gen}=H/{\equiv}$ off the discrete set $\Lambda_{\mathcal P}$ (24.5). On $\Lambda_{\mathcal P}$ points only merge, into the strict coarsening $\mathcal P_{\rm gen}/{\approx_\lambda}$; they never split (24.6). **(c)** $N_V$ changes only on the discrete set $\Lambda_N\subseteq Z_{\mathcal F}$. No condition beyond A1 and $\lambda$-independent records is needed (24.7)–(24.8). **(d)** $\lambda_*=\min(\Lambda_{\mathcal P}\cup\Lambda_N)>0$ always. On $(0,\lambda_*)$ the background is the small-$\lambda$ one, equal to (11.11) under the assumptions of 11 (24.9).
2. The trajectory is defined for every $\lambda\notin\Lambda_{\mathcal P}$ (indeed for every $\lambda>0$) and is analytic (24.10). It joins across $\Lambda_{\mathcal P}$ unconditionally, with the coarse-graining (24.11) at the merger. At $\Lambda_N$ only the neighbour relation changes, not the trajectory.
3. **(a)** $\Lambda_{\mathcal P}$ (24.13), $\Lambda_N\cap(0,\pi]=\{\lambda_*,\pi-\lambda_*\}$ (24.20), full background (24.17), (24.19), $\lambda_*=\pi/(\lfloor3n/2\rfloor-1)$ (24.18), $\delta$ (24.21). **(b)** $\Lambda_{\mathcal P}$ is nonempty iff the spectral support of $X$ in $\chi$ has commensurable differences (24.22)–(24.23). $s_m/(n-1)\le\lambda_*\le s_c/(n-1)$ with $\lambda_*\in\Lambda_N$ (24.24), and $\delta$ is bounded by (24.25). $\Lambda_N$ consists exactly of the candidates $\lambda_0\in Z_b\setminus\Lambda_{\mathcal P}$ at which the edge indicators, computed from the values $F(k\lambda)$, are not all equal at $\lambda_0^-$, $\lambda_0$ and $\lambda_0^+$ (24.30). **(c)** Background for all $\lambda$ (24.27); $\Lambda_{\mathcal P}=\pi\mathbb Z_{>0}$, $\Lambda_N=\pi/2+\pi\mathbb Z_{\ge0}$, $\lambda_*=\pi/2$ independent of $n$ (24.28); $\delta$ (24.29).

## Consistency checks

1. **$\lambda\to0$.** In (a), $\alpha/\lambda=\lvert i-j\rvert=d_0$ exactly for $\lambda\le\pi/(2(n-1))$, matching (5.10) and (5.15) ($\delta=0$). $N_V$ is the path, which is (11.11) for $d_0=\lvert i-j\rvert$; its assumptions hold, since $\max(\lvert i-l\rvert,\lvert l-j\rvert)\neq\lvert i-j\rvert$. In (c), $\delta\to0$.
2. **Periodicity and mirror symmetry.** $\Lambda_{\mathcal P}$ in (24.13) ($p/q\to(q-p)/q$) and $\Lambda_N=\{\lambda_*,\pi-\lambda_*\}$ are invariant under $\lambda\to\pi-\lambda$, as (24.14) requires. In (c), $\pi\mathbb Z$ and $\pi/2+\pi\mathbb Z$ are invariant under $\lambda\to\pi-\lambda$ and $\lambda\to\lambda+\pi$.
3. **Direct evaluation.**
   - $n=3$, $\lambda=\pi/3$: $\alpha(h_1,h_2)=\alpha(h_2,h_3)=\alpha(h_1,h_3)=\pi/3$, so by the tie rule the graph is a triangle. For $\pi/4<\lambda<\pi/3$, $h_2$ blocks $(h_1,h_3)$ since $\lambda<\pi-2\lambda$. So $\lambda_*(3)=\pi/3$, as in (24.18).
   - $n=4$, $\lambda=\pi/5$: $\alpha(h_1,h_4)=2\pi/5=\max(\alpha(h_1,h_2),\alpha(h_2,h_4))$, so the tie keeps the edge. Below $\pi/5$, $2\lambda<\pi-3\lambda$ blocks it. So $\lambda_*(4)=\pi/5$.

## Open issues

- (b): $\Lambda_N$ is characterized exactly by (24.30), but only implicitly. The candidates $Z_b$ are roots of $F(k\lambda)=F(k'\lambda)$ and $F(k\lambda)=0$, which have no closed form for general $X$. $s_m$, $s_c$, $\tau$ are defined from $F$ but are not in closed form in general.

## Methods used

- Identity theorem for real-analytic functions; finite exponential sums
- Bohr almost periodicity
- Partial order of partitions (coarsening)
- Sign patterns of analytic families; intermediate value theorem
- Circle (arc-metric) embedding; ball intersections on $\mathbb R/\pi\mathbb Z$
- Farey neighbours; linear gap functions
- Spectral decomposition; triangle-inequality equality
- Power series of $\ln\cos$