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1 \section{One Hilbert space behind all cylinder depths}
2 \label{a-limit-hilbert}
3 
4 An isometric refinement map compares two finite registers. An infinite
5 quantum dynamics requires more: a common completed Hilbert space, a
6 specified operator domain, and a topology in which the cutoffs converge.
7 These requirements can all be met for the cylinder tower. They are separate
8 requirements for a walk on a growing finite tree.
9 
10 Fix a prime $p$. The compact additive group $\Z_p$ has normalized Haar
11 measure $\mu_p$, with
12 \[
13  B_{m,a}=a+p^m\Z_p,\qquad 0\leq a<p^m,\qquad
14  \mu_p(B_{m,a})=p^{-m}.
15 \]
16 Write $\HH_p=L^2(\Z_p,\mu_p;\C)$. We use the complex inner product
17 $\inner{f}{g}=\int\overline f g\,d\mu_p$, linear in its second argument.
18 The $p$-adic coordinate belongs to the measurable space. Amplitudes and the
19 Born probabilities use ordinary complex Hilbert space.
20 
21 \subsection{Counting amplitudes and Haar densities}
22 
23 At depth $m$, the finite space $\HH_{p,m}=\ell^2(\Z/p^m\Z)$ has counting
24 inner product and orthonormal basis $e_a$. Define
25 \begin{equation}
26  \mathcal U_{p,m}e_a=p^{m/2}\mathbf1_{B_{m,a}},
27  \qquad \mathcal V_m=\operatorname{span}
28  \{\mathbf1_{B_{m,a}}:0\leq a<p^m\}.
29  \label{a-limit-cylinder-embedding}
30 \end{equation}
31 The cylinders are disjoint. Consequently
32 \[
33  \inner{\mathcal U_{p,m}e_a}{\mathcal U_{p,m}e_b}
34  =p^m\mu_p(B_{m,a}\cap B_{m,b})=\delta_{ab}.
35 \]
36 The same indicators span $\mathcal V_m$, so $\mathcal U_{p,m}$ is unitary
37 onto this subspace. For $c=\sum_a c_ae_a$, its image has constant value
38 $p^{m/2}c_a$ on $B_{m,a}$. In particular,
39 \begin{equation}
40  \int_{B_{m,a}}|\mathcal U_{p,m}c|^2\,d\mu_p=|c_a|^2.
41  \label{a-limit-born-normalization}
42 \end{equation}
43 There is no extra probability factor after this normalization.
44 
45 The finite refinement map is
46 \[
47  V_{p,m}e_a=\frac1{\sqrt p}
48  \sum_{d=0}^{p-1}e_{a+d p^m}.
49 \]
50 Since $B_{m,a}=\bigsqcup_d B_{m+1,a+d p^m}$, it satisfies
51 \begin{equation}
52  \mathcal U_{p,m+1}V_{p,m}=\mathcal U_{p,m}.
53  \label{a-limit-common-embedding}
54 \end{equation}
55 Indeed, the coefficient on each child indicator is
56 $p^{-1/2}p^{(m+1)/2}=p^{m/2}$, precisely the coefficient on the parent.
57 Thus the refinement changes the address resolution of an embedded state
58 without changing the function in $\HH_p$.
59 
60 An equivalent finite convention uses the weighted norm
61 \[
62  \norm{f}_{m,\mathrm{Haar}}^2=p^{-m}\sum_a|f(a)|^2.
63 \]
64 Here $f(a)$ is a density amplitude rather than a counting amplitude, and
65 the embedding into $\HH_p$ sends $f$ to
66 $\sum_a f(a)\mathbf1_{B_{m,a}}$. Its refinement repeats the parent value
67 on all children, without $p^{-1/2}$. The coordinate change
68 $f(a)=p^{m/2}c_a$ identifies the two conventions. The factors
69 $p^{-m}$ in the measure and $p^{m/2}$ in the counting embedding are a
70 single normalization expressed in two places.
71 
72 \subsection{Cylinder averaging as an orthogonal projection}
73 
74 For $f\in\HH_p$, define
75 \begin{equation}
76  (\mathsf E_m f)(x)
77  =p^m\int_{x+p^m\Z_p} f(y)\,d\mu_p(y).
78  \label{a-limit-expectation}
79 \end{equation}
80 The integral is defined since $L^2$ is contained in $L^1$ on this
81 probability space. It produces a function constant on each depth-$m$
82 cylinder.
83 
84 \begin{proposition}[Nested cylinder projections]
85 \label{a-limit-expectation-algebra}
86 The operator $\mathsf E_m$ is the orthogonal projection onto $\mathcal V_m$.
87 For $m,n\geq0$,
88 \begin{equation}
89  \mathsf E_m^*=\mathsf E_m,\qquad
90  \mathsf E_m^2=\mathsf E_m,\qquad
91  \mathsf E_m\mathsf E_n=\mathsf E_{\min(m,n)}.
92  \label{a-limit-projection-product}
93 \end{equation}
94 Moreover $\mathsf E_m f\to f$ in $L^2$ for every $f\in\HH_p$.
95 \end{proposition}
96 \begin{proof}
97 For a cylinder-constant $g$, the integral of $f-\mathsf E_m f$ over each
98 depth-$m$ cylinder is zero. Multiplying by the constant value of
99 $\overline g$ there and summing shows
100 $\inner{g}{f-\mathsf E_mf}=0$. The image is in $\mathcal V_m$, and averaging
101 a function in $\mathcal V_m$ returns the function. These statements
102 characterize the orthogonal projection and establish the first two
103 identities. The spaces are nested, because every cylinder partitions into
104 its children. Orthogonal projections onto nested subspaces satisfy the
105 third identity: the larger projection fixes the smaller range, and the
106 reverse order follows by taking adjoints.
107 
108 For completeness, the union of these ranges is dense. Cylinders form a
109 countable clopen base for the topology on $\Z_p$ and generate its Borel
110 sigma-algebra. If $f\in L^2$ is orthogonal to every cylinder indicator,
111 the finite complex measure $\nu(A)=\int_A f\,d\mu_p$ vanishes on
112 cylinders and their finite unions. The class of sets on which $\nu$ is
113 zero contains the whole space, is closed under complements, and is
114 closed under countable disjoint unions. The elementary monotone-class
115 extension from the cylinder algebra therefore gives $\nu(A)=0$ for every
116 Borel set. Applying this to the positive and negative level sets of the
117 real and imaginary parts of $f$ yields $f=0$ almost everywhere.
118 The orthogonal complement of $\bigcup_m\mathcal V_m$ is thus zero.
119 
120 Given $\varepsilon>0$, choose $g\in\mathcal V_M$ with
121 $\norm{f-g}_2<\varepsilon$. For $m\geq M$, the projection fixes $g$, so
122 $\norm{f-\mathsf E_m f}_2
123 =\norm{(I-\mathsf E_m)(f-g)}_2\leq\norm{f-g}_2<\varepsilon$.
124 This proves strong convergence without exchanging an infinite integral
125 with a limit.
126 \end{proof}
127 
128 The completion of the isometric direct system
129 $(\HH_{p,m},V_{p,m})$ is therefore $\HH_p$: equation
130 \eqref{a-limit-common-embedding} gives a single isometric map on the
131 algebraic direct limit, and its image is the dense union of the
132 $\mathcal V_m$. For $m\leq M$, with
133 $V_{p,m\to M}=V_{p,M-1}\cdots V_{p,m}$, one also has
134 \begin{equation}
135  \mathsf E_m\big|_{\mathcal V_M}
136  =\mathcal U_{p,M}V_{p,m\to M}V_{p,m\to M}^*
137  \mathcal U_{p,M}^*.
138  \label{a-limit-finite-projection}
139 \end{equation}
140 Both sides are the orthogonal projection in $\mathcal V_M$ onto
141 $\mathcal V_m$. This is the exact relation between the finite matrices
142 and the common infinite Hilbert space.
143 
144 \subsection{Orthogonal details and a complete Haar basis}
145 
146 Use $\mathsf F_0=\mathsf E_0$ and
147 $\mathsf F_m=\mathsf E_m-\mathsf E_{m-1}$ for $m\geq1$. These are the
148 infinite depth details called $\mathsf D_m$ in the archived chapter.
149 They correspond to the finite $D_m^{(M)}$ after the embedding
150 $\mathcal U_{p,M}$. Equation \eqref{a-limit-projection-product} gives
151 \begin{equation}
152  \mathsf F_m\mathsf F_n=\delta_{mn}\mathsf F_m,
153  \qquad \sum_{m=0}^M\mathsf F_m=\mathsf E_M.
154  \label{a-limit-detail-algebra}
155 \end{equation}
156 For example, if $1\leq m<n$, the four products in
157 $(\mathsf E_m-\mathsf E_{m-1})(\mathsf E_n-\mathsf E_{n-1})$
158 are $\mathsf E_m,\mathsf E_m,\mathsf E_{m-1},\mathsf E_{m-1}$ and cancel.
159 For $m=n$ the same expansion is
160 $\mathsf E_m-\mathsf E_{m-1}$. The cases involving $m=0$ use
161 $\mathsf E_0\mathsf E_n=\mathsf E_0$.
162 
163 Consequently
164 \begin{equation}
165  \HH_p=\bigoplus_{m=0}^{\infty}\operatorname{Ran}\mathsf F_m,
166  \qquad \norm{f}_2^2=\sum_{m=0}^{\infty}\norm{\mathsf F_mf}_2^2.
167  \label{a-limit-parseval}
168 \end{equation}
169 The sum of the projections is $I$ in the strong operator topology.
170 The detail ranks are $d_0=1$ and $d_m=(p-1)p^{m-1}$ for $m\geq1$,
171 the differences of the dimensions $p^m-p^{m-1}$.
172 
173 For an explicit basis, put $\omega_p=\exp(2\pi i/p)$ and define
174 \begin{equation}
175  \psi_{m,a,q}=p^{(m-1)/2}\sum_{d=0}^{p-1}
176  \omega_p^{qd}\mathbf1_{B_{m,a+d p^{m-1}}},
177  \quad
178  \begin{matrix}m\geq1,\quad 0\leq a<p^{m-1},\\1\leq q<p.\end{matrix}
179  \label{a-limit-haar-wavelets}
180 \end{equation}
181 Each has zero average over its parent, since
182 $\sum_d\omega_p^{qd}=0$, and lies in $\operatorname{Ran}\mathsf F_m$.
183 Its norm squared is $p^{m-1}p\,p^{-m}=1$. Distinct parents have disjoint
184 support, while two characters at the same parent have inner product
185 $p^{-1}\sum_d\omega_p^{(q'-q)d}=\delta_{qq'}$. The number of these
186 orthonormal functions is exactly $d_m$, so they span that level.
187 Together with $\psi_0=1$, they form a complete orthonormal basis by
188 \eqref{a-limit-parseval}.
189 
190 \subsection{A statewise approximation rate}
191 
192 For $s>0$, define the Haar regularity norm
193 \[
194  \norm{f}_{\mathfrak H_p^s}^2
195  =\sum_{m\geq0}p^{2sm}\norm{\mathsf F_mf}_2^2,
196  \qquad
197  \mathfrak H_p^s=\{f\in\HH_p:\norm{f}_{\mathfrak H_p^s}<\infty\}.
198 \]
199 No identification with another Sobolev convention is required here.
200 The definition itself gives the quantitative cutoff estimate.
201 
202 \begin{proposition}[Exact tail and regularity bound]
203 \label{a-limit-tail}
204 For $f\in\HH_p$ and every $M\geq0$,
205 \begin{equation}
206  \norm{(I-\mathsf E_M)f}_2^2
207  =\sum_{m>M}\norm{\mathsf F_mf}_2^2.
208  \label{a-limit-tail-identity}
209 \end{equation}
210 If $f\in\mathfrak H_p^s$, then
211 \begin{equation}
212  \norm{(I-\mathsf E_M)f}_2
213  \leq p^{-s(M+1)}\norm{f}_{\mathfrak H_p^s}.
214  \label{a-limit-tail-bound}
215 \end{equation}
216 The convergence $\mathsf E_M\to I$ is strong and is not convergence in
217 operator norm.
218 \end{proposition}
219 \begin{proof}
220 The omitted details are orthogonal, so Parseval gives the equality.
221 For $m>M$, $p^{2sm}\geq p^{2s(M+1)}$. Multiplying the omitted summands
222 by this lower bound proves the squared estimate, and taking square roots
223 gives \eqref{a-limit-tail-bound}. At every finite $M$ there is a unit
224 vector in $\operatorname{Ran}\mathsf F_{M+1}$, on which
225 $I-\mathsf E_M$ acts as the identity. Thus
226 $\norm{I-\mathsf E_M}=1$ for every $M$.
227 \end{proof}
228 
229 The rate is a bound on states in the displayed regularity class. For an
230 arbitrary $L^2$ state the tail still tends to zero, but no common geometric
231 rate follows. A finite array already in $\mathcal V_L$ has zero tail once
232 $M\geq L$, which is a different statement from testing an infinite tail.
233 
234 
235 \section{Self-adjoint dynamics and what a cutoff converges to}
236 \label{a-limit-dynamics}
237 
238 \subsection{The maximal domain for level energies}
239 
240 Let $(\epsilon_m)_{m\geq0}$ be any sequence of real numbers. Define
241 \begin{align}
242  \Dom(H_\epsilon)
243  &=\left\{f\in\HH_p:
244  \sum_{m\geq0}|\epsilon_m|^2\norm{\mathsf F_mf}_2^2<\infty\right\},
245  \nonumber\\
246  H_\epsilon f&=\sum_{m\geq0}\epsilon_m\mathsf F_mf.
247  \label{a-limit-hamiltonian-domain}
248 \end{align}
249 The series converges in $L^2$ exactly on this domain. It is not necessary
250 to bound the energies uniformly, but it is necessary to state the domain.
251 
252 \begin{theorem}[Real level energies define a self-adjoint operator]
253 \label{a-limit-selfadjoint}
254 The operator in \eqref{a-limit-hamiltonian-domain} is densely defined,
255 closed, and self-adjoint. The finite cylinder functions form an operator
256 core. It is bounded if and only if $\sup_m|\epsilon_m|<\infty$, and its
257 spectrum is
258 \begin{equation}
259  \sigma(H_\epsilon)=\overline{\{\epsilon_m:m\geq0\}}.
260  \label{a-limit-spectrum}
261 \end{equation}
262 \end{theorem}
263 \begin{proof}
264 The orthogonal decomposition \eqref{a-limit-parseval} identifies $f$ with
265 the square-summable sequence $f_m=\mathsf F_mf$. Finite sequences belong
266 to the domain, so it is dense. The real multipliers imply symmetry.
267 To identify the adjoint, suppose $g\in\Dom(H_\epsilon^*)$ and write
268 $k=H_\epsilon^*g$. Testing the adjoint identity against an arbitrary
269 vector supported on level $m$ gives $k_m=\epsilon_m g_m$. Since $k$ is
270 square summable, $\sum_m|\epsilon_m|^2\norm{g_m}^2<\infty$.
271 Thus $g\in\Dom(H_\epsilon)$ and the adjoint agrees with $H_\epsilon$.
272 The reverse inclusion is symmetry. This proves self-adjointness and
273 therefore closedness.
274 
275 For $f\in\Dom(H_\epsilon)$, the cylinder truncation $\mathsf E_M f$
276 satisfies both
277 $\norm{f-\mathsf E_M f}_2\to0$ and
278 $\norm{H_\epsilon(f-\mathsf E_M f)}_2\to0$, because the two squared norms
279 are the tails of convergent nonnegative series. This is precisely density
280 in the graph norm and proves the core assertion.
281 If the energies have supremum $C<\infty$, Parseval gives
282 $\norm{H_\epsilon f}_2\leq C\norm f_2$. Conversely, a unit vector on
283 level $m$ has image norm $|\epsilon_m|$, so boundedness forces a finite
284 supremum.
285 
286 If $z$ is outside the closed energy set, the sequence
287 $(\epsilon_m-z)^{-1}$ is bounded and defines an inverse for
288 $H_\epsilon-z$. Its image lies in the domain because
289 $\epsilon_m/(\epsilon_m-z)=1+z/(\epsilon_m-z)$ is bounded as well.
290 This proves that such $z$ is in the resolvent set. An energy
291 $\epsilon_m$ is an eigenvalue, while a limit point $z$ has unit level
292 vectors $u_j$ with $\norm{(H_\epsilon-z)u_j}_2\to0$.
293 A bounded inverse would contradict this limit. Hence the closed energy
294 set is exactly the spectrum.
295 \end{proof}
296 
297 This proof also excludes a hidden choice of boundary condition at infinite
298 depth. On the finite-cylinder core, the real level operator has the unique
299 self-adjoint closure just described.
300 
301 \subsection{Explicit spectral calculus and strong continuity}
302 
303 For bounded Borel $F:\R\to\C$, define
304 \begin{equation}
305  F(H_\epsilon)f=\sum_{m\geq0}F(\epsilon_m)\mathsf F_mf.
306  \label{a-limit-borel-calculus}
307 \end{equation}
308 The series converges by Parseval and
309 $\norm{F(H_\epsilon)f}_2\leq\sup_m|F(\epsilon_m)|\norm f_2$.
310 The same level decomposition verifies sums, products, and adjoints of
311 these multipliers directly. In particular,
312 \begin{align}
313  U(t)&=e^{-itH_\epsilon}
314  =\sum_{m\geq0}e^{-it\epsilon_m}\mathsf F_m,
315  \label{a-limit-unitary}\\
316  (z-H_\epsilon)^{-1}
317  &=\sum_{m\geq0}(z-\epsilon_m)^{-1}\mathsf F_m,
318  \quad z\notin\sigma(H_\epsilon).
319  \label{a-limit-resolvent}
320 \end{align}
321 The projections in these sums converge strongly, not absolutely in
322 operator norm in general.
323 
324 The modulus-one multipliers give $U(t)^*U(t)=I$ and
325 $U(t+s)=U(t)U(s)$. They also give strong continuity:
326 \[
327  \norm{(U(t)-U(t_0))f}_2^2
328  =\sum_m|e^{-it\epsilon_m}-e^{-it_0\epsilon_m}|^2
329  \norm{\mathsf F_mf}_2^2\longrightarrow0.
330 \]
331 To justify this limit, first bound the omitted summands by
332 $4\sum_{m>M}\norm{\mathsf F_mf}_2^2$, then use ordinary continuity in
333 the finite sum up to $M$. On $\Dom(H_\epsilon)$ the same argument,
334 using $|(e^{-it\epsilon}-1)/t|\leq|\epsilon|$, proves
335 \[
336  \lim_{t\to0}\frac{U(t)f-f}{t}=-iH_\epsilon f.
337 \]
338 Thus the declared self-adjoint operator is the generator. A general
339 $L^2$ state evolves unitarily even if this derivative does not exist.
340 
341 \subsection{Two distinct extensions of the finite dynamics}
342 
343 Let $H_\epsilon^{(M)}$ be the finite restriction of $H_\epsilon$ to
344 $\mathcal V_M$. One useful approximation to $U(t)$ is its embedded
345 finite propagator, extended by zero on the omitted subspace:
346 \begin{equation}
347  U_M^{\mathrm{emb}}(t)
348  =\sum_{m=0}^{M}e^{-it\epsilon_m}\mathsf F_m
349  =U(t)\mathsf E_M.
350  \label{a-limit-embedded-propagator}
351 \end{equation}
352 It is unitary on $\mathcal V_M$ and has kernel
353 $\mathcal V_M^\perp$ as an operator on $\HH_p$.
354 
355 \Needspace{15\baselineskip}
356 \begin{proposition}[Exact state error for a compatible cutoff]
357 \label{a-limit-unitary-error}
358 For every $f\in\HH_p$,
359 \begin{equation}
360  \norm{U(t)f-U_M^{\mathrm{emb}}(t)f}_2
361  =\norm{(I-\mathsf E_M)f}_2.
362  \label{a-limit-unitary-tail}
363 \end{equation}
364 This equality is uniform for all real $t$. For
365 $f\in\mathfrak H_p^s$, the right side is bounded by
366 $p^{-s(M+1)}\norm f_{\mathfrak H_p^s}$.
367 For $f\in\Dom(H_\epsilon)$ there is also the exact graph-norm identity
368 \begin{align}
369  &\norm{U(t)f-U_M^{\mathrm{emb}}(t)f}_2^2
370  +\norm{H_\epsilon(U(t)f-U_M^{\mathrm{emb}}(t)f)}_2^2
371  \nonumber\\
372  &\hspace{2em}=
373  \sum_{m>M}(1+|\epsilon_m|^2)\norm{\mathsf F_mf}_2^2.
374  \label{a-limit-graph-tail}
375 \end{align}
376 \end{proposition}
377 \begin{proof}
378 The difference is the omitted orthogonal sum
379 $\sum_{m>M}e^{-it\epsilon_m}\mathsf F_mf$. Its multipliers have modulus
380 one, so Parseval gives \eqref{a-limit-unitary-tail}. Applying
381 $H_\epsilon$ multiplies the omitted level by $\epsilon_m$ and gives the
382 second identity. The domain hypothesis ensures that this second series
383 is finite. The regularity estimate is Proposition \ref{a-limit-tail}.
384 \end{proof}
385 
386 An alternative is to extend the finite Hamiltonian by the zero
387 Hamiltonian on the omitted subspace:
388 \[
389  \widetilde H_M=H_\epsilon\mathsf E_M
390  =\sum_{m\leq M}\epsilon_m\mathsf F_m.
391 \]
392 This is a bounded self-adjoint operator on all of $\HH_p$. Its
393 propagator is a full-space unitary, but has a different tail:
394 \begin{equation}
395  e^{-it\widetilde H_M}
396  =U_M^{\mathrm{emb}}(t)+(I-\mathsf E_M),
397  \qquad
398  \norm{(e^{-it\widetilde H_M}-U(t))f}_2
399  \leq2\norm{(I-\mathsf E_M)f}_2.
400  \label{a-limit-zero-hamiltonian-extension}
401 \end{equation}
402 Likewise, for nonreal $z$,
403 \begin{align}
404  (z-\widetilde H_M)^{-1}
405  &=\mathsf E_M(z-H_\epsilon)^{-1}
406    +z^{-1}(I-\mathsf E_M),\nonumber\\
407  \norm{((z-\widetilde H_M)^{-1}-(z-H_\epsilon)^{-1})f}_2
408  &\leq\left(|z|^{-1}+|\operatorname{Im}z|^{-1}\right)
409  \norm{(I-\mathsf E_M)f}_2.
410  \label{a-limit-strong-resolvent}
411 \end{align}
412 Thus $\widetilde H_M$ converges to $H_\epsilon$ in the strong resolvent
413 sense. These statements specify operators on one common Hilbert space.
414 They do not compare matrices of changing dimensions without an embedding.
415 
416 The embedded propagators converge strongly, uniformly in time on each
417 fixed state. Their operator norm error nevertheless satisfies
418 \[
419  \norm{U(t)-U_M^{\mathrm{emb}}(t)}=1
420  \qquad\text{for every }M,t.
421 \]
422 The fully unitary extensions also converge strongly with the bound in
423 \eqref{a-limit-zero-hamiltonian-extension}; norm convergence of these
424 extensions needs a separate condition on the omitted phases. Neither
425 assertion is a general norm-limit theorem for arbitrary quantum dynamics.
426 
427 \subsection{A graph-domain example with exact infinite tails}
428 
429 \begin{example}[Convergent states and finite energy derivatives]
430 \label{a-limit-geometric-example}
431 Choose one unit vector $u_m\in\operatorname{Ran}\mathsf F_m$ at every
432 level, let $0<q<1$, and set
433 \[
434  f_q=\sum_{m\geq0}\sqrt{(1-q)q^m}\,u_m,
435  \qquad \epsilon_m=2^m.
436 \]
437 Then $\norm{f_q}_2=1$ and
438 \begin{equation}
439  \norm{(I-\mathsf E_M)f_q}_2^2=q^{M+1}.
440  \label{a-limit-geometric-state-tail}
441 \end{equation}
442 However, $f_q\in\Dom(H_\epsilon)$ exactly when $4q<1$:
443 \[
444  \norm{H_\epsilon f_q}_2^2
445  =(1-q)\sum_{m\geq0}(4q)^m.
446 \]
447 For $q<1/4$, its graph tail is exactly
448 \begin{equation}
449  q^{M+1}+\frac{(1-q)(4q)^{M+1}}{1-4q}.
450  \label{a-limit-geometric-graph-tail}
451 \end{equation}
452 For $q=1/8$, the state norm tail decays faster than the graph norm tail.
453 For $q=1/2$, the normalized state still has a convergent $L^2$ cylinder
454 approximation and a well-defined unitary evolution, but is outside the
455 Hamiltonian domain. All these conclusions concern genuinely infinite
456 series, rather than a finite last level with its tail set to zero.
457 \end{example}
458 
459 \adicfigure{10_tail_graph_norm}{Exact infinite tails for the geometric
460 level masses of Example \ref{a-limit-geometric-example}. The operator
461 domain controls a weighted tail in addition to the state norm tail.
462 The parameters are declared in the plate and its reproducible Python
463 source.}
464 
465 \subsection{Compact resolvents and thermal weights}
466 
467 The omitted embedded resolvent has the exact norm
468 \begin{equation}
469  \norm{(I-\mathsf E_M)(z-H_\epsilon)^{-1}}
470  =\sup_{m>M}|z-\epsilon_m|^{-1}.
471  \label{a-limit-resolvent-tail-norm}
472 \end{equation}
473 The upper bound follows by Parseval. For the lower bound, test on a unit
474 vector at any omitted level and take the supremum. Since every level is
475 nonzero and finite dimensional, this proves the following equivalence.
476 
477 \begin{proposition}[Exact compactness criterion]
478 \label{a-limit-compact-resolvent}
479 The operator $H_\epsilon$ has compact resolvent if and only if
480 $|\epsilon_m|\to\infty$. In that case the embedded finite resolvents
481 converge in operator norm. This is an assertion about the embedded
482 resolvents, rather than the zero-Hamiltonian extensions of
483 \eqref{a-limit-strong-resolvent}.
484 \end{proposition}
485 \begin{proof}
486 If the absolute energies tend to infinity, the right side of
487 \eqref{a-limit-resolvent-tail-norm} tends to zero. The retained part of the
488 resolvent has finite rank, so the resolvent is a norm limit of finite-rank
489 operators and is compact. Conversely, if the absolute energies do not
490 tend to infinity, there are infinitely many distinct levels $m_j$ with
491 $|\epsilon_{m_j}|\leq C$ for some $C$. Unit vectors $u_j$ on these levels
492 are orthogonal. For a fixed nonreal $z$, their resolvent images are also
493 orthogonal and have norm at least $(|z|+C)^{-1}$.
494 They have no convergent subsequence. The resolvent is therefore not
495 compact.
496 \end{proof}
497 
498 When $|\epsilon_m|\to\infty$, the omitted part of the resolvent for
499 $\widetilde H_M$ in \eqref{a-limit-strong-resolvent} still carries the
500 constant multiplier $z^{-1}$. It does not converge in norm to the omitted
501 resolvent of $H_\epsilon$. Strong resolvent convergence and norm
502 convergence of an embedded finite-rank resolvent are distinct claims.
503 
504 For $\beta>0$, the multiplier $e^{-\beta H_\epsilon}$ is bounded precisely
505 when the energies are bounded below. If there is no lower bound, the
506 levelwise exponential is an unbounded operator with domain
507 $\{f:\sum_m e^{-2\beta\epsilon_m}\norm{\mathsf F_mf}_2^2<\infty\}$.
508 Thus a finite Euclidean intertwining identity does not by itself supply a
509 bounded infinite heat semigroup.
510 When bounded below, the positive operator has trace
511 \begin{equation}
512  \operatorname{Tr}(e^{-\beta H_\epsilon})
513  =e^{-\beta\epsilon_0}
514  +\sum_{m\geq1}(p-1)p^{m-1}e^{-\beta\epsilon_m},
515  \label{a-limit-thermal-trace}
516 \end{equation}
517 possibly infinite. This follows by summing its nonnegative diagonal
518 entries in the complete Haar basis. It is trace class exactly when this
519 series converges. For $\epsilon_m=c m$ with $c>0$, the condition is
520 $\beta c>\log p$, by the geometric-series test. For
521 $\epsilon_m=p^{\alpha m}$ with $\alpha>0$, it converges for every
522 $\beta>0$, since the exponential energy term eventually dominates the
523 linear-in-$m$ logarithm of the multiplicity. A thermodynamic partition
524 function therefore needs an energy choice and a summability condition in
525 addition to Hilbert-space reconstruction.
526 
527 
528 \section{Compatible interactions across finitely many primes}
529 \label{a-limit-multiprime}
530 
531 \subsection{Tensor embeddings and rectangular cutoffs}
532 
533 Let $\mathcal P$ be a finite prime set and let $\KK_{\mathrm{tor}}$ be a
534 finite-dimensional complex space for the retained torsion and dyadic sign
535 registers. Set
536 \begin{equation}
537  \HH_{\mathcal P}^{\infty}
538  =\KK_{\mathrm{tor}}\otimes
539  \bigotimes_{p\in\mathcal P}L^2(\Z_p,\mu_p).
540  \label{a-limit-multiprime-space}
541 \end{equation}
542 The corresponding finite register embeds by
543 $I_{\KK_{\mathrm{tor}}}\otimes\bigotimes_p\mathcal U_{p,h_p}$.
544 For a multi-depth $\mathbf h=(h_p)$, define
545 \[
546  \mathsf E_{\mathbf h}
547  =I_{\KK_{\mathrm{tor}}}\otimes\bigotimes_p\mathsf E_{p,h_p}.
548 \]
549 Factorwise projection identities give
550 \begin{equation}
551  \mathsf E_{\mathbf h}\mathsf E_{\mathbf k}
552  =\mathsf E_{\min(\mathbf h,\mathbf k)},
553  \qquad \mathsf E_{\mathbf h}f\to f
554  \quad\text{as every }h_p\to\infty.
555  \label{a-limit-rectangular-projections}
556 \end{equation}
557 To justify the second statement, first take an elementary tensor. The
558 difference of the tensor projections and the identity is a finite
559 telescoping sum, one factor of which tends strongly to zero in each term,
560 while all other projection factors have norm at most one. Finite sums of
561 elementary tensors are dense, and the projections are contractions. This
562 extends convergence to every $f$.
563 
564 Refinement in distinct coordinates is path independent because
565 \[
566  (V_p\otimes I)(I\otimes V_q)
567  =(I\otimes V_q)(V_p\otimes I)=V_p\otimes V_q.
568 \]
569 The identity on $\KK_{\mathrm{tor}}$ leaves the separately retained sign
570 channel untouched by the averaging. This is a finite set of primes.
571 An infinite tensor product, a restricted adelic product, and a model
572 including the Archimedean place each require further choices.
573 
574 Define the joint detail projections
575 \begin{equation}
576  \mathsf F_{\mathbf r}
577  =I_{\KK_{\mathrm{tor}}}\otimes
578  \bigotimes_{p\in\mathcal P}\mathsf F_{p,r_p},
579  \qquad \mathbf r\in\N^{\mathcal P}.
580  \label{a-limit-joint-details}
581 \end{equation}
582 They are pairwise orthogonal and have finite-dimensional ranges.
583 Rectangular sums equal $\mathsf E_{\mathbf h}$, so their strong sum is
584 the identity. This permits interactions between primes while preserving
585 the exact depth filtration.
586 
587 \subsection{A complete block-domain theorem}
588 
589 The scalar weights are one particularly simple choice. The following
590 editorial extension states the exact additional input for interacting
591 blocks. It is useful because the torsion operators inside a joint packet
592 need not commute with one another.
593 
594 \begin{theorem}[Self-adjoint packet blocks and compatible finite dynamics]
595 \label{a-limit-block-hamiltonian}
596 For every $\mathbf r$, let $A_{\mathbf r}$ be a self-adjoint operator on
597 the finite-dimensional space
598 $\KK_{\mathbf r}=\operatorname{Ran}\mathsf F_{\mathbf r}$.
599 On $\HH_{\mathcal P}^{\infty}=\bigoplus_{\mathbf r}\KK_{\mathbf r}$ define
600 \begin{align}
601  \Dom(H_A)&=\left\{f:\sum_{\mathbf r}
602  \norm{A_{\mathbf r}\mathsf F_{\mathbf r}f}^2<\infty\right\},
603  \nonumber\\
604  H_A f&=\sum_{\mathbf r}A_{\mathbf r}\mathsf F_{\mathbf r}f.
605  \label{a-limit-block-domain}
606 \end{align}
607 Then $H_A$ is self-adjoint, and finite rectangular packet sums form a
608 core. Every rectangular range reduces $H_A$, and its finite restriction
609 $H_A^{(\mathbf h)}$ satisfies exact embedding intertwining with every
610 larger rectangular restriction. For bounded Borel $F$,
611 \begin{equation}
612  F(H_A)f=\sum_{\mathbf r}F(A_{\mathbf r})\mathsf F_{\mathbf r}f,
613  \qquad
614  F(H_A)\mathsf E_{\mathbf h}\to F(H_A)
615  \quad\text{strongly}.
616  \label{a-limit-block-calculus}
617 \end{equation}
618 In particular,
619 \begin{equation}
620  \norm{e^{-itH_A}(I-\mathsf E_{\mathbf h})f}
621  =\norm{(I-\mathsf E_{\mathbf h})f}.
622  \label{a-limit-block-unitary-error}
623 \end{equation}
624 \end{theorem}
625 \begin{proof}
626 Finite rectangular sums belong to the domain and are dense. Symmetry
627 holds by symmetry on every block. If $g$ belongs to the adjoint domain
628 and $k=H_A^*g$, testing against an arbitrary vector in
629 $\KK_{\mathbf r}$ gives
630 $\mathsf F_{\mathbf r}k=A_{\mathbf r}\mathsf F_{\mathbf r}g$.
631 Square summability of $k$ is precisely the domain condition in
632 \eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two
633 square-summable sequences $f_{\mathbf r}$ and
634 $A_{\mathbf r}f_{\mathbf r}$ proves the graph-core assertion.
635 
636 The rectangular projection selects a collection of whole blocks, so it
637 commutes with $H_A$ on the domain and reduces the operator.
638 The restriction to each retained block is literally the same matrix at
639 every subsequent depth. This proves the finite intertwining, and
640 finite-dimensional spectral calculus on each block gives
641 \eqref{a-limit-block-calculus}. Boundedness of $F$ bounds its omitted
642 image by $\norm F_\infty\norm{(I-\mathsf E_{\mathbf h})f}$, proving
643 strong convergence. Finally, each block exponential is unitary, so
644 orthogonality of the omitted blocks gives
645 \eqref{a-limit-block-unitary-error}.
646 \end{proof}
647 
648 For example, on a joint depth packet one may take
649 \[
650  A_{\mathbf r}
651  =\left(\sum_a\epsilon_{a,\mathbf r}T_a\right)
652  \otimes I_{\mathrm{phase},\mathbf r},
653  \qquad \epsilon_{a,\mathbf r}\in\R,\quad T_a=T_a^*.
654 \]
655 The matrices $T_a$ act on the sign and torsion factors. Their mutual
656 commutation is unnecessary because their sum is self-adjoint on the
657 finite block. The coefficients assigned to an existing packet are fixed
658 under refinement. An extension acting wholly inside a new detail
659 complement has no matrix element between that complement and the old
660 range. A cross term between them would be a different model.
661 
662 There is also an exact construction for any \emph{specified}
663 self-adjoint compatible finite family along a cofinal nested sequence.
664 Suppose finite-dimensional spaces $\mathcal V_0\subset\mathcal V_1
665 \subset\cdots$ have dense union, and self-adjoint $K_M$ satisfy
666 $K_{M+1}f=K_Mf$ for $f\in\mathcal V_M$.
667 Self-adjointness implies that $\mathcal V_M^\perp\cap\mathcal V_{M+1}$
668 is invariant: for $g$ there and $f\in\mathcal V_M$,
669 $\inner{f}{K_{M+1}g}=\inner{K_{M+1}f}{g}=0$.
670 The initial space $\mathcal V_0$ and the successive difference spaces
671 therefore have stable self-adjoint finite blocks.
672 Their orthogonal direct sum with the maximal square-summability domain
673 is self-adjoint by the same adjoint proof above. This is an editorial
674 compatible-family construction, not an assertion that arbitrary finite
675 Hamiltonians are compatible. The identities $K_{M+1}|_{\mathcal V_M}=K_M$
676 are its substantive hypothesis.
677 
678 \subsection{An exact cross-prime entangling generator}
679 
680 Let $Z_{\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,
681 and let $B=B^*$ act on the remaining factors. Assume there are orthonormal
682 states $|b_0\rangle,|b_1\rangle$ with
683 $B|b_0\rangle=0$ and $B|b_1\rangle=|b_1\rangle$.
684 One concrete choice is an odd-prime torsion involution times a fixed
685 odd-prime detail projector, with the torsion state in its $+1$ eigenspace.
686 For real $g\neq0$, use
687 \[
688  H_{\mathrm{int}}=gZ_{\mathrm{sign}}\otimes B,
689  \qquad
690  |\Psi(0)\rangle=|+_x\rangle\otimes
691  \frac{|b_0\rangle+|b_1\rangle}{\sqrt2}.
692 \]
693 Evolution under this explicit interaction alone gives
694 \begin{equation}
695  |\Psi(t)\rangle=\frac1{\sqrt2}
696  \left(|+_x\rangle\otimes|b_0\rangle
697  +e^{-igtZ_{\mathrm{sign}}}|+_x\rangle\otimes|b_1\rangle\right).
698  \label{a-limit-entangled-state}
699 \end{equation}
700 Tracing out the orthogonal $b$ states yields the equally weighted mixture
701 of the two displayed sign states. Their overlap is $\cos(gt)$.
702 A mixture of two unit vectors with equal weights and overlap magnitude
703 $c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its
704 determinant on their two-dimensional span is $(1-c^2)/4$.
705 Thus the sign entropy in bits is
706 \begin{equation}
707  S_{\mathrm{sign}}(t)
708  =h_2\left(\frac{1+|\cos(gt)|}{2}\right),
709  \quad h_2(u)=-u\log_2u-(1-u)\log_2(1-u),
710  \label{a-limit-entanglement-entropy}
711 \end{equation}
712 with $0\log_2 0=0$. At $t_* =\pi/(2|g|)$ the two sign states are
713 orthogonal and the entropy is one bit. A fixed level projector is
714 compatible under refinement, so this interaction can entangle the prime
715 factors while preserving the cylinder embedding.
716 The tensor factorization alone creates no entanglement. It is this
717 chosen generator and initial superposition that produce the result.
718 
719 
720 \section{Controlled leakage when compatibility is broken}
721 \label{a-limit-leakage}
722 
723 The preceding identities are exact because the coarse range reduces the
724 Hamiltonian. A weighted coupling to its complement has a different,
725 quantifiable behavior. This section concerns finite-dimensional
726 self-adjoint operators; no unbounded perturbation theorem is implicit.
727 
728 Let $V$ be an isometry from a coarse space to a finite fine space, set
729 $P=VV^*$ and $Q=I-P$, and let $H_0=H_0^*$ satisfy
730 $H_0V=VH_c$. Let $B=B^*$ be purely off diagonal:
731 \[
732  PBP=0,\qquad QBQ=0.
733 \]
734 For a real $\lambda$, define $H_\lambda=H_0+\lambda B$. In the source
735 model $\lambda=p^{-w}$ for an assigned weight $w\geq1$.
736 
737 \begin{proposition}[Defect, leakage, and a second-order self-energy]
738 \label{a-limit-leakage-bounds}
739 Under these hypotheses,
740 \begin{align}
741  \norm{H_\lambda V-VH_c}&=|\lambda|\norm{BV},
742  \label{a-limit-defect-norm}\\
743  \norm{Qe^{-itH_\lambda}P}
744  &\leq\min\{1,|t\lambda|\norm{QBP}\}.
745  \label{a-limit-duhamel-bound}
746 \end{align}
747 For $z\notin\sigma(QH_\lambda Q|_{Q\HH})$, the coarse self-energy is
748 \begin{equation}
749  \Sigma_P(z)=PH_\lambda Q
750  (z-QH_\lambda Q|_{Q\HH})^{-1}QH_\lambda P,
751  \label{a-limit-feshbach-definition}
752 \end{equation}
753 where the inverse acts only on $Q\HH$, and
754 \begin{equation}
755  \norm{\Sigma_P(z)}
756  \leq\frac{|\lambda|^2\norm{PBQ}^2}
757  {\operatorname{dist}(z,\sigma(QH_\lambda Q|_{Q\HH}))}.
758  \label{a-limit-feshbach-bound}
759 \end{equation}
760 \end{proposition}
761 \begin{proof}
762 The defect is $\lambda BV$, so the first identity is immediate.
763 Self-adjointness of $H_0$ makes its invariant coarse range reducing.
764 Hence $Qe^{-itH_0}P=0$, and Duhamel's identity gives
765 \[
766  Qe^{-itH_\lambda}P
767  =-i\lambda\int_0^t
768  Qe^{-i(t-s)H_\lambda}B e^{-isH_0}P\,ds.
769 \]
770 The off-diagonal assumption and reduction by $H_0$ give
771 $B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior
772 projections are contractions. Integrating the norm bound over an
773 interval of length $|t|$ gives the second expression in
774 \eqref{a-limit-duhamel-bound}. The bound one follows directly from
775 unitarity and the norms of the projections.
776 
777 The coarse/detail blocks of $H_0$ vanish, so
778 $PH_\lambda Q=\lambda PBQ$ and $QH_\lambda P=\lambda QBP$.
779 The detail compression is self-adjoint. Diagonalizing it bounds the
780 resolvent norm by the reciprocal spectral distance. Since
781 $QBP=(PBQ)^*$, submultiplicativity gives
782 \eqref{a-limit-feshbach-bound}.
783 \end{proof}
784 
785 The small-time leading term is $-it\lambda QBP$. More precisely,
786 \begin{equation}
787  \norm{Qe^{-itH_\lambda}P+it\lambda QBP}
788  \leq\frac{|\lambda|t^2}{2}
789  \bigl(\norm{H_\lambda}+\norm{H_0}\bigr)\norm{QBP}.
790  \label{a-limit-shorttime-remainder}
791 \end{equation}
792 To verify this, subtract the constant integrand $QBP$ in Duhamel's
793 formula and use $\norm{e^{-iuK}-I}\leq|u|\norm K$ for a finite
794 self-adjoint $K$. The two deviations are bounded by
795 $(|t-s|\norm{H_\lambda}+|s|\norm{H_0})\norm{QBP}$ on the oriented
796 integration interval. Their integrals are both $t^2/2$.
797 For $|\lambda|\leq1$, this is a uniform $O(t^2|\lambda|)$ remainder.
798 A generic $O(t^2\lambda^2)$ remainder would omit the first-order
799 interaction with the background Hamiltonian.
800 
801 For the stipulated assignment $\lambda=p^{-w}$, the operator defect and
802 this leakage upper bound scale linearly with $p^{-w}$. The self-energy
803 bound is quadratic in $p^{-w}$ away from the detail spectrum. Under the
804 purely off-diagonal hypothesis, $QH_\lambda Q=QH_0Q$, so the denominator
805 does not depend on $\lambda$. Approaching a detail resonance removes
806 that uniform estimate. These are exact finite quantum statements for a
807 chosen coupling law. An integral valuation packet does not, by itself,
808 force every physical interaction strength to equal $p^{-w}$.
809 
810 
811 \section{A local walk on physical position and refinement depth}
812 \label{a-limit-product-walk}
813 
814 \subsection{The finite graph and its unitary tick}
815 
816 Let $N\geq3$ and $h\geq0$. Let $C_N$ be the physical cycle and let
817 $T_{p,h}$ be the rooted finite $p$-ary tree of prefixes of length at most
818 $h$. The product graph is
819 \[
820  \mathcal G=C_N\square T_{p,h}.
821 \]
822 A vertex is $v=(x,\mathbf d)$, with $x\in\Z/N\Z$ and $\mathbf d$ a
823 prefix. An edge changes either $x$ by one on the cycle or the prefix by
824 one parent-child step. The graph has
825 $N\sum_{r=0}^h p^r$ vertices. It includes all tree depths, unlike a
826 cylinder register which contains only the residues at its last depth.
827 
828 Let $\vec E$ denote the directed arcs and let $\tau_2=2$,
829 $\tau_p=p-1$ for odd $p$. The finite arc Hilbert space is
830 \[
831  \HH_{\mathrm{arc}}=\ell^2(\vec E)\otimes\C^{\tau_p}.
832 \]
833 For each vertex of degree $d_v$, define the normalized outgoing vector
834 $s_v=d_v^{-1/2}(1,\ldots,1)$ and the Grover coin
835 $G_v=2|s_v\rangle\langle s_v|-I$.
836 Its square is $I$, because $|s_v\rangle\langle s_v|$ is a projection.
837 Take any local unitary $R_v$ on the torsion or sign factor and put
838 \[
839  C=\bigoplus_v(G_v\otimes R_v).
840 \]
841 The reversal shift $S$ sends the arc $v\to u$ to $u\to v$, retaining
842 the internal register. It is a permutation with $S^*=S$ and $S^2=I$.
843 The one-tick walk
844 \begin{equation}
845  U_{p,h,N}=SC
846  \label{a-limit-walk-unitary}
847 \end{equation}
848 is unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.
849 Here locality concerns the graph edges, with no continuum limit assumed.
850 
851 \subsection{Local conservation and the product cone}
852 
853 Write $\psi^n(v,u)$ for the internal amplitude on an arc leaving $v$
854 before the coin, and $\phi^n=C\psi^n$ for its post-coin value. Define
855 \[
856  \rho_v^n=\sum_{u\sim v}\norm{\psi^n(v,u)}^2,
857  \qquad F_{v\to u}^n=\norm{\phi^n(v,u)}^2.
858 \]
859 The local unitary coin gives
860 $\sum_uF_{v\to u}^n=\rho_v^n$.
861 The reversal shift gives
862 $\rho_v^{n+1}=\sum_u F_{u\to v}^n$. Subtraction proves
863 \begin{equation}
864  \rho_v^{n+1}-\rho_v^n
865  =\sum_{u\sim v}(F_{u\to v}^n-F_{v\to u}^n).
866  \label{a-limit-walk-continuity}
867 \end{equation}
868 For every set of vertices $A$, summing this equation cancels every
869 internal directed flux once with its opposite occurrence. Hence
870 \begin{equation}
871  \sum_{v\in A}(\rho_v^{n+1}-\rho_v^n)
872  =\sum_{\substack{u\notin A,\ v\in A\\u\sim v}}F_{u\to v}^n
873  -\sum_{\substack{v\in A,\ u\notin A\\u\sim v}}F_{v\to u}^n.
874  \label{a-limit-walk-coarse-balance}
875 \end{equation}
876 Taking $A$ to be all vertices gives conservation of total probability.
877 The coarse identity uses a subset of vertices and does not presume
878 amplitude-level Haar refinement.
879 
880 \begin{proposition}[Exact finite propagation]
881 \label{a-limit-walk-cone}
882 If the initial outgoing amplitudes are supported at $v_0$, then
883 \begin{equation}
884  d_{C_N}(x,x_0)+d_T(\mathbf d,\mathbf d_0)>n
885  \quad\Longrightarrow\quad\rho_{(x,\mathbf d)}^n=0.
886  \label{a-limit-product-cone}
887 \end{equation}
888 For two prefixes, writing $\operatorname{lcp}$ for the length of their
889 longest common prefix,
890 \[
891  d_T(\mathbf d,\mathbf d_0)
892  =|\mathbf d|+|\mathbf d_0|-2\operatorname{lcp}(\mathbf d,\mathbf d_0).
893 \]
894 In particular, each tick allows at most one physical edge or one tree
895 edge, and $\bigl||\mathbf d|-|\mathbf d_0|\bigr|\leq n$.
896 \end{proposition}
897 \begin{proof}
898 Every path in a Cartesian product graph uses a number of physical edges
899 at least the physical distance and a number of tree edges at least the
900 tree distance. Their sum is a lower bound on path length. Following a
901 shortest physical path and then a shortest tree path attains that sum,
902 so it is the product graph distance. In a rooted tree the shortest path
903 rises to the longest common prefix and then descends, giving the
904 displayed formula.
905 
906 The claim at $n=0$ is the support hypothesis. A coin mixes outgoing
907 arcs only at their common source, so it does not enlarge the vertex
908 support. The shift moves the source across one edge. A state supported
909 in the radius-$n$ ball is therefore supported in the radius-$(n+1)$
910 ball after a tick. Induction proves the result. Since each graph edge
911 changes only one factor, the individual distance and depth bounds follow.
912 \end{proof}
913 
914 The depth difference alone is weaker than the full tree distance.
915 Different prefixes at the same depth can have positive distance, so a
916 plot aggregated over depth does not show the complete graph geometry.
917 The cone is an exact support property of this finite coined model.
918 A Lorentzian continuum dispersion law or a relativistic limiting field
919 equation requires a separate scaling analysis.
920 
921 \adicfigure{07_product_walk}{A finite cycle--tree walk, aggregated over
922 vertices at the same tree depth. The graph-distance support check uses
923 the full product graph, including prefix labels. Probability plotted
924 against depth alone displays only an aggregation of that geometry.}
925 
926 \FloatBarrier
927 \Needspace{17\baselineskip}
928 \subsection{Why growing this tree is not automatically Haar refinement}
929 
930 \begin{example}[A boundary coin breaks the canonical arc embedding]
931 \label{a-limit-walk-refinement-counterexample}
932 Take $h\geq1$ and $N\geq3$, and embed the old arcs of
933 $C_N\square T_{p,h}$ in $C_N\square T_{p,h+1}$ by identical labels,
934 with zero amplitude on new arcs. Denote this isometry by $J$.
935 At an old leaf, the old degree is $d=3$: two cycle neighbors and one
936 parent. The larger graph adds $p$ children, changing the degree to $d+p$.
937 For an old outgoing arc basis vector $e_j$, the new Grover coin gives
938 amplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An
939 internal unitary preserves its internal norm. The subsequent reversal
940 shift sends these amplitudes to new arcs at the children, outside the old
941 arc range. Therefore, for a unit internal state $\chi$,
942 \begin{equation}
943  \norm{(I-JJ^*)U_{p,h+1,N}J(e_j\otimes\chi)}
944  =\frac{2\sqrt p}{3+p}>0.
945  \label{a-limit-walk-refinement-leakage}
946 \end{equation}
947 The state $JU_{p,h,N}(e_j\otimes\chi)$ has zero amplitude there, so
948 $U_{p,h+1,N}J\neq JU_{p,h,N}$.
949 \end{example}
950 
951 Each walk remains unitary, locally conservative, and confined to its
952 product graph cone. A different coin family, embedding, or infinite-tree
953 construction could be investigated. The canonical old-arc inclusion
954 fails the compatibility hypothesis of
955 Theorem \ref{a-limit-block-hamiltonian}. The cylinder limit and the tree
956 walk are exact models with distinct state spaces.
957 
958 
959 \section{The boundary of the infinite-depth result}
960 \label{a-limit-scope}
961 
962 The exact result comprises complex $L^2$ cylinder reconstruction and
963 dynamics for declared real level energies or self-adjoint packet blocks.
964 State convergence uses an $L^2$ tail; graph norm convergence uses an
965 operator-domain tail. Compact resolvents require energies escaping every
966 bounded set, and finite thermal traces require multiplicity summability.
967 Each claim has a distinct hypothesis.
968 
969 The source programs check finite restrictions of these formulas. Code 45
970 uses three towers,
971 \[
972  (p,M)=(2,7),\ (3,5),\ (5,3),
973 \]
974 and a fixed random seed. Their terminal zero tails do not certify an
975 infinite domain or compactness; the proofs supply those conclusions.
976 Code 40 tests specified dyadic/ternary interactions and weighted
977 off-diagonal couplings. Code 42 tests finite product walks, whose
978 unitarity does not imply compatibility with the Haar tower.
979 
980 C15 records finite refinement identities and C16 the conditional
981 infinite-depth construction. Their archived chapter numbers predate the
982 current arrangement. Exact members are chapter 31 for multi-prime
983 dynamics, chapter 33 for product walks, and chapter 36 for the infinite
984 Haar limit. Prime-polyphase semigroups appear in the spectral part.
985 No fresh Lean compilation, unrestricted quantum-field completion,
986 infinite-prime adelic construction, or experimental identification is
987 claimed here.

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