spectral.tex
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spectral.tex
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1 \section{Polyphase coordinates and folded bands}
2 \label{a-spectral-polyphase}
3
4 The finite Haar decomposition isolates an average and its new detail
5 channels. A standard polyphase decomposition instead groups the entries
6 of a cyclic lattice by their residue class. Both transformations preserve
7 the full state space. An elimination begins only when some coordinates
8 are solved for and removed. The spectral parameter then enters the
9 remaining operator, and the eliminated degrees of freedom return as
10 memory in time.
11
12 We use complex Hilbert spaces with inner product linear in the second
13 argument and set $\hbar=1$. All matrices in the finite sections act on
14 their stated coordinate spaces. In particular, the inverse of a low
15 channel block is a low channel operator, rather than a full space matrix.
16
17 \subsection{Residue coordinates and exact block reconstruction}
18
19 Fix an integer $p\geq2$ and $M\geq3$, and put $N=pM$. Primality is not
20 needed for the identities in this section. For $0\leq r<p$, define the
21 isometry $J_r:\C^M\to\C^N$ by
22 \[
23 J_r|k\rangle=|pk+r\rangle,\qquad 0\leq k<M.
24 \]
25 Then $J_r^*J_s=\delta_{rs}I_M$ and $\sum_rJ_rJ_r^*=I_N$. The standard
26 analysis map $\mathcal A$ sends $|pk+r\rangle$ to
27 $|r\rangle\otimes|k\rangle$. This is a permutation unitary. For any
28 operator $X$ on $\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:
29 \begin{equation}
30 X=\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\qquad
31 (\mathcal A X\mathcal A^*)_{rs}=X_{rs}.
32 \label{a-spectral-block-reconstruction}
33 \end{equation}
34 The first equality follows by inserting $\sum_rJ_rJ_r^*$ on both sides
35 of $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is
36 unitary, the same insertion gives
37 \begin{align}
38 \sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\delta_{st}I_M,\nonumber\\
39 \sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\delta_{rt}I_M.
40 \label{a-spectral-paraunitarity}
41 \end{align}
42 These are the block unitary identities. Every coordinate is retained.
43
44 For finite $H$, expansion of the matrix exponential proves
45 \[
46 e^{-it\mathcal A H\mathcal A^*}
47 =\mathcal A e^{-itH}\mathcal A^*.
48 \]
49 Indeed $(\mathcal A H\mathcal A^*)^n=\mathcal A H^n\mathcal A^*$,
50 and the series converges in matrix norm. The same transformation of
51 each Kraus operator gives a unitarily equivalent quantum channel:
52 if $\Phi(\rho)=\sum_\mu K_\mu\rho K_\mu^*$ with
53 $\sum_\mu K_\mu^*K_\mu=I$, then
54 \[
55 K_\mu^{\rm poly}=\mathcal A K_\mu\mathcal A^*,\qquad
56 \sum_\mu(K_\mu^{\rm poly})^*K_\mu^{\rm poly}=I.
57 \]
58 This is a change of coordinates in a given channel. It does not supply
59 a new dissipative dynamics.
60
61 \Needspace{16\baselineskip}
62 \begin{proposition}[Probability currents between residue channels]
63 \label{a-spectral-channel-current}
64 Suppose $i\dot\psi_r=\sum_sH_{rs}\psi_s$ with $H_{rs}^*=H_{sr}$.
65 The channel weights $w_r=\norm{\psi_r}^2$ satisfy
66 \[
67 \dot w_r=\sum_s\mathcal J_{r\leftarrow s},\qquad
68 \mathcal J_{r\leftarrow s}
69 =2\operatorname{Im}\langle\psi_r,H_{rs}\psi_s\rangle,
70 \qquad
71 \mathcal J_{r\leftarrow s}=-\mathcal J_{s\leftarrow r}.
72 \]
73 Consequently $\sum_rw_r$ is constant.
74 \end{proposition}
75 \begin{proof}
76 Differentiating the norm gives
77 $\dot w_r=2\operatorname{Re}\langle\psi_r,-i\sum_sH_{rs}\psi_s\rangle$,
78 which is the stated sum of imaginary parts. Self-adjoint block symmetry
79 implies
80 $\langle\psi_s,H_{sr}\psi_r\rangle
81 =\overline{\langle\psi_r,H_{rs}\psi_s\rangle}$.
82 The currents therefore cancel in pairs, including a zero diagonal current.
83 \end{proof}
84 Here $w_r$ is a probability or squared amplitude weight. Calling it an
85 energy would require an additional energy observable and its balance law.
86
87 \subsection{A Bloch symbol with a fixed shift convention}
88
89 Let $S|n\rangle=|n+1\pmod N\rangle$ and
90 \begin{equation}
91 H=\mu I+\kappa(S+S^*),\qquad \mu,\kappa\in\R.
92 \label{a-spectral-free-cycle}
93 \end{equation}
94 Use the coarse Fourier matrix
95 $(F_M)_{jk}=M^{-1/2}e^{-2\pi ijk/M}$ and
96 $\theta_j=2\pi j/M$. After $\mathcal A$, Fourier transformation in $k$,
97 and the permutation from residue major to momentum major ordering, the
98 operator becomes a direct sum of $p\times p$ symbols. To see their entries,
99 note that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends
100 $(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue
101 $e^{-i\theta_j}$. Thus define
102 \begin{equation}
103 P_p(\theta)e_r=
104 \begin{cases}
105 e_{r+1},&r<p-1,\\
106 e^{-i\theta}e_0,&r=p-1.
107 \end{cases}
108 \qquad
109 \mathsf H_p(\theta)=\mu I_p+\kappa(P_p(\theta)+P_p(\theta)^*).
110 \label{a-spectral-bloch-symbol}
111 \end{equation}
112 The transformed $H$ is $\bigoplus_{j=0}^{M-1}\mathsf H_p(\theta_j)$.
113 In particular, $P_p(\theta)^p=e^{-i\theta}I_p$.
114
115 \begin{proposition}[Exact folded cyclic spectrum]
116 \label{a-spectral-folded-bands}
117 For $q=0,\ldots,p-1$, the eigenvalues of
118 $\mathsf H_p(\theta)$ are
119 \begin{equation}
120 E_q(\theta)=\mu+2\kappa
121 \cos\!\left(\frac{\theta+2\pi q}{p}\right).
122 \label{a-spectral-band-formula}
123 \end{equation}
124 At $\theta=\theta_j$, the union of these $pM$ eigenvalues, counting
125 multiplicity, is exactly the spectrum of \eqref{a-spectral-free-cycle}.
126 \end{proposition}
127 \begin{proof}
128 Put $\alpha_q=(\theta+2\pi q)/p$ and let the $r$th component of
129 $v_q$ be $p^{-1/2}e^{i\alpha_qr}$. On the interior components,
130 $P_p(\theta)v_q=e^{-i\alpha_q}v_q$. The wrapped component has the
131 same value because $e^{ip\alpha_q}=e^{i\theta}$. The $v_q$ are
132 orthonormal by the finite character sum. Applying
133 $\mu I+\kappa(P_p+P_p^*)$ gives \eqref{a-spectral-band-formula}.
134 Every fine Fourier index is uniquely $k=j+qM$. Its angle satisfies
135 $2\pi k/(pM)=(\theta_j+2\pi q)/p$. Substitution into the fine cyclic
136 eigenvalue $\mu+2\kappa\cos(2\pi k/N)$ proves the multiplicity statement.
137 \end{proof}
138
139 The same construction works for any circulant polynomial
140 $H=\sum_\ell c_\ell S^\ell$, with symbol
141 $\sum_\ell c_\ell P_p(\theta)^\ell$. It preserves all bands, whether or
142 not their energies coincide. For $N=p^h$, digit reversal relates this
143 standard residue split to the newest-digit adic split developed earlier.
144 It is still necessary to distinguish the retained residue-zero sites
145 from the retained Haar average channel. They are different subspaces.
146
147 \adicfigure{09_polyphase_symbol}{Three exact folded bands for the
148 nearest-neighbor cycle. The discrete markers represent the finite cyclic
149 spectrum. Band folding reorganizes its multiplicities without discarding
150 any eigenstate.}
151
152
153 \section{Regular Schur reduction and causal memory}
154 \label{a-spectral-feshbach}
155
156 Let a finite Hilbert space be the orthogonal sum
157 $\HH_L\oplus\HH_D$, and write a self-adjoint Hamiltonian as
158 \begin{equation}
159 H=\begin{pmatrix}A&B\\B^*&D\end{pmatrix},\qquad A=A^*,\quad D=D^*.
160 \label{a-spectral-block-hamiltonian}
161 \end{equation}
162 Here $B:\HH_D\to\HH_L$. Let $\iota_L$ embed $\HH_L$ into the full
163 space. This notation fixes the coordinate type of every compression.
164
165 \subsection{The resolvent chart and its exceptional energies}
166
167 \begin{theorem}[Finite Feshbach identity]
168 \label{a-spectral-schur-theorem}
169 If $z\notin\sigma(D)$, define
170 \begin{equation}
171 \Sigma(z)=B(z-D)^{-1}B^*,\qquad
172 \mathcal S(z)=z-A-\Sigma(z).
173 \label{a-spectral-schur-definition}
174 \end{equation}
175 Then $z-H$ is invertible exactly when $\mathcal S(z)$ is invertible.
176 On this common regular set,
177 \begin{align}
178 \iota_L^*(z-H)^{-1}\iota_L&=\mathcal S(z)^{-1},\nonumber\\
179 \det(z-H)&=\det(z-D)\det\mathcal S(z).
180 \label{a-spectral-schur-identities}
181 \end{align}
182 For $E\in\R\setminus\sigma(D)$, an eigenvector of $H$ is equivalent to
183 a nonzero solution of $\mathcal S(E)u=0$, with hidden component
184 $v=(E-D)^{-1}B^*u$.
185 \end{theorem}
186 \begin{proof}
187 Put $C=z-D$. Direct block multiplication gives
188 \[
189 z-H=
190 \begin{pmatrix}I&-BC^{-1}\\0&I\end{pmatrix}
191 \begin{pmatrix}\mathcal S(z)&0\\0&C\end{pmatrix}
192 \begin{pmatrix}I&0\\-C^{-1}B^*&I\end{pmatrix}.
193 \]
194 Both outer factors are invertible and have determinant one. This proves
195 the invertibility equivalence and determinant formula. Inverting the
196 factors gives the stated upper left block. Alternatively, in
197 $(z-H)(u,v)^{\mathsf T}=(f,0)^{\mathsf T}$, the lower equation gives
198 $v=C^{-1}B^*u$ and the upper equation becomes $\mathcal S(z)u=f$.
199 The same elimination with $f=0$ proves the eigenvector statement.
200 If $u=0$ at such an $E$, invertibility of $E-D$ forces $v=0$, so a
201 nonzero full eigenvector indeed has $u\ne0$.
202 \end{proof}
203
204 If $P_L=\iota_L\iota_L^*$, the full space compression is
205 $P_L(z-H)^{-1}P_L=\iota_L\mathcal S(z)^{-1}\iota_L^*$.
206 When a Haar transform $\mathcal W$ first exposes low and detail channels,
207 the embedding in the original coordinates is
208 $\iota_L=\mathcal W^*(|0\rangle\otimes I)$.
209 The inverse on $\HH_L$ must be embedded in this way before it is compared
210 with a full space matrix.
211
212 The determinant equality is initially a rational identity on the regular
213 set. Multiplication by $\det(z-D)$ cancels its denominators and yields the
214 full characteristic polynomial, which extends across $\sigma(D)$.
215 This extension does not make the displayed inverse of $z-D$ exist at
216 a pole. At a resonant energy the uneliminated equations determine which
217 hidden modes couple, survive, or cancel. A resonant subspace can be kept
218 as part of an enlarged visible block.
219
220 \subsection{Spectral residues, analytic sign, and branch weights}
221
222 Let $D=\sum_\lambda\lambda E_\lambda$ be its spectral resolution. Then
223 \begin{equation}
224 \Sigma(z)=\sum_\lambda\frac{B E_\lambda B^*}{z-\lambda}.
225 \label{a-spectral-self-energy-residues}
226 \end{equation}
227 Every residue $B E_\lambda B^*$ is positive semidefinite. Its rank records
228 which directions of the hidden eigenspace reach the visible space.
229 A vanishing residue removes the corresponding pole of $\Sigma$, even
230 though $z-D$ is still singular there.
231
232 \begin{proposition}[Upper half-plane sign and the exact visible norm]
233 \label{a-spectral-resolvent-sign}
234 For $z=x+iy$ with $y>0$,
235 \[
236 \operatorname{Im}\Sigma(z)
237 =-yB\bigl((x-D)^2+y^2I\bigr)^{-1}B^*\leq0,
238 \qquad
239 \operatorname{Im}\mathcal S(z)\geq yI.
240 \]
241 In particular $\mathcal S(z)$ is invertible and
242 $\norm{\mathcal S(z)^{-1}}\leq y^{-1}$.
243 For real $E\notin\sigma(D)$,
244 \[
245 \Sigma'(E)=-B(E-D)^{-2}B^*.
246 \]
247 If $(u,v)$ is a normalized full eigenvector at $E$, then
248 \begin{equation}
249 1=\langle u,(I-\Sigma'(E))u\rangle.
250 \label{a-spectral-visible-norm}
251 \end{equation}
252 \end{proposition}
253 \begin{proof}
254 For real $\lambda$,
255 $\operatorname{Im}(z-\lambda)^{-1}=-y/((x-\lambda)^2+y^2)$.
256 Applying this identity to the spectral sum gives the first formula and
257 then the bound on $\operatorname{Im}\mathcal S$.
258 For any $u$, Cauchy--Schwarz implies
259 $y\norm{u}^2\leq|\langle u,\mathcal S(z)u\rangle|
260 \leq\norm{u}\norm{\mathcal S(z)u}$.
261 Thus $\mathcal S(z)$ has trivial kernel and an inverse of the stated
262 norm in finite dimension. Differentiating the resolvent gives the minus
263 sign in $\Sigma'$. The reconstructed hidden norm is
264 $\norm{v}^2=\langle u,B(E-D)^{-2}B^*u\rangle$.
265 Adding $\norm{u}^2$ proves \eqref{a-spectral-visible-norm}.
266 \end{proof}
267
268 For a one-dimensional visible block this says that an eigenstate's
269 visible probability is $[1-\Sigma'(E)]^{-1}$. It is at most one, as
270 required. Eliminating a normalized eigenvector therefore changes its
271 normalization metric as well as its spectral equation. Solving only a
272 nonlinear energy equation does not provide that probability weight.
273
274 \subsection{Exact elimination in the time domain}
275
276 \begin{theorem}[Causal memory with the initial hidden forcing]
277 \label{a-spectral-memory-theorem}
278 The finite system
279 $i\dot u=Au+Bv$, $i\dot v=Dv+B^*u$, with initial data $(u_0,v_0)$,
280 is equivalent to
281 \begin{align}
282 i\dot u(t)&=Au(t)+B e^{-itD}v_0
283 -i\int_0^tK(t-s)u(s)\,ds,\nonumber\\
284 K(\tau)&=B e^{-i\tau D}B^*,\qquad \tau\geq0,
285 \label{a-spectral-memory-equation}
286 \end{align}
287 together with reconstruction
288 \begin{equation}
289 v(t)=e^{-itD}v_0-i\int_0^te^{-i(t-s)D}B^*u(s)\,ds.
290 \label{a-spectral-hidden-reconstruction}
291 \end{equation}
292 For $\operatorname{Im}z>0$, the self-energy is its exact one-sided transform:
293 \begin{equation}
294 \Sigma(z)=-i\int_0^\infty e^{iz\tau}K(\tau)\,d\tau.
295 \label{a-spectral-memory-transform}
296 \end{equation}
297 \end{theorem}
298 \begin{proof}
299 Differentiate $e^{itD}v(t)$ and integrate the hidden equation. This gives
300 \eqref{a-spectral-hidden-reconstruction}, whose substitution in the
301 visible equation gives \eqref{a-spectral-memory-equation}. Conversely
302 the reconstruction differentiates to the hidden equation and supplies
303 the prescribed initial value, so it recovers the full evolution.
304 Since $D$ is self-adjoint, $\norm{K(\tau)}\leq\norm{B}^2$.
305 The factor $e^{-\operatorname{Im}z\,\tau}$ makes the transform absolutely
306 convergent. On a spectral eigenvalue of $D$,
307 $\int_0^\infty e^{i(z-\lambda)\tau}\,d\tau=i/(z-\lambda)$.
308 The finite spectral sum then proves \eqref{a-spectral-memory-transform}.
309 \end{proof}
310
311 The memory is causal because it uses only $s\leq t$. An initially empty
312 hidden sector removes the forcing $B e^{-itD}v_0$, not the convolution.
313 In finite dimension $K$ is a finite sum of oscillatory terms. Its decay,
314 a Markov approximation, or irreversible damping requires additional
315 limits or assumptions. The closed full dynamics preserves
316 $\norm{u(t)}^2+\norm{v(t)}^2$, while the visible norm alone can decrease
317 and later return. Energy-dependent Schur reduction is consequently
318 different from replacing $H$ by a fixed smaller self-adjoint matrix.
319
320
321 \section{Rooted trees and radial Jacobi response}
322 \label{a-spectral-radial}
323
324 Let $T_{p,h}$ be a rooted $p$-ary tree of depth $h\geq1$. Vertices at
325 depth $r$ have onsite energy $\epsilon_r\in\R$. Every parent-to-child
326 edge from level $r$ to $r+1$ has real coupling $\gamma_r/\sqrt p$,
327 where $\gamma_r\in\R$. The Hamiltonian is the corresponding real
328 symmetric adjacency matrix with onsite diagonal. The level symmetry
329 and this branch normalization are structural assumptions of the model.
330
331 \subsection{The radial reducing subspace}
332
333 Define normalized level states
334 \[
335 |r\rangle=p^{-r/2}\sum_{|v|=r}|v\rangle,
336 \qquad r=0,\ldots,h.
337 \]
338 They are orthonormal because different levels are disjoint and level
339 $r$ contains $p^r$ vertices. A direct action calculation gives
340 \begin{equation}
341 H|r\rangle=\gamma_{r-1}|r-1\rangle
342 +\epsilon_r|r\rangle+\gamma_r|r+1\rangle,
343 \label{a-spectral-radial-action}
344 \end{equation}
345 with omitted terms at the endpoints. For example, the coefficient at
346 each child at level $r+1$ is
347 $p^{-r/2}\gamma_r/\sqrt p=\gamma_r p^{-(r+1)/2}$.
348 At each parent at level $r-1$, the $p$ child contributions instead give
349 $p\,p^{-r/2}\gamma_{r-1}/\sqrt p
350 =\gamma_{r-1}p^{-(r-1)/2}$.
351
352 Let $R:\C^{h+1}\to\ell^2(T_{p,h})$ send $e_r$ to $|r\rangle$.
353 Equation \eqref{a-spectral-radial-action} proves $HR=RJ_h$, where
354 \begin{equation}
355 J_h=\begin{pmatrix}
356 \epsilon_0&\gamma_0&&&\\
357 \gamma_0&\epsilon_1&\gamma_1&&\\
358 &\gamma_1&\epsilon_2&\ddots&\\
359 &&\ddots&\ddots&\gamma_{h-1}\\
360 &&&\gamma_{h-1}&\epsilon_h
361 \end{pmatrix}.
362 \label{a-spectral-jacobi-matrix}
363 \end{equation}
364 Because $H$ is self-adjoint, the orthogonal complement of $\Ran R$ is
365 also invariant: for $w\perp\Ran R$,
366 $\langle Re_r,Hw\rangle=\langle HRe_r,w\rangle=0$.
367 Hence this is a reducing decomposition, rather than only a compressed
368 approximation. The root belongs to the radial block.
369
370 \subsection{Angular chains and their multiplicities}
371
372 There is a more precise form of the complementary decomposition. For
373 each vertex $u$ at depth $a<h$, choose an orthonormal basis of vectors
374 $c=(c_0,\ldots,c_{p-1})\in\C^p$ with $\sum_dc_d=0$. For
375 $s=0,\ldots,h-a-1$, set
376 \begin{equation}
377 |u,c;s\rangle
378 =p^{-s/2}\sum_{d=0}^{p-1}c_d
379 \sum_{w\in\{0,\ldots,p-1\}^{s}}|u d w\rangle.
380 \label{a-spectral-angular-chain}
381 \end{equation}
382 For fixed $u,c$, these are unit vectors on successive levels. They obey
383 the Jacobi recurrence with onsite energies
384 $\epsilon_{a+1},\ldots,\epsilon_h$ and couplings
385 $\gamma_{a+1},\ldots,\gamma_{h-1}$. At $s=0$ the upward amplitude into
386 $u$ is proportional to $\sum_dc_d$, so it vanishes. At $s>0$ the same
387 branch-counting calculation as in \eqref{a-spectral-radial-action}
388 produces the neighboring chain vectors.
389
390 \begin{proposition}[Complete angular-chain decomposition]
391 \label{a-spectral-angular-completeness}
392 The radial chain and all the chains \eqref{a-spectral-angular-chain}
393 form an orthogonal decomposition of $\ell^2(T_{p,h})$. A chain starting
394 at depth $r$ has length $h-r+1$ and appears with multiplicity
395 $(p-1)p^{r-1}$, for $1\leq r\leq h$.
396 \end{proposition}
397 \begin{proof}
398 Within one chain, different $s$ have disjoint level supports. Chains
399 based at the same $u$ and different contrast vectors have inner product
400 $\langle c,c'\rangle=0$ at a common level. If two vertices are in disjoint
401 subtrees, their chains have disjoint support. If one vertex lies below
402 another, the higher chain is constant, up to its fixed ancestral
403 coefficient, on all children below the lower vertex at a common level.
404 The lower chain has zero sum over its own child contrasts, so their
405 inner product is zero. The same zero-sum calculation makes every
406 angular vector orthogonal to the uniform radial level vector.
407 At a fixed depth $b$, the radial vector contributes one dimension.
408 The angular vectors present there contribute
409 $\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and
410 therefore form a basis of that $p^b$-dimensional level space. Taking
411 all levels proves completeness. There are $p^{r-1}$ choices of the
412 vertex $u$ at depth $r-1$ and $p-1$ contrasts at each, giving the
413 stated multiplicity.
414 \end{proof}
415
416 The rank $(p-1)p^{r-1}$ thus counts chains born at level $r$ or
417 new Haar directions on a level. A single level's detail space is
418 usually transported to the next level; it is the entire angular
419 chain that is invariant under the tree Hamiltonian. The full dark
420 dimension is
421 \[
422 \sum_{r=1}^h(p-1)p^{r-1}(h-r+1)
423 =\sum_{b=0}^hp^b-(h+1).
424 \]
425 Every one of these states is invisible to a root-localized symmetric
426 source because it is orthogonal to the radial reducing block.
427
428 \subsection{Continued fractions and their meromorphic reading}
429
430 For a tail $J_{r:h}$, put
431 $g_r(z)=\langle r,(z-J_{r:h})^{-1}r\rangle$. On a common regular
432 domain, scalar Schur elimination gives
433 \begin{align}
434 g_h(z)&=(z-\epsilon_h)^{-1},\nonumber\\
435 g_r(z)&=\frac1{z-\epsilon_r-\gamma_r^2g_{r+1}(z)},
436 \qquad r<h.
437 \label{a-spectral-continued-fraction}
438 \end{align}
439 Defining $\Sigma_h=0$ and
440 $\Sigma_r=\gamma_r^2/(z-\epsilon_{r+1}-\Sigma_{r+1})$ gives
441 \begin{equation}
442 \langle\mathrm{root},(z-H)^{-1}\mathrm{root}\rangle
443 =g_0(z)=\frac1{z-\epsilon_0-\Sigma_0(z)}.
444 \label{a-spectral-root-green}
445 \end{equation}
446 The equality to the tree resolvent follows from the exact reducing
447 decomposition. Equations \eqref{a-spectral-continued-fraction} are
448 rational identities. Nested denominators can become singular at a
449 real energy where a canceled final rational expression is finite.
450 One should use the resolvent or the canceled rational function at
451 such a point, rather than perform division by zero in the nested chart.
452
453 The Jacobi coefficients in \eqref{a-spectral-jacobi-matrix} do not
454 depend on $p$. Thus the root Green function and radial propagator
455 are prime independent when the same $\epsilon_r,\gamma_r$ are used.
456 The complete tree spectrum retains the $p$-dependent angular
457 multiplicities. If the per-edge coupling is instead a fixed $g_r$,
458 the radial coupling is $\sqrt p\,g_r$ and its response depends on $p$.
459 Prime independence is therefore a consequence of the declared
460 normalization, not a universality statement about arbitrary trees.
461
462 \subsection{A visible two-component block}
463
464 As a conditional finite model, take
465 $A=H_D(k)=k\sigma_z+m_0\sigma_x$ with $k,m_0\in\R$ and
466 $D=J_{1:h}\otimes I_2$. Couple the visible spinor identically to the
467 first site of the tail by $B=\gamma_0(\langle1|\otimes I_2)$.
468 Its self-energy is scalar in spin space:
469 \[
470 \Sigma_D(z)=\sigma(z)I_2,\qquad
471 \sigma(z)=\gamma_0^2\langle1,(z-J_{1:h})^{-1}1\rangle.
472 \]
473 The determinant formula from Theorem~\ref{a-spectral-schur-theorem} is
474 \begin{equation}
475 \det(z-H_{\rm full})
476 =\det(z-D)\bigl((z-\sigma(z))^2-k^2-m_0^2\bigr).
477 \label{a-spectral-visible-determinant}
478 \end{equation}
479 Consequently any eigenvalue in the regular chart obeys
480 \begin{equation}
481 (E-\sigma(E))^2=k^2+m_0^2.
482 \label{a-spectral-visible-branch}
483 \end{equation}
484 The formula is a dressed dispersion relation of this specific model.
485 Its mass, couplings, and hidden sector have been supplied. No physical
486 interaction or measured dispersion follows solely from the tree split.
487
488 For a nondegenerate visible spin direction with normalized low-space
489 direction $u_*$, the visible weight of its normalized full eigenstate is
490 $[1-\sigma'(E)]^{-1}$. If $k=m_0=0$, the spin eigenspace is degenerate,
491 but the same normalization relation holds for each chosen unit vector
492 in that eigenspace.
493
494 If every internal tail hopping is nonzero, the first tail site is cyclic.
495 One can see this without an abstract cyclicity theorem: a tail eigenvector
496 with zero first component has zero second component by the first row,
497 and successive rows force all later components to vanish. Therefore
498 every tail eigenvector has nonzero first component. At an eigenvalue
499 $\lambda$ of that tail, solvability of
500 $(\lambda-D)v=B^*u$ requires its right side to be orthogonal to the tail
501 eigenspace. With $\gamma_0\ne0$ and identical spin coupling this forces
502 $u=0$. The visible equation would then require $Bv=0$, which forces
503 $v=0$ on the tail eigenspace. Thus in this irreducible case the full
504 coupled spectrum avoids the tail eigenvalues. With a vanishing hopping,
505 coincident visible and hidden energies can occur and the regular-chart
506 qualification in \eqref{a-spectral-visible-branch} is essential.
507
508 The scalar causal kernel is
509 $K(\tau)=\gamma_0^2\langle1,e^{-i\tau J_{1:h}}1\rangle$, multiplying
510 $I_2$. It uses the same radial coefficients as the rational self-energy.
511 Its exact causal relation is \eqref{a-spectral-memory-transform}.
512 \adicfigure{08_radial_self_energy}{A branch-normalized finite tree has
513 the same root response as its radial Jacobi chain. The complex-energy
514 resolvent avoids real poles. The oscillatory memory is the return
515 amplitude of a finite hidden chain.}
516
517
518 \section{Chebyshev decimation and resonant channels}
519 \label{a-spectral-chebyshev}
520
521 We now eliminate intermediate residue sites in
522 \eqref{a-spectral-free-cycle}, assuming $\kappa\ne0$. The retained
523 sites are $C=\{0,p,\ldots,(M-1)p\}$ and the eliminated sites comprise
524 the $p-1$ interior sites between adjacent retained sites. Identify
525 $\ell^2(C)$ with $\C^M$ and let $T|j\rangle=|j+1\pmod M\rangle$.
526 Set
527 \begin{equation}
528 x=\frac{z-\mu}{\kappa},\qquad
529 R_n(x)=2T_n(x/2),
530 \label{a-spectral-normalized-energy}
531 \end{equation}
532 where $T_n$ is the first-kind Chebyshev polynomial. The second-kind
533 polynomials $U_n$ satisfy $U_{-1}=0$, $U_0=1$ and
534 $U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)$.
535
536 \subsection{The eliminated path fixes the Schur prefactor}
537
538 The stationary equation on an interior site is
539 $\psi_{n+1}=x\psi_n-\psi_{n-1}$. The eliminated block of $z-H$ in
540 one cell is $\kappa K_{p-1}(x)$, where $K_n(x)$ has diagonal $x$
541 and neighboring off-diagonal entries $-1$.
542
543 \begin{lemma}[Dirichlet path determinant and endpoint entries]
544 \label{a-spectral-path-inverse}
545 For $n\geq1$, $\det K_n(x)=U_n(x/2)$. If this determinant is nonzero,
546 \[
547 (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}
548 =\frac{U_{n-1}(x/2)}{U_n(x/2)},\qquad
549 (K_n(x)^{-1})_{1n}=\frac1{U_n(x/2)}.
550 \]
551 For $n=1$ these entries all denote the same value $1/x$.
552 \end{lemma}
553 \begin{proof}
554 Expansion along the last row gives the determinant recurrence
555 $d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.
556 This is exactly the recurrence and initial data for $U_n(x/2)$.
557 Deleting the first or last row and column leaves $K_{n-1}$, proving
558 the diagonal inverse entries by cofactors. For the endpoint entry,
559 the cofactor consists of the product of $n-1$ entries $-1$ along the
560 unique path from the first endpoint to the last. Its cofactor sign is
561 $(-1)^{n+1}$, so the product has total sign
562 $(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.
563 \end{proof}
564
565 \begin{theorem}[Exact free-cycle decimation]
566 \label{a-spectral-chebyshev-schur}
567 For $U_{p-1}(x/2)\ne0$, the Schur operator on the retained sublattice is
568 \begin{equation}
569 \mathcal S_p(z)=\frac{\kappa}{U_{p-1}(x/2)}
570 \bigl[R_p(x)I_M-(T+T^*)\bigr].
571 \label{a-spectral-decimation-formula}
572 \end{equation}
573 Equivalently, $\mathcal S_p(z)=zI-H_{\rm eff}^{(p)}(z)$, where
574 \begin{align}
575 H_{\rm eff}^{(p)}(z)&=\mu_p(z)I+\kappa_p(z)(T+T^*),\nonumber\\
576 \kappa_p(z)&=\frac{\kappa}{U_{p-1}(x/2)},\qquad
577 \mu_p(z)=z-\frac{\kappa R_p(x)}{U_{p-1}(x/2)}.
578 \label{a-spectral-effective-coefficients}
579 \end{align}
580 \end{theorem}
581 \begin{proof}
582 The eliminated sites split into $M$ disjoint Dirichlet paths of length
583 $p-1$. Hence their resolvent exists exactly when
584 $U_{p-1}(x/2)\ne0$. A retained site touches one endpoint on each of
585 its two neighboring paths. Each path contributes the onsite
586 self-energy $\kappa U_{p-2}/U_{p-1}$ by
587 Lemma~\ref{a-spectral-path-inverse}; the two contributions add.
588 The path joining two adjacent retained sites contributes off-diagonal
589 self-energy $\kappa/U_{p-1}$. Thus
590 \[
591 \mathcal S_p(z)
592 =\kappa\left(x-\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\right)I
593 -\frac{\kappa}{U_{p-1}(x/2)}(T+T^*).
594 \]
595 The identity
596 $xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the
597 three-term recurrence and $U_p-U_{p-2}=2T_p$.
598 For the latter identity, substituting $x/2=\cos\theta$ gives
599 $[\sin((p+1)\theta)-\sin((p-1)\theta)]/\sin\theta
600 =2\cos(p\theta)$, and polynomial equality extends it to all $x$.
601 This proves \eqref{a-spectral-decimation-formula} including its
602 prefactor. Rearranging its diagonal gives
603 \eqref{a-spectral-effective-coefficients}.
604 \end{proof}
605
606 The normalized effective energy is
607 $(z-\mu_p(z))/\kappa_p(z)=R_p(x)$. The parameter $z$ remains fixed
608 through the elimination. Since $\mu_p$ and $\kappa_p$ depend on it,
609 this is an energy-dependent spectral closure, not an energy-independent
610 Hamiltonian flow. For $p=2$, $R_2(x)=x^2-2$ and $U_1(x/2)=x$, giving
611 \begin{equation}
612 H_{\rm eff}^{(2)}(z)
613 =\mu I+\frac{\kappa^2}{z-\mu}(2I+T+T^*),\qquad z\ne\mu.
614 \label{a-spectral-dyadic-effective}
615 \end{equation}
616 This decimation retains every other site. It does not retain the
617 average of each neighboring pair.
618
619 \subsection{The poles have an exact Dirichlet interpretation}
620
621 The sine formula $U_{p-1}(\cos\theta)=\sin(p\theta)/\sin\theta$
622 shows that the excluded normalized energies are
623 \begin{equation}
624 x_j=2\cos(j\pi/p),\qquad
625 z_j=\mu+2\kappa\cos(j\pi/p),\quad 1\leq j<p.
626 \label{a-spectral-dirichlet-resonances}
627 \end{equation}
628 These are the energies of one eliminated path. Indeed its $j$th
629 eigenvector has interior components $\sin(j\pi r/p)$,
630 $r=1,\ldots,p-1$, and the nearest-neighbor recurrence gives precisely
631 $z_j$. Both boundary values at $r=0,p$ vanish. The eigenvectors are
632 orthogonal and there are $p-1$ of them, so these exhaust the path spectrum.
633
634 The polynomial map $R_p$ is defined at every $x_j$, while the Schur
635 coefficients in \eqref{a-spectral-effective-coefficients} are not.
636 Cancelling denominators in a determinant can cross this chart boundary;
637 dividing by them in an eigenvector reconstruction cannot. A correct
638 description there keeps the resonant cell modes or solves the full
639 finite system. The fixed poles of the eliminated path are not all
640 automatically eigenvalues of the full cyclic Hamiltonian.
641
642 Let $Q$ project onto the eliminated sites. On the regular chart,
643 \[
644 \det(z-H)=\det(z-QHQ)\det\mathcal S_p(z),\qquad
645 \det(z-QHQ)=\bigl[\kappa^{p-1}U_{p-1}(x/2)\bigr]^M.
646 \]
647 Their product is a polynomial identity after cancellation. It records
648 both the hidden path factor and the coarse Schur factor, including
649 their cancellations at the resonant energies.
650
651 \subsection{Transfer coordinates and many-to-one branch maps}
652
653 Define $A(x)=\begin{pmatrix}x&-1\\1&0\end{pmatrix}$. Its determinant
654 is one and induction gives
655 \begin{equation}
656 A(x)^n=\begin{pmatrix}
657 U_n(x/2)&-U_{n-1}(x/2)\\
658 U_{n-1}(x/2)&-U_{n-2}(x/2)
659 \end{pmatrix},\quad n\geq1.
660 \label{a-spectral-transfer-power}
661 \end{equation}
662 For $n=1$ this uses $U_{-1}=0$. Multiplication by $A(x)$ advances
663 the first column through the Chebyshev recurrence and places the
664 negative previous first column in the second. The trace is $R_n(x)$.
665 Sampling the fine recurrence every $p$ steps therefore uses the
666 unimodular transfer matrix $A(x)^p$. Cayley--Hamilton gives the coarse
667 recurrence $\chi_{j+1}-R_p(x)\chi_j+\chi_{j-1}=0$ for
668 $\chi_j=\psi_{pj}$. It explains the normalized spectral polynomial.
669 The path-inverse proof above additionally supplies the prefactor needed
670 for the resolvent.
671
672 The folded branches from Proposition~\ref{a-spectral-folded-bands},
673 in normalized coordinates, satisfy
674 \begin{equation}
675 x_{p,q}(\theta)=2\cos\!\left(\frac{\theta+2\pi q}{p}\right),
676 \qquad R_p(x_{p,q}(\theta))=2\cos\theta.
677 \label{a-spectral-branch-relation}
678 \end{equation}
679 The second equality is just $T_p(\cos\alpha)=\cos(p\alpha)$.
680 For a coarse normalized energy in the open interval $(-2,2)$,
681 choose its angle in $(0,\pi)$. These formulas supply the $p$ distinct
682 real roots of $R_p(x)=2\cos\theta$. At the endpoints, some roots
683 coincide and algebraic multiplicities must be retained. Globally the
684 branches exchange labels when the angle crosses a cell boundary.
685 There is no single-valued inverse of $R_p$ on $[-2,2]$.
686
687 For the fine cycle, $x_k=2\cos(2\pi k/(pM))$ maps to
688 $2\cos(2\pi k/M)$. Each coarse Fourier label has exactly $p$ fine
689 preimages. If two labels already have the same numerical energy,
690 their multiplicities add as usual. Band folding retains the branch
691 label and its vector, whereas normalized decimation identifies all
692 of these labels at the same coarse energy. Their spectral relation
693 does not make elimination an invertible operator change of basis.
694
695
696 \section{Composition, transfer traces, and the free spectral measure}
697 \label{a-spectral-semigroup}
698
699 The spectral maps compose more simply than the energy-dependent
700 operators that produce them. Keeping the normalized polynomial map,
701 the rational Schur chart, and the transfer matrix distinct prevents an
702 exact free-chain identity from being used outside its model.
703
704 \subsection{Polynomial composition and iterated regular elimination}
705
706 \begin{theorem}[Chebyshev composition and Schur composition]
707 \label{a-spectral-semigroup-theorem}
708 For all positive integers $m,n$, the polynomial identities
709 \begin{align}
710 R_n(R_m(x))&=R_{mn}(x)=R_m(R_n(x)),\nonumber\\
711 U_{mn-1}(t)&=U_{n-1}(T_m(t))\,U_{m-1}(t)
712 \label{a-spectral-polynomial-semigroup}
713 \end{align}
714 hold over $\C$. In a free cycle with $N=mnL$, $L\geq3$, successive
715 $m$- then $n$-site decimation at a fixed $z$ agrees with direct
716 $mn$-site decimation whenever its intermediate Schur inverses exist.
717 Its final coefficients are
718 \begin{equation}
719 \kappa_{mn}(z)=\frac{\kappa}{U_{mn-1}(x/2)},\qquad
720 \mu_{mn}(z)=z-\frac{\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.
721 \label{a-spectral-iterated-coefficients}
722 \end{equation}
723 \end{theorem}
724 \begin{proof}
725 For $x=2\cos\theta$, first-kind composition gives
726 $R_n(R_m(x))=2\cos(mn\theta)=R_{mn}(x)$.
727 It holds on an interval and is therefore an identity of polynomials,
728 including complex $x$. For $t=\cos\theta$ away from zeros of the sine
729 denominators,
730 \[
731 U_{n-1}(T_m(t))U_{m-1}(t)
732 =\frac{\sin(mn\theta)}{\sin(m\theta)}
733 \frac{\sin(m\theta)}{\sin\theta}
734 =U_{mn-1}(t).
735 \]
736 Polynomial equality extends it through those zeros and to all $t$.
737 After the first regular elimination, the normalized variable is
738 $x_m=(z-\mu_m)/\kappa_m=R_m(x)$. A second one gives
739 \[
740 \kappa_{m,n}
741 =\frac{\kappa_m}{U_{n-1}(x_m/2)}
742 =\frac{\kappa}
743 {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}
744 =\frac{\kappa}{U_{mn-1}(x/2)}.
745 \]
746 Similarly
747 $\mu_{m,n}=z-\kappa_{m,n}R_n(R_m(x))$, which gives
748 \eqref{a-spectral-iterated-coefficients}.
749 At each stage the matrices remain scalar nearest-neighbor circulants
750 with the displayed complex coefficients, so their algebraic Schur
751 formula continues to apply. These are rational identities on a common
752 regular chart. They need not be interpreted as self-adjoint matrices at
753 complex $z$.
754 \end{proof}
755
756 The product identity in \eqref{a-spectral-polynomial-semigroup} makes
757 the exceptional denominators explicit. A nonzero final denominator
758 implies both denominators along that two-stage chart are nonzero.
759 At a zero of the final denominator the polynomial trace composition
760 is still valid, but the divided Schur formulas require a new chart or
761 retained resonant modes. Iterating one prime $p$ through depth $h$,
762 where the cycle has the required divisibility, replaces $p$ by $p^h$
763 in \eqref{a-spectral-iterated-coefficients}. Different prime orders
764 give the same regular free-chain result. A nontranslation-invariant
765 onsite potential, an interacting Hamiltonian, or a field-dependent
766 constraint elimination needs its own closure analysis.
767
768 \subsection{What remains true for a general unimodular transfer matrix}
769
770 Let $M_0\in SL_2(\C)$ and $x=\operatorname{tr}M_0$. No
771 diagonalizability assumption is needed. Cayley--Hamilton gives
772 $M_0^2-xM_0+I=0$, hence
773 \begin{equation}
774 M_0^n=U_{n-1}(x/2)M_0-U_{n-2}(x/2)I,\qquad n\geq1,
775 \label{a-spectral-sl2-power}
776 \end{equation}
777 with $U_{-1}=0$. For $n=1$ it is the identity $M_0=M_0$, and for
778 $n=2$ it is Cayley--Hamilton. Multiplying by $M_0$ and replacing
779 $M_0^2$ by $xM_0-I$ proves the induction step. Taking traces yields
780 \begin{equation}
781 \operatorname{tr}(M_0^n)=R_n(\operatorname{tr}M_0).
782 \label{a-spectral-sl2-trace}
783 \end{equation}
784 The coefficient identity is the same
785 $xU_{n-1}-2U_{n-2}=2T_n$ used above.
786
787 If $w_{k+1}=M_0w_k$, each residue subsequence satisfies
788 \[
789 w_{nq+j}=M_0^j(M_0^n)^q w_0,\qquad 0\leq j<n.
790 \]
791 Thus ordinary matrix powering produces the normalized trace semigroup,
792 including matrices with repeated eigenvalues at trace $\pm2$.
793 The trace alone does not determine $M_0$, its eigenvectors, or a
794 local Hamiltonian realization. In the free chain, $M_0=A(x)$ has
795 the required nearest-neighbor realization. For a general transfer
796 problem only the matrix power and trace statements have been proved.
797
798 \subsection{The arcsine measure and the Koopman operator}
799
800 On $[-2,2]$ define the probability measure
801 \begin{equation}
802 d\nu(x)=\frac{dx}{\pi\sqrt{4-x^2}}.
803 \label{a-spectral-arcsine-measure}
804 \end{equation}
805 It is the distribution of $2\cos\theta$ for a uniform angle on the
806 circle, or on $[0,\pi]$. It is also the limiting normalized spectral
807 measure of the free cycles: for a continuous test function $f$,
808 \[
809 \frac1N\sum_{k=0}^{N-1}f(2\cos(2\pi k/N))
810 \longrightarrow\frac1{2\pi}\int_0^{2\pi}f(2\cos\theta)\,d\theta.
811 \]
812 This is the Riemann-sum limit of a continuous periodic function.
813
814 \Needspace{17\baselineskip}
815 \begin{proposition}[Invariant free measure and exact mode routing]
816 \label{a-spectral-arcsine-invariance}
817 For every integer $p\geq2$, $(R_p)_*\nu=\nu$. The map
818 $\mathcal K_pf=f\circ R_p$ is an isometry on $L^2(\nu)$. With
819 \[
820 \phi_0=1,\qquad \phi_n(x)=\sqrt2\,T_n(x/2),\quad n\geq1,
821 \]
822 its action and adjoint on an orthonormal basis are
823 \begin{equation}
824 \mathcal K_p\phi_n=\phi_{pn},\qquad
825 \mathcal K_p^*\phi_m=
826 \begin{cases}\phi_{m/p},&p\mid m,\\0,&p\nmid m.\end{cases}
827 \label{a-spectral-koopman-routing}
828 \end{equation}
829 In the first case $m=0$ is included, so $\mathcal K_p^*\phi_0=\phi_0$.
830 \end{proposition}
831 \begin{proof}
832 For a bounded measurable $f$, use the full circle parameterization:
833 \begin{align*}
834 \int f(R_p(x))\,d\nu(x)
835 &=\frac1{2\pi}\int_0^{2\pi}f(2\cos(p\theta))\,d\theta\\
836 &=\frac1{2\pi p}\int_0^{2\pi p}f(2\cos u)\,du
837 =\int f(x)\,d\nu(x).
838 \end{align*}
839 The last equality uses periodicity. Applying this equality to $|f|^2$
840 proves the isometry, first on bounded functions and then by truncation
841 on all of $L^2(\nu)$. Pulling back to $[0,\pi]$ makes the $\phi_n$
842 the normalized cosine functions, and elementary cosine orthogonality
843 proves their orthonormality. Their span is dense: Chebyshev polynomials
844 span ordinary polynomials; those approximate continuous functions
845 uniformly, for example by Bernstein polynomials after rescaling to
846 $[0,1]$. The approximation follows from uniform continuity and the
847 binomial variance bound, by separating $|k/n-t|\leq\delta$ from its
848 complement and then letting $n\to\infty$, $\delta\to0$.
849 Continuous functions are dense for this finite measure, as one sees by
850 approximating interval steps in the angle coordinate by continuous
851 ramps. This supplies completeness.
852 Finally, Chebyshev composition gives $\mathcal K_p\phi_n=\phi_{pn}$.
853 For every $n,m\geq0$,
854 $\langle\phi_n,\mathcal K_p^*\phi_m\rangle
855 =\langle\phi_{pn},\phi_m\rangle=\delta_{pn,m}$.
856 Completeness yields the adjoint formula.
857 \end{proof}
858
859 $\mathcal K_p$ is an isometry with proper range, rather than a unitary:
860 for example $\phi_1$ is orthogonal to its range. Its adjoint discards
861 nondivisible modes. Invariance of $\nu$ is a statement about the
862 normalized free spectral distribution under $R_p$. It is not a proof
863 that a sequence of growing Hilbert spaces has converged to an interacting
864 Hamiltonian or that its quantum state converges. The direct-limit
865 operator construction has separate hypotheses in the final part.
866
867 The mean logarithmic expansion of this map is also exact:
868 \begin{equation}
869 \int_{-2}^2\log|R_p'(x)|\,d\nu(x)=\log p.
870 \label{a-spectral-log-expansion}
871 \end{equation}
872 Indeed for $x=2\cos\theta$,
873 $R_p'(x)=p\sin(p\theta)/\sin\theta$. Logarithms at its finitely
874 many zeros are integrable, since they have the local form
875 $\log|\theta-\theta_0|$ plus a bounded term. Hence the integral is
876 $\log p$ plus the difference of the integrals of
877 $\log|\sin(p\theta)|$ and $\log\sin\theta$ over $[0,\pi]$.
878 The substitution $u=p\theta$ and $\pi$-periodicity of
879 $\log|\sin u|$ make these two integrals equal. This calculation
880 does not assert the entropy of a different physical dynamics.
881
882 \adicfigure{11_prime_renormalization}{The normalized maps
883 $R_2$, $R_3$, and $R_5$ preserve the same arcsine spectral measure.
884 The accompanying bounded moment checks illustrate the exact measure
885 identity, while each Schur realization retains its own excluded poles.}
886
887
888 \section{Analytic residue lanes and hypergeometric functions}
889 \label{a-spectral-hypergeometric}
890
891 Residue filtering also acts on coefficients of a convergent analytic
892 series. Here the input is a function, not a quantum state, and the
893 projector acts on its coefficient sequence. Evaluating one resulting
894 lane at an operator does not generally turn that lane into a Hilbert
895 space projector.
896
897 \subsection{Character filtering and factorial separation}
898
899 Fix an integer $L\geq2$, write $\omega_L=e^{2\pi i/L}$, and let
900 $F(z)=\sum_{n=0}^\infty a_nz^n$ be analytic for $|z|<\rho$.
901 The whole disk is invariant under the rotations used below. Define
902 \begin{equation}
903 (\mathfrak R_{L,j}F)(z)
904 =\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}F(\omega_L^\ell z),
905 \quad 0\leq j<L.
906 \label{a-spectral-analytic-filter}
907 \end{equation}
908 Absolute convergence for $|z|<\rho$ and the finite character sum give
909 \begin{equation}
910 (\mathfrak R_{L,j}F)(z)=\sum_{q=0}^\infty a_{Lq+j}z^{Lq+j},
911 \quad
912 \mathfrak R_{L,j}\mathfrak R_{L,k}
913 =\delta_{jk}\mathfrak R_{L,j},\quad
914 \sum_j\mathfrak R_{L,j}=I.
915 \label{a-spectral-analytic-lanes}
916 \end{equation}
917 To prove the coefficient formula, the coefficient of $a_nz^n$ in
918 \eqref{a-spectral-analytic-filter} is
919 $L^{-1}\sum_\ell\omega_L^{(n-j)\ell}$, equal to one exactly when
920 $n\equiv j\pmod L$. The idempotent and reconstruction laws follow
921 by filtering these coefficients again. For an entire $F$ the formulas
922 hold everywhere. For a function continued outside its disk, the
923 domain and any chosen branches must also support all rotations; the
924 local power-series identity does not remove such analytic restrictions.
925
926 Apply this filter to $e^z$ and put
927 \begin{equation}
928 E_{L,j}(z)=\sum_{q=0}^\infty\frac{z^{Lq+j}}{(Lq+j)!}.
929 \label{a-spectral-factorial-lane}
930 \end{equation}
931 For $\alpha\in\C$, write
932 $(\alpha)_q=\alpha(\alpha+1)\cdots(\alpha+q-1)$ and
933 $(\alpha)_0=1$. Splitting the factorial into step-$L$ factors gives
934 \begin{align}
935 \frac{(Lq+j)!}{j!}
936 &=\prod_{r=1}^{L}\prod_{k=0}^{q-1}(j+r+Lk)\nonumber\\
937 &=L^{Lq}\prod_{r=1}^{L}\left(\frac{j+r}{L}\right)_q.
938 \label{a-spectral-factorial-split}
939 \end{align}
940 Exactly one bottom parameter is one, at $r=L-j$. Define the multiset
941 \[
942 \mathcal A_{L,j}
943 =\left\{\frac{j+r}{L}:1\leq r\leq L\right\}\setminus\{1\}.
944 \]
945 Every member is positive, so none is a forbidden nonpositive integer
946 in a hypergeometric denominator. Since $(1)_q=q!$,
947 \begin{equation}
948 E_{L,j}(z)=\frac{z^j}{j!}\,
949 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
950 \frac{z^L}{L^L}\right).
951 \label{a-spectral-hypergeometric-lane}
952 \end{equation}
953 Here ${}_0F_s(;b_1,\ldots,b_s;w)$ is defined by
954 $\sum_{q\geq0}w^q/[q!\prod_r(b_r)_q]$. The ratio of successive
955 terms has magnitude $|w|/[(q+1)\prod_r|q+b_r|]$, which tends to
956 zero. Its series is entire in $w$, so \eqref{a-spectral-hypergeometric-lane}
957 is an everywhere convergent identity with the factorial series.
958 For example $\mathcal A_{3,1}=\{2/3,4/3\}$ and
959 $E_{3,1}(z)=z\,{}_0F_2(;2/3,4/3;z^3/27)$.
960
961 \subsection{The cyclic companion system}
962
963 Termwise differentiation on compact subsets gives
964 $E'_{L,j}=E_{L,j-1}$ for $j\geq1$. For $j=0$, the constant term
965 vanishes under differentiation and reindexing gives
966 $E'_{L,0}=E_{L,L-1}$. Let
967 $\mathbf E_L=(E_{L,0},\ldots,E_{L,L-1})^{\mathsf T}$ and
968 define the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and
969 $(C_Lv)_j=v_{j-1}$ for $j\geq1$. Then
970 \begin{equation}
971 \mathbf E_L'=C_L\mathbf E_L,\qquad
972 \mathbf E_L(0)=e_0,\qquad
973 C_L^L=I,\qquad
974 \mathbf E_L(z)=e^{zC_L}e_0.
975 \label{a-spectral-companion-system}
976 \end{equation}
977 One can verify the last equality directly: $C_L^ne_0=e_{n\bmod L}$,
978 so its $j$th exponential coordinate is exactly
979 \eqref{a-spectral-factorial-lane}. Character filtering also gives
980 \begin{equation}
981 E_{L,j}(z)=\frac1L\sum_{\ell=0}^{L-1}
982 \omega_L^{-j\ell}e^{\omega_L^\ell z}.
983 \label{a-spectral-root-exponential}
984 \end{equation}
985 Thus factorial, hypergeometric, companion-state, and root-filter forms
986 describe the same analytic function. Each form has a different numerical
987 conditioning: a root sum can contain cancellations even when the
988 defining series has positive terms.
989
990 For a general finite transfer matrix $M_1$, the residue-state identity
991 $v_{Lq+j}=M_1^j(M_1^L)^qv_0$ follows by adding exponents. It needs
992 no hypergeometric coefficients. A factorial lane provides a particular
993 analytic coordinate system; it does not impose a unique microscopic
994 Hamiltonian on every such transfer problem.
995
996 \Needspace{24\baselineskip}
997 \subsection{Bounded operators and controlled series errors}
998
999 \begin{theorem}[Entire lane calculus for a bounded operator]
1000 \label{a-spectral-bounded-operator-lanes}
1001 Let $A_0\in\mathcal B(\HH)$ and $t\in\C$. The series
1002 $E_{L,j}(tA_0)$ converges absolutely in operator norm and satisfies
1003 \begin{align}
1004 E_{L,j}(tA_0)
1005 &=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}
1006 e^{\omega_L^\ell tA_0},\nonumber\\
1007 &=\frac{(tA_0)^j}{j!}\,
1008 {}_0F_{L-1}\!\left(\begin{matrix}-\\\mathcal A_{L,j}\end{matrix};
1009 \frac{(tA_0)^L}{L^L}\right),\nonumber\\
1010 \sum_{j=0}^{L-1}E_{L,j}(tA_0)&=e^{tA_0}.
1011 \label{a-spectral-operator-lane-identities}
1012 \end{align}
1013 If all terms through degree $Q$ are retained in a lane, the omitted
1014 operator norm is at most
1015 \begin{equation}
1016 e^a\frac{a^{Q+1}}{(Q+1)!},\qquad a=|t|\norm{A_0}.
1017 \label{a-spectral-lane-tail-bound}
1018 \end{equation}
1019 \end{theorem}
1020 \begin{proof}
1021 The sum of norms of all exponential terms is at most $e^a$. A lane
1022 is a subseries of this sum, so it converges absolutely. The character
1023 calculation and factorial split can therefore be applied in operator
1024 norm, giving the first two identities. Partitioning the nonnegative
1025 degrees by their residues proves the third. All products are powers
1026 of the same operator, so there is no commuting-operator assumption.
1027 For the tail, bound a lane by the full tail. Since
1028 $(Q+1+k)!\geq(Q+1)!k!$,
1029 \[
1030 \sum_{n=Q+1}^\infty\frac{a^n}{n!}
1031 \leq\frac{a^{Q+1}}{(Q+1)!}\sum_{k=0}^\infty\frac{a^k}{k!},
1032 \]
1033 which proves \eqref{a-spectral-lane-tail-bound}.
1034 \end{proof}
1035
1036 If $A_0^L=\alpha I$, the hypergeometric expression reduces every
1037 lane to a scalar function multiplying $A_0^j$. The propagator lies
1038 in the finite span of $I,A_0,\ldots,A_0^{L-1}$. A more general
1039 polynomial identity gives the corresponding finite algebra of powers.
1040 These statements follow from the supplied algebraic relation; they
1041 do not assert one for an arbitrary Hamiltonian.
1042
1043 The coefficient maps $\mathfrak R_{L,j}$ in
1044 \eqref{a-spectral-analytic-lanes} are idempotents on analytic functions.
1045 The evaluated operators $E_{L,j}(tA_0)$ generally are not.
1046 For $L=2$, $E_{2,0}(tA_0)=\cosh(tA_0)$ and
1047 $E_{2,1}(tA_0)=\sinh(tA_0)$, which are neither an orthogonal
1048 projector family nor a pair of independent unitary evolutions.
1049 The sum reconstructs an evolution; an individual lane is an analytic
1050 contribution to it.
1051
1052 \subsection{A squared-factorial lane for Bessel functions}
1053
1054 The same separation gives the entire function
1055 \[
1056 B(y)=\sum_{n=0}^\infty\frac{(-1)^ny^n}{(n!)^2}
1057 ={}_0F_1(;1;-y).
1058 \]
1059 For $y\geq0$, this is $J_0(2\sqrt y)$; the entire series itself
1060 does not need a square-root branch. Define
1061 $B_{L,j}(y)=\sum_{q\geq0}(-1)^{Lq+j}y^{Lq+j}/[(Lq+j)!]^2$.
1062 Squaring \eqref{a-spectral-factorial-split} gives
1063 \begin{equation}
1064 B_{L,j}(y)
1065 =\frac{(-1)^jy^j}{(j!)^2}\,
1066 {}_0F_{2L-1}\!\left(
1067 \begin{matrix}-\\1,\mathcal A_{L,j},\mathcal A_{L,j}\end{matrix};
1068 \frac{(-1)^Ly^L}{L^{2L}}\right).
1069 \label{a-spectral-bessel-lane}
1070 \end{equation}
1071 One of the two $q!$ factors is the conventional hypergeometric
1072 factorial and the other supplies the explicit bottom parameter $1$.
1073 All bottom parameters are positive and the ratio test again gives
1074 entire convergence. Character filtering reconstructs $B=\sum_jB_{L,j}$.
1075 With the Euler derivative $\Theta=y\,d/dy$, termwise differentiation
1076 also gives the cyclic system
1077 \begin{equation}
1078 \Theta^2B_{L,j}(y)=-yB_{L,j-1\bmod L}(y).
1079 \label{a-spectral-bessel-euler}
1080 \end{equation}
1081 For a term of positive degree $n$, its new coefficient is
1082 $n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.
1083 The degree-zero term disappears under $\Theta^2$, which also verifies
1084 the wraparound case $j=0$. The scalar Gowdy Bessel mode therefore
1085 admits these analytic coordinates. Filtering that already-defined
1086 solution does not add a gravitational field equation.
1087
1088
1089
1090 \section{Spectral domains and number-residue packets}
1091 \label{a-spectral-domains-packets}
1092
1093 The bounded calculus has two useful continuations. An unbounded
1094 Hamiltonian admits spectral functions with their exact domains.
1095 An oscillator admits actual residue projectors on its number spectrum.
1096 These are distinct constructions, and only the latter is a family of
1097 orthogonal quantum packet projectors.
1098
1099 \subsection{An unbounded propagator does not make every lane bounded}
1100
1101 Let $H_\infty$ be self-adjoint with spectral measure $E$. For real $t$
1102 define the scalar functions
1103 $f_j(\lambda)=E_{L,j}(-it\lambda)$.
1104 The operator $f_j(H_\infty)$ has domain
1105 \begin{equation}
1106 \Dom(f_j(H_\infty))
1107 =\left\{\psi:\int_{\R}|f_j(\lambda)|^2
1108 \,d\langle\psi,E(\lambda)\psi\rangle<\infty\right\}.
1109 \label{a-spectral-lane-domain}
1110 \end{equation}
1111 The scalar identity $\sum_jf_j(\lambda)=e^{-it\lambda}$ gives
1112 \begin{equation}
1113 \sum_jf_j(H_\infty)\psi=e^{-itH_\infty}\psi,
1114 \qquad \psi\in\bigcap_j\Dom(f_j(H_\infty)).
1115 \label{a-spectral-unbounded-reconstruction}
1116 \end{equation}
1117 The equality holds on the stated common domain because it is a finite
1118 sum of spectral functions there. It does not say that each lane acts
1119 on every vector.
1120
1121 For an explicit example, take $L=4$. The scalar formula gives
1122 \[
1123 f_0(\lambda)=\tfrac12\bigl(\cosh(t\lambda)+\cos(t\lambda)\bigr),
1124 \quad
1125 f_2(\lambda)=\tfrac12\bigl(\cos(t\lambda)-\cosh(t\lambda)\bigr).
1126 \]
1127 If $H_\infty$ is multiplication by $\lambda$ on $L^2(\R)$ and
1128 $t\ne0$, these functions define unbounded operators. Their sum
1129 cancels the growing hyperbolic contribution. The full propagator
1130 $e^{-itH_\infty}$ is nevertheless unitary on all of $L^2(\R)$.
1131 For $L=2$, by contrast, the two Schr\"odinger lanes are
1132 $\cos(tH_\infty)$ and $-i\sin(tH_\infty)$, both bounded. A larger
1133 lane count must not inherit this special dyadic boundedness without proof.
1134
1135 A sufficient common-domain condition is
1136 \begin{equation}
1137 \int_{\R} e^{2|t||\lambda|}
1138 \,d\langle\psi,E(\lambda)\psi\rangle<\infty,
1139 \label{a-spectral-exponential-domain}
1140 \end{equation}
1141 since $|f_j(\lambda)|\leq e^{|t||\lambda|}$. Spectral cutoffs
1142 $P_R=E([-R,R])$ supply a dense set satisfying this condition. On
1143 $\Ran P_R$, the restricted Hamiltonian is bounded, the norm-series
1144 identities apply, and
1145 \[
1146 \sum_jf_j(H_\infty)P_R\psi=e^{-itH_\infty}P_R\psi
1147 \longrightarrow e^{-itH_\infty}\psi.
1148 \]
1149 On the common domain of the individual lanes, each term also converges
1150 to $f_j(H_\infty)\psi$ by its spectral-domain integrability.
1151 For an arbitrary vector only convergence of the summed cutoff expression
1152 is asserted. The domain question for an interacting Hamiltonian therefore
1153 remains even when a bounded surrogate has excellent numerical residuals.
1154
1155 \subsection{A short factorial tail and its adic unit window}
1156
1157 Let $p$ be prime, let $L=mp$ for a positive integer $m$, and take
1158 $0\leq j<p$. For every $q\geq0$,
1159 \begin{equation}
1160 g_q=\frac{j!(Lq)!}{(Lq+j)!}
1161 =\binom{Lq+j}{j}^{-1}\in\Z_p^\times.
1162 \label{a-spectral-unit-window}
1163 \end{equation}
1164 Indeed
1165 $\binom{Lq+j}{j}=\prod_{r=1}^j(Lq+r)/r$.
1166 Each numerator and denominator is a $p$-adic unit because
1167 $Lq\equiv0\pmod p$ and $1\leq r<p$. For $j=0$ the empty product
1168 is one. The proof also covers $p=2$ in its short window. Outside this
1169 window, a factor can cross a multiple of $p$ and the same proof gives
1170 no unit guarantee. This rational coefficient statement supplies no
1171 claim about the $p$-adic convergence of the complex exponential series.
1172
1173 Analytic refinement of a factorial lane is independent of this unit
1174 condition. Splitting $q=pr+a$, $0\leq a<p$, gives the exact identity
1175 \begin{equation}
1176 E_{L,j}(z)=\sum_{a=0}^{p-1}E_{pL,j+aL}(z),\qquad 0\leq j<L.
1177 \label{a-spectral-analytic-refinement}
1178 \end{equation}
1179 Absolute convergence permits this partition of the summation index.
1180 The labels $j+aL$ all lie in $[0,pL)$, but need not lie in the
1181 parent's short unit window. Exact complex analytic refinement and
1182 the local $p$-adic unit assertion therefore have separate hypotheses.
1183
1184 \subsection{Actual orthogonal packets on the oscillator number spectrum}
1185
1186 On the oscillator Fock space, let
1187 $\widehat N|n\rangle=n|n\rangle$ and define the bounded unitary
1188 $V=e^{2\pi i\widehat N/L}$. Its spectral residues are
1189 \begin{equation}
1190 Q_j^{(L)}=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}V^\ell
1191 =\sum_{n\equiv j\;({\rm mod}\;L)}|n\rangle\langle n|.
1192 \label{a-spectral-number-projectors}
1193 \end{equation}
1194 On every number basis vector, the same finite character sum proves the
1195 second equality. These $Q_j^{(L)}$ are bounded orthogonal projectors,
1196 $Q_jQ_k=\delta_{jk}Q_j$, and $\sum_jQ_j=I$. No exponential norm series
1197 for the unbounded number operator is needed; its spectral calculus
1198 defines $V$.
1199
1200 For a coherent state
1201 \[
1202 |\alpha\rangle=e^{-|\alpha|^2/2}
1203 \sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}}|n\rangle,
1204 \qquad \mu=|\alpha|^2\geq0,
1205 \]
1206 the packet probability is the Poisson residue sum
1207 \begin{align}
1208 p_{L,j}(\mu)
1209 &=\langle\alpha,Q_j^{(L)}\alpha\rangle
1210 =e^{-\mu}E_{L,j}(\mu)\nonumber\\
1211 &=\frac1L\sum_{\ell=0}^{L-1}\omega_L^{-j\ell}
1212 e^{\mu(\omega_L^\ell-1)}.
1213 \label{a-spectral-coherent-packets}
1214 \end{align}
1215 The factorial series proves nonnegativity, including the limiting values
1216 $p_{L,0}(0)=1$ and $p_{L,j}(0)=0$ for $j>0$. Partition of the
1217 Poisson series proves $\sum_jp_{L,j}=1$. Their equivalent hypergeometric
1218 form is $e^{-\mu}$ times \eqref{a-spectral-hypergeometric-lane} at $z=\mu$.
1219
1220 \begin{proposition}[Approach to uniform residue probability]
1221 \label{a-spectral-coherent-bound}
1222 For fixed $L\geq2$ and $\mu\geq0$,
1223 \begin{equation}
1224 \left|p_{L,j}(\mu)-\frac1L\right|
1225 \leq\frac1L\sum_{\ell=1}^{L-1}
1226 e^{-\mu(1-\cos(2\pi\ell/L))}
1227 \leq\frac{L-1}{L}
1228 e^{-\mu(1-\cos(2\pi/L))}.
1229 \label{a-spectral-packet-mixing-bound}
1230 \end{equation}
1231 \end{proposition}
1232 \begin{proof}
1233 In \eqref{a-spectral-coherent-packets}, the $\ell=0$ term is $1/L$.
1234 The modulus of every other term is
1235 $e^{\mu(\operatorname{Re}\omega_L^\ell-1)}/L$.
1236 The triangle inequality gives the first bound. Among the nonzero
1237 residues, the largest cosine is $\cos(2\pi/L)$, giving the second.
1238 \end{proof}
1239
1240 The exponent is positive for each fixed $L$, but decreases as
1241 $2\pi^2/L^2$ when $L$ grows. The conclusion is therefore uniform in
1242 the finite packet label $j$ at fixed $L$; it is not a depth-uniform
1243 estimate at arbitrarily large residue modulus.
1244
1245 Finally the annihilation operator routes adjacent residues:
1246 \begin{equation}
1247 Q_r^{(L)}aQ_s^{(L)}=0\quad\text{unless }r\equiv s-1\pmod L,
1248 \qquad
1249 Q_r^{(L)}a^\dagger Q_s^{(L)}=0\quad\text{unless }r\equiv s+1\pmod L.
1250 \label{a-spectral-ladder-routing}
1251 \end{equation}
1252 On a finite number-state combination this follows from
1253 $a|n\rangle=\sqrt n\,|n-1\rangle$ and
1254 $a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle$.
1255 Each residue projector preserves $\Dom(a)$ and $\Dom(a^\dagger)$,
1256 because it only removes terms from their weighted number norm.
1257 Truncation of a vector in either domain converges in the corresponding
1258 graph norm, so the identities extend to those domains.
1259 For the isolated oscillator Hamiltonian, which is a function of
1260 $\widehat N$, the packet projectors commute with the evolution and
1261 the weights stay constant. A driving term containing ladder operators
1262 can transfer weight according to the routing rule. Selecting such a
1263 drive is further model data.
1264
1265 These packet probabilities are directly interpretable within the
1266 ordinary oscillator model. A physical interpretation of adic digits,
1267 hidden tree levels, or a preferred hypergeometric residue split requires
1268 an independently specified coupling and observable. The exact analytic
1269 and spectral coordinates derived here remain useful without making
1270 that identification.
1271
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