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1 \hypertarget{conventions-and-finite-packet-space}{%
2 \section{Conventions and finite packet
3 space}\label{conventions-and-finite-packet-space}}
4 
5 We use natural units \(\hbar=c=1\). Space is the flat unit torus
6 
7 \[
8 \mathbb T^3=(\mathbb R/\mathbb Z)^3,
9 \]
10 
11 equipped with normalized Haar measure \(dx\), so
12 \(\int_{\mathbb T^3}1\,dx=1\). Integration of a periodic representative
13 may equivalently be taken over \([0,1)^3\) with Lebesgue measure. A
14 Fourier mode is \(n=(n_1,n_2,n_3)\in\mathbb Z^3\), its momentum is
15 \(p(n)=2\pi n\), and its scalar plane wave is
16 
17 \[
18 e_n(x)=e^{2\pi i n\cdot x}=e^{i p(n)\cdot x}.
19 \]
20 
21 Components of a spinor are numbered \(0,1,2,3\), in the exact order of
22 the source representation. For \(v,w\in\mathbb C^4\),
23 
24 \[
25 \langle v,w\rangle=\sum_{r=0}^{3}\overline{v_r}w_r,
26 \qquad
27 \|v\|_2^2=\sum_{r=0}^{3}|v_r|^2.
28 \]
29 
30 The inner product is conjugate-linear in its first argument and linear
31 in its second. The adjoint is therefore the conjugate transpose. Unless
32 indicated otherwise, a spinor norm is this Euclidean norm and an
33 operator norm is its induced operator norm. The field pairing is
34 
35 \[
36 \langle f,g\rangle_{L^2}
37 =\int_{\mathbb T^3}\langle f(x),g(x)\rangle\,dx
38 \]
39 
40 on \(L^2(\mathbb T^3;\mathbb C^4)\). Fix a finite set
41 \(S\subset\mathbb Z^3\). A packet with amplitudes \(a_n\in\mathbb C^4\)
42 is
43 
44 \[
45 f_a(x)=\sum_{n\in S}e_n(x)a_n.
46 \]
47 
48 The associated coefficient space is the finite Hilbert direct sum
49 
50 \[
51 \mathcal K_S=\bigoplus_{n\in S}\mathbb C^4,
52 \qquad
53 \|a\|_{\mathcal K_S}^2=\sum_{n\in S}\|a_n\|_2^2.
54 \]
55 
56 The packet space \(\mathcal F_S=\{f_a:a\in\mathcal K_S\}\) will be
57 identified isometrically with \(\mathcal K_S\). Its elements are smooth
58 periodic functions, so the differential and centered-translation
59 calculations below raise no regularity question. Empty support is
60 allowed: both spaces then contain only zero and all packet identities
61 reduce to \(0=0\).
62 
63 The parameters \(m,h,t\) are arbitrary real numbers. The mathematics
64 does not require \(m\ge0\); the usual physical rest-mass interpretation
65 uses that additional convention. The shell value \(E\) is always a
66 nonnegative square root, even when \(m<0\). Mesh limits in the checked
67 statements are \(h\to0^+\). Some explicit estimates proved here hold for
68 every real \(h\), since the centered symbol is even in \(h\).
69 
70 \hypertarget{the-coordinate-clifford-representation}{%
71 \section{The coordinate Clifford
72 representation}\label{the-coordinate-clifford-representation}}
73 
74 The spatial generators and mass generator are
75 
76 \[
77 \alpha_x=
78 \begin{pmatrix}
79 0&0&0&1\\
80 0&0&1&0\\
81 0&1&0&0\\
82 1&0&0&0
83 \end{pmatrix},
84 \qquad
85 \alpha_y=
86 \begin{pmatrix}
87 0&0&0&-i\\
88 0&0&i&0\\
89 0&-i&0&0\\
90 i&0&0&0
91 \end{pmatrix},
92 \]
93 
94 \[
95 \alpha_z=
96 \begin{pmatrix}
97 0&0&1&0\\
98 0&0&0&-1\\
99 1&0&0&0\\
100 0&-1&0&0
101 \end{pmatrix},
102 \qquad
103 \beta=
104 \begin{pmatrix}
105 1&0&0&0\\
106 0&1&0&0\\
107 0&0&-1&0\\
108 0&0&0&-1
109 \end{pmatrix}.
110 \]
111 
112 In particular, \(\alpha_yv=(-iv_3,iv_2,-iv_1,iv_0)\); this fixes a sign
113 that cannot be changed silently in a worked example. The matrices act on
114 column spinors. Grouping the first two and last two components gives
115 their block form without any coordinate permutation:
116 
117 \[
118 \alpha_j=
119 \begin{pmatrix}0&\sigma_j\\\sigma_j&0\end{pmatrix},
120 \qquad
121 \beta=\begin{pmatrix}I_2&0\\0&-I_2\end{pmatrix},
122 \]
123 
124 where
125 
126 \[
127 \sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},
128 \quad
129 \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},
130 \quad
131 \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
132 \]
133 
134 \needspace{4\baselineskip}
135 \textbf{Proposition PH01 (Hermitian Clifford relations).} The four
136 displayed matrices are Hermitian involutions. Distinct spatial
137 generators anticommute, and each spatial generator anticommutes with
138 \(\beta\):
139 
140 \[
141 \alpha_j^*=\alpha_j,\quad \beta^*=\beta,\quad
142 \alpha_j^2=\beta^2=I_4,
143 \]
144 
145 \[
146 \alpha_j\alpha_k+\alpha_k\alpha_j=0\quad(j\ne k),
147 \qquad
148 \alpha_j\beta+\beta\alpha_j=0.
149 \]
150 
151 \emph{Proof.} The real diagonal entries of \(\beta\) are
152 \(1,1,-1,-1\), so its adjoint and square have the stated forms. Each
153 \(\sigma_j\) is Hermitian and squares to \(I_2\), by direct
154 multiplication. Their products are
155 
156 \[
157 \sigma_x\sigma_y=i\sigma_z,\qquad
158 \sigma_y\sigma_z=i\sigma_x,\qquad
159 \sigma_z\sigma_x=i\sigma_y,
160 \]
161 
162 with reversed products equal to their negatives. For example,
163 
164 \[
165 \begin{pmatrix}0&1\\1&0\end{pmatrix}
166 \begin{pmatrix}0&-i\\i&0\end{pmatrix}
167 =\begin{pmatrix}i&0\\0&-i\end{pmatrix}=i\sigma_z.
168 \]
169 
170 The other two follow by the same two-by-two multiplication; reversal
171 follows by taking adjoints. The block form gives
172 
173 \[
174 \alpha_j^*=
175 \begin{pmatrix}0&\sigma_j^*\\\sigma_j^*&0\end{pmatrix}
176 =\alpha_j,\qquad
177 \alpha_j\alpha_k=
178 \begin{pmatrix}\sigma_j\sigma_k&0\\0&\sigma_j\sigma_k\end{pmatrix}.
179 \]
180 
181 Thus \(\alpha_j^2=I_4\) and the spatial anticommutators vanish. Finally,
182 
183 \[
184 \alpha_j\beta=
185 \begin{pmatrix}0&-\sigma_j\\\sigma_j&0\end{pmatrix},
186 \qquad
187 \beta\alpha_j=
188 \begin{pmatrix}0&\sigma_j\\-\sigma_j&0\end{pmatrix},
189 \]
190 
191 whose sum is zero. Every identity is an identity of the exact source
192 matrices. \(\square\)
193 
194 \hypertarget{continuum-and-centered-mode-hamiltonians}{%
195 \section{Continuum and centered mode
196 Hamiltonians}\label{continuum-and-centered-mode-hamiltonians}}
197 
198 For \(p\in\mathbb R^3\), define
199 
200 \[
201 H(m,p)=m\beta+p_x\alpha_x+p_y\alpha_y+p_z\alpha_z.
202 \]
203 
204 Its coordinates are
205 
206 \[
207 H(m,p)=
208 \begin{pmatrix}
209 m&0&p_z&p_x-ip_y\\
210 0&m&p_x+ip_y&-p_z\\
211 p_z&p_x-ip_y&-m&0\\
212 p_x+ip_y&-p_z&0&-m
213 \end{pmatrix}.
214 \]
215 
216 On mode \(n\), the continuum Hamiltonian is \(H_n=H(m,p(n))\). It is the
217 symbol of the differential expression
218 
219 \[
220 -i\sum_{j=x,y,z}\alpha_j\partial_j+m\beta
221 \]
222 
223 on the finite packet space: \(\partial_je_n=ip_j(n)e_n\), so applying
224 the expression to \(e_na_n\) gives \(e_nH_na_n\). For real spacing
225 \(h\), set
226 
227 \[
228 q_j(h,p)=
229 \begin{cases}
230 \sin(hp_j)/h,&h\ne0,\\
231 p_j,&h=0.
232 \end{cases}
233 \]
234 
235 Write \(q_h(n)=q(h,p(n))\) and \(H_{h,n}=H(m,q_h(n))\). The function
236 \(q_j\) is even in \(h\), and its value at zero is justified by
237 \(\sin z/z\to1\). For \(h\ne0\), a centered translation difference on a
238 periodic smooth function is
239 
240 \[
241 (D_{h,j}f)(x)=\frac{f(x+he_j)-f(x-he_j)}{2h}.
242 \]
243 
244 Here \(e_j\) is the j-th spatial unit vector, distinct from the Fourier
245 characters \(e_n\) and spinor vectors. Its action on a plane wave is
246 exactly
247 
248 \[
249 D_{h,j}e_n
250 =e_n\frac{e^{ihp_j(n)}-e^{-ihp_j(n)}}{2h}
251 =i\frac{\sin(hp_j(n))}{h}e_n.
252 \]
253 
254 The centered differential expression \(-i\sum_j\alpha_jD_{h,j}+m\beta\)
255 therefore has symbol \(H_{h,n}\). Its signs follow from \(-i\cdot i=1\).
256 At \(h=0\), the symbol equals \(H_n\) by definition. These formulas take
257 place on the continuum torus and use no sampled spatial data.
258 
259 \needspace{4\baselineskip}
260 \textbf{Proposition PH02 (Hermiticity and the mass shell).} For every
261 real \(m\) and \(p\in\mathbb R^3\),
262 
263 \[
264 H(m,p)^*=H(m,p),\qquad
265 H(m,p)^2=(m^2+|p|^2)I_4.
266 \]
267 
268 In particular, \(H_n,H_{h,n}\) are Hermitian for all real \(m,h\), and
269 
270 \[
271 H_n^2=E_n^2I_4,\quad E_n=\sqrt{m^2+|p(n)|^2},
272 \]
273 
274 \[
275 H_{h,n}^2=E_{h,n}^2I_4,\quad
276 E_{h,n}=\sqrt{m^2+|q_h(n)|^2}.
277 \]
278 
279 \emph{Proof.} Adjoint is additive, and a real multiple of a Hermitian
280 matrix is Hermitian. PH01 therefore proves the first identity. Expanding
281 the square gives the square terms \(m^2\beta^2+\sum_jp_j^2\alpha_j^2\),
282 the terms \(mp_j(\beta\alpha_j+\alpha_j\beta)\), and the terms
283 \(p_jp_k(\alpha_j\alpha_k+\alpha_k\alpha_j)\) for \(j<k\). The
284 anticommutators vanish and all four squares are \(I_4\), giving
285 \((m^2+\sum_jp_j^2)I_4\). Substitution proves both mode statements.
286 \(\square\)
287 
288 Hermiticity here is a statement about matrices acting on all of
289 \(\mathbb C^4\). Every linear map on this finite space is bounded and
290 everywhere defined. This is not a construction of the adjoint or domain
291 of an unbounded differential operator on the full field space.
292 
293 \hypertarget{spectral-projectors-and-exact-mode-evolution}{%
294 \section{Spectral projectors and exact mode
295 evolution}\label{spectral-projectors-and-exact-mode-evolution}}
296 
297 Let \(H=H(m,p)\) and \(E=\sqrt{m^2+|p|^2}\). Define the exact propagator
298 
299 \[
300 U_H(t)=e^{-itH}=\sum_{k=0}^{\infty}\frac{(-itH)^k}{k!}.
301 \]
302 
303 The series converges absolutely in operator norm, being dominated by
304 \(\sum_k(|t|\|H\|)^k/k!\). This is exact continuous-time evolution, with
305 no time step or series truncation.
306 
307 \needspace{4\baselineskip}
308 \textbf{Proposition PH03 (spectral splitting and closed propagator).} If
309 \(E>0\), define
310 
311 \[
312 P_+=\frac12\left(I_4+\frac HE\right),\qquad
313 P_-=\frac12\left(I_4-\frac HE\right).
314 \]
315 
316 Then \(P_\pm^*=P_\pm\), \(P_\pm^2=P_\pm\), \(P_+P_-=P_-P_+=0\),
317 \(P_++P_-=I_4\), and \(HP_\pm=\pm EP_\pm\). Both projectors have rank
318 two, and
319 
320 \[
321 U_H(t)=e^{-itE}P_++e^{itE}P_-
322 =\cos(tE)I_4-i\frac{\sin(tE)}{E}H.
323 \]
324 
325 If \(E=0\), then \(H=0\) and \(U_H(t)=I_4\). Equivalently, the closed
326 expression is valid with the continuous convention \(\sin(tE)/E=t\) at
327 \(E=0\).
328 
329 \emph{Proof.} For \(E>0\), the Hermitian matrix \(K=H/E\) satisfies
330 \(K^2=I_4\). Thus
331 
332 \[
333 P_\pm^2=\tfrac14(I_4\pm2K+K^2)=\tfrac12(I_4\pm K)=P_\pm,
334 \]
335 
336 and \(P_+P_-=\tfrac14(I_4-K^2)=0\); the reversed product is the same.
337 Their adjoints equal themselves and their sum is \(I_4\). Also
338 \(K(I_4\pm K)=K\pm I_4=\pm(I_4\pm K)\), proving \(HP_\pm=\pm EP_\pm\).
339 Each displayed Clifford matrix has trace zero. Hence
340 \(\operatorname{tr}H=0\) and \(\operatorname{tr}P_\pm=2\). An idempotent
341 splits the space into its range and kernel, acting as identity and zero
342 respectively. Its trace is therefore its rank, proving rank two.
343 
344 The identity \(I_4=P_++P_-\) and the two eigenvalue identities give, by
345 induction, \(H^k=E^kP_++(-E)^kP_-\) for every \(k\ge0\). Substitute into
346 the absolutely convergent series to obtain the spectral expression.
347 Expanding its projectors gives the cosine and sine expression.
348 
349 When \(E=0\), the sum \(m^2+|p|^2\) of nonnegative squares is zero.
350 Therefore \(m=0,p=0\), and \(H=0\). Its exponential is \(I_4\). The
351 quotient \(\sin(tE)/E\) tends to \(t\) at zero; multiplying its filled
352 value by \(H=0\) gives the same identity. No division by zero is used.
353 \(\square\)
354 
355 The proof applies unchanged to \(H_{h,n}\), with energy \(E_{h,n}\). A
356 positive-energy continuum mode can have \(E_{h,n}=0\) at a nonzero
357 spacing when \(m=0\) and all centered components vanish. That mode uses
358 the zero-energy branch. Positivity of the continuum energy does not
359 license division by the centered energy at every spacing.
360 
361 \needspace{4\baselineskip}
362 \textbf{Proposition PH04 (group law, unitarity, and stability).} For
363 real \(s,t\),
364 
365 \[
366 U_H(0)=I_4,\qquad U_H(s+t)=U_H(s)U_H(t),
367 \]
368 
369 \[
370 U_H(-t)=U_H(t)^{-1}=U_H(t)^*,
371 \qquad U_H(t)^*U_H(t)=U_H(t)U_H(t)^*=I_4.
372 \]
373 
374 Consequently,
375 
376 \[
377 \langle U_H(t)v,U_H(t)w\rangle=\langle v,w\rangle,\qquad
378 \|U_H(t)v\|_2=\|v\|_2
379 \]
380 
381 for all spinors. These statements hold for both mode Hamiltonians at
382 every real \(m,h,t\).
383 
384 \emph{Proof.} When \(E>0\), the spectral form and orthogonality of
385 \(P_\pm\) give
386 
387 \[
388 U_H(s)U_H(t)
389 =e^{-i(s+t)E}P_++e^{i(s+t)E}P_-=U_H(s+t).
390 \]
391 
392 At zero time the sum is \(I_4\). Hermiticity of the projectors and
393 reality of \(tE\) give \(U_H(t)^*=e^{itE}P_++e^{-itE}P_-=U_H(-t)\).
394 Setting \(s=-t\) proves the inverse identities. When \(E=0\), every
395 assertion follows from \(U_H(t)=I_4\).
396 
397 For any matrix \(A\), expansion and interchange of two finite sums gives
398 \(\langle Av,w\rangle=\langle v,A^*w\rangle\). Thus
399 
400 \[
401 \langle U_H(t)v,U_H(t)w\rangle
402 =\langle v,U_H(t)^*U_H(t)w\rangle=\langle v,w\rangle.
403 \]
404 
405 Take \(w=v\) and then nonnegative square roots to obtain norm
406 preservation. PH02 supplies the mode cases. \(\square\)
407 
408 An equivalent bounded-operator argument is represented in the formal source:
409 \(-itH\) is skew-adjoint, and its exponential is unitary. Our spectral
410 proof supplies additional explicit projectors and formulas.
411 
412 \hypertarget{fourier-orthogonality-and-packet-conservation}{%
413 \section{Fourier orthogonality and packet
414 conservation}\label{fourier-orthogonality-and-packet-conservation}}
415 
416 Let \(J_n:\mathbb C^4\to L^2(\mathbb T^3;\mathbb C^4)\) be
417 \(J_nv=e_nv\).
418 
419 \needspace{4\baselineskip}
420 \textbf{Proposition PH05 (mode isometry and finite Parseval identity).}
421 For modes \(n,k\) and spinors \(v,w\),
422 
423 \[
424 \langle J_nv,J_kw\rangle_{L^2}
425 =\begin{cases}\langle v,w\rangle,&n=k,\\0,&n\ne k.\end{cases}
426 \]
427 
428 In particular, \(J_n\) is a linear isometry and
429 
430 \[
431 \left\|\sum_{n\in S}J_na_n\right\|_{L^2}^2
432 =\sum_{n\in S}\|a_n\|_2^2.
433 \]
434 
435 \emph{Proof.} The constant spinors factor out of the scalar integral:
436 
437 \[
438 \langle J_nv,J_kw\rangle_{L^2}
439 =\langle v,w\rangle\int_{\mathbb T^3}e^{2\pi i(k-n)\cdot x}\,dx.
440 \]
441 
442 The integral is the product of three one-dimensional integrals. For
443 integer \(r\), the integral of \(e^{2\pi irx}\) over \([0,1)\) is \(1\)
444 when \(r=0\); for \(r\ne0\), it is \((e^{2\pi ir}-1)/(2\pi ir)=0\). The
445 product is therefore \(1\) exactly when \(n=k\). Linearity is immediate,
446 and setting \(k=n,w=v\) proves isometry. Expansion of the squared norm
447 of the finite sum gives a double sum of pairings. The off-diagonal
448 pairings vanish and the diagonal pairings are \(\|a_n\|_2^2\).
449 \(\square\)
450 
451 Define the evolved packets
452 
453 \[
454 f(t,x)=\sum_{n\in S}e_n(x)U_{H_n}(t)a_n,\qquad
455 f_h(t,x)=\sum_{n\in S}e_n(x)U_{H_{h,n}}(t)a_n.
456 \]
457 
458 Both remain in \(\mathcal F_S\). Their operators \(U_S(t),U_{h,S}(t)\)
459 have the finite block diagonal coefficient actions with blocks
460 \(U_{H_n}(t),U_{H_{h,n}}(t)\).
461 
462 \needspace{4\baselineskip}
463 \textbf{Theorem PH06 (exact finite-packet norm conservation).} For every
464 real \(m,h,t\), finite \(S\), and coefficient family \(a\),
465 
466 \[
467 \|f(t)\|_{L^2}^2=\|f_h(t)\|_{L^2}^2
468 =\sum_{n\in S}\|a_n\|_2^2=\|f_a\|_{L^2}^2.
469 \]
470 
471 Both packet evolutions preserve the \(L^2\) inner product between two
472 packets in \(\mathcal F_S\), and each forms a one-parameter unitary
473 group there.
474 
475 \emph{Proof.} Apply PH05 to the evolved coefficients and PH04 on each
476 mode:
477 
478 \[
479 \|f_h(t)\|_{L^2}^2
480 =\sum_{n\in S}\|U_{H_{h,n}}(t)a_n\|_2^2
481 =\sum_{n\in S}\|a_n\|_2^2.
482 \]
483 
484 The continuum argument is identical. For coefficient families \(a,b\),
485 the same two propositions give
486 
487 \[
488 \langle f_{h,a}(t),f_{h,b}(t)\rangle_{L^2}
489 =\sum_{n\in S}
490 \langle U_{H_{h,n}}(t)a_n,U_{H_{h,n}}(t)b_n\rangle
491 =\sum_{n\in S}\langle a_n,b_n\rangle.
492 \]
493 
494 Composition of coefficient blocks proves the packet group law, and
495 inverse time proves surjectivity. This establishes unitarity on
496 \(\mathcal F_S\) using only finite sums. \(\square\)
497 
498 The quantity conserved is squared Hilbert norm, or total probability for
499 a normalized state. The parameter \(m\), the positive shell value
500 \(E_n\), and Hamiltonian expectation are different quantities. Calling
501 squared norm a packet mass does not identify it with the rest mass or
502 physical Hamiltonian energy.
503 
504 \hypertarget{generator-consistency-and-dispersion-error}{%
505 \section{Generator consistency and dispersion
506 error}\label{generator-consistency-and-dispersion-error}}
507 
508 The mode Hamiltonian difference contains no mass term:
509 
510 \[
511 \Delta_{h,n}:=H_{h,n}-H_n
512 =\sum_{j=x,y,z}\delta_j(h,n)\alpha_j,
513 \qquad
514 \delta_j(h,n)=q_j(h,p(n))-p_j(n).
515 \]
516 
517 Define
518 
519 \[
520 C(n)=\frac16\left(\sum_{j=x,y,z}|p_j(n)|^6\right)^{1/2}.
521 \]
522 
523 This constant depends on the fixed mode and increases cubically with
524 momentum scale. That dependence is essential in a fixed-cutoff result.
525 
526 \needspace{4\baselineskip}
527 \textbf{Proposition PH07 (quadratic generator consistency).} For every
528 real \(h\) and fixed mode \(n\),
529 
530 \[
531 \|\Delta_{h,n}\|=|\delta(h,n)|\le h^2C(n).
532 \]
533 
534 For every fixed finite packet,
535 
536 \[
537 \left\|\sum_{n\in S}e_n\Delta_{h,n}a_n\right\|_{L^2}^2
538 =\sum_{n\in S}|\delta(h,n)|^2\|a_n\|_2^2,
539 \]
540 
541 and
542 
543 \[
544 \left\|\sum_{n\in S}e_n\Delta_{h,n}a_n\right\|_{L^2}
545 \le h^2\left(\sum_{n\in S}C(n)^2\|a_n\|_2^2\right)^{1/2}.
546 \]
547 
548 \emph{Proof.} First prove the scalar estimate without a sign
549 restriction on \(h\). The fundamental theorem of calculus gives
550 \(|\sin u|\le|u|\), since \(\sin u=\int_0^u\cos v\,dv\) and
551 \(|\cos v|\le1\). Therefore
552 
553 \[
554 |1-\cos u|=2\sin^2(u/2)\le u^2/2.
555 \]
556 
557 For \(z\ge0\), integrate this bound in
558 \(\sin z-z=\int_0^z(\cos u-1)\,du\) to obtain \(|\sin z-z|\le z^3/6\).
559 Oddness of the residual proves \(|\sin z-z|\le|z|^3/6\) for negative
560 \(z\). For \(h\ne0\), substitute \(z=hp_j(n)\) and divide by \(|h|\):
561 
562 \[
563 |\delta_j(h,n)|\le \frac{h^2|p_j(n)|^3}{6}.
564 \]
565 
566 For \(h=0\), the residual is zero by definition and the bound still
567 holds. Squaring and adding proves \(|\delta(h,n)|\le h^2C(n)\).
568 
569 The Hermitian matrix \(\Delta_{h,n}\) satisfies
570 \(\Delta_{h,n}^2=|\delta(h,n)|^2I_4\) by PH01. Hence
571 
572 \[
573 \|\Delta_{h,n}v\|_2^2
574 =\langle v,\Delta_{h,n}^*\Delta_{h,n}v\rangle
575 =|\delta(h,n)|^2\|v\|_2^2.
576 \]
577 
578 Its induced operator norm is exactly \(|\delta(h,n)|\): the upper bound
579 holds on every vector and equality holds on every unit vector. Apply
580 PH05 to the residual coefficients for the packet equality, substitute
581 the scalar bounds, and take nonnegative square roots. \(\square\)
582 
583 The checked generator results state corresponding \(O(h^2)\) estimates
584 as \(h\to0^+\). The explicit constant and all-real inequality above are
585 traditional deductions. Precisely, \(r(h)=O(h^2)\) means that some
586 \(M,\varepsilon>0\) satisfy \(\|r(h)\|\le Mh^2\) for
587 \(0<h<\varepsilon\). Fixed support and amplitudes permit finite
588 summation of such bounds; no estimate uniform over all integer modes
589 follows.
590 
591 The reverse triangle inequality in \(\mathbb R^4\), applied to
592 \((m,q_h(n))\) and \((m,p(n))\), gives
593 
594 \[
595 |E_{h,n}-E_n|\le|q_h(n)-p(n)|\le h^2C(n).
596 \]
597 
598 Moreover, \(|\sin(hp_j)/h|\le|p_j|\) for \(h\ne0\), and the filled value
599 is equal at zero. Thus \(0\le E_{h,n}\le E_n\). For a fixed mode,
600 Taylor's formula gives
601 
602 \[
603 q_j(h,p(n))=p_j(n)-\frac{h^2p_j(n)^3}{6}+O(h^4),
604 \]
605 
606 \[
607 E_{h,n}^2=E_n^2-\frac{h^2}{3}\sum_jp_j(n)^4+O(h^4).
608 \]
609 
610 For clarity, Taylor's theorem through degree four has zero even
611 coefficients for sine, and its fifth-derivative remainder is at most
612 \(|hp_j|^5/120\). Division by \(|h|\) gives \(O(h^4)\); squaring the
613 three expansions and summing preserves that order for a fixed mode. If
614 \(E_n>0\), the square root is smooth near \(E_n^2\), so
615 
616 \[
617 E_{h,n}=E_n-\frac{h^2}{6E_n}\sum_jp_j(n)^4+O(h^4).
618 \]
619 
620 If \(E_n=0\), then \(m=0,n=0\) and \(E_{h,n}=0\) exactly for every
621 \(h\). The expansion containing \(1/E_n\) is not used.
622 
623 For fixed time, the phase error on either energy branch obeys
624 
625 \[
626 |e^{\pm itE_{h,n}}-e^{\pm itE_n}|
627 \le |t|\,|E_{h,n}-E_n|,
628 \]
629 
630 by integrating the derivative of \(s\mapsto e^{\pm its}\) between the
631 two real energies. Phase comparison alone does not compare full spinor
632 evolution: the eigenspaces also depend on the kinetic vector. The
633 operator argument below controls both effects.
634 
635 \needspace{10\baselineskip}
636 \atlasfigure{04_dispersion}{The continuum and centered shell branches along one momentum axis, with $m=1$, $h=1/12$. Marked points are integer torus modes; intermediate points use the real-symbol extension.}
637 
638 \hypertarget{fixed-time-propagator-consistency}{%
639 \section{Fixed-time propagator
640 consistency}\label{fixed-time-propagator-consistency}}
641 
642 \needspace{4\baselineskip}
643 \textbf{Theorem PH08 (fixed-mode and fixed-packet convergence).} For
644 fixed real \(m,t\) and fixed \(n\),
645 
646 \[
647 U_{H_{h,n}}(t)-U_{H_n}(t)=O(h^2)\quad(h\to0^+).
648 \]
649 
650 For fixed finite \(S\) and amplitudes \(a\),
651 
652 \[
653 f_h(t)-f(t)=O(h^2)
654 \quad\hbox{in }L^2(\mathbb T^3;\mathbb C^4),
655 \]
656 
657 and in particular \(\|f_h(t)-f(t)\|_{L^2}\to0\). The exact residual
658 identity is
659 
660 \[
661 \|f_h(t)-f(t)\|_{L^2}^2
662 =\sum_{n\in S}\|(U_{H_{h,n}}(t)-U_{H_n}(t))a_n\|_2^2.
663 \]
664 
665 \emph{Proof.} Give a fixed-time proof independent of the later
666 compact-time extension. For arbitrary matrices \(X,Y\), telescoping
667 products give
668 
669 \[
670 X^k-Y^k=\sum_{r=0}^{k-1}X^{k-1-r}(X-Y)Y^r
671 \quad(k\ge1).
672 \]
673 
674 No commutation is assumed. Expanding the summands gives successive
675 differences \(X^{k-r}Y^r-X^{k-1-r}Y^{r+1}\), which cancel. If
676 \(\|X\|,\|Y\|\le R\), then
677 
678 \[
679 \|X^k-Y^k\|\le kR^{k-1}\|X-Y\|.
680 \]
681 
682 Subtract the absolutely convergent exponential series and sum:
683 
684 \[
685 \|e^X-e^Y\|
686 \le\|X-Y\|\sum_{k=1}^{\infty}\frac{kR^{k-1}}{k!}
687 =e^R\|X-Y\|.
688 \]
689 
690 PH02 gives \(\|H_n\|=E_n,\|H_{h,n}\|=E_{h,n}\), by the norm-squared
691 argument of PH07. Since \(E_{h,n}\le E_n\), take \(X=-itH_{h,n}\),
692 \(Y=-itH_n\), and \(R=|t|E_n\). The result is
693 
694 \[
695 \|U_{H_{h,n}}(t)-U_{H_n}(t)\|
696 \le |t|e^{|t|E_n}h^2C(n).
697 \]
698 
699 This proves the modewise order, including \(t=0\). The difference of
700 packets is the finite sum of the mode residuals, so PH05 gives its exact
701 residual identity. Thus
702 
703 \[
704 \|f_h(t)-f(t)\|_{L^2}
705 \le h^2\left(
706 \sum_{n\in S}|t|^2e^{2|t|E_n}C(n)^2\|a_n\|_2^2
707 \right)^{1/2}.
708 \]
709 
710 The constant is finite for fixed \(t,S,a\), proving the order and
711 convergence. \(\square\)
712 
713 The formal algebraic-coordinate proof obtains this order from continuity and local
714 differentiability of the Banach-algebra exponential. Its operator norm
715 belongs to the algebraic coordinate spinor type. The coefficient
716 conversion in its packet definitions is continuous, so its checked
717 torus-\(L^2\) convergence is genuine. Our Euclidean PH08 also has the
718 direct proof above. The following bridge explains why its coordinates
719 agree with the separately defined Hilbert propagators.
720 
721 \hypertarget{the-algebraic-to-hilbert-coordinate-bridge}{%
722 \section{The algebraic-to-Hilbert coordinate
723 bridge}\label{the-algebraic-to-hilbert-coordinate-bridge}}
724 
725 Let \(\mathcal A\) be the coordinate space \(\{0,1,2,3\}\to\mathbb C\)
726 with maximum norm \(\|v\|_\infty=\max_r|v_r|\), and let \(\mathcal H\)
727 be the same coordinates with the Euclidean inner product. The map
728 
729 \[
730 T:\mathcal A\longrightarrow\mathcal H,\qquad (Tv)_r=v_r
731 \]
732 
733 is complex-linear and bijective; its inverse also retains the
734 coordinates. It is not generally an isometry between these norms. The
735 inequalities
736 
737 \[
738 \|v\|_\infty\le\|Tv\|_2\le2\|v\|_\infty
739 \]
740 
741 show that both directions are continuous. The first follows because each
742 coordinate square is bounded by the sum; the second because each of four
743 squares is at most \(\|v\|_\infty^2\).
744 
745 \needspace{4\baselineskip}
746 \textbf{Proposition PH09 (coordinate conjugacy of generators and
747 exponentials).} Let \(A\) be a complex-linear endomorphism of
748 \(\mathcal A\), and let \(A^{\mathcal H}=TAT^{-1}\). For every complex
749 \(z\),
750 
751 \[
752 e^{zA^{\mathcal H}}=Te^{zA}T^{-1}.
753 \]
754 
755 For the source Clifford actions, the continuum and centered Hilbert
756 generators are exactly these conjugates of their algebraic generators.
757 Their exact propagators therefore have the same coordinates after
758 applying \(T\). The algebraic-coordinate packet, transported coefficientwise by
759 \(T\), agrees with the Hilbert packet for the corresponding
760 amplitudes.
761 
762 \emph{Proof.} Induction gives \((TAT^{-1})^k=TA^kT^{-1}\). For
763 \(k=0\), use \(TT^{-1}=I\); for the induction step, cancel the adjacent
764 \(T^{-1}T\). Every exponential partial sum therefore satisfies
765 
766 \[
767 \sum_{k=0}^{N}\frac{z^k(A^{\mathcal H})^k}{k!}
768 =T\left(\sum_{k=0}^{N}\frac{z^kA^k}{k!}\right)T^{-1}.
769 \]
770 
771 The exponential series converges in both operator norms. Conjugation is
772 continuous: writing induced norms in the relevant spaces gives
773 
774 \[
775 \|TBT^{-1}\|_2\le\|T\|\,\|B\|_\infty\,\|T^{-1}\|
776 \le2\|B\|_\infty.
777 \]
778 
779 Pass to the limit to obtain the identity.
780 
781 The source's transported Clifford action is defined by
782 \(v\mapsto T(A(T^{-1}v))\). The same is true for the displayed diagonal
783 beta action. Conjugation preserves real coefficients and finite sums,
784 proving the Hamiltonian equality. Apply the exponential identity with
785 \(z=-it\). Substituting each coefficient equality into the finite
786 Fourier sum then proves equality of the packets. \(\square\)
787 
788 The norm bound also transports quadratic operator errors between
789 realizations. The bridge is a traditional proof in this paper. The
790 checked source contains inverse coordinate identities and the pairing
791 identification, but no complete generator-and-exponential conjugacy
792 theorem. PH09's conjugacy itself remains an ordinary proof in this text.
793 Blocks 44 and 45 separately establish Hilbert stability and consistency
794 directly.
795 
796 \hypertarget{an-exact-three-mode-packet}{%
797 \section{An exact three-mode packet}\label{an-exact-three-mode-packet}}
798 
799 \needspace{4\baselineskip}
800 \textbf{Example PH10 (axis and diagonal modes at \(h=1/12\)).} Set
801 \(m=1\), \(h=1/12\), and
802 
803 \[
804 S=\{n_0,n_A,n_D\},\qquad
805 n_0=(0,0,0),\quad n_A=(1,0,0),\quad n_D=(1,1,1).
806 \]
807 
808 Let \(e_0=(1,0,0,0)^T\) denote the first spinor basis vector, and choose
809 
810 \[
811 a_{n_0}=\frac1{\sqrt2}e_0,\qquad
812 a_{n_A}=\frac12e_0,\qquad
813 a_{n_D}=\frac12e_0.
814 \]
815 
816 Their squared norms are \(1/2,1/4,1/4\), so the packet is normalized.
817 Its continuum and centered momenta are
818 
819 \needspace{8\baselineskip}
820 \begin{center}
821 \begin{tabular}{@{}lll@{}}
822 \toprule
823 Mode & \(p(n)\) & \(q_h(n)\) \\
824 \midrule
825 
826 \(n_0\) & \((0,0,0)\) & \((0,0,0)\) \\
827 \(n_A\) & \((2\pi,0,0)\) & \((6,0,0)\) \\
828 \(n_D\) & \((2\pi,2\pi,2\pi)\) & \((6,6,6)\) \\
829 \bottomrule
830 \end{tabular}
831 \end{center}
832 
833 Each nonzero component has \(hp_j=\pi/6\), so
834 \(\sin(hp_j)/h=(1/2)/(1/12)=6\). Thus
835 
836 \needspace{8\baselineskip}
837 \begin{center}
838 \begin{tabular}{@{}lll@{}}
839 \toprule
840 Mode & \(E_n\) & \(E_{h,n}\) \\
841 \midrule
842 
843 \(n_0\) & \(1\) & \(1\) \\
844 \(n_A\) & \(\sqrt{1+4\pi^2}\) & \(\sqrt{37}\) \\
845 \(n_D\) & \(\sqrt{1+12\pi^2}\) & \(\sqrt{109}\) \\
846 \bottomrule
847 \end{tabular}
848 \end{center}
849 
850 These are exact symbolic equalities. The centered component \(6\) is
851 distinct from the integer mode label \(1\) and continuum momentum
852 \(2\pi\).
853 
854 For any real kinetic vector \(r\), the exact matrix gives
855 
856 \[
857 H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.
858 \]
859 
860 Set \(E_r=\sqrt{1+|r|^2}\), \(c_r(t)=\cos(E_rt)\), and
861 \(s_r(t)=\sin(E_rt)/E_r\). PH03 gives
862 
863 \[
864 u_r(t)=U_{H(1,r)}(t)e_0=
865 \begin{pmatrix}
866 c_r(t)-is_r(t)\\
867 0\\
868 -ir_zs_r(t)\\
869 (r_y-ir_x)s_r(t)
870 \end{pmatrix}.
871 \]
872 
873 In particular, \(u_0(t)=e^{-it}e_0\). For the centered axis and diagonal
874 modes,
875 
876 \[
877 u_A(t)=
878 \begin{pmatrix}
879 \cos(\sqrt{37}t)-i\sin(\sqrt{37}t)/\sqrt{37}\\
880 0\\
881 0\\
882 -6i\sin(\sqrt{37}t)/\sqrt{37}
883 \end{pmatrix},
884 \]
885 
886 \[
887 u_D(t)=
888 \begin{pmatrix}
889 \cos(\sqrt{109}t)-i\sin(\sqrt{109}t)/\sqrt{109}\\
890 0\\
891 -6i\sin(\sqrt{109}t)/\sqrt{109}\\
892 6(1-i)\sin(\sqrt{109}t)/\sqrt{109}
893 \end{pmatrix}.
894 \]
895 
896 The complete centered packet is
897 
898 \[
899 f_h(t,x)=\frac{e^{-it}}{\sqrt2}e_0
900 +\frac12e^{2\pi ix_1}u_A(t)
901 +\frac12e^{2\pi i(x_1+x_2+x_3)}u_D(t).
902 \]
903 
904 The continuum packet has the same form with \(u_A,u_D\) replaced by
905 \(u_{(2\pi,0,0)}\) and \(u_{(2\pi,2\pi,2\pi)}\). At \(t=0\), both equal
906 
907 \[
908 f_a(x)=\left(\frac1{\sqrt2}+\frac12e^{2\pi ix_1}
909 +\frac12e^{2\pi i(x_1+x_2+x_3)}\right)e_0.
910 \]
911 
912 The spinor formula verifies stability directly:
913 
914 \[
915 \|u_r(t)\|_2^2
916 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2)
917 =\cos^2(E_rt)+\sin^2(E_rt)=1.
918 \]
919 
920 Here \(|c-is|^2=c^2+s^2\) and \(|r_y-ir_x|^2=r_x^2+r_y^2\), since the
921 coefficients are real. Fourier orthogonality gives
922 
923 \[
924 \|f_h(t)\|_{L^2}^2=\tfrac12+\tfrac14+\tfrac14=1,\qquad
925 \|f(t)\|_{L^2}^2=1.
926 \]
927 
928 This is an integrated statement. The initial pointwise density is not
929 constant: at \(x=0\), it is \((1+1/\sqrt2)^2=3/2+\sqrt2\). Cross terms
930 can change local density while their torus integrals vanish.
931 Normalization does not mean that each point has density one.
932 
933 The error also has an exact expression. For fixed mode, let \(r=p(n)\),
934 \(q=q_h(n)\), with the corresponding \(c_r,s_r,c_q,s_q\). Coordinate
935 subtraction gives
936 
937 \[
938 \|u_q(t)-u_r(t)\|_2^2
939 =(c_q-c_r)^2+(s_q-s_r)^2
940 +\sum_{j=x,y,z}(q_js_q-r_js_r)^2.
941 \]
942 
943 For the selected packet, the zero mode contributes no error, so
944 
945 \[
946 \begin{aligned}
947 \|f_h(t)-f(t)\|_{L^2}^2
948 ={}&\frac14\|u_{(6,0,0)}(t)-u_{(2\pi,0,0)}(t)\|_2^2\\
949 &+\frac14\|u_{(6,6,6)}(t)-u_{(2\pi,2\pi,2\pi)}(t)\|_2^2.
950 \end{aligned}
951 \]
952 
953 A plot or numerical evaluation illustrates this exact expression;
954 conservation and normalization have already been proved symbolically.
955 
956 \needspace{10\baselineskip}
957 \atlasfigure{06_packet_evolution}{The centered three-mode packet at $m=1$, $h=1/12$ changes local density and spinor composition while keeping total squared norm one. The image evaluates the closed formula at finitely many times.}
958 
959 \hypertarget{a-finite-matrix-compact-time-extension}{%
960 \section{A finite-matrix compact-time
961 extension}\label{a-finite-matrix-compact-time-extension}}
962 
963 The following proof gives an explicit Duhamel bound, including its sharp coefficient. This is an ordinary mathematical deduction with a complete proof. The formal compact-time convergence argument uses exponential continuity and is identified separately in the verification appendix.
964 
965 \needspace{4\baselineskip}
966 \textbf{Lemma (finite Hermitian Duhamel estimate).} Let \(A,B\) be
967 Hermitian matrices on a nonzero finite-dimensional complex Hilbert
968 space. For every real \(t\),
969 
970 \[
971 \|e^{-itA}-e^{-itB}\|\le |t|\,\|A-B\|.
972 \]
973 
974 The same inequality holds trivially on the zero space.
975 
976 \emph{Proof.} First justify differentiation. On \(|s|\le R\), the
977 series for \(e^{-isB}\) converges uniformly, and its derivative series
978 is norm-dominated by
979 
980 \[
981 \sum_{k=1}^{\infty}\frac{kR^{k-1}\|B\|^k}{k!}
982 =\|B\|e^{R\|B\|}.
983 \]
984 
985 Termwise differentiation therefore gives
986 
987 \[
988 \frac{d}{ds}e^{-isB}=-iB e^{-isB}=-ie^{-isB}B.
989 \]
990 
991 The same holds for \(A\). Since \(A\) commutes with every power of
992 itself, it commutes with \(e^{-iuA}\). Fix \(t\) and define
993 \(F(s)=e^{-i(t-s)A}e^{-isB}\). The product rule gives
994 
995 \[
996 F'(s)=i e^{-i(t-s)A}(A-B)e^{-isB}.
997 \]
998 
999 The endpoints are \(F(0)=e^{-itA}\) and \(F(t)=e^{-itB}\). Integrating
1000 entrywise in the finite matrix space gives the oriented-integral
1001 identity
1002 
1003 \[
1004 e^{-itB}-e^{-itA}
1005 =i\int_0^t e^{-i(t-s)A}(A-B)e^{-isB}\,ds.
1006 \]
1007 
1008 For Hermitian matrices the two exponentials in the integrand are
1009 unitary. This does not rely on the special Clifford algebra: the series
1010 gives \((e^{-iuA})^*=e^{iuA}\), and the Cauchy product of the two
1011 absolutely convergent series gives \(e^{iuA}e^{-iuA}=I\). Indeed, the
1012 coefficient of \(A^k\) for \(k>0\) is
1013 \((iu)^k\sum_{r=0}^{k}\binom{k}{r}(-1)^r/k!=0\), and the constant
1014 coefficient is \(I\). A unitary matrix on a nonzero space has norm one.
1015 Submultiplicativity therefore bounds the integrand norm by \(\|A-B\|\).
1016 For \(t\ge0\), integration gives \(t\|A-B\|\); for \(t<0\), reversing
1017 the interval gives \((-t)\|A-B\|\). This proves the estimate for every
1018 real time. \(\square\)
1019 
1020 \needspace{4\baselineskip}
1021 \textbf{Theorem PH11 (time-explicit finite-packet bound).} For every
1022 real \(m,h,t\), fixed mode \(n\), and fixed finite packet,
1023 
1024 \[
1025 \|U_{H_{h,n}}(t)-U_{H_n}(t)\|
1026 \le |t|\,|\delta(h,n)|\le |t|h^2C(n),
1027 \]
1028 
1029 \[
1030 \begin{aligned}
1031 \|f_h(t)-f(t)\|_{L^2}
1032 &\le |t|\left(\sum_{n\in S}|\delta(h,n)|^2\|a_n\|_2^2\right)^{1/2}\\
1033 &\le |t|h^2\left(\sum_{n\in S}C(n)^2\|a_n\|_2^2\right)^{1/2}.
1034 \end{aligned}
1035 \]
1036 
1037 Consequently, for every \(T\ge0\),
1038 
1039 \[
1040 \sup_{|t|\le T}\|f_h(t)-f(t)\|_{L^2}
1041 \le Th^2\left(\sum_{n\in S}C(n)^2\|a_n\|_2^2\right)^{1/2},
1042 \]
1043 
1044 and evolution converges uniformly on every compact time interval for
1045 fixed support and initial amplitudes.
1046 
1047 \emph{Proof.} Apply the Duhamel lemma with \(A=H_{h,n}\), \(B=H_n\).
1048 Both are Hermitian by PH02, and PH07 gives the residual norm and bound.
1049 For each amplitude,
1050 
1051 \[
1052 \|(U_{H_{h,n}}(t)-U_{H_n}(t))a_n\|_2
1053 \le |t|\,|\delta(h,n)|\,\|a_n\|_2
1054 \le |t|h^2C(n)\|a_n\|_2.
1055 \]
1056 
1057 Square and sum over \(S\), apply the exact residual Parseval identity of
1058 PH08, and take the nonnegative square root. The middle coefficient is
1059 exactly the initial generator-residual norm in PH07 because these
1060 Clifford differences satisfy
1061 \(\Delta_{h,n}^*\Delta_{h,n}=|\delta(h,n)|^2I_4\). This identification
1062 is specific to the displayed symbol differences; the general Duhamel
1063 lemma is an operator-norm estimate. Replacing \(|t|\) by \(T\) gives the
1064 compact-time bound. Its right side tends to zero as \(h\to0\).
1065 \(\square\)
1066 
1067 For PH10, \(C(n_0)=0\), \(C(n_A)=(2\pi)^3/6\), and
1068 \(C(n_D)=\sqrt3(2\pi)^3/6\), so
1069 
1070 \[
1071 \left(\sum_{n\in S}C(n)^2\|a_n\|_2^2\right)^{1/2}
1072 =\frac{(2\pi)^3}{6}=\frac{4\pi^3}{3}.
1073 \]
1074 
1075 At its spacing, put \(d=2\pi-6>0\). The axis residual has magnitude
1076 \(d\), the diagonal residual has magnitude \(\sqrt3d\), and their
1077 squared amplitude weights are \(1/4,1/4\). Hence the exact middle
1078 coefficient is
1079 
1080 \[
1081 \left(\tfrac14d^2+\tfrac14\,3d^2\right)^{1/2}=d,
1082 \]
1083 
1084 and the worked bound is
1085 
1086 \[
1087 \|f_{1/12}(t)-f(t)\|_{L^2}
1088 \le(2\pi-6)|t|\le\frac{\pi^3}{108}|t|.
1089 \]
1090 
1091 The triangle inequality also gives \(\|f_h(t)-f(t)\|_{L^2}\le2\) for
1092 this normalized packet, so one may use the smaller bound. Neither says
1093 that a fixed nonzero spacing has small error for arbitrarily large time.
1094 Nor does the time bound remove the cutoff: its coefficient contains high
1095 powers of mode momenta. For arbitrary \(L^2\) data, the Fourier-tail argument in the following part
1096 provides convergence without this uniform quadratic coefficient.
1097 
1098 \hypertarget{probability-hamiltonian-expectation-and-physical-scope}{%
1099 \section{Probability, Hamiltonian expectation, and physical
1100 scope}\label{probability-hamiltonian-expectation-and-physical-scope}}
1101 
1102 Let \(H_S,H_{h,S}\) denote the finite block diagonal Hamiltonians. In
1103 coefficient coordinates, squared norm and continuum Hamiltonian
1104 expectation are
1105 
1106 \[
1107 N(a)=\sum_{n\in S}\|a_n\|_2^2,\qquad
1108 \mathcal E(a)=\sum_{n\in S}\langle a_n,H_na_n\rangle,
1109 \]
1110 
1111 with centered expectation \(\mathcal E_h(a)\) obtained by replacing
1112 \(H_n\) with \(H_{h,n}\). Hermiticity makes each expectation real: its
1113 complex conjugate is
1114 \(\langle H_na_n,a_n\rangle=\langle a_n,H_na_n\rangle\).
1115 
1116 \needspace{4\baselineskip}
1117 \textbf{Proposition PH12 (separate conservation of Hamiltonian
1118 expectation).} Each exact packet evolution preserves the expectation of
1119 its own time-independent Hamiltonian:
1120 
1121 \[
1122 \mathcal E(U_S(t)a)=\mathcal E(a),\qquad
1123 \mathcal E_h(U_{h,S}(t)a)=\mathcal E_h(a).
1124 \]
1125 
1126 Here the packet operators act on coefficients through the isometry
1127 \(\mathcal K_S\cong\mathcal F_S\). Squared-norm conservation does not
1128 identify either expectation with \(N(a)\). At common initial data,
1129 
1130 \[
1131 |\mathcal E_h(a)-\mathcal E(a)|
1132 \le h^2\sum_{n\in S}C(n)\|a_n\|_2^2.
1133 \]
1134 
1135 \emph{Proof.} The exponential of a fixed matrix commutes with that
1136 matrix, term by term in its power series. Unitarity gives on each mode
1137 
1138 \[
1139 \langle U_H(t)a,H U_H(t)a\rangle
1140 =\langle a,U_H(t)^*HU_H(t)a\rangle
1141 =\langle a,Ha\rangle.
1142 \]
1143 
1144 Sum to prove both conservation laws. For common initial coefficients,
1145 Cauchy--Schwarz and PH07 give
1146 
1147 \[
1148 |\langle a_n,(H_{h,n}-H_n)a_n\rangle|
1149 \le\|a_n\|_2\|\Delta_{h,n}a_n\|_2
1150 \le h^2C(n)\|a_n\|_2^2.
1151 \]
1152 
1153 The finite-sum triangle inequality proves the comparison. \(\square\)
1154 
1155 A unit vector in the negative-energy range of \(P_-\) has squared norm
1156 one but expectation \(-E\); a unit vector in the positive-energy range
1157 has expectation \(E\). Norm and energy are therefore different even for
1158 one mode. For \(e_0\), the matrix gives
1159 \(\langle e_0,H(m,p)e_0\rangle=m\), since the kinetic terms have zero
1160 diagonal entry in that component. PH10 has expectation \(1\) for both
1161 Hamiltonians. Its values \(\sqrt{37},\sqrt{109}\) are shell magnitudes,
1162 and its spinors contain both energy signs. Their expectations need not
1163 equal a positive shell value.
1164 
1165 The analysis is free and linear. No gauge field, forcing, interaction,
1166 or time-dependent Hamiltonian occurs in these symbols. Other project
1167 modules treat different transport structures; importing them does not
1168 convert these statements into theorems about interacting fields. The
1169 centered propagator is the exact exponential of a modified spatial
1170 symbol. A numerical approximation to that exponential must be checked separately
1171 for its own norm error.
1172 

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