Stratified Relational Workspace Theory

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1 % This file is part of the modular source for Stratified Relational Workspace Theory.
2 % !TeX root = ../Stratified_Relational_Workspace_Theory.tex
3 \theorypart{IV}{Approximate Workspaces}
4 
5 The exact theory asks whether local core states agree perfectly under every
6 transfer. Approximate theory asks three more realistic questions:
7 \begin{enumerate}
8 \item how strongly a candidate state violates edgewise compatibility;
9 \item whether a low-defect subspace is spectrally separated from incoherent
10  directions; and
11 \item how that subspace changes when transfers, metrics, or operations are
12  perturbed.
13 \end{enumerate}
14 Throughout Parts~IV--V, set
15 \[
16 \mathcal C_V=\bigoplus_{i\in V}C_i,
17 \qquad
18 \mathcal C_E=\bigoplus_{e\in E}C_{t(e)},
19 \]
20 with their Hilbert direct-sum inner products, and retain strictly positive
21 weights \(w_e\).
22 
23 \topic{11}{Transfer Discrepancy Operator}
24 
25 \begin{definition}[Transfer discrepancy]\label{def:transfer-discrepancy}
26 The transfer discrepancy operator is
27 \[
28 D_H:\mathcal C_V\longrightarrow\mathcal C_E,
29 \qquad
30 (D_Hc)_e
31 =
32 \sqrt{w_e}\bigl(c_{t(e)}-H_ec_{s(e)}\bigr).
33 \]
34 Its squared norm is the coherence energy
35 \[
36 \boxed{
37 \mathcal E_H(c)
38 =
39 \|D_Hc\|^2
40 =
41 \sum_{e\in E}w_e
42 \left\|c_{t(e)}-H_ec_{s(e)}\right\|^2.
43 }
44 \]
45 The transfer Laplacian is
46 \[
47 \boxed{L_H=D_H^*D_H.}
48 \]
49 \end{definition}
50 
51 The operator \(D_H\) is an incidence-type derivative twisted by the edge
52 transfers. It compares a state already present at the target with the state
53 predicted there by translation from the source. A direction can therefore
54 have low energy even when its coordinates differ substantially across nodes,
55 provided those coordinates differ exactly as the transfer maps prescribe.
56 
57 For \(z=(z_e)_{e\in E}\in\mathcal C_E\), the adjoint is
58 \[
59 (D_H^*z)_i
60 =
61 \sum_{e:t(e)=i}\sqrt{w_e}\,z_e
62 -
63 \sum_{e:s(e)=i}\sqrt{w_e}\,H_e^*z_e.
64 \]
65 Consequently,
66 \[
67 \boxed{
68 \begin{aligned}
69 (L_Hc)_i
70 ={}&
71 \sum_{e:t(e)=i}w_e\bigl(c_i-H_ec_{s(e)}\bigr)\\
72 &+
73 \sum_{e:s(e)=i}w_eH_e^*
74 \bigl(H_ec_i-c_{t(e)}\bigr).
75 \end{aligned}
76 }
77 \]
78 This formula makes the two contributions at a vertex explicit: incoming
79 prediction error and adjoint-propagated outgoing error.
80 
81 \begin{proposition}[Positivity and variational identity]\label{prop:laplacian-positive}
82 The transfer Laplacian is self-adjoint and positive semidefinite. For every
83 \(c\in\mathcal C_V\),
84 \[
85 \langle c,L_Hc\rangle=\mathcal E_H(c)\geq0.
86 \]
87 \end{proposition}
88 
89 \begin{proof}
90 Self-adjointness and positivity follow from \(L_H=D_H^*D_H\). The displayed
91 identity is
92 \[
93 \langle c,D_H^*D_Hc\rangle
94 =
95 \langle D_Hc,D_Hc\rangle.
96 \]
97 \end{proof}
98 
99 \begin{remark}[Weight dependence]
100 Replacing \(w_e\) by another positive family does not alter \(\ker D_H\), but
101 it generally changes every positive eigenvalue and therefore changes the
102 approximate geometry. Edge weights are part of the metric model, not part of
103 the exact compatibility relation.
104 \end{remark}
105 
106 \topic{12}{Exact Kernel Theorem}
107 
108 \begin{theorem}[Transfer-Laplacian kernel theorem]\label{thm:laplacian-kernel}
109 For an arbitrary finite directed core diagram, without assuming invertibility,
110 \[
111 \boxed{
112 \ker L_H=\ker D_H=\Gamma(H).
113 }
114 \]
115 Thus the exact compatible sections are precisely the zero-energy modes of the
116 transfer Laplacian.
117 \end{theorem}
118 
119 \begin{proof}
120 If \(D_Hc=0\), then \(L_Hc=D_H^*D_Hc=0\). Conversely, if \(L_Hc=0\), then
121 \[
122 0=\langle c,L_Hc\rangle=\|D_Hc\|^2,
123 \]
124 so \(D_Hc=0\). Finally, \(D_Hc=0\) if and only if
125 \[
126 c_{t(e)}=H_ec_{s(e)}
127 \]
128 for every edge, which is exactly the definition of \(\Gamma(H)\).
129 \end{proof}
130 
131 \begin{corollary}[Finite rank computation]\label{cor:workspace-rank-D}
132 Let
133 \[
134 n_C=\sum_{i\in V}\dim C_i.
135 \]
136 Then
137 \[
138 \boxed{
139 \dim\Gamma(H)=n_C-\operatorname{rank}(D_H).
140 }
141 \]
142 Hence the exact workspace dimension can be obtained by a single rank or
143 singular-value computation, even when holonomy is unavailable because some
144 transfers are noninvertible.
145 \end{corollary}
146 
147 \subtopic{12.1 Relation to the holonomy computation}
148 
149 In the local-system regime, Theorems~\ref{thm:limit-holonomy} and
150 \ref{thm:laplacian-kernel} give two equivalent constructions:
151 \[
152 \Gamma(H)
153 \cong
154 C_o^{\Hol_o}
155 \qquad\text{and}\qquad
156 \Gamma(H)=\ker L_H.
157 \]
158 The first reduces the problem to cycle invariants in a root fiber. The second
159 works directly on all vertex coordinates and extends to noninvertible diagrams.
160 The two computations therefore serve different practical regimes rather than
161 competing definitions.
162 
163 \topic{13}{Approximate Structural Workspace}
164 
165 Let
166 \[
167 0\leq\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_N,
168 \qquad
169 N=\dim\mathcal C_V,
170 \]
171 be the eigenvalues of \(L_H\), repeated with multiplicity, and let
172 \(P_I(L_H)\) denote the spectral projector associated with an interval \(I\).
173 
174 \begin{definition}[Threshold workspace]\label{def:threshold-workspace}
175 For \(\varepsilon\geq0\), define
176 \[
177 \boxed{
178 \Gamma_\varepsilon(H)
179 =
180 \ran P_{[0,\varepsilon]}(L_H).
181 }
182 \]
183 Its dimension is
184 \[
185 d_\varepsilon
186 =
187 \#\{k:\lambda_k\leq\varepsilon\}.
188 \]
189 When \(0<d_\varepsilon<N\), define the boundary gap
190 \[
191 \boxed{
192 g_\varepsilon
193 =
194 \lambda_{d_\varepsilon+1}-\lambda_{d_\varepsilon}.
195 }
196 \]
197 \end{definition}
198 
199 The threshold \(\varepsilon\) declares which discrepancy scale is treated as
200 structurally negligible. The dimension \(d_\varepsilon\) alone is not a
201 robust certificate: a cluster is stable only when it is separated by a
202 positive boundary gap.
203 
204 An equivalent index-based description fixes \(d\) and writes
205 \[
206 W_d=\ran P_d,
207 \qquad
208 P_d=P_{\{\lambda_1,\ldots,\lambda_d\}}(L_H),
209 \]
210 with cluster gap
211 \[
212 g_d=\lambda_{d+1}-\lambda_d>0.
213 \]
214 The index-based form is convenient for perturbation theorems because the
215 workspace rank is declared explicitly.
216 
217 \begin{proposition}[Min--max characterization]\label{prop:minmax}
218 For \(1\leq k\leq N\),
219 \[
220 \lambda_k
221 =
222 \min_{\substack{S\subseteq\mathcal C_V\\\dim S=k}}
223 \ \max_{\substack{c\in S\\\|c\|=1}}
224 \mathcal E_H(c).
225 \]
226 Consequently, the span of the first \(k\) eigenvectors minimizes the worst-case
227 coherence energy among all \(k\)-dimensional candidate subspaces.
228 \end{proposition}
229 
230 \begin{proof}
231 Expand every vector in an orthonormal eigenbasis of \(L_H\). The span of the
232 first \(k\) eigenvectors has maximal Rayleigh quotient \(\lambda_k\). Every
233 \(k\)-dimensional subspace intersects the orthogonal complement of the first
234 \(k-1\) eigenvectors nontrivially, and any unit vector in that intersection has
235 Rayleigh quotient at least \(\lambda_k\).
236 \end{proof}
237 
238 \subtopic{13.1 Exact, soft, and spurious low modes}
239 
240 The spectrum supports three distinct interpretations:
241 \[
242 \begin{array}{rcl}
243 \lambda_k=0
244 &\Longleftrightarrow&
245 \text{exact compatible direction},\\
246 0<\lambda_k\ll g_d
247 &\Longrightarrow&
248 \text{approximately transported direction},\\
249 \lambda_k\approx\lambda_{k+1}
250 &\Longrightarrow&
251 \text{rank selection is unstable at that boundary}.
252 \end{array}
253 \]
254 A low eigenvalue can still be scientifically spurious if it arises from a
255 poorly chosen weight, a nearly disconnected graph, or a noncausal encoding.
256 The causal and null gates in later parts remain necessary.
257 
258 \topic{14}{Normalized Coherence Defect}
259 
260 For \(c\neq0\), define the Rayleigh defect
261 \[
262 \delta_{\mathrm{coh}}(c)
263 =
264 \frac{\mathcal E_H(c)}{\|c\|^2}.
265 \]
266 For a nonzero subspace \(W\subseteq\mathcal C_V\), define
267 \[
268 \delta_{\mathrm{coh}}(W)
269 =
270 \sup_{\substack{c\in W\\\|c\|=1}}
271 \mathcal E_H(c).
272 \]
273 If \(W\) is \(L_H\)-invariant, then
274 \[
275 \delta_{\mathrm{coh}}(W)
276 =
277 \|L_H|_W\|.
278 \]
279 In particular,
280 \[
281 \delta_{\mathrm{coh}}(\Gamma_\varepsilon(H))
282 \leq\varepsilon.
283 \]
284 
285 \begin{theorem}[Coherence-to-core distance estimate]\label{thm:poincare-workspace}
286 Let
287 \[
288 W_0=\Gamma(H)=\ker L_H,
289 \qquad
290 r=\dim W_0,
291 \]
292 and suppose \(r<N\). Set
293 \[
294 \gamma_H=\lambda_{r+1}>0.
295 \]
296 Then every \(c\in\mathcal C_V\) satisfies
297 \[
298 \boxed{
299 \operatorname{dist}(c,W_0)^2
300 \leq
301 \frac{\mathcal E_H(c)}{\gamma_H}.
302 }
303 \]
304 Moreover,
305 \[
306 \mathcal E_H(c)
307 \leq
308 \|L_H\|\operatorname{dist}(c,W_0)^2.
309 \]
310 \end{theorem}
311 
312 \begin{proof}
313 Write \(c=P_0c+Q_0c\), where \(P_0\) projects onto \(W_0\) and
314 \(Q_0=I-P_0\). Since \(L_HP_0=0\),
315 \[
316 \mathcal E_H(c)
317 =
318 \langle Q_0c,L_HQ_0c\rangle.
319 \]
320 The spectrum of \(L_H\) on \(W_0^\perp\) lies in
321 \([\gamma_H,\|L_H\|]\). Hence
322 \[
323 \gamma_H\|Q_0c\|^2
324 \leq
325 \mathcal E_H(c)
326 \leq
327 \|L_H\|\|Q_0c\|^2.
328 \]
329 Finally, \(\|Q_0c\|=\operatorname{dist}(c,W_0)\).
330 \end{proof}
331 
332 This estimate is the exact sense in which a spectral gap turns small edgewise
333 disagreement into closeness to a genuinely compatible global section.
334 
335 \subtopic{14.1 Two-node consistency cycle}
336 
337 Let \(C_1=C_2=\mathbb R\), use unit weights, and take two directed transfers
338 \[
339 H_{12}=a,
340 \qquad
341 H_{21}=b.
342 \]
343 Then
344 \[
345 D_H
346 =
347 \begin{pmatrix}
348 -a&1\\
349 1&-b
350 \end{pmatrix},
351 \qquad
352 L_H
353 =
354 \begin{pmatrix}
355 a^2+1&-(a+b)\\
356 -(a+b)&b^2+1
357 \end{pmatrix}.
358 \]
359 Writing
360 \[
361 \tau=a^2+b^2+2,
362 \]
363 the eigenvalues are
364 \[
365 \lambda_\pm
366 =
367 \frac{1}{2}
368 \left(
369 \tau\pm\sqrt{\tau^2-4(ab-1)^2}
370 \right),
371 \]
372 and
373 \[
374 \det L_H=(ab-1)^2.
375 \]
376 Thus a nonzero exact shared core exists precisely when
377 \[
378 ab=1,
379 \]
380 which is exactly the trivial-holonomy condition. Away from exact consistency,
381 \[
382 \lambda_-\lambda_+=(ab-1)^2.
383 \]
384 When \(|ab-1|\) is small while \(\lambda_+\) stays bounded away from zero, the
385 soft workspace eigenvalue is quadratic in the cycle mismatch:
386 \[
387 \lambda_-
388 =
389 \frac{(ab-1)^2}{\lambda_+}.
390 \]
391 This example connects approximate spectral coherence directly to approximate
392 holonomy closure.
393 
394 \topic{15}{Perturbation Stability Theorem}
395 
396 The central robustness question is whether a selected low-energy subspace
397 persists when the estimated transfer maps change. The next lemma isolates the
398 finite-dimensional spectral fact needed for the answer.
399 
400 \begin{lemma}[Separated spectral subspaces]\label{lem:separated-subspaces}
401 Let \(A\) and \(B=A+E\) be self-adjoint operators. Let \(P\) be an
402 \(A\)-spectral projector whose eigenvalues lie in \(( -\infty,a]\), and let
403 \(\widetilde Q\) be a \(B\)-spectral projector whose eigenvalues lie in
404 \([b,\infty)\), with \(b>a\). Then
405 \[
406 \|P\widetilde Q\|
407 \leq
408 \frac{\|E\|}{b-a}.
409 \]
410 \end{lemma}
411 
412 \begin{proof}
413 On \(\ran P\) and \(\ran\widetilde Q\), set
414 \[
415 A_P=PAP|_{\ran P},
416 \qquad
417 B_Q=\widetilde QB\widetilde Q|_{\ran\widetilde Q}.
418 \]
419 The operator \(X=P\widetilde Q\) satisfies the Sylvester equation
420 \[
421 A_PX-XB_Q=-PE\widetilde Q.
422 \]
423 Let \(\mu=(a+b)/2\). Since
424 \[
425 \|e^{t(A_P-\mu I)}\|\leq e^{-(b-a)t/2},
426 \qquad
427 \|e^{-t(B_Q-\mu I)}\|\leq e^{-(b-a)t/2},
428 \]
429 the inverse Sylvester operator has the convergent integral representation
430 \[
431 X
432 =
433 \int_0^\infty
434  e^{t(A_P-\mu I)}
435  (PE\widetilde Q)
436  e^{-t(B_Q-\mu I)}\,\dd t.
437 \]
438 Taking norms and integrating gives the result.
439 \end{proof}
440 
441 \begin{theorem}[Spectral workspace stability]\label{thm:spectral-stability}
442 Let \(L\) and \(\widetilde L=L+E\) be self-adjoint positive semidefinite
443 operators on the same \(N\)-dimensional space. Suppose
444 \[
445 \lambda_d(L)<\lambda_{d+1}(L),
446 \qquad
447 g_d=\lambda_{d+1}(L)-\lambda_d(L)>0,
448 \]
449 and
450 \[
451 \eta=\|E\|<\frac{g_d}{2}.
452 \]
453 Let \(P_d\) and \(\widetilde P_d\) project onto the first \(d\) eigenvectors
454 of \(L\) and \(\widetilde L\), respectively. Then the rank-\(d\) cluster
455 remains separated and
456 \[
457 \boxed{
458 \|P_d-\widetilde P_d\|
459 \leq
460 \frac{\eta}{g_d-\eta}
461 \leq
462 \frac{2\eta}{g_d}.
463 }
464 \]
465 \end{theorem}
466 
467 \begin{proof}
468 The min--max characterization gives the eigenvalue perturbation bound
469 \[
470 |\lambda_k(\widetilde L)-\lambda_k(L)|\leq\eta.
471 \]
472 Hence
473 \[
474 \lambda_{d+1}(\widetilde L)-\lambda_d(L)
475 \geq g_d-\eta
476 \]
477 and
478 \[
479 \lambda_{d+1}(L)-\lambda_d(\widetilde L)
480 \geq g_d-\eta.
481 \]
482 Apply Lemma~\ref{lem:separated-subspaces} first to
483 \(P_d(I-\widetilde P_d)\) and then, with the roles of \(L\) and
484 \(\widetilde L\) exchanged, to \((I-P_d)\widetilde P_d\). Orthogonal
485 projections of equal rank satisfy
486 \[
487 \|P_d-\widetilde P_d\|
488 =
489 \max\left\{
490 \|P_d(I-\widetilde P_d)\|,
491 \|(I-P_d)\widetilde P_d\|
492 \right\}.
493 \]
494 This yields the first bound. The second follows from \(\eta<g_d/2\).
495 \end{proof}
496 
497 \subtopic{15.1 From transfer-map error to Laplacian error}
498 
499 Suppose the edge maps are perturbed to
500 \[
501 \widetilde H_e=H_e+\Delta_e
502 \]
503 while the weights and vertex metrics remain fixed. Define
504 \[
505 \boxed{
506 \delta_H^2
507 =
508 \max_{i\in V}
509 \sum_{e:s(e)=i}w_e\|\Delta_e\|^2.
510 }
511 \]
512 Let \(F=D_{\widetilde H}-D_H\). Then
513 \[
514 (Fc)_e=-\sqrt{w_e}\,\Delta_ec_{s(e)},
515 \qquad
516 \|F\|\leq\delta_H.
517 \]
518 Since
519 \[
520 L_{\widetilde H}-L_H
521 =D_H^*F+F^*D_H+F^*F,
522 \]
523 we obtain
524 \[
525 \boxed{
526 \|L_{\widetilde H}-L_H\|
527 \leq
528 2\|D_H\|\delta_H+\delta_H^2.
529 }
530 \]
531 
532 \begin{corollary}[Edgewise stability certificate]\label{cor:edge-stability}
533 If
534 \[
535 \eta_H=2\|D_H\|\delta_H+\delta_H^2
536 <\frac{g_d}{2},
537 \]
538 then
539 \[
540 \boxed{
541 \|P_d(H)-P_d(\widetilde H)\|
542 \leq
543 \frac{\eta_H}{g_d-\eta_H}.
544 }
545 \]
546 \end{corollary}
547 
548 This converts declared edge-map uncertainty into an explicit bound on the
549 motion of the extracted workspace subspace.
550 
551 \subtopic{15.2 Approximate operation closure}
552 
553 Define the vertex and edge direct-sum representations
554 \[
555 \rho_V(a)=\bigoplus_{i\in V}\rho_i(a),
556 \qquad
557 \rho_E(a)=\bigoplus_{e\in E}\rho_{t(e)}(a).
558 \]
559 Lift the edgewise intertwining defects to
560 \[
561 (\mathfrak X_a c)_e
562 =
563 \sqrt{w_e}\,\Xi_e(a)c_{s(e)}.
564 \]
565 Then the exact commutator identity is
566 \[
567 \boxed{
568 D_H\rho_V(a)-\rho_E(a)D_H=-\mathfrak X_a.
569 }
570 \]
571 Furthermore,
572 \[
573 \|\mathfrak X_a\|
574 \leq
575 \beta(a),
576 \qquad
577 \beta(a)^2
578 =
579 \max_{i\in V}
580 \sum_{e:s(e)=i}w_e\|\Xi_e(a)\|^2.
581 \]
582 
583 \begin{theorem}[Approximate operation-closure theorem]\label{thm:approx-operation-closure}
584 Let \(P_d\) project onto the first \(d\) eigenvectors of \(L_H\), set
585 \[
586 \lambda_d\leq\varepsilon,
587 \qquad
588 \mu=\lambda_{d+1}>0,
589 \]
590 and let \(Q_d=I-P_d\). Then every \(a\in\mathcal A\) satisfies
591 \[
592 \boxed{
593 \|Q_d\rho_V(a)P_d\|
594 \leq
595 \frac{
596 \|\rho_E(a)\|\sqrt{\varepsilon}+\beta(a)
597 }{\sqrt{\mu}}.
598 }
599 \]
600 \end{theorem}
601 
602 \begin{proof}
603 For \(x\in\ran P_d\),
604 \[
605 \|D_Hx\|^2
606 =\langle x,L_Hx\rangle
607 \leq\varepsilon\|x\|^2.
608 \]
609 The commutator identity gives
610 \[
611 \|D_H\rho_V(a)x\|
612 \leq
613 \|\rho_E(a)\|\sqrt{\varepsilon}\|x\|
614 +
615 \beta(a)\|x\|.
616 \]
617 Because \(P_d\) and \(Q_d\) reduce \(L_H\), the two energy components are
618 orthogonal and
619 \[
620 \|D_HQ_dy\|\leq\|D_Hy\|.
621 \]
622 On \(\ran Q_d\), the spectrum of \(L_H\) is bounded below by \(\mu\), so
623 \[
624 \sqrt{\mu}\|Q_dy\|
625 \leq
626 \|D_HQ_dy\|.
627 \]
628 Apply these inequalities to \(y=\rho_V(a)x\), then take the supremum over unit
629 \(x\in\ran P_d\).
630 \end{proof}
631 
632 \begin{corollary}[Recovery of exact operation closure]
633 If \(d=\dim\ker L_H\), \(\varepsilon=0\), and every edge intertwines \(a\)
634 exactly, then
635 \[
636 Q_d\rho_V(a)P_d=0.
637 \]
638 Thus Theorem~\ref{thm:approx-operation-closure} reduces to the exact
639 operation-closure result of Part~III.
640 \end{corollary}
641 
642 \begin{corollary}[Closure under projector drift]\label{cor:closure-drift}
643 Let \(\widetilde P_d\) be another rank-\(d\) orthogonal projector and set
644 \[
645 \delta_P=\|P_d-\widetilde P_d\|.
646 \]
647 Then
648 \[
649 \boxed{
650 \|(I-\widetilde P_d)\rho_V(a)\widetilde P_d\|
651 \leq
652 \|Q_d\rho_V(a)P_d\|
653 +2\|\rho_V(a)\|\delta_P.
654 }
655 \]
656 Combining this estimate with Theorem~\ref{thm:spectral-stability} controls
657 operation leakage after noisy workspace extraction.
658 \end{corollary}
659 
660 \begin{proof}
661 Subtract \(Q_d\rho_V(a)P_d\) and use
662 \[
663 I-\widetilde P_d-Q_d=P_d-\widetilde P_d.
664 \]
665 The two resulting perturbation terms are each bounded by
666 \(\|\rho_V(a)\|\delta_P\).
667 \end{proof}
668 
669 \subtopic{15.3 Core--residual path accumulation}
670 
671 Let a path have vertices
672 \[
673 i_0\xrightarrow{e_1}i_1\xrightarrow{e_2}\cdots
674 \xrightarrow{e_m}i_m.
675 \]
676 Write
677 \[
678 T_r=T_{e_r},
679 \quad
680 H_r=H_{e_r},
681 \quad
682 V_r=V_{e_r},
683 \]
684 and let
685 \[
686 \iota_i^C:C_i\hookrightarrow X_i,
687 \qquad
688 \iota_i^R:R_i\hookrightarrow X_i
689 \]
690 be the canonical inclusions. Empty products below are identities.
691 
692 \begin{theorem}[Pathwise core--residual Duhamel formula]\label{thm:residual-duhamel}
693 The deviation of full path transport from pure core transport is exactly
694 \[
695 \boxed{
696 \begin{aligned}
697 &T_m\cdots T_1\iota_{i_0}^C
698 -
699 \iota_{i_m}^C H_m\cdots H_1\\
700 &\qquad=
701 \sum_{r=1}^{m}
702 T_m\cdots T_{r+1}
703 \iota_{i_r}^R V_r
704 H_{r-1}\cdots H_1.
705 \end{aligned}
706 }
707 \]
708 Consequently,
709 \[
710 \boxed{
711 \begin{aligned}
712 \|Q_{i_m}T_m\cdots T_1P_{i_0}\|
713 \leq
714 \sum_{r=1}^{m}
715 &\left(\prod_{\ell=r+1}^{m}\|T_\ell\|\right)
716 \|V_r\|\\
717 &\times
718 \left(\prod_{\ell=1}^{r-1}\|H_\ell\|\right),
719 \end{aligned}
720 }
721 \]
722 and the same right-hand side bounds
723 \[
724 \|P_{i_m}T_m\cdots T_1P_{i_0}-H_m\cdots H_1\|.
725 \]
726 \end{theorem}
727 
728 \begin{proof}
729 For every step,
730 \[
731 T_r\iota_{i_{r-1}}^C
732 =
733 \iota_{i_r}^CH_r+
734 \iota_{i_r}^RV_r.
735 \]
736 Telescope the sequence
737 \[
738 T_m\cdots T_{r+1}\iota_{i_r}^C H_r\cdots H_1,
739 \qquad
740 r=0,1,\ldots,m.
741 \]
742 The difference between consecutive terms is exactly the \(r\)-th summand in
743 the displayed formula. Projection onto \(R_{i_m}\) or \(C_{i_m}\), followed
744 by submultiplicativity of the operator norm, gives the two bounds.
745 \end{proof}
746 
747 \begin{corollary}[Uniform path bound]\label{cor:uniform-path-bound}
748 If along the path
749 \[
750 \|T_r\|\leq\tau,
751 \qquad
752 \|H_r\|\leq h,
753 \qquad
754 \|V_r\|\leq\nu,
755 \]
756 then the leakage and core-transport defects are bounded by
757 \[
758 \boxed{
759 \nu\sum_{r=1}^{m}\tau^{m-r}h^{r-1}
760 =
761 \begin{cases}
762 \displaystyle
763 \nu\frac{\tau^m-h^m}{\tau-h},&\tau\neq h,\\[1.1em]
764 \displaystyle
765 m\nu\tau^{m-1},&\tau=h.
766 \end{cases}
767 }
768 \]
769 In particular, if \(\tau\leq1\) and \(h\leq1\), the bound is at most
770 \(m\nu\).
771 \end{corollary}
772 
773 The linear factor \(m\) in the merely nonexpansive regime is important. Small
774 single-edge leakage does not by itself justify arbitrarily long transfer
775 chains. Long-horizon stability requires contraction, cancellation, error
776 correction, or leakage that decreases with path length.
777 
778 \subtopic{15.4 Residual-mediated reentry}
779 
780 For two consecutive transfers,
781 \[
782 P_{i_2}T_2T_1P_{i_0}
783 =
784 H_2H_1+U_2V_1,
785 \]
786 \[
787 Q_{i_2}T_2T_1P_{i_0}
788 =
789 V_2H_1+K_2V_1.
790 \]
791 Thus leakage out of the core is first order in the blocks \(V_r\), while an
792 excursion that leaves and then returns to the core contains at least one
793 \(V\)-block and one later \(U\)-block. For a fixed path length, if
794 \[
795 \|U_r\|,\|V_r\|=O(\delta)
796 \]
797 and the diagonal blocks remain bounded, the residual-mediated correction to
798 pure core-to-core transport is \(O(\delta^2)\), whereas terminal residual
799 leakage can be \(O(\delta)\). This distinction separates loss of core
800 invariance from distortion caused by temporary excursions through local
801 residual degrees of freedom.

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