Large Signal Model

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1 # Large Signal Model
2 
3 ## Foundations of realization, refinement and predictive closure
4 
5 Komorebi P.
6 
7 This mathematical body contains 87 numbered propositions with written proofs. The complete publication also includes its proof register, formal companion, glossary and references. The formal companion records 87 complete mappings, no partial mappings, and 0 propositions without a formalization. The exact scope and accepted execution evidence are recorded separately in the formal coverage map.
8 
9 # 1. Three objects that must remain distinct
10 
11 An **observation** is an element $x$ of a declared space $X$. Examples include a finite real signal, a complex field on a finite grid, or an array of categorical labels. Its type includes shape, scalar field or palette, and boundary convention. A zero-extended finite sequence and a periodic sequence are different objects even when their displayed samples agree.
12 
13 An **archive** is a lossless representation of the accepted observation. In a mathematical model this means an injective analysis map with an explicit left inverse. In an implementation, conversion and floating-point error must be accounted for separately. An archive of a measured signal does not recover physical variables that were never observed.
14 
15 A **predictive state** is a summary $z=\gamma(x)$, or more generally a summary of an observation history. It is sufficient only relative to a specified family of future quantities, actions and dynamics. There is no implication from exact storage of $x$ to sufficiency of $\gamma(x)$.
16 
17 The historical model's macro, meso and micro vectors are nested projections of a feature vector. They are not, merely by being nested, state variables on which the signal dynamics close. The distinction is retained in the present theory.
18 
19 The term **large** denotes the structured hypothesis family: representations, resolutions, typed operators, compositions, memory mechanisms and their parameters. It is not a theorem about capacity, sample complexity or scale. Such a theorem needs a declared grammar, a complexity measure and a learning protocol.
20 
21 # 2. Exact realization and atlas consistency
22 
23 ## 2.1 A left inverse is the reconstruction contract
24 
25 Let $X$ and $C_a$ be finite-dimensional Hilbert spaces. A linear atlas consists of analysis and synthesis maps
26 
27 $$
28 A_a:X\longrightarrow C_a,\qquad S_a:C_a\longrightarrow X,
29 \qquad S_aA_a=I_X.
30 $$
31 
32 The coefficients representing actual signals form $\mathcal C_a=\operatorname{im}A_a$. This distinction matters for redundant representations, where $\mathcal C_a$ is a proper subspace of the ambient coefficient space.
33 
34 **Proposition 1. Exact realization and consistency.** The analysis map is injective, $\Pi_a=A_aS_a$ is an idempotent with image $\mathcal C_a$, and $\Pi_ac=c$ exactly when $c$ is a realizable coefficient vector. Moreover,
35 
36 $$
37 \|A_ax\|\geq \frac{\|x\|}{\|S_a\|},\qquad
38 \|S_a(c+e)-S_ac\|\leq\|S_a\|\,\|e\|.
39 $$
40 
41 The first inequality is understood for nonzero $X$. No assertion of orthogonality follows from a left inverse alone.
42 
43 **Proof.** If $A_ax=0$, then $x=S_aA_ax=0$. Also $\Pi_a^2=A_a(S_aA_a)S_a=\Pi_a$. Its image is contained in $\mathcal C_a$ and it fixes every $A_ax$, so its image is exactly $\mathcal C_a$. Both norm inequalities follow from the operator-norm inequality applied to $S_a$. $\square$
44 
45 For a unitary atlas $U_a$, the synthesis is $U_a^*$ and the condition number is one. For a learned invertible atlas, exact invertibility is not enough: the synthesis norm controls amplification of coefficient errors. An arbitrarily ill-conditioned exact transform is not a stable representation.
46 
47 ## 2.2 Several exact atlases do not create new observations
48 
49 For unitary $U_a$ and positive weights $w_a$, put
50 
51 $$
52 \mathcal Ax=(w_aU_ax)_a,\qquad
53 \mathcal S(c_a)_a=\frac{1}{c}\sum_aw_aU_a^*c_a,
54 \qquad c=\sum_aw_a^2.
55 $$
56 
57 Then $\mathcal S\mathcal A=I$ and $\|\mathcal Ax\|^2=c\|x\|^2$, as proved in [K1]. Here $\mathcal A\mathcal S=c^{-1}\mathcal A\mathcal A^*$ is the orthogonal projection onto the consistent coefficient subspace.
58 
59 Each complete atlas contains the same observed information because it has a left inverse. Multiple atlases offer different sparsity, locality, conditioning and operator descriptions. They do not provide independent observations of the world. In particular, multiplying likelihoods from deterministic recodings as though they were independent measurements generally counts the same evidence more than once.
60 
61 Categorical one-hot encoding is a separately typed injective map of sets. It is not a unitary map on the scalar-valued signal space. Its exact decoder is valid on one-hot arrays, not on arbitrary coefficient vectors. The linear tight-frame theorem does not apply to it without a separately specified linear embedding and inner product. This separation is already explicit in the capstone's categorical section.
62 
63 # 3. Coordinate transport and one underlying operator
64 
65 For a nonempty atlas family, define transport on realizable coefficients from atlas $a$ to atlas $b$ by
66 
67 $$
68 T_{ba}=A_bS_a:\mathcal C_a\longrightarrow\mathcal C_b.
69 $$
70 
71 For an endomorphism $F:X\to X$, define its coordinate realization by $F_a=A_aFS_a$ on $\mathcal C_a$. In this paragraph $F$ may be nonlinear. For a partial map, every formula is restricted to the transported admissible domain. Maps between different signal types use separate input and output atlases.
72 
73 **Proposition 2. Coherent transport and operator realization.** On the consistent coefficient spaces,
74 
75 $$
76 T_{aa}=I,\qquad T_{cb}T_{ba}=T_{ca},\qquad
77 F_bT_{ba}=T_{ba}F_a.
78 $$
79 
80 Conversely, suppose maps $G_a:\mathcal C_a\to\mathcal C_a$ satisfy $G_bT_{ba}=T_{ba}G_a$ for every pair. Then there is a unique map $F:X\to X$ with $G_a=A_aFS_a$ for every $a$.
81 
82 **Proof.** The transport identities follow by inserting $S_bA_b=I$. The operator identity follows because either path is $A_bFS_a$. For the converse, choose a reference atlas $r$ and put $F=S_rG_rA_r$. Compatibility gives $G_aA_a=A_aF$. Every consistent coefficient is $A_ax$, which proves the asserted realization. Applying $S_a$ proves uniqueness. $\square$
83 
84 An independently fitted family of atlas predictors need not satisfy this proposition. Its compatibility residual is a well-defined quantity,
85 
86 $$
87 d_{ba}(x)=G_b(A_bx)-T_{ba}G_a(A_ax).
88 $$
89 
90 After synthesis it becomes an object-space discrepancy:
91 
92 $$
93 S_bd_{ba}(x)=S_bG_bA_bx-S_aG_aA_ax.
94 $$
95 
96 Thus $\|S_bd_{ba}(x)\|\leq\|S_b\|\,\|d_{ba}(x)\|$. Zero discrepancy establishes that the chart descriptions agree, not that their common prediction is correct. All charts can agree on the same wrong operator.
97 
98 Redundant coefficient outputs must also be tested for consistency. Synthesizing an arbitrary coefficient vector always produces some signal, but reanalysis returns $\Pi_ac$, not necessarily the original $c$. Reconstruction and coefficient consistency are different checks.
99 
100 # 4. Packets, refinement and transfer blocks
101 
102 ## 4.1 Packets belong to a declared resolution
103 
104 Within one orthogonal atlas, let $E_i$ be pairwise orthogonal coordinate selectors with $\sum_iE_i=I$. Pull them back by a unitary map:
105 
106 $$
107 P_i=U^*E_iU,\qquad P_iP_j=\delta_{ij}P_i,
108 \qquad P_i^*=P_i,\qquad\sum_iP_i=I_X.
109 $$
110 
111 A packet is the component $P_ix$ together with its projector, domain, atlas and resolution. A bare coefficient block without this information is not a fully specified packet.
112 
113 Projectors from different atlases need not commute and must not be treated as one larger mutually orthogonal family. Weighted frame packets can sum to the identity while failing to be idempotent. These are different structures.
114 
115 **Proposition 3. Complete refinement preserves reconstruction.** Suppose a finite rooted tree has root projector $I$, and every internal projector is the sum of pairwise orthogonal child projectors. Every complete antichain $\Lambda$, meeting each root-to-leaf path exactly once, satisfies
116 
117 $$
118 \sum_{v\in\Lambda}P_v=I,\qquad
119 x=\sum_{v\in\Lambda}P_vx,\qquad
120 \|x\|^2=\sum_{v\in\Lambda}\|P_vx\|^2.
121 $$
122 
123 **Proof.** Replace an internal node by the sum of its children. Repeated replacement reaches any complete antichain and preserves the total projector. Different selected nodes have disjoint descendant branches, hence orthogonal ranges. The norm identity is Pythagoras. $\square$
124 
125 This is a conservation law for information under complete refinement. Deleting a child, storing only its energy, or imposing a coefficient threshold changes the contract. For a retained subset $J$ of an orthogonal decomposition, the lost signal is exactly $\sum_{i\notin J}P_ix$, and its squared norm is the sum of the omitted packet energies.
126 
127 The finite best-basis recursion
128 
129 $$
130 V(v)=\min\left(c(v),\sum_{u\in\operatorname{ch}(v)}V(u)\right)
131 $$
132 
133 minimizes an additive cost over the declared finite tree [K1, R1]. The recurrence itself does not make a nonadditive predictive objective additive. If a prediction cost couples two leaves, a scalar cost at each node is generally insufficient for global optimization. Packet construction and cost evaluation must also be included in an implementation's complexity account.
134 
135 ## 4.2 Operations, not just signals, decompose
136 
137 Let $L:X\to Y$ be linear, with orthogonal input projectors $P_i$ and output projectors $Q_j$. Its transfer blocks are $L_{ji}=Q_jLP_i$.
138 
139 **Proposition 4. Transfer reconstruction and refinement.** One has
140 
141 $$
142 L=\sum_{j,i}Q_jLP_i.
143 $$
144 
145 If $P_i=\sum_rP_{i,r}$ and $Q_j=\sum_sQ_{j,s}$ are complete refinements, then
146 
147 $$
148 Q_jLP_i=\sum_{s,r}Q_{j,s}LP_{i,r}.
149 $$
150 
151 For another linear map $M:Y\to Z$ and output projectors $R_k$,
152 
153 $$
154 R_kMLP_i=\sum_j(R_kMQ_j)(Q_jLP_i).
155 $$
156 
157 **Proof.** Insert the appropriate identity decomposition on the input, output or intermediate space. Use $Q_j^2=Q_j$ in the composition formula. $\square$
158 
159 This is the algebraic reason to make transfer blocks first-class objects. The same operation can be expressed at several resolutions, while the coarse and fine expressions remain linked by exact identities. Approximations acquire explicit omitted-block terms rather than an unnamed representation error.
160 
161 ## 4.3 Nonlinear operations need interactions
162 
163 For a nonlinear map, $F(\sum_iP_ix)$ generally differs from $\sum_iF(P_ix)$. Applying a linear transfer formula to such a map silently loses interactions.
164 
165 **Proposition 5. Polynomial packet interactions.** Let
166 
167 $$
168 F(x)=b+\sum_{m=1}^{p}B_m(x,\ldots,x),
169 $$
170 
171 where each $B_m:X^m\to Y$ is multilinear. Then
172 
173 $$
174 Q_jF(x)=Q_jb+
175 \sum_{m=1}^{p}\ \sum_{i_1,\ldots,i_m}
176 Q_jB_m(P_{i_1}x,\ldots,P_{i_m}x).
177 $$
178 
179 **Proof.** Substitute $x=\sum_iP_ix$ in every argument and distribute by multilinearity. $\square$
180 
181 For example, $(u+v)^2=u^2+2uv+v^2$. A grammar containing only independent packet squares misses $2uv$. For nonpolynomial maps, one needs direct nonlinear semantics or a local expansion with a stated remainder bound. The word algebraic does not remove that obligation.
182 
183 # 5. Predictive closure is the central contract
184 
185 Let $\gamma:X\to Z=\gamma(X)$ be a summary and let $F_u:X\to X$ describe a deterministic transition for action $u$. The reduced description closes when there is $\bar F_u:Z\to Z$ such that
186 
187 $$
188 \gamma F_u=\bar F_u\gamma.
189 $$
190 
191 This is an equality of functions on the declared domain, not a fit statistic.
192 
193 **Proposition 6. Exact quotient criterion.** Such a reduced transition exists if and only if
194 
195 $$
196 \gamma(x)=\gamma(x')\ \Longrightarrow\
197 \gamma(F_u(x))=\gamma(F_u(x'))
198 $$
199 
200 for every admissible $x,x'$ and every action. When it exists it is unique on $\gamma(X)$.
201 
202 **Proof.** Necessity follows by applying $\bar F_u$ to equal summaries. For sufficiency, define $\bar F_u(z)=\gamma(F_u(x))$ for any representative with $\gamma(x)=z$. The stated condition makes this well-defined and also proves uniqueness. $\square$
203 
204 This criterion also applies when $X$ is a latent physical state and $\gamma$ is an observation map. Two physically different states can produce identical images and different futures. Exact image reconstruction does not resolve that hidden-state ambiguity. A history, a belief state or additional observations may be needed.
205 
206 For a stochastic kernel, the corresponding condition is equality of the distributions of the next summary for states with the same current summary. Section 7 makes this finite criterion explicit.
207 
208 ## 5.1 The obstruction in the linear case
209 
210 Suppose $X$ is a Hilbert space, $P$ is an orthogonal projection, $Q=I-P$ and $x_{t+1}=Lx_t$. The retained state is $z_t=Px_t$. Then
211 
212 $$
213 z_{t+1}=PLPz_t+PLQx_t.
214 $$
215 
216 **Proposition 7. Closure defect and compositional defect.** The retained state closes for every initial signal if and only if $PLQ=0$. If it does not close, the error in using only $PLP$ is
217 
218 $$
219 e_t=PLQx_t,\qquad
220 \|e_t\|\leq\|PLQ\|\,\|Qx_t\|.
221 $$
222 
223 For two linear operators $L$ and $M$,
224 
225 $$
226 \boxed{PLMP-(PLP)(PMP)=PLQMP.}
227 $$
228 
229 Consequently, compressing two operations separately can discard a path that leaves the retained subspace and later returns.
230 
231 **Proof.** If $PLQ=0$, the displayed recurrence depends only on $z_t$. Conversely, apply Proposition 6 to $x$ and $Px$, which have the same retained state, to obtain $PLQx=0$ for every $x$. The norm bound is submultiplicativity. The compositional identity follows by inserting $I=P+Q$ between $L$ and $M$. $\square$
232 
233 The relevant quantity is not omitted energy alone. A very small omitted packet can have a large effect if its transfer into retained packets is strongly amplified. Conversely, an energetic packet can be irrelevant to a specified prediction if every corresponding return block is zero.
234 
235 A residual is therefore typed. Reconstruction residual, coefficient-consistency residual, predictive residual, closure residual and statistical uncertainty should not be collapsed into one scalar without a specified objective.
236 
237 # 6. Constructing the smallest closed linear refinement
238 
239 There is a constructive finite solution when the dynamics are a known finite family of linear maps $L_u$ on a $d$-dimensional space. It identifies which additional observables are required, rather than merely requesting a larger state vector.
240 
241 Let the initial summary be $C_0x$. Regard its rows as linear observables. Starting from $W_0=\operatorname{row}(C_0)$, define
242 
243 $$
244 W_{n+1}=W_n+\sum_uW_nL_u,
245 \qquad W_nL_u=\{wL_u:w\in W_n\}.
246 $$
247 
248 **Proposition 8. Minimal linear predictive completion.** The sequence stabilizes after at most $d-\dim W_0$ strict enlargements. Its limit $W_*$ is the smallest row space containing $W_0$ and invariant under right multiplication by every $L_u$. If $R$ is any full-row-rank matrix whose rows form a basis of $W_*$, then
249 
250 $$
251 C_0=HR,\qquad RL_u=B_uR
252 $$
253 
254 for suitable matrices $H$ and $B_u$. Therefore $z=Rx$ retains the original summary and has exact reduced dynamics $z_{t+1}=B_uz_t$. Every linear summary with those two properties has row space containing $W_*$.
255 
256 **Proof.** The spaces are nested and have dimension at most $d$. Equality at one step means invariance under every $L_u$, so it is a fixed point. Containment in any invariant row space containing $W_0$ follows by induction. The matrix identities express inclusion of the relevant rows in the basis $R$. Conversely, if $C_0=HR$ and $RL_u=B_uR$, the row space of $R$ is invariant and contains $W_0$, so it contains every $W_n$. $\square$
257 
258 For one transition matrix this reduces to the span of the rows of
259 
260 $$
261 C_0,\ C_0L,\ \ldots,\ C_0L^{d-1}.
262 $$
263 
264 Its annihilator is the largest $L$-invariant subspace inside $\ker C_0$, the standard unobservable-subspace construction [R2]. With several actions, observables pulled back through action words are required, but the linear span still stabilizes in finite dimension.
265 
266 This gives a precise role for obstruction-driven refinement: add observable directions created by the dynamics until the chosen prediction variables close. It does not guarantee that the completion is small. In some systems $W_*=X^*$ and exact memoryless closure needs the full state.
267 
268 The theorem assumes the operators and relevant state are available. Applying it to estimated operators constructs closure for those estimates. It is not a proof that the real system has the same closure. If only $C_0x$ is measured and the full $x$ is unavailable, the augmented state $Rx$ is not automatically accessible.
269 
270 # 7. Joint closure of prediction and Bayesian inference
271 
272 The finite categorical case admits an equally explicit construction that joins dynamics with Paper VI's inference refinement. This is useful because a state can be closed for prediction but too coarse for a new observation, or sufficient for an update but not for the next transition.
273 
274 Let $\Omega$ be a finite state set. Let $K_u(i,j)$ be known row-stochastic transition matrices. Let $\ell_y(i)>0$ be a finite family of observation likelihoods. A partition $\pi$ has aggregation matrix
275 
276 $$
277 C_\pi(a,i)=\mathbf 1_{i\in B_a}.
278 $$
279 
280 Distributions are column vectors. Prediction is $p\mapsto K_u^\top p$, and a Bayesian update is
281 
282 $$
283 \mathcal B_{\ell_y}(p)
284 =\frac{p\odot\ell_y}{\ell_y^\top p}.
285 $$
286 
287 The coarse state must commute with both operations:
288 
289 $$
290 C_\pi K_u^\top=\bar K_u^\top C_\pi,
291 \qquad
292 C_\pi\mathcal B_{\ell_y}(p)
293 =\mathcal B_{\bar\ell_y}(C_\pi p).
294 $$
295 
296 Fix a mandatory initial partition $\pi_0$ with at least two blocks. This preserves distinctions that the intended application requires and excludes the trivial one-state quotient.
297 
298 **Proposition 9. Coarsest joint finite closure.** First split $\pi_0$ by the likelihood vectors $(\ell_y(i))_y$. Then repeatedly split each current block by the signatures
299 
300 $$
301 \left(\sum_{j\in B}K_u(i,j)\right)_{u,B},
302 $$
303 
304 where $B$ ranges over the current partition. The process terminates after at most $|\Omega|-|\pi_{\mathrm{init}}|$ strict refinement rounds. The final partition is the unique coarsest refinement of $\pi_0$ on which all declared predictions and Bayesian updates close for every prior.
305 
306 **Proof.** At a stable partition, states in a common block have the same transition probability into every coarse block. Those probabilities define $\bar K_u$ and give the first commutation identity, by checking it on point-mass priors. Likelihood constancy on each block defines $\bar\ell_y$ and gives the second identity by direct summation.
307 
308 For minimality, every admissible partition refining $\pi_0$ has at least two blocks. By Proposition 12 below, exact updating for all priors requires each fine likelihood to be constant within each block. Thus it refines the first likelihood split. Suppose it refines a current partition. Each current block is a union of its blocks. Exact prediction makes the probability of reaching each such union constant within any admissible block, so it also refines the next split. Induction proves it refines the final partition. Each strict refinement increases the number of blocks, which proves termination. $\square$
309 
310 The transition condition is the finite lumpability criterion [R4], expressed directly as a commutation identity. The joint construction adds the explicitly declared likelihood distinctions. Its output is a quotient of a known finite model, not an automatically learned model of an unknown world.
311 
312 The accompanying rational example starts with six states. Likelihoods create three blocks, transition distinctions force two further rounds of splitting, and the final model has five states. Two original states remain genuinely equivalent for every action and observation in the declared family. The example verifies both commuting identities exactly.
313 
314 # 8. Memory is an exact alternative to adding state
315 
316 Refinement and memory are two ways to represent the influence of unresolved components. This can be derived rather than merely proposed as an architectural option.
317 
318 For fixed linear dynamics and an orthogonal split, write
319 
320 $$
321 z_t=Px_t,\qquad q_t=Qx_t,
322 $$
323 
324 and define the restricted blocks
325 
326 $$
327 A=PLP,\quad B=PLQ,\quad C=QLP,\quad D=QLQ.
328 $$
329 
330 Thus $z_{t+1}=Az_t+Bq_t$ and $q_{t+1}=Cz_t+Dq_t$.
331 
332 **Proposition 10. Exact projected memory and a conditional tail bound.** For every $t\geq0$,
333 
334 $$
335 \boxed{
336 z_{t+1}=Az_t+
337 \sum_{k=0}^{t-1}BD^kCz_{t-1-k}+BD^tq_0.
338 }
339 $$
340 
341 If $\|D\|\leq\rho<1$ and $\|z_s\|\leq Z$ over the relevant history, omitting terms with $k\geq H$ changes the one-step right-hand side by at most
342 
343 $$
344 \|B\|\,\|C\|Z\frac{\rho^H}{1-\rho}.
345 $$
346 
347 Dropping the initial unresolved-state term adds at most $\|B\|\rho^t\|q_0\|$.
348 
349 **Proof.** Iteration of the unresolved recurrence gives
350 
351 $$
352 q_t=D^tq_0+\sum_{s=0}^{t-1}D^{t-1-s}Cz_s.
353 $$
354 
355 Substitute this into the retained recurrence and change indices. Bound the omitted terms by the geometric series. $\square$
356 
357 The memory kernel is $BD^kC$. It records a path from retained variables into unresolved variables, evolution there, and return. The first kernel $BC$ is precisely the return path appearing in the two-step composition defect. The term $BD^tq_0$ is deterministic when $q_0$ is specified. It becomes uncertain only relative to incomplete information or a probability model for $q_0$.
358 
359 This is the elementary linear instance of projection-memory reduction, associated with Mori-Zwanzig theory [R3]. A finite hand-chosen power-law or exponential memory feature is an approximation class for such effects, not a general derivation of the exact kernel.
360 
361 The tail bound compares right-hand sides using the same resolved history. Replacing that history by recursively generated predictions introduces additional rollout error. Stability of that recursion needs its own bound. Also, a spectral-radius condition alone does not imply $\|D\|\leq\rho$ in the norm already being used, especially for nonnormal operators.
362 
363 # 9. Statistical refinement needs its own theorem
364 
365 Nested feature vectors are algebraically nested when the smaller vector can be recovered from the larger by a fixed map. Predictive risk is a separate matter.
366 
367 **Proposition 11. Ideal statistical benefit of refinement.** Let a target $Y$ take values in a finite-dimensional Hilbert space with $\mathbb E\|Y\|^2<\infty$. Let $\mathcal F\subseteq\mathcal G$ be information sets and define $m_{\mathcal F}=\mathbb E[Y\mid\mathcal F]$ and $m_{\mathcal G}=\mathbb E[Y\mid\mathcal G]$. Then
368 
369 $$
370 \mathbb E\|Y-m_{\mathcal F}\|^2
371 =
372 \mathbb E\|Y-m_{\mathcal G}\|^2
373 +\mathbb E\|m_{\mathcal G}-m_{\mathcal F}\|^2.
374 $$
375 
376 **Proof.** Expand $Y-m_{\mathcal F}=(Y-m_{\mathcal G})+(m_{\mathcal G}-m_{\mathcal F})$. The cross term has zero conditional expectation given $\mathcal G$. $\square$
377 
378 Thus finer information cannot worsen the optimal squared-error predictor under a fixed distribution. It can worsen an estimated predictor with finite data, restricted function classes, changed regularization or shifted distributions. The theorem supplies the ideal target for refinement, not a guarantee for every fitting procedure.
379 
380 An exact archive makes later reanalysis possible without reconstructing discarded observations from a lossy state. It does not eliminate estimation error or distribution shift.
381 
382 # 10. Bayesian coarsening and the information-loss packet
383 
384 This section imports a specific existing Komorebi foundation rather than inventing another inference heuristic. Let $p_i$ be a prior on a finite fine state set and let $B_a$ be coarse blocks. Define
385 
386 $$
387 P_a=\sum_{i\in B_a}p_i,\qquad
388 r_a(p)=\frac{\sum_{i\in B_a}p_i\ell_i}{P_a}
389 \quad(P_a>0).
390 $$
391 
392 On zero-mass blocks the value of $r_a$ is immaterial. Updating finely and then aggregating is exactly updating $P$ with the effective likelihood $r(p)$. The effective likelihood generally depends on the fine prior inside each coarse block, information that $P$ alone does not retain.
393 
394 **Proposition 12. Bayesian closure and its exact defect.** Suppose there are at least two coarse blocks and the fine and declared coarse likelihoods are strictly positive. Updating and coarsening commute for every prior if and only if
395 
396 $$
397 \ell_i=c\bar\ell_a\quad\text{for every }i\in B_a
398 $$
399 
400 with one common $c>0$. For a fixed prior, put $\delta=r(p)-\bar\ell$, $\bar b=\mathcal B_{\bar\ell}(P)$ and $Z_r=r(p)^\top P$. Then
401 
402 $$
403 \boxed{
404 C_\pi\mathcal B_{\ell}(p)-\bar b
405 =
406 \frac{1}{Z_r}\left(\operatorname{diag}P-\bar bP^\top\right)\delta.
407 }
408 $$
409 
410 The fixed-prior identity also holds for nonnegative likelihoods when both normalizing evidences are positive.
411 
412 **Proof.** Constancy up to one global factor gives commutation by summing the fine posterior. Conversely, compare priors supported on one state in each of two distinct coarse blocks. Equality of coarse posterior odds forces the ratio of those states' likelihoods to equal the ratio of the coarse likelihoods. Varying the representatives proves constancy within each block and the common factor across blocks. For the defect, use $Z_r=\bar\ell^\top P+P^\top\delta$ and subtract the two normalized vectors. $\square$
413 
414 With only one coarse block, both posteriors aggregate to one regardless of the likelihood, so the necessity criterion is false there. This boundary is explicitly retained from the corrected Paper VI statement [K2].
415 
416 The matrix in the defect formula annihilates $\bar\ell$, recording the irrelevant common-scale direction. It does not make every likelihood discrepancy irrelevant. The construction in Proposition 9 refines until both the likelihood and transition defects vanish for a specified finite family.
417 
418 ## 10.1 Parameter inference is not program truth
419 
420 Within one declared regression model, a normal-inverse-gamma prior gives a standard conditional Bayesian parameter posterior. The software's program weights instead use penalized held-out predictive scores and then may refit coefficients on all available training records. These are finite-candidate predictive weights. Without a corresponding generative derivation, they should not be conflated with the marginal-likelihood posterior over programs under one fixed prior.
421 
422 Every candidate must be scored on the same target event and base measure. Densities on differently sized coordinate spaces are not directly comparable. An invertible nonlinear coordinate change requires the appropriate Jacobian. A redundant deterministic embedding induces a measure on its consistent subspace, not an independent full-dimensional density on all coefficient entries.
423 
424 The law of total variance separates within-candidate uncertainty from disagreement of candidate means. It does not supply a probability that the truth lies outside the grammar. An explicit outside-grammar component needs its own prior and predictive distribution. Assigning an arbitrary leftover mass does not solve open-world inference.
425 
426 # 11. What a typed operator language must guarantee
427 
428 A program has a source type, target type, admissible domain, evaluation rule and declared approximation status. A nonlinear partial operation is not a total matrix merely because its inputs are stored in a vector. The domain of a composite is
429 
430 $$
431 \operatorname{dom}(g\circ f)
432 =\{x\in\operatorname{dom}f:f(x)\in\operatorname{dom}g\}.
433 $$
434 
435 Composition must preserve those types and domains. Coordinate changes must preserve the represented operation as in Proposition 2. Packet refinement must preserve exact transfer semantics as in Proposition 4, or carry an explicit approximation residual. Nonlinear interactions must retain their cross-packet terms as in Proposition 5.
436 
437 A grammar can be closed under a specified set of constructors without containing every possible physical law. It can also be expressive while computationally intractable. Universal discovery, practical search and semantic closure are distinct claims.
438 
439 For a known operator, Proposition 8 can build its minimal linear summary. For an unknown operator, learning must address identifiability: which different candidates make identical predictions under the available observations and actions? More coordinate systems alone do not distinguish candidates that agree on the entire observed signal. Additional interventions or data may be needed.
440 
441 The foundation should therefore treat a learned hypothesis as an operator with an evidence record, not just a matrix with a high normalized weight. Its record includes fitting assumptions, admissible inputs, compared alternatives, unresolved equivalences and the representation in which its residual is measured.
442 
443 # 12. A bounded infinite-resolution bridge
444 
445 Finite reconstruction identities do not alone prove convergence of an unbounded or infinite-dimensional model. A useful bounded extension can nevertheless be established directly.
446 
447 **Proposition 13. Strong fixed-horizon refinement limit.** Let $X$ be a Hilbert space, $L:X\to X$ bounded, and $P_n$ orthogonal projections converging strongly to $I$. Put $L_n=P_nLP_n$. For every fixed $x\in X$ and every fixed nonnegative integer $k$,
448 
449 $$
450 L_nx\longrightarrow Lx,\qquad
451 L_n^kx\longrightarrow L^kx.
452 $$
453 
454 **Proof.** Since $\|P_n\|\leq1$,
455 
456 $$
457 \|(L_n-L)x\|
458 \leq\|L\|\,\|(P_n-I)x\|+\|(P_n-I)Lx\|
459 \longrightarrow0.
460 $$
461 
462 Also $\|L_n\|\leq\|L\|$. Induct on $k$, using
463 
464 $$
465 (L_n^{k+1}-L^{k+1})x
466 =L_n(L_n^k-L^k)x+(L_n-L)L^kx.
467 $$
468 
469 The first term tends to zero by the uniform norm bound and induction, and the second by strong convergence on the fixed vector $L^kx$. $\square$
470 
471 If estimated operators $\widehat L_n$ are uniformly bounded and $(\widehat L_n-L_n)x\to0$ for every fixed $x$, the same argument gives fixed-horizon convergence for $\widehat L_n$. These are explicit assumptions, not conclusions of the finite learning procedure.
472 
473 This result is strong convergence, not operator-norm convergence and not a bound uniform in time. On $\ell^2$, finite coordinate projections converge strongly to $I$ while $\|I-P_n\|=1$ for every $n$. Unbounded operators require domain control. Adaptive data-dependent partitions and fitted coefficients require additional consistency arguments. These remain separate mathematical tasks.
474 
475 # 13. A complete small signal example
476 
477 Consider the rational linear signal dynamics
478 
479 $$
480 x_{t+1}=Lx_t,\qquad
481 L=\begin{pmatrix}\tfrac12&1\\[2pt]\tfrac14&\tfrac13\end{pmatrix},
482 \qquad x_0=\begin{pmatrix}1\\0\end{pmatrix}.
483 $$
484 
485 The spatial archive stores $x_t$. A second atlas uses the orthogonal Haar matrix
486 
487 $$
488 U=\frac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}.
489 $$
490 
491 Its coefficients $c_t=Ux_t$ reconstruct as $x_t=U^*c_t$. The same dynamics in that atlas are $ULU^*$, not a separate physical law. Thus coordinate transport preserves the full dynamics.
492 
493 Now retain only the first spatial coordinate. With $P=\operatorname{diag}(1,0)$, the retained scalar $z_t$ and discarded scalar $q_t$ obey
494 
495 $$
496 z_{t+1}=\tfrac12z_t+q_t,\qquad
497 q_{t+1}=\tfrac14z_t+\tfrac13q_t.
498 $$
499 
500 The exact first-coordinate sequence begins
501 
502 $$
503 z_0=1,\qquad z_1=\tfrac12,\qquad z_2=\tfrac12,
504 \qquad z_3=\tfrac{11}{24}.
505 $$
506 
507 The compressed one-step matrix predicts $z_2=1/4$ instead of $1/2$. The lost return path contributes exactly
508 
509 $$
510 BC=1\cdot\tfrac14=\tfrac14.
511 $$
512 
513 There are two exact repairs. One is state refinement: start with $C_0=(1,0)$ and add $C_0L=(1/2,1)$. The resulting row space is the full two-dimensional space, so it closes. The other is memory:
514 
515 $$
516 z_{t+1}=\tfrac12z_t+
517 \tfrac14\sum_{k=0}^{t-1}\left(\tfrac13\right)^kz_{t-1-k},
518 $$
519 
520 since $q_0=0$. This is the exact same system expressed without retaining $q_t$ as a separate current-state variable. At $t=2$, it yields
521 
522 $$
523 z_3=\tfrac12\cdot\tfrac12
524 +\tfrac14\cdot\tfrac12
525 +\tfrac1{12}\cdot1
526 =\tfrac{11}{24}.
527 $$
528 
529 To see a nontrivial reduced closure, adjoin a third independent coordinate with multiplier $5/6$. The smallest closed refinement of the first coordinate then contains the first two coordinates, not the third. Exact prediction of the chosen summary does not require reconstructing every component of the enlarged signal.
530 
531 Finally, low energy is not the same as low predictive importance. For
532 
533 $$
534 L'=\begin{pmatrix}0&1000\\0&0\end{pmatrix},
535 \qquad x=(0,1/1000)^\top,
536 $$
537 
538 omitting the second coordinate loses norm $1/1000$, but changes the next first coordinate by exactly one. The correct certificate is the product of omitted magnitude and transfer amplification.
539 
540 # 14. Connection to observable learning
541 
542 Exact realization, complete decomposition and operator transport provide the representation laws. Propositions 6–10 separate predictive closure, constructive refinement and memory. The remaining requirement for learning is access to a sufficient predictive state. Part II addresses a finite linear observable sector; it does not infer its rank or coverage assumptions from a successful fit.
543 
544 \newpage
545 
546 # Part II. Learning, stability and coherent expansion
547 
548 The first stage constructed predictive closures for declared operators. The second stage asks how those constructions relate to what a learner can actually observe. A closure that depends on an unavailable hidden coordinate is a mathematical description, not yet an accessible state. A fitted recurrence can be accessible but wrong outside the span represented by its data. A one-step approximation can be accurate and still fail under repeated composition.
549 
550 The results below connect these distinctions. They give an exact learning result for a finite linear observable sector, deterministic perturbation bounds, a propagation law for typed defects, a sufficient stability condition for truncated memory, controlled nonlinear interaction bounds, coherent atlas reconciliation and posterior accounting for an expanding grammar. These are finite or explicitly bounded results. They do not assume that every unknown process has a short linear state.
551 
552 The observable-state perspective has classical antecedents in predictive state representations [R5] and observable-operator learning [R7]. Finite invariant observable spaces also underlie Koopman reductions [R6]. The present development uses elementary statements proved below and identifies their specific place in Komorebi's realization and obstruction calculus. It does not import those papers' sample-complexity guarantees into this model.
553 
554 
555 
556 # 15. From hidden state to an accessible predictive state
557 
558 Let a finite-dimensional autonomous linear process have hidden state $x_t\in\mathbb R^d$ and observed output $y_t\in\mathbb R^p$:
559 
560 $$
561 x_{t+1}=Lx_t,\qquad y_t=Cx_t.
562 $$
563 
564 The observation map need not be injective. For a positive integer $h$, define the observation map
565 
566 $$
567 O_h=\begin{pmatrix}C\\CL\\\vdots\\CL^{h-1}\end{pmatrix},
568 \qquad V_h=\operatorname{im}O_h.
569 $$
570 
571 The coordinates of $O_hx_s$ are future observations relative to time $s$. They are not available at time $s$ merely because this matrix is written down. At time $t\geq h-1$, however, the same vector is the observed history
572 
573 $$
574 z_t=\begin{pmatrix}y_{t-h+1}\\\vdots\\y_t\end{pmatrix}
575 =O_hx_{t-h+1}.
576 $$
577 
578 Thus a forward observation map becomes a causal state after a declared warm-up interval. The time index is part of the contract.
579 
580 **Proposition 14. Exact causal delay closure.** The following statements are equivalent:
581 
582 1. There is a linear map $D_h:V_h\to V_h$ satisfying $O_hL=D_hO_h$.
583 2. $\ker O_h\subseteq\ker(O_hL)$.
584 3. $\operatorname{rank}O_{h+1}=\operatorname{rank}O_h$.
585 
586 When these statements hold, $D_h$ is unique on $V_h$. The history state obeys $z_{t+1}=D_hz_t$, and its last output block is $y_t$. Its iterates therefore determine every subsequent observed output. If a positive integer $d$ bounds the hidden dimension, $h=d$ is sufficient for closure, though a smaller positive $h$ may suffice.
587 
588 **Proof.** The equivalence of the first two statements is the linear factorization criterion: define $D_h(O_hx)=O_hLx$ and require independence from the representative $x$. The rows of $O_hL$ are $CL,\ldots,CL^h$. All except the final block already occur in $O_h$. The kernel condition is therefore equivalent to $CL^h$ being in the row space of $O_h$, which is exactly the rank equality. It also implies $O_hLx\in V_h$ because this vector is $O_h$ applied to $Lx$. Uniqueness follows because every element of $V_h$ has a representative. The history recurrence follows directly. Cayley-Hamilton expresses $L^d$ as a linear combination of lower powers, so $CL^d$ is in the row space of $O_d$. $\square$
589 
590 This does not recover hidden variables in $\ker O_h$. It identifies the quotient needed for the declared outputs. An independent hidden component can remain unobservable forever without preventing exact output prediction.
591 
592 The rank equality here is a property of the *true observation maps*. A plateau in a finite empirical data matrix is not the same statement. It may reflect unexcited directions rather than a closed observation space. Section 17 makes the difference exact.
593 
594 For several actions $L_u$, a fixed delay vector does not automatically have one autonomous transition matrix. Products of the actual action matrices determine its observation map, which can change with the action history. One needs an action-aware state, a common closed observable family, or a separately justified belief-state construction. Proposition 14 is not silently generalized to arbitrary controlled or stochastic dynamics.
595 
596 # 16. What finite data identify
597 
598 Suppose candidate states and their successors have been recorded as columns of matrices $Z_-,Z_+\in\mathbb R^{m\times N}$. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.
599 
600 Write $W=\operatorname{im}Z_-$. Exact interpolation of these columns is a question about a linear map on $W$, not yet about every physically realizable state.
601 
602 **Proposition 15. Data-span identification and its boundary.** A linear map $B$ on the ambient space with $BZ_-=Z_+$ exists if and only if
603 
604 $$
605 \ker Z_-\subseteq\ker Z_+.
606 $$
607 
608 Its restriction to $W$ is unique. It is an endomorphism of $W$ if and only if $\operatorname{im}Z_+\subseteq W$. A particular ambient extension is
609 
610 $$
611 B_0=Z_+Z_-^\dagger,
612 $$
613 
614 where $\dagger$ denotes the Moore-Penrose inverse. Every other fitted extension agrees with $B_0$ on $W$. If $W$ is a proper subspace of the true realizable state space $V$, the data alone do not identify the operator on $V\setminus W$.
615 
616 **Proof.** Necessity follows by applying $B$ to every linear dependence among the input columns. Under the kernel condition, define $B(Z_-a)=Z_+a$. Two choices of $a$ differ by a kernel vector and give the same answer. This defines and uniquely determines the map on $W$. It maps $W$ into $W$ exactly under the image condition. The pseudoinverse formula gives that map because $a-Z_-^\dagger Z_-a\in\ker Z_-$. If $W\subsetneq V$, choose a nonzero linear functional $\varphi$ on $V$ vanishing on $W$ and a nonzero $v\in V$. The operators $B$ and $B+v\varphi$ agree on every observed column and differ elsewhere. $\square$
617 
618 The three checks are different. The kernel test checks consistency of a deterministic linear fit. The image test checks whether the observed span is closed under the fitted transitions. Equality $W=V$ is a *coverage condition*. Neither of the first two checks establishes it.
619 
620 For a known fixed feature space of dimension $r$, coverage is witnessed by $r$ independent input states. For an unknown larger feature space, the same rank only gives a lower bound on dimension. A model card must say which interpretation is being used.
621 
622 If several action-specific operators act on a common declared $r$-dimensional state, each action's paired data must determine that operator on the required span. Pooling all actions into one high-rank matrix does not identify an action that has never been tried on some direction.
623 
624 ## 16.1 Perturbations have an explicit amplification factor
625 
626 A noisy rank test is not an exact certificate. Even after the state space is fixed, fitting can be ill-conditioned. The following deterministic bound isolates that issue without introducing a probability model for noise.
627 
628 Work in a fixed $r$-dimensional coordinate system, where $Z_-\in\mathbb R^{r\times N}$ has full row rank. Let
629 
630 $$
631 Z_+=BZ_-+R,\qquad
632 \widetilde Z_-=Z_-+E,\qquad
633 \widetilde Z_+=Z_++F.
634 $$
635 
636 The matrix $R$ records deviations from the declared linear recurrence on these samples. The matrices $E$ and $F$ record input and successor perturbations. All norms in the next proposition are spectral norms.
637 
638 **Proposition 16. Fixed-space identification bound.** Put $s=\sigma_{\min}(Z_-)>0$. If $\|E\|<s$, then $\widetilde Z_-$ has full row rank, and
639 
640 $$
641 \widehat B=\widetilde Z_+\widetilde Z_-^\dagger
642 $$
643 
644 satisfies the exact error identity and bound
645 
646 $$
647 \begin{aligned}
648 \widehat B-B
649 &=(R+F-BE)\widetilde Z_-^\dagger,\\
650 \|\widehat B-B\|
651 &\leq\frac{\|R\|+\|F\|+\|B\|\,\|E\|}
652 {s-\|E\|}.
653 \end{aligned}
654 $$
655 
656 **Proof.** For every unit $v\in\mathbb R^r$,
657 $\|\widetilde Z_-^\top v\|\geq\|Z_-^\top v\|-\|E^\top v\|\geq s-\|E\|$.
658 Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most $(s-\|E\|)^{-1}$. Since $\widetilde Z_-\widetilde Z_-^\dagger=I_r$, subtract $B$ from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. $\square$
659 
660 This is a bound for a fixed state space and a fixed coordinate basis. If the basis or its dimension is also estimated from noisy data, subspace-estimation error must be added. No such error is hidden inside $R$ without a derivation. Correlated and adversarial perturbations are allowed by the displayed deterministic inequality, but there is no sample-complexity claim until their magnitudes are bounded under a sampling model.
661 
662 The denominator explains why numerical rank alone is insufficient. Full rank can coexist with severe noise amplification. The reference calculations include examples attaining the $1/s$ dependence and a rank-collapse example at the excluded boundary $\|E\|=s$.
663 
664 # 17. Why finite agreement is not unrestricted discovery
665 
666 **Proposition 17. Finite-prefix indistinguishability.** For every integer $T\geq0$, there are two finite-dimensional autonomous rational linear systems with identical scalar observations at times $0,\ldots,T$ and different observations at time $T+1$. Both transition matrices can be chosen nilpotent.
667 
668 **Proof.** One system has scalar output identically zero. For the other, take dimension $T+2$, initial state $x_0=e_{T+2}$, observation row $C=e_1^\top$, and shift $Le_j=e_{j-1}$ for $j\geq2$, $Le_1=0$. Then $CL^tx_0=0$ for $0\leq t\leq T$ and $CL^{T+1}x_0=1$. The shift is rational and nilpotent. $\square$
669 
670 An invisible delayed component of this kind can also be adjoined to any fixed base model. The combined output agrees with the base output on the entire inspected prefix and then differs. This is not an argument against learning. It identifies why a claimed guarantee must state a dimension bound, a model class, a coverage condition, an intervention protocol or a probabilistic assumption.
671 
672 Even nilpotent operators have only zero eigenvalues, so stability language alone does not remove this ambiguity. A finite rank plateau cannot certify the absence of a delayed mode when arbitrarily large hidden dimensions are admitted.
673 
674 # 18. A complete output-only identification example
675 
676 Return to the rational system from Section 13, now with an additional independent hidden coordinate:
677 
678 $$
679 L=\begin{pmatrix}
680 1/2&1&0\\
681 1/4&1/3&0\\
682 0&0&5/6
683 \end{pmatrix},
684 \qquad C=\begin{pmatrix}1&0&0\end{pmatrix}.
685 $$
686 
687 The third coordinate never influences the observed output. The first coordinate alone is not closed, but
688 
689 $$
690 O_2=\begin{pmatrix}1&0&0\\1/2&1&0\end{pmatrix}
691 $$
692 
693 has rank two and $O_3$ has the same rank. Proposition 14 gives an exact two-dimensional delay state. Choose $x_0=(1,0,q)^\top$ for any $q$. The first four observations are
694 
695 $$
696 y_0=1,\quad y_1=1/2,\quad y_2=1/2,\quad y_3=11/24.
697 $$
698 
699 A learner receives those four outputs and the declared delay length $h=2$. It does **not** receive $L$, $C$, $O_2$ or $q$. From consecutive observed windows it forms
700 
701 $$
702 Z_-=\begin{pmatrix}1&1/2\\1/2&1/2\end{pmatrix},
703 \qquad
704 Z_+=\begin{pmatrix}1/2&1/2\\1/2&11/24\end{pmatrix}.
705 $$
706 
707 Since $\det Z_-=1/4$, these windows span the declared two-dimensional state. Exact arithmetic gives
708 
709 $$
710 \boxed{
711 D_2=Z_+Z_-^{-1}
712 =\begin{pmatrix}0&1\\1/12&5/6\end{pmatrix}.
713 }
714 $$
715 
716 The learned recurrence is
717 
718 $$
719 \boxed{y_{t+1}=\frac56y_t+\frac1{12}y_{t-1},\qquad t\geq1.}
720 $$
721 
722 It predicts $y_4=61/144$, which was not used to identify the coefficients. Proposition 14 and the verified rank condition on the true observation maps establish why this recurrence continues for every initial state of this particular system. The four samples alone establish the fitted recurrence only within the declared closed two-state model class. Proposition 17 forbids confusing those two conclusions.
723 
724 The reference script separates the roles. One routine fits only the observed samples. A different check, with access to the generating system, verifies the theoretical assumptions and compares subsequent outputs. This prevents hidden access to the true operator from being presented as learning.
725 
726 The same example connects back to the exact memory kernel. Section 13 expressed the output through a decaying convolution of its history. Here the history is summarized by a two-component recursive state. Neither representation stores the independent third coordinate. Memory and state realization describe the same observable behavior in two different ways.
727 
728 \begin{figure}[htbp]
729 \centering
730 \includegraphics[width=0.95\textwidth]{../assets/figures/01_delay_prediction.pdf}
731 \caption{Output-only identification in the declared rational example. Four observations determine the delay recurrence. Later outputs are calculated independently from the hidden system. The one-coordinate compression omits the return path. This is a mathematical example, not a task-suite score.}
732 \end{figure}
733 
734 # 19. A single propagation law for typed defects
735 
736 A reconstruction error, an operator-fit error and a refinement error have different origins. Once each has been translated into an error in the state used by the next operation, their propagation can be analyzed by the same identity.
737 
738 Allow the representation to change with time. Let $\gamma_t:X\to Z_t$ be the declared representation at step $t$, with a norm on each $Z_t$. Let $x_{t+1}=F_t(x_t)$ be the true transition and let $\widehat f_t:Z_t\to Z_{t+1}$ be the implemented step. Its defect is
739 
740 $$
741 r_t(x)=\gamma_{t+1}(F_t(x))-\widehat f_t(\gamma_t(x)).
742 $$
743 
744 This definition already includes a representation switch. It is not restricted to one fixed feature vector.
745 
746 **Proposition 18. Compositional defect propagation.** Suppose $\widehat f_t$ is $\lambda_t$-Lipschitz on a domain containing both $\gamma_t(x_t)$ and the predicted state $\widehat z_t$. Suppose
747 $\|r_t(x_t)\|\leq\delta_t$ and
748 $\widehat z_{t+1}=\widehat f_t(\widehat z_t)$.
749 Then, with $e_t=\|\gamma_t(x_t)-\widehat z_t\|$,
750 
751 $$
752 \boxed{
753 e_T\leq
754 \left(\prod_{j=0}^{T-1}\lambda_j\right)e_0
755 +\sum_{k=0}^{T-1}
756 \left(\prod_{j=k+1}^{T-1}\lambda_j\right)\delta_k.
757 }
758 $$
759 
760 Empty products are one. If the output $h_t(x)$ has a decoder $H_t$ with Lipschitz constant $\kappa_t$ and
761 $\|h_t(x_t)-H_t(\gamma_t(x_t))\|\leq\eta_t$, then
762 
763 $$
764 \|h_t(x_t)-H_t(\widehat z_t)\|
765 \leq\eta_t+\kappa_te_t.
766 $$
767 
768 **Proof.** Add and subtract $\widehat f_t(\gamma_t(x_t))$ to obtain
769 $e_{t+1}\leq\delta_t+\lambda_te_t$.
770 Repeated substitution yields the product-sum formula. Add and subtract $H_t(\gamma_t(x_t))$ for the output inequality. $\square$
771 
772 For a fixed linear representation $R$ and an estimated transition $\widehat B_u$, the defect is $(RL_u-\widehat B_uR)x$. For a reference reduced operator $B_u$, it splits exactly as
773 
774 $$
775 RL_u-\widehat B_uR
776 =(RL_u-B_uR)+(B_u-\widehat B_u)R.
777 $$
778 
779 The first term is representational nonclosure. The second is operator-estimation error. A fit statistic for the second cannot certify that the first vanishes. Proposition 16 supplies a conditional bound on the second in its declared fixed-space setting.
780 
781 If $\lambda_t\leq\rho<1$ and $\delta_t\leq\delta$, then
782 
783 $$
784 e_T\leq\rho^Te_0+\delta\frac{1-\rho^T}{1-\rho}.
785 $$
786 
787 This is a genuinely recursive prediction bound. It is not merely a one-step comparison made using the true history at every step. If the Lipschitz constants exceed one, the same theorem remains valid but can permit error growth.
788 
789 ## 19.1 Stability under changing operations
790 
791 For linear candidate operators, one sufficient way to obtain a common contraction norm is to exhibit a positive-definite matrix $M$ and $0\leq\rho<1$ with
792 
793 $$
794 \widehat B_u^\top M\widehat B_u\preceq\rho^2M
795 \quad\text{for every declared action }u.
796 $$
797 
798 Then $\|z\|_M=(z^\top Mz)^{1/2}$ gives
799 $\|\widehat B_uz\|_M\leq\rho\|z\|_M$ for all actions, directly from the displayed inequality. Proposition 18 therefore applies under arbitrary switching among them. This is the common quadratic Lyapunov construction from control theory [R8], used here as an explicit certificate rather than inferred from a list of eigenvalues.
800 
801 Individual spectral stability does not suffice. The matrices
802 
803 $$
804 A=\begin{pmatrix}0&2\\0&0\end{pmatrix},\qquad
805 B=\begin{pmatrix}0&0\\2&0\end{pmatrix}
806 $$
807 
808 are both nilpotent, but $BA=\operatorname{diag}(0,4)$. Alternating them amplifies an appropriate state by four every two steps. Even a single stable nonnormal matrix can produce large transient amplification in a specified norm.
809 
810 If a learned policy chooses actions from predicted states while the reference process uses actions chosen from true states, the two action sequences may differ. Proposition 18 then needs a bound for that action mismatch, or a Lipschitz bound for the full closed-loop map. The result is not an automatic control-policy guarantee.
811 
812 # 20. Truncating memory without losing the prediction bound
813 
814 Section 8 bounded the omitted tail of the exact linear memory expression using the same resolved history. We now close the separate recursion argument.
815 
816 Write the exact relation as
817 
818 $$
819 z_{t+1}=Az_t+\sum_{k=0}^{t-1}K_kz_{t-1-k}+f_t,
820 \qquad K_k=BD^kC,\quad f_t=BD^tq_0.
821 $$
822 
823 For negative indices set $z_s=0$, solely as a convention for missing prehistory in this initialized formula. A length-$H$ approximate model, with $H\geq0$, is
824 
825 $$
826 \widehat z_{t+1}=\widehat A\widehat z_t+
827 \sum_{k=0}^{H-1}\widehat K_k\widehat z_{t-1-k}.
828 $$
829 
830 An empty sum is zero. Let $d_t$ be the exact right-hand side minus this truncated right-hand side evaluated on the **true** history.
831 
832 **Proposition 19. Uniform prediction bound for a stable finite-memory truncation.** Suppose
833 
834 $$
835 a=\|\widehat A\|+\sum_{k=0}^{H-1}\|\widehat K_k\|<1,
836 \qquad\sup_{t\geq0}\|d_t\|\leq\Delta.
837 $$
838 
839 Let $E_0$ bound the initialization errors at indices $-H,\ldots,0$. Then the recursively generated predictions satisfy
840 
841 $$
842 \boxed{
843 \sup_{t\geq0}\|z_t-\widehat z_t\|
844 \leq\max\left(E_0,\frac{\Delta}{1-a}\right).
845 }
846 $$
847 
848 If $\|z_t\|\leq Z$ for $t\geq0$ and $\|D\|\leq\rho<1$, a valid choice of defect bound is
849 
850 $$
851 \begin{aligned}
852 \Delta={}&Z\left(\|A-\widehat A\|
853 +\sum_{k=0}^{H-1}\|K_k-\widehat K_k\|\right)\\
854 &+\|B\|\,\|C\|Z\frac{\rho^H}{1-\rho}
855 +\|B\|\,\|q_0\|.
856 \end{aligned}
857 $$
858 
859 The last term can be removed if the exact initialized forcing $f_t$ is retained in the approximate recurrence.
860 
861 **Proof.** Subtract the two recurrences and bound every retained history error by the largest of the errors at indices $t-H,\ldots,t$. This gives
862 $e_{t+1}\leq a\max_{t-H\leq s\leq t}e_s+\Delta$.
863 The ball of radius $E=\max(E_0,\Delta/(1-a))$ is invariant because $aE+\Delta\leq E$. Induction proves the bound. The estimate for $\Delta$ separates coefficient errors, the geometric kernel tail and the initialized forcing, using Proposition 10. $\square$
864 
865 This norm-sum condition is sufficient, not necessary. A stable recurrence with cancellation can fail this test. Failure to find this certificate is not proof of instability.
866 
867 For the scalar example with $q_0=0$, take exact retained coefficients and $z_0=1$. All outputs lie in $[0,1]$: the full nonnegative memory coefficients sum to at most $7/8$, so induction bounds every new output by the previous maximum. Thus $Z=1$ is valid. With
868 
869 $$
870 A=1/2,\quad K_k=\frac14\left(\frac13\right)^k,
871 $$
872 
873 one obtains
874 
875 $$
876 a_H=\frac78-\frac38\,3^{-H},\qquad
877 \Delta_H=\frac38\,3^{-H},\qquad
878 \boxed{\sup_t|z_t-\widehat z_t|\leq
879 \frac{3\,3^{-H}}{1+3\,3^{-H}}.}
880 $$
881 
882 The right-hand side is a theorem for every time in this initialized example. The figure compares it with a separately computed finite-horizon error. The computed maximum is not substituted for the uniform bound.
883 
884 \begin{figure}[htbp]
885 \centering
886 \includegraphics[width=0.95\textwidth]{../assets/figures/02_memory_certificate.pdf}
887 \caption{A memory-tail estimate becomes a recursive prediction certificate only after a stability margin is supplied. The measured curve is the maximum over 80 calculated steps. The upper curve is the proved bound for all steps under the stated scalar initialization.}
888 \end{figure}
889 
890 # 21. Nonlinear packet truncation with an explicit remainder
891 
892 Polynomial interactions were expanded exactly in Proposition 5. A computational model may retain only some ordered interaction tuples. It then needs a bound on what was omitted, rather than an assumption that packetwise evaluation is exact.
893 
894 Let $P_i$ be an orthogonal resolution of a finite-dimensional real Hilbert space. Suppose
895 
896 $$
897 F(x)=b+\sum_{m=1}^{p}B_m(x,\ldots,x),
898 $$
899 
900 with bounded multilinear maps. For each ordered tuple $\boldsymbol i=(i_1,\ldots,i_m)$, let $\beta_{m,\boldsymbol i}$ bound the norm of $B_m$ restricted to
901 $\operatorname{im}P_{i_1}\times\cdots\times\operatorname{im}P_{i_m}$.
902 Let $\mathcal I_m$ be the retained tuple set, and let $F_{\mathcal I}$ retain the constant term and exactly those interactions.
903 
904 **Proposition 20. Packet-interaction truncation bound.** Put $a_i=\|P_ix\|$. Then
905 
906 $$
907 \boxed{
908 \|F(x)-F_{\mathcal I}(x)\|
909 \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\notin\mathcal I_m}
910 \beta_{m,\boldsymbol i}\prod_{j=1}^{m}a_{i_j}.
911 }
912 $$
913 
914 On the ball $\|x\|\leq R$, a valid Lipschitz bound for $F_{\mathcal I}$ is
915 
916 $$
917 \operatorname{Lip}(F_{\mathcal I})
918 \leq\sum_{m=1}^{p}\ \sum_{\boldsymbol i\in\mathcal I_m}
919 m\beta_{m,\boldsymbol i}R^{m-1}.
920 $$
921 
922 **Proof.** Subtract the retained terms from the exact multilinear expansion. Bound each omitted term by its restricted multilinear norm and sum. For the Lipschitz estimate, telescope the difference between two $m$-fold evaluations by replacing one argument at a time. Each changed argument has norm at most $\|x-y\|$ after projection, and each unchanged argument has norm at most $R$. There are $m$ terms for each retained tuple. $\square$
923 
924 If the declared polynomial itself approximates a nonpolynomial map with remainder at most $\varepsilon_R$ on the ball, add $\varepsilon_R$ to the defect bound. Without such a remainder estimate, the result is only a theorem for the polynomial, not for the unexpanded physical law.
925 
926 For a retained state $z=Px$ and omitted state $q=(I-P)x$, dropping every term that uses $q$ gives the coarser majorant
927 
928 $$
929 \|F(z+q)-F(z)\|
930 \leq\sum_{m=1}^{p}\|B_m\|
931 \left((\|z\|+\|q\|)^m-\|z\|^m\right).
932 $$
933 
934 The tuple-level result is finer because it can distinguish a harmless omitted packet from a strongly coupled one. Ordered tuples must be counted with their multiplicities unless symmetry has explicitly combined them.
935 
936 A prediction guarantee follows by inserting these defect and Lipschitz bounds into Proposition 18, provided the true and predicted trajectories remain in the declared domain. That domain condition cannot be inferred from a small initial error. One possible construction is to assume that the true map preserves a closed convex set $K$ and to project every approximate output back onto $K$. Euclidean projection is nonexpansive and fixes the true output, so it does not increase the one-step approximation bound. This is a declared modification of the predictor, not an identity for an unmodified recurrence.
937 
938 For the scalar quadratic $(u+v)^2$, retaining only $u^2+v^2$ omits $2uv$. The packet bound is $2|u||v|$ and is exact in magnitude. Small energy in one packet matters in proportion to both its amplitude and the cross-interaction coefficient.
939 
940 # 22. Learning several charts without inventing several worlds
941 
942 The exact atlas identities constrain the meaning of a fitted model, but independently trained predictors need not satisfy them. Here is a simple coherent reconciliation with a precise objective.
943 
944 Fix a finite-dimensional represented space $X$. It may be the observed signal space or a predictive quotient, but it must be the *same* space for all atlases in this section. Let $A_a:X\to C_a$ and $S_a:C_a\to X$ satisfy $S_aA_a=I$. Let $G_a:C_a\to C_a$ be linear fitted maps. Pull them back to object space:
945 
946 $$
947 L_a=S_aG_aA_a.
948 $$
949 
950 For positive weights summing to one, define
951 
952 $$
953 \overline L=\sum_aw_aL_a,
954 \qquad\widetilde G_a=A_a\overline LS_a
955 \quad\text{on }\mathcal C_a=\operatorname{im}A_a.
956 $$
957 
958 **Proposition 21. Coherent reconciliation and coordinate-stable transport.** The family $\widetilde G_a$ is consistent and satisfies every atlas intertwining identity. The operator $\overline L$ is the unique minimizer of
959 
960 $$
961 \sum_aw_a\|L-L_a\|_F^2
962 $$
963 
964 over linear maps on $X$, where $\|\cdot\|_F$ is the Frobenius norm. For any target operator $L_*$ and any chosen operator norm,
965 
966 $$
967 \|\overline L-L_*\|\leq\sum_aw_a\|L_a-L_*\|.
968 $$
969 
970 Further, the norm $\|c\|_a=\|S_ac\|$ on $\mathcal C_a$ makes every exact transport $T_{ba}=A_bS_a$ an isometry. A represented transition $A_bLS_a$ has exactly the same induced Lipschitz constant in these norms as $L$ has in object space.
971 
972 **Proof.** Coherence follows from Proposition 2 because every chart realizes the same $\overline L$. Expanding the squares gives
973 
974 $$
975 \sum_aw_a\|L-L_a\|_F^2
976 =\|L-\overline L\|_F^2+
977 \sum_aw_a\|L_a-\overline L\|_F^2.
978 $$
979 
980 The weighted cross term vanishes, proving the unique minimizer. The target bound follows from the triangle inequality. The pullback expression is a norm because $S_a$ is inverse to $A_a$ on $\mathcal C_a$. Finally,
981 $\|T_{ba}c\|_b=\|S_bA_bS_ac\|=\|S_ac\|$.
982 The same identity for a transition, and the bijection between $X$ and $\mathcal C_a$, establish equality of Lipschitz constants. $\square$
983 
984 This reconciliation is ordinary weighted least squares in a common semantic space. It does not prove that consensus is correct, reduce statistical error under arbitrary dependence, or turn deterministic recodings into independent evidence. The output must also be reanalyzed into a realizable coefficient vector. An independently fitted redundant output can have an additional inconsistency residual $(I-A_aS_a)G_aA_ax$ before reconciliation.
985 
986 For Euclidean coefficient errors, conditioning remains important. If $A_a$ has smallest singular value at least $m_a>0$ and largest singular value at most $M_a$, its canonical synthesis $A_a^\dagger$ has norm at most $1/m_a$, and
987 $\|T_{ba}\|\leq M_b/m_a$.
988 An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis qualification matters.
989 
990 A learned atlas can therefore be parameterized within prescribed singular-value bounds, with synthesis tied to its inverse or pseudoinverse. This gives a construction contract. It is not a claim that every unconstrained training procedure respects those bounds.
991 
992 When atlases change over time but represent the same state, using the pullback norms avoids multiplying arbitrary chart condition numbers at every step. Exact transport telescopes to the direct transport. Numerical synthesis errors and genuinely inconsistent predictions remain defects and must still be bounded.
993 
994 # 23. Refinement cannot manufacture missing information
995 
996 An atlas change is an invertible recoding of the same represented information. Predictive refinement usually adds information. Treating these as identical operations would undo the central distinction of the foundation.
997 
998 **Proposition 22. The upward-lift obstruction.** Let $\gamma:X\to Z$ be a coarse representation and $\eta:X\to W$ a proposed refined representation. There is a deterministic map $J:\gamma(X)\to\eta(X)$ satisfying $\eta=J\gamma$ if and only if
999 
1000 $$
1001 \gamma(x)=\gamma(x')\ \Longrightarrow\ \eta(x)=\eta(x')
1002 \quad\text{for every }x,x'.
1003 $$
1004 
1005 For linear representations, this is equivalent to $\ker\gamma\subseteq\ker\eta$. If two states share a coarse representation but their refined representations differ, any predicted refined value $w$ has worst-case error at least
1006 
1007 $$
1008 \max\bigl(\|w-\eta(x)\|,\|w-\eta(x')\|\bigr)
1009 \geq\tfrac12\|\eta(x)-\eta(x')\|.
1010 $$
1011 
1012 **Proof.** A lift must be constant on each coarse fiber. Conversely, the stated condition defines it unambiguously on every fiber. The linear version is the kernel factorization criterion. The lower bound follows from the triangle inequality between the two refined values through $w$. $\square$
1013 
1014 For a strict refinement, the coarse state is recoverable from the refined one but not conversely. In the linear case this typically means $\ker\eta\subsetneq\ker\gamma$, exactly the opposite of the condition required for an upward lift.
1015 
1016 For example, $\gamma(x_1,x_2)=x_1$ and $\eta(x_1,x_2)=(x_1,x_2)$. The states $(0,1)$ and $(0,-1)$ have the same coarse state, but any common point initialization has worst-case error at least one in Euclidean norm. Padding a vector with zeros is a choice, not information recovery.
1017 
1018 A refinement step therefore needs one of three explicitly distinguished inputs: an archive that already contains the relevant observations, new observations, or a set-valued or probabilistic state retaining uncertainty about the missing details. An exact archive of measured outputs can initialize a longer history when those samples were stored. It cannot recover a latent coordinate that never affected any measurement.
1019 
1020 For adaptive prediction, an attempted lift $J_t$ contributes its own initialization defect
1021 
1022 $$
1023 \eta_t(x_t)-J_t\gamma_t(x_t).
1024 $$
1025 
1026 It belongs in Proposition 18's initial error or in the defect of the composed transition. It cannot be set to zero merely because a finer packet tree has been allocated.
1027 
1028 A data-dependent representation schedule also requires care. The pathwise bound applies to the realized schedule when all of its hypotheses hold along that schedule. Comparing two different schedules chosen by true and predicted histories introduces a selection discrepancy. No uniform statistical consistency of adaptive selection follows solely from the deterministic propagation law.
1029 
1030 # 24. A growing operator grammar with coherent evidence
1031 
1032 A large algebraic signal model should be able to add operator programs without changing the meaning of its existing evidence. This requires more than assigning a new softmax weight when a candidate is introduced.
1033 
1034 Let $\mathcal G$ be a countable declared program set, with one fixed probability mass function $\pi(g)>0$ satisfying $\sum_g\pi(g)=1$. Every program specifies a likelihood $L_g(D)$ for the **same** observed data event or density relative to a common base measure. Assume
1035 
1036 $$
1037 0<Z(D)=\sum_{g\in\mathcal G}\pi(g)L_g(D)<\infty.
1038 $$
1039 
1040 Let finite active sets $\mathcal G_n$ increase to $\mathcal G$. Their unnormalized evidence is $Z_n=\sum_{g\in\mathcal G_n}\pi(g)L_g(D)$. Only stages with $Z_n>0$ are considered.
1041 
1042 **Proposition 23. Exact accounting for an enumerated grammar.** Define the full posterior $P(g\mid D)=\pi(g)L_g(D)/Z$ and extend the active posterior $P_n(g\mid D)=\pi(g)L_g(D)/Z_n$ by zero outside $\mathcal G_n$. Then
1043 
1044 $$
1045 \boxed{
1046 \|P-P_n\|_{\mathrm{TV}}=\tau_n,
1047 \qquad \tau_n=1-\frac{Z_n}{Z}.
1048 }
1049 $$
1050 
1051 For any common family of candidate predictive probability kernels, the corresponding predictive mixtures differ in total variation by at most $\tau_n$. If the omitted unnormalized evidence has a bound
1052 
1053 $$
1054 \sum_{g\notin\mathcal G_n}\pi(g)L_g(D)\leq b_n,
1055 $$
1056 
1057 then
1058 
1059 $$
1060 \tau_n\leq\frac{b_n}{Z_n+b_n}.
1061 $$
1062 
1063 **Proof.** On the active set, $P=(Z_n/Z)P_n$, so the total mass of the difference there is $\tau_n$. The mass outside is also $\tau_n$. Half the sum of absolute differences is therefore $\tau_n$. Applying the same probability kernel to two mixing measures cannot increase total variation: integrate any event probability, a function taking values in $[0,1]$, against their difference. For the final bound, write $Z=Z_n+W_n$ with $0\leq W_n\leq b_n$ and use that $W_n/(Z_n+W_n)$ increases with $W_n$. $\square$
1064 
1065 Since the active sets exhaust the grammar and the evidence sum is finite, $Z_n\uparrow Z$ and $\tau_n\downarrow0$. This is posterior approximation *inside the declared grammar*. It does not establish that the real signal-generating mechanism belongs to it.
1066 
1067 For a discrete data sequence, a probability likelihood is at most one, so the omitted prior mass is a valid $b_n$. A continuous-data density need not be at most one. It requires a density envelope or another justified bound on omitted evidence. A small unnormalized tail can also carry substantial posterior mass when the active evidence is very small.
1068 
1069 The fixed prior need not be uniform. For an explicit enumeration $g_1,g_2,\ldots$, $\pi(g_k)=2^{-k}$ is a normalized example, with omitted prior mass $2^{-n}$ after $n$ programs. It expresses a particular enumeration bias, not a canonical prior on mathematical meaning.
1070 
1071 A program outside the enumerated prefix can be accounted for only when its prior and likelihood are defined or bounded. A genuinely outside-grammar component needs its own model. Assigning an unexplained residual probability is not a Bayesian treatment of missing mechanisms.
1072 
1073 ## 24.1 Refining a program without double counting it
1074 
1075 A representation-rich language can contain many syntactic versions of one mechanism. It can also split one broad mechanism into more specific children. These operations must conserve prior mass when they are intended as a re-expression of the same model.
1076 
1077 **Proposition 24. Prior-preserving program refinement.** Suppose a parent program $g$ is replaced by children $g_j$ with weights $\alpha_j\geq0$, $\sum_j\alpha_j=1$, and suppose its likelihood is the mixture
1078 
1079 $$
1080 L_g(D)=\sum_j\alpha_jL_{g_j}(D).
1081 $$
1082 
1083 Give child $j$ prior mass $\pi(g)\alpha_j$ and leave all other programs unchanged. Then total evidence is unchanged and the sum of the children's posterior masses equals the former parent's posterior mass. Posterior predictive distributions are also unchanged when the same mixture relation holds for the joint likelihood of $(D,y)$ for every predicted event $y$.
1084 
1085 **Proof.** The children's total unnormalized evidence is
1086 $\sum_j\pi(g)\alpha_jL_{g_j}(D)=\pi(g)L_g(D)$.
1087 Thus both the total normalizer and every other program's posterior are unchanged. The same sum for joint evidence gives the predictive statement after normalization by the unchanged evidence of $D$. $\square$
1088 
1089 The mixture identity must hold for the **whole data record**. Averaging one-step likelihoods independently at each time generally describes a switching mechanism, not a fixed child selected once for the whole sequence.
1090 
1091 If the children are only exact atlas recodings of the same operator and data model, their likelihoods are identical. Splitting the original prior mass leaves every prediction unchanged. Giving each duplicate a fresh full prior mass instead changes the model's prior preference. That may be an intentional new prior, but it is not invariance under representation.
1092 
1093 The implementation's existing held-out predictive weights, described in Section 10, are not replaced by these identities merely by calling them posterior weights. The proposition supplies the probabilistic semantics that a future grammar-weighting implementation would need to enforce.
1094 
1095 # 25. The finite learned-realization theorem
1096 
1097 The preceding results combine into a concrete completion of one part of the learning problem. This is the integrated identification claim for the autonomous linear sector.
1098 
1099 **Proposition 25. Observationally grounded finite linear realization.** Assume all of the following:
1100 
1101 1. The data come from an autonomous finite-dimensional linear process with declared outputs $y_t=Cx_t$.
1102 2. A delay length $h$ satisfies $\operatorname{rank}O_{h+1}=\operatorname{rank}O_h$.
1103 3. Exact observed input windows in $Z_-$ span $V_h=\operatorname{im}O_h$, and $Z_+$ records their true one-step successor windows.
1104 
1105 Then an algorithm using only $(Z_-,Z_+)$ recovers the unique transition on $V_h$. Initialized with a fully observed length-$h$ window, it predicts every later output exactly. Any lossless atlas of $V_h$, any complete packet decomposition of that space, and every coherently transported realization of the learned transition preserve the same predictions. Hidden variables outside the observable quotient need not be recovered.
1106 
1107 If the fixed-space assumptions of Proposition 16 replace exact observations, that proposition bounds operator error. Whenever the resulting transition defects and Lipschitz constants satisfy Proposition 18, its product-sum bound controls the output prediction error. A common contraction margin yields the stated uniform bound.
1108 
1109 **Proof.** Proposition 14 gives the unique true transition $D_h$ on $V_h$. Because the observed windows span $V_h$, Proposition 15 identifies exactly its restriction, which is the entire required operator. Equality of the learned and true states follows by induction after initialization. The current output is the last block of the history state. Propositions 1--4 preserve state reconstruction, transport and complete decomposition, so they preserve that output sequence. The perturbation statement is the direct composition of Propositions 16 and 18 with the declared output decoder. $\square$
1110 
1111 This establishes an actual route from observations to a composable algebraic signal model under explicit finite assumptions. It is more than a fit on a few points. It is also less than unrestricted nonlinear world-model discovery. In particular, the algorithm does not infer assumptions 1--3 merely by returning a matrix. The accompanying certificate keeps the data-verified identities separate from those model assumptions.
1112 
1113 # 26. Connection to controlled prediction
1114 
1115 Proposition 25 completes the autonomous observable linear sector under its declared coverage assumptions. It does not establish a common delay operator for arbitrary actions. Part III constructs controlled predictive states from interventional experiment probabilities. The two results are compatible routes, not interchangeable learning assumptions.
1116 
1117 \newpage
1118 
1119 # Part III. Controlled stochastic realization
1120 
1121 The next step is not to rename a deterministic predictor as a probabilistic one. A stochastic signal model must say what its state predicts, which actions intervene on the process, how observations update the state, and why every resulting number is a probability. These are separate obligations. A low dimensional linear representation can satisfy its reconstruction equations while failing probability positivity.
1122 
1123 Part II established observable learning for an autonomous linear signal. Here the observable objects are probabilities of finite action and observation experiments. We derive a finite realization from them, show when it is minimal, give an explicit probability-valid state domain, and separate exact reconstruction from noisy estimation. The final construction connects back to Komorebi's packets through a normalized closure-defect identity.
1124 
1125 The classical antecedents are predictive state representations, observable operator models and finite-rank realization [R5, R7, R9]. Those traditions already use probabilities of tests and products of operators. The purpose here is to make their assumptions and consequences explicit within the realization calculus. The twelve propositions below are a connected development, not twelve claims of historical novelty. No statement here replaces a proof with a numerical check.
1126 
1127 
1128 
1129 # 27. Actions and observations are different inputs
1130 
1131 Let $\mathcal A$ and $\mathcal O$ be finite nonempty action and observation sets. A symbol $s=(a,o)$ records an action followed by an observation. A word $w=s_1\cdots s_k$ has temporal order from left to right. The empty word is $\epsilon$.
1132 
1133 Write
1134 
1135 $$
1136 f(w)=\Pr(o_1,\ldots,o_k\mid \operatorname{do}(a_1,\ldots,a_k)).
1137 $$
1138 
1139 The process is reset to one specified initial distribution before this experiment. The actions in this definition are fixed externally. This is not, in general, the distribution obtained by conditioning a passive record on an adaptively selected sequence of actions. We assume the consistency laws
1140 
1141 $$
1142 f(\epsilon)=1,\qquad f(w)\geq0,\qquad
1143 \sum_{o\in\mathcal O}f(w(a,o))=f(w)
1144 \quad\text{for every }w,a.
1145 $$
1146 
1147 These laws specify causal probabilities on finite observation trees. A fixed history-dependent policy can be evaluated by selecting the action on each branch and multiplying the appropriate conditional probabilities. This does not establish optimality of that policy or its identification from insufficient data.
1148 
1149 ## 27.1 A finite hidden-state construction
1150 
1151 A concrete realization has a belief $p\in\mathbb R^n_+$ with $\mathbf1^\top p=1$. For each $s=(a,o)$, let $M_s\in\mathbb R^{n\times n}$ be entrywise nonnegative, with
1152 
1153 $$
1154 \mathbf1^\top\sum_oM_{a,o}=\mathbf1^\top.
1155 $$
1156 
1157 The matrix entry $M_{a,o}(j,i)$ is the probability of next hidden state $j$ and observation $o$, given current hidden state $i$ and chosen action $a$. This is a discrete probability instrument. It includes an action-specific transition followed by an observation, but does not require that factorization.
1158 
1159 Put $M_w=M_{s_k}\cdots M_{s_1}$. Then
1160 
1161 $$
1162 f(w)=\mathbf1^\top M_wp_0,\qquad
1163 p^+=\frac{M_sp}{\mathbf1^\top M_sp}
1164 $$
1165 
1166 on branches of positive probability. An observation with zero probability has no conditional posterior under that model. It must not be handled by silently dividing by an arbitrary numerical floor.
1167 
1168 **Proposition 26. Finite instrument closure.** Starting with $W_0=\operatorname{span}\{\mathbf1^\top\}$, define
1169 
1170 $$
1171 W_{j+1}=W_j+\sum_{s\in\mathcal A\times\mathcal O}W_jM_s.
1172 $$
1173 
1174 This stabilizes after at most $n-1$ strict enlargements at
1175 
1176 $$
1177 W_* = \operatorname{span}\{\mathbf1^\top M_w:w\text{ a finite word}\}.
1178 $$
1179 
1180 Choose a full row rank matrix $R$ with row space $W_*$ and first row $\mathbf1^\top$. There are unique matrices $B_s$ with
1181 
1182 $$
1183 RM_s=B_sR,\qquad \lambda=e_1^\top,\qquad \lambda R=\mathbf1^\top.
1184 $$
1185 
1186 For $z=Rp$, both the next-observation likelihood and the posterior predictive state close:
1187 
1188 $$
1189 \Pr(o\mid z,a)=\lambda B_{a,o}z,\qquad
1190 z^+=\frac{B_{a,o}z}{\lambda B_{a,o}z}.
1191 $$
1192 
1193 Every linear summary determining all word probabilities by linear readouts for every initial belief has row space containing $W_*$.
1194 
1195 **Proof.** The nested-space argument from Proposition 8 applies to the finite family $M_s$. Multiplication of a test row by $M_s$ prepends the symbol $s$ to that test, so the stable span is exactly the displayed word span. Invariance supplies the matrices $B_s$, and full row rank makes them unique. Apply $R$ to the unnormalized and normalized hidden-state updates. If all test probabilities are linear functions of another summary, then each row $\mathbf1^\top M_w$ lies in its row space. Equality on all beliefs implies equality on their linear span, which is $\mathbb R^n$. $\square$
1196 
1197 This proposition uses a known hidden-state model. It is not yet an observable-data learner. It also concerns all initial beliefs, whereas a model initialized at one fixed $p_0$ can have a smaller reachable predictive dimension. The next construction handles that distinction without requiring recovery of hidden states.
1198 
1199 
1200 
1201 # 28. The state is a function of possible future experiments
1202 
1203 For a history $h$, define its unnormalized future-behavior column
1204 
1205 $$
1206 v_h(t)=f(ht),
1207 $$
1208 
1209 where $t$ ranges over every finite test word. Form the infinite history/test matrix
1210 
1211 $$
1212 \mathscr H_{t,h}=f(ht).
1213 $$
1214 
1215 Its rank means the dimension of the span of its columns, equivalently the supremum of the ranks of finite submatrices. The columns are functions, not vectors obtained by physically observing every future. Their finite coordinate realization becomes learnable only when enough experiment probabilities are available.
1216 
1217 **Proposition 27. Minimal finite behavior realization.** If $\operatorname{rank}\mathscr H=r<\infty$, the vector space $V=\operatorname{span}\{v_h\}$ has dimension $r$. There are well-defined linear maps $B_s:V\to V$, an initial vector $v_\epsilon$, and a linear mass functional $\lambda(v)=v(\epsilon)$ satisfying
1218 
1219 $$
1220 B_sv_h=v_{hs},\qquad
1221 f(s_1\cdots s_k)=\lambda B_{s_k}\cdots B_{s_1}v_\epsilon.
1222 $$
1223 
1224 Every linear realization of $f$ has dimension at least $r$. Any two reachable and observable realizations of dimension $r$ differ by an invertible change of coordinates. For $f(h)>0$, the normalized state $v_h/f(h)$ determines every future test probability conditional on $h$ and the declared future interventions.
1225 
1226 **Proof.** If $\sum_hc_hv_h=0$, then for every test $t$,
1227 
1228 $$
1229 \sum_hc_hv_{hs}(t)=\sum_hc_hf(hst)=0,
1230 $$
1231 
1232 because $st$ is another test. Thus shifting columns preserves all linear relations and defines $B_s$. The probability formula follows by induction. For any $d$-dimensional realization, the history/test matrix factors into test rows and reachable state columns, so its rank is at most $d$.
1233 
1234 For uniqueness, map the reachable vector after each history in one minimal realization to the corresponding vector in the other. If a linear combination vanishes in the first, every future test evaluates to zero on it. Those same evaluations vanish in the second, where observability makes the combination zero. The map is therefore well-defined and bijective, and intertwines every symbol operator and both boundary vectors. Finally $f(ht)/f(h)$ is exactly the conditional experiment probability. $\square$
1235 
1236 The coordinate dimension $r$ includes probability mass. Normalized states lie in the affine hyperplane $\lambda z=1$, so their affine dimension is at most $r-1$. A finite hidden-state model of size $n$ has behavior rank at most $n$. The theorem itself requires finite behavior rank, not a preferred hidden-state ontology.
1237 
1238 Two realized histories with equal normalized future columns are equivalent for every declared future test. This is a predictive equivalence, not equality of their physical pasts or lossless archives.
1239 
1240 # 29. Recovering the operators from an observable probability table
1241 
1242 Select $r$ tests $t_1,\ldots,t_r$ and $r$ histories $h_1,\ldots,h_r$, with $t_1=\epsilon$, for which the square matrix
1243 
1244 $$
1245 H_{ij}=f(h_jt_i)
1246 $$
1247 
1248 is invertible. Such a choice exists under the rank hypothesis of Proposition 27. The nonzero empty-test row can be extended to a row basis. Select independent history columns for that basis.
1249 
1250 For each symbol $s$, also obtain the shifted table and initial test vector
1251 
1252 $$
1253 (G_s)_{ij}=f(h_jst_i),\qquad (b_0)_i=f(t_i).
1254 $$
1255 
1256 **Proposition 28. Observable table reconstruction.** Assume the full behavior rank is at most $r$ and the displayed $r\times r$ matrix $H$ is invertible. Then the full rank is exactly $r$, and
1257 
1258 $$
1259 \boxed{B_s=G_sH^{-1},\qquad \lambda=e_1^\top,\qquad b_0=(f(t_i))_i}
1260 $$
1261 
1262 is a minimal realization of every word probability. Its state after history $h$ is
1263 
1264 $$
1265 b(h)=(f(ht_i))_i=B_hb_0.
1266 $$
1267 
1268 The normalized state is $b(h)/f(h)$ when $f(h)>0$. Its update uses the formula in Proposition 26, with no hidden-state coordinates supplied to the reconstruction algorithm.
1269 
1270 **Proof.** Invertibility gives rank at least $r$, so the assumed upper bound is attained. Evaluation on the selected tests is an isomorphism $V\to\mathbb R^r$, because it maps the chosen basis of history columns to the invertible matrix $H$. The shifted images of those basis columns are precisely the columns of $G_s$, so $B_sH=G_s$. The first selected test evaluates the mass. Initial evaluation gives $b_0$. Proposition 27 supplies correctness for every history, not only the selected columns. $\square$
1271 
1272 There are two ways to justify coverage in this statement. One can know that the selected row and column families span the true behavior space. Alternatively, a justified upper bound $r$ on the process rank, together with the invertible $r\times r$ block, proves that coverage. The observed invertibility by itself only gives a lower bound. An unknown higher-rank process can agree on a finite table and differ later.
1273 
1274 Exact experiment probabilities are population quantities. The reference example below supplies rational tables from a declared process and keeps that table generator separate from the learner. Section 36 supplies one finite sampling protocol for estimating those entries. Neither a hidden matrix nor its probability cone is secretly passed to the observable learner.
1275 
1276 
1277 
1278 # 30. Probability validity is not a rank condition
1279 
1280 A real matrix product can yield a negative number. Therefore a fitted factorization is not a stochastic model merely because it is low rank or exactly interpolates a probability table. Spectral estimators can encounter negative-probability problems in practice [R10]. We address the structural issue directly.
1281 
1282 In the canonical behavior space $V$, define
1283 
1284 $$
1285 \mathcal C=\{v\in V:v(t)\geq0\text{ for every finite test }t\}.
1286 $$
1287 
1288 **Proposition 29. Canonical probability cone.** The set $\mathcal C$ is a closed convex pointed cone that spans $V$. Every $B_s$ preserves it. The mass functional is strictly positive on its nonzero elements, and
1289 
1290 $$
1291 \sum_o\lambda B_{a,o}=\lambda.
1292 $$
1293 
1294 Consequently the normalized base
1295 
1296 $$
1297 \mathcal K=\{v\in\mathcal C:\lambda v=1\}
1298 $$
1299 
1300 is a compact convex probability-valid state space. Every positive-probability branch maps $\mathcal K$ into itself by normalized updating. In the core-test coordinates of Proposition 28, $\mathcal K\subseteq\{1\}\times[0,1]^{r-1}$.
1301 
1302 **Proof.** Each test evaluation is a continuous linear functional on finite-dimensional $V$, so the cone is closed and convex. If both $v$ and $-v$ belong to it, every test value is zero and $v=0$. It spans $V$ because it contains every column $v_h$. Also $(B_sv)(t)=v(st)$, proving invariance.
1303 
1304 The consistency law $\sum_ov(t(a,o))=v(t)$ holds for each history column and hence throughout $V$. Iterating it along any fixed action schedule expresses $v(\epsilon)$ as a sum of nonnegative values for all observation words on that schedule. If $v(\epsilon)=0$, each summand is zero, for every schedule and length, so $v=0$. The same sums show $0\leq v(t)\leq1$ when $\lambda v=1$. They also give mass conservation. The base is closed and bounded in the finite core coordinates and thus compact. Invariance and positive normalization prove the posterior assertion. $\square$
1305 
1306 This cone is defined by all finite tests. Its existence is not an algorithm for finding a finite list of inequalities. A finite list gives an outer approximation and can miss a longer-test violation. The cone also need not equal the convex hull of the states reachable from the fixed initial distribution.
1307 
1308 A practical sufficient certificate can instead use a known closed cone $C_0$ with nonempty normalized base, a strictly positive mass functional, $b_0\in C_0$, $B_sC_0\subseteq C_0$, and mass conservation. A nonnegative hidden-state realization supplies one after a coordinate change. Such a certificate may use information unavailable to an observable-only learner and must be labeled accordingly.
1309 
1310 Negative entries of $B_s$ do not, by themselves, prove invalidity. Coordinates can be signed while the represented state remains in the correct cone. Conversely, clipping negative coefficients can destroy the operator identities. Validity belongs to the pair consisting of the operators and their admissible cone, not to a coordinatewise heuristic.
1311 
1312 # 31. Packets can reconstruct a belief without being beliefs
1313 
1314 Let $P_1,\ldots,P_m$ be a complete family of linear packet maps on a predictive space, with $\sum_iP_i=I$. Algebraic reconstruction gives $z=\sum_iP_iz$. It does not imply that $P_iz$ is an unnormalized probability state.
1315 
1316 **Proposition 30. Positive packet decomposition and its boundary.** Suppose a cone $C$ has a strictly positive mass functional $\lambda$. If every packet map preserves $C$, then each normalized $z\in C$ decomposes as a genuine mixture
1317 
1318 $$
1319 z=\sum_{i:w_i>0} w_i z_i,\qquad
1320 w_i=\lambda P_iz\geq0,\qquad
1321 z_i=P_iz/w_i,\qquad\sum_iw_i=1.
1322 $$
1323 
1324 A zero weight has $P_iz=0$. In the standard cone $\mathbb R^n_+$, however, a complete family of cone-preserving idempotents must consist of diagonal coordinate-selection matrices. In particular, a nontrivial Fourier or Haar packet family cannot generally be reinterpreted as a mixture of probability states merely because it is orthogonal and complete.
1325 
1326 **Proof.** Cone preservation and strict positivity give the weights and the zero-weight claim. Apply $\lambda$ to the complete decomposition to obtain their sum. Normalization of each nonzero term gives the mixture.
1327 
1328 On $\mathbb R^n_+$, cone preservation is entrywise nonnegativity. Since the matrices sum to $I$, every off-diagonal entry in every matrix is zero. Idempotence forces each diagonal entry to be zero or one, and completeness makes the selections disjoint and exhaustive. $\square$
1329 
1330 For a two-state belief $p=(3/4,1/4)^\top$, the Haar detail projector
1331 
1332 $$
1333 P_{\mathrm d}=\frac12\begin{pmatrix}1&-1\\-1&1\end{pmatrix}
1334 $$
1335 
1336 produces $(1/4,-1/4)^\top$. It is a valid signed detail packet, but not a subprobability distribution. Its total mass is zero despite being nonzero, which cannot happen inside a cone with a strictly positive mass functional.
1337 
1338 This does not prohibit useful spectral packets. It fixes their type. Signed algebraic packets can be used for transfer, approximation and diagnostics. They must be synthesized before probability interpretation unless a separate positive decomposition has been established. That distinction prevents a reconstruction identity from being misused as a Bayesian mixture identity.
1339 
1340 
1341 
1342 # 32. Action coverage is a mathematical requirement
1343 
1344 **Proposition 31. Missing-intervention nonidentifiability.** There are finite controlled stochastic models that agree on every possible observation record generated using only action $a$, but disagree under another action $b$. Therefore even infinitely many passive records confined to $a$ cannot identify the unrestricted controlled model.
1345 
1346 **Proof.** Use one hidden state and two observations. In both models, action $a$ gives probabilities $(1/2,1/2)$. Under $b$, let the first model give $(1,0)$ and the second $(0,1)$. Every length-$k$ record under $a$ has probability $2^{-k}$ in both models. Their first observation under $b$ differs with probability one. $\square$
1347 
1348 This is a coverage obstruction, not an optimization problem. It remains even when the hidden dimension is known exactly. Richer transforms of the same action-$a$ observations do not create data about $b$.
1349 
1350 Action selection based on past observations requires a further distinction. Under a known policy $\pi$, an action/observation word has joint probability
1351 
1352 $$
1353 \Pr_\pi(w)=f(w)\prod_{j=1}^{|w|}\pi(a_j\mid h_{j-1}),
1354 $$
1355 
1356 provided the action randomization has no access to hidden information beyond the declared history. The policy factors depend on the observed branch. Conditioning on a selected action sequence generally reweights branches rather than removing these factors. Recovering intervention probabilities from such data requires an appropriate coverage and reweighting argument. The reset experiments in Section 36 avoid that additional inference problem.
1357 
1358 # 33. Learning the coordinate space under bounded noise
1359 
1360 Part II bounded operator error after a state space and basis were fixed. Here we also bound one step of data-dependent space selection. This remains a finite-matrix statement. It does not certify that a chosen finite history/test table captures the full process.
1361 
1362 Let $X\in\mathbb R^{m\times n}$ have rank $r\geq1$, let $s=\sigma_r(X)>0$, and observe $Y=X+E$ with $\|E\|_2\leq\varepsilon<s/2$. Let $P$ project onto $\operatorname{im}X$, and let $\widehat P$ project onto the leading $r$ left singular vectors of $Y$.
1363 
1364 **Proposition 32. Finite-table rank separation and subspace stability.** One has
1365 
1366 $$
1367 \sigma_r(Y)\geq s-\varepsilon,\qquad
1368 \sigma_{r+1}(Y)\leq\varepsilon,\qquad
1369 \|P-\widehat P\|_2\leq\frac{2\varepsilon}{s}<1.
1370 $$
1371 
1372 The $(r+1)$st singular value is interpreted as zero when $r$ is the maximum possible rank. If $U$ and $\widehat U$ are orthonormal bases of the two ranges, then
1373 
1374 $$
1375 \sigma_{\min}(\widehat U^\top U)
1376 \geq\sqrt{1-(2\varepsilon/s)^2}.
1377 $$
1378 
1379 Thus $x\mapsto\widehat U^\top x$ is injective on the true column space. Its inverse on that space is
1380 
1381 $$
1382 y\longmapsto U(\widehat U^\top U)^{-1}y,
1383 $$
1384 
1385 with norm at most $[1-(2\varepsilon/s)^2]^{-1/2}$.
1386 
1387 **Proof.** The min-max characterization of singular values and the triangle inequality imply that adding $E$ changes each singular value by at most $\varepsilon$. The leading-$r$ singular subspace gives $\|(I-\widehat P)Y\|_2=\sigma_{r+1}(Y)\leq\varepsilon$. Hence
1388 
1389 $$
1390 \|(I-\widehat P)X\|_2\leq2\varepsilon,
1391 \qquad
1392 \|(I-\widehat P)P\|_2
1393 =\|(I-\widehat P)XX^\dagger\|_2\leq2\varepsilon/s.
1394 $$
1395 
1396 For equal-rank orthogonal projections, the latter norm equals $\|P-\widehat P\|_2$. To see the equality, diagonalize $U^\top\widehat U$ by singular value decomposition. Each nontrivial paired principal plane contributes a sine to both norms, while common directions contribute zero. On the same planes the singular values of $\widehat U^\top U$ are the corresponding cosines. This proves the last bound and the inverse formula. $\square$
1397 
1398 A threshold $\tau$ with $\varepsilon<\tau<s-\varepsilon$ recovers the rank of this finite matrix. The unknown true gap must be justified or replaced by a valid confidence interval. Rank thresholds do not exclude modes absent from the selected tests or unexcited by the selected histories.
1399 
1400 The inverse formula contains the unknown true basis $U$. It proves identifiability and conditioning relative to that space, not that a learner already knows a perfect decoder. When using the estimated subspace as coordinates for a realization, all boundary vectors, symbol operators and probability constraints must be transformed together. Signed singular-vector coordinates are not new independent observations.
1401 
1402 
1403 
1404 # 34. Noisy operator reconstruction in a fixed core
1405 
1406 After choosing a core block, the square reconstruction formula yields a particularly direct error identity. It is useful even when the core was chosen on a separate discovery data set.
1407 
1408 Let $G_s=B_sH$ with $s_H=\sigma_{\min}(H)>0$. Observe
1409 
1410 $$
1411 \widetilde H=H+E,\qquad
1412 \widetilde G_s=G_s+F_s,\qquad
1413 \|E\|_2<s_H.
1414 $$
1415 
1416 **Proposition 33. Noisy core-block realization.** The estimate $\widehat B_s=\widetilde G_s\widetilde H^{-1}$ satisfies
1417 
1418 $$
1419 \widehat B_s-B_s=(F_s-B_sE)\widetilde H^{-1},
1420 $$
1421 
1422 and therefore
1423 
1424 $$
1425 \boxed{
1426 \|\widehat B_s-B_s\|_2
1427 \leq\frac{\|F_s\|_2+\|B_s\|_2\|E\|_2}{s_H-\|E\|_2}.
1428 }
1429 $$
1430 
1431 A bound using only the fitted operator and the same perturbation budgets is
1432 
1433 $$
1434 \|\widehat B_s-B_s\|_2
1435 \leq
1436 \frac{\|F_s\|_2+\|\widehat B_s\|_2\|E\|_2}{s_H}.
1437 $$
1438 
1439 If $\widetilde s=\sigma_{\min}(\widetilde H)$ is known and $\|E\|_2\leq e<\widetilde s$, the last denominator may be replaced by $\widetilde s-e$.
1440 
1441 **Proof.** Since $\widetilde H$ is invertible, subtract $B_s$ from its fitted expression and collect the terms. The least singular value of $\widetilde H$ is at least $s_H-\|E\|_2$. For the second identity, use
1442 
1443 $$
1444 (\widehat B_s-B_s)H=F_s-\widehat B_sE
1445 $$
1446 
1447 and multiply by $H^{-1}$. Finally $s_H\geq\widetilde s-e$ by singular-value perturbation. $\square$
1448 
1449 When the first test is empty, the mass row $\lambda=e_1^\top$ is fixed rather than estimated. Set the first component of the initial vector to one as required by $f(\epsilon)=1$. Errors in other initial test probabilities remain initial-state error. This convention removes an avoidable normalizer-estimation ambiguity but does not enforce cone preservation or mass conservation for the fitted symbol family.
1450 
1451 Numerical implementation should solve the linear systems rather than explicitly form a floating-point inverse. The theorem describes the exact algebraic estimator. Finite-precision solve error is an additional residual, which the reference script measures separately.
1452 
1453 # 35. Conditioning a prediction can amplify its error
1454 
1455 Let $C$ be a probability cone with compact normalized base $K$ and mass row $\lambda$. Work in a declared norm and its dual. Suppose true normalized states have norm at most $R$. Let a true branch have
1456 
1457 $$
1458 u=B_sz,\qquad p=\lambda u\geq\eta>0,\qquad z^+=u/p\in K.
1459 $$
1460 
1461 For a predicted state $\widehat z$, form $\widehat u=\widehat B_s\widehat z$, $\widehat p=\lambda\widehat u$, and define $d=\widehat u-u$. Write $e=\|\widehat z-z\|$, $\|\widehat B_s\|\leq L$, and $\|\widehat B_s-B_s\|\leq\epsilon_B$. Then
1462 
1463 $$
1464 \|d\|\leq Le+\epsilon_BR=:D.
1465 $$
1466 
1467 **Proposition 34. Normalization-aware filtering stability.** If $\|\lambda\|_*D<\eta$, then $\widehat p>0$ and
1468 
1469 $$
1470 \widehat z^+-z^+
1471 =\frac{(I-z^+\lambda)d}{\widehat p},
1472 $$
1473 
1474 so
1475 
1476 $$
1477 \boxed{
1478 \|\widehat z^+-z^+\|
1479 \leq\frac{(1+R\|\lambda\|_*)D}{\eta-\|\lambda\|_*D}.
1480 }
1481 $$
1482 
1483 Alternatively, if each realized fitted denominator is checked to satisfy $\widehat p_t\geq\widehat\eta_t>0$, the pathwise recursion obeys
1484 
1485 $$
1486 e_{t+1}\leq\alpha_te_t+\beta_t,\qquad
1487 \alpha_t=\frac{(1+R\|\lambda\|_*)L_t}{\widehat\eta_t},\qquad
1488 \beta_t=\frac{(1+R\|\lambda\|_*)\epsilon_{B,t}R}{\widehat\eta_t}.
1489 $$
1490 
1491 The product-sum bound of Proposition 18 applies. A uniform bound follows if these constants have a common contraction margin and bounded forcing. The algebraic bound alone does not certify that a fitted state or every unchosen branch is probability-valid.
1492 
1493 **Proof.** The branch likelihood differs by $\lambda d$, giving the lower bound on $\widehat p$. Subtract $u/p$ from $(u+d)/(p+\lambda d)$ and use $u=pz^+$. The induced norm of $I-z^+\lambda$ is at most $1+R\|\lambda\|_*$. Applying that identity at each step gives the recursion. $\square$
1494 
1495 Rare evidence can create a large posterior change from a small absolute error. For example, take a true unnormalized vector $(\eta,0)$ and a perturbed vector $(\eta,\eta)$. Their $\ell^1$ difference is $\eta$, while their normalized vectors differ by one in $\ell^1$. Both arise as possible branch outputs of valid probability instruments when $0<\eta\leq1/2$, after assigning the remaining mass to another branch. There is no uniform posterior Lipschitz constant independent of a probability margin.
1496 
1497 When a known closed convex feasible base is available, Euclidean projection of a tentative posterior back onto that base cannot increase its distance from the true posterior. This is a declared repair to the estimator, not an identity for an unmodified update. It also does not repair negative branch probabilities of arbitrary fitted operators before normalization.
1498 
1499 
1500 
1501 # 36. A finite experimental route to the probability table
1502 
1503 The following deliberately simple protocol makes a finite-sample statement possible without confusing a deterministic identity with a statistical guarantee.
1504 
1505 Fix positive integers $M$ and $N$ and choose $M$ finite words before estimation. For each word, reset independently to the same initial distribution, execute its prescribed action sequence, and record the indicator that all observations match. Obtain $N$ independent trials per word. Dependencies between different words' estimates are allowed, but the $N$ trials used for each individual mean must be independent with the declared success probability.
1506 
1507 **Proposition 35. Reset-experiment probability certificate.** Let $\widehat f(w)$ be the sample mean for each selected word. For $0<\delta<1$, with probability at least $1-\delta$, all $M$ estimates satisfy
1508 
1509 $$
1510 |\widehat f(w)-f(w)|\leq
1511 \epsilon_N,\qquad
1512 \epsilon_N=\sqrt{\frac{\log(2M/\delta)}{2N}}.
1513 $$
1514 
1515 If the selected words cover every entry of $H$, every $G_s$, and $b_0$, then on the same event
1516 
1517 $$
1518 \|\widetilde H-H\|_2\leq r\epsilon_N,\qquad
1519 \|\widetilde G_s-G_s\|_2\leq r\epsilon_N,\qquad
1520 \|\widetilde b_0-b_0\|_2\leq\sqrt r\epsilon_N.
1521 $$
1522 
1523 These bounds feed Propositions 32 and 33 whenever their respective gap conditions hold. The empty-word probability may be set exactly to one instead of sampled.
1524 
1525 **Proof.** For a centered Bernoulli variable $X-p$, write $g(t)=\log\mathbb E\exp(t(X-p))$. Then $g(0)=g'(0)=0$, and $g''(t)$ is the variance of a Bernoulli under exponential tilting, at most $1/4$. Integrating twice gives $g(t)\leq t^2/8$. Independence and the exponential Markov inequality give an upper-tail bound $\exp(-2N\epsilon^2)$ after optimizing $t=4\epsilon$. The lower tail has the same bound. A union bound over the $M$ words proves the simultaneous result. The spectral norm is bounded by the Frobenius norm, yielding the matrix and vector estimates. $\square$
1526 
1527 This is the Bernoulli form of Hoeffding's inequality [R11], proved here for the protocol used. It is a conservative entrywise-to-spectral bound, not a claim of optimal sample complexity. The number and lengths of chosen experiments are part of its cost.
1528 
1529 If core tests and histories were selected from data, use a separate independent discovery sample and freeze the selected finite experiment list before these estimation trials. Conditional on that selection, the proposition applies. Alternatively, a uniform bound over a fixed larger candidate list can support data-dependent selection within that list. Reusing a post-selected table without either argument changes the claim.
1530 
1531 A single dependent trajectory, an adaptive stopping rule, unknown resets, drifting sensors, or experiments chosen without coverage do not satisfy this proposition as stated. Those settings require their own concentration and identifiability arguments. Synthetic reset sampling in the reference example demonstrates the protocol, not its validity for every measurement source.
1532 
1533 # 37. A filtering defect with a packet interpretation
1534 
1535 The relation to the realization calculus is particularly direct before normalization. Let $R$ be a linear summary of a hidden probability space, let $\lambda R=\mathbf1^\top$, and let $B_s$ be a proposed reduced branch operator. For a true belief $p$, put
1536 
1537 $$
1538 z=Rp,\qquad
1539 \delta_s=(RM_s-B_sR)p.
1540 $$
1541 
1542 The defect $\delta_s$ is a vector in the reduced representation, not a probability of model failure. It records the failure of the unnormalized instrument to commute with the summary.
1543 
1544 **Proposition 36. Exact normalized closure defect.** Suppose the true and proposed branch likelihoods
1545 
1546 $$
1547 a=\mathbf1^\top M_sp>0,\qquad
1548 b=\lambda B_sz>0
1549 $$
1550 
1551 are positive, and let $\widehat z^+=B_sz/b$. Then
1552 
1553 $$
1554 \boxed{
1555 R\left(\frac{M_sp}{a}\right)-\widehat z^+
1556 =\frac{(I-\widehat z^+\lambda)\delta_s}{a},
1557 \qquad a-b=\lambda\delta_s.
1558 }
1559 $$
1560 
1561 If $M_s\neq0$ and both likelihoods and normalized updates agree on every strictly positive belief, then $RM_s=B_sR$. Agreement of normalized updates alone need not force this equality.
1562 
1563 **Proof.** The true unnormalized summary is $B_sz+\delta_s$, with mass $a=b+\lambda\delta_s$. Subtract the two normalized vectors and collect terms. Since a nonzero nonnegative branch has positive likelihood on every strictly positive belief, agreement permits multiplication by the common mass. This gives equality of unnormalized summaries on the interior of the probability simplex, whose linear span is the whole hidden vector space. For the last claim, a one-state model can assign different positive branch probabilities while its normalized state always remains the scalar one. $\square$
1564 
1565 For an orthogonal retained projector $P$ preserving the mass row, put $Q=I-P$, $R=P$ on its image and $B_s=PM_sP$. Then
1566 
1567 $$
1568 \delta_s=PM_sQp.
1569 $$
1570 
1571 This is the unresolved-to-retained transfer block from Part I. Its posterior effect is not simply its norm: normalization applies the oblique map $I-\widehat z^+\lambda$ and divides by the true branch probability. A component proportional to $\widehat z^+$ changes only the branch mass and disappears from the normalized defect, while still changing likelihood-based inference over mechanisms.
1572 
1573 A certificate should therefore retain both $\lambda\delta_s$ and the posterior defect. A posterior that looks correct can hide a wrong evidence model. This is the stochastic counterpart of the distinction between reconstruction and predictive closure.
1574 
1575 
1576 
1577 # 38. A complete two-action rational example
1578 
1579 Use two hidden states only to generate a reference process. The learner does not receive these matrices. Let
1580 
1581 $$
1582 T_0=\begin{pmatrix}3/4&1/4\\1/4&3/4\end{pmatrix},\qquad
1583 T_1=\begin{pmatrix}1/4&2/3\\3/4&1/3\end{pmatrix},\qquad
1584 p_0=\begin{pmatrix}2/3\\1/3\end{pmatrix}.
1585 $$
1586 
1587 The observation matrices are
1588 
1589 $$
1590 O_0=\operatorname{diag}(1/4,3/4),\qquad
1591 O_1=\operatorname{diag}(3/4,1/4),\qquad
1592 M_{a,o}=O_oT_a.
1593 $$
1594 
1595 The two transitions do not commute. Thus the process is genuinely action-dependent rather than one autonomous recurrence with an action label attached.
1596 
1597 Choose tests $(\epsilon,(0,1))$ and histories $(\epsilon,(0,1))$. The observable tables give
1598 
1599 $$
1600 H=\begin{pmatrix}1&13/24\\13/24&5/16\end{pmatrix},\qquad
1601 \det H=11/576,\qquad
1602 b_0=\begin{pmatrix}1\\13/24\end{pmatrix}.
1603 $$
1604 
1605 The selected coordinate $q$ is the probability of seeing observation one after choosing action zero. In the hidden realization,
1606 
1607 $$
1608 R=\begin{pmatrix}1&1\\5/8&3/8\end{pmatrix},\qquad
1609 z=Rp=(1,q)^\top.
1610 $$
1611 
1612 The observable reconstruction $B_s=G_sH^{-1}$ yields
1613 
1614 $$
1615 \begin{aligned}
1616 B_{0,0}&=\begin{pmatrix}1&-1\\11/32&-1/4\end{pmatrix},&
1617 B_{0,1}&=\begin{pmatrix}0&1\\-3/32&3/4\end{pmatrix},\\[4pt]
1618 B_{1,0}&=\begin{pmatrix}5/48&5/6\\23/192&5/24\end{pmatrix},&
1619 B_{1,1}&=\begin{pmatrix}43/48&-5/6\\37/64&-5/8\end{pmatrix}.
1620 \end{aligned}
1621 $$
1622 
1623 The coefficients include negatives. Nevertheless, every operator preserves the cone $R\mathbb R^2_+$, because $B_sR=RM_s$. A sufficient normalized base is
1624 
1625 $$
1626 K_0=\{(1,q)^\top:3/8\leq q\leq5/8\}.
1627 $$
1628 
1629 This cone certificate uses the known generating model and is checked separately. It is not inferred merely from the finite probability table, and $K_0$ is not claimed to be the maximal canonical base of Proposition 29.
1630 
1631 The likelihoods and posterior maps can now be read from the matrices. For action zero and observation one,
1632 
1633 $$
1634 p(1\mid z,0)=q,\qquad
1635 q^+=\frac{-3/32+(3/4)q}{q}
1636 =\frac34-\frac{3}{32q}.
1637 $$
1638 
1639 Starting at $q_0=13/24$, this gives probability $13/24$ and $q_1=15/26$. If action one then produces observation zero, its probability is
1640 
1641 $$
1642 \frac5{48}+\frac56\frac{15}{26}=\frac{365}{624},
1643 $$
1644 
1645 and the new predictive coordinate is
1646 
1647 $$
1648 q_2=\frac{23/192+(5/24)(15/26)}{365/624}=\frac{599}{1460}.
1649 $$
1650 
1651 The joint probability of this two-step action/observation word is $365/1152$. The reference checks compare these values and every word up to a declared finite test length against the hidden generator in exact rational arithmetic. The proposition, under its rank and coverage assumptions, is what extends the result to all word lengths.
1652 
1653 The model's online state is not a sequence of newly measured future probabilities. It is a recursively maintained prediction initialized from learned experiment probabilities and updated after each actual action and observation. Uncertainty in those learned parameters is separate from the stochastic uncertainty in the next observation.
1654 
1655 
1656 
1657 # 39. Integrated controlled stochastic realization
1658 
1659 **Proposition 37. Conditional observation-based stochastic completion.** Suppose a causal finite-alphabet signal process has behavior rank at most $r$, and a selected $r\times r$ history/test block containing the empty test is invertible. Exact values of that block, every one-symbol shifted block, and the initial core-test vector determine a minimal realization of every future experiment probability. Its normalized state is recursively accessible after each positive-probability history, without reconstructing the hidden physical state.
1660 
1661 The exact realization has the canonical probability cone of Proposition 29. Under any invertible atlas, transform the state, symbol operators, mass row and cone together:
1662 
1663 $$
1664 z'=Tz,\qquad B_s'=TB_sT^{-1},\qquad
1665 \lambda'=\lambda T^{-1},\qquad C'=TC.
1666 $$
1667 
1668 Every likelihood and posterior prediction is then unchanged. A complete packet decomposition preserves these predictions after full synthesis. Packetwise probability interpretation requires the extra positivity condition of Proposition 30.
1669 
1670 For estimates obtained under Proposition 35's reset protocol, the simultaneous event there bounds the finite table perturbations. A separated finite-table spectrum yields Proposition 32's subspace certificate. A well-conditioned fixed core yields Proposition 33's operator certificate. On branches satisfying Proposition 34's likelihood margins, the resulting normalized state errors obey its recursive bound. These perturbation statements alone do not certify a learned cone or global positivity of an estimated model.
1671 
1672 **Proof.** Proposition 28 reconstructs the unique minimal behavior realization, and Proposition 27 gives its conditional interpretation. Proposition 29 supplies probability validity for the exact realization. Substituting the transformed quantities cancels every $T^{-1}T$, including in the scalar normalization. Full packet synthesis is the identity, whereas Proposition 30 characterizes the additional probability-mixture condition. The perturbation conclusions are the stated chain of propositions with their separate assumptions retained. No missing gap, coverage or positivity assumption follows from matrix inversion alone. $\square$
1673 
1674 This theorem closes a finite controlled and stochastic sector of the foundational model. It does not claim unrestricted discovery of a nonlinear environment from a single trajectory. It does identify what a learned predictive state represents, why its operations compose, where its evidence comes from, and which defects change likelihoods or posteriors.
1675 
1676 # 40. From realization to admissible domains
1677 
1678 Proposition 37 identifies a finite controlled realization from a covered population table. Existence of its canonical probability cone does not provide a finite certificate for a fitted model. Part IV supplies such certificates in finite positive-lift sectors, while keeping internal validity distinct from statistical coverage.
1679 
1680 \newpage
1681 
1682 # Part IV. Certified probability domains and robust learning
1683 
1684 A matrix learned from observations is not yet a probability model. Part III made that distinction exact: a finite predictive realization needs an admissible cone preserved by its action and observation operators. It supplied a canonical cone defined by all possible future tests, but did not supply a finite way to verify that cone for an estimated model.
1685 
1686 This block develops a constructive finite sector of that missing step. It gives a certificate that can be checked by rational arithmetic, a search procedure that does not need hidden-state matrices, and a robust version that covers an entire declared uncertainty set. It also proves why a finite polyhedral certificate is not available for every valid finite-dimensional process. The resulting distinction is between an exact finite certificate, an uncertified candidate, and an explicitly modified approximation. Those are different mathematical outcomes.
1687 
1688 The positive-realization and invariant-cone ingredients are classical. Jaeger's account includes the polyhedral-cone characterization and an irrational-rotation obstruction [R12]. Polyhedral invariance is also a standard subject of linear systems and optimization [R13]. The present contribution is a self-contained connection of these ingredients to Komorebi's observable learner, probability packets, confidence sets and approximation defects. The controlled rational rotation example is an adaptation of that classical obstruction, not a claim of discovering the phenomenon.
1689 
1690 The next twelve propositions continue the numbering from the preceding volume. All have proofs below. The exact computations check stated finite instances and counterexamples. They are not a Lean formalization and do not change the production PR-LASM 2.3.0 runtime.
1691 
1692 
1693 
1694 # 41. The object to be certified
1695 
1696 Fix a real predictive space $E=\mathbb R^r$, a nonzero row $\lambda$, and a finite family $B_s$, indexed by $s=(a,o)$. The action and observation alphabets are finite and nonempty. Every declared observation branch is present, including branches represented by the zero operator. Temporal words act from right to left:
1697 
1698 $$
1699 B_{s_1\cdots s_k}=B_{s_k}\cdots B_{s_1},\qquad B_\epsilon=I.
1700 $$
1701 
1702 The mass identity is an exact constraint,
1703 
1704 $$
1705 \lambda\sum_o B_{a,o}=\lambda \quad\text{for every action }a.
1706 $$
1707 
1708 An initial state has $\lambda z_0=1$. The proposed probability of a word is $\lambda B_wz_0$. An invertible coordinate change must transform $z_0$, $B_s$, $\lambda$ and the admissible cone together, as in Proposition 37.
1709 
1710 A **probability-domain certificate** specifies a closed, pointed, generating cone $C\subset E$, with $\lambda$ strictly positive on $C\setminus\{0\}$, such that
1711 
1712 $$
1713 z_0\in C,\qquad B_sC\subseteq C.
1714 $$
1715 
1716 Its normalized base is $K=\{z\in C:\lambda z=1\}$. These conditions imply nonnegative word probabilities and their per-action normalization at every depth. Positive-probability updates remain in $K$. Zero-probability branches have no conditional posterior and must not be assigned one by denominator flooring.
1717 
1718 There are three separate claims. **Internal validity** says the fitted model is a stochastic model. **Data compatibility** says it agrees with the observations within a specified error budget. **Coverage of the generating process** says the assumptions and uncertainty set actually include the process that generated those observations. The first can have a deterministic certificate even when the other two remain conditional.
1719 
1720 # 42. Finite tests give an outer hierarchy, not automatic closure
1721 
1722 Define the test cones
1723 
1724 $$
1725 C_h=\{z:\lambda B_wz\geq0\text{ for every }|w|\leq h\},
1726 \qquad C_0=\{z:\lambda z\geq0\}.
1727 $$
1728 
1729 Each $C_h$ is polyhedral because it is an intersection of finitely many homogeneous half-spaces. It need not be a pointed cone or have a bounded normalized base.
1730 
1731 **Proposition 38. Test-cone hierarchy and a finite stopping certificate.** The cones satisfy
1732 
1733 $$
1734 C_{h+1}=C_h\cap\bigcap_s B_s^{-1}C_h,
1735 \qquad C_{h+1}\subseteq C_h.
1736 $$
1737 
1738 Their intersection $C_\infty$ is the largest cone contained in $C_0$ and invariant under all $B_s$. If an exact equality $C_{h+1}=C_h$ occurs, then $C_h=C_\infty$ and the finite inequalities certify all future word probabilities. If the representation is observable, meaning that the rows $\lambda B_w$ span $E^*$, then $C_\infty$ is pointed and $\lambda$ is strictly positive on its nonzero elements. If reachable vectors $B_wz_0\in C_\infty$ span $E$, the cone is also generating.
1739 
1740 **Proof.** For a first symbol $s$ followed by a word $v$, $B_{sv}=B_vB_s$. The inequalities for all words of length at most $h+1$ are therefore exactly the displayed recurrence. If $z\in C_\infty$, every test on $B_sz$ is another test on $z$, so $C_\infty$ is invariant. Conversely, invariance of a cone contained in $C_0$ makes every successive test nonnegative, hence places it in $C_\infty$. Equality at one finite step establishes invariance immediately and therefore gives the intersection.
1741 
1742 If both $z$ and $-z$ lie in $C_\infty$, every test row annihilates $z$, so observability gives $z=0$. For $z\in C_\infty$ with $\lambda z=0$, sum word probabilities along any fixed action sequence. Mass conservation makes the sum zero. Every term is nonnegative, so every word test vanishes and again $z=0$. Spanning by reachable cone elements gives the final assertion. $\square$
1743 
1744 A useful negative witness is a state admitted by the current inequalities and a specific longer word with negative proposed probability. It demonstrates that this state needs to be excluded. It does not prove that the initial state, or every reachable state, is invalid.
1745 
1746 Finite stabilization is sufficient, not promised. Even three predictive coordinates can require a nonpolyhedral domain, as Section 51 shows. Checking a long list of trajectories is not the same as checking the invariance identity that makes every trajectory valid.
1747 
1748 # 43. A finite certificate with a positive lift
1749 
1750 Let $V=[v_1\ \cdots\ v_m]\in\mathbb R^{r\times m}$ have full row rank and satisfy
1751 
1752 $$
1753 \lambda V=\mathbf1_m^\top.
1754 $$
1755 
1756 Its columns generate $C=V\mathbb R_+^m$ and its normalized base is $K=\operatorname{conv}\{v_1,\ldots,v_m\}$. Finite generation makes the cone closed. Its normalized generators and full row rank make it pointed and generating, with a strictly positive mass functional.
1757 
1758 **Proposition 39. Finite positive-lift certificate.** This cone is invariant under $B_s$ if and only if there is an entrywise nonnegative matrix $A_s\in\mathbb R^{m\times m}$ such that
1759 
1760 $$
1761 \boxed{B_sV=VA_s,\qquad A_s\geq0.}
1762 $$
1763 
1764 If these identities hold for every symbol and $z_0=V\alpha_0$ for $\alpha_0\geq0$ with $\mathbf1^\top\alpha_0=1$, then
1765 
1766 $$
1767 \mathbf1^\top\sum_o A_{a,o}=\mathbf1^\top,
1768 \qquad
1769 \lambda B_wz_0=\mathbf1^\top A_w\alpha_0.
1770 $$
1771 
1772 Thus the finite witness defines a nonnegative probability instrument whose image under $V$ is the predictive model. It certifies all word probabilities without enumerating them.
1773 
1774 **Proof.** Invariance places each $B_sv_j$ in the conical hull of the columns of $V$. Choose its nonnegative coefficients as column $j$ of $A_s$. Conversely, $B_sV\alpha=VA_s\alpha\in C$ for every $\alpha\geq0$. Multiplying the intertwining identity by $\lambda$ and summing over observations gives
1775 
1776 $$
1777 \mathbf1^\top\sum_oA_{a,o}
1778 =\lambda\sum_oB_{a,o}V
1779 =\lambda V=\mathbf1^\top.
1780 $$
1781 
1782 Iterating $B_sV=VA_s$ gives $B_wV=VA_w$. Applying the mass row and the initial coefficients proves the word identity. Strict positivity of $\lambda$ follows because a nonzero conical combination has positive total coefficient mass. $\square$
1783 
1784 The $m$ nonnegative coordinates need not be physical hidden states, and $m$ need not equal the minimal predictive dimension $r$. They are one finite realization witnessing validity. When $V$ has dependent columns, the lift can be nonunique. Those redundant coordinates do not change the predictive state after synthesis.
1785 
1786 There is a dual finite test. If a complete inequality description of the same cone is $C=\{z:Gz\geq0\}$, then $GB_sv_j\geq0$ for every generator $j$ is equivalent to invariance. One may also certify $GB_s=H_sG$ with $H_s\geq0$. The latter certificate is sufficient by direct multiplication and necessary by the polyhedral cone form of Farkas' lemma [R13]. The reference calculation uses the direct generator test, not an unreported numerical optimization certificate.
1787 
1788 # 44. Robustness is a quantified invariance statement
1789 
1790 A certificate for one fitted operator does not automatically cover neighboring operators. For uncertainty, the quantifier must range over the whole proposed model set, including correlations required by mass conservation.
1791 
1792 Let $\mathcal U$ be the convex hull of finitely many **complete operator families** $\{B_s^{(j)}\}_s$. Every family must satisfy the exact per-action mass identities. The convex coefficient is shared across that family's symbols. A product of independent uncertainty sets can instead be handled by its joint vertices.
1793 
1794 **Proposition 40. Robust finite-domain certificate.** For a fixed normalized generator matrix $V$, suppose every vertex family has witnesses
1795 
1796 $$
1797 B_s^{(j)}V=VA_s^{(j)},\qquad A_s^{(j)}\geq0.
1798 $$
1799 
1800 Then every family in $\mathcal U$ preserves $C=V\mathbb R_+^m$ and is mass conserving. If its initial state lies in $K$, it defines valid finite observation-tree probabilities at every horizon. This remains true when a family in $\mathcal U$ is selected separately at each history.
1801 
1802 For a complete facet description $C=\{z:Gz\geq0\}$, let $g_i^\top$ be its rows. A sufficient norm-ball certificate around $\widehat B_s$ is
1803 
1804 $$
1805 \boxed{
1806  g_i^\top\widehat B_sv_j
1807  \geq\epsilon_s\|g_i\|_2\|v_j\|_2
1808  \quad\text{for all }s,i,j.
1809 }
1810 $$
1811 
1812 It covers all perturbations with $\|B_s-\widehat B_s\|_2\leq\epsilon_s$ that also satisfy the exact mass constraints.
1813 
1814 **Proof.** For convex weights $\theta_j$, take $A_s=\sum_j\theta_jA_s^{(j)}$. The equality and nonnegativity survive convex combination, as does mass conservation. At each history, every selected family maps the current cone state back into the cone. Applying the mass identity at that node proves recursive normalization, regardless of how the family was selected. For the norm ball,
1815 
1816 $$
1817 |g_i^\top(B_s-\widehat B_s)v_j|
1818 \leq\epsilon_s\|g_i\|_2\|v_j\|_2.
1819 $$
1820 
1821 The margin makes every perturbed generator image satisfy all facets. Proposition 39, or the direct generator test, then supplies invariance. $\square$
1822 
1823 The switching conclusion is structural validity, not an assertion that time-varying selection represents one fixed unknown parameter. A likelihood for a fixed mechanism and a likelihood for a mechanism selected anew at each history are different statistical models.
1824 
1825 The norm-ball inequality is sufficient and can be conservative, especially at boundary images with zero slack. Failure of this margin is not a proof that no robust certificate exists. The exact vertex test retains uncertainty correlations that a separate bound for each entry may discard.
1826 
1827 # 45. Fitting inside a certified domain
1828 
1829 Once $V$ is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of $\widetilde B_s$, $A_s$, $\widetilde z_0$ and $\alpha_0$. The constraints
1830 
1831 $$
1832 \begin{gathered}
1833 \widetilde B_sV=VA_s,\quad A_s\geq0,
1834 \quad \lambda\sum_o\widetilde B_{a,o}=\lambda,\\
1835 \widetilde z_0=V\alpha_0,\quad\alpha_0\geq0,
1836 \quad\mathbf1^\top\alpha_0=1
1837 \end{gathered}
1838 $$
1839 
1840 are linear. Entrywise confidence intervals and fixed-input regression tolerances add linear inequalities. Norm balls add convex constraints. Jointly optimizing $V$ and $A_s$ is different: the product $VA_s$ is then bilinear. No general convexity is claimed for learning the domain and dynamics together.
1841 
1842 **Proposition 41. Certified correction and its error budget.** For fixed $V$, every feasible solution of the displayed constraints is an internally valid model. Suppose a center estimate $\widehat B_s$ and a feasible corrected estimate $\widetilde B_s$ satisfy
1843 
1844 $$
1845 \|B_s^*-\widehat B_s\|\leq\epsilon_s,
1846 \qquad
1847 \|\widetilde B_s-\widehat B_s\|\leq\epsilon_s.
1848 $$
1849 
1850 Then $\|\widetilde B_s-B_s^*\|\leq2\epsilon_s$. The same conclusion holds for initial states in their specified norm. If the true model lies in these confidence sets and itself admits the fixed-domain certificate, the convex constraints are feasible. Feasibility alone does not establish that the true model preserves this domain.
1851 
1852 **Proof.** Validity is Proposition 39. The error bound is the triangle inequality through the center estimate. Under the final assumptions, insert the true operators, true initial state and their nonnegative witnesses into the constraints. They provide a feasible point. The converse fails because existence of one valid point does not say that every point in a confidence set, or its true member, has the same invariant domain. $\square$
1853 
1854 This is why existential correction and robust certification must be reported separately. A corrected model can be a well-defined approximation even when no common true-model cone has been proved. The trajectory bounds below require both models to be valid on a common specified cone.
1855 
1856 ## 45.1 An explicit positivity repair in a simplicial cone
1857 
1858 When $V$ is square and invertible, put $A_s=V^{-1}\widehat B_sV$. Assume mass conservation has already been imposed. Let $N_s$ be an entrywise positive reference instrument with the same per-action mass identities. For $0\leq\eta\leq1$, the mixture
1859 
1860 $$
1861 \widetilde A_s=(1-\eta)A_s+\eta N_s
1862 $$
1863 
1864 is nonnegative provided
1865 
1866 $$
1867 \eta\geq\max_{s,i,j:(A_s)_{ij}<0}
1868 \frac{-(A_s)_{ij}}{(N_s)_{ij}-(A_s)_{ij}}.
1869 $$
1870 
1871 An empty maximum is zero. If every negative entry has magnitude at most $d$ and every reference entry is at least $m_0>0$, the simpler sufficient choice is $\eta=d/(m_0+d)$. These statements follow by solving each scalar inequality. Mass conservation is unchanged by convex mixing.
1872 
1873 This is an explicit model modification, not coefficient clipping or an exact coordinate change. Relative to a true instrument on the same cone, the one-step error is at most $(1-\eta)\epsilon+2\eta$ in the summed base-norm metric defined next, when the original error was at most $\epsilon$. The factor two follows because two valid instruments each distribute unit total mass. The reference example retains the repair amount as part of its approximation record.
1874 
1875 # 46. A norm that measures probability mass before conditioning
1876 
1877 Assume $C$ is a closed, pointed, generating cone and $\lambda$ is strictly positive on its nonzero elements. Its base $K$ is compact. Define the base norm
1878 
1879 $$
1880 \|x\|_C=\inf\{\lambda u+\lambda v:x=u-v,\ u,v\in C\}.
1881 $$
1882 
1883 The infimum is finite because the cone is generating. Compactness of $K$ makes this a genuine norm. Its unit ball is $\operatorname{conv}(K\cup(-K))$. In particular $\|u\|_C=\lambda u$ on $C$, and $|\lambda x|\leq\|x\|_C$. On the ordinary probability orthant this is the $\ell^1$ norm. These are standard ordered-space constructions [R15], used here to avoid mixing Euclidean coefficient magnitude with probability mass.
1884 
1885 **Proposition 42. Instrument contraction before normalization.** If $\{B_{a,o}\}_o$ is cone preserving and mass conserving for a fixed action, then
1886 
1887 $$
1888 \boxed{\sum_o\|B_{a,o}x\|_C\leq\|x\|_C.}
1889 $$
1890 
1891 For a polyhedral base $K=\operatorname{conv}\{v_j\}$, define the discrepancy between two instruments by
1892 
1893 $$
1894 \epsilon_a=\sup_{z\in K}\sum_o
1895 \|(B_{a,o}-\widetilde B_{a,o})z\|_C.
1896 $$
1897 
1898 The supremum equals the maximum of the same expression at the finitely many generators $v_j$. When both instruments are valid on $C$, $\epsilon_a\leq2$.
1899 
1900 **Proof.** For any $x=u-v$ with $u,v\in C$, cone preservation gives
1901 
1902 $$
1903 \sum_o\|B_{a,o}x\|_C
1904 \leq\sum_o\lambda B_{a,o}(u+v)
1905 =\lambda u+\lambda v.
1906 $$
1907 
1908 Take the infimum. For the generator statement, the discrepancy is convex in $z$. At a convex combination its value is at most the same combination of generator values, hence at most their maximum. The generators themselves belong to $K$, proving equality. Finally, for $z\in K$, the triangle inequality bounds the sum by the total masses of both instruments, which are one each. $\square$
1909 
1910 For a polyhedral certificate, the norm itself has a finite linear-program formulation:
1911 
1912 $$
1913 \|x\|_C=\min_{a,b\geq0}
1914 \{\mathbf1^\top(a+b):V(a-b)=x\}.
1915 $$
1916 
1917 If $V$ is square and invertible, this reduces to $\|V^{-1}x\|_1$. The exact worked interval certificate is of this type. With dependent generators, choosing one pseudoinverse does not in general compute the base norm.
1918 
1919 # 47. Whole observation laws can be stable without rare-event margins
1920 
1921 The posterior after one rare observation can be sensitive, but that does not force the entire distribution of observation records to be equally sensitive. Work before normalization and retain the probability of each branch.
1922 
1923 Let two models share $(C,\lambda)$ and have initial states $z_0,\widetilde z_0\in K$. At each time use the same policy, a specified function or randomization depending only on the observed history. Thus on any common history the action probabilities are identical in the two comparisons. This is not a comparison of two policies optimized separately for the two models.
1924 
1925 **Proposition 43. Observation-tree total-variation bound.** If both instruments preserve $C$ and conserve mass, and if every action at time $t$ has the discrepancy in Proposition 42 bounded by $\epsilon_t$, then the distributions $P_T,\widetilde P_T$ of length-$T$ observation records under the common policy satisfy
1926 
1927 $$
1928 \boxed{
1929 \|P_T-\widetilde P_T\|_{\mathrm{TV}}
1930 \leq\min\left(1,\frac12\|z_0-\widetilde z_0\|_C
1931 +\frac12\sum_{t=0}^{T-1}\epsilon_t\right).
1932 }
1933 $$
1934 
1935 The same bound holds for joint action and observation records when the common policy is randomized. No positive lower bound on individual branch probabilities is required.
1936 
1937 **Proof.** Let $u_h,\widetilde u_h\in C$ be the unnormalized states on each history at depth $t$, including the common policy factors. Their masses sum to one in each model. Define $D_t=\sum_h\|u_h-\widetilde u_h\|_C$. At a history with selected action $a$, write
1938 
1939 $$
1940 B_{a,o}u_h-\widetilde B_{a,o}\widetilde u_h
1941 =B_{a,o}(u_h-\widetilde u_h)
1942 +(B_{a,o}-\widetilde B_{a,o})\widetilde u_h.
1943 $$
1944 
1945 Sum the first term over observations and use Proposition 42. When $\lambda\widetilde u_h>0$, normalize only to apply the definition of $\epsilon_t$ to the second term. Its sum is bounded by $\epsilon_t\lambda\widetilde u_h$. If the mass is zero, strict positivity implies $\widetilde u_h=0$. Summing histories yields $D_{t+1}\leq D_t+\epsilon_t$. Averaging over common policy probabilities gives the same inequality for randomized actions. Finally each word-probability difference is bounded by its state difference because $|\lambda x|\leq\|x\|_C$. Total variation is half the sum of those absolute differences. $\square$
1946 
1947 This bound separates total law error from conditional-state error. It can grow linearly with the horizon and eventually become uninformative. It is not a uniform-in-time approximation theorem. It also does not control the regret of a changed planning policy without additional decision-theoretic assumptions.
1948 
1949 # 48. Conditioning still has an unavoidable cost
1950 
1951 Let $P,Q$ be probability distributions on a finite record space. A record event $H$ might be a particular observed prefix. Conditional continuation laws are obtained by restriction to $H$ and normalization.
1952 
1953 **Proposition 44. Sharp event-probability dependence of conditioning.** If $\|P-Q\|_{\mathrm{TV}}\leq\delta$, $p=P(H)>0$ and $q=Q(H)>0$, then
1954 
1955 $$
1956 \boxed{\|P(\cdot\mid H)-Q(\cdot\mid H)\|_{\mathrm{TV}}
1957 \leq\min(1,\delta/p).}
1958 $$
1959 
1960 When $p>\delta$, positivity of $q$ follows automatically. The factor $1/p$ cannot be removed in general.
1961 
1962 **Proof.** Write the subprobability restrictions on $H$ as $u,v$, of masses $p,q$. On the complementary event, the sum of absolute probability differences is at least $|p-q|$. Therefore
1963 
1964 $$
1965 \|u-v\|_1+|p-q|\leq2\delta.
1966 $$
1967 
1968 Adding and subtracting $v/p$ gives
1969 
1970 $$
1971 \frac12\|u/p-v/q\|_1
1972 \leq\frac{\|u-v\|_1+|p-q|}{2p}
1973 \leq\delta/p.
1974 $$
1975 
1976 Also $|p-q|\leq\delta$, establishing the positivity condition. For sharpness, put total mass $p$ on an event with two outcomes. Let $P$ assign all of it to the first, and let $Q$ move mass $\delta\leq p$ to the second, leaving the complement unchanged. The unconditional total variation is $\delta$ and the conditional distance is $\delta/p$. $\square$
1977 
1978 Combining this proposition with Proposition 43 gives a conditional continuation bound when the evidence event has adequate probability. It does not give a coordinate-state error bound unless test probabilities control that coordinate norm. Validity at every branch and stable inference at every branch remain different properties.
1979 
1980 # 49. Searching for a domain from observable operators
1981 
1982 A cone certificate need not be imported from a hidden-state generator. With exact rational operators it can be searched for directly, then checked independently.
1983 
1984 **Proposition 45. Rational finite-certificate search.** Suppose $\lambda$, $B_s$ and $z_0$ are rational and satisfy the exact mass identities. There is an enumeration procedure that returns a valid finite certificate whenever a certificate of Proposition 39 exists with a rational normalized full-row-rank generator matrix $V$. The procedure is sound whenever it returns. No termination guarantee is asserted when no such rational certificate exists.
1985 
1986 **Proof.** Enumerate all finite rational matrices $V$ in an order that eventually includes every size and every bounded numerator and denominator range. Retain those of full row rank with $\lambda V=\mathbf1^\top$. For each retained matrix, the requirements $B_sV=VA_s$, $A_s\geq0$, $V\alpha_0=z_0$, $\alpha_0\geq0$ and $\mathbf1^\top\alpha_0=1$ are rational linear feasibility problems. A nonempty rational polyhedron has a rational feasible point, as follows from elimination of rational linear inequalities. These problems can therefore be decided by exact rational linear programming, or witnesses can be enumerated in a dovetailed rational search. Recheck every returned equality and sign exactly. Proposition 39 proves soundness. If a rational generator certificate exists, its finite representation is eventually considered and its feasible witnesses are found. $\square$
1987 
1988 The general enumeration is a mathematical existence algorithm, not a practical complexity claim. A bounded candidate family is often the useful reference experiment. Failure within that family means only that the chosen family did not produce a certificate.
1989 
1990 ## 49.1 A certificate learned without hidden matrices
1991 
1992 The observable tables of Section 38 determine the two-dimensional operators $B_s$ and initial vector $(1,13/24)^\top$. The new reference procedure receives only these fitted operators and that vector. It enumerates intervals with rational endpoints in $[0,1]$ of reduced denominator at most eight, ordered by increasing width and then by endpoints. The coordinate restriction $[0,1]$ comes from its declared test-probability meaning.
1993 
1994 The first accepted interval is
1995 
1996 $$
1997 \boxed{K=\{(1,q)^\top:2/5\leq q\leq3/5\}.}
1998 $$
1999 
2000 It uses
2001 
2002 $$
2003 V=\begin{pmatrix}1&1\\2/5&3/5\end{pmatrix},
2004 \qquad \alpha_0=\begin{pmatrix}7/24\\17/24\end{pmatrix},
2005 \qquad A_s=V^{-1}B_sV.
2006 $$
2007 
2008 Every entry of every $A_s$ is at least $1/64$. The full per-action column sums are exactly one. These rational identities certify all word probabilities of the fitted model, not only the words used in the fit. The certificate was not supplied to the search, and no hidden generator is imported by the standalone example.
2009 
2010 The interval is the narrowest accepted interval **in this specified finite grid**, not a claim of the smallest invariant interval among all real endpoints. It also need not contain every predictive state associated with an arbitrary initialization of the old hidden model. It contains the declared initial state and preserves every subsequent branch of this fitted realization, which is the requirement being certified.
2011 
2012 The positive slack gives a robust statement: any perturbation of the lifted entries by at most $1/64$, coupled to retain the exact per-action column sums, remains nonnegative. Interior perturbations of size $1/128$ are used in the worked comparison. Repeated composition is valid for all horizons, while Proposition 43 bounds the changed observation law for each fixed horizon.
2013 
2014 # 50. Selection and noise do not invalidate an exact witness
2015 
2016 A data-dependent cone proposal is not itself a statistical confidence statement. Once the matrices and witness are fixed, its exact invariance identities hold or fail independently of how it was proposed. What remains statistical is whether the chosen uncertainty set covers the generating model.
2017 
2018 For a finite-sample interpretation, first obtain a valid uncertainty set for a fixed identifiable model class. Proposition 35 supplies one route under independent reset experiments and declared rank coverage. Proposition 33 translates those probability bounds into a fixed-core operator bound. A validity witness can then be sought for a fitted member or robustly for every member. Only the latter identifies a common true-model domain on the coverage event.
2019 
2020 The admissible family must enforce mass conservation exactly. A free box around unconstrained matrices generally contains nonnormalized families. Discarding those families is legitimate only because the generating process is already assumed to obey the exact mass law. Other cuts on a confidence set need their own justification.
2021 
2022 # 51. Why finite predictive rank need not mean finitely many positive states
2023 
2024 The following controlled rational example makes the boundary explicit. It is a worked adaptation of the irrational-rotation obstruction in [R12, Section 7]. Related distinctions among finite linear, positive and quantum realizations are discussed in [R14]. Only the elementary cone argument below is used.
2025 
2026 Let $z=(t,x,y)^\top$, $\lambda=(1,0,0)$, $z_0=(1,1,0)^\top$, and
2027 
2028 $$
2029 R=\begin{pmatrix}3/5&-4/5\\4/5&3/5\end{pmatrix},
2030 \qquad T=\operatorname{diag}(1,R).
2031 $$
2032 
2033 There are two actions and two observations. A rotate action has branches
2034 
2035 $$
2036 B_{\mathrm r,0}=T,\qquad B_{\mathrm r,1}=0.
2037 $$
2038 
2039 A probe action has branches
2040 
2041 $$
2042 B_{\mathrm p,0}=z_0(1/2,1/2,0),\qquad
2043 B_{\mathrm p,1}=z_0(1/2,-1/2,0).
2044 $$
2045 
2046 The rotate outcome zero is certain. The probe has probabilities $(t+x)/2$ and $(t-x)/2$ and resets every nonzero branch to $z_0$ after normalization. Both actions conserve the mass $t$.
2047 
2048 **Proposition 46. A rational rank-three model with no polyhedral probability-domain certificate.** The Lorentz cone
2049 
2050 $$
2051 C_{\mathrm L}=\{(t,x,y):t\geq\sqrt{x^2+y^2}\}
2052 $$
2053 
2054 is probability valid for these operators. It is the canonical all-test cone. Every closed convex invariant cone containing $z_0$ and admitting the nonnegative mass interpretation equals $C_{\mathrm L}$. In particular no finite normalized generator certificate exists for this exact model. Its reachable and observable predictive dimension is three.
2055 
2056 **Proof.** Rotation preserves $t$ and $x^2+y^2$. On $C_{\mathrm L}$ both probe effects are nonnegative and each probe image is a nonnegative multiple of $z_0$. Thus the displayed cone is invariant and valid.
2057 
2058 Write the rotation angle as $\theta$. It is not a rational multiple of $2\pi$. Otherwise $\zeta=e^{i\theta}$ would be a root of unity, so $\zeta+\zeta^{-1}=6/5$ would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting $6/5\notin\mathbb Z$. The forward powers of an irrational circle rotation are dense. Compactness supplies powers approaching the identity, so their closure contains the inverse rotation and is a closed subgroup. An infinite closed subgroup has arbitrarily small positive angles. Integer multiples of such angles approximate every point on the circle, so that subgroup is the entire circle.
2059 
2060 A state passing every test must pass a probe after any number $k$ of rotations. Hence
2061 
2062 $$
2063 t\pm e_1^\top R^k(x,y)^\top\geq0
2064 \quad\text{for every }k\geq0.
2065 $$
2066 
2067 Density gives $t\geq\sqrt{x^2+y^2}$. The reverse inclusion was already proved, so the canonical cone is exactly $C_{\mathrm L}$.
2068 
2069 Any invariant cone with nonnegative mass lies inside this canonical cone, by Proposition 38. If it is closed, convex and contains $z_0$, it contains the orbit $T^kz_0$ and the closed convex hull of that orbit. On $t=1$, this hull is the full unit disk. Conical scaling gives all of $C_{\mathrm L}$. The unit disk has infinitely many extreme points, so it is not the convex hull of finitely many points.
2070 
2071 Finally the reachable columns $z_0,Tz_0,T^2z_0$ are independent. The mass row and probe rows before zero and one rotations span the dual space because the rotation has nonzero sine. Reachability and observability establish predictive dimension three. $\square$
2072 
2073 This is not a failure of finite-dimensional prediction. The exact model has only three coordinates and a simple quadratic cone. It is a failure of the narrower requirement that every valid model admit a finite *polyhedral* domain. A certificate language limited to finitely many positive generators would exclude a legitimate low-rank signal model.
2074 
2075 No general impossibility of all finite certificates follows. This very example has a finite quadratic certificate. The obstruction concerns polyhedral certificates, not invariant cones described by other mathematics.
2076 
2077 
2078 
2079 # 52. A finite approximation with an explicit price
2080 
2081 A curved probability domain can sometimes be replaced by a finite one after changing the dynamics. That change must be quantified, not described as exact refinement.
2082 
2083 Let $P_n$ be the regular $n$-gon inscribed in the unit disk, with a vertex at $(1,0)$ and $n\geq3$. Replace the rotate branch by
2084 
2085 $$
2086 T_\alpha=\operatorname{diag}(1,\alpha R),\qquad 0\leq\alpha<1,
2087 $$
2088 
2089 while retaining the same probe branches. The spatial coordinates are attenuated, but total probability mass remains unchanged.
2090 
2091 **Proposition 47. Polyhedral approximation of the rotation model.** If
2092 
2093 $$
2094 \alpha\leq\cos(\pi/n),
2095 $$
2096 
2097 then the cone over $P_n$ is invariant under all modified operators and contains $z_0$. Consequently the modified model admits a finite positive lift. The original and modified models are both valid on the larger cone $C_{\mathrm L}$, and for every common policy their length-$T$ laws satisfy
2098 
2099 $$
2100 \boxed{\|P_T-P_T^{(\alpha)}\|_{\mathrm{TV}}
2101 \leq\min(1,T(1-\alpha)/2).}
2102 $$
2103 
2104 The angle-independent sufficient condition can be improved for a fixed polygon and rotation. For the rational diamond $|x|+|y|\leq t$, the present rotation is invariant after attenuation whenever $\alpha\leq5/7$. Thus $\alpha=2/3$ gives an entirely rational four-generator certificate.
2105 
2106 **Proof.** The regular polygon contains the disk of radius $\cos(\pi/n)$ and lies in the unit disk. Every $\alpha Rv$ with $v\in P_n$ has Euclidean norm at most $\alpha$, hence lies in $P_n$ under the stated condition. The probe effects remain nonnegative because $P_n$ lies in the disk, and their normalized images are the retained vertex $(1,0)$.
2107 
2108 The base norm of $C_{\mathrm L}$ is
2109 
2110 $$
2111 \|(t,x,y)\|_{C_{\mathrm L}}=\max(|t|,\sqrt{x^2+y^2}).
2112 $$
2113 
2114 Indeed the convex hull of the unit disks in the planes $t=1$ and $t=-1$ is the cylinder with this unit ball. The rotate-branch discrepancy on a normalized state is at most $1-\alpha$ in this norm. Probe discrepancies are zero. Proposition 43 gives the law bound. For the diamond, the induced $\ell^1$ norm of $R$ is $7/5$, so $\alpha\|R\|_1\leq1$ proves invariance. $\square$
2115 
2116 For a target attenuation loss $0<\eta\leq1$, choosing $\alpha=1-\eta$ and sufficiently many polygon vertices satisfies the certificate. The inequality $1-\cos u\leq u^2/2$ shows that $n\geq\pi/\sqrt{2\eta}$ is sufficient, with an integer ceiling and $n\geq3$. This is a constructive upper bound on certificate size, not a lower bound or a proof of optimality.
2117 
2118 The distinction is exact. At $\alpha=1$ there is no finite polyhedral certificate for this model. At $\alpha<1$ a sufficiently fine polygon works, at the cost of a declared change in the observation law. The reference figures show that tradeoff rather than labeling the approximation an identical model.
2119 
2120 # 53. Repeated proposal and verification with a confidence budget
2121 
2122 Model development may choose another core, propose another domain or collect another independent batch after an earlier certificate is withheld. Statistical coverage must survive that selection process.
2123 
2124 **Proposition 48. Sequential confidence accounting for certified proposals.** At stage $n$, let the proposal be measurable from previous data, and let a verification procedure return a set $\mathcal U_n$ for the true model in the declared identifiable coordinate system. Assume conditional coverage
2125 
2126 $$
2127 \Pr(\text{true model}\notin\mathcal U_n\mid\text{previous data})
2128 \leq\delta_n,
2129 \qquad\sum_{n\geq1}\delta_n\leq\delta.
2130 $$
2131 
2132 Then with probability at least $1-\delta$, every returned uncertainty set covers its true model. On this event, any data-selected domain with an exactly verified robust certificate for **all** members of its corresponding $\mathcal U_n$ is also valid for the true model, provided the true initial state is covered by the certified base. The conclusion remains valid at any stopping time chosen from the recorded results.
2133 
2134 **Proof.** Take expectations of the conditional failure bounds, then apply the countable union bound. Off the union of failure events, all coverages hold simultaneously. Robust invariance over an entire set then applies to its true member by deterministic implication, independently of how its domain was chosen. A stopped stage is one of the already covered stages. $\square$
2135 
2136 For example, $\delta_n=\delta 2^{-n}$ spends a finite total confidence budget. Fresh reset trials for a proposal fixed by past data can use Proposition 35 conditionally, with this stage budget. An exact witness needs no statistical correction for how many candidate cones were tried. The coverage statement does need correction for repeatedly constructing confidence sets or inspecting post-selected data.
2137 
2138 The proposition does not infer a global rank bound, validate an unknown reset operation, or transform dependent observations into independent trials. Those are assumptions of the coverage procedure. A known coordinate transformation can transport a confidence set exactly. An estimated transformation requires the subspace and conditioning controls already separated in Part III.
2139 
2140 # 54. Integrated certificate-carrying realization
2141 
2142 **Proposition 49. Conditional learning with an all-horizon probability certificate.** Suppose the observable identification assumptions of Proposition 37 hold for a declared finite-rank controlled process. Suppose a justified confidence procedure produces an operator and initial-state set containing the true realization on an event of probability at least $1-\delta$. Assume the per-action mass constraints are exact.
2143 
2144 If a fitted realization admits a checked finite witness of Proposition 39, it is an internally valid stochastic model at every horizon. If the witness is robust over the whole confidence set and covers its initial states, then on the coverage event the same probability domain is valid for both the fitted and true models. When their summed base-norm instrument discrepancies are at most $\epsilon_t$ and their initial discrepancy is at most $e_0$, every common policy has the bound
2145 
2146 $$
2147 \|P_T-\widehat P_T\|_{\mathrm{TV}}
2148 \leq\min\left(1,\frac{e_0+\sum_{t<T}\epsilon_t}{2}\right).
2149 $$
2150 
2151 Conditioning on a positive-probability event then obeys Proposition 44. Exact coordinate transport preserves the certificate. A signed packet decomposition preserves it after complete synthesis but does not turn individual packets into probability states without Proposition 30's extra positivity requirement.
2152 
2153 **Proof.** Observable identification supplies the specified realization and the confidence procedure supplies the stated event, not an inferred event of its own. Proposition 39 supplies internal validity for the fitted model. Proposition 40 extends the same domain to every member of the uncertainty set, hence to its true member on that event. Propositions 42 and 43 give the total-variation bound, and Proposition 44 gives the conditional bound. For a coordinate change $T$, use $V'=TV$, $B_s'=TB_sT^{-1}$ and $\lambda'=\lambda T^{-1}$. Then $B_s'V'=V'A_s$ and $\lambda'V'=\mathbf1^\top$ with the unchanged nonnegative witnesses. Full packet synthesis is the identity. $\square$
2154 
2155 This closes a concrete foundational gap. A learner can now return not just a matrix family but a verifiable domain in which that family has probability meaning. It can attach a robust uncertainty statement when one is justified, or return an internally valid approximation with its correction budget explicitly recorded.
2156 
2157 It does not turn a successful fit into unrestricted knowledge of the world. The model class, observation protocol, rank coverage, exact mass law, domain witness and approximation budget remain visible parts of the result. Failure of a finite search is a withheld certificate, not a proof of invalidity. The nonpolyhedral example explains why this distinction is necessary even for very small exact models.
2158 
2159 # 55. From polyhedral to curved certificates
2160 
2161 A finite positive lift is one sufficient certificate language. Proposition 46 shows that a low-dimensional model can require a nonpolyhedral domain. Part V represents that same model exactly through a curved matrix domain. A failed finite-generator search is therefore not a general invalidity theorem.
2162 
2163 \newpage
2164 
2165 # Part V. Curved domains and continuous observations
2166 
2167 Part IV separated a valid predictive model from a successful fit. Its finite positive lift gives a useful certificate, but its rank-three rotation example has no finite polyhedral probability domain. The same block left continuous observations outside the finite-alphabet construction. This part addresses both boundaries in a declared finite sector.
2168 
2169 A curved domain can have a short matrix certificate even when it has infinitely many extreme states. A continuous observation can have a finite operator description even when its possible values are uncountable. Neither statement follows from finite predictive dimension alone. The constructions below require an explicit domain, a reference measure and a restricted observation-kernel family.
2170 
2171 The block connects matrix positivity, integral observation semantics, population-moment identification, exact coarsening and approximation. It also identifies a defect that is easy to overlook: averaging a matrix transformation before applying its quadratic action does not equal averaging that action. The difference is a positive covariance contribution. Omitting it can destroy normalization even in a very small example.
2172 
2173 There are established antecedents. Congruence representations and their positive semidefinite Gram matrices belong to the theory of completely positive maps [R16]. Polynomial Gram certificates are standard sum-of-squares machinery [R17]. Continuous-valued observable operator models already occur in Jaeger's work [R18]. The proofs here develop their specific connection to the realization and defect contracts of the large signal model. They do not claim to originate those ingredients.
2174 
2175 This is a discrete-time theory with continuous-valued observations. It is not a continuous-time stochastic dynamics theorem. Production PR-LASM 2.3.0 is unchanged.
2176 
2177 
2178 
2179 # 56. A matrix certificate for a curved domain
2180 
2181 Let $E=\mathbb R^r$ and let $\mathbb S^d$ be the real symmetric $d\times d$ matrices. Fix an injective linear map
2182 
2183 $$
2184 \mathcal J:E\longrightarrow\mathbb S^d,
2185 \qquad \lambda z=\operatorname{tr}\mathcal J(z).
2186 $$
2187 
2188 Suppose some $e\in E$ satisfies $\mathcal J(e)\succ0$. Define
2189 
2190 $$
2191 C_{\mathcal J}=\{z:\mathcal J(z)\succeq0\},
2192 \qquad K_{\mathcal J}=\{z\in C_{\mathcal J}:\lambda z=1\}.
2193 $$
2194 
2195 The map $\mathcal J$ is an embedding, not a many-to-one decoder of a larger positive coordinate space. The cone is an inverse image of the positive semidefinite cone. This is different from the image cone $V\mathbb R_+^m$ used in Proposition 39.
2196 
2197 **Proposition 50. Matrix-embedding probability certificate.** The cone $C_{\mathcal J}$ is closed, pointed and generating, and $\lambda$ is strictly positive on its nonzero elements. Its normalized base is compact. Suppose a finite-alphabet predictive model has $z_0\in K_{\mathcal J}$, exact mass conservation and identities
2198 
2199 $$
2200 \boxed{\mathcal J B_s=\Phi_s\mathcal J,}
2201 \qquad
2202 \Phi_s(X)=\sum_{j=1}^{m_s} w_{sj}K_{sj}XK_{sj}^{\mathsf T},
2203 \quad w_{sj}\geq0.
2204 $$
2205 
2206 Then every $B_s$ preserves $C_{\mathcal J}$, and every finite observation word has a nonnegative, normalized probability. Zero maps may use an empty sum. An invertible change of predictive coordinates transports the certificate by
2207 
2208 $$
2209 B'_s=TB_sT^{-1},\qquad
2210 \mathcal J'=\mathcal JT^{-1},\qquad
2211 \lambda'=\lambda T^{-1}.
2212 $$
2213 
2214 **Proof.** Closedness follows from continuity. If both $z$ and $-z$ are in the cone, $\mathcal J(z)$ is both positive and negative semidefinite, hence zero. Injectivity gives $z=0$. A sufficiently small neighborhood of $e$ maps into positive definite matrices, so the cone has an interior point and is generating. A nonzero positive semidefinite matrix has positive trace. The trace-one positive semidefinite matrices form a compact set. The image of $\mathcal J$ is a closed finite-dimensional subspace and its inverse on that image is continuous, so $K_{\mathcal J}$ is compact.
2215 
2216 Every term $wKXK^{\mathsf T}$ is positive semidefinite when $X$ is. The intertwining identity therefore proves invariance. Iteration gives invariance of every branch state, and the mass identity normalizes every finite observation tree. Substituting the transported quantities proves the final statement. $\square$
2217 
2218 This is a sufficient certificate language. It is not asserted that every invariant cone has this form, or that every positive map on a matrix subspace extends through the displayed congruence representation. The extension requirement is stronger than positivity on the represented states alone.
2219 
2220 The base norm of Section 46 is also not automatically the ambient trace norm. One always has
2221 
2222 $$
2223 \|\mathcal J(z)\|_1\leq\|z\|_{C_{\mathcal J}},
2224 $$
2225 
2226 because every positive decomposition in $E$ gives one in the matrix space. Equality holds when $\mathcal J$ maps onto all of $\mathbb S^d$, by the positive and negative spectral parts. For a proper matrix subspace, those spectral parts may leave the subspace. Norm transport therefore needs its own stated hypothesis.
2227 
2228 # 57. The earlier disk obstruction has an exact finite matrix witness
2229 
2230 For the rank-three model of Section 51, put
2231 
2232 $$
2233 \mathcal J(m,x,y)=\frac12
2234 \begin{pmatrix}m+x&y\\y&m-x\end{pmatrix}.
2235 $$
2236 
2237 Then $\operatorname{tr}\mathcal J=m$ and
2238 
2239 $$
2240 \det\mathcal J=\frac{m^2-x^2-y^2}{4}.
2241 $$
2242 
2243 Its eigenvalues are $(m\pm\sqrt{x^2+y^2})/2$, so positivity is exactly the Lorentz-cone condition. The normalized base is the unit disk.
2244 
2245 **Proposition 51. Exact rational matrix witness for the rotation model.** Let $T=\operatorname{diag}(1,R)$ with the unchanged rotation
2246 
2247 $$
2248 R=\begin{pmatrix}3/5&-4/5\\4/5&3/5\end{pmatrix},
2249 \qquad W=\begin{pmatrix}2&-1\\1&2\end{pmatrix}.
2250 $$
2251 
2252 The rotate branch and the two probe branches from Proposition 46 have the identities
2253 
2254 $$
2255 \begin{aligned}
2256 \mathcal J(Tz)&=\frac15W\mathcal J(z)W^{\mathsf T},\\
2257 \mathcal J(B_{\mathrm p,0}z)&=E_{00}\mathcal J(z)E_{00}^{\mathsf T},\\
2258 \mathcal J(B_{\mathrm p,1}z)&=E_{01}\mathcal J(z)E_{01}^{\mathsf T},
2259 \end{aligned}
2260 $$
2261 
2262 where $E_{ij}$ is the corresponding $2\times2$ matrix unit. The other rotate branch is zero. Thus this exact model has a finite rational matrix certificate without attenuation or a polygonal replacement.
2263 
2264 **Proof.** Direct multiplication gives the rotate identity. The two matrix-unit congruences retain, respectively, the upper and lower diagonal entries of $\mathcal J(z)$ and place them in its upper diagonal entry. These are $(m+x)/2$ and $(m-x)/2$, exactly the two probe masses resetting to $(1,1,0)$. Also $W^{\mathsf T}W=5I$ and
2265 
2266 $$
2267 E_{00}^{\mathsf T}E_{00}+E_{01}^{\mathsf T}E_{01}=I.
2268 $$
2269 
2270 The actions conserve trace. The initial state embeds as $\operatorname{diag}(1,0)$, and $(1,0,0)$ is a strictly positive interior witness. Proposition 50 applies. $\square$
2271 
2272 Nothing has been changed in the observation law. Proposition 46 still rules out a finite positive-generator certificate for this model. What changes is the certificate language. A curved matrix domain and a finite set of positive coordinate states are not the same representation class.
2273 
2274 This use of symmetric positive semidefinite matrices is mathematical. It neither requires quantum hardware nor claims a quantum description of the measured system. It borrows a well-defined operator representation rather than a physical interpretation.
2275 
2276 
2277 
2278 # 58. An observation density is not a branch probability
2279 
2280 Let $(Y,\mathcal Y)$ be a standard Borel observation space and let $\mu$ be a fixed probability reference measure. Keep a finite action set. For each action $a$, let $B_a(y):E\to E$ be measurable and integrable in operator norm. Let $(C,\lambda)$ be a proper probability cone and mass functional as in Section 46. Assume that, outside one $\mu$-null set for each action,
2281 
2282 $$
2283 B_a(y)C\subseteq C,
2284 \qquad
2285 \lambda\int_Y B_a(y)\,\mu(dy)=\lambda.
2286 $$
2287 
2288 The second identity is an equality of linear functionals. The first is a statement on the whole cone, not only on a finite selection of states.
2289 
2290 **Proposition 52. Continuous-observation instrument semantics.** For $z\in K=\{z\in C:\lambda z=1\}$,
2291 
2292 $$
2293 \boxed{
2294 P_a(A\mid z)=\int_A\lambda B_a(y)z\,\mu(dy),
2295 \qquad
2296 z^+(y)=\frac{B_a(y)z}{\lambda B_a(y)z}
2297 }
2298 $$
2299 
2300 define a probability kernel and, for $\mu$-almost every observation with positive denominator, a conditional state. They generate consistent finite record laws under any common measurable history-dependent policy. For prescribed actions, the joint observation density is
2301 
2302 $$
2303 \lambda B_{a_T}(y_T)\cdots B_{a_1}(y_1)z_0
2304 $$
2305 
2306 with respect to $\mu^{\otimes T}$. Replacing the reference measure by $\mu'=r\mu$, with $r>0$ almost everywhere and $\int r\,d\mu=1$, requires $B'_a(y)=B_a(y)/r(y)$. Event probabilities and normalized states then remain unchanged.
2307 
2308 **Proof.** Cone positivity makes the density nonnegative and the integrated mass identity makes its integral one. A positive-density update lies in $K$. Assign any fixed state of $K$ on the common exceptional null set and at zero-density points to obtain a measurable version of the update. This arbitrary choice has no effect on the probability kernel. Repeated integration, or conditioning one step at a time, proves normalization and consistency of the finite laws. Measurable policy factors are nonnegative and sum to one at each history. The reference-measure transformation cancels $r$ in each integral and cancels $1/r$ in the normalized update. $\square$
2309 
2310 When $\mu$ has no atoms, an individual point has probability zero even when its density is positive. A density may exceed one. Neither fact is a numerical pathology. Flooring a vanishing denominator changes the model and is not part of this theorem.
2311 
2312 A continuous density at an exact point is a version-dependent object on null sets. The update formula specifies a convenient version wherever its denominator is positive. Event probabilities are the invariant statistical objects. Comparing model densities requires a common measure or a declared change-of-measure calculation.
2313 
2314 A representation that lacks an operator-norm integrable density needs a more general operator-valued measure. Such representations are not silently covered by the density hypothesis above.
2315 
2316 # 59. A finite certificate over an entire observation interval
2317 
2318 Specialize to $Y=[-1,1]$ and $\mu(du)=du/2$. Suppose $B_a(u)$ is a matrix polynomial. A finite congruence certificate can prove positivity for every $u$, not merely for values on a grid.
2319 
2320 For a real $d^2\times d^2$ symmetric matrix $F$, define
2321 
2322 $$
2323 [\Phi_F(X)]_{ij}=\sum_{p,q=1}^d F_{(i,p),(j,q)}X_{pq}.
2324 $$
2325 
2326 Vectorization orders the pairs with output index first inside each input column. If
2327 
2328 $$
2329 F=\sum_\nu w_\nu\operatorname{vec}(K_\nu)
2330 \operatorname{vec}(K_\nu)^{\mathsf T},\qquad w_\nu\geq0,
2331 $$
2332 
2333 then $\Phi_F(X)=\sum_\nu w_\nu K_\nu X K_\nu^{\mathsf T}$. Every positive semidefinite $F$ has a real factorization of this kind. The matrix is the usual Choi/Gram representation, with an explicit index convention [R16].
2334 
2335 **Proposition 53. Polynomial Gram certificate for interval observations.** Fix the injective embedding $\mathcal J$ and its strict interior witness from Proposition 50. Set
2336 
2337 $$
2338 Z_q(u)=\begin{pmatrix}I_{d^2}\\uI_{d^2}\\\vdots\\u^qI_{d^2}\end{pmatrix}.
2339 $$
2340 
2341 Suppose each action has constant positive semidefinite matrices $G_{a,0},G_{a,1}$, with the second term omitted when $q=0$, such that
2342 
2343 $$
2344 \begin{aligned}
2345 F_a(u)&=Z_q(u)^{\mathsf T}G_{a,0}Z_q(u)\\
2346 &\quad +(1-u^2)Z_{q-1}(u)^{\mathsf T}G_{a,1}Z_{q-1}(u),\\
2347 \mathcal J B_a(u)&=\Phi_{F_a(u)}\mathcal J.
2348 \end{aligned}
2349 $$
2350 
2351 Assume also $z_0\in K_{\mathcal J}$ and the exact integrated mass identity. Then the model is internally probability-valid for every observation value in $[-1,1]$ and every finite horizon. The intertwining condition is finitely many coefficient equalities. The normalization condition uses only
2352 
2353 $$
2354 \int_{-1}^1u^k\,\frac{du}{2}
2355 =\begin{cases}0&k\text{ odd},\\1/(k+1)&k\text{ even}.\end{cases}
2356 $$
2357 
2358 **Proof.** Both Gram expressions are positive semidefinite on the interval because $1-u^2\geq0$ there. Factoring $F_a(u)$ at a fixed $u$ gives a positive congruence map, whether or not its individual factors depend polynomially on $u$. The intertwining identity therefore proves $B_a(u)C_{\mathcal J}\subseteq C_{\mathcal J}$. Polynomials on a compact interval are integrable. The mass identity and Proposition 52 prove the assertion. Equality of two finite matrix polynomials is equivalent to equality of every coefficient. The moment values follow by elementary integration. $\square$
2359 
2360 For fixed $\mathcal J$ and fixed degree bounds, unknown entries of $B_a$ and the Gram matrices obey linear equalities and positive semidefinite constraints. A convex data-fit objective therefore gives a semidefinite feasibility or optimization problem in this fixed representation. Jointly choosing $\mathcal J$ changes the problem: the intertwining equation then contains products of unknown coefficients.
2361 
2362 This is a sufficient certificate, not a completeness theorem for all positive kernels or all curved domains. A negative answer at one degree is not a proof of model invalidity. Approximate Gram feasibility is also not an exact witness. Rational factorizations with nonnegative rational weights, or rigorous matrix-positivity bounds, must survive a separate replay.
2363 
2364 The reference checker accepts a factored version:
2365 
2366 $$
2367 \Phi_u(X)=\sum_\nu w_\nu g_\nu(u)K_\nu(u)XK_\nu(u)^{\mathsf T},
2368 \qquad g_\nu(u)\in\{1,1-u^2\}.
2369 $$
2370 
2371 It checks the polynomial identities, weights, support, embedding, initial state and normalization exactly. It does not run a general semidefinite optimizer. The operator identity holds on the image of $\mathcal J$; trace preservation of an arbitrary extension outside that image is not required.
2372 
2373 # 60. Learning from integral observations rather than point probabilities
2374 
2375 A finite polynomial family can be identified through moment operators. This gives a continuous-observation analogue of the core-table construction without pretending that a point has positive probability.
2376 
2377 Assume the true operator density belongs to the declared class
2378 
2379 $$
2380 B_a(u)=\sum_{k=0}^D u^k C_{a,k},
2381 \qquad
2382 M_{a,j}=\int_{-1}^1 u^j B_a(u)\,\frac{du}{2}.
2383 $$
2384 
2385 Define the moment Gram matrix $W_D$ by
2386 
2387 $$
2388 (W_D)_{jk}=\int_{-1}^1u^{j+k}\,\frac{du}{2},
2389 \qquad 0\leq j,k\leq D.
2390 $$
2391 
2392 A history is now a prescribed action sequence with observations constrained to measurable events of positive or zero probability. It produces an unnormalized predictive vector $h_j$. A future event experiment produces a row functional $t_i$. Choose $r$ of each and form
2393 
2394 $$
2395 H_{ij}=t_i h_j,\qquad
2396 (G_{a,j})_{i\ell}=t_i M_{a,j}h_\ell.
2397 $$
2398 
2399 The index $j$ in $G_{a,j}$ is a moment order. Odd-order entries can be signed expectations. They are not branch probabilities and are never normalized as if they were.
2400 
2401 **Proposition 54. Conditional finite-moment identification.** Suppose the true predictive dimension is $r$, the degree bound $D$ is correct, and the selected $r\times r$ core $H$ is invertible. Exact population values of $H$, every $G_{a,j}$ and the initial test vector determine the complete polynomial operator family in test coordinates:
2402 
2403 $$
2404 \boxed{
2405 \widehat M_{a,j}=G_{a,j}H^{-1},
2406 \qquad
2407 \widehat C_{a,k}=\sum_{j=0}^D(W_D^{-1})_{kj}\widehat M_{a,j}.
2408 }
2409 $$
2410 
2411 If every moment-operator error is at most $\eta$ in one fixed induced norm, then coefficient reconstruction alone obeys
2412 
2413 $$
2414 \|\widetilde C_{a,k}-\widehat C_{a,k}\|
2415 \leq\eta\sum_j|(W_D^{-1})_{kj}|.
2416 $$
2417 
2418 **Proof.** A nonzero polynomial has strictly positive squared integral, so its coefficient vector $c$ satisfies $c^{\mathsf T}W_Dc>0$. Thus $W_D$ is invertible. Let $T$ have rows $t_i$ and $S$ have columns $h_j$. Since $H=TS$ is an invertible square core in an $r$-dimensional space, both $T$ and $S$ are invertible. Therefore $G_{a,j}H^{-1}=TM_{a,j}T^{-1}$. The relation $M_{a,j}=\sum_k(W_D)_{jk}C_{a,k}$ is exact integration of the polynomial expansion. Invert it and conjugate by $T$ to obtain the displayed reconstruction. The error bound is the triangle inequality for that same fixed linear inversion. $\square$
2419 
2420 The rank and degree assumptions are not learned from the determinant of $H$. The determinant establishes coverage only after the ambient predictive dimension has been justified. The moment Gram can become poorly conditioned at higher degree. An orthogonal polynomial basis can improve this representation without removing an incorrect degree assumption.
2421 
2422 The entries have an operational population meaning. A reset experiment for an event word, with one observation weighted by $u^j$, returns a bounded random variable in $[-1,1]$. Its expectation is the corresponding entry of $G_{a,j}$. With $N$ independent trials for each of $M$ predeclared entries, the same exponential-moment argument as Proposition 35 gives the simultaneous bound
2423 
2424 $$
2425 \epsilon_N=\sqrt{\frac{2\log(2M/\delta)}{N}}
2426 $$
2427 
2428 with failure probability at most $\delta$. The factor differs from the Bernoulli bound because the value range is two. Dependent trajectories, unknown resets and post-selection are not covered by that observation protocol. This edition uses exact constructed population tables, not a new sampling experiment.
2429 
2430 # 61. Finite moments cannot validate an unrestricted density class
2431 
2432 A finite observation basis is a modeling assumption. Positivity certification does not turn that assumption into a consequence of data.
2433 
2434 **Proposition 55. A valid continuous counterexample to finite-moment identification.** Let
2435 
2436 $$
2437 L_3(u)=\frac{5u^3-3u}{2},\qquad
2438 p_\pm(u)=1\pm\epsilon L_3(u),\qquad 0<\epsilon<1.
2439 $$
2440 
2441 Relative to $du/2$, these are distinct strictly positive normalized densities with the same moments of orders zero, one and two. Their total-variation distance is
2442 
2443 $$
2444 \boxed{\|p_+\mu-p_-\mu\|_{\mathrm{TV}}=\frac{13\epsilon}{40}.}
2445 $$
2446 
2447 Taking independent observations gives two rank-one processes whose products of these low-order moments agree at every sequence length.
2448 
2449 **Proof.** The endpoints and stationary points of $L_3$ show $|L_3(u)|\leq1$ on the interval. Direct integration gives $\int u^jL_3(u)\,du/2=0$ for $j=0,1,2$. This proves normalization and moment agreement, while strict positivity follows from $\epsilon<1$. Its nonnegative root other than zero is $\sqrt{3/5}$. Integrating $|L_3|$ on the two subintervals of $[0,1]$ and using symmetry gives $\int|L_3|\,du/2=13/40$. The density difference is $2\epsilon L_3$, so its half-integral is the stated distance. Independence makes every product moment factor into the identical one-step moments. $\square$
2450 
2451 This is not a defect in Proposition 54. These densities lie outside its quadratic class. It is a concrete reason to retain an explicit approximation or misspecification term when increasing the observation basis. Internal probability validity and correct density identification remain different claims even in one predictive dimension.
2452 
2453 # 62. Exact observation bins preserve their own event law
2454 
2455 Let $A_1,\ldots,A_n$ be a measurable partition of $Y$. Define the cell operators
2456 
2457 $$
2458 B_{a,j}=\int_{A_j}B_a(y)\,\mu(dy).
2459 $$
2460 
2461 Their role is to represent observed events. They are not point samples of the original density.
2462 
2463 **Proposition 56. Exact bin coarsening and its posterior.** Every $B_{a,j}$ preserves the probability cone, and $\lambda\sum_jB_{a,j}=\lambda$. For a policy measurable with respect to bin histories, the finite model $\{B_{a,j}\}$ reproduces exactly the binned observation law of the continuous model at every finite horizon. On an observed cell with $p_j=\lambda B_{a,j}z>0$,
2464 
2465 $$
2466 \frac{B_{a,j}z}{p_j}
2467 =\frac1{p_j}\int_{A_j}z^+(y)\lambda B_a(y)z\,\mu(dy).
2468 $$
2469 
2470 **Proof.** Integrals of vectors in a closed convex cone remain in the cone. Summing over a partition gives the original mass identity. For a fixed bin history, its policy factors are constant across the corresponding cells. Successive integration of the operator product therefore gives the product of the cell operators, in the original temporal order. This establishes every finite binned record probability. In the posterior identity, substitute $z^+(y)\lambda B_a(y)z=B_a(y)z$ on the positive-density set. The remaining outputs are zero because they lie in the cone and have zero mass. $\square$
2471 
2472 The coarsened posterior is a conditional average, not the posterior at a representative observation value. It preserves exactly the information retained by the event history. An arbitrary policy that reacts differently to two observations inside the same cell is not bin-measurable, so the preceding exact-policy conclusion does not apply to it.
2473 
2474 Cell refinement is compatible with this semantics: a parent operator is the sum of its children. In this setting those cell operators are positive probability branches. This positivity is an extra property of event integration, not a property of every signed spectral packet in the signal archive.
2475 
2476 # 63. Averaging transformations has a covariance defect
2477 
2478 A common discretization replaces a family of matrices $K(y)$ by its mean and then applies $X\mapsto KXK^{\mathsf T}$. This is generally the wrong mean operation.
2479 
2480 **Proposition 57. Exact covariance correction for an averaged congruence.** Let $\nu$ be a probability measure and suppose $K(y)$ has square-integrable entries. Put $\bar K=\int K(y)\,\nu(dy)$. For every symmetric $X$,
2481 
2482 $$
2483 \boxed{
2484 \int K(y)XK(y)^{\mathsf T}\,\nu(dy)
2485 =\bar KX\bar K^{\mathsf T}
2486 +\int (K(y)-\bar K)X(K(y)-\bar K)^{\mathsf T}\,\nu(dy).
2487 }
2488 $$
2489 
2490 The correction is positive semidefinite when $X\succeq0$. For an affine matrix function $K(u)=K_0+uK_1$ with scalar $u$ of finite second moment, it equals $\operatorname{Var}_\nu(u)K_1XK_1^{\mathsf T}$. On a uniform interval of width $h$, the variance is $h^2/12$.
2491 
2492 **Proof.** Substitute $K=\bar K+(K-\bar K)$ and expand. The two cross terms integrate to zero. The remaining integrand is a positive congruence when $X$ is positive. For an affine matrix function the centered term is $(u-\mathbb E_\nu u)K_1$, giving the variance formula. Integrating the square of a centered uniform variable gives $h^2/12$. $\square$
2493 
2494 This is a defect between two operations: averaging a transformation parameter and applying its quadratic action. It is not the same as the residual from dropping a signal packet, although both fit the framework's requirement to record failures of commutation.
2495 
2496 For the example in Section 66, equal bins with midpoint evaluation lose total mass
2497 
2498 $$
2499 \boxed{\frac{3}{4n^2}}
2500 $$
2501 
2502 when there are $n$ bins. With one bin, midpoint evaluation retains only one quarter of the required mass. Exact integration or the covariance correction restores the missing contribution. Dividing an inaccurate approximation by a guessed normalizer would define another model rather than establish this identity.
2503 
2504 # 64. A valid density approximation with a resolution bound
2505 
2506 Exact coarsening gives a finite event model. To compare complete continuous observation laws, lift that model back to a density on the same observation space. Assume all cells have positive reference mass and put
2507 
2508 $$
2509 B_a^\Pi(y)=\frac{B_{a,j}}{\mu(A_j)}\quad(y\in A_j).
2510 $$
2511 
2512 This uses the normalized reference measure inside each cell. It is an explicit density approximation, not an assertion that the original conditional density was constant there.
2513 
2514 **Proposition 58. Positive cell-average approximation.** The family $B_a^\Pi$ is cone preserving and mass conserving. It has exactly the same cell operators as $B_a$. Let $Y$ have a metric and suppose
2515 
2516 $$
2517 \|B_a(y)-B_a(v)\|_{C\to C}\leq L\,d(y,v)
2518 $$
2519 
2520 in the induced base norm, with every cell of diameter at most $h$. Then
2521 
2522 $$
2523 \epsilon_\Pi:=\sup_{a,z\in K}\int
2524 \|(B_a(y)-B_a^\Pi(y))z\|_C\,\mu(dy)
2525 \leq Lh.
2526 $$
2527 
2528 For equal intervals of width $h$ under the uniform measure on $[-1,1]$, the sharper bound is $\epsilon_\Pi\leq Lh/3$.
2529 
2530 **Proof.** Each cell average is a positive scalar multiple of a positive cell operator. Its integral over that cell is $B_{a,j}$, proving both normalization and exact cell agreement. For $y\in A_j$, subtract the cell average and use
2531 
2532 $$
2533 \|(B_a(y)-B_a^\Pi(y))z\|_C
2534 \leq\frac{L}{\mu(A_j)}\int_{A_j}d(y,v)\,\mu(dv),
2535 $$
2536 
2537 because $\|z\|_C=1$. Integrating over $y$ proves the diameter bound. Two independent uniform points in an interval of width $h$ have mean absolute separation $h/3$, obtained by integrating over its two triangles. Applying this inside every equal interval gives the sharper bound. $\square$
2538 
2539 The Lipschitz constant is a declared operator bound. It is not a fit residual inferred from adjacent samples. The comparison needs a common domain and a common reference measure. If two kernels are only close in Euclidean coefficient norm, the relevant norm-conversion constant must be included.
2540 
2541 In the full $2\times2$ symmetric-matrix example, the pullback base norm is
2542 
2543 $$
2544 \|(m,x,y)\|_C=\max\{|m|,\sqrt{x^2+y^2}\}=\|\mathcal J(m,x,y)\|_1.
2545 $$
2546 
2547 There $\|K(u)\|_2\leq2$ and $\|K(u)-K(v)\|_2\leq(3/2)|u-v|$. The identity
2548 
2549 $$
2550 K(u)XK(u)^{\mathsf T}-K(v)XK(v)^{\mathsf T}
2551 =(K(u)-K(v))XK(u)^{\mathsf T}
2552 +K(v)X(K(u)-K(v))^{\mathsf T}
2553 $$
2554 
2555 and $\|AXB\|_1\leq\|A\|_2\|X\|_1\|B\|_2$ give $L=6$. The rotated action uses an orthogonal matrix before this map and has the same bound. Equal bins therefore give $\epsilon_\Pi\leq4/n$, capped by two for two valid instruments. The inequality is sufficient, not asserted sharp.
2556 
2557 # 65. Full continuous record laws inherit the stability argument
2558 
2559 The base-norm proof from Part IV has an integral version. Its hypotheses must be supplied rather than inferred by replacing a finite sum with an integral in a program.
2560 
2561 **Proposition 59. Continuous-record total-variation bound.** Let $B_a(y)$ and $\widetilde B_a(y)$ satisfy Proposition 52 on a common $(C,\lambda,\mu)$, and let the two models use the same measurable policy on every common observation history. Suppose their initial states satisfy $\|z_0-\widetilde z_0\|_C\leq e_0$ and their time-$t$ instruments satisfy
2562 
2563 $$
2564 \sup_{a,z\in K}\int\|(B_{a,t}(y)-\widetilde B_{a,t}(y))z\|_C\,\mu(dy)
2565 \leq\epsilon_t.
2566 $$
2567 
2568 The length-$T$ record laws, including actions when the policy is randomized, obey
2569 
2570 $$
2571 \boxed{
2572 \|P_T-\widetilde P_T\|_{\mathrm{TV}}
2573 \leq\min\left\{1,\frac{e_0+\sum_{t=0}^{T-1}\epsilon_t}{2}\right\}.
2574 }
2575 $$
2576 
2577 No lower bound on each observation density is required. For identical initial states and the equal-bin approximation in Section 64, the example has the sufficient bound $\min(1,2T/n)$.
2578 
2579 **Proof.** For any positive decomposition $x=u-v$, cone preservation and mass conservation give
2580 
2581 $$
2582 \int\|B_a(y)x\|_C\,\mu(dy)
2583 \leq\int\lambda B_a(y)(u+v)\,\mu(dy)
2584 =\lambda u+\lambda v.
2585 $$
2586 
2587 Taking the infimum proves integrated instrument contraction. At each continuous history let $v_h,\widetilde v_h$ be the unnormalized state densities, including the policy factors. Split the next difference as
2588 
2589 $$
2590 B_a(y)(v_h-\widetilde v_h)
2591 +(B_a(y)-\widetilde B_a(y))\widetilde v_h.
2592 $$
2593 
2594 Integrate the first term using contraction. The second is bounded by $\epsilon_t\lambda\widetilde v_h$ after normalizing only its nonzero positive mass. Its integral over histories is at most $\epsilon_t$. Tonelli's theorem applies to these nonnegative norm bounds, and operator integrability supplies well-defined finite-dimensional integrals. The integrated state discrepancy therefore satisfies $D_{t+1}\leq D_t+\epsilon_t$. Finally $|\lambda x|\leq\|x\|_C$ bounds the record-density difference, and total variation is half its absolute integral. $\square$
2595 
2596 This result compares one common policy, not two policies optimized separately. It concerns the whole law and can become uninformative at long horizons. It is not a uniform posterior contraction theorem.
2597 
2598 For a measurable event $H$ of probability $p=P_T(H)>0$, the restriction-and-normalization argument of Proposition 44 works for measures as well as finite vectors. When both conditional laws exist, their total-variation distance is at most $\min(1,\delta/p)$ if the full-law distance is at most $\delta$. The event probability is invariant under a reference-measure change. A density value at one point is not a substitute for that event probability.
2599 
2600 The full-law comparison between $B$ and $B^\Pi$ permits any common measurable policy because both are continuous-output models on the same space. The exact equivalence to the finite bin model in Proposition 56 instead requires a bin-measurable policy. These are different conclusions with different observation information.
2601 
2602 # 66. A complete continuous-observation example
2603 
2604 Use the same three-coordinate matrix embedding as Section 57. Set
2605 
2606 $$
2607 K(u)=\frac12\begin{pmatrix}1+3u&0\\0&1-3u\end{pmatrix},
2608 \qquad -1\leq u\leq1,
2609 \qquad \mu(du)=du/2.
2610 $$
2611 
2612 The direct action uses $X\mapsto K(u)XK(u)^{\mathsf T}$. The rotated action first applies the unchanged rotation congruence from Section 57, giving
2613 
2614 $$
2615 X\longmapsto\frac15 K(u)WXW^{\mathsf T}K(u)^{\mathsf T}.
2616 $$
2617 
2618 Both are exactly normalized because $\int K(u)^{\mathsf T}K(u)\,du/2=I$ and $W^{\mathsf T}W=5I$. In predictive coordinates, write
2619 
2620 $$
2621 \begin{gathered}
2622 a(u)=\frac{1+9u^2}{4},\qquad
2623 b(u)=\frac{3u}{2},\qquad
2624 c(u)=\frac{1-9u^2}{4},\\[3pt]
2625 B_0(u)=\begin{pmatrix}a(u)&b(u)&0\\b(u)&a(u)&0\\0&0&c(u)\end{pmatrix},
2626 \qquad B_1(u)=B_0(u)T.
2627 \end{gathered}
2628 $$
2629 
2630 Although $c(u)$ is negative for part of the interval, both actions preserve the disk cone. Entrywise sign is not the positivity criterion in these coordinates. The initial state is $z_0=(1,0,0)^{\mathsf T}$, or the matrix $I/2$.
2631 
2632 ## 66.1 Event tables and moment reconstruction
2633 
2634 Let the three future tests be the empty test, a nonnegative observation after action zero, and a nonnegative observation after action one. Their rows form
2635 
2636 $$
2637 Q=\begin{pmatrix}
2638 1&0&0\\
2639 1/2&3/8&0\\
2640 1/2&9/40&-3/10
2641 \end{pmatrix}.
2642 $$
2643 
2644 Use histories consisting of the empty history, a nonnegative observation under action zero, and that history followed by a nonnegative observation under action one. Their unnormalized columns are
2645 
2646 $$
2647 S=\begin{pmatrix}
2648 1&1/2&107/320\\
2649 0&3/8&3/10\\
2650 0&0&-3/40
2651 \end{pmatrix}.
2652 $$
2653 
2654 The learner is supplied only the resulting integral tables and the declared class. Its core is
2655 
2656 $$
2657 \boxed{
2658 H=\begin{pmatrix}
2659 1&1/2&107/320\\
2660 1/2&25/64&179/640\\
2661 1/2&107/320&823/3200
2662 \end{pmatrix},\qquad\det H=\frac{81}{25600}.
2663 }
2664 $$
2665 
2666 The core is not required to be symmetric. Its rows are tests and its columns are histories, which are different experiments. The three moment orders use
2667 
2668 $$
2669 W_2=\begin{pmatrix}1&0&1/3\\0&1/3&0\\1/3&0&1/5\end{pmatrix},
2670 \qquad
2671 W_2^{-1}=\begin{pmatrix}9/4&0&-15/4\\0&3&0\\-15/4&0&45/4\end{pmatrix},
2672 \qquad\det W_2=\frac4{135}.
2673 $$
2674 
2675 Proposition 54 reconstructs $QB_a(u)Q^{-1}$ from $H$ and the six inserted-moment tables. The code that performs this reconstruction imports no generating matrices. The separate generator produces the tables and is used only in the verification suite. These are population values from an exact example, not learned sample estimates.
2676 
2677 ## 66.2 Two densities and a genuine event probability
2678 
2679 For a direct observation value $u=1/2$, the density and updated state are
2680 
2681 $$
2682 p_1=\frac{13}{16},\qquad z_1=(1,12/13,0)^{\mathsf T}.
2683 $$
2684 
2685 If the rotated action next gives $u=-1/2$, they are
2686 
2687 $$
2688 p_2=\frac{413}{1040},\qquad
2689 z_2=(1,-312/413,-240/413)^{\mathsf T}.
2690 $$
2691 
2692 Their product $413/1280$ is a joint density with respect to $(du/2)^{\otimes2}$, not the probability of those two exact values. The two point values each have zero probability in this nonatomic model.
2693 
2694 By contrast, the event $u_1\geq0$ under action zero followed by $u_2\geq0$ under action one has actual probability
2695 
2696 $$
2697 \boxed{\frac{107}{320},}
2698 $$
2699 
2700 and its conditional state is $(1,96/107,-24/107)^{\mathsf T}$. Exact bin operators produce this value without quadrature.
2701 
2702 The final point-conditioned state remains strictly inside the disk:
2703 
2704 $$
2705 \det\mathcal J(z_2)=\frac{15625}{682276}>0.
2706 $$
2707 
2708 A pure boundary state can instead have zero density at isolated observations. The checker rejects conditioning on such a zero-density branch. It does not manufacture a posterior by flooring its denominator.
2709 
2710 ## 66.3 The missing covariance is exactly calculable
2711 
2712 On an interval with midpoint $v$ and width $h$, the average matrix instrument is
2713 
2714 $$
2715 \mathbb E[K(u)XK(u)^{\mathsf T}]
2716 =K(v)XK(v)^{\mathsf T}
2717 +\frac{h^2}{12}K_1XK_1^{\mathsf T},
2718 \qquad K_1=\frac32\operatorname{diag}(1,-1).
2719 $$
2720 
2721 The correction is itself a positive congruence. The cell operator multiplies this mean by the cell mass $h/2$. Equal bins have width $2/n$, so omitting the covariance loses trace $3/(4n^2)$ over the full partition. The reference calculation checks both the exact matrix identity and this mass defect for several rational partitions.
2722 
2723 # 67. A declared template can be recognized without being rediscovered
2724 
2725 General curved-domain learning remains difficult. There is a useful finite case between supplying a finished coordinate certificate and claiming unrestricted domain discovery.
2726 
2727 Suppose a structural library supplies a certified template with polynomial coefficients $C_{a,k}$, mass row $\lambda$ and embedding $\mathcal J$. Suppose observed population tables produce coefficients $\widehat C_{a,k}$ and mass row $\widehat\lambda$ of the same dimension.
2728 
2729 **Proposition 60. Exact recognition of a fixed certified template.** The equations
2730 
2731 $$
2732 \boxed{
2733 \widehat C_{a,k}T=TC_{a,k},\qquad
2734 \widehat\lambda T=\lambda
2735 }
2736 $$
2737 
2738 are linear in $T$. Every invertible solution transports the template certificate to the fitted coordinates by $\widehat{\mathcal J}=\mathcal JT^{-1}$. If the fitted coefficient family is observable, meaning the rows $\widehat\lambda\widehat C_w$ span the dual space, the solution is unique. The fitted initial state must still be tested in the transported normalized cone.
2739 
2740 **Proof.** All coefficient matrices are fixed, so the displayed constraints are linear. An invertible solution yields $\widehat B_a(u)=TB_a(u)T^{-1}$ coefficient by coefficient. Substitution in the template intertwining identity proves the transported certificate. To prove uniqueness, let $D$ be the difference of two solutions. It satisfies $\widehat C_{a,k}D=DC_{a,k}$ and $\widehat\lambda D=0$. Therefore $\widehat\lambda\widehat C_wD=0$ for every coefficient word. Observability makes every column of $D$ zero. An initial state outside the transported cone is not repaired by this operator identity, which explains the separate check. $\square$
2741 
2742 For the worked example, the linear system has the unique solution $T=Q$ from Section 66. The learner receives the population tables and a fixed structural template. It reconstructs the operators, solves the linear match, transports the embedding and exports a certificate. Independent replay then needs only that certificate. It does not refit, search, or import the template or generator.
2743 
2744 This is template recognition, not unrestricted geometry discovery. The supplied template includes the specific operator family. An incompatible model is rejected rather than changed to fit the template. The reference routine also withholds a result for an underdetermined or singular match. Noise would require a declared approximate match and a separate validity or correction argument. This edition's recognition example is exact.
2745 
2746 For fixed embeddings, polynomial Gram optimization offers a broader construction route than fixed-template matching. That convex proposal problem is stated in Section 59 but is not implemented as a general solver here. The independent checker remains useful regardless of whether a proposal came from a template, symbolic derivation or an optimization routine.
2747 
2748 # 68. The combined continuous polynomial sector
2749 
2750 **Proposition 61. Conditional continuous realization with a finite domain certificate.** Assume a controlled continuous-observation process has the declared finite predictive dimension and polynomial operator-density degree, a fixed uniform reference measure on $[-1,1]$, and an invertible complete event-test core. Exact moment tables identify its operator density in test coordinates by Proposition 54.
2751 
2752 If the fitted family and initial state admit a checked matrix/Gram certificate of Proposition 53, the supplied realization is internally valid at every observation value and finite horizon. A successful invertible fixed-template recognition is one sufficient way to construct such a certificate. Exact coordinate transport and complete packet synthesis preserve the represented event probabilities. Individual signed state packets still need the separate positivity condition before probabilistic interpretation.
2753 
2754 Exact observation-cell integration gives the precise finite event model for bin-measurable policies. Positive cell-average density approximations have the resolution bound of Proposition 58. Two certified models on a common cone and common reference measure satisfy Proposition 59's full-law bound whenever the declared instrument discrepancy is controlled.
2755 
2756 **Proof.** Moment identification uses the true dimension and degree hypotheses to apply Proposition 54. The independent Gram identity establishes positivity on the whole declared interval, while the exact integral mass identity supplies normalization through Proposition 52. Proposition 60 supplies a transported witness in the template case. Coordinate cancellation and full packet synthesis are exact identities. Proposition 56 proves the cell-model statement. Propositions 58 and 59 provide the two approximation conclusions with their common-domain and common-policy assumptions unchanged. $\square$
2757 
2758 For a noisy fitted model, the certificate alone proves only internal validity. A guarantee about the generating process requires a valid uncertainty set covering it. A common-domain certificate over a convex uncertainty family can be checked at complete-family vertices with the same embedding, because the operator identities, Gram positivity and mass constraints survive convex combination. This does not justify an uncertainty set that excludes the true degree, rank or dynamics.
2759 
2760 The construction leaves visible what is often collapsed into the phrase signal model: a represented state, an operator-valued observation density, a reference measure, a valid probability domain, an observation protocol and an approximation budget. Each is needed for a different reason.
2761 
2762 # 69. From continuous semantics to discovery
2763 
2764 Part V gives event-based probability semantics, exact binning and conditional finite-moment reconstruction. Fixed-template recognition in Proposition 60 is not unrestricted geometry discovery. Parts VI–VII add restricted discovery and robust verification with their own assumptions. The continuous reference measure and the observation basis remain explicit inputs.
2765 
2766 \newpage
2767 
2768 # Part VI. Certificate discovery, approximation and sequential evidence
2769 
2770 A signal model must distinguish a representation that preserves probability from a representation that predicts the observed process accurately. Part V gave an exact curved certificate and a way to recognize a supplied operator template. It also made the polynomial-degree assumption explicit. This block advances both boundaries without identifying them with each other.
2771 
2772 The first development is a restricted discovery procedure. It recovers an ellipsoidal probability domain from fitted polynomial operators, without a supplied template, matrix embedding or hidden generator. The second is a positive approximation theorem for operator densities outside a finite polynomial class. The third is sequential evidence under dependent observations, including the cost of action coverage. Together they separate structural validity, approximation error and observable disagreement.
2773 
2774 The invariant-form and quadratic-certificate ingredients belong to classical systems theory [R8, R13]. Positive polynomial averaging uses the classical Bernstein--Durrmeyer kernel [R19]. The sequential argument is an elementary martingale confidence sequence, not a new concentration principle [R20]. The contribution of this block is their connection to the large signal model's explicit representation contracts and to independently replayable examples. Propositions 62--70 have proofs below. No proof kernel or production training system is invoked.
2775 
2776 # 70. An ellipsoidal certificate with an orientation test
2777 
2778 Work in mass-first coordinates $z=(m,x)\in\mathbb R\times\mathbb R^d$, with $\lambda=(1,0,\ldots,0)$. For a center $c\in\mathbb R^d$ and a symmetric positive definite matrix $H$, define
2779 
2780 $$
2781 C_{c,H}=\{(m,x):m\geq0,\ (x-mc)^{\mathsf T}H(x-mc)\leq m^2\}.
2782 $$
2783 
2784 Its normalized base is an ellipsoid. Set
2785 
2786 $$
2787 S_c=\begin{pmatrix}1&0\\-c&I\end{pmatrix},\qquad
2788 Q=S_c^{\mathsf T}\begin{pmatrix}1&0\\0&-H\end{pmatrix}S_c.
2789 $$
2790 
2791 Then $C_{c,H}=\{z:m\geq0,\ z^{\mathsf T}Qz\geq0\}$. The cone is proper and generating, and $m$ is strictly positive on every nonzero cone element. These facts follow by an invertible linear transformation to the Lorentz cone. In particular, an output with zero mass in the cone is the zero vector.
2792 
2793 Let $B_a(u)$ be polynomial operator densities on $[-1,1]$, relative to $du/2$. Write
2794 
2795 $$
2796 \lambda B_a(u)=(b_{a,0}(u),b_a(u)^{\mathsf T}),\qquad
2797 \beta_a(u)=b_{a,0}(u)+b_a(u)^{\mathsf T}c.
2798 $$
2799 
2800 **Proposition 62. Oriented quadratic probability certificate.** Suppose that for each action and every $u\in[-1,1]$ there are $\tau_a(u)\geq0$ and $R_a(u)\succeq0$ satisfying
2801 
2802 $$
2803 \boxed{B_a(u)^{\mathsf T}QB_a(u)-\tau_a(u)Q=R_a(u),}
2804 $$
2805 
2806 and that
2807 
2808 $$
2809 \boxed{\beta_a(u)\geq0,\qquad
2810 \beta_a(u)^2-b_a(u)^{\mathsf T}H^{-1}b_a(u)\geq0.}
2811 $$
2812 
2813 If $z_0\in C_{c,H}$ has mass one and $\lambda\int B_a(u)\,du/2=\lambda$, then every operator preserves $C_{c,H}$. The supplied model has valid probabilities at every finite horizon. Polynomial sum-of-squares witnesses for the scalar inequalities and for $R_a$ make this a finite certificate.
2814 
2815 **Proof.** For $z\in C_{c,H}$, the quadratic identity gives
2816 
2817 $$
2818 (B_a z)^{\mathsf T}Q(B_a z)
2819 =\tau_a z^{\mathsf T}Qz+z^{\mathsf T}R_a z\geq0.
2820 $$
2821 
2822 For a nonzero $z$, write $x=m(c+w)$ with $w^{\mathsf T}Hw\leq1$. Its output mass is $m(\beta_a+b_a^{\mathsf T}w)$. Cauchy--Schwarz in the $H$ inner product bounds this below by
2823 
2824 $$
2825 m\left(\beta_a-\sqrt{b_a^{\mathsf T}H^{-1}b_a}\right)\geq0.
2826 $$
2827 
2828 Thus both inequalities defining the desired sheet of the quadratic cone hold. The zero input causes no difficulty. Integrated mass conservation gives normalization, and induction gives the finite-horizon conclusion. For polynomial data, identities between coefficients and nonnegative sums of polynomial squares establish the displayed hypotheses throughout the interval. Terms multiplied by $1-u^2$ are also nonnegative on that interval. $\square$
2829 
2830 The orientation test is essential. The matrix $-I$ preserves every quadratic form, but maps positive mass to negative mass. A quadratic inequality without its mass orientation can certify the wrong sheet. Likewise, testing finitely many values of $u$ is not a substitute for a polynomial identity and its positivity witness.
2831 
2832 This is a sufficient language, not a completeness theorem for all cones. The reference checker accepts rational scalar and matrix sum-of-squares witnesses. Its discovery routine uses the narrower conformal case $R_a=0$, but the checker does not confuse that search restriction with the theorem's scope.
2833 
2834 # 71. Geometry can be proposed from the operators themselves
2835 
2836 An operator identity can reveal a useful coordinate geometry without specifying that geometry in advance. Suppose two fitted operators evaluated at an anchor $u_*$ give an invertible ratio
2837 
2838 $$
2839 F=B_b(u_*)B_a(u_*)^{-1},\qquad \lambda F=\lambda.
2840 $$
2841 
2842 This ratio is an algebraic object. It need not be a physically available action or a positive map by itself. In mass-first coordinates write
2843 
2844 $$
2845 F=\begin{pmatrix}1&0\\b&A\end{pmatrix}.
2846 $$
2847 
2848 If $I-A$ is invertible, its fixed center is $c=(I-A)^{-1}b$. The centered spatial action is $A$. A candidate invariant metric obeys
2849 
2850 $$
2851 A^{\mathsf T}HA=H.
2852 $$
2853 
2854 With $A$ fixed, this is a linear equation in the independent entries of symmetric $H$. No coordinate template appears in that equation.
2855 
2856 **Proposition 63. Conditional operator-only ellipsoid discovery.** Suppose the symmetric solution space of $A^{\mathsf T}HA=H$ is one-dimensional and contains a positive definite matrix $H_0$. Let a rank-one fitted branch at a declared observation have a nonzero-mass image, normalized to $v=(1,x_v)$. If $x_v\neq c$, define
2857 
2858 $$
2859 H=\frac{H_0}{(x_v-c)^{\mathsf T}H_0(x_v-c)}.
2860 $$
2861 
2862 This gives a unique positive definite metric in that solution space for which $v$ is on the normalized boundary. If this proposed geometry passes every hypothesis of Proposition 62 for the complete fitted family and its initial state, it is an internally valid probability-domain certificate. The proposal requires no supplied template. Failure at any step withholds this certificate and does not establish model invalidity.
2863 
2864 **Proof.** Invertibility of $I-A$ gives the unique center. One-dimensionality makes every symmetric invariant form a scalar multiple of $H_0$. The nonzero displacement has strictly positive squared $H_0$ length, so the normalization is well-defined, positive and independent of the initially chosen positive scale. The final acceptance test is exactly Proposition 62. Neither invariance under the ratio nor the choice of a boundary image alone is asserted to certify the full family. $\square$
2865 
2866 The rank-one image proposes a scale. It does not prove that every possible probability domain must place that image on its boundary. Some valid models have many admissible domains. The procedure can also fail because there is no suitable ratio, because invariant forms are not unique, because no rational rank-one anchor is found, or because the chosen ellipsoid fails another branch. These are explicit search outcomes.
2867 
2868 ## 71.1 The continuous example no longer needs its template
2869 
2870 Use the same exact population moment tables as Section 66. The existing moment learner first reconstructs the two polynomial operators in test coordinates. The new discovery routine receives those operators, their mass row and initial vector. It does not read `continuous_template.json` or import the population generator.
2871 
2872 At $u_*=0$, the operator ratio gives the required invariant-form equation. A rational rank-one observation supplies a boundary image. The recovered center and metric are
2873 
2874 $$
2875 \boxed{
2876 c=\begin{pmatrix}1/2\\1/2\end{pmatrix},\qquad
2877 H=\begin{pmatrix}100/9&-20/3\\-20/3&100/9\end{pmatrix}.
2878 }
2879 $$
2880 
2881 The two metric eigenvalues are $40/9$ and $160/9$. For both actions the exact certificate is
2882 
2883 $$
2884 \begin{aligned}
2885 B_a(u)^{\mathsf T}QB_a(u)&=\frac{(1-9u^2)^2}{16}Q,\\
2886 \beta_a(u)&=\frac{1+9u^2}{4},\\
2887 \beta_a(u)^2-b_a(u)^{\mathsf T}H^{-1}b_a(u)
2888 &=\frac{(1-9u^2)^2}{16}.
2889 \end{aligned}
2890 $$
2891 
2892 Every scalar on the right has an exact rational sum-of-squares representation. Integrated mass is unchanged. The fitted initial state is the ellipsoid center, and the two densities $13/16$ and $413/1040$ from Section 66 are unchanged. This is a new certificate construction for the same model, not a change to its predictions.
2893 
2894 The exported JSON contains the fitted operators, center, metric, initial state and all witnesses. Independent replay needs only that JSON. The discovery calculation is also checked on three additional rational coefficient families and affine coordinate changes. Those finite examples exercise the stated sector and do not establish unrestricted discovery.
2895 
2896 The procedure currently uses three predictive coordinates, a two-dimensional ellipsoid, a finite rational anchor list, a one-dimensional invariant-form space and rational rank-one roots. It proposes only exactly conformal polynomial branches whose scalar nonnegativity can be resolved by its factor-based sum-of-squares routine. General noisy or jointly optimized curved-domain learning remains outside this implementation.
2897 
2898 # 72. Approximate an unknown density without assuming its exact degree
2899 
2900 Part V identified a polynomial operator density under an exact degree assumption. A different question is how to approximate a valid density that is not a polynomial at all.
2901 
2902 In this section observations lie in $[0,1]$ with Lebesgue reference measure. This is an explicit change from the preceding $[-1,1]$ convention. Under $v=(u+1)/2$, the probability measure $du/2$ becomes $dv$. A Lipschitz constant with respect to $u$ becomes twice as large with respect to $v$.
2903 
2904 Write the Bernstein basis and the classical Bernstein--Durrmeyer kernel as
2905 
2906 $$
2907 b_{n,k}(x)=\binom nk x^k(1-x)^{n-k},\qquad
2908 D_n(x,v)=(n+1)\sum_{k=0}^n b_{n,k}(x)b_{n,k}(v).
2909 $$
2910 
2911 Both variables range over $[0,1]$. The kernel is nonnegative and integrates to one in either variable. Let $B_a(v)$ be any operator-norm integrable density preserving a common probability cone $C$, with mass functional $\lambda$ and normalized base $K$. Define
2912 
2913 $$
2914 B_a^{[n]}(x)=\int_0^1D_n(x,v)B_a(v)\,dv
2915 =\sum_{k=0}^n b_{n,k}(x)C_{a,n,k},
2916 $$
2917 
2918 where
2919 
2920 $$
2921 C_{a,n,k}=(n+1)\int_0^1b_{n,k}(v)B_a(v)\,dv.
2922 $$
2923 
2924 **Proposition 64. Valid positive polynomial approximation.** Every $B_a^{[n]}$ preserves $C$, has degree at most $n$, and satisfies exact integrated mass conservation. If
2925 
2926 $$
2927 \|B_a(v)-B_a(x)\|_{C\to C}\leq L|v-x|
2928 $$
2929 
2930 for all actions, then
2931 
2932 $$
2933 \boxed{
2934 \sup_{a,z\in K}\int_0^1
2935 \|(B_a^{[n]}(x)-B_a(x))z\|_C\,dx
2936 \leq\frac{L}{\sqrt{3(n+2)}}.
2937 }
2938 $$
2939 
2940 With identical initial states, the corresponding length-$T$ laws under a common policy differ in total variation by at most
2941 
2942 $$
2943 \boxed{\min\left(1,\frac{TL}{2\sqrt{3(n+2)}}\right).}
2944 $$
2945 
2946 No finite true-degree hypothesis is needed.
2947 
2948 **Proof.** Nonnegative integration preserves the closed convex cone. Since $\int_0^1b_{n,k}=1/(n+1)$ and $\sum_k b_{n,k}=1$, the kernel is doubly stochastic. Its integral in $x$ gives $\int B_a^{[n]}=\int B_a$, proving exact mass conservation.
2949 
2950 For the error, subtract $B_a(x)$ inside the kernel integral, apply the Lipschitz inequality and use $\|z\|_C=1$. The integrated discrepancy is at most $L\mathbb E|V-X|$ under the joint probability density $D_n(x,v)$. This joint law can be constructed by first choosing $J$ uniformly from $\{0,\ldots,n\}$, and then drawing $X,V$ independently, conditionally on $J=k$, with density $(n+1)b_{n,k}$.
2951 
2952 That conditional beta law has variance
2953 
2954 $$
2955 \operatorname{Var}(X\mid J=k)
2956 =\frac{(k+1)(n-k+1)}{(n+2)^2(n+3)}.
2957 $$
2958 
2959 The average of $(k+1)(n-k+1)$ over $k=0,\ldots,n$ is $(n+2)(n+3)/6$, obtained by summing the first two integer powers. Hence
2960 
2961 $$
2962 \mathbb E(V-X)^2
2963 =2\mathbb E\operatorname{Var}(X\mid J)
2964 =\frac1{3(n+2)}.
2965 $$
2966 
2967 Cauchy--Schwarz gives the claimed integrated bound. The whole-law conclusion is Proposition 59 applied to the two valid instruments on the same domain. $\square$
2968 
2969 This construction approximates the operator density itself. It does not average a congruence factor and omit its covariance. Its coefficients are positive operators even when their entries in predictive coordinates are signed.
2970 
2971 Operationally, it replaces the raw observation $v$ by a noisy observation $x$ drawn from $D_n(x,v)$. It preserves total mass and underlying unconditioned evolution, not the entire original observation law. That changed law is what the bound measures. A prediction policy must be compared as the same function of the reported history.
2972 
2973 ## 72.1 A nonpolynomial density and a failed ordinary truncation
2974 
2975 Consider the valid scalar density
2976 
2977 $$
2978 p(v)=\frac12+2\left|v-\frac12\right|.
2979 $$
2980 
2981 It integrates to one, takes values between $1/2$ and $3/2$, and has Lipschitz constant two. It is not a single polynomial on $[0,1]$. Proposition 64 nevertheless supplies normalized positive polynomial approximations at every degree, with an explicit approximation budget.
2982 
2983 An ordinary orthogonal truncation need not preserve that validity. The valid density $p(v)=3v^2$ has degree-one $L^2$ projection
2984 
2985 $$
2986 q(v)=3v-\frac12,
2987 $$
2988 
2989 which is negative near zero. The positive construction at degree one instead gives
2990 
2991 $$
2992 p^{[1]}(v)=v+\frac12.
2993 $$
2994 
2995 The two approximations solve different mathematical problems. Neither has permission to call its approximation error zero. The reference checks derive both expressions exactly.
2996 
2997 # 73. Keep approximation bias separate from coefficient error
2998 
2999 The positive approximation theorem assumes access to exact integral coefficients. A fitted model introduces another discrepancy. Its coefficient estimates need their own validity constraints.
3000 
3001 **Proposition 65. Certified coefficient error and approximation bias.** Suppose every fitted coefficient $\widetilde C_{a,n,k}$ preserves the same cone $C$, and
3002 
3003 $$
3004 \lambda\frac1{n+1}\sum_{k=0}^n\widetilde C_{a,n,k}=\lambda.
3005 $$
3006 
3007 Then $\widetilde B_a^{[n]}(x)=\sum_k b_{n,k}(x)\widetilde C_{a,n,k}$ is internally valid. If
3008 
3009 $$
3010 \sup_{z\in K}\|(C_{a,n,k}-\widetilde C_{a,n,k})z\|_C
3011 \leq\eta_{a,k},
3012 $$
3013 
3014 its integrated discrepancy from the original Lipschitz density is at most
3015 
3016 $$
3017 \boxed{
3018 \frac{L}{\sqrt{3(n+2)}}+
3019 \max_a\frac1{n+1}\sum_{k=0}^n\eta_{a,k}.
3020 }
3021 $$
3022 
3023 A known additional model-correction discrepancy is added to this budget, not absorbed into the approximation term.
3024 
3025 **Proof.** Nonnegative Bernstein weights preserve cone positivity, and their integrals prove normalization. For any $z\in K$, the triangle inequality gives
3026 
3027 $$
3028 \int\|\textstyle\sum_k b_{n,k}(x)(C_{a,n,k}-\widetilde C_{a,n,k})z\|_C\,dx
3029 \leq\frac1{n+1}\sum_k\eta_{a,k}.
3030 $$
3031 
3032 Add the approximation error from Proposition 64. A separate correction is handled by another application of the triangle inequality. $\square$
3033 
3034 This is a conditional approximation-and-estimation theorem, not a claim that more coefficients always improve a finite-data estimator. Increasing $n$ reduces the stated bias bound while changing the number and conditioning of the quantities to be estimated. A cone witness proves that the resulting object is a probability model. It does not prove that the coefficient error budgets cover the generating process.
3035 
3036 The moments needed for $C_{a,n,k}$ are bounded Bernstein-weighted experiments. Their interpretation as full operator coefficients still requires suitable predictive-state coverage or a justified observable core. The next section provides sequential evidence for conditional experiments, not an automatic replacement for those identification assumptions.
3037 
3038 # 74. Evidence from dependent observations
3039 
3040 Let $\mathcal F_{t-1}$ denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function $\phi_{t,j}$ of the upcoming measurement, chosen from this information, with values in $[0,1]$. Put
3041 
3042 $$
3043 X_{t,j}=\phi_{t,j}(Y_t),\quad
3044 m_{t,j}=\mathbb E[X_{t,j}\mid\mathcal F_{t-1}],\quad
3045 p_{t,j}=\text{the model's prediction for that test}.
3046 $$
3047 
3048 The model prediction is also predictable. It may change as the model learns. The true conditional mean $m_{t,j}$ may depend on the entire past. No independence, stationarity or known reset state is assumed in this paragraph.
3049 
3050 Choose budgets $\alpha_j>0$ for a finite or countable prospectively specified family of test rules, with $\sum_j\alpha_j\leq\delta$. Define
3051 
3052 $$
3053 R_{t,j}=\sum_{s=1}^t(X_{s,j}-p_{s,j}),\qquad
3054 D_{t,j}=\sum_{s=1}^t(m_{s,j}-p_{s,j}).
3055 $$
3056 
3057 **Proposition 66. Time-uniform predictable-residual certificate.** With probability at least $1-\delta$, for every time $t\geq1$ and every registered rule $j$,
3058 
3059 $$
3060 \boxed{
3061 |R_{t,j}-D_{t,j}|\leq
3062 b_{t,j},\qquad
3063 b_{t,j}=\sqrt{\frac t2\log\frac{2t(t+1)}{\alpha_j}}.
3064 }
3065 $$
3066 
3067 For variables taking values in an interval of width $c$, multiply the boundary by $c$. The simultaneous guarantee remains valid when the observation time is chosen from the data.
3068 
3069 **Proof.** For a random variable $X\in[0,1]$ with conditional mean $m$, its conditional centered logarithmic moment-generating function has value and first derivative zero at the origin. Its second derivative is the variance under exponential tilting. A variable supported on $[0,1]$ has variance at most $1/4$, because its squared distance from $1/2$ is at most $1/4$. Integrating twice gives
3070 
3071 $$
3072 \mathbb E[e^{\theta(X-m)}\mid\mathcal F]\leq e^{\theta^2/8}.
3073 $$
3074 
3075 Iterating this conditional inequality gives the same exponential bound for the martingale sum $\sum_{s\leq t}(X_{s,j}-m_{s,j})$. Optimizing the exponential Markov inequality gives the two-sided bound $2\exp(-2b^2/t)$. At the displayed $b_{t,j}$ this is $\alpha_j/[t(t+1)]$. Sum over $t$, using $\sum_{t\geq1}1/[t(t+1)]=1$, and then over $j$. The resulting event holds at every time, so it also holds at a data-selected time. Scaling the variable proves the range version. $\square$
3076 
3077 This elementary boundary pays a logarithmic time-union cost. It is not claimed to be optimal. Stronger confidence-sequence constructions are available in the literature [R20]. The proof here is supplied because the exact assumptions and the estimated object matter more than a sharp constant in this foundation.
3078 
3079 The certificate estimates accumulated **conditional predictive drift**, not one fixed unconditional population probability. A prospectively chosen rule may use history, but an analyst cannot inspect an entire data sequence and then retroactively choose an unrestricted test with the same budget. New rules can be registered with fresh budgets and applied to future observations, using elapsed time after registration.
3080 
3081 # 75. Actions need coverage, not only more observations
3082 
3083 For intervention evidence use the information $\mathcal H_{t-1}$ just before choosing the action. Let the action be randomized from a known history-dependent policy $\pi_t$. The true next-observation kernel under action $a$ is $P_t(\cdot\mid\mathcal H_{t-1},a)$. Randomization uses no unrecorded future information. For a predictable test $\phi_t\in[0,1]$, define
3084 
3085 $$
3086 \mu_t^a=\int\phi_t(y)P_t(dy\mid\mathcal H_{t-1},a),\qquad
3087 W_t^a=\frac{\mathbf1_{\{A_t=a\}}\phi_t(Y_t)}{\pi_t(a\mid\mathcal H_{t-1})}.
3088 $$
3089 
3090 **Proposition 67. Covered adaptive-action evidence.** If the target action has $\pi_t(a\mid\mathcal H_{t-1})\geq\pi_{\min}>0$ at every time, then
3091 
3092 $$
3093 \mathbb E[W_t^a\mid\mathcal H_{t-1}]=\mu_t^a,
3094 \qquad 0\leq W_t^a\leq1/\pi_{\min}.
3095 $$
3096 
3097 Consequently Proposition 66 gives a time-uniform certificate for $\sum_{s\leq t}\mu_s^a$ from $\sum_{s\leq t}W_s^a$, with its boundary multiplied by $1/\pi_{\min}$. The target is the average action-conditioned quantity along the encountered contexts, not a reset-population table or the state distribution under a different long-term policy.
3098 
3099 **Proof.** Condition first on the action and then on its observation. Only action $a$ contributes, with probability $\pi_t(a\mid\mathcal H_{t-1})$, cancelling the denominator. The range bound follows directly. Apply the scaled conditional moment-generating-function argument to the resulting martingale differences. $\square$
3100 
3101 This simple bound is conservative. Variance-aware weighting can improve it, but cannot remove the need for information about the action of interest. When $\pi_{\min}=0$, two models can give exactly the same law under the observed policy and disagree on a never-selected action. For example, both can emit a Bernoulli observation with parameter $1/2$ under action zero, while their action-one parameters are $1/10$ and $9/10$. Repeating action zero cannot distinguish them.
3102 
3103 The action randomization and its recorded probabilities are part of the experimental assumption. An observational association with a hidden selection mechanism is not automatically an intervention estimate. Long-horizon off-policy predictions require additional occupancy or trajectory-coverage arguments that this one-step statement does not provide.
3104 
3105 # 76. What a residual can prove about misspecification
3106 
3107 A certified operator family can still be the wrong model for the observed process. Predictable residuals offer an explicit way to find that mismatch without pretending that internal validity proves empirical adequacy.
3108 
3109 Let $P_{t,j}$ and $Q_{t,j}$ denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since $0\leq\phi_{t,j}\leq1$,
3110 
3111 $$
3112 |m_{t,j}-p_{t,j}|\leq\|P_{t,j}-Q_{t,j}\|_{\mathrm{TV}}.
3113 $$
3114 
3115 **Proposition 68. Sequential evidence gives a lower error certificate.** On the simultaneous event of Proposition 66,
3116 
3117 $$
3118 \boxed{
3119 \frac1t\sum_{s=1}^t\|P_{s,j}-Q_{s,j}\|_{\mathrm{TV}}
3120 \geq\frac{(|R_{t,j}|-b_{t,j})_+}{t}.
3121 }
3122 $$
3123 
3124 Thus crossing the boundary rejects zero conditional drift for that rule, with the registered false-rejection budget. Failing to cross supplies no general upper bound on total variation, no rank certificate and no guarantee for untested actions.
3125 
3126 **Proof.** The triangle inequality gives $|D_{t,j}|\geq(|R_{t,j}|-b_{t,j})_+$. Also $|D_{t,j}|\leq\sum_s|m_{s,j}-p_{s,j}|\leq\sum_s\|P_{s,j}-Q_{s,j}\|_{\mathrm{TV}}$. Combine these inequalities. $\square$
3127 
3128 ## 76.1 Why one aggregate residual can miss conditional errors
3129 
3130 The reference path has binary observations. Its true conditional probability of one is $4/5$ after a one and $1/5$ after a zero. A deliberately inadequate model predicts $1/2$ at every step. Its long-run aggregate frequency can be near one half even though its history-conditioned predictions are wrong.
3131 
3132 Three tests are registered before the path is generated. One tests the raw observation. A second tests it only after the preceding observation was one. The third applies the same gate but uses the correct conditional predictor. Gates are functions of past data, not selected after seeing the current measurement.
3133 
3134 For the supplied 4,096-step seeded path, the aggregate mean residual is approximately $0.00732$. The gated mean residual is approximately $0.14917$, while its time-uniform mean boundary is approximately $0.05114$. The lower certificate for average conditional total variation is approximately $0.09803$. The correctly predicted gated residual is approximately $-0.00303$.
3135 
3136 These are illustrative path calculations, not estimated coverage rates. The theorem supplies the probability guarantee. The example supplies an explicit reason to test conditional structure rather than relying only on cancellation-prone aggregate residuals.
3137 
3138 The exact reference suite also checks a dependent likelihood-ratio construction on every binary path through depth eight. Its expectation under the declared null is exactly one. This is an additional finite martingale calculation, not a replacement for the general argument or a claim that every possible test has been run.
3139 
3140 # 77. Comparing a model with a process outside its representation
3141 
3142 The earlier operator-norm law bounds compare two models on one common probability cone. A misspecified generating process may not have a representation in the fitted cone. It is useful to state the more basic comparison at the level of conditional observation laws.
3143 
3144 Let $P$ be any causal process on a standard Borel observation space with finite actions. Let $Q$ be a proposed process. Both use the same measurable policy. Write $P_t(\cdot\mid h,a)$ and $Q_t(\cdot\mid h,a)$ for their next-observation kernels, and define
3145 
3146 $$
3147 d_t(h,a)=\|P_t(\cdot\mid h,a)-Q_t(\cdot\mid h,a)\|_{\mathrm{TV}}.
3148 $$
3149 
3150 The proposed kernel must be specified on every history needed for the comparison, including histories impossible under $Q$ but possible under $P$. A declared fallback on a model-null history completes a causal kernel without changing $Q$'s own law. An implementation that rejects such a history must report that rejection rather than silently claim to have filtered it.
3151 
3152 **Proposition 69. Observation-law error without a shared state space.** With the same initial history, the length-$T$ joint action-and-observation laws satisfy
3153 
3154 $$
3155 \boxed{
3156 \|P_T-Q_T\|_{\mathrm{TV}}
3157 \leq\min\left(1,\sum_{t=0}^{T-1}
3158 \mathbb E_{P,\pi}[d_t(H_t,A_t)]\right).
3159 }
3160 $$
3161 
3162 An initial-history discrepancy adds its total variation. In particular, uniform bounds $d_t(h,a)\leq\epsilon_t$ give the upper bound $\min(1,\sum_{t<T}\epsilon_t)$.
3163 
3164 **Proof.** Introduce hybrid laws that follow $P$ for the first $t$ transitions and $Q$ for the rest, with the common policy throughout. Two successive hybrids have the same $P$ prefix distribution up to time $t$ and the same future $Q$ kernel after the next transition. Conditional on that prefix and its policy-chosen action, changing the next kernel costs $d_t(h,a)$ in total variation. A common continuation kernel cannot increase total variation, because integration of a test bounded between zero and one produces another such test. Average over the shared prefix, sum the adjacent-hybrid bounds, and cap by one. The same contraction argument handles a different initial law. $\square$
3165 
3166 This theorem does not make the right-hand side observable without further assumptions. Proposition 68 supplies lower evidence about visited conditional discrepancies, not an upper estimate for every future history. Turning a finite collection of residuals into an upper law guarantee requires additional regularity, coverage or a restricted class with a separating test family.
3167 
3168 The distinction prevents a common logical jump: a model can pass several tests and still have a large untested discrepancy. Conversely, a statistically significant residual need not mean that its internal probability algebra is invalid. The model can be a valid distribution and an inadequate approximation at the same time.
3169 
3170 # 78. An integrated theory of construction and correction
3171 
3172 **Proposition 70. Separate structural acceptance, approximation control and sequential rejection.** Consider a fitted polynomial model in a declared predictive coordinate system, and prospectively registered observation tests with total budget $\delta$.
3173 
3174 A successfully replayed quadratic certificate from Proposition 62 establishes internal all-horizon validity. A successful operator-only proposal from Proposition 63 is one way to obtain it, not an assumption that all valid models have such a proposal.
3175 
3176 For any generating density on a common probability cone with a justified Lipschitz constant, a positive degree-$n$ approximation and certified coefficient budgets satisfy Propositions 64--65, hence a fixed-horizon upper law bound. If a generating process lies outside that representation, a separately justified bound on its conditional kernels instead invokes Proposition 69.
3177 
3178 For predictable tests on actual dependent observations, Propositions 66 and 68 give simultaneous lower discrepancy certificates, including at data-selected stopping times. Covered randomized actions admit the contextual evidence of Proposition 67. These conclusions coexist without implying that structural acceptance or statistical nonrejection proves the generating process belongs to the model class.
3179 
3180 **Proof.** Each structural statement is a deterministic consequence of its replayed equalities and inequalities. The approximation statements require their declared common-domain, regularity and coefficient conditions, or alternatively the conditional-kernel discrepancy assumptions. The statistical conclusions hold on the simultaneous probability event of Proposition 66. None of these implications uses nonrejection as an upper bound or infers coverage from a finite determinant. Their conjunction therefore retains the separate hypotheses and conclusions exactly as stated. $\square$
3181 
3182 The central object is now more than a fitted operator. It includes an admissible domain with a replayable witness, an explicit approximation class, and an evidence rule specifying which observations could contradict its predictions. Geometry discovery proposes a representation. Verification checks its algebra. Approximation theory accounts for information lost by a chosen finite description. Sequential tests expose conditional discrepancies in observed signals.
3183 
3184 Those roles are related but not interchangeable. A failed search means that this construction did not provide a witness. A failed witness means its submitted identities or signs do not establish the claimed validity. A detected residual means a registered predictive condition is inconsistent with the observations at the stated error budget. These three outcomes should be reported differently.
3185 
3186 # 79. From exact geometry to robustness
3187 
3188 Propositions 62–70 distinguish structural acceptance, controlled approximation and sequential evidence. Exact zero-margin preservation does not supply a positive noise radius. Part VII examines that obstruction and gives strict-margin witnesses for selected perturbed families. Statistical nonrejection is not used as an upper accuracy bound.
3189 
3190 \newpage
3191 
3192 # Part VII. Robust curved certificates and noisy geometry
3193 
3194 The preceding block discovered a curved probability domain from exact operators. Exact preservation is not the same as stability under uncertainty. In the worked continuous model, every pure boundary state can remain on the boundary. A perturbation too small to be conspicuous in a coefficient table can then move a predicted state outside the probability domain.
3195 
3196 This block separates three operations. A numerical procedure proposes a geometry. An exact checker decides whether the submitted polynomial family has a declared margin on that geometry. A separately stated uncertainty set determines which other families share the guarantee. If a repair changes the operators, its prediction cost is retained rather than called an exact coordinate change.
3197 
3198 The resulting construction is broader than the preceding exact conformal search. It accepts nonconformal polynomial families and certifies a full coefficient box around a repaired estimate. The reference example uses a deterministic perturbation, not an empirical confidence region. Statistical coverage is a conditional theorem, not a conclusion of that example.
3199 
3200 Lorentz-positive maps, robust convex constraints and Bernstein positivity certificates have classical antecedents [R8, R13, R21, R22]. The following arguments are self-contained specializations connecting them to the signal model's representation and evidence contracts. The nine propositions continue the numbering at 71. Their count is not a novelty claim or a count of Lean proofs.
3201 
3202 # 80. A margin that has probability meaning
3203 
3204 Let $\mathcal L_d=\{(m,v):m\geq\|v\|_2\}$, with mass row $\lambda=(1,0,\ldots,0)$. A mass preserving invertible matrix $T$ transports a proposed predictive domain to this cone. Its normalized base in the original coordinates is
3205 
3206 $$
3207 K_T=\{T^{-1}(1,v):\|v\|_2\leq1\}.
3208 $$
3209 
3210 Write $F_a(u)=TB_a(u)T^{-1}$ for the transported operator density. Observations lie in $[-1,1]$ with reference measure $du/2$. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
3211 
3212 $$
3213 \sigma_a(u)=\min_{\|v\|_2\leq1}
3214 \left[(F_a(u)(1,v))_0-\|(F_a(u)(1,v))_{\rm sp}\|_2\right].
3215 $$
3216 
3217 The subscript $\rm sp$ denotes spatial coordinates, not a separate measured signal. This margin is expressed before normalization. It simultaneously controls mass orientation and distance from the curved boundary. It is coordinate dependent, so its uncertainty budget must use the same $T$.
3218 
3219 **Proposition 71. Robust oriented margin.** Suppose $\sigma_a(u)\geq\kappa_a(u)$ and an uncertain family has transported errors $\Delta_a(u)$ satisfying, for every normalized input,
3220 
3221 $$
3222 \begin{aligned}
3223 |[\Delta_a(u)(1,v)]_0|&\leq e_{a,0}(u),\\
3224 \|[\Delta_a(u)(1,v)]_{\rm sp}\|_2&\leq e_{a,1}(u).
3225 \end{aligned}
3226 $$
3227 
3228 If $\kappa_a(u)\geq e_{a,0}(u)+e_{a,1}(u)$, every uncertain branch preserves $\mathcal L_d$. If the uncertain family has exact integrated mass and its initial state belongs to $K_T$, it defines a valid probability model at every finite horizon. With a strict inequality, the residual slack is at least $\kappa_a-e_{a,0}-e_{a,1}$.
3229 
3230 **Proof.** For $w=F_a(u)(1,v)$ and $d=\Delta_a(u)(1,v)$,
3231 
3232 $$
3233 (w_0+d_0)-\|w_{\rm sp}+d_{\rm sp}\|_2
3234 \geq w_0-\|w_{\rm sp}\|_2-|d_0|-\|d_{\rm sp}\|_2.
3235 $$
3236 
3237 The right side is nonnegative under the hypothesis. Homogeneity extends the assertion from normalized inputs to the entire cone. The only zero-mass cone element is zero. Integrated mass conservation and successive cone preservation give nonnegative normalized observation laws by the same induction as in Proposition 52. A positive-density branch may be normalized. A zero-density branch is not assigned a posterior. $\square$
3238 
3239 This is a uniform deterministic guarantee. The inequalities must cover the whole observation interval and the whole normalized domain. A noise scale estimated from a few residuals does not automatically provide those inequalities. For continuous observations the displayed margin is a density margin relative to the declared measure, not a probability assigned to a single observation value.
3240 
3241 # 81. Why exact boundary preservation can have zero noise radius
3242 
3243 An exact certificate can be correct while having no tolerance to unrestricted perturbations. This is a geometrical fact, not a failure of the certificate checker.
3244 
3245 **Proposition 72. Boundary fragility under mass preserving perturbations.** Assume $d\geq1$. Let $F$ preserve $\mathcal L_d$. Suppose a normalized input $z=(1,v)$ has a boundary image $Fz=(m,w)$ with $m=\|w\|_2$. For every $\epsilon>0$, there is a linear perturbation $\Delta$ with $\lambda\Delta=0$ and
3246 
3247 $$
3248 \sup_{\|v'\|_2\leq1}\|\Delta(1,v')\|_{\mathcal L_d}=\epsilon
3249 $$
3250 
3251 such that $(F+\Delta)z\notin\mathcal L_d$. Here the base norm is $\|(m,w)\|_{\mathcal L_d}=\max(|m|,\|w\|_2)$.
3252 
3253 **Proof.** If $w\ne0$, put $e=w/\|w\|_2$. If $w=0$, choose any spatial unit vector $e$. Define $\Delta=\epsilon(0,e)\lambda$. Its mass row vanishes and its value on every normalized input is $(0,\epsilon e)$, of base norm $\epsilon$. If $w\ne0$, the new spatial norm is $m+\epsilon$. If $w=0$, the original output mass is zero and the new spatial norm is $\epsilon$. Either image lies outside the cone. $\square$
3254 
3255 The positive spatial dimension is essential. When $d=0$, the state is mass alone and $\lambda$ is the identity on $\mathbb R$. The condition $\lambda\Delta=0$ forces $\Delta=0$, so a perturbation of prescribed norm $\epsilon>0$ cannot exist. This explicit boundary was missing from the earlier wording. All previously worked examples have two spatial coordinates and are unaffected.
3256 
3257 
3258 For a continuous operator family, a constant spatial perturbation of this form preserves every integrated mass row and produces the same obstruction at an observation with a boundary image. It need not preserve the algebraic restrictions of a narrower uncertainty class. Perturbing a positive congruence factor and rebuilding its congruence, for example, is different from perturbing an arbitrary predictive matrix.
3259 
3260 In the preceding exact continuous example, the direct operator at $u=0$ is $I/4$. Every normalized boundary state has a boundary image. There is therefore no positive unrestricted matrix-error radius on that domain, even though the original family is exactly valid. A robust theorem that silently treats its zero slack as positive would be false.
3261 
3262 # 82. An explicit repair and the price of its slack
3263 
3264 In Lorentz coordinates let
3265 
3266 $$
3267 N_a(u)=g_a(u)e_0\lambda,\qquad
3268  g_a(u)\geq g_{\min}>0,\qquad \int g_a(u)\,du/2=1.
3269 $$
3270 
3271 This reference instrument reports a state-independent density and resets to the center. It is a mathematical model modification, not a new observation. Given an estimate $\widehat F$, define
3272 
3273 $$
3274 \widetilde F_a(u)=(1-\eta)\widehat F_a(u)+\eta N_a(u),
3275 \qquad 0\leq\eta\leq1.
3276 $$
3277 
3278 **Proposition 73. Interior repair with an explicit error budget.** Suppose a reference family $F^*$ is valid on $\mathcal L_d$, and the combined mass and spatial error of $\widehat F-F^*$ is at most $d_a(u)$ on normalized inputs. Then the margin of $\widetilde F_a(u)$ is at least
3279 
3280 $$
3281 \boxed{\eta g_a(u)-(1-\eta)d_a(u).}
3282 $$
3283 
3284 An additional uncertainty family with combined output error at most $\nu_a(u)$ is certified whenever
3285 
3286 $$
3287 \eta g_a(u)\geq(1-\eta)d_a(u)+\nu_a(u).
3288 $$
3289 
3290 Assume exact integrated mass for the estimate. If its integrated base-norm discrepancy from $F^*$ is at most $\varepsilon$, the corrected discrepancy is at most
3291 
3292 $$
3293 \boxed{(1-\eta)\varepsilon+2\eta.}
3294 $$
3295 
3296 When the corrected model is valid on the same cone, with the same initial state, its length-$T$ law under any common policy differs from the reference law by at most
3297 
3298 $$
3299 \min\left(1,\frac T2[(1-\eta)\varepsilon+2\eta]\right).
3300 $$
3301 
3302 **Proof.** The margin of $\widehat F$ is at least $-d_a$ by Proposition 71's inequality applied to $F^*$. Adding the reference changes only output mass, by $\eta g_a$, while scaling the remaining output by $1-\eta$. This proves the first bound and the robust implication. Integrated mass is preserved by mixing.
3303 
3304 For the error, write
3305 
3306 $$
3307 \widetilde F-F^*=(1-\eta)(\widehat F-F^*)+\eta(N-F^*).
3308 $$
3309 
3310 On a normalized input, both $N$ and $F^*$ distribute total cone mass one. Their integrated difference has base norm at most two. Integrate the triangle inequality to obtain the stated budget. The final implication is the continuous observation-law bound of Proposition 59. It requires validity of both comparison models on the same cone. $\square$
3311 
3312 For constant budgets, a sufficient repair weight is $(d+\nu)/(g_{\min}+d)$ when this number is at most one. This is not an optimality claim. The reference example instead fixes a repair weight and verifies the result independently. If an exact model already has a robust structured certificate, adding a reset may be unnecessary and would introduce avoidable bias.
3313 
3314 The term $2\eta$ is deliberately retained. A larger repair creates more room for uncertainty but can change future predictions more. The worst-case law bound can become uninformative at long horizons. Neither positivity nor mixing alone proves long-horizon predictive accuracy.
3315 
3316 # 83. Noisy invariant forms are proposals, not certificates
3317 
3318 The exact identity $A^{\mathsf T}HA=H$ need not survive estimation error. A useful proposal can still be obtained from a least-squares invariant form, but the residual must not be interpreted as a positive-cone guarantee.
3319 
3320 Let $\mathcal M_A$ map symmetric matrices into a Euclidean product space by
3321 
3322 $$
3323 \mathcal M_A(H)=(A^{\mathsf T}HA-H,\operatorname{tr}H).
3324 $$
3325 
3326 Use the Frobenius norm on symmetric matrices and the induced product norm. Assume $\mathcal M_{A^*}$ is injective and $\mathcal M_{A^*}(H^*)=(0,1)=b$. This fixes the scale of a one-dimensional invariant-form space.
3327 
3328 **Proposition 74. A posteriori proposal error.** Given $\widehat H$, let $r=\mathcal M_{\widehat A}(\widehat H)-b$. Suppose
3329 
3330 $$
3331 \|\mathcal M_{\widehat A}-\mathcal M_{A^*}\|\leq e_M,
3332 \qquad \sigma_{\min}(\mathcal M_{\widehat A})>e_M.
3333 $$
3334 
3335 Then
3336 
3337 $$
3338 \boxed{
3339 \|\widehat H-H^*\|_F
3340 \leq
3341 \frac{\|r\|+e_M\|\widehat H\|_F}
3342 {\sigma_{\min}(\mathcal M_{\widehat A})-e_M}.
3343 }
3344 $$
3345 
3346 One may take $e_M$ from the bound
3347 
3348 $$
3349 \|\mathcal M_{\widehat A}-\mathcal M_{A^*}\|\leq(\|\widehat A\|_2+\|A^*\|_2)\|\widehat A-A^*\|_2.
3350 $$
3351 
3352 For fixed centers $c^*=(I-A^*)^{-1}b^*$ and $\widehat c=(I-\widehat A)^{-1}\widehat b$, with $\|\widehat A-A^*\|_2\leq e_A$ and $\|\widehat b-b^*\|_2\leq e_b$, one has
3353 
3354 $$
3355 \|\widehat c-c^*\|_2
3356 \leq\frac{e_b+e_A\|\widehat c\|_2}
3357 {\sigma_{\min}(I-\widehat A)-e_A}
3358 $$
3359 
3360 when the denominator is positive.
3361 
3362 **Proof.** The invariant-form difference satisfies
3363 
3364 $$
3365 \mathcal M_{A^*}(\widehat H-H^*)
3366 =r-(\mathcal M_{\widehat A}-\mathcal M_{A^*})\widehat H.
3367 $$
3368 
3369 The least singular value of $\mathcal M_{A^*}$ is at least that of $\mathcal M_{\widehat A}$ minus $e_M$. Apply this lower bound and the triangle inequality. To bound the map perturbation, insert and subtract $\widehat A^{\mathsf T}HA^*$ and apply the Frobenius product inequality. The trace component is unchanged. Finally,
3370 
3371 $$
3372 (I-A^*)(\widehat c-c^*)=(\widehat b-b^*)+(\widehat A-A^*)\widehat c.
3373 $$
3374 
3375 The same singular-value argument proves the center bound. $\square$
3376 
3377 The theorem assumes an actual perturbation budget and a corresponding exact reference problem. The example reports numerical residuals but does not claim to have statistically certified these budgets. It uses a trace-normalized least-squares shape, a nearly rank-one image for scale, and rationalized center and factor. It accepts none of those computations as a proof of validity. The next checker uses only the submitted rational geometry and the complete polynomial operators.
3378 
3379 A rationalized invertible factor $L$ specifies $H=L^{\mathsf T}L$ exactly. Positive definiteness is then an algebraic fact even when the factor was suggested by floating-point calculations. It still does not establish operator invariance.
3380 
3381 # 84. Finite exact checking on a continuum
3382 
3383 For three predictive coordinates, the spatial unit circle admits two rational charts. For $s\in\{1,-1\}$ and $t\in[-1,1]$, put
3384 
3385 $$
3386 d(t)=1+t^2,\qquad
3387 r_s(t)=\frac{s}{d(t)}(1-t^2,2t).
3388 $$
3389 
3390 The charts cover the two semicircles, including their endpoints. For an operator $F_a(u)$ and required margin $\kappa\geq0$, form polynomial vectors and scalars
3391 
3392 $$
3393 \begin{aligned}
3394 w_{a,s}(u,t)&=F_a(u)(d(t),s(1-t^2),2st)^{\mathsf T},\\
3395 f_{a,s}(u,t)&=w_{a,s,0}(u,t)-\kappa d(t),\\
3396 h_{a,s}(u,t)&=f_{a,s}(u,t)^2-w_{a,s,1}(u,t)^2-w_{a,s,2}(u,t)^2.
3397 \end{aligned}
3398 $$
3399 
3400 No square root occurs in these polynomials.
3401 
3402 **Proposition 75. Rational semicircle and Bernstein certificate.** If $f_{a,s}$ and $h_{a,s}$ are nonnegative on $[-1,1]^2$ for both charts and every action, then $\sigma_a(u)\geq\kappa$ for every observation. For rational operators and geometry, nonnegative tensor-Bernstein coefficients on a complete rational subdivision of that square give a finite exact witness. If the actual margin is uniformly strictly greater than $\kappa$, sufficiently fine dyadic subdivision succeeds.
3403 
3404 **Proof.** The first polynomial inequality establishes output orientation after subtracting $\kappa d$. The second then gives
3405 
3406 $$
3407 w_{a,s,0}-\|(w_{a,s,1},w_{a,s,2})\|_2\geq\kappa d.
3408 $$
3409 
3410 Divide by $d>0$ to obtain the required margin on the circle. For fixed $u$, the function consisting of affine output mass minus the norm of affine spatial output is concave in the input. Every point in the closed disk is a convex combination of circle points. Concavity therefore extends the lower bound to the disk.
3411 
3412 On any rectangle mapped affinely to $[0,1]^2$, the tensor-Bernstein basis functions are nonnegative and sum to one. A polynomial with nonnegative coefficients in that basis is nonnegative throughout the rectangle. If the rectangles cover the square, their inequalities prove the global statement. The reference verifier checks coverage through a complete subdivision tree, not by accepting an arbitrary list of rectangles.
3413 
3414 For termination under strict margin, both $f$ and $h$ are strictly positive on the compact square. Their minima are positive. For a fixed-degree polynomial on a small cell of side at most $\delta$, its Taylor expansion at one corner has constant term equal to its value there and nonconstant coefficients bounded uniformly by $O(\delta)$. The linear change from the fixed-degree power basis to the Bernstein basis is fixed and bounded. Thus every Bernstein coefficient differs from that constant value by $O(\delta)$, uniformly over cells. At sufficiently small $\delta$ all coefficients are positive. A finite dyadic subdivision attains that size. $\square$
3415 
3416 This proof is a specialized use of classical Bernstein range bounds [R22]. It does not claim that a negative Bernstein coefficient proves the polynomial negative. Such a coefficient can merely mean that further subdivision is needed. A negative evaluated corner, by contrast, is an actual counterexample. At a zero-margin boundary, finite termination of this particular witness language is not promised. The exact conformal and matrix certificates of preceding blocks remain useful there.
3417 
3418 The two tests are both necessary to this sufficient certificate. The squared inequality alone also admits reversed mass orientation. Both charts are required. A verifier that omits one chart has not covered the circle.
3419 
3420 # 85. A finite uncertainty box and the role of convexity
3421 
3422 A robust witness should identify its uncertainty family rather than supply an unexplained tolerance. Suppose the transformed error has polynomial coefficients
3423 
3424 $$
3425 \Delta_a(u)=\sum_{k=0}^{D}u^kE_{a,k},\qquad
3426 |(E_{a,k})_{ij}|\leq r_k.
3427 $$
3428 
3429 All bounds are in the submitted Lorentz coordinates. For three coordinates, normalized inputs have each coordinate of magnitude at most one.
3430 
3431 **Proposition 76. Coefficient boxes and convex uncertainty.** The coefficient box satisfies the output bounds of Proposition 71 with
3432 
3433 $$
3434 \boxed{e_0=3\sum_{k=0}^{D}r_k,\qquad e_1=6\sum_{k=0}^{D}r_k.}
3435 $$
3436 
3437 Only coefficient arrays obeying $\lambda\int\Delta_a(u)\,du/2=0$ are included in the normalized uncertainty family. A checked margin at least $e_0+e_1$ certifies every such array.
3438 
3439 Alternatively, if finitely many complete polynomial operator families satisfy a common margin $\kappa$ on the same domain, every convex combination of those families satisfies that margin. Initial states may also range over a certified convex hull.
3440 
3441 For a fixed geometry, the constraints $\sigma_a(u)\geq\kappa$, exact mass, and membership in a convex data-consistency set are convex constraints on the operator coefficients and initial state. Joint optimization of geometry and operators is not asserted to be convex.
3442 
3443 **Proof.** Each output scalar is bounded by $3r_k$ on a normalized input. Bound the spatial Euclidean norm by the sum of the absolute values of its two coordinates, obtaining $6r_k$. Sum over coefficients using $|u|\leq1$. Mass conservation is a separate linear constraint and follows for every family satisfying it.
3444 
3445 For convex combinations, output mass is affine and the spatial norm is convex. Hence the margin of the combined output is at least the same combination of margins. A convex cone contains convex combinations of admissible initial states. For fixed geometry and input, the inequality affine mass minus a norm at least $\kappa$ is convex in the coefficients. Intersect over all inputs and observations, then add the linear mass constraints and the convex data set. $\square$
3446 
3447 The squared polynomial witness in Proposition 75 is a convenient verification language. Its coefficient constraints are not automatically a convex optimization formulation in unknown operators. Convexity of the underlying admissibility set must not be confused with convexity of every certificate parameterization.
3448 
3449 Likewise, a free uncertainty box in all entries contains families that violate integrated mass. The affine restriction is essential. Removing physically valid candidates from a statistical confidence set for some other reason requires its own coverage argument.
3450 
3451 # 86. What a search can guarantee
3452 
3453 The numerical proposal is one practical route. There is also a precise conditional search theorem showing why a broad class of strict curved certificates can be found without an exact conformal identity.
3454 
3455 **Proposition 77. Conditional search for strict rational curved certificates.** Consider finitely many rational polynomial operator families in three coordinates, each with exact mass, and finitely many rational normalized initial states. Suppose there exists a real mass preserving invertible $T_*$ such that every transformed initial state lies strictly inside the Lorentz cone and every transformed operator family has a common positive margin on the observation interval. Then a rational mass preserving invertible $T$, a positive rational margin, and finite rational Bernstein witnesses exist. A dovetailed enumeration of rational transforms, positive rational margins and subdivision depths eventually finds such a witness. Whenever it returns, the witness is sound for the convex hull of the supplied families.
3456 
3457 **Proof.** Inversion and matrix multiplication are continuous near $T_*$. Initial membership is strict, and the operator margin is continuous jointly in the transform, the observation, and the normalized input. The latter two range over compact sets and there are finitely many families. Thus the strict inequalities persist in a neighborhood of $T_*$. Rational mass preserving matrices are dense in that affine matrix space, and invertibility persists in a neighborhood. Choose a rational transform there and a rational positive margin smaller than the remaining minimum. Proposition 75 supplies finite subdivisions because there is still a strict gap above the chosen margin. Enumerating all finite rational descriptions and depths eventually reaches this finite witness. Proposition 76 proves the convex-hull conclusion. $\square$
3458 
3459 This is a conditional semidecision result, not a polynomial-time complexity theorem. It does not cover every valid model. Initial states on an essential boundary and families with zero margin may violate its assumptions. The implemented procedure uses a finite rationalized proposal and bounded subdivision, not this unbounded enumeration. Failure of that bounded procedure is a withheld certificate rather than a proof that no other domain works.
3460 
3461 The theorem's uncertainty families are supplied in fixed predictive coordinates. It does not establish that those families cover an unknown generating process. It demonstrates a route from a strict robust model sector to finite machine-checkable witnesses.
3462 
3463 # 87. Selection from data does not remove the uncertainty quantifier
3464 
3465 A domain can be selected after examining fitted operators without a separate statistical penalty for each geometry tried. That statement holds only when the final verification ranges over an already justified confidence set.
3466 
3467 **Proposition 78. Data-selected geometry with honest coverage.** Suppose a random set $\mathcal U$ of operators and initial states, in a declared fixed predictive coordinate system, contains the true model with probability at least $1-\delta$. A data-dependent procedure may propose any transform $T$ and any domain. If its exact verifier proves domain membership and invariance for every member of the same set $\mathcal U$, then on that coverage event the true model has the verified domain. No union bound over geometry proposals is needed for this deterministic implication.
3468 
3469 If original-coordinate perturbations satisfy $\|\Delta\|_2\leq e$, their combined mass and spatial output error after a known transform $T$ is at most
3470 
3471 $$
3472 \boxed{2\|T\|_2\|T^{-1}\|_2 e.}
3473 $$
3474 
3475 The corresponding transformed uncertainty budget must be used even when $T$ was chosen from the data.
3476 
3477 **Proof.** On the coverage event the true model is a member of the set quantified by the verifier. Its domain properties follow by direct inclusion, regardless of how $T$ was selected. For the error, a normalized Lorentz input has Euclidean norm at most $\sqrt2$. Thus the Euclidean norm of $T\Delta T^{-1}z$ is at most $\sqrt2\|T\|_2\|T^{-1}\|_2e$. The sum of its mass magnitude and spatial norm is at most $\sqrt2$ times its Euclidean norm. Multiply the two bounds. $\square$
3478 
3479 The set must not be shrunk around a convenient candidate without preserving coverage. Moving the center by a reset mixture changes the family being certified. A tube around the repaired family need not contain the original true family. Repeatedly estimating new confidence sets still requires the stage accounting of Proposition 48. Statistical rank coverage and observation identifiability remain assumptions of the upstream procedure.
3480 
3481 The final reference certificate explicitly says that no statistical coverage is claimed. Its uncertainty box is a deterministic, exactly described set around the repaired operator family. This supports a reproducible robustness statement without pretending that the example has estimated an unknown noise distribution.
3482 
3483 # 88. A complete perturbed-operator construction
3484 
3485 The input to the new proposer contains only two perturbed polynomial operators and the initial vector. A separate fixture generator creates those inputs from the preceding continuous model by adding $\pm\delta E(u)$, with opposite signs for the two actions and
3486 
3487 $$
3488 \delta=\frac1{2000},\qquad
3489 E(u)=E_0+uE_1+u^2E_2,
3490 $$
3491 
3492 where
3493 
3494 $$
3495 \begin{aligned}
3496 E_0&=\begin{pmatrix}0&0&0\\-1&2&0\\0&1&-1\end{pmatrix},&
3497 E_1&=\begin{pmatrix}0&1&-1\\1&0&1\\1&-1&0\end{pmatrix},&
3498 E_2&=\begin{pmatrix}0&0&0\\0&-1&1\\1&0&1\end{pmatrix}.
3499 \end{aligned}
3500 $$
3501 
3502 The perturbation preserves integrated mass. It is not a statistical sample. The proposer does not receive the predecessor metric or a hidden generator.
3503 
3504 A noisy operator ratio supplies a center and a least-squares shape. A finite observation-anchor list supplies a scale proposal. Rationalizing the center and a triangular metric factor with denominator limit 48 gives
3505 
3506 $$
3507 c=\begin{pmatrix}1/2\\1/2\end{pmatrix},\qquad
3508 L=\begin{pmatrix}123/37&-2\\0&109/41\end{pmatrix},
3509 $$
3510 
3511 and hence
3512 
3513 $$
3514 \boxed{
3515 H=L^{\mathsf T}L=
3516 \begin{pmatrix}15129/1369&-246/37\\-246/37&18605/1681\end{pmatrix}.
3517 }
3518 $$
3519 
3520 This is not the exact predecessor metric. The transformed family is not assumed conformal. The raw fitted family actually fails the proposed domain: for the direct action at $u=-1$ and Lorentz input $(1,1,0)$, its output mass is $106321/82000>0$, while the output quadratic form is
3521 
3522 $$
3523 -\frac{12009932651317769}{557059220496000000}<0.
3524 $$
3525 
3526 This is a genuine rejected input state, not a negative Bernstein coefficient of uncertain meaning. The normalized output lies outside the ellipse. No positivity certificate is claimed for the raw fit.
3527 
3528 The construction next mixes each operator with the center-reset reference using $\eta=1/8$. The exact checker proves a margin of at least $1/16$ for the resulting family. It verifies eight polynomial inequalities across both actions and both semicircle charts. The witness contains 116 subdivision leaves and 1,780 exactly recomputed nonnegative Bernstein coefficients. These are arithmetic certificate sizes, not separate theorems.
3529 
3530 The exported family admits mass and spatial perturbation budgets each equal to $1/128$, leaving certified slack $3/64$. In particular, it covers every degree-two transformed coefficient perturbation with each entry bounded by $1/4096$, subject to the exact integrated mass constraints. Proposition 76 verifies that the whole coefficient box lies inside the declared output budgets. No uncertainty vertices or future trajectories are sampled to establish that inclusion.
3531 
3532 The certificate is replayed independently from its JSON record. The record contains rational operators, initial state, transform, margin, uncertainty definition and complete subdivision trees. Floating-point proposal diagnostics are stored elsewhere and are not trusted by the replay routine.
3533 
3534 # 89. The prediction cost needs a common comparison domain
3535 
3536 A successful discovered domain does not imply that the known reference model has that same domain. To invoke a common-cone law bound, the example separately checks the corrected family on the predecessor's exact ellipse. This is a second certificate used only for comparison. It is not supplied to geometry discovery.
3537 
3538 In that comparison domain, the coefficient-wise upper bounds on the raw fitting discrepancy are
3539 
3540 $$
3541 \frac{11}{5000},\qquad \frac{137}{30000},\qquad \frac{37}{8000}.
3542 $$
3543 
3544 Using $\int_{-1}^1|u|^kdu/2=1/(k+1)$ gives the integrated upper budget $241/40000$. The same center-reset reference is valid there. Both models have the same initial state.
3545 
3546 **Proposition 79. Certified repair with a retained comparison budget.** For the exact construction of Section 88, independent replay establishes internal validity of the repaired model and its declared coefficient uncertainty family. Independent comparison-domain replay and the coefficient bounds above give
3547 
3548 $$
3549 \varepsilon_{\rm repaired}
3550 \leq\frac78\frac{241}{40000}+\frac14
3551 =\frac{81687}{320000}.
3552 $$
3553 
3554 Consequently, for any common policy and any finite horizon $T$,
3555 
3556 $$
3557 \boxed{
3558 \|P_T-\widetilde P_T\|_{\rm TV}
3559 \leq\min\left(1,\frac{81687T}{640000}\right).
3560 }
3561 $$
3562 
3563 This compares the unmodified reference process and the nominal repaired process. It is not a guarantee that the raw noisy model was valid or that the repaired uncertainty family contains the original reference.
3564 
3565 **Proof.** The polynomial witnesses and exact mass identities imply the nominal and robust validity statements by Propositions 71, 75 and 76. The separate comparison witness establishes the required common domain for the nominal repaired and reference operators. The absolute coefficient bounds dominate the integrated raw error by the triangle inequality. Proposition 73 then gives the displayed repair budget and law bound. All hypotheses are either exact identities in the supplied witnesses or direct norm bounds from the declared rational coefficients. $\square$
3566 
3567 The bound is conservative. It becomes one at horizon eight, so it makes no nontrivial long-horizon accuracy claim. The example's point is the separation of a verified all-horizon probability model from a finite-horizon approximation guarantee. A probability-valid model is not necessarily a close model.
3568 
3569 The noisy proposal, the failed raw fit, the repair and the comparison bound all appear in the release records. None is replaced by a more favorable trajectory score. The coefficient box is around the repaired family in the specified coordinates, not an inferred distribution of sensor noise.
3570 
3571 # 90. From deterministic robustness to data confidence
3572 
3573 Part VII gives exact continuum certificates for deterministic uncertainty families. The worked repaired model is internally valid at every finite horizon, but its original comparison bound becomes trivial at horizon eight. Part VIII reduces that fixed repair and separately constructs a statistically covered domain for a supplied observable family. Its confidence statement is not transferred to the earlier repaired operators.
3574 
3575 \newpage
3576 
3577 # Part VIII. Data-supported domains and smaller repairs
3578 
3579 The preceding block certified a perturbed operator family after mixing it with a center reset. Its all-horizon validity guarantee was exact, but its error bound became trivial at horizon eight. It also kept deterministic coefficient boxes separate from statistical confidence sets. This block addresses both limitations in declared sectors rather than weakening either distinction.
3580 
3581 First, the same fitted family and previously proposed geometry admit a smaller, independently checked repair. Second, a self-normalized martingale argument produces operator uncertainty sets from a dependent trajectory when the predictive state is an accessible, fixed observable. A nonlinear three-coordinate example connects those sets to a curved domain without a reset correction. Its geometry and feature dictionary are specified in advance. This is not a solution to discovering an unknown hidden state from the same data.
3582 
3583 The sequential-regression result is a classical self-normalized concentration theorem [R23], proved here in the form used by the signal model. The contraction argument uses standard Markov-kernel and total-variation ideas [R24]. The new block's contribution is the explicit sequence of assumptions, finite witnesses, structured uncertainty, observation-measure distinctions and executable examples connecting these tools to the realization calculus. Proposition counts are not historical-novelty counts.
3584 
3585 
3586 
3587 # 91. A smaller repair can be bracketed rather than guessed
3588 
3589 Keep the fitted operators, proposed transform and initial state from Section 88 unchanged. In Lorentz coordinates use the center-reset operator $N=e_0\lambda$, of constant density one under $du/2$. Write
3590 
3591 $$
3592 F_\eta=(1-\eta)\widehat F+\eta N,
3593 \qquad 0\leq\eta\leq1.
3594 $$
3595 
3596 Fix a required margin $0\leq\kappa\leq1$. Feasibility means that every action, observation and normalized input has margin at least $\kappa$. Integrated mass and initial membership remain separate requirements.
3597 
3598 **Proposition 80. Convex repair bracketing.** The feasible repair weights form a closed interval $[\eta_*,1]$. If a cone-positive functional $\ell=(1,-v^{\mathsf T})$, with $\|v\|_2\leq1$, and a normalized input $z$ give $c=\ell\widehat F_a(u)z<\kappa$, then every feasible weight satisfies
3599 
3600 $$
3601 \boxed{\eta\geq\frac{\kappa-c}{1-c}.}
3602 $$
3603 
3604 A complete certificate at a weight $\eta_U$ gives an upper bound on $\eta_*$. A negative Bernstein coefficient without a negative evaluated point does not give a lower bound.
3605 
3606 **Proof.** For a fixed input and observation, output mass minus spatial norm is a concave continuous function of the operator, hence of $\eta$. Its superlevel set is closed and convex. Intersecting these sets preserves those properties. The reset has margin one, so the intersection contains one and is an interval ending there. If $F_\eta$ has margin at least $\kappa$, every such functional satisfies $\ell F_\eta z\geq\kappa$. Since $\ell Nz=1$, this requires $(1-\eta)c+\eta\geq\kappa$. Here $1-c>0$, and solving gives the bound. A complete positive witness proves feasibility. A negative coefficient only says that a particular sufficient polynomial bound failed. $\square$
3607 
3608 ## 91.1 The unchanged predecessor example
3609 
3610 Use the previous transform with center $(1/2,1/2)$ and triangular factor
3611 
3612 $$
3613 L=\begin{pmatrix}123/37&-2\\0&109/41\end{pmatrix}.
3614 $$
3615 
3616 At the direct action, observation $u=-1$, normalized input $(1,1,0)$ and functional $\ell=(1,7/25,-24/25)$, exact evaluation gives
3617 
3618 $$
3619 c=-\frac{17590411}{2332387500}.
3620 $$
3621 
3622 For the required margin $\kappa=1/64$, this proves
3623 
3624 $$
3625 \frac{864543451}{37599646576}
3626 \leq\eta_*\leq\frac1{32}.
3627 $$
3628 
3629 The lower endpoint is approximately $0.02299$. The upper endpoint is not a sampled estimate. A complete exact Bernstein certificate proves that $\eta=1/32$ works. The weight $1/64$ fails the requested $1/64$ margin at the displayed separating functional. This does not by itself prove that its nominal model is probability-invalid.
3630 
3631 The new certificate uses 194 leaves and 3,058 recomputed coefficients. A separate comparison-domain certificate uses 185 leaves and 2,833 coefficients. Both retain the same degree-two coefficient box of radius $1/4096$ in their respective declared Lorentz coordinates, subject to exact integrated mass. The more general mass and spatial output budgets remain $1/128$ each. Their sum equals the certified margin, so that larger output-error family has a nonnegative lower margin, not a claimed strictly positive residual margin.
3632 
3633 The reset weight has fallen from $1/8$ to $1/32$, while the deterministic uncertainty box is unchanged. No global optimization over geometry or repair operators is claimed. The current bracket concerns this fixed geometry, this reset path and this required margin.
3634 
3635 The comparison domain is checked again before invoking the law theorem. Keeping the same raw discrepancy budget $241/40000$ from Section 89 gives
3636 
3637 $$
3638 \varepsilon_{\mathrm{repair}}
3639 \leq\frac{31}{32}\frac{241}{40000}+\frac1{16}
3640 =\frac{87471}{1280000}.
3641 $$
3642 
3643 Consequently, under a common policy and identical initial states,
3644 
3645 $$
3646 \boxed{
3647 \|P_T-\widetilde P_T\|_{\mathrm{TV}}
3648 \leq\min\left(1,\frac{87471T}{2560000}\right).
3649 }
3650 $$
3651 
3652 The first integer horizon where this upper bound reaches one is now 30 rather than eight. The bound remains conservative. The repaired uncertainty tube is still not a confidence set for the unmodified generating process. This improvement is deterministic certificate refinement, not a new statistical experiment.
3653 
3654 
3655 
3656 # 92. Confidence from dependent observations
3657 
3658 Let $\mathcal F_{t-1}$ contain all information available before observation $t$, including its already selected action. Let $\phi_t\in\mathbb R^d$ be measurable with respect to that information. Assume
3659 
3660 $$
3661 Y_t=\theta_*^{\mathsf T}\phi_t+\xi_t,
3662 \qquad
3663 \mathbb E[e^{h\xi_t}\mid\mathcal F_{t-1}]
3664 \leq e^{h^2\sigma^2/2}
3665 $$
3666 
3667 for every real $h$, with $\sigma>0$, and a declared bound $\|\theta_*\|_2\leq S$. With ridge $\gamma>0$, set
3668 
3669 $$
3670 V_t=\gamma I+\sum_{s\leq t}\phi_s\phi_s^{\mathsf T},
3671 \qquad
3672 \widehat\theta_t=V_t^{-1}\sum_{s\leq t}\phi_sY_s.
3673 $$
3674 
3675 **Proposition 81. Time-uniform regression confidence.** For $0<\delta<1$, with probability at least $1-\delta$, simultaneously for every finite $t$,
3676 
3677 $$
3678 \boxed{
3679 \|\widehat\theta_t-\theta_*\|_{V_t}
3680 \leq
3681 \sigma\sqrt{
3682 \log\frac{\det V_t}{\det(\gamma I)}+2\log\frac1\delta
3683 }
3684 +\sqrt\gamma S.
3685 }
3686 $$
3687 
3688 No independence of successive observations or of the regressors is required. The conditional mean, conditional noise and predictability hypotheses are required. The confidence set remains valid at a stopping time selected from the data.
3689 
3690 **Proof.** Put $S_t=\sum_{s\leq t}\phi_s\xi_s$. For every fixed $q$, the conditional moment bound makes
3691 
3692 $$
3693 \exp\left(q^{\mathsf T}S_t/\sigma
3694 -\tfrac12q^{\mathsf T}(V_t-\gamma I)q\right)
3695 $$
3696 
3697 a nonnegative supermartingale. Integrate against the fixed Gaussian law $q\sim N(0,(\gamma I)^{-1})$. Conditional Tonelli and completing the square give the nonnegative supermartingale
3698 
3699 $$
3700 M_t=
3701 \left(\frac{\det(\gamma I)}{\det V_t}\right)^{1/2}
3702 \exp\left(\frac{\|S_t\|_{V_t^{-1}}^2}{2\sigma^2}\right),
3703 \qquad M_0=1.
3704 $$
3705 
3706 For completeness, stop at the first crossing of $1/\delta$, then at any deterministic finite horizon. The supermartingale expectation is at most one, so the crossing probability up to that horizon is at most $\delta$. Increasing the horizon proves the same bound for any finite crossing. Off this event, taking logarithms bounds $\|S_t\|_{V_t^{-1}}$ by the square-root term. Finally,
3707 
3708 $$
3709 \widehat\theta_t-\theta_*
3710 =V_t^{-1}(S_t-\gamma\theta_*).
3711 $$
3712 
3713 The triangle inequality and $V_t\succeq\gamma I$ bound the ridge term by $\sqrt\gamma S$. The simultaneous event already covers every selected finite stopping time. $\square$
3714 
3715 This is the self-normalized mixture argument in [R23], not a new concentration principle. A bounded response in an interval of length $b-a$ has conditionally sub-Gaussian centered noise with $\sigma=(b-a)/2$. One direct proof differentiates its tilted log moment-generating function twice. The second derivative is a variance bounded by $(b-a)^2/4$. Integrating from zero gives the stated moment bound.
3716 
3717 The determinant term accounts for the observed design, but does not create excitation. If $\phi_t$ never probes a direction, the set need not shrink in that direction. Regressors extracted after inspecting the same response are not predictable. An estimated hidden state cannot be substituted for $\phi_t$ while silently retaining the theorem's true conditional-mean equation.
3718 
3719 # 93. From observable moment regression to operator sets
3720 
3721 There is a direct connection to the unnormalized instrument. Assume that the normalized predictive state $Z_t$ is an exact, fixed, measurable function of the available observation history. For action $a$, suppose
3722 
3723 $$
3724 P(U_t\in du\mid\mathcal F_{t-1})
3725 =\lambda B_a(u)Z_t\,\mu(du),
3726 \qquad
3727 Z_{t+1}=\frac{B_a(U_t)Z_t}{\lambda B_a(U_t)Z_t}
3728 $$
3729 
3730 on realized positive-density branches. Crucially, $Z_{t+1}$ must also be available from the observations using the specified state map. This is an accessibility assumption, not a consequence of successful regression.
3731 
3732 Let $B_a(u)=\sum_{k=0}^{D}\psi_k(u)C_{a,k}$. Choose functions $g_j$ so that the known matrix $W_{jk}=\int g_j(u)\psi_k(u)\,\mu(du)$ is invertible.
3733 
3734 **Proposition 82. Sequential observable-moment operator sets.**
3735 
3736 Define $M_{a,j}=\int g_j(u)B_a(u)\,\mu(du)$. Then
3737 
3738 $$
3739 \boxed{
3740 \mathbb E[g_j(U_t)Z_{t+1}\mid\mathcal F_{t-1}]
3741 =M_{a,j}Z_t,
3742 \qquad
3743 C_{a,k}=\sum_j(W^{-1})_{kj}M_{a,j}.
3744 }
3745 $$
3746 
3747 Under the rowwise noise and parameter-bound assumptions of Proposition 81, simultaneous confidence ellipsoids for the rows of $M_{a,j}$ give a confidence set for every $C_{a,k}$ by this known linear map. If row $i$ of $\widehat M_{a,j}-M_{a,j}$ has $V_{a,t}$-norm at most $\beta_{a,j,i}$, then for any direction $x$,
3748 
3749 $$
3750 \left|e_i^{\mathsf T}(\widehat C_{a,k}-C_{a,k})x\right|
3751 \leq
3752 \sum_j |(W^{-1})_{kj}|\,
3753 \beta_{a,j,i}\,\|x\|_{V_{a,t}^{-1}}.
3754 $$
3755 
3756 **Proof.** In the conditional integral, multiply the normalized state by the observation density. The denominator cancels, giving $\int g_j(u)B_a(u)Z_t\,\mu(du)$. The polynomial expansion gives $M_{a,j}=\sum_kW_{jk}C_{a,k}$ and hence the inverse formula. Apply Proposition 81 to each row, using the predictable regressor $Z_t$ on visits to action $a$, or its action-gated version. A union bound over the predeclared finite rows and moments gives simultaneous coverage. Independence between these response coordinates is unnecessary. Weighted Cauchy--Schwarz and the triangle inequality give the displayed directional bound. $\square$
3757 
3758 Exact mass constraints can be imposed by intersecting the confidence set with the known mass affine subspace. This preserves coverage because the true model is assumed to obey that law. Arbitrarily shrinking a set around a convenient corrected operator does not have the same justification. Data-selected geometry can be checked against the whole resulting set using Proposition 78.
3759 
3760 The theorem is conditional on a correct finite observation basis, accessible predictive state and parameter bounds. It does not obtain a hidden predictive state from a single dependent path without additional assumptions. Unknown dictionary discovery, state-estimation error and polynomial misspecification remain distinct problems. The following example makes the state accessible by construction rather than concealing that requirement.
3761 
3762 
3763 
3764 # 94. A nonlinear observable state with a curved moment domain
3765 
3766 For $u\in[-1,1]$, put
3767 
3768 $$
3769 v(u)=(1,u,u^2)^{\mathsf T},
3770 \qquad L_2(u)=\frac{3u^2-1}{2}.
3771 $$
3772 
3773 The normalized states are moments of a scalar signal:
3774 
3775 $$
3776 K=\{(1,x,y):x^2\leq y\leq1\}.
3777 $$
3778 
3779 Their cone has the finite curved description
3780 
3781 $$
3782 C=\left\{(m,x,y):
3783 \begin{pmatrix}m&x\\x&y\end{pmatrix}\succeq0,
3784 \quad m-y\geq0\right\},
3785 \qquad\lambda=(1,0,0).
3786 $$
3787 
3788 Assume that the declared probability measure $\mu$ on $[-1,1]$ has moments zero at orders one and three, $1/3$ at order two and $1/5$ at order four. Define a two-parameter degree-four operator density
3789 
3790 $$
3791 \boxed{
3792 B_{a,b}(u)=v(u)\,(1,au,bL_2(u)).
3793 }
3794 $$
3795 
3796 Here $a,b$ are parameters, not action labels. This example is autonomous. Its dependence on the observed signal is nonlinear because the state includes its square.
3797 
3798 **Proposition 83. Structured curved-domain certificate.** The set $K$ is the convex hull of all $v(u)$. If $|a|+|b|\leq1-\rho$ for $0\leq\rho\leq1$, then $B_{a,b}(u)$ preserves $C$ and has exact integrated mass. On a normalized state, its density is
3799 
3800 $$
3801 p_{a,b}(u\mid x,y)=1+a x u+b yL_2(u)\geq\rho.
3802 $$
3803 
3804 Every positive-density posterior is exactly $v(u)$. In particular, starting from $v(s)$ produces the observable Markov law
3805 
3806 $$
3807 p_{a,b}(u\mid s)=1+a s u+b s^2L_2(u).
3808 $$
3809 
3810 Its next conditional moments satisfy
3811 
3812 $$
3813 \boxed{
3814 \mathbb E[U_t\mid U_{t-1}=s]=\frac a3s,
3815 \qquad
3816 \mathbb E[L_2(U_t)\mid U_{t-1}=s]=\frac b5s^2.
3817 }
3818 $$
3819 
3820 **Proof.** Every distribution on $[-1,1]$ has $x^2\leq y\leq1$. Conversely, fix $|x|<1$. A point mass at $x$ gives $(x,x^2)$, while the endpoint mixture with mean $x$ gives $(x,1)$. Their convex combinations give every intermediate $y$. The cases $x=\pm1$ are immediate. Homogenizing yields the matrix and scalar inequalities, which also force the only mass-zero cone element to be zero.
3821 
3822 On $K$, $|x|\leq1$, $0\leq y\leq1$, $|u|\leq1$ and $|L_2(u)|\leq1$. Thus the density is at least $1-|a|-|b|$. Each operator output is that nonnegative scalar times $v(u)$, so lies in $C$. The first and second reference moments imply $\lambda\int B_{a,b}(u)\,\mu(du)=\lambda$. Normalization cancels the scalar density, yielding $v(u)$. Symmetry through order three gives $\int uL_2(u)d\mu=0$. The second and fourth moments give $\int L_2(u)^2d\mu=1/5$. Integrating $u$ and $L_2$ against the density proves the last two formulas. $\square$
3823 
3824 The posterior ray lies on a curved boundary. Arbitrary perturbations of its output coordinates can destroy validity, as in Proposition 72. Parameter uncertainty inside the declared family behaves differently: it changes the likelihood but not the posterior ray. A certified rectangle in $(a,b)$ can therefore be robust even though a free matrix-entry ball has no positive validity radius.
3825 
3826 This is an example of uncertainty that preserves the model's algebraic structure. It is not a claim that every real system obeys the two-parameter family. The feature dictionary $1,u,u^2$, its geometry, the reference measure and the parameter model are declared inputs to learning.
3827 
3828 # 95. A fully rational dependent-data reference experiment
3829 
3830 Both the continuous uniform measure $du/2$ and the following five-node measure satisfy the moment assumptions:
3831 
3832 $$
3833 \begin{array}{c|rrrrr}
3834 u&-1&-1/2&0&1/2&1\\\hline
3835 \mu\{u\}&7/90&16/45&2/15&16/45&7/90.
3836 \end{array}
3837 $$
3838 
3839 The reference experiment uses the **five-node measure**, not rounded observations from the continuous model. At each step, the conditional probability of a node is its reference weight times $p_{a,b}(u\mid s)$. Exact rational probabilities are sampled by integer draws. The learner receives only the node sequence, the measure and the declared model. The true parameters are present only in the separate generator and verification code.
3840 
3841 Matching four moments does not identify the two observation laws. The continuous model gives the set of five nodes probability zero, while the discrete model gives it probability one. Their single-observation total variation is therefore one. The continuous theorems and the exact discrete reference calculation are deliberately distinguished.
3842 
3843 For a path of $n$ transitions, use two scalar regressions,
3844 
3845 $$
3846 \begin{aligned}
3847 \phi_{a,t}&=U_{t-1}/3,&Y_{a,t}&=U_t,&\sigma_a&=1,\\
3848 \phi_{b,t}&=U_{t-1}^2/5,&Y_{b,t}&=L_2(U_t),&\sigma_b&=3/4.
3849 \end{aligned}
3850 $$
3851 
3852 The noise scales follow from response ranges $[-1,1]$ and $[-1/2,1]$. The prior parameter bounds are $|a|,|b|\leq1$. Each regression uses ridge one and failure probability $1/40$, giving a total failure budget $1/20$ for both parameters at every time.
3853 
3854 **Proposition 84. Exact rational outer confidence intervals.** Write $V=1+\sum\phi_t^2$ and $\widehat\theta=\sum\phi_tY_t/V$. Let $k=\lceil\log_2 V\rceil$ and $h=\lceil\log_2(1/\delta)\rceil$. A rational upper enclosure of
3855 
3856 $$
3857 \sigma\sqrt{\frac{(7/10)(k+2h)}{V}}+\sqrt{\frac1V}
3858 $$
3859 
3860 is an upper bound on Proposition 81's scalar radius. Rounding the lower interval endpoint downward and the upper endpoint upward, then intersecting with $[-1,1]$, preserves its conditional coverage. Both intervals cover their true parameters simultaneously at every time with probability at least $19/20$.
3861 
3862 **Proof.** The positive exponential series gives $e^{7/10}>1+7/10+(7/10)^2/2+(7/10)^3/6+(7/10)^4/24>2$. Hence $\log 2<7/10$, so $\log V+2\log(1/\delta)\leq(7/10)(k+2h)$. Divide the scalar regression bound by $\sqrt V$. For a nonnegative rational $x$, the smallest integer $j$ with $(j/10^{12})^2\geq x$ supplies a rational upper square-root enclosure. Integer square root and an exact final comparison compute it. Outward rounding only enlarges the interval. Intersecting with a true prior constraint preserves coverage. Apply the two-regression union bound on their time-uniform events. $\square$
3863 
3864 All counts, sums, radius enclosures and interval endpoints in this experiment are rational. No floating-point logarithm is trusted in the confidence arithmetic. An exact replay verifies the arithmetic and the declared protocol. It does not authenticate the observations, prove that a physical process satisfies the model, or measure the procedure's coverage from one pseudorandom run.
3865 
3866 ## 95.1 Executed result
3867 
3868 The seeded reference path uses $a=2/5$, $b=1/5$ and initial observation zero. It was first run to 65,536 transitions and then extended along the same seeded path to 262,144. Time-uniform coverage permits that extension without spending another time-selection budget. This is not a new independent coverage experiment.
3869 
3870 At the final checkpoint, outward-rounded intervals are
3871 
3872 $$
3873 \boxed{
3874 \frac{89}{250}\leq a\leq\frac{231}{500},
3875 \qquad
3876 \frac{29}{250}\leq b\leq\frac{29}{100}.
3877 }
3878 $$
3879 
3880 Every member of this rectangle has density at least
3881 
3882 $$
3883 \rho=1-\frac{231}{500}-\frac{29}{100}=\frac{31}{125}.
3884 $$
3885 
3886 The rounded fitted center is $(409/1000,203/1000)$. It lies inside the rectangle. No reset or other coefficient correction is applied.
3887 
3888 At earlier recorded checkpoints the sufficient density-floor test withheld a whole-box certificate. Its first successful recorded checkpoint was 32,768 transitions, with floor $1/100$. This is not a claim that 32,768 is the smallest possible sample size, or even the first passing individual step. Only the declared checkpoints were evaluated. A withheld whole-box certificate does not declare the nominal fit or true process invalid.
3889 
3890 
3891 
3892 # 96. Structured uncertainty has an observable error bound
3893 
3894 Consider two parameters in a certified family. For normalized $z=(1,x,y)$, their density difference is
3895 
3896 $$
3897 \Delta p(u)=\Delta a\,x u+\Delta b\,yL_2(u).
3898 $$
3899 
3900 The state-update vector is the same $v(u)$ in both models after observing $u$. This does not make the models equivalent. Their likelihoods and hence their predicted observation laws can differ.
3901 
3902 **Proposition 85. Parameter-to-law error without a state-estimation term.** Let
3903 
3904 $$
3905 c_1=\int|u|\,d\mu,
3906 \qquad c_2=\int|L_2(u)|\,d\mu.
3907 $$
3908 
3909 If $|\Delta a|\leq d_a$ and $|\Delta b|\leq d_b$, the conditional observation laws on any common normalized state obey
3910 
3911 $$
3912 \boxed{
3913 \|P(\cdot\mid z)-\widehat P(\cdot\mid z)\|_{\mathrm{TV}}
3914 \leq\Delta:=\frac{c_1d_a+c_2d_b}{2}.
3915 }
3916 $$
3917 
3918 The same bound holds on every realized common observation history. With the same initial observation, complete future records of length $T$ have total variation at most $\min(1,T\Delta)$. For the five-node measure,
3919 
3920 $$
3921 c_1=\frac{23}{45},\qquad c_2=\frac{14}{45}.
3922 $$
3923 
3924 For $du/2$, the respective constants are $1/2$ and $2/(3\sqrt3)$.
3925 
3926 **Proof.** The first claim follows by integrating $|\Delta p|/2$, using $|x|\leq1$ and $y\leq1$. After the same observed value both models have exactly $v(u)$, so the conditional bound applies again without accumulated coordinate-state error. Telescoping conditional kernels gives the record bound, equivalently the common-history coupling argument of Proposition 69. Direct summation gives the discrete constants. For the continuous constant $c_2$, split the integral at $u=\pm1/\sqrt3$ and integrate the quadratic polynomial with its respective signs. $\square$
3927 
3928 The use of $\Delta$ here is an upper bound proved from a confidence set and a model class. It is not the lower discrepancy bound from a rejected residual test in Part VI. A test that does not reject cannot supply the displayed upper bound by itself.
3929 
3930 For the final reference rectangle and fitted center,
3931 
3932 $$
3933 d_a=\frac{53}{1000},\qquad d_b=\frac{87}{1000},
3934 \qquad
3935 \boxed{\Delta=\frac{2437}{90000}.}
3936 $$
3937 
3938 These bounds compare the fitted model with every member of the rectangle. They apply to the true process on the conditional coverage event. Even when that event fails, the rectangle's internal probability certificate remains an exact statement about its members.
3939 
3940 # 97. Long-horizon marginal control is not full-record control
3941 
3942 The previous record bound grows with $T$. A different question can have a uniform answer: how far apart are the distributions of the signal at a single future time? A lower density bound permits this distinction to be made exactly.
3943 
3944 Let $P,Q$ be Markov kernels on a common state space. Suppose $P(s,\cdot)\geq\rho\mu(\cdot)$ for every $s$, where $\mu$ is one probability measure and $0<\rho\leq1$. Suppose also $\sup_s\|P(s,\cdot)-Q(s,\cdot)\|_{\mathrm{TV}}\leq\Delta$.
3945 
3946 **Proposition 86. Uniform marginal and rolling-window stability.** For initial laws at distance $e_0$, the one-time marginal laws satisfy
3947 
3948 $$
3949 \boxed{
3950 e_t\leq\min\left(1,
3951 (1-\rho)^te_0+
3952 \frac{\Delta[1-(1-\rho)^t]}{\rho}\right).
3953 }
3954 $$
3955 
3956 For $k$ consecutive observations starting at time $t$, their joint laws are at distance at most $\min(1,e_t+(k-1)\Delta)$. A uniform small marginal bound does not in general bound the total variation of arbitrarily long complete records away from one.
3957 
3958 **Proof.** For $\rho<1$, write $P(s,\cdot)=\rho\mu+(1-\rho)R(s,\cdot)$, where $R$ is a Markov kernel. Applying a Markov kernel cannot increase total variation, by integration of bounded test functions. Thus $\|\nu P-\widetilde\nu P\|_{\mathrm{TV}}\leq(1-\rho)\|\nu-\widetilde\nu\|_{\mathrm{TV}}$. Adding and subtracting $\widetilde\nu P$ yields $e_{t+1}\leq(1-\rho)e_t+\Delta$. Iterate this inequality. The case $\rho=1$ follows directly. Applying the finite-record telescoping argument from the initial time $t$ gives the rolling-window statement.
3959 
3960 For the final assertion, take two distinct independent Bernoulli laws with parameters strictly between zero and one. Each is a Markov kernel with a positive common minorization. At any time their marginal distance is the fixed parameter difference. For a full record, test whether its empirical mean lies above the midpoint between the parameters. Under each law, Chebyshev's inequality makes the error probability tend to zero as the record length grows. The total variation of the record laws therefore tends to one. $\square$
3961 
3962 This is the standard minorization/contraction mechanism [R24], with its error accounting made explicit. It is not a stability theorem for arbitrary feedback policies that remember the full history. The reference model is autonomous and its current scalar observation is the Markov state. It also does not bound long products of unnormalized likelihood ratios.
3963 
3964 With $e_0=0$, $\rho=31/125$ and $\Delta=2437/90000$, the reference confidence rectangle gives
3965 
3966 $$
3967 \boxed{
3968 \sup_{t\geq0}e_t\leq\frac{2437}{22320}
3969 \approx0.10919.
3970 }
3971 $$
3972 
3973 This is a nontrivial all-time bound on **one-time marginal observation distributions**. The complete-record bound is still $\min(1,2437T/90000)$, which becomes trivial at integer horizon 37. The model does not exchange those two different claims. In live filtering after a shared observed value, its state is exactly known and the next-observation law has the separate uniform upper bound $2437/90000$.
3974 
3975 
3976 
3977 # 98. A conditional data-to-domain theorem
3978 
3979 **Proposition 87. Observable structured learning with a covered domain.** Suppose the scalar process obeys the conditional law of Proposition 83 with the declared reference measure and fixed parameters satisfying $|a|,|b|\leq1$. Use the predictable regressions, fixed dictionary and confidence protocol of Proposition 84. At a data-selected finite time, form the full outward-rounded rectangle. If its density floor $\rho$ is nonnegative, that rectangle has an exact all-horizon probability certificate on $C$. If $\rho>0$, every chosen center inside it has the marginal and record bounds of Propositions 85--86 relative to every member.
3980 
3981 With probability at least $19/20$ over the specified stochastic observation process, these implications cover its true parameter at every checked time, including a stopping time. Internal validity of a verified rectangle is deterministic. Statistical coverage remains conditional on the declared process, moment, parameter and observation assumptions.
3982 
3983 **Proof.** Proposition 84 provides one simultaneous event on which both true parameters belong to all returned intervals. Proposition 83 supplies the domain certificate for the entire rectangle whenever its sufficient density floor is nonnegative. The selected center is one of its members, and the maximum distances to the endpoints bound each parameter difference. Proposition 85 gives conditional and record error. If the floor is positive, all true conditional rows have the same lower reference component, so Proposition 86 applies to the scalar Markov process. Selection of the stopping time does not alter the already simultaneous event. None of these implications verifies the assumptions from a single sample path. $\square$
3984 
3985 No claim of empirical coverage follows from the saved run. It is one seeded finite realization of the exactly declared discrete analogue. The general confidence result is a theorem, with assumptions recorded in the certificate. Its independent checker rederives the sufficient statistics, rational radii, outer endpoints, density floor and error budgets from the saved path. It rejects a changed reference measure, a narrower claimed interval, an altered path checksum or an unsupported claim that the assumptions were independently verified.
3986 
3987 This is the point where probability-domain certification and data uncertainty become connected in an executed dependent-signal example. The family is more restrictive than an arbitrary learned operator. That restriction is visible and does useful work: it preserves the nonlinear posterior curve while uncertainty changes its likelihoods.
3988 
3989 # 99. Two distinct completed reference sectors
3990 
3991 The smaller-repair construction retains the earlier operator family and geometry. Its coefficient box is deterministic, and its all-horizon validity is distinct from its finite-horizon comparison bound. The dependent-data construction learns two parameters of a different, supplied nonlinear observable family. It certifies a whole confidence rectangle without reset correction and distinguishes next-step, record and marginal bounds. Neither example establishes unrestricted representation discovery.
3992 
3993 # 100. Scope and research boundary
3994 
3995 The finite scope contains exact realization and transfer, closure and memory, conditional observable identification, controlled probability instruments, finite polyhedral and curved certificates, explicit approximation budgets, and a restricted dependent-data learning sector. The reference code and its executed examples are catalogued in the Proof and Evidence Register. General nonlinear dictionary discovery, unknown hidden-state learning, empirical sensor modeling and broad deployment remain research directions, not unstated conclusions or release prerequisites.
3996 
3997 \newpage
3998 
3999 # References and source lineage
4000 
4001 **[K1]** Komorebi P. *Projector-Resolved Large Algebraic Signal Model*. Historical 38-page capstone preserved in `Komorebi_PR_LASM_2.3.0_Source.zip`, `paper/Projector_Resolved_Large_Algebraic_Signal_Model.tex`. In particular, Sections 2–7 and the categorical-atlas distinction. Historical experimental claims are not used here.
4002 
4003 **[K2]** Komorebi P. *Projector-Resolved Seed Calculus VI*, corrected presentation in `Paper_VI_With_Visual_Appendix.pdf`, Section 8, Theorem 8.4 and the adjacent coarsening criterion and one-packet boundary. This supplement restates and proves the finite result it uses. It does not claim a fresh Lean audit of that paper.
4004 
4005 **[K3]** Komorebi P. *PR-LASM 2.3.0 Release Audit*, 9 October 2026, and the accompanying source. The source inspected includes `exact_atlas.py`, `lasm_packetizer.py`, `lasm_model.py`, `docs/RELEASE_GUIDE.md` and `docs/claim-boundary.md`.
4006 
4007 **[R1]** Ronald R. Coifman and Mladen Victor Wickerhauser. “Entropy-based algorithms for best basis selection.” *IEEE Transactions on Information Theory* 38(2), 713–718, 1992. DOI: 10.1109/18.119732. Source checked: the authors' Washington University research record. The finite additive tree optimization is classical background.
4008 
4009 **[R2]** MIT OpenCourseWare. *6.241J Dynamic Systems and Control*, Chapter 24, “Observability,” Spring 2011, especially Section 24.5 on the largest invariant subspace contained in the observation kernel. Source checked: the MIT course notes, PDF page 6. Used to identify the classical control-theoretic antecedent of Proposition 8.
4010 
4011 **[R3]** Michael Woodward, Yifeng Tian, Arvind Mohan, Yen Ting Lin, Christoph Hader, Hermann Fasel, Misha Chertkov and Daniel Livescu. *Data-Driven Mori-Zwanzig: Approaching a Reduced Order Model for Hypersonic Boundary Layer Transition*, arXiv:2301.07203, version 1, 2023, introduction and description of resolved, memory and unresolved contributions. Cited for the established projection-memory context, not as evidence for PR-LASM's performance or for any application discussed in that paper. Proposition 10 is proved directly by finite linear recurrence expansion here.
4012 
4013 **[R4]** Peter Buchholz. “Exact and ordinary lumpability in finite Markov chains.” *Journal of Applied Probability* 31(1), 59–75, 1994. DOI: 10.2307/3215235. Source checked: Cambridge University Press publication record. Used for the classical finite Markov aggregation context.
4014 
4015 
4016 **[R5]** Michael L. Littman, Richard S. Sutton and Satinder Singh. “Predictive Representations of State.” *Advances in Neural Information Processing Systems* 14, NIPS 2001. The paper connects state with predictions of future observations conditional on actions. Cited for conceptual lineage, not as a proof of the finite delay theorem or of this implementation. The original paper and proceedings record are the primary sources. The scanned paper carries all three author names.
4017 
4018 **[R6]** Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor and J. Nathan Kutz. “Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control.” *PLOS ONE* 11(2), e0150171, 2016. DOI: 10.1371/journal.pone.0150171. Used for the established role and limitations of finite invariant observable families in nonlinear modeling. No general nonlinear feature-discovery guarantee is imported.
4019 
4020 **[R7]** Daniel Hsu, Sham M. Kakade and Tong Zhang. “A Spectral Algorithm for Learning Hidden Markov Models.” *Journal of Computer and System Sciences* 78(5), 1460–1480, 2012. Primary author manuscript inspected: `hmm-jcss.pdf`, hosted by Daniel Hsu at Columbia University, especially its abstract and rank assumptions. The result illustrates observable-operator learning under explicit spectral hypotheses. Its statistical guarantees are not claimed for PR-LASM.
4021 
4022 **[R8]** Stephen Boyd, Laurent El Ghaoui, Eric Feron and Venkataramanan Balakrishnan. *Linear Matrix Inequalities in System and Control Theory*. SIAM Studies in Applied Mathematics 15, 1994. The authors' Stanford publication page identifies the book and provides the primary text. Used to identify the classical common quadratic Lyapunov and matrix-inequality setting. The discrete contraction implication used in Section 19 is proved directly there.
4023 
4024 
4025 **[R9]** Byron Boots, Sajid M. Siddiqi and Geoffrey J. Gordon. “Closing the Learning-Planning Loop with Predictive State Representations.” *Robotics: Science and Systems VI*, 2010. DOI: 10.15607/RSS.2010.VI.036. Primary eight-page proceedings paper inspected at the authors' Carnegie Mellon host, especially Section II on interventional tests and Section III on observable matrices. A journal version appeared in *The International Journal of Robotics Research* 30(7), 954–966, 2011. Used for the established PSR lineage, not as an assertion that this supplement implements that paper's complete algorithm or planning guarantees. Source: `https://www.cs.cmu.edu/~ggordon/boots-siddiqi-gordon-closing-loop-psrs.pdf`.
4026 
4027 **[R10]** Han Zhao and Pascal Poupart. “A Sober Look at Spectral Learning.” *ICML 2014 Workshop on Spectral Learning*, 2014. Primary author-hosted manuscript inspected, including its discussion of negative probabilities in spectral estimates. Used only to identify that practical failure mode. The cone and packet statements in this supplement are proved independently. Source: `https://hanzhaoml.github.io/papers/ICML2014/spectral-main.pdf`.
4028 
4029 **[R11]** Wassily Hoeffding. “Probability Inequalities for Sums of Bounded Random Variables.” *Journal of the American Statistical Association* 58(301), 13–30, 1963. DOI: 10.1080/01621459.1963.10500830. Publisher record checked. Proposition 35 supplies a self-contained Bernoulli moment-generating-function proof for the protocol it uses. Source: `https://www.tandfonline.com/doi/abs/10.1080/01621459.1963.10500830`.
4030 
4031 **[R7a]** Daniel Hsu, Sham M. Kakade and Tong Zhang. “A Spectral Algorithm for Learning Hidden Markov Models.” *COLT*, 2009. Original conference manuscript inspected at the official Learning Theory proceedings site, particularly Sections 2.3–3.1 on rank, sampling and observable representations. The later journal version is [R7]. No statistical guarantee from either version is imported without its assumptions. Source: `https://www.learningtheory.org/colt2009/papers/011.pdf`.
4032 
4033 **[R12]** Herbert Jaeger. “Observable operator models for discrete stochastic time series.” *Neural Computation* 12(6), 1371–1398, 2000. Author-hosted manuscript with the August 2017 erratum. Section 7 gives the polyhedral-cone connection to finite positive realizations and an irrational-rotation obstruction. The controlled rational example in Section 51 is an independently worked adaptation of that established obstruction, not a claim of discovering it. The corrected manuscript and its geometric argument were inspected. Source: `https://www.ai.rug.nl/minds/uploads/oom_neco00.pdf`.
4034 
4035 **[R13]** Zoltán Horváth, Yunfei Song and Tamás Terlaky. *A Novel Unified Approach to Invariance Conditions for a Linear Dynamical System*. arXiv:1405.5167, version 2, 5 July 2016. Primary author manuscript, particularly the polyhedral-invariance discussion and use of Farkas' lemma. Used for the classical invariant-cone and optimization setting. The generator and robust-vertex statements used here are proved directly. Source: `https://arxiv.org/html/1405.5167v2`.
4036 
4037 **[R14]** Marco Fanizza, Josep Lumbreras and Andreas Winter. *Quantum theory in finite dimension cannot explain every general process with finite memory*. arXiv:2209.11225, version 2, 5 May 2023. Primary manuscript, especially its discussion of finite linear realizations, positive realizations and polyhedral cones. Cited for the distinction between these representation classes. No quantum implementation or quantum advantage is claimed for PR-LASM. The present block uses a real three-dimensional cone construction proved in Section 51. Source: `https://arxiv.org/html/2209.11225v2`.
4038 
4039 **[R15]** Anna Jenčová. *Base norms and discrimination of generalized quantum channels*. arXiv:1308.4030, version 1, 19 August 2013. Primary manuscript, Section 1.2, printed page 5, on finite-dimensional proper cones, normalized bases and base norms. That ordered-vector-space definition is used in Section 46. The classical instrument contraction and observation-tree bound needed here are derived in the text, without importing quantum-channel assumptions. Source: `https://arxiv.org/pdf/1308.4030`.
4040 
4041 
4042 **[R16]** John Watrous. *The Theory of Quantum Information*. Cambridge University Press, 2018. Author-hosted prepublication text, Chapter 2, Theorem 2.22, printed page 82 (PDF page 90), and the trace-preserving characterization in Theorem 2.26. Used for the established Choi and congruence representations of completely positive maps, not to assign quantum physics to the signal model. The relevant theorem page was visually inspected. Source: `https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf`. This third-party text is not redistributed in the bundle.
4043 
4044 **[R17]** Pablo A. Parrilo. *Algebraic Techniques and Semidefinite Optimization*, MIT 6.972, Lecture 10, 14 March 2006. MIT OpenCourseWare notes, pages 10-1 through 10-3, on polynomial Gram matrices, coefficient constraints and interval positivity. Used for the standard sum-of-squares certificate method. Proposition 53 proves only the sufficient matrix-valued certificate used here and does not import scalar completeness as a matrix theorem. The formula pages were visually inspected. Source: `https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/resources/lecture_10/`.
4045 
4046 **[R18]** Herbert Jaeger. *Modeling and learning continuous-valued stochastic processes with OOMs*. GMD technical report, 8 June 2000. Author-hosted report, title and abstract inspected, establishing that continuous-valued observable-operator modeling and learning are prior work. The present polynomial, certificate and binning arguments are proved within this volume. No guarantee from the report is assigned to the reference implementation. Source: `https://www.ai.rug.nl/minds/uploads/jaeger.00.tr.contoom.pdf`.
4047 
4048 
4049 **[R19]** Ulrich Abel, Ana Maria Acu, Margareta Heilmann and Ioan Raşa. *A representation for the integral kernel of the composition of multivariate Bernstein--Durrmeyer operators*. arXiv:2512.17297v1, 19 December 2025. The authors' primary manuscript, Introduction, gives the classical univariate Bernstein--Durrmeyer kernel used in Section 72. Its composition theorems are not needed or claimed here. Proposition 64 derives the operator-valued positivity and error statements directly. Source: `https://arxiv.org/html/2512.17297v1`.
4050 
4051 **[R20]** Steven R. Howard, Aaditya Ramdas, Jon McAuliffe and Jasjeet Sekhon. *Time-uniform, nonparametric, nonasymptotic confidence sequences*. *Annals of Statistics* 49(2), 1055--1080, 2021. DOI: 10.1214/20-AOS1991. The primary arXiv record and the conditional exponential-supermartingale discussion in the v4 HTML manuscript were inspected. Proposition 66 proves a simpler union-bound confidence sequence, not the paper's sharper rates. Source: `https://arxiv.org/abs/1810.08240` and `https://arxiv.org/html/1810.08240v4`.
4052 
4053 **[R21]** Roland Hildebrand. *An LMI description for the cone of Lorentz-positive maps*. Author manuscript, 17 January 2006. Primary manuscript inspected, especially the introduction and definitions on pages 1--2. It supplies classical context for Lorentz-positive maps and the existence of convex descriptions. The present margin and semicircle certificate arguments are proved directly and do not claim to implement the manuscript's full matrix-inequality construction. Source: `https://optimization-online.org/wp-content/uploads/2005/12/1260.pdf`.
4054 
4055 **[R22]** Tareq Hamadneh, Hassan Al-Zoubi, Hamza Alzaareer and Rafael Wisniewski. *Optimization and Positivity Certificates of Rational Functions using Bernstein Form*. arXiv:1906.11037, 2019. Primary manuscript inspected, including its Bernstein range enclosure discussion on printed pages 3--4. It supplies established positivity-certificate context. Proposition 75 gives its own tensor-product rectangle argument and strict-subdivision proof. Source: `https://arxiv.org/abs/1906.11037`.
4056 
4057 **Inherited dependencies.** Proposition 52 defines continuous instrument semantics. Proposition 59 supplies the common-domain continuous-law bound. Propositions 62--63 distinguish exact ellipsoid proposals from verification. Proposition 48 supplies repeated confidence-set accounting. The new block restates the specific repair and uncertainty inequalities it uses. The cited earlier results are present in this cumulative volume.
4058 
4059 **[R23]** Yasin Abbasi-Yadkori, David Pál and Csaba Szepesvári. *Online Least Squares Estimation with Self-Normalized Processes: An Application to Bandit Problems*. arXiv:1102.2670, version 1, 14 February 2011. The primary manuscript's Section 2, Theorem 3, and Section 5 supply the classical martingale-mixture and adaptive-regression context. Proposition 81 includes its own proof, retains conditional noise and design assumptions, and does not claim a new concentration inequality. Source checked: `https://arxiv.org/pdf/1102.2670`.
4060 
4061 **[R24]** David A. Levin and Yuval Peres, with contributions by Elizabeth L. Wilmer. *Markov Chains and Mixing Times*, second edition, 2017. Chapters 4 and 5 develop total variation and coupling. The present minorization argument is proved directly and does not import a finite-state assumption into its general kernel statement. The worked reference chain is finite. Source checked: the author-hosted second-edition text, `https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf`.
4062 
4063 **Internal dependencies.** Part VII, especially Propositions 71, 73, 75, 76 and 79, supplies the robust checker and predecessor repair comparison. Propositions 59 and 69 supply the continuous and conditional-kernel full-law comparisons. Their mathematical source passages are retained in this cumulative volume; prior archive identifiers remain in the provenance records.