Paper XXV.md

Transformation Laws and Localized Stability of Generator-Resolved Diagnostics

Exact Transport Covariance, Carrier-Localized Perturbation, and Minimax Information Bounds

WuJun Chen

Independent Researcher | RIME Program | 2026

Paper XXV of the RIME program. This paper develops a finite-dimensional methods layer for aligned generator-resolved diagnostics, separating exact transport covariance, carrier-local perturbation, and application-specific semantics.

Abstract

Problem. A generator-resolved block $Q_iY_aQ_j$ depends on the labelled operator, the marked source and target carriers, and the frame in which those data are compared. Global norm budgets remain portable but can erase exactly the localization information needed to decide whether a thresholded block is stable.

Approach. A typed finite-dimensional diagnostic instance separates representation-intrinsic, labelled-generator-relative, and marked-sectorization-relative data. After two instances are aligned by a unitary, each block difference resolves into left-carrier motion, additive operator error, and right-carrier motion. Global, semi-local, and local error summaries are ordered by an explicit information lattice.

Results. Simultaneous unitary transport preserves weighted averages, commutants, labelled block norms, and thresholded activity relations. The aligned block difference has an exact three-term decomposition with a portable global bound and a sharper carrier-localized bound. All three one-axis global constants are sharp. In the fixed-projector additive-error submodel, global norms cannot detect localization, and every node of the carrier-information lattice has an exact minimax envelope. Additive error has an exact conditional completion for fixed carrier motion, and the localized triangle bound has a complete equality criterion. Margin theorems separate stable activity, stable inactivity, and an information-theoretically complete unresolved interval. The algebra and localized bound extend to bounded oblique carrier resolutions, with explicit projector-conditioning factors. Exact rational controls and bounded float64 observations are kept distinct: Rubik supplies the simultaneous-transport hostile control, while the two-state Markov construction supplies a portability example.

Boundary. The elementary conjugation and telescoping identities are not claimed as new in isolation. The results control declared finite-dimensional blocks; they do not reconstruct a representation, decide survival of an arbitrary routed product, estimate distance to the incidence locus $AB=0$, or establish general spectral, pseudospectral, Markov-mixing, continuum, or open-quantum-channel stability.

Keywords: operator perturbation; sector projectors; localized matrix blocks; minimax bounds; oblique projections; threshold stability; typed diagnostics.

Notation Table {.unnumbered}

Symbol	Meaning
$V$	finite-dimensional complex Hilbert space
$A$, $I$	labelled generator and sector index sets
$Y_a$, $Q_i$	represented operator and marked sector projector
$\mu_a$, $M_\mu(Y)$	labelled weight and weighted generator average
$B_a(i,j)=Q_iY_aQ_j$	generator-resolved block
$\mathcal R_\tau(a;i,j)$	thresholded labelled activity predicate
$U$, $\bar Y_a$, $\bar Q_i$	alignment unitary and transported reference data
$E_a$, $\Delta_i$	operator and carrier perturbations
$D_a(i,j)$	aligned generator-resolved block difference
$b_{\rm loc}$, $b_{\rm glob}$, $b_{\rm obl}$	localized, global, and oblique-carrier bounds
$\mathcal I_G,\mathcal I_L,\mathcal I_R,\mathcal I_{LR},\mathcal I_{\rm loc}$	cumulative carrier-information levels
$U_G,U_L,U_R,U_{LR},U_{\rm loc}$	exact minimax envelopes at those levels
$\tau$, $b$	activity threshold and certified error radius

Introduction

This paper studies one concrete comparison question: which properties of a labelled operator-block diagnostic survive aligned change of frame, and what is the sharpest block-stability statement justified by a declared level of global or carrier-local information?

The basic coordinate is

$$ B_a(i,j)=Q_iY_aQ_j. $$

It has three owners. The represented family controls $Y_a$; the labelled generator or measure policy controls which operators are retained and how they are averaged; and the marked sectorization controls the source and target carriers. A comparison that transports only one owner is not an aligned comparison. A bound that retains only $\lVert Y_a\rVert$ and $\lVert E_a\rVert$ is valid but may discard whether the perturbation reaches the selected carriers at all.

The theorem spine follows this distinction. Section 2 introduces the typed diagnostic instance, proves simultaneous-transport covariance, and states its fixed-frame and similarity boundaries. Section 3 proves the exact three-term perturbation identity, global and localized bounds, the bounded oblique extension, sharpness, the carrier-information lattice, exact minimax envelopes in the fixed-projector additive-error submodel, conditional additive completion for fixed carrier geometry, and the equality criterion. Section 4 converts certified block radii into a complete threshold trichotomy. Section 5 gives a deliberately narrow two-state Markov portability lift. Section 6 separates exact certificates from bounded observations and states their frozen numerical protocols.

The central distinction is

$$ \text{portable global control} \ne \text{carrier-local information} \ne \text{application-specific semantics}. $$

The first inequality creates the minimax information problem. The second prevents a Hilbert-space block estimate from being renamed a probability, support, reconstruction, or routed-composition theorem without additional hypotheses.

Related Work and Novelty Boundary {.unnumbered}

Norm inequalities, invariant subspaces, oblique projections, and matrix perturbation theory are standard background \cite{hornJohnson2013,stewartSun1990,kato1995perturbation}. Finite Markov-chain hitting times, regeneration, and stationary distributions are standard tools \cite{kemenySnell1960,norris1997,levinPeresWilmer2017}; the narrow lift in Section 5 is stated only under its explicit two-state hypotheses. The ideal-property estimate for the Frobenius norm, unitary invariance, and the algebraic telescoping of a three-factor difference are standard ingredients; the telescoping inequality alone is not the principal novelty. The contribution is their typed ownership interface: transformation laws for labelled diagnostics, localized evidence resolution, exact minimax envelopes, sharpness and equality theorems, and a complete threshold-uncertainty boundary.

Within the RIME program, Paper II introduced the direct sector block $Q_iY_aQ_j$ as the carrier of generator transport and related off-diagonal Frobenius mass to sector non-invariance \cite{paper2}. Paper III separated direct support from projected composition and isolated the static obstruction

$$ AB=0 \quad\Longleftrightarrow\quad \operatorname{im}B\subseteq\ker A $$

for a two-block route \cite{paper3}. Paper VII developed that obstruction into rank-stratified, carrier-resolved incidence geometry; its transported-frame covariance proposition also records covariance of routed products, Frobenius norms, and profile counts under simultaneous transport of operators, sectors, and carriers \cite{paper7}. Paper XX organizes the fixed-carrier geometry at every finite route depth through carrierwise factorization, cutwise image--kernel tests, and survivor recursion \cite{paper20}.

Theorem 2.1 packages the covariance of Paper II's direct block at the typed diagnostic level, including labelled weights, weighted averages, commutants, unitarily invariant block norms, and thresholded labelled activity. Its elementary block-conjugation identity is not reclaimed as a new fact. Theorem 3.1 has a different role: it compares one block in two aligned frames and resolves its finite difference into motion of the left carrier, operator, and right carrier. It is a metric finite-difference counterpart of the earlier static block-incidence geometry, not a derivation from the image--kernel criterion.

The perturbation theorem therefore does not decide survival of a routed product, estimate distance to ${(A,B):AB=0}$, or propagate error through Paper XX's all-depth survivor recursion. Those extensions require productwise, transversality, or carrier-preservation hypotheses not assumed here.

Typed Unitary Transport

Typed diagnostic instances

Let $V$ be a finite-dimensional complex Hilbert space. Let $A$ and $I$ be finite label sets. A typed generator-resolved diagnostic instance consists of

$$ \mathcal D=(V,A,I,Y,Q,\mu,\lVert\cdot\rVert_F,\mathsf{zero}), $$

where

$$ Y:A\longrightarrow\operatorname{End}(V),\quad a\longmapsto Y_a, \qquad Q:I\longrightarrow\operatorname{End}(V),\quad i\longmapsto Q_i, $$

the $Q_i$ are nonzero mutually orthogonal projectors with $\sum_iQ_i=I_V$, and $\mu$ is a declared labelled weight family. Labels are part of the data: equal represented operators do not identify distinct source labels.

Define the weighted generator average and labelled blocks by

$$ M_\mu(Y)=\sum_{a\in A}\mu_aY_a, \qquad B_a(i,j)=Q_iY_aQ_j. $$

For a declared threshold $\tau\ge0$, define the generator-resolved activity predicate

$$ \mathcal R_\tau(a;i,j) \quad\Longleftrightarrow\quad \lVert B_a(i,j)\rVert_F>\tau. $$

At $\tau=0$, this is exact operator nonvanishing. At positive $\tau$, it is a strength policy and must not be renamed exact support.

Typed Unitary Transport Theorem

Theorem 2.1 (Typed Unitary Transport). Let

$$ \begin{aligned} \mathcal D&=(V,A,I,Y,Q,\mu,\lVert\cdot\rVert_F,\mathsf{zero}),\\ \mathcal D'&=(V',A,I,Y',Q',\mu,\lVert\cdot\rVert_F,\mathsf{zero}) \end{aligned} $$

have the same generator labels, sector labels, labelled weights, and zero policy. Suppose $U:V\to V'$ is unitary and, for every $a\in A$ and $i\in I$,

$$ Y'_a=UY_aU^\ast, \qquad Q'_i=UQ_iU^\ast. $$

Then:

1. $Q'$ is again a complete orthogonal sectorization with the same labelled sector dimensions.

2. The weighted averages satisfy

   $$ M_\mu(Y')=UM_\mu(Y)U^\ast. $$

   Consequently their characteristic polynomials, spectra with algebraic multiplicity, and trace moments agree.

3. Every labelled block satisfies

   $$ Q'_iY'_aQ'_j=U(Q_iY_aQ_j)U^\ast. $$

4. For every unitarily invariant matrix norm $\lVert!\lvert\cdot\rvert!\rVert$,

   $$ \lVert!\lvert Q'_iY'_aQ'_j\rvert!\rVert =\lVert!\lvert Q_iY_aQ_j\rvert!\rVert. $$

   In particular, the full Frobenius block profile and every thresholded labelled activity relation $\mathcal R_\tau$ are preserved.

5. Conjugation by $U$ is a vector-space isomorphism

   $$ \operatorname{Comm}(Y)\longrightarrow\operatorname{Comm}(Y'), \qquad T\longmapsto UTU^\ast, $$

   so the simultaneous commutant dimensions agree. If the declared family is accompanied by a completeness witness for a represented action, this also preserves the corresponding representation-intrinsic commutant dimension.

Proof. Unitary conjugation preserves adjoints, products, sums, identity, rank, and orthogonality. Therefore

$$ Q'_iQ'_j=UQ_iQ_jU^\ast, \qquad \sum_iQ'_i=U\left(\sum_iQ_i\right)U^\ast=I_{V'}. $$

Linearity gives the weighted-average identity. Associativity and $U^\ast U=I$ give

$$ Q'_iY'_aQ'_j =UQ_iU^\ast UY_aU^\ast UQ_jU^\ast =U(Q_iY_aQ_j)U^\ast. $$

Unitary invariance gives the norm equality. Hence zero/nonzero and any common threshold comparison are preserved coordinatewise. Finally,

$$ TY_a=Y_aT \quad\Longleftrightarrow\quad (UTU^\ast)Y'_a=Y'_a(UTU^\ast) $$

for every label $a$, and the inverse commutant map is conjugation by $U^\ast$. This proves all claims. $\square$

Corollary 2.2 (Aggregate activity covariance). Any aggregate relation formed from the labelled predicates by fixed Boolean operations, including

$$ (i,j)\in\mathcal R_\tau^{\rm agg} \quad\Longleftrightarrow\quad \exists a\in A:\ \mathcal R_\tau(a;i,j), $$

is preserved under the simultaneous transport in Theorem 2.1.

Boundary 2.3 (Fixed-frame hostile control). If the operators are transported but the projectors are held fixed, the hypotheses of Theorem 2.1 are not satisfied. No block-profile or activity invariance follows. This is an alignment failure, not a counterexample to the theorem.

Boundary 2.4 (General similarity is algebraic, not metric). If $S$ is only invertible, the algebraic identities still hold with $S^{-1}$ in place of $U^\ast$:

$$ Y'_a=SY_aS^{-1}, \qquad Q'_i=SQ_iS^{-1}, \qquad B'_a(i,j)=SB_a(i,j)S^{-1}. $$

But the transported idempotents need not be self-adjoint, and Frobenius block norms need not be preserved. Thus general similarity does not inhabit the metric conclusion of Theorem 2.1 without an additional transported metric.

Carrier-Localized Perturbation Theorem

Transport the reference instance into one common Hilbert space and write

$$ \bar Y_a=UY_aU^\ast, \qquad \bar Q_i=UQ_iU^\ast. $$

Let the observed operators and projectors be

$$ Y'_a=\bar Y_a+E_a, \qquad Q'_i=\bar Q_i+\Delta_i, $$

where both $\bar Q$ and $Q'$ are complete orthogonal sectorizations. No normality or unitarity assumption is imposed on $Y_a$ or $Y'_a$.

Theorem 3.1 (Carrier-Localized Perturbation Bound). For every $a\in A$ and $i,j\in I$, define

$$ D_a(i,j)=Q'_iY'_aQ'_j-\bar Q_i\bar Y_a\bar Q_j. $$

Then the exact three-term identity

$$ D_a(i,j) =\Delta_i\bar Y_aQ'_j +Q'_iE_aQ'_j +\bar Q_i\bar Y_a\Delta_j $$

holds. Consequently,

$$ \lVert D_a(i,j)\rVert_F\le b_{\rm loc}(a;i,j), $$

where

$$ \begin{aligned} b_{\rm loc}(a;i,j) ={}&\lVert\Delta_i\bar Y_aQ'_j\rVert_F +\lVert Q'_iE_aQ'_j\rVert_F\\ &+\lVert\bar Q_i\bar Y_a\Delta_j\rVert_F. \end{aligned} $$

If declared bounds satisfy

$$ \lVert\Delta_k\rVert_F\le\delta_k, \qquad \lVert E_a\rVert_F\le\varepsilon_{a,F}, $$

then also

$$ \lVert D_a(i,j)\rVert_F\le b_{\rm glob}(a;i,j), $$

with

$$ b_{\rm glob}(a;i,j) =\lVert Y_a\rVert_2(\delta_i+\delta_j)+\varepsilon_{a,F}. $$

Moreover, when the right-hand quantities are evaluated from the same exact operators and declared upper bounds,

$$ b_{\rm loc}(a;i,j)\le b_{\rm glob}(a;i,j). $$

Proof. First separate the additive error:

$$ \begin{aligned} Q'_iY'_aQ'_j-\bar Q_i\bar Y_a\bar Q_j ={}&\bigl[Q'_i\bar Y_aQ'_j-\bar Q_i\bar Y_a\bar Q_j\bigr] +Q'_iE_aQ'_j. \end{aligned} $$

Insert and subtract $\bar Q_i\bar Y_aQ'j$ in the bracket. The three differences are exactly the displayed terms. The triangle inequality gives $b{\rm loc}$.

For the global estimate, orthogonal projectors have operator norm at most one, unitary transport preserves $\lVert Y_a\rVert_2$, and compression by two orthogonal projectors cannot increase Frobenius norm. Using $\lVert AB\rVert_F\le\lVert A\rVert_F\lVert B\rVert_2$ and $\lVert AB\rVert_F\le\lVert A\rVert_2\lVert B\rVert_F$ bounds the three localized terms respectively by

$$ \delta_i\lVert Y_a\rVert_2, \qquad \varepsilon_{a,F}, \qquad \lVert Y_a\rVert_2\delta_j. $$

Their sum is $b_{\rm glob}$, and the same termwise estimates prove $b_{\rm loc}\le b_{\rm glob}$. $\square$

Extension 3.1a (Bounded oblique carriers). The exact three-term identity and the inequality

$$ \lVert D_a(i,j)\rVert_F\le b_{\rm loc}(a;i,j) $$

do not require self-adjointness, mutual orthogonality, or idempotence. They only use

$$ Q'_k=\bar Q_k+\Delta_k, \qquad Y'_a=\bar Y_a+E_a. $$

To retain carrier semantics, suppose now that $\bar Q_k$ and $Q'_k$ are bounded, possibly oblique idempotents belonging to two declared resolutions of the identity. Put

$$ \bar q_k=\lVert\bar Q_k\rVert_2, \qquad q'_k=\lVert Q'_k\rVert_2, \qquad \eta_k=\lVert\Delta_k\rVert_2, \qquad F_a=\lVert\bar Y_a\rVert_F, \qquad \varepsilon_{a,F}=\lVert E_a\rVert_F. $$

Then the same identity gives the portable oblique estimate

$$ \begin{aligned} \lVert D_a(i,j)\rVert_F \le b_{\rm obl}(a;i,j) ={}&\eta_iF_aq'_j +q'_i\varepsilon_{a,F}q'_j +\bar q_iF_a\eta_j. \end{aligned} $$

In particular, if all four relevant carrier idempotents have operator norm at most $K$, then

$$ b_{\rm obl}(a;i,j) \le KF_a(\eta_i+\eta_j)+K^2\varepsilon_{a,F}. $$

If the reference carriers are orthogonal, then $\bar q_i=\bar q_j=1$ and $q'_k\le1+\eta_k$, which gives the fully delta-controlled specialization

$$ b_{\rm obl}(a;i,j) \le F_a\bigl[\eta_i(1+\eta_j)+\eta_j\bigr] +(1+\eta_i)(1+\eta_j)\varepsilon_{a,F}. $$

Proof. The algebraic decomposition and triangle inequality are unchanged. Apply the ideal-property estimate

$$ \lVert ABC\rVert_F \le\lVert A\rVert_2\lVert B\rVert_F\lVert C\rVert_2 $$

to the three summands. This gives, in order,

$$ \eta_iF_aq'_j, \qquad q'_i\varepsilon_{a,F}q'_j, \qquad \bar q_iF_a\eta_j. $$

The two specializations follow by substitution and by $\lVert\bar Q_k+\Delta_k\rVert_2\le1+\eta_k$ in the orthogonal-reference case. $\square$

This extension is a stability statement, not a similarity invariance statement: unlike Theorem 2.1, it does not assert preservation of Frobenius block norms. The factors $\bar q_k,q'_k$ explicitly record the conditioning cost of passing from orthogonal sectors to oblique carrier coordinates.

Remark 3.2 (Why the additive error must be combined first). The alternative split

$$ \Delta_iY'_aQ'_j+\bar Q_iE_aQ'_j $$

is algebraically valid, but separately bounding its two terms double-counts the part $\Delta_iE_aQ'_j$. It yields the weaker global expression

$$ \delta_i(\lVert Y_a\rVert_2+\varepsilon_{a,2}) +\varepsilon_{a,F}+\lVert Y_a\rVert_2\delta_j. $$

Theorem 3.1 combines those terms as $Q'_iE_aQ'j$ and removes the spurious $\delta_i\varepsilon{a,2}$ contribution. The weaker expression is therefore not minimax optimal under the stated global information.

Theorem 3.3 (One-Axis Sharpness of the Global Constants). Each of the three coefficients in

$$ b_{\rm glob} =\lVert Y_a\rVert_2\delta_i+\varepsilon_{a,F} +\lVert Y_a\rVert_2\delta_j $$

is best possible when the other two perturbation axes are zero. More precisely:

1. for every $\varepsilon\ge0$, there is a fixed-sectorization additive perturbation with $\lVert E_a\rVert_F=\varepsilon$ and block difference exactly $\varepsilon$;
2. for every $M\ge0$ and $\delta\ge0$, there is a finite-dimensional left-projector perturbation with $\lVert Y_a\rVert_2=M$, $\lVert\Delta_i\rVert_F=\delta$, and block difference exactly $M\delta$;
3. the analogous statement holds for a right-projector perturbation.

Consequently no one of the constants $1,1,1$ can be uniformly decreased in a global-budget theorem, even though no claim is made that the sum is the exact joint minimax value when several axes are simultaneously nonzero.

Proof. For the additive axis, take fixed coordinate projectors onto $\mathbb C e_1$ and $\mathbb C e_2$, let $\bar Y_a=0$, and set $E_a=\varepsilon\lvert e_1\rangle\langle e_2\rvert$.

For the left-projector axis, choose an integer $r$ with $\delta\le\sqrt{2r}$. Let $A,B,C$ be mutually orthogonal subspaces with dimensions $r,r,2r$. Let $\bar Q_i$ project onto $A$, let an auxiliary sector project onto $B$, and let $\bar Q_j=Q'_j$ project onto $C$. Rotate each paired coordinate plane of $A\oplus B$ through an angle $\theta$ satisfying

$$ \sin\theta=\frac{\delta}{\sqrt{2r}}. $$

The resulting rank-$r$ projector $Q'_i$ satisfies $\lVert Q'_i-\bar Q_i\rVert_F=\sqrt{2r}\sin\theta=\delta$. The self-adjoint projector difference has $2r$ singular values equal to $\sin\theta$. Choose $\bar Y_a$ to be $M$ times an isometry from $C$ onto a complete right-singular basis of that difference and zero on $C^\perp$. Then

$$ \lVert(Q'_i-\bar Q_i)\bar Y_aQ'_j\rVert_F=M\delta. $$

Completing $Q'_i$ by the rotated auxiliary projector gives a valid complete orthogonal sectorization. Taking adjoints gives the right-projector example. This proves all three assertions. $\square$

Theorem 3.4 (Global Information Cannot Detect Carrier Localization). Fix coordinate projectors $Q_i=\lvert e_1\rangle\langle e_1\rvert$ and $Q_j=\lvert e_2\rangle\langle e_2\rvert$ in a carrier of dimension at least three, and set $\bar Y_a=0$. For $\alpha\in[0,1]$, define

$$ u_\alpha=\alpha e_1+\sqrt{1-\alpha^2},e_3, \qquad E_\alpha=\varepsilon\lvert u_\alpha\rangle\langle e_2\rvert. $$

All members of this family have exactly the same global data

$$ \lVert E_\alpha\rVert_2=\lVert E_\alpha\rVert_F=\varepsilon, \qquad \lVert Y_a\rVert_2=\delta_i=\delta_j=0, $$

but

$$ \lVert Q_iE_\alpha Q_j\rVert_F=\alpha\varepsilon, \qquad \frac{b_{\rm loc}}{b_{\rm glob}}=\alpha \quad(\varepsilon>0). $$

Thus $b_{\rm loc}/b_{\rm glob}$ can be arbitrarily small and can equal zero. In the fixed-projector additive-error submodel, the single localized seminorm

$$ \lambda_{ij}(E_a)=\lVert Q_iE_aQ_j\rVert_F $$

is an exact certificate for the block error. Conversely, any information summary that certifies a bound smaller than $\varepsilon$ for $E_0$ must distinguish $E_0$ from $E_1$; the two global norms alone cannot do so.

Proof. $E_\alpha$ is rank one with its two displayed global norms equal to $\varepsilon$. Its marked compression is $\alpha\varepsilon\lvert e_1\rangle\langle e_2\rvert$. Since the projectors are fixed and the reference operator vanishes, this compression is the complete block difference and is also the sole nonzero term in $b_{\rm loc}$. At $\alpha=0$ and $\alpha=1$ the global records are identical while the exact local errors are respectively zero and $\varepsilon$. $\square$

Remark 3.5 (Information ownership). $b_{\rm glob}$ needs only global norms. $b_{\rm loc}$ needs the three carrier-restricted actions in Theorem 3.1, or just their three Frobenius norms if the actions have already been evaluated. Theorem 3.4 proves a necessity statement in the operator-only submodel: some statistic that separates on-carrier from off-carrier perturbations is required for a strict localized improvement. It does not claim that one unique data structure is minimal among all equivalent encodings.

Definition 3.6 (Cumulative Carrier-Information Lattice). In the fixed-projector additive-error submodel, put $P=Q_i$, $Q=Q_j$, $p=\operatorname{rank}P$, $q=\operatorname{rank}Q$, and $k=\min(p,q)$. Consider certified upper budgets

$$ \lVert E\rVert_2\le\varepsilon_2, \quad \lVert E\rVert_F\le\varepsilon_F, \quad \lVert PE\rVert_F\le\ell, \quad \lVert EQ\rVert_F\le r, \quad \lambda=\lVert PEQ\rVert_F. $$

The cumulative information levels are

$$ \begin{aligned} I_G&=(\varepsilon_2,\varepsilon_F),\\ I_L&=(I_G,\ell),& I_R&=(I_G,r),\\ I_{LR}&=(I_G,\ell,r),\\ I_{\rm loc}&=(I_G,\ell,r,\lambda). \end{aligned} $$

They form the refinement lattice

$$ \begin{array}{ccccc} &&I_{\rm loc}&&\\ &&\downarrow&&\\ &&I_{LR}&&\\ &\swarrow&&\searrow&\\ I_L&&&&I_R\\ &\searrow&&\swarrow&\\ &&I_G&& \end{array} $$

The two semi-local nodes are incomparable: one resolves the target carrier of the perturbation and the other resolves its source carrier.

Theorem 3.7 (Exact Minimax Envelope at Every Information Level). Among all operators satisfying the budgets visible at the declared node, the best possible universal upper bounds for $\lambda=\lVert PEQ\rVert_F$ are

$$ \begin{aligned} U_G&=\min(\varepsilon_F,\sqrt{k},\varepsilon_2),\\ U_L&=\min(U_G,\ell),&U_R&=\min(U_G,r),\\ U_{LR}&=\min(U_G,\ell,r),\\ U_{\rm loc}&=\lambda. \end{aligned} $$

Every displayed envelope is minimax sharp. When $\dim V\ge2$ and the marked carriers and their complements are nontrivial, every edge is strict as an information refinement: the statistic added at the upper node is not determined by the lower node. The nodes $I_L$ and $I_R$ are incomparable, and even $I_{LR}$ does not determine $\lambda$. This information-theoretic strictness does not assert a strict pointwise inequality between the two envelope values for every numerical budget tuple.

Proof. Compression gives

$$ \lambda\le\lVert E\rVert_F, \qquad \lambda\le\lVert PE\rVert_F, \qquad \lambda\le\lVert EQ\rVert_F. $$

The block $PEQ$ has rank at most $k$, so

$$ \lambda\le\sqrt{k},\lVert PEQ\rVert_2 \le\sqrt{k},\lVert E\rVert_2. $$

Taking the minimum of the inequalities available at each node proves the upper bounds. For sharpness, let $m$ be the displayed minimum and choose orthonormal vectors $p_1,\ldots,p_k$ in $\operatorname{ran}P$ and $q_1,\ldots,q_k$ in $\operatorname{ran}Q$. Then

$$ E=\frac{m}{\sqrt{k}}\sum_{t=1}^k \lvert p_t\rangle\langle q_t\rvert $$

has $\lVert E\rVert_F=\lambda=m$, $\lVert E\rVert_2=m/\sqrt{k}$, and both semi-local norms equal to $m$. It therefore satisfies every visible upper budget and attains the corresponding envelope.

For strictness, choose unit vectors $p_1\in\operatorname{ran}P$, $p_0\in\ker P$, $q_1\in\operatorname{ran}Q$, and $q_0\in\ker Q$. The rank-one operators

$$ E_{11}=\lvert p_1\rangle\langle q_1\rvert, \qquad E_{10}=\lvert p_1\rangle\langle q_0\rvert, \qquad E_{01}=\lvert p_0\rangle\langle q_1\rvert $$

all have global operator and Frobenius norms one, while their left and right semi-local norms distinguish the two directions independently. Finally, put

$$ E_{\rm diag}=E_{11}+\lvert p_0\rangle\langle q_0\rvert, \qquad E_{\rm cross}=E_{10}+E_{01}. $$

These two operators have the same exact statistics

$$ \lVert E\rVert_2=1, \qquad \lVert E\rVert_F=\sqrt2, \qquad \lVert PE\rVert_F=\lVert EQ\rVert_F=1, $$

but their local block norms are respectively $1$ and $0$. Hence every claimed strictness and incomparability already occurs inside the declared nontrivial carrier and complement directions. In particular, the two semi-local statistics together still do not determine $\lambda$; the local node is a genuine strict refinement. $\square$

Theorem 3.8 (Exact Additive Completion of Fixed Carrier Motion). Fix orthogonal projectors $\bar Q_i,Q'_i,\bar Q_j,Q'_j$ and a reference operator $\bar Y_a$. Put

$$ A=Q'_i\bar Y_aQ'_j-\bar Q_i\bar Y_a\bar Q_j $$

and let

$$ \mathcal S'_{ij}={Q'_iXQ'_j:X\in\operatorname{End}(V)}. $$

Write the Hilbert--Schmidt orthogonal decomposition

$$ A_\parallel=Q'_iAQ'_j, \qquad A_\perp=A-A_\parallel. $$

Then, for every $\varepsilon\ge0$, the exact worst-case block error over additive perturbations with $\lVert E_a\rVert_F\le\varepsilon$ is

$$ \sup_{\lVert E_a\rVert_F\le\varepsilon} \bigl\lVert Q'_i(\bar Y_a+E_a)Q'_j -\bar Q_i\bar Y_a\bar Q_j\bigr\rVert_F =\sqrt{\lVert A_\perp\rVert_F^2 +(\lVert A_\parallel\rVert_F+\varepsilon)^2}. $$

Thus the additive budget combines linearly only with the component of the carrier-motion error already lying in the final block space. It combines orthogonally with the remaining component.

Proof. The map

$$ \Pi'_{ij}(X)=Q'_iXQ'_j $$

is the Hilbert--Schmidt orthogonal projector onto $\mathcal S'{ij}$. Moreover, the set of compressed additive errors with $\lVert E_a\rVert_F\le\varepsilon$ is exactly the closed Frobenius ball of radius $\varepsilon$ in $\mathcal S'{ij}$: compression cannot increase the norm, while every $C\in\mathcal S'_{ij}$ is realized by taking $E_a=C$. Therefore, for $C=Q'_iE_aQ'_j$, orthogonality gives

$$ \lVert A+C\rVert_F^2 =\lVert A_\perp\rVert_F^2+\lVert A_\parallel+C\rVert_F^2. $$

The second term is maximized by choosing $C$ parallel to $A_\parallel$; if that component vanishes, any $C\in\mathcal S'_{ij}$ of norm $\varepsilon$ attains the same value. This proves the formula. $\square$

Corollary 3.9 (Explicit Three-Axis Strictness). The three-term localized sum and the scalar global sum are not, in general, the exact joint optimum for a fixed carrier geometry. Let $e_1,e_2,e_3,e_4$ be orthonormal, choose $c,s\in(0,1)$ with $c^2+s^2=1$, and set

$$ \begin{aligned} \bar Q_i&=\lvert e_1\rangle\langle e_1\rvert, &Q'_i&=\lvert u\rangle\langle u\rvert, &u&=ce_1+se_3,\\ \bar Q_j&=\lvert e_2\rangle\langle e_2\rvert, &Q'_j&=\lvert v\rangle\langle v\rvert, &v&=ce_2+se_4,\\ \bar Y_a&=\lvert e_1\rangle\langle e_2\rvert. \end{aligned} $$

Complete both pairs to orthogonal sectorizations using their orthogonal rotated complements. Then

$$ \lVert\Delta_i\rVert_F=\lVert\Delta_j\rVert_F=\sqrt2,s, \qquad \lVert\bar Y_a\rVert_2=1. $$

For the carrier-motion error $A$ of Theorem 3.8,

$$ A_\parallel=0, \qquad \lVert A\rVert_F^2=1-c^4=s^2(1+c^2). $$

Hence, with any $\varepsilon>0$, all three perturbation axes are nonzero and the true additive worst case is

$$ \sqrt{s^2(1+c^2)+\varepsilon^2}. $$

By contrast, the three-term localized and global bounds are respectively

$$ b_{\rm loc}=cs+s+\varepsilon, \qquad b_{\rm glob}=2\sqrt2,s+\varepsilon. $$

Both inequalities are strict. In particular, the sum in Theorem 3.1 is a portable certificate, not a claim of joint minimax optimality.

Proof. Direct multiplication gives $Q'_i\bar Y_aQ'j=c^2\lvert u\rangle\langle v\rvert$. Its projection against the final block space cancels the projection of $\lvert e_1\rangle\langle e_2\rvert$, so $A\parallel=0$. The two rank-one operators have Hilbert--Schmidt inner product $c^2$, giving

$$ \lVert A\rVert_F^2=c^4+1-2c^4=1-c^4. $$

Theorem 3.8 gives the exact supremum. The first and third localized terms have norms $cs$ and $s$, and the additive term can have norm $\varepsilon$. Finally, $\sqrt{x^2+y^2}<x+y$ for positive $x,y$, while $\sqrt{1+c^2}<1+c<2\sqrt2$. This proves both strict comparisons. $\square$

Theorem 3.10 (Equality Criterion for the Localized Bound). Write the three terms in Theorem 3.1 as

$$ T_L=\Delta_i\bar Y_aQ'_j, \qquad T_E=Q'_iE_aQ'_j, \qquad T_R=\bar Q_i\bar Y_a\Delta_j. $$

Then

$$ \lVert D_a(i,j)\rVert_F=b_{\rm loc}(a;i,j) $$

if and only if there are a matrix $H$ with $\lVert H\rVert_F=1$ and nonnegative real numbers $\alpha_L,\alpha_E,\alpha_R$ such that

$$ T_L=\alpha_LH, \qquad T_E=\alpha_EH, \qquad T_R=\alpha_RH. $$

Zero terms correspond to zero coefficients. Equivalently, every two nonzero terms have Hilbert--Schmidt inner product equal to the product of their norms, with positive real phase.

For differences of orthogonal projectors, this condition forces

$$ \alpha_L\alpha_E\alpha_R=0. $$

Thus all three localized terms can never be simultaneously nonzero at an equality point. Equality is possible with one active term, or with two active terms that lie on the same nonnegative ray.

Proof. The Frobenius norm is the Hilbert-space norm associated with the Hilbert--Schmidt inner product. Equality in its finite triangle inequality holds exactly when all nonzero summands lie on one nonnegative ray. This proves the first equivalence.

It remains to use projector geometry. Suppose all three coefficients were positive. Because $T_E=Q'_iT_EQ'_j$, the range of $H$ lies in $\operatorname{ran}Q'_i$. Because $T_R=\bar Q_iT_R$, it also lies in $\operatorname{ran}\bar Q_i$. Hence

$$ \operatorname{ran}H \subseteq\operatorname{ran}Q'_i\cap\operatorname{ran}\bar Q_i. $$

The self-adjoint difference $\Delta_i=Q'_i-\bar Q_i$ annihilates this intersection. Consequently its range is orthogonal to the intersection:

$$ \langle h,\Delta_i x\rangle=\langle\Delta_i h,x\rangle=0 $$

for every $h$ in the intersection. But the range of $T_L$ lies in the range of $\Delta_i$, while $T_L=\alpha_LH$ would put the same nonzero range inside the intersection, a contradiction. Therefore at least one coefficient is zero. The converse follows immediately from the common-ray condition. $\square$

Remark 3.11 (When the global bound is also attained). Equality $\lVert D_a(i,j)\rVert_F=b_{\rm loc}=b_{\rm glob}$ additionally requires equality in every active submultiplicative estimate used in Theorem 3.1, and requires every declared upper budget appearing in $b_{\rm glob}$ to be saturated:

$$ \begin{aligned} \lVert\Delta_i\bar Y_aQ'_j\rVert_F &=\lVert\Delta_i\rVert_F\lVert Y_a\rVert_2,\\ \lVert Q'_iE_aQ'_j\rVert_F&=\lVert E_a\rVert_F,\\ \lVert\bar Q_i\bar Y_a\Delta_j\rVert_F &=\lVert Y_a\rVert_2\lVert\Delta_j\rVert_F, \end{aligned} $$

together with the common-ray criterion above. Theorem 3.10 shows that the three displayed equalities cannot all contribute positively to one joint equality point.

Margin consequences

Corollary 4.1 (Typed activity margin). Let

$$ n=\lVert\bar Q_i\bar Y_a\bar Q_j\rVert_F, \qquad n'=\lVert Q'_iY'_aQ'_j\rVert_F, $$

and let $b$ be either valid bound from Theorem 3.1. Then

$$ \lvert n'-n\rvert\le b. $$

For a declared threshold $\tau\ge0$:

$$ \begin{array}{rcll} n-b>\tau&\Longrightarrow&n'>\tau & (\mathsf{STABLE_ACTIVE}),\\ n+b\le\tau&\Longrightarrow&n'\le\tau & (\mathsf{STABLE_INACTIVE}),\\ \text{otherwise}&&&(\mathsf{UNRESOLVED}). \end{array} $$

$\mathsf{UNRESOLVED}$ is a proof-status outcome. It is not evidence that the perturbed block is zero, nonzero, active, or inactive.

Theorem 4.2 (Completeness of the Unresolved Interval). For $n,b,\tau\ge0$, let the only available scalar error information be

$$ n'\ge0, \qquad \lvert n'-n\rvert\le b. $$

Then the margin policy of Corollary 4.1 returns $\mathsf{UNRESOLVED}$ if and only if the admissible interval contains both an inactive value and an active value:

$$ \begin{aligned} &\exists x_-:\quad \lvert x_--n\rvert\le b,\quad x_-\le\tau,\\ &\exists x_+:\quad \lvert x_+-n\rvert\le b,\quad x_+>\tau. \end{aligned} $$

Both alternatives are realized by finite-dimensional operator-only perturbations with fixed coordinate projectors. Hence no sound classifier using only $n$, $b$, and $\tau$ can refine $\mathsf{UNRESOLVED}$ to either stable status.

Proof. The policy is unresolved exactly when

$$ n-b\le\tau<n+b. $$

In that case $x_-=\max(0,n-b)$ is admissible and inactive, while any $x_+$ strictly between $\tau$ and $n+b$, for example their midpoint, is admissible and active. Conversely, the existence of an admissible inactive value rules out $n-b&gt;\tau$, and the existence of an admissible active value rules out $n+b\le\tau$.

For realization, use fixed projectors onto $\mathbb C e_1$ and $\mathbb C e_2$, take $\bar Y_a=n\lvert e_1\rangle\langle e_2\rvert$, and set $Y'a=x\lvert e_1\rangle\langle e_2\rvert$. Then $E_a=(x-n)\lvert e_1\rangle\langle e_2\rvert$ has Frobenius norm at most $b$ and the perturbed block norm is exactly $x$. Applying this once with $x=x-$ and once with $x=x_+$ gives the two hostile realizations. $\square$

Corollary 4.3 (Finite aggregate margin). For a finite coordinate family $K$, suppose $\lvert n'_k-n_k\rvert\le b_k$ for every $k\in K$. Then

$$ \max_{k\in K}\max(0,n_k-b_k) \le\max_{k\in K}n'_k \le\max_{k\in K}(n_k+b_k). $$

The same threshold trichotomy therefore applies to an aggregate coordinate using these lower and upper bounds. Section 6.2 applies this aggregate policy to the Rubik observation; the Markov example in Section 6.3 remains a coordinatewise semantic lift.

Two-state Markov probability lift

The following is an application with additional structure, not a conclusion about arbitrary Markov systems.

Corollary 5.1 (Singleton Markov Lift). Let

$$ P=\begin{pmatrix}1-a&a\ \kappa&1-\kappa\end{pmatrix} $$

be row-stochastic with $a,\kappa&gt;0$, and let $Q_0,Q_1$ be the coordinate singleton projectors. Under the convention $P[\mathrm{source},\mathrm{target}]$,

$$ \lVert Q_0PQ_1\rVert_F=a, \qquad \lVert Q_1PQ_0\rVert_F=\kappa. $$

Suppose a perturbation preserves this two-state form and the reverse probability $\kappa$, while Theorem 3.1 gives

$$ a'\in[\ell,u], \qquad \ell=\max(0,a-\beta), \qquad u=\min(1,a+\beta). $$

If $\ell&gt;0$, then throughout the admitted interval:

$$ \begin{aligned} &\text{the positive edge }0\to1\text{ persists},\\ &\Pr_0(T_1<\infty)=1,\\ &\mathbb E_0[T_1]\in[1/u,1/\ell],\\ &\pi_1\in\left[\frac{\ell}{\ell+\kappa},\frac{u}{u+\kappa}\right],\\ &\operatorname{sep}_1=\lvert1-\lambda_2\rvert \in[\kappa+\ell,\kappa+u]. \end{aligned} $$

Proof. The coordinate block is $a e_0e_1^\ast$, whose Frobenius norm is $a$ because $a\ge0$. For $a'&gt;0$, the first transition from state $0$ to state $1$ has a geometric waiting time with parameter $a'$, hence eventual hit probability one and expectation $1/a'$. Solving $\pi P=\pi$ gives $\pi_1=a'/(a'+\kappa)$. The eigenvalues are $1$ and $\lambda_2=1-a'-\kappa$, so the separation from the stationary eigenvalue is $\lvert1-\lambda_2\rvert=a'+\kappa$. All displayed functions are monotone on the admitted interval, yielding the endpoint bounds. $\square$

If $\ell=0$, the block theorem alone does not certify edge persistence, almost-sure hitting, or a finite upper hitting-time bound. A localized bound may make $\ell$ positive when the global bound does not, as in the two-state control registered in Section 6.3.

Boundary 5.2 (Partition and threshold discipline). Corollary 5.1 is not available after continuously rotating the coordinate projectors: the result is still an orthogonal Hilbert frame but no longer a set-valued partition of the Markov states. Extension 3.1a remains algebraically and metrically valid for a bounded oblique carrier resolution, but such a resolution is even further from a state partition and therefore does not restore the Markov probability interpretation. Also, for any positive threshold $\tau$,

$$ 0<a\le\tau $$

is a positive Markov edge that remains inactive under the block-strength policy. Positive support and thresholded block activity are different types.

Supplementary Technical Note S1

The separately packaged note proves one nontrivial proportional-row preservation contract for finite chains: expected hitting time decreases and stationary target mass increases as the direct target probability grows, while the absolute spectral gap need not be monotone. The classification of general preservation contracts remains open.

Exact Controls and Bounded Realizations

The manuscript proofs are theorem authority. The computational records are registered in two distinct layers. The exact layer uses integer and \texttt{Fraction} arithmetic and literal exact-zero semantics. The bounded layer uses the frozen Python/NumPy float64/complex128 protocols stated below; its records are observations for declared fixtures, not proofs of the general theorems.

Exact rational controls

The exact certificate records one-axis equality witnesses for all three global constants, the fixed-global-data localization family of Theorem 3.4, the carrier-information lattice and its strict hostile pairs, an exact three-axis witness for Corollary 3.9, a localized-bound equality witness, and two-sided realizations of the unresolved interval. All matrix entries are rational, and the registry records the integer/Fraction backend, norm protocol, threshold policy, and exact-zero policy. It is a finite certificate for the declared fixtures, not a second proof of the general theorems.

The frozen exact protocol uses Python $3.13.0$, Python integers and \texttt{fractions.Fraction}, exact Frobenius norms through squared entries, and exact witness identities for the operator-norm fields. Its threshold fixture uses $n=b=\tau=1$, with $\mathsf{UNRESOLVED}$ defined by $n-b\le\tau&lt;n+b$. Zero means literal rational equality to $0$; no numerical tolerance or near-zero substitution is admitted.

Rubik simultaneous-transport hostile control

The Rubik control uses the canonical $228$-dimensional cubie carrier, all $18$ labelled generators, and $9\times9$ ordered sector coordinates. It tests simultaneous transport of operators and labelled carriers against the hostile fixed-frame control. It does not serve as a general perturbation distribution or Markov example. The frozen bounded protocol uses Python $3.13.0$, NumPy $2.1.3$, SciPy $1.18.0$, complex128/float64 arithmetic, Frobenius block norms, spectral operator norms, Frobenius projector deltas, support thresholds $10^{-10}$, $10^{-6}$, and $0.05$, activity threshold $0.05$, transport residual tolerance $10^{-10}$, and perturbation comparison tolerance $2\times10^{-10}$. A localized/global ratio at most the latter tolerance is reported as zero; a threshold-inactive coordinate is not thereby an exact-zero certificate. The generic near-zero control uses a separately declared error radius $10^{-9}$.

At the largest declared perturbation level, $\varepsilon=\eta=0.05$, the following table reports the coordinatewise ratio $b_{\rm loc}/b_{\rm glob}$ only where $b_{\rm glob}&gt;0$. The two bound maxima are separate maxima over all coordinates; their quotient is not used as a coordinatewise statistic.

axis	positive-global coordinates	zero ratio	median ratio	mean ratio	maximum ratio
operator	$1458$	$1440$	$0$	$0.01235$	$1.00000$
sector	$576$	$408$	$<10^{-15}$	$0.11972$	$0.70711$
coupled	$1458$	$1290$	$0$	$0.03553$	$0.77006$

The corresponding bound maxima and aggregate margin transitions are:

axis	$\max b_{\rm glob}$	$\max b_{\rm loc}$	aggregate unresolved, global $\to$ local
operator	$0.05000$	$0.05000$	$24\to1$
sector	$0.14136$	$0.09996$	$13\to1$
coupled	$0.19490$	$0.15008$	$24\to1$

Thus the localized theorem is not merely a modest improvement of the largest global scalar. In the operator-only control, only the $18$ coordinates hit by the rank-one generator errors retain ratio one; $1440$ other coordinates have the same positive global budget but localized bound reported as zero under the frozen comparison tolerance. On the coupled axis every coordinate is strictly improved, and the aggregate unresolved census drops from $24$ to $1$. These are bounded float64 observations for the declared carrier and perturbation family, not universal distribution laws.

Markov portability and cross-layer example

The Markov record is deliberately narrower. It uses the two-state singleton partition to illustrate how a certified block interval lifts to positive-edge, hitting-time, stationary-mass, and eigenvalue-separation statements under the additional hypotheses of Corollary 5.1. Rotated Hilbert frames appear only as a cross-layer boundary: they are not state partitions, so no Markov probability conclusion is admitted. The frozen bounded protocol uses Python $3.13.0$, NumPy $2.1.3$, float64/complex128 arithmetic, block-activity threshold $0.15$, positive-support threshold $10^{-12}$, and comparison tolerance $2\times10^{-12}$. Positive support and thresholded block activity remain distinct policies; $\mathsf{UNRESOLVED}$ is not an exact-zero status.

Claim Status and Boundary

The theorem and evidence levels are:

Surface	Claim level	Registered control
Theorem 2.1, Corollary 2.2, and Boundary 2.3	Theorem/boundary	exact integer/Fraction transport fixture and Rubik simultaneous-transport hostile control
Theorem 3.1 and Extension 3.1a	Theorem	Rubik orthogonal-specialization observation; Extension 3.1a is proof-only
Theorems 3.3--3.4	Theorem	exact rational sharpness and separation controls
Definition 3.6 and Theorem 3.7	Theorem	exact rational lattice grid and hostile pairs
Theorem 3.8 and Corollary 3.9	Theorem	exact rational conditional-minimax and three-axis controls
Theorem 3.10	Theorem	exact rational equality witness
Corollary 4.1, Theorem 4.2, and Corollary 4.3	Theorem	exact two-sided hostile realizations and bounded Rubik aggregate sweep
Corollary 5.1	Theorem under its stated two-state hypotheses	bounded Markov portability example

Supplementary Technical Note S1 has its own theorem/boundary ledger and local validator under experiments/paper25/notes/. It is part of the Paper XXV release package but not part of the main theorem numbering above.

No claim is made that aggregate block profiles reconstruct the labelled operator family or its representation; that global norms determine carrier localization; that thresholded activity equals exact support; that an orthogonal Hilbert frame or oblique carrier resolution is a set-valued state partition; or that a one-block perturbation estimate controls arbitrary-depth routed composition. Theorem 3.8 is conditional on fixed carrier geometry and does not determine the joint minimax envelope from $\delta_i,\delta_j,\varepsilon_{a,F},\lVert Y_a\rVert_2$ alone across all sectorizations. The Rubik table is a bounded float64 observation for one declared carrier and perturbation family, not a universal distribution law; the Markov observation is confined to the stated two-state model.

Conclusion

Aligned unitary transport preserves the complete typed block profile because the represented operators and marked carriers move together. After alignment, the change of one generator-resolved block separates exactly into left-carrier motion, additive operator error, and right-carrier motion. The resulting localized bound retains information that every portable global norm budget forgets.

These results make that information loss quantitative. The global coefficients are sharp one axis at a time. In the fixed-projector additive-error submodel, global norms cannot detect localization and the global/semi-local/local information lattice has an exact minimax envelope at every node. For fixed carrier motion, additive error admits an exact conditional completion, while the localized triangle bound has a complete equality criterion. The threshold interval left unresolved by a certified radius is complete: the admitted information genuinely permits realizations on both sides of the threshold.

The algebra extends beyond orthogonal sectors to bounded oblique carrier resolutions, with explicit conditioning factors. Application semantics do not extend automatically. In particular, the two-state Markov lift requires a genuine singleton state partition, and the one-block theorem does not replace the image--kernel or survivor tests needed for routed composition.

Supplementary Technical Note S1 supplies one positive n-state preservation contract for hitting time and stationary mass together with a negative absolute-gap boundary. It does not close the broader classification problem.

Open Problems

The established boundary leaves the following questions open:

1. Joint oblique minimax geometry. Determine the exact information lattice and minimax envelopes when both carrier resolutions are oblique and only projector-condition and operator-norm budgets are declared.
2. Incidence-locus stability. Relate principal angles, small singular values, and carrier perturbations to quantitative distance from ${(A,B):AB=0}$ without identifying near incidence with exact route failure.
3. Productwise propagation. Establish conditions under which one-block bounds compose along a finite routed product or Paper XX survivor recursion without exponential loss.
4. Semantic lifts. Supplementary Technical Note S1 establishes one proportional-row contract for hitting time and stationary mass. Classify broader preservation contracts under which block intervals imply probability, mixing, or channel statements for finite Markov partitions and open quantum channels.

Appendix A: Computational Artifacts {.unnumbered}

All listed artifacts are available in the RIME repository under \path{experiments/paper25/}; short paths below are relative to that directory.

Surface	Role	Short path
diagnostic sources	transport, localized perturbation, and two-state Markov controls	\path{*.py}
retained evidence	exact certificates, bounded observations, and receipts	\path{results/}
validators	exact replay, producer replay, claim alignment, and release closure	\path{validation/}
evidence metadata	claim/evidence alignment and exact release inventory	\path{claim-surface-map.json}; \path{release-manifest.json}
Supplement S1	proportional-row Markov theorems, hostile matrices, and bounded audit	\path{notes/proportional_markov_semantic_lift/}
partial Lean surface	typed transport and scalar-margin declarations	\path{lean/}

The exact and bounded-observation receipts have separate scopes and neither occurs in its own transitive closure. They record local closure verification, not independent validation or proof of the manuscript theorems. Reproduction commands and exact artifact paths are maintained in the experiment README and release manifest.