Observation-Determined Normal Forms: Stability, Obstructions, and Refinement in Constraint and Rewrite Systems

Keywords: abstract rewriting, observation-determined normal forms, observational confluence, error bounds, metric subregularity, boundary reconstruction, refinement towers, local repair.

Mathematics Subject Classification (2020): Primary 68Q42; Secondary 49J53, 68Q25, 68Q85, 94B05.

ACM CCS (2012): Theory of computation \(\rightarrow\) Rewrite systems; Logic and verification; Complexity classes. Computing methodologies \(\rightarrow\) Theorem proving algorithms.

Introduction

A confluent rewrite system has a familiar geometry: paths may split, but paths from the same source eventually rejoin. In many applications the source itself is hidden. What is fixed is an interface, input, conserved sector, sensor record, or boundary condition. The relevant question is then not \[x\to^*y,\quad x\to^*z\quad\Longrightarrow\quad y\downarrow z,\] but rather \[B(x)=B(x')\quad\Longrightarrow\quad \text{the settled observable outputs of }x\text{ and }x'\text{ agree}.\] The two statements have different proof obligations. Confluence is a property of paths. Observable reconstruction is a property of the consistent set and of the observation map on that set.

We use observation-determined normal form for a settled state determined through an observation map on the consistent set. This is distinct from the control-theory use of “observable normal form” for a state-space form obtained by coordinate transformation, including in Boolean control networks [source].

Let \(Q\) be a state space after quotienting representation changes that are regarded as silent, let \(C\subseteq Q\) be the consistent states, and let \[B:Q\longrightarrow\mathcal B\] be protected observable data. For \(b\in\mathcal B\), set \[C_b:=C\cap B^{-1}(b).\] The three possibilities \[|C_b|=0,\qquad |C_b|=1,\qquad |C_b|>1\] mean, respectively, that the observation is unrealizable, reconstructing, or ambiguous. Termination and confluence do not alter this trichotomy. They control how a computation reaches \(C\); they do not determine how many points of \(C\) share one observation.

The distinction becomes more consequential under perturbation. Exact injectivity of \(B|_C\) says nothing about conditioning. A sequence of finite systems may have injective observation maps at every stage and nevertheless become arbitrarily unstable under refinement. To capture the missing information, we use two moduli:

1. an error-bound modulus carrying a constraint residual to distance from \(C\);

2. an inverse-observation modulus carrying observation discrepancy within \(C\) to state discrepancy.

The first is an error-bound or metric-subregularity object in the sense of variational analysis [sources]. The second is the modulus of continuity of the inverse of \(B|_C\). Both notions are standard. Their use as an observation-determined normal-form certificate for constraint and rewrite systems yields the uniform and multi-stage refinement calculus developed here.

Main contributions

The manuscript has six technical contributions.

1. Correct exact interface. We state a canonical partial normalizer and a boundary-indexed unique-normal-form theorem with the empty-fiber case separated from weak normalization. An audited-terminal theorem further separates computational quiescence from semantic consistency. Identification modulo a silent equivalence is characterized exactly by cross-source endpoint agreement. For a separately declared finite scheduler, we also distinguish the static fiber question from positive first hit, almost-sure first hit, and endpoint-class selection. The exact confluence implications are explicitly treated as classical.

2. Sharp observational stability. For a residual \(\Phi\) and an observation map \(B\), we define intrinsic moduli \(\eta_\Phi\) and \(\omega_B\). For arbitrary residual levels \(\delta_x,\delta_y\), \[\begin{equation} \label{eq:intro-master} \begin{split} d_Q(x,y)\le{}&\eta_\Phi(\delta_x)+\eta_\Phi(\delta_y)\\ &+\omega_B\!\left( d_{\mathcal B}(Bx,By)+L_B[\eta_\Phi(\delta_x)+\eta_\Phi(\delta_y)] \right). \end{split} \end{equation}\] At zero residual the bound is exact by definition. Across all finite metric systems, the coefficient \(2\) in the equal-residual specialization cannot be lowered.

3. Uniform refinement towers. One-step approximate naturality follows from consistency and observation defects. We distinguish same-level receipt-conditioned agreement from genuinely cross-level comparison. An anchor-optimized theorem compares independently computed outputs at arbitrary refinement levels after restriction to a common level. Compatible nested inputs leave only the intervening defect segment, and summable pathwise defects yield an implementation-independent projective limit. A counterexample shows that stagewise injectivity is not enough without a uniform inverse modulus.

4. Repair and selection obstructions. A total exact local repair preserving an exterior collar exists if and only if the local consistency relation projects surjectively onto that collar. For a finite scheduler, four nested protected-behavior sets and their complementary cuts isolate fiber, positive-reach, almost-sure, and endpoint-selection obstructions. On the canonical OPH adaptive branch, every attempt stream stabilizes after finitely many genuine changes, pathwise weak fairness is sufficient for normality, and work conservation gives an initial-mismatch bound on the normal horizon. Counting stutter as positive attempt cost separates that bound from cumulative capacity: no single mismatch-only attempt horizon uniformly bounds all normalizing schedulers, bounded waste gives a quantitative upper horizon, and two nonisomorphic canonical finite sources make the work-conserving threshold sharp. Completeness and confluence remain explicit requirements for a consistent canonical endpoint. The underlying finite Markov, weighted-rewriting, and fair-termination facts are standard; the contribution is the source-bound protected-observation certificate, the adaptive attempt/cost classification, and their exact connection to the existing behavior-cut and public-endpoint interfaces. On a finite full-support probability space, observation-fiber resampling is the weighted orthogonal conditional-expectation projector and has a finite matrix recognition receipt. A stabilizer obstruction prevents an equivariant selector on ambiguous fibers.

5. Constructive and linear classes. Ranked functional-audit systems have singleton-or-empty fibers, terminate under every effective asynchronous schedule with an explicit update bound, and settle synchronously in dependency depth; acyclic Boolean circuits compile directly into this class. Linear quotient systems admit rank tests and singular-value bounds that instantiate eq:intro-master; finite linear dynamics yield the usual observability matrix as a special case.

6. Receipts and complexity. A standard finite Markov drift lemma turns a row-wise kernel inequality into endpoint, stationary, and expected-occupation receipts, without asserting infinite-horizon confinement under persistent noise. Machine-checked first-hit laws, exact finite-state transport, and native TwoBit, fixed-relation, and proper-target counterexamples provide separate receipts for the protected-behavior profile. For succinct Boolean systems, we record tight coNP- and \(\Pi_2^P\)-completeness classifications in this formulation; these supporting results are not part of the principal novelty claim.

Relation to existing theory

The exact theory overlaps several established areas. Abstract rewriting supplies Newman’s lemma, Church–Rosser properties, unique normal forms, and normalization theory [sources]. Rewriting modulo equivalence and confluence under invariants address quotient and relevant-state restrictions [sources]. Quantitative rewriting enriches rewrite relations themselves and proves metric analogues of classical confluence theorems [source]. Error bounds and inverse stability are mature subjects in variational analysis [sources]. Functoriality of rewriting is also established in categorical graph transformation [source].

Cumulative and nonuniform rewrite costs, derivation-height bounds, failed-attempt accounting, and self-stabilizing enabled-move convergence are established theory [sources]. The contribution below is not a new weighted-rewriting formalism. It locates the quantitative boundary on the repository’s broader chosen-site attempt semantics, where a scheduler may spend capacity on a quiescent site, and then proves the exact additional condition and finite OPH witnesses that recover a sharp bound.

The triangle inequality, error bounds, inverse continuity, section criteria, boundary-indexed unique normal forms, and the Lipschitz telescope are standard. The integrated certificate architecture combines them with an explicit anchor-optimized cross-level receipt comparison and its cofinal-limit consequence. Approximate inverse-system theory supplies the neighboring framework, while the result here is an organizing synthesis for automated reasoning, rewriting, and semantics.

Relation to existing theories

Rewriting, invariants, and quotients

The classical properties of an abstract reduction system are standard [sources]. Confluence restricted to an invariant or taken modulo an equivalence has been developed in Constraint Handling Rules and graph transformation [sources]. Those frameworks are close to the first arrow in \[\text{presentations}\longrightarrow Q\longrightarrow\mathcal B:\] they identify states that should count as the same or restrict attention to relevant states. Our observation map is the second arrow. It may be strictly coarser than equality in \(Q\), and it is required to be injective only on the consistent subset when exact reconstruction is desired.

Unique-normal-form properties weaker than confluence are well known and have been automated for term rewriting [source]. The fiber trichotomy and partial-normalizer universal property here are elementary set theory; only the reachable-normal-form and fiberwise-confluence corollaries specialize classical rewriting theory.

Quantitative rewriting versus output stability

Gavazzo and Di Florio enrich rewriting relations over quantales and derive quantitative Newman, Church–Rosser, and critical-pair theorems [source]. Here the rewrite relation remains ordinary. Distances are attached to two semantic interfaces: residual-to-solution error and inversion of observable data on the solution set. Thus the theory does not estimate a quantitative rewrite path; it estimates disagreement between settled outputs of possibly unrelated schedules or initial interiors.

Error bounds and inverse mappings

The inequality \[\operatorname{dist}(x,C)\le \eta(\Phi(x))\] is an error-bound property. In variational analysis, local linear versions are equivalent to metric subregularity or calmness of suitable set-valued mappings [sources]. The inverse-observation modulus is a global finite or compact analogue of inverse Lipschitz/calmness information. Our use of these notions is intentionally transparent: the novelty claim concerns their operational consequences for schedules and refinement towers, not their analytic origin.

Approximate inverse systems

Approximate inverse systems and almost-commuting diagrams have a substantial topological literature, including stability and resolution theorems for compacta [sources]. Those works perturb or relax compatibility of bonding data in topological or cover terms. Our setting keeps the ambient inverse system exact and instead gives pointwise metric bounds for normalizer defects, solver receipts, and protected-observation mismatch. The connection limits the novelty claim: the contribution is an explicit certificate adapted to observation-determined normalizers, not the invention of approximate inverse systems.

Static reconstruction and functional observability

Control-theoretic observability reconstructs a state, or a prescribed function of state, from an output history. Linear observability and functional observability supply rank and conditioning tests; nonlinear work characterizes when a nonlinear functional can be reconstructed from measurements and studies the quality of the inverse map [sources]. This is the closest neighboring language for the map \(B|_C\). Our setting is nevertheless static at the theorem interface: \(C\) is the settled consistent set, \(B\) may be any protected quotient observation, and the questions are how residual error and inverse conditioning control schedule disagreement and how these certificates propagate through refinement towers.

Constraints, databases, and global sections

Acyclic database schemes and positive-semiring relations provide local-to-global consistency results [sources]. Constraint satisfaction studies local consistency as an algorithmic reduction [source]. Database repair semantics distinguishes the existence of many consistent repairs from the answers invariant across them [source]. Sheaf-theoretic contextuality similarly turns failure of a global section into an obstruction [source]. Our empty-fiber certificate belongs to this broad lineage. The additional layer is the interaction with normal-form stability and refinement.

Fair scheduling and asynchronous Boolean-network fixing

Fairness is a qualitative liveness condition rather than a latency bound; Lamport formulates the general closure issue explicitly [source], while the self-stabilization daemon taxonomy separates fairness, boundedness, and enabledness as independent scheduling axes [source]. Asynchronous Boolean-network work studies fixing words and their minimal lengths, including monotone, acyclic, tree, and conjunctive families [sources]. We assign no novelty credit to weak fairness, round robin, fixing words, or quadratic fixing phenomena themselves. The source-specific contribution below is the joint machine-checked classification for the actual nonmonotone go compiler: its first emission-order cycle, taken in the order in which the compiler emits nodes, is the existing sweep and therefore has a linear exact-output horizon, its adversarial genuine accepted-step paths are order-sharp quadratic in the same emitted-node metric, and its Tower endpoint distinguishes both from arbitrary weak-fair attempts and from a separately supplied bounded-waste contract. General well-formed federations remain sharply exponential, in line with the order of the acyclic fixing-length results [source]; the source-specific delta is that the compiler’s single-consumer shape is exactly what removes the exponent.

Asynchronous functional networks

Kahn process networks provide timing-independent determinacy for networks of functional components, while generalized Kahn principles treat broader abstract asynchronous, including nondeterministic, models under their stated hypotheses [sources]. Ranked functional systems below are a finite terminating special case with residual audits. Their purpose is to give a class in which the exact boundary hypotheses, update bound, and obstruction output are explicit. They do not reprove those principles.

Finite Markov reachability and lumping

First-hitting probabilities, the decomposition into transient and closed classes in a finite chain, least nonnegative solutions of reachability Bellman equations, and strong lumpability are standard [sources]. Markov functions and intertwining provide neighboring language for law pushforward [source]. We use these facts without a novelty claim. Their role here is architectural: declared active sources and a protected observation index four separate failure modes; those modes recover the pre-existing OPH behavior cuts through explicit supported traces; and exact transport retains the active-source, target, observation, and silent-quotient data in addition to the kernel law.

Exact observable fibers

Let \(\to\) be a relation on \(Q\), write \(\to^*\) for its reflexive transitive closure, and let \(\operatorname{NF}(\to)\) be the normal forms.

The completeness hypothesis \(\operatorname{NF}(\to)=C\) is sometimes too strong: a computation may halt uniquely and then fail a separate semantic audit. The next theorem separates those two layers.

When \(T=C\), 13 reduces to the realizable part of 12. When \(T\supsetneq C\), it prevents an audit failure from being misreported as nontermination or as a consistent normal form.

Observational stability moduli

Let \((Q,d_Q)\) be a finite metric space, let \((\mathcal B,d_{\mathcal B})\) be any metric space, let \(\varnothing\ne C\subseteq Q\), and let \(\Phi:Q\to\mathbb R_{\ge0}\) satisfy \[C=\Phi^{-1}(0).\] All maxima below therefore exist. The same formulas extend to compact spaces by replacing maxima with suprema and imposing the continuity hypotheses stated in 27; the displayed Lipschitz estimates continue to assume a finite Lipschitz constant for \(B\).

Let \(L_B\) be any Lipschitz constant for \(B\) on \(Q\).

The symmetric form used most often is \[\begin{equation} \label{eq:symmetric-bound} d_Q(x,y)\le 2\eta_\Phi(\delta)+ \omega_B\!\left(\varepsilon+2L_B\eta_\Phi(\delta)\right). \end{equation}\]

Optimality and the exact observational modulus

Approximate schedule independence

A finite execution is a sequence \(x=x_0\to x_1\to\cdots\to x_n\). It is \((\delta,\beta)\)-settled if its endpoint \(z\) satisfies \[\Phi(z)\le\delta, \qquad d_{\mathcal B}(Bz,Bx)\le\beta.\] No exact observation-preservation assumption is made.

The preceding result consumes a probabilistic settling receipt. The following standard drift calculation is one finite mechanism for producing such a receipt.

Compact and uniform families

Let \(\{\mathfrak S_i\}_{i\in I}\) be a nonempty family of compact or finite metric systems with a common Lipschitz bound \(L\) for the maps \(B_i\). Define \[H(t):=\sup_i\eta_i(t),\qquad \Omega(r):=\sup_i\omega_i(r).\] We assume these suprema are finite on the ranges under consideration; this uniform boundedness is not automatic when the diameters of the systems are unbounded.

A uniform settled-output modulus is a function \(\Psi(t,r)\) such that \(\Theta_{\Phi_i,B_i}(t,r)\le\Psi(t,r)\) for every \(i\in I\).

Executable uniformity controls.

The accompanying finite proof-obligation certificate stores descriptions of both infinite counterfamilies and an executable witness constructor: for every positive integer \(m\), choose stage \(n=m\) and radius or residual threshold \(1/m\). The exact witness has observation or residual gap \(1/m\) and state error one. Since \(1/m\to0\), the quantified examples above, rather than any fixed finite prefix, prove that the family moduli do not vanish. The receipt regression-checks exact instances and records the cross-multiplied null-sequence proof schema.

Further controls describe the infinite unit-defect tower \(p_j=j,a_j=1\), whose tail from \(n\) to \(n+M\) is \(M\), and a restriction map that amplifies fine distance \(1/8\) to coarse distance \(1\) when its Lipschitz receipt is removed. A three-point parametric family independently shows why the observation maps need one common Lipschitz bound: their stagewise constants grow like \(n\), while residuals and observation gaps vanish and the state gap tends to one. The cofinal-limit corollary has separate controls for incomplete coordinate spaces, incompatible inputs, and nonvanishing solver errors. The arbitrary-depth positive telescope is the mathematical theorem above and its Lean counterpart; the family-uniform modulus and cofinal-limit quantifiers are not claimed as Lean formalizations.

Parallel composition and observation design

Naturality and refinement towers

For an observable system define its reconstructing, realizable domain by \[Q^{\mathrm{adm}} := \{q\in Q:\text{there exists a unique }c\in C \text{ with }B(c)=B(q)\}.\] Every normalizer in this section is understood to act on a chosen metric subspace of \(Q^{\mathrm{adm}}\), with the consistent set restricted to the corresponding normalizer image; every coarse map is assumed to preserve the chosen admissible subspaces. To avoid carrying restriction notation through every formula, we relabel those subspaces and restricted consistent sets as \(Q_s,Q_r,Q_n\) and \(C_s,C_r,C_n\). Thus \(B_n|_{C_n}\) is injective on the declared domain. Obstructed and ambiguous fibers are governed by 3; they are not silently assigned a finite normalizer value in the metric estimates below.

Let \[\mathfrak S_s=(Q_s,C_s,\mathcal B_s,B_s,\Phi_s), \qquad \mathfrak S_r=(Q_r,C_r,\mathcal B_r,B_r,\Phi_r)\] be finite or compact observable systems on the admissible domains declared above. Their normalizers \(N_s\) and \(N_r\) are therefore total on these domains. Let \(f:Q_s\to Q_r\) be a coarse state map and \(h:\mathcal B_s\to\mathcal B_r\) a coarse observation map.

Accumulation through arbitrary depth

Let \[Q_0\xleftarrow{\rho_{1,0}}Q_1 \xleftarrow{\rho_{2,1}}Q_2\xleftarrow{}\cdots\] be an inverse sequence of nonempty metric spaces with Lipschitz restriction maps, with \(\rho_{m,n}=\rho_{n+1,n}\circ\cdots\circ\rho_{m,m-1}\). Let \(N_n:Q_n\to Q_n\) be normalizers. This subsection deliberately restarts with general metric spaces; the finite-or-compact ambient assumption used for the preceding uniform-family examples is not retained. Completeness is imposed only where a Cauchy estimate is converted into a coordinate limit. Assume the following pathwise and uniform one-step normalization defects are finite: \[\alpha_n(q) := d_n(\rho_{n+1,n}N_{n+1}q,N_n\rho_{n+1,n}q), \qquad a_n:=\sup_{q\in Q_{n+1}}\alpha_n(q).\] For \(j\ge n\), write \[K_{j,n}:=\operatorname{Lip}(\rho_{j,n}),\qquad K_{n,n}=1.\]

Combining 34 with 38 gives an explicit tower certificate in terms of residual, observation, and Lipschitz data at each stage.

For \(n\le h\le r\) and \(q_r\in Q_r\), define the amplified pathwise defect segment \[\mathsf A_{r,h}^{(n)}(q_r) := \sum_{j=h}^{r-1} K_{j,n}\, \alpha_j\!\left(\rho_{r,j+1}q_r\right),\] with an empty sum equal to zero. By definition of \(a_j\), \[\mathsf A_{r,h}^{(n)}(q_r) \le \sum_{j=h}^{r-1}K_{j,n}a_j.\]

The uniform hypotheses used earlier imply \(e_m^{(i)}\to0\) whenever \(\delta_m^{(i)},\beta_m^{(i)}\to0\), the observation maps have a common Lipschitz bound, and \(H(t),\Omega(t)\to0\) at zero. Uniform summability of \(\sum_j a_j\) implies the pathwise summability condition. Thus raw receipt parameters alone are not enough: uniform conditioning and the amplified restriction factors are load-bearing.

Exact inverse limits

Assume that all one-step defects vanish and that \(\rho_{n+1,n}(C_{n+1})\subseteq C_n\). Let \[Q_\infty:=\varprojlim Q_n =\{(q_n):\rho_{n+1,n}q_{n+1}=q_n\},\] and similarly let \(C_\infty=\varprojlim C_n\). Here \(Q_\infty\) is the inverse limit of the admissible domains fixed at the start of the section. It is the full ambient inverse limit only when every observation value is reconstructing and realizable.

Repairability and selection obstructions

The existence of a unique consistent state with a given observation does not imply that a prescribed local write support can construct it. This section separates extension, uniqueness, and repair support.

Let \(X_W\) be an interior write space, let \(X_D\) be a protected collar space, and let \[R\subseteq X_W\times X_D\] be the locally consistent relation.

If \(R=\varnothing\) while \(X_W\times X_D\ne\varnothing\), nonexistence of a repair is immediate; when the domain is empty, the unique empty map is a vacuous repair. The next proposition assumes \(R\ne\varnothing\) because ordinary real-valued distance to the empty set is convention-dependent; an extended-distance formulation may instead assign that case margin \(+\infty\).

The definition prevents a one-site no-go from being misreported as a global inconsistency: increasing the transaction support may restore surjectivity.

A protected-behavior first-hit profile

This section adds stochastic data without identifying it with the static repair question. Let \(Q\) be finite and let \(K:Q\times Q\to[0,1]\) be a row-stochastic kernel. Fix a protected carrier \(P\subseteq\mathcal B\). For every \(b\in P\), let \(I_b\subseteq Q\) be a nonempty set of declared active sources and put \(T_b=C_b\). Let \(\approx_b\) be an equivalence relation on \(T_b\). The first entrance time \[\tau_b:=\inf\{n\geq0:X_n\in T_b\}\] includes time zero. For \(q\in I_b\) and \(c\in T_b\), define the endpoint mass \[h_b(q,c):=\Pr_q[X_{\tau_b}=c,\ \tau_b<\infty]\] and its total \(h_b(q)=\sum_{c\in T_b}h_b(q,c)\). These are determined by the increasing finite-horizon sums of first-hit path weights; no arbitrary initial probability law is part of the qualitative definition.

The all-source quantifier here means all members of the declared \(I_b\), not all states of \(Q\). Positive reach, almost-sure hit, and endpoint selection are different predicates. In particular, \(h_b(q)=1\) supplies neither a rate nor an expected hitting-time bound, and singleton support modulo \(\approx_b\) need not mean literal endpoint equality.

Consequently, variant comparison is coordinatewise: \((K_0,K_1,K_2,K_3)\leq(K'_0,K'_1,K'_2,K'_3)\) when every coordinate is included. This is a product preorder, not a scalar score or total order.

Finite observation-conditioned resampling

There is nevertheless a canonical averaging operation once a reference measure is supplied; unlike a section, it does not select one state from an ambiguous fiber.

Symmetry and hidden tie-breaking

Let a group \(G\) act on \(C\) and \(\mathcal B\), and assume \(B|_C\) is equivariant.

Ranked functional-audit systems

We give a broad constructive class in which the partial normalizer, asynchronous dynamics, and obstruction output are all explicit.

Write \[Q=\prod_{v\in V}X_v, \qquad B:Q\longrightarrow\prod_{v\in V_0}X_v\] for the full state space and the boundary-coordinate projection. For a boundary value \(b\in\prod_{v\in V_0}X_v\), define \(E(b)\) recursively by rank.

Define recursively \[M(v)=0\quad(v\in V_0), \qquad M(v)=1+\sum_{u\in P(v)}M(u)\quad(v\notin V_0).\]

Linear quotient systems and observability

Let \(V,W,Z\) be finite-dimensional real inner-product spaces. Let \[A:V\to Z, \qquad C=\ker A, \qquad G\subseteq C\] be a gauge subspace. Let \(B:V\to W\) vanish on \(G\), so it descends to \(V/G\). Equip \(V/G\) with the quotient norm.

Let \[H=C\cap G^\perp.\] Under the injectivity hypothesis, \(B|_H\) is injective. Let \[\kappa_A:=\|A^\dagger\|, \qquad \kappa_B:=\|(B|_H)^\dagger\|,\] where \(\dagger\) denotes the Moore–Penrose pseudoinverse. Thus, when \(H\ne\{0\}\), \(\kappa_B\) is the reciprocal of the smallest singular value of \(B|_H\); on the zero space the pseudoinverse convention gives \(\kappa_B=0\). The constant \(\kappa_A\) is the global linear error-bound constant for \(C=\ker A\).

The finite-field version retains the rank criterion but not singular values. It is the quotient form of an information-set condition from coding theory [source].

Finite linear dynamics

Let \(T:U\to U\) be a linear update, let \(S:U\to Y\) be a sensor, and let \(G\subseteq U\) satisfy \(T(G)\subseteq G\) and \(S(G)=0\). Equip \(Y^{k+1}\) with its product inner product. The length-\(k\) observation map is \[\mathcal O_kx=(Sx,STx,\ldots,ST^kx).\]

This is the classical observability matrix viewed through the quotient-boundary language; no novelty is claimed for the control-theoretic rank test [source].

Complexity of succinct boundary problems

The finite theory is elementary when \(Q\) and \(C\) are listed explicitly. It becomes computationally nontrivial when consistency is given by a Boolean circuit or formula.

Examples and separation results

Rule 90 as a regression test

No cellular automaton is used in the general theory. Rule 90 is retained because a three-cell instance simultaneously exhibits a hidden kernel, a proper information boundary, a deficient boundary, and a reverse-repair obstruction.

Over \(\mathbb F_2\), define \[F(a,b,c)=(b,a+c,b).\] A one-step spacetime record is a pair \((s,t)\) satisfying \(t=F(s)\).

The width-three example is a regression test, not a rule selected by the theory or a load-bearing assumption.

Application: set-valued public-capacity readback

The empty/singleton/ambiguous fiber distinction has a direct application to finite observer-record capacity. This subsection records the application; it does not make capacity an implication of confluence. Fix a supplied positive carrier dimension \(D\), a trial system with terminal physical quotient fiber \(\widetilde\Omega_{r,D}\), and a separately constructed functional \[M_\varepsilon:\widetilde\Omega_{r,D}\longrightarrow\mathbb N_{>0}.\] In the OPH capacity application, \(M_0(q)\) is the independence number of the compound confusability graph built from reachable compatible public record atoms and globally coupled checkpoint kernels. That construction, its publicness policy, and its physical carrier are external data for the present normal-form theorem.

Define the capacity readback fiber \[\mathfrak F_{r,\varepsilon}(D)= \{M_\varepsilon(q):q\in\widetilde\Omega_{r,D}\}.\]

If an implementation isomorphism preserves observer lineage, record atoms, interface atom maps, reachable public sections, publicness policy, every global checkpoint kernel, and capacity-carrier projection ranks, then it transports the compound confusability graph and every approximate decoder. The functional \(M_\varepsilon\) and the readback set are therefore invariant under that full packet isomorphism. A normal-form hash, stagewise singleton, or implementation relabeling alone supplies no such theorem. A refinement-limit statement additionally requires uniform stable separation or an explicit capacity-packet naturality receipt, in accord with the general refinement results above.

Discussion

Why the factorization is useful

The estimate \[\text{residual}\xrightarrow{\ \eta\ } \text{distance to consistency} \xrightarrow{\ \omega\ } \text{observable uniqueness}\] separates two burdens that are often mixed. A rewrite algorithm, numerical solver, or physical implementation controls \(\eta\) by showing that a small residual means closeness to \(C\). The observation architecture controls \(\omega\) by showing that \(B|_C\) is well conditioned, a property strictly stronger than injectivity of \(B|_C\). Critical-pair arguments may help establish exact completion or small residuals, and they are not required in the final output comparison.

Why finite uniqueness is not enough

In a refinement program, proving \(B_r|_{C_r}\) injective for every \(r\) does not prevent the inverse condition number from diverging. 29 is deliberately minimal. The correct limit-facing statement is uniform stable separation, not a list of stagewise singleton claims. This is the main stability requirement emphasized for refinement-sensitive applications.

The protected first-hit profile has the same discipline at a different interface. Finite hitting, closed-class, Bellman, and lumpability arguments are standard; the OPH-specific step is to assemble them around protected observations and declared sources, recover the existing behavior cuts, and retain every source, target, quotient, and scheduler premise during transport.

A declared scheduler class and its capacity split

The machine-checked development fixes one declared admissible scheduler class on the two-bit repair carrier. A scheduler in the class is a finite Markov kernel whose supported steps do not increase a declared defect mismatch, write at most one of the two registers, and spend at most one unit of expected declared capacity per step. Reading the observed register as the shared network record and the defect flag as a private register splits each step cost into a shared and a private part and cuts out two sub-classes: the self-update-only schedulers, which never write the shared register, and the network-update-only schedulers, which never write the private one. Three results hold on this carrier. Under every scheduler in the class the defective fiber is unreachable from every consistent state, while an explicit mismatch-violating kernel outside the class reaches it with probability one. The fiber with the shared register flipped is unreachable under every self-update-only scheduler and is hit with probability one by an explicit network-update-only member, so a positive hit prices at least one shared write. The almost-sure hitting hypothesis of the fine two-bit fixture is discharged by the declared repair kernel, and the in-class identity scheduler witnesses that the class constraints do not force it. The class is a declaration on this finite carrier: the locality and capacity clauses are typed and unconsumed by the obstruction theorems, and no scheduler-selection claim is made.

Adaptive canonical repair: stutter, fairness, and endpoint promotion

The declared finite Markov profile above and the repository’s canonical adaptive repair answer different questions. Let \(C\) be a finite OPH carrier, let \(\sigma\) choose one site from the active record at each attempt, and write \[x_n=\operatorname{adaptiveRun}_C(n,\sigma,x_0), \qquad r_n=\operatorname{mismatchCount}(x_n).\] An attempt may stutter when the chosen site has no effective repair. Every nonstuttering attempt is an accepted repair step and strictly lowers \(r_n\), while every stuttering attempt leaves the record unchanged.

The machine-checked provider also proves that every \(x_n\) is reachable from \(x_0\) by accepted repair steps. A downstream endpoint theorem then adds the separately named completeness and confluence hypotheses: every pathwise weak-fair adaptive run reaches the same raw endpoint as the canonical repair, and therefore the same public fixed-point object. This promotion supplies no stochastic kernel, active-source law, hitting probability, rate, physical scheduler, or refinement theorem.

Cumulative attempt capacity and sharpness

Every scheduler invocation receives one attempt unit, including an equality stutter. Let \(A_N=N\) be this cumulative attempt cost, let \(G_N\) count the genuine changes among the first \(N\) attempts, and let \(S_N=N-G_N\). Thus \(A_N=G_N+S_N\); the initial mismatch rank controls \(G_N\), but it does not control \(S_N\).

Under the same budget and waste premises, the downstream Tower theorem adds the separately named completeness and confluence hypotheses and returns the existing canonical raw and public fixed-point endpoint at an index \(N\le B\). The capacity is a mathematical count of scheduler invocations. It is not a physical clock, rate, energy, bandwidth allocation, hardware quota, fee, or stochastic policy.

Fixed computation federations

The computation compiler has a second, distinct asynchronous surface. For a Boolean formula \(\varphi\) on \(k\) inputs, the generated node list \(\operatorname{go}(\varphi,k)\) is independent of the input. Its registers start at \(k\); the initial state supplies the lower input ports. A canonical accepted step chooses a failing node and performs exactly that node’s declared write. The historical patch-frame relation remains unchanged and is not used as the accepted relation.

Read the node defects in increasing register order, with every earlier bit more significant than all later bits together. This superincreasing natural rank strictly decreases at every canonical accepted step. An upstream repair may expose several later defects: the machine-checked fanout control raises the raw failing-node count from one to two, so ordinary defect count is not the measure. The dependency rank nevertheless falls, and the accepted relation is well founded without a fairness premise.

The fairness premise removes attempt stutter; it does not prove accepted-step termination. Its positive singleton instance and a root-only starvation schedule inhabit the two sides. The broader historical relation admits a sharper one-member control: the only member and site are selected at every attempt and remain continuously enabled, yet the run stutters forever outside consensus because the relation does not require the declared repair to execute. Thus fairness cannot compensate for a step semantics that counts a no-op.

The bounded-waste counting theorem is shared with the canonical adaptive repair route through a neutral ranked-attempt interface. Each domain proves its own run and quiescence equivalence, and each Tower receiver calls the shared theorem. This is code and proof reuse, not a claim that adaptive scheduler waste and fixed-compiler repair work are the same scientific quantity.

The endpoint statement uses the immutable input prefix as the protected observation and output-register equality as the endpoint equivalence. Unused natural-number registers are not identified. The selected round-robin object is a mathematical function, not a physical scheduler source. No physical device, finite PublicWorld presentation, clock, rate, energy, bandwidth, fee, hardware realization, or raw-state uniqueness is constructed.

Limitations

The theory assumes a declared meaningful quotient, consistent set, residual, observation metric, and refinement map. It does not select those objects. In particular:

• a normalizer is not a probability measure;

• an inverse-limit normalizer is not a continuum field theory;

• a finite update bound is not a wall-clock liveness theorem;

• a boundary pseudometric does not acquire additional semantics without a separate interpretation theorem;

• a small Rule-90 example does not establish cellular-automaton universality or generality.

• a finite protected profile is relative to one declared scheduler and active-source family; it gives no rate, expected hitting time, mixing bound, stationary law, infinite-state extension, or refinement-tower conclusion; results quantified over the declared admissible class are relative to that class declaration and to the two-register capacity-split reading;

• canonical adaptive repair is eventually constant without a fairness premise, but normality requires pathwise weak fairness and a canonical consistent endpoint requires completeness and confluence;

• cumulative attempt capacity counts every chosen-site invocation, including equality stutter; bounded waste and the budget are supplied premises, weak fairness alone gives no uniform mismatch-only finite attempt bound, and the sharp lower bound is family-specific rather than universal over all carriers;

• fixed-federation pathwise weak fairness is exactly stable consensus only relative to one attempt run and the source-derived eventual-constancy theorem; its reverse is vacuous after stable consensus and does not give recurrence, weak fairness alone has no uniform finite attempt horizon, the accepted-step result is order-sharp rather than constant-sharp, and bounded-waste attempts have no physical unit;

• coordinatewise cut comparison is not a scalar score, total order, or family-wide optimum, and exact finite transport is not implementation or product refinement.

These exclusions are necessary for both mathematical correctness and journal review.

Data availability

The repository contains the PDF, LaTeX source, bibliography, reproducible build receipt, archive manifest, and SHA-256 checksums. The mathematical proofs in this manuscript are self-contained. No empirical dataset is used.

Conclusion

Confluence is about paths from one state. Observable normalization is about what partial data determine after consistency is enforced. Their exact intersection is classical unique-normal-form theory: under observation preservation and \(\operatorname{NF}(\to)=C\), identification modulo any silent equivalence is equivalent to agreement of normal endpoints across all sources with the same observation. This does not by itself provide normalization or all-schedule settlement. Their stable intersection is governed by two analytic objects: an error bound to the consistent set and an inverse modulus for the observation map on that set.

On the canonical OPH adaptive branch, finite genuine-change stabilization is source-derived rather than postulated as a scheduler property. The remaining steps are deliberately separate: pathwise weak fairness supplies normality, while completeness and confluence promote the stable state to one consistent canonical raw and public endpoint.

On the separate fixed compiler branch, the emission-order round-robin schedule reaches the exact output in one \(n\)-attempt cycle, arbitrary genuine accepted order has an order-sharp \(\Theta(n^2)\) worst-case family, weak fairness alone has no uniform finite attempt horizon, and bounded waste supplies the general conditional attempt bound. These four statements classify distinct scheduler and cost regimes rather than attaching a physical duration to an attempt.

The resulting estimate is universal and sharp in its two unavoidable components. It survives refinement: one-step defects telescope, same-level solver agreement needs no tower term, and compatible cross-level computations differ by their receipt errors and the intervening normalizer-defect tail. For a compatible admissible input in complete spaces, nonexpansive restrictions, vanishing receipt errors, and summable pathwise tails give a common projective limit; exact compatibility recovers the inverse-limit normalizer. On finite probability spaces, fiber averaging supplies the canonical conditional-expectation projector, while its matrix receipt keeps that averaging distinct from representative selection or a spectral-gap claim. Empty fibers and ambiguous fibers stay visible; a repair operator does not absorb them. The theory therefore supplies the determination and stability question, a four-stage protected-behavior ladder, and an orthogonal local-support question: \[\boxed{ \begin{array}{c} \text{Does a consistent extension exist?}\\ \text{Is it observably unique and stable?}\\ \text{Does some declared source reach it with positive probability?}\\ \text{Do all declared sources reach it almost surely?}\\ \text{Do first-hit endpoints select one silent-equivalence class?}\\ \text{Can the declared local support construct it?} \end{array}}\] No one of these questions answers the others. Keeping them separate is the central mathematical lesson and the reason the framework applies across rewrite, reconstruction, and distributed-inference models.

Statements and Declarations

Funding declaration: The authors declare that no external grants or dedicated third-party research funding were received for the preparation of this manuscript.

Author af f iliations declaration: Bernhard Mueller is af f iliated with Pragma Research Inc. No institutional af f iliations were declared for David Matscheko or Jonathan Hill in this submission. The corresponding-author email address is provided in the correspondence declaration.

Consent to Participate declaration: not applicable.

Consent to Publish declaration: not applicable.

Author Contribution declaration: Bernhard Mueller developed the observation-determined normal-form framework, the stability, refinement, repair, and complexity results and the manuscript text. David Matscheko and Jonathan Hill contributed proof review, checking of the exact and stability results, and consistency review of the manuscript. All authors reviewed and approved the manuscript.

Correspondence declaration: Correspondence should be addressed to Bernhard Mueller, bernhard@floatingpragma.ai.

Data Availability declaration: No empirical dataset is used in this work. The LaTeX source, bibliography, reproducible build receipt, and SHA-256 checksums are available through the project repository.

Ethics declaration: not applicable.

Competing Interests declaration: The authors declare no competing financial or non-financial interests for this manuscript.

Use of AI-assisted tools declaration: AI-assisted tools were used for language editing, drafting assistance, formatting support, and submission-package preparation. The authors reviewed and remain responsible for all scientific content, claims, references, and declarations.

AI Assistance Disclosure

This research project used research-grade commercial models, including Anthropic’s Fable and OpenAI’s GPT-5.6-Sol, for research support, software development, editing, and synthesis. The authors are responsible for the paper’s claims, methods, and final text.