Devin Bostick

What can a bounded system justify from its evidence, establish within its resources, and safely claim about its own verification? This paper separates identification, hypothesis generation, computation, implementation and correspondence, giving each boundary explicit hypotheses and distinguishing impossibility from an unfinished search. A claim is determined by available evidence exactly when it is constant on the evidence fibers. A generator computed from that evidence, including randomized generation with an independent seed, cannot distinguish situations the evidence identifies. These identification limits remain separate from computational attainment: a semantically determined answer need not be obtainable within a declared machine and resource contract. Finite, effectively presented classes support exact attainment constructions and feasible capability sets; counterexamples show why a single greatest bounded evaluator need not exist. The paper develops exact adaptive query and relational-selection results, resource-accounted finite-machine and revision constructions, and distinct representations of posterior probabilities, environmental stochastic assumptions and randomized worst-case guarantees. Finite guarantee families admit canonical antichain representations, while sequential composition requires additional structure. Self-inspection cannot settle a property that differs between evidence-equivalent possibilities, although particular self-properties remain verifiable under explicit contracts. These results support a typed routing protocol. Accepted certificates justify only their stated conclusions; simultaneous obstruction causes remain visible; and interrupted or unaffordable diagnosis returns unresolved rather than an impossibility claim. Probabilistic continuation requires additional declared premises and is never an automatic fallback. The contribution is a mathematical specification of scoped verification limits and attainable responses. Implemented execution, physical correspondence and general justified recursive improvement retain their separate obligations.

What can a bounded observer distinguish, discover, construct, justify, adopt, preserve and revise with the evidence, authority and resources actually available to it? This work proposes the bounded-observer lens: treating these capacities as a connected object of analysis rather than assessing each in isolation. The starting distinction is between information an observer lacks and information it possesses but cannot afford to extract or use. Further distinctions separate an adequate representation’s existence from its accessible discovery, certification, construction and adoption. The account follows these dependencies through action, interruption and revision, including the work needed to sustain the observer’s own methods. Drawing on scoped mathematical results and worked examples, the book explains how informational limits, computational costs, retained obligations and authority interact. It applies the lens to measurement, AI systems, institutions and biological perception while distinguishing epistemic claims from claims about reality’s ultimate nature. It also examines how the research process can apply the same standards to its own constructions. This revised edition corrects earlier universal claims, compares the approach with related work, and identifies the remaining mathematical connections. Its central proposal is a connected account of attainable, justified capacity; the full characterization remains an open research program.

This paper studies authorized realization as a sequential process. In a finite declared system, machine motion consists of productive acquisitions under a portfolio-permission predicate. A maximal run either realizes the original demand or stalls with a computed least supported narrowing; adoption and refusal remain owner decisions outside the acquisition dynamics. Restricted permissions can make terminal offers—and even realizability itself—depend on acquisition order. Under unrestricted permission, by contrast, every maximal run produces the same terminal demand label, expressible as the join of the demand kernel and the full-pool kernel. Nevertheless, demand confluence can coexist with revelation nonconfluence: different runs may support the same verdict while acquiring different portfolios and disclosing different distinctions. The terminal demand label is a function of the terminal revelation label. Revelation confluence therefore implies demand confluence, making the opposite combination impossible. For any declared revelation-audit demand, a target-relative confluence gap gives the exact obstruction to answering that audit from verdict outcomes alone. The associated kernel intersection supplies the coarsest record preserving both verdict and audit, thereby indexing provenance requirements to the audit actually demanded.

This paper formalizes the supply side of informational repair for bounded evaluators. Given a finite carrier, a current information channel, a demanded verdict, a restricted language of acquisition channels, and a declared family of authorized portfolios, it determines whether the distinction required by the verdict is attainable. Realizability is characterized both as factorization through an authorized enrichment and as coverage of the target-obstructing pairs. The framework distinguishes exact realization from sufficient acquisition with excess revelation; separates an attained resolution from an ideal envelope that no authorized portfolio reaches; and computes the least portfolio-supported coarsening of an unrealizable demand. When authorized portfolios are incomparable, the supported coarsenings form a relaxation frontier rather than a uniquely privileged replacement. Authorization remains a declared input throughout. Informational sufficiency therefore does not imply authority, and a canonical informational repair does not imply a canonical realization. The formal outputs are exact acquisition, excess-revealing acquisition, supported demand narrowing, or refusal. Results are proved for finite declared systems; infinite, stochastic, continuous, adaptive, and general complexity questions remain open.

When may the concrete realization of a network be forgotten without changing what the network observably does? This paper studies that question for finite networks of bounded evaluators whose information states are represented by kernels and whose interfaces act by pullback. I first show that licensed disclosure and transport are one mechanism at kernel semantics: every licensed disclosure channel is kernel-equivalent to a suitable self-transport. The general closure is then characterized by a walk-meet law over all finite transport walks. However, literal paths are not the canonical object. Closure factors through the induced effective operator action, and all transports between an ordered pair admit a joint transport normal form whose induced operator is canonical even when its component realization is not. This yields a collapse hierarchy. Under suitable coherence, path detail may be forgotten down to one transported contribution per reachable ancestor; under identity transport, capability factors through bare reachability. At the intermediate level I establish fixed-profile meet-redundancy and a rigidity theorem showing that the pointwise order on partition-pullback operators admits essentially no nontrivial uniform domination. The same quotient hierarchy does not govern every observable. Revelation can vary inside a common reachability fiber. I give an exact criterion for zero-excess realization by multiple senders and prove that this generation property is independent of transport coherence. The underlying operator action is therefore common, but quotient exactness is observable-relative. The paper develops a general distinction between fine realization structure, induced action, and coarse observable quotients, and locates precisely when reachability is sufficient and when implementation detail remains irreducible.

This paper studies networks of bounded evaluators in which the interface graph itself is not fixed but may be synthesized subject to declared communication permissions. Building on prior fixed-point composition and interface-commensuration results, it asks four ordered questions: whether declared demands are realizable, what capability closure follows from a chosen augmentation, which augmentations minimally realize the demands, and how a separately declared objective may select among those realizations. For finite fixed-carrier systems with reusable full-disclosure interfaces, capability closure is shown to be exactly the meet of the initial kernels of all reachable ancestors. Consequently, capability factors through the network’s reachability preorder. Existence of a realizing augmentation reduces to a single reachability computation followed by the corresponding realizability test. The central separation is that attainability does not determine realization. Minimal realizing augmentations may form a nontrivial antichain, their intersection may realize nothing, and distinct minimal realizations may induce different excess capability even while satisfying the same declared demand. Thus feasibility and the consequences of any fixed realization are canonical, while the realizing network generally is not. Selection among realizations therefore requires an independently declared objective rather than following from capability closure itself. The paper further proves objective-relativity, gives a displayed counterexample to locally greedy interface synthesis, and separates reusable-edge confluence, relay depth, single-use order dependence, and cyclic iteration cost. The results locate a formal boundary between what network structure determines and what remains a matter of declared policy, objective, and authority. Scope is finite, deterministic, static, set-level, and fixed-carrier.

Bounded Judgment is a thirteen-lecture conceptual course through the Identity-Persistence Program, a formal research program on the structural preconditions of coherent judgment by bounded observers. Its generative question is: what licenses sameness through change? The course develops a dependency architecture in which a sufficiently declared regime fixes the distinctions relevant to a judgment; those distinctions induce identity conditions and quotient structure; identity makes admissible transformation and persistence meaningful; and only then do observables, evidence, probability, consequence, and verification acquire determinate roles. Where a demanded judgment does not factor through the available resolution, the framework separates the exact obstruction, minimal informational repair, realizability question, authorization boundary, and any declaration required to proceed. The central philosophical discipline is non-promotion. State is not object-as-judged; representation is not identity; evidence accumulation is not unique identification; mathematical sufficiency is not realization; realization is not authorization; declaration is not derivation; and verification under a declared regime is not final ontology. A bounded evaluator may produce judgments that another bounded evaluator can verify exactly, up to the capacity of the declared channels, without certifying its regime as reality’s final constitution. This position asserts neither realism nor anti-realism about what reality is “in itself”; application never required deciding that question. The standard is uniform across applied theories. Every application operates through indexed questions, distinctions, access conditions, inference rules, and resolution, whether fully declared or partly tacit. Making that index explicit neither uniquely demotes this program nor grants competing theories ontological privilege. Nor does equal treatment imply relativism: applications may differ substantially in empirical reach, predictive accuracy, robustness, resolution, reproducibility, and practical success. Worked finite examples develop equivalence, quotienting, admissible transformation, evidence collapse, factorization, minimal repair, declaration, verification, and lawful retirement. The course introduces no new theorem ownership; it provides a pedagogical and operational map of results proved elsewhere in the Identity-Persistence corpus while preserving their exact scopes and claim types.

This paper formalizes the boundary between derivation and declaration in bounded judgment. Derivation is characterized by factorization through the information quotient licensed by the current regime. When a desired output does not factor through that quotient, the failure identifies an exact informational obstruction rather than licensing arbitrary choice. At fixed-carrier set scope, the paper derives the canonical minimal refinement sufficient for a target and shows that it coincides with the resolution obtained by formally adjoining that target to the demand. For finite carriers, a specialization of an existing Minimum Repair Alphabet theorem gives the minimum auxiliary alphabet width required by any sufficient informational realization. These results distinguish four questions: which current identifications obstruct a target, what minimal refinement would remove them, how much finite discriminating capacity any realization must carry, and who is authorized to supply such a realization. A resolving declaration is therefore treated not as an alternative to reason but as an authorized introduction of distinctions unavailable to the current inferential regime. The paper separates declaration content, informational effect, quotient refinement, derivable consequence, and authority; develops recursive provenance and declaration relativity; and connects declaration to lawful retirement. Higher-order rules may relocate the boundary by supplying additional information, but cannot make the provenance of that information disappear. The resulting principle is that bounded judgment should derive what can be derived, expose rather than disguise unresolved content, declare only through authorized provenance, and verify the consequences of the resulting regime without treating operational closure as self-certifying finality.

This paper studies when distinct finite erasure orders are semantically distinguishable and, more specifically, which apparent semantic histories correspond to realizable erasure histories. Erasure orders are quotiented by equality of their trajectories under an antitone semantic map, producing a trajectory quotient represented by a layered graph of rank fibers. For antitone Galois connections, every source-to-sink path in this graph is realizable: semantic histories are exactly the paths of the rank-fiber graph. The result is driven by the meet structure induced by the Galois connection, but the underlying mechanism is more general. Backward stability of fiber partitions suffices for path exactness, Galois fibers satisfy backward stability, and both implications in the resulting hierarchy are strict. Forward stability provides a second sufficient condition, while neither stability condition is necessary. These separations distinguish algebraic structure, local lifting conditions, and global realizability rather than treating them as equivalent notions. The analysis provides a finite structural account of semantic histories under erasure while deliberately introducing no metric, probability, ranking, or optimization over those histories. The exact characterization of path exactness beyond the sufficient stability conditions remains open.

Suppose that some requirements or distinctions are withdrawn from a specification. When does their removal preserve exactly the same semantic content, and why can individually valid removal certificates fail when used together? I formulate the problem for an arbitrary antitone Galois connection. The central Endpoint Criterion states that erasing S ⊆ Q is conservative exactly when every erased item is certified by the closure generated by the final remainder: F(Q ∖ S) = F(Q) ⇔ S ⊆ G(F(Q ∖ S)). Conservative erasure is therefore determined by the remainder that actually survives, not by independent certificates issued against larger intermediate contexts. This yields a Certificate Transport law. The composite G ∘ F is a monotone closure operator, so certificates issued by a smaller remainder remain valid in larger remainders, while no corresponding transport principle holds generally in the reverse direction. Apparent failures of composition can therefore be understood as context-indexing failures: a certificate remains tied to the remainder against which it was established. The same structure distinguishes endpoint semantics from trajectory semantics. The final semantic value and the conservativity verdict depend only on the endpoint pair. If the endpoint is conservative, every intermediate value along every removal order collapses to the same value. Outside the conservative case, however, intermediate trajectories may differ; a minimal polarity provides a counterexample to unconditional history-independence. The result abstracts a retirement calculus for declared specifications while separating its generic order-theoretic component from domain-specific constraints such as forced obligation floors. No novelty claim is made for classical Galois or closure theory; the emphasis is on the semantics and typing of certificates under changing contexts.

Suppose that some requirements or distinctions are withdrawn from a specification. When does their removal preserve exactly the same semantic content, and why can individually valid removal certificates fail when used together? I formulate the problem for an arbitrary antitone Galois connection. The central endpoint criterion states that erasing S ⊆ Q is conservative exactly when every erased item is certified by the closure generated by the final remainder: F(Q ∖ S) = F(Q) ⇔ S ⊆ G(F(Q ∖ S)). Conservative erasure is therefore determined by the remainder that actually survives, not by independent certificates issued against larger intermediate contexts. This yields a certificate-transport law. The composite G ∘ F is a monotone closure operator, so certificates issued by a smaller remainder remain valid in larger remainders, while no corresponding transport principle holds generally in the reverse direction. Apparent failures of composition can therefore be understood as context-indexing failures: a certificate remains tied to the remainder against which it was established. The same structure distinguishes endpoint semantics from trajectory semantics. The final semantic value and the conservativity verdict depend only on the endpoint pair. If the endpoint is conservative, every intermediate value along every removal order collapses to the same value. Outside the conservative case, however, intermediate trajectories may differ; a minimal polarity provides a counterexample to unconditional history-independence. The result abstracts a retirement calculus for declared specifications while separating its generic order-theoretic component from domain-specific constraints such as forced obligation floors. No novelty claim is made for classical Galois or closure theory; the emphasis is on the semantics and typing of certificates under changing contexts.

What structural form must an answerable world take for a bounded evaluator if judgments of persistence are to be meaningful, non-arbitrary, and non-trivial? This paper composes results from the Identity-Persistence Program into a Partial Order of Reason. It does not derive the constitution of reality simpliciter; it concerns reality insofar as it becomes answerable to a bounded method. Answerability under a co-declared standard grounds a verdict demand and sufficiently declared regime R with demand Q. Independently, conditions on meaningful and non-arbitrary identity judgment supply the relevant identity and admissibility structure. At their convergence, Q induces static answer-equivalence K0(R,Q), its greatest structure-compatible congruence Theta(R,Q), and the invariant quotient core N_Q(R). Safe evidence channels canonically forget down to this core, while declared verdicts factor out of it. Together with the static answer quotient A_Q(R), this gives a Kernel Sandwich separating retained evidence, operable structural identity, static answer-equivalence, and final judgment. Structural existence of the core is thereby distinguished from epistemic attainment of it. The core is demand-relative rather than a question-independent essence. Structural Transparency holds exactly when Theta(R,Q) equals K0(R,Q): static answer-equivalence is then already structurally transportable and the operable and static quotients coincide. The second edition shows that this architecture is governed by contextual saturation. Closing Q under the lawful translations generated by R defines a closure operator on extensional demands, whose variance-reversed image on answer kernels is the congruence interior. Hence operability is static answerability after contextual closure. The resulting Contextual Indiscernibility Characterization says that two states are operably identical exactly when no declared question distinguishes them after any lawful regime-generated context. This is interpreted as a Bounded Leibniz Principle: identity of indiscernibles made regime- and demand-relative, without assuming an exhaustive predicate universe or promoting relative identity to identity simpliciter. The same adjoint structure yields a general Demand-Meet Law under which joins of demands correspond to meets of their operable equivalences, while infinite inverse-limit realization remains a separate completeness question. The discrepancy between static and operable equivalence is isolated as a contextual defect whose witnessing contexts form a dependency-incidence structure. Residual symmetry, decomposition, governance, capacity, observables, probability, and verification are located downstream. Within the additive governance subclass, the representation-independent governance object is a positive projective ray, with PAS_h as a harmonic coordinate realization. Canonical weight ratios exist exactly when the admissible projective ray set is a singleton.

This paper develops a finite structural theory of condensation across nested declared regimes. A declared question family determines a maximal safe congruence and a terminal safe quotient; the paper asks how much distinction may be condensed along the way, how condensation composes across stages, when interacting condensations admit canonical allocation, and under what conditions a higher-level persistent unit can emerge. For a finite regime, structural condensation is defined as the logarithmic reduction in distinguishable states under a safe quotient. The resulting condensation potential is an exact path potential on the quotient order: every quotient-stage amount is a potential difference, every finite chain telescopes, and every maximal safe tower spends the same total condensation budget regardless of staging. Demand refinement is antitone in that budget: requiring more future distinctions permits weakly less safe condensation. Adding a declared clock yields a structural condensation rate. Serial nesting obeys a no-free-acceleration theorem: the total rate is a duration-weighted average of the stage rates and cannot exceed the fastest serial stage. Independent product channels, by contrast, exhibit exact parallel amplification. On shared carriers, additivity fails in both directions. The paper introduces an interaction defect that measures exactly the path-dependent allocation discrepancy across a congruence-lattice square. The defect can have either sign. For permuting congruences it admits an exact covariance formula in terms of class counts inside join blocks. Uniform and permuting congruence structure gives a broad zero-defect class; in particular, the condensation potential is a lattice valuation on the congruence lattice of every finite congruence-uniform, congruence-permutable algebra, including finite groups. The paper also distinguishes two independent forms of canonicity. In finite distributive safe-congruence intervals, Birkhoff representation yields a canonical join-irreducible condensation skeleton, but the amount assigned to an irreducible increment can remain path-dependent. Conversely, finite groups can have canonical pairwise allocation while admitting non-unique irreducible factorizations. Thus canonical total budget, canonical allocation, and canonical factorization are separate structural questions. Finally, the paper gives a regime-relative theory of emergence. A nontrivial semantic condensation of an already reduced lower-level normal form is possible exactly when the transported demand is strictly weaker; the released condensation budget bounds the amount available to the emerging level, with equality at its corresponding normal form. Persistence verdicts do not automatically transport between levels. Biological evolution, boreal ecosystems, culture, and networked information systems are therefore treated only as candidate declared instantiations. The theory does not claim that nesting causes historical acceleration; it specifies the structural quantities and common-clock conditions required for such a claim to become testable.

This paper develops a compositional theory for declared identity-persistence regimes. It asks when a certified output under one regime may become the identity-bearing input of another, which obligations must propagate backward through a chain or network of regimes, and which distinctions may be compressed into certificates that move forward. Its governing law is that compression is demand-indexed while soundness need not be: a demand-independent identity interface can always preserve everything, but canonical maximal compression is defined only relative to what downstream evaluation requires. For each certified interface, the paper defines a contravariant weakest-obligation transformer on Galois-closed demand families. These transformers compose laxly, which is sufficient to form a category of demand-indexed regimes: an interface from (R,Q_R) to (S,Q_S) is admissible when the upstream declared demand is strong enough to support the pullback of the downstream demand. Composition therefore closes without knowledge of future extensions to the network; a later stronger promise is represented by a new indexed object and a fresh interface obligation rather than by retroactively invalidating an earlier morphism. The corresponding normal forms form a coherent inverse system. If Q is refined to Q′, the finer certificate N_Q′(R) maps canonically onto N_Q(R). The obstruction to reversing that map is typed by the existing migration calculus: newly required distinctions, distinctions already lost by the old certificate, and the residual deficiency not repaired by an added channel. For fixed acyclic diagrams, backward obligation propagation computes the least admissible demand profile and the forward normal-form pass is maximally compressed among structurally admissible sound towers supporting the same ultimate promises. The network result is not limited to acyclic diagrams. On any fixed small directed network, the product of the closed-demand lattices is complete and the monotone network-obligation operator has a least fixed point. Ordinal iteration from the bottom converges abstractly to that least fixed point on any set-sized complete lattice. Cycles, infinite chains, and infinite branching therefore create no existence obstruction at this level; effective computation, closure-ordinal bounds, complexity, constructive implementation, and dynamic topology remain separate questions. The paper also gives exact laws for interface soundness, joint masking, locality, repair, and entitlement. At symmetric interfaces it imports the Orbit-Congruence Criterion and distinguishes three requirements: downstream demand must respect the declared symmetry, the symmetry must be compatible with the declared operations for the quotient to be operable, and the composite signature must be adequate for regime-level certification. Under these conditions, the semantic channel of any given sound symmetry-respecting interface descends through the orbit quotient, and every such interface canonically forgets down to the same terminal normal form. The resulting architecture is a dual flow: demands move backward; certificates move forward.

This paper studies the invariant layer of identity persistence under declared symmetry: which representation-dependent distinctions may be removed, when the resulting orbit quotient remains operable under the regime's declared operations, and how much further compression is permitted by the questions the regime is actually required to answer. The central result separates three conditions that must not be conflated. Orbit invariance licenses a semantic quotient; orbit congruence licenses an operable quotient algebra; signature adequacy licenses the quotient as a valid regime. Ordinary diagonal equivariance is shown not to imply orbit congruence by an SO(2) vector-addition counterexample, and the Orbit-Congruence Criterion gives the exact replacement: a declared operation descends to X/G exactly when its quotient value depends only on the orbit classes of its arguments. Under that condition, representation reduction and purpose-relative compression form a staged factorization. The orbit quotient X/G is the finest representation-independent description, while the declared normal form N_Q(R) is the coarsest structurally admissible description sufficient for the required question family. The normal form is therefore a quotient of the orbit quotient, with equality exactly when the declared query universe supplies an orbit-separating invariant family at the relevant Galois closure. Symmetry reduction and normal-form reduction commute at their general algebraic scope; compactness is required only for the finite-coordinate invariant-theory results. The paper then develops quotient-in-stages for residual symmetry. Chirality is typed as a reflection-odd semi-invariant under the residual Z₂ action after quotienting rotations. For finite abelian residual symmetry, prime-primary decomposition and the Chinese remainder theorem yield an exact residual-label reconstruction law: preserving every primary component of a declared residual label jointly preserves the full label. This finite mechanism is explicitly separated from continuous Fourier-phase reconstruction, which remains governed by relation-lattice rank and saturation. Finally, scalar governance is recast on the additive invariant cone. PAS_h is represented by a positive linear functional whose weight ratios are its projective content. Automorphisms of the declared pre-governance structure force equal weights on coordinates they exchange. The admissible PAS weight rays are exactly the projective classes of owner-admissible, automorphism-fixed positive functionals; canonical ratios exist exactly when this projective set contains a single ray. Thus the remaining weighting freedom is characterized structurally rather than left as an unspecified parameter-selection problem.

This paper develops recurrence normal forms for identity persistence under declared regimes. After quotienting recurrence distinctions irrelevant to required identity behavior, the surviving one-degree homogeneous recurrence structure is classified as ℝ/H, where the closed subgroup H ≤ ℝ yields exactly three possibilities: point, line, or circle. Non-triviality excludes the point. Finite comparison covering radius is shown to be equivalent to cocompactness of the return subgroup and, together with non-triviality, forces the circle S¹. Applying this result to the program's normative Bounded Re-Comparability condition is isolated to one explicit owner-interface identification; an independent forward-semigroup derivation is established conditional on strong forward connectivity. The paper also distinguishes finite eventual periodicity from genuine cyclicity and characterizes the boundary to higher-dimensional recurrence. Downstream, it repairs the invariant layer on S¹: Fourier magnitudes are shown not to be complete rotation invariants, the finite-bandlimit weight-zero invariant algebra is finitely generated and orbit-separating, and bispectral completeness is characterized by support-sensitive rank and lattice-saturation conditions. The result separates recurrence, return, compactness, and invariant canonicalization while locating the precise assumptions under which each follows.

This paper develops a theory of independent reconstruction for formal theorem corpora. Its starting distinction is between proof-architecture verification and proof verification itself: a dependency graph may be acyclic, complete at the declared route level, and machine-audited without thereby establishing that the mathematical derivations represented by its nodes are correct. For a finite acyclic registered architecture, the paper defines a base-relative reconstructibility closure \(S_B^\star\). Each theorem has one or more admissible conjunctive support routes, and a target enters the closure exactly when at least one complete route is supported by the verified base and kernel-checked derivations. This closure is represented more generally by a symbolic provenance object \(P_t\), built from unresolved obligations using conjunction for joint requirements and disjunction for alternative routes. Evaluating \(P_t\) under a declared set of resolved obligations exactly determines whether the target is reconstructible. The provenance calculus yields two dual certificate systems. Minimal completion sets identify the least unresolved obligation sets sufficient to close a target through some route. Minimal blocking cuts identify the least sets intersecting every possible completion route. These are related by finite hypergraph blocker duality. The distinction prevents a common error: a minimal obstruction to every route is not itself necessarily a sufficient repair. The paper also gives an exact factorization criterion for source-to-formal correspondence. A correspondence claim is certifiable from a declared evidence interface if and only if the correspondence predicate factors through that interface, equivalently if the interface kernel is contained in the correspondence kernel. Mechanization therefore has no fixed human/machine boundary: expanding the formally verified interface may enlarge the reconstructible claim set, while downstream processing cannot recreate distinctions the interface does not represent. Finally, the paper separates logical from operational trusted bases and characterizes the computational tasks induced by the reconstruction calculus. Closure under a fixed resolved set is linear in the route DAG. Certificate-family enumeration may require exponentially many outputs. Minimum Verification Cost is NP-hard in the unrestricted finite provenance class by reduction from weighted hitting set, while read-once proof architectures form a syntactically recognizable subclass admitting exact linear min-plus evaluation. The result is a theory of mechanized reconstruction rather than a report that the Identity-Persistence Program has already been fully machine-verified. The theory specifies exactly what the eventual artifact must compute: which claims are reconstructible from the verified base, which obligation sets complete them, which cuts block them, and which source-to-formal correspondences remain outside the current machine-certified interface.

What does it mean to withdraw a previously declared demand without changing what a system must preserve? This paper develops a formal calculus of retirement as the reverse direction of demand refinement. For a fixed declared regime, removing demands coarsens contextual equivalence. Retirement is conservative precisely when that coarsening leaves the governing congruence unchanged. Using the underlying antitone Galois connection, the paper proves an exact redundancy criterion: a promise may be withdrawn without loss exactly when the promises that remain still definably entail it. For joint retirement, all retired promises must be certified against the final remainder. This explains why independently valid retirement certificates need not compose and why conservative retirement is nevertheless order-independent once its endpoint is fixed. A second result identifies the limit of retirement. The mathematical zero-demand endpoint collapses each inhabited sort to a point, but a sufficient specification cannot generally retire to that endpoint. Forced obligations survive under-declaration. Thus lawful retirement within a fixed problem class is bounded by the core induced by its forced obligation floor. Going below that floor produces an inadmissible declaration; it does not, by itself, constitute a change of problem class or regime. The resulting picture distinguishes contingent promises that may be forgotten, forced obligations that cannot be retired within the problem class, and changes to the problem class or regime itself. Retirement therefore supplies a precise semantics for bounded forgetting while preserving the distinction between specification weakening, inadmissibility, and regime change.

What must a recorded artifact reveal for an independent evaluator to determine whether a consequential action was entitled under a declared governance contract? This paper separates four requirements: assessment, sound positive certification, enforcement, and completeness. For a declared target on histories and an observation map, it characterizes exact assessment by kernel inclusion: histories indistinguishable through the artifact must have the same target value. The target quotient consequently gives the coarsest sufficient observation at the semantic level, without establishing its computability, affordability, independence, or authenticity. A separate theorem characterizes preservation of a static self-licensing predicate under capability-preserving quotients with transported coalition permissions. Non-retroactivity is declared as a governance premise rather than derived from recording requirements. Enforcement requires sound certification together with an explicit execution-gate contract; accurate assessment alone does not prevent prohibited actions. Finite executable models exhibit the distinctions and counterexamples, with exhaustive enumeration, constructed witnesses, randomized checks, and arithmetic examples separately identified. The paper contributes scoped results toward a general theory of justified epistemic activity for bounded observers. It establishes neither a universally necessary artifact-field list nor authentic history, real-system enforcement, or complete resource-bounded self-revision.

This paper studies an interpreted reading of declarations as boundary conditions for recognition within the Identity-Persistence Program. Using a fully synthetic declaration fixture, it evaluates whether this interpretation makes falsifiable predictions about redundancy, ambiguity reduction, obstruction behavior, and recognition under the published declaration framework. The paper introduces no new mathematical results. All computational findings are explicitly restricted to the synthetic fixture and are distinguished from corpus-level conjectures and undeclared systems through the program's claim-status discipline. Its contribution is methodological: demonstrating how an interpreted reading can be converted into executable predictions and independently replayed within the published framework. The executable companion artifacts—including the declaration fixture, deterministic audit runner, replayable audit outputs, artifact manifest, and supporting documentation—are available in the associated Zenodo record accompanying this publication.

This paper presents a complete worked instance of the declaration-to-recognition procedure developed within the Identity-Persistence Program. Rather than introducing new mathematics, it executes the published methodology end-to-end on a fully synthetic corpus, demonstrating the declaration language, translation layer, conformance evaluation, recognition procedure, validation stages, and typed evidence artifacts. The worked instance exhibits all three terminal outcomes of the procedure: successful realization, semantic obstruction, and pre-semantic validation failure. Every stage is replayable from the accompanying declaration package and bound artifacts, providing an executable conformance fixture for the formal specification itself rather than a proof of additional mathematical results. The paper makes no claims about the actual fifteen-role corpus, undeclared regimes, or production execution. Its purpose is methodological: to demonstrate that the published declaration language, recognition mathematics, and artifact contracts compose into a coherent executable workflow before the real declaration process begins. The complete companion corpus—including the supporting theorem papers, declaration schemas, executable fixtures, certificates, manifests, and reproducibility artifacts associated with this worked instance—is available in the corresponding Zenodo record. This paper should therefore be read as the executable methods companion to the broader Identity-Persistence Program rather than as a standalone mathematical treatment.

This paper studies the floor-relative intrinsic-image problem for finite declared regimes: which compatible feasibility–consequence pairs arise from regimes satisfying a specified floor of identity, admissibility, evidence, authority, and verification conditions? The unrestricted representation layer is characterized as a Moore reflection from raw profile families to their nonempty-intersection completions. Compatible pairs correspond exactly to nonempty saturated families, while observational equivalence is equality of Moore completion. Each semantic fiber is an inclusion interval between its unique irreducible basis of essential profiles and its complete family of feasible closed profiles. The reflection preserves arbitrary joins but not meets, producing a complete join-semilattice quotient rather than a lattice congruence. Floor-relative realizability is reduced to an interval-incidence problem: a pair belongs to the intrinsic image exactly when its semantic fiber contains a floor-conforming representative. The paper derives a minimal-completion criterion, a pair-relative attainment theorem, a four-part obstruction classification, a contamination hypergraph, and a Reflection Transfer Principle connecting declared operations on the floor-family locus to structural properties of its semantic image. It also distinguishes the actual nonempty semantic domains from their free formal-bottom completions and identifies null-regime admission as the exact condition under which the formal bottom enters the augmented image. The actual floor-relative image is not computed before the author declarations, floor-profile translation, and corpus-native dependency system exist. The paper instead establishes the strongest universal recognition and transfer apparatus available before those inputs and proves that no further nontrivial restriction follows from unrestricted representation alone. The archival version is available in the corresponding Zenodo record.

This paper presents the machine-readable proof dependency index accompanying Closing Algebra v18. It describes the canonical dependency graph, a deterministic checker, a machine-generated audit, a cyclic negative test, documentation, and a cryptographic manifest binding the companion publication. The graph records strict prior logical dependencies among the named results of the four closure documents. The checker validates graph consistency, rejects cycles, computes dependency statistics, transitive reduction, support cones, profile equivalence classes, import signatures, and the paper-level quotient, reproducing the published audit deterministically. The companion artifacts provide a reproducible dependency index for the published proof presentation. They certify structural properties of the declared dependency graph rather than the mathematical correctness of the underlying proofs. The complete machine-readable companion artifacts (dependency graph, checker, audit, test fixtures, manifests, and supporting documentation) are archived separately in the corresponding Zenodo record.

This record contains the companion artifacts for The Declaration Dossier v8. It provides the machine-readable declaration template, formal schemas governing release and validation packages, a validator-output template, documentation describing the declaration workflow, and a manifest binding the published artifacts. The declaration file records only author declarations. All computed relations, semantic tests, gate evaluations, and validation results are produced separately by the validator and are never author-authored. The package therefore separates declarations from derived computation by construction. The artifacts define a versioned declaration interface for corpus computation and are intended as reproducible infrastructure accompanying the paper rather than as executable implementations of the validator itself. Companion machine-readable artifacts for this paper are archived in the corresponding Zenodo record.

The unrestricted representation problem for finite dependency systems is vacuous: when coordinate domains, admissibility, and semantics may be freely declared, every finite system realizes itself. This paper proves that all nontrivial representation content must therefore come from restrictions on the regime class and classifies the unrestricted external realization layer exactly. Relative to a finite axiom set, the realizations of a dependency system determine a simplicial feasibility complex whose minimal nonfaces are the canonical incompatibility obstructions and whose maximal faces are the strongest jointly realizable profiles. The associated satisfaction polarity induces a closure operator recording fiber-relative consequence. A feasibility complex and consequence closure arise together from a finite realization-profile family exactly when feasible sets close to feasible sets and infeasible sets collapse to the full axiom set. Such compatible pairs possess a canonical realizing family and a unique reduced normal form given by their essential profiles. Compatible consequence theories over a fixed complex are classified by admissible feasible Moore families and form a finite lattice dual to the lattice of those families. The paper further identifies safe record omission with the feasibility–obstruction layer under an omitted-sets-as-faces correspondence. Universality results show that every finite simplicial complex and every compatible consequence structure occurs in the unrestricted class. Hence the remaining representation problem is not representability in general but the floor-relative intrinsic-image problem: characterizing which compatible feasibility–consequence pairs are induced by the substantive axioms of a declared regime.

This paper develops an exact finite theory of joint omission in declared-regime decision records. A record consists of values assigned to declared semantic roles, and replay semantics comprises the verdict, reason path, and obligations at a fixed grain. A set of roles is safely omitted exactly when replay semantics is functionally determined by the retained roles. The central difficulty is that separately valid role-wise omission certificates need not compose. To explain this failure, the paper constructs a reconfiguration graph on each retained-coordinate fiber. Its vertices are admitted records with the same retained values, and its edges are admitted variations in one omitted role. Role-wise certificates imply semantic constancy along every edge and therefore on every connected component. They do not imply agreement between distinct components. Joint omission is safe exactly when the component labels agree throughout every retained fiber. This gives an exact decomposition theorem, a sufficient connectivity criterion, and a canonical obstruction consisting of pairs of records that agree on every retained role but differ in replay semantics. The obstruction vanishes exactly when omission is safe and supplies complete two-point failure certificates. The paper further constructs a universal omission quotient. This quotient identifies precisely the records connected by licensed single-role variations and is universal among maps invariant under those variations. Valid local certificates force replay semantics to factor through this quotient. The retained-coordinate projection then factors through a second map, and safe joint omission is equivalent to the induced semantics factoring through that map. Local certification therefore determines semantics only up to a component quotient; joint omission requires the remaining component identity to be semantically irrelevant. A second construction characterizes minimal safe records. For every pair of records with unequal replay semantics, form the set of roles on which they differ. These difference sets define a semantic conflict hypergraph. A retained role set is replay-sufficient exactly when it intersects every conflict hyperedge. Inclusion-minimal sufficient records are therefore the minimal transversals of the conflict hypergraph, and each retained role in such a record has a private conflict witness establishing its necessity. Minimum-cardinality safe retention is equivalent to minimum hitting set and is NP-hard. The paper thereby closes joint omission at schema level as a theory of correctness, certification, and minimal compression. It also provides the exact condition under which the joint-omission restriction in a bounded deployment interface may be lifted. It does not claim that the fifteen-role corpus relation has been instantiated or that its proofs have been independently verified.

This paper develops a finite dependency atlas for sufficient declared-regime specifications. Its governing question is whether the fifteen semantic roles required by the Identity-Persistence Program can be faithfully represented by a finite typed diagram of quotient maps and kernel inclusions, and which obligations—such as interaction, existence, priority, or global assembly—necessarily escape that local representation. The paper first establishes the structural instruments that a corpus-native computation requires: a complete lattice of semantically closed role sets, a canonical implication basis, and a quotient-dependency diagram in which arrows are kernel inclusions and absent arrows carry two-point certificates. It then develops a closed interaction theory for finite deterministic semantics at any declared finite partition. The theory is organized by a Galois connection. Equal semantic outputs generate coordinate equivalences, and product formation reconstructs the least product relation containing the semantic kernel. Noninteraction is exact reconstruction; interaction is failure of exact reconstruction. This characterization yields a polynomial-time decision procedure, independently checkable chain certificates, a canonical closure defect, and a proof that interaction admits no uniformly bounded induced-substructure certificates. The generated-equivalence domain is canonically the congruence lattice of the many-sorted empty-signature algebra on the declared components. The paper uses this structure to characterize transformer-relative completeness, categorical transport, partition-indexed closure monotonicity, merge exactness, and the interaction frontier—the antichain of minimal partitions at which the semantics becomes noninteractive. It also proves that interaction is asymptotically generic on total full grids with fixed finite output alphabet and, under a fixed finite relational encoding, is definable with transitive closure but not in first-order logic. Interaction is distinguished from global assembly. Interaction asks whether an existing semantic relation factors across a declared decomposition; assembly asks whether compatible local relations possess any global realization. The latter is governed jointly by context-hypergraph structure and the polymorphisms of the admitted relation language. The paper does not claim that the fifteen-role corpus relation has been instantiated or computed. Seven declarations remain deliberately blank. The result is therefore a draft whose abstract instruments and finite interaction theory are complete within their declared scope, while corpus computation and independent proof verification remain outstanding.

This paper develops an indexed algebra for the internal structure of sufficient declared-regime specifications. Its starting point is a bounded-verification problem: once a regime has declared the identity conditions, admissible transformations, evidence boundaries, authority, outcome types, and replay obligations governing a verdict, what dependency structure exists among those declarations, and which parts of that structure are recoverable, minimal, interactive, or realizable? A specification at a declared index is represented by a finite quotient-level dependency system, D_I, consisting of admissible role assignments paired with replay semantics at a declared grain. Quotient-level recoverability induces a semantic closure operator; functional determination characterizes replay sufficiency; minimal semantic keys characterize record sufficiency modulo mandatory provenance and version fields; and refinement is represented by dependency-preserving factorization. A consolidation theorem proves that factorization-based recoverability and quotient-level semantic closure are the same operator. Minimal sufficiency and interaction are shown to be distinct. Minimal keys describe which information jointly determines a result, whereas interaction describes failure of the semantic kernel to decompose across declared coordinate blocks. The paper establishes their nonidentity and gives separate fiberwise and section-level results, including a connectivity obstruction limiting where masking can occur. The realizability analysis distinguishes arbitrary finite dependency systems, systems representable in the declared presentation language, floor-gated systems, and systems induced by corpus-sufficient regimes. The presented class receives a sound and complete representation theorem. A fixed pre-verdict signature yields a strict gate separation and a universal gate quotient, characterized as the least-informative quotient sufficient for gate status. Every declared factorization obligation f = f̄ ∘ q is equivalent to a kernel inclusion and admits a labeled two-point obstruction certificate. In finite systems, these obligations compile into definite Horn implications. Countermodels further show that mixed raw/quotient determination need not form a closure operator, that the quotient-level closure is not generally matroidal, and that minimal generating sets may have unequal cardinalities. The paper does not claim to characterize corpus sufficiency. Its remaining realizability question is whether regime-induced systems coincide with systems satisfying the complete relational translation of the specification floor: Reg_I ?= FloorDep_I

This paper studies a deployment question for a bounded formal research program: when the program’s specification floor, declared-class capacity results, and identification ceiling are closed, but parts of its internal specification algebra remain unresolved at the corpus level, must deployment of the closed results wait? The answer is given relative to a declared verdict demand, D₀. The demand fixes one governing regime and version, deterministic typed evaluation, independent replay at a declared grain, a role-wise decision record, and omission licensed only by role-wise certificates. The associated interface consists of loading a specification, evaluating a canonical candidate, recording the typed verdict and obligations, and independently reproducing them. A finite, versioned ledger of unresolved mathematical questions is classified through four mechanisms. Quotient applies when alternative resolutions are observationally equivalent at the declared interface. Wall applies when the demand excludes the operation whose correctness would depend on the unresolved question. Fence applies when the runtime records a constitutive selection and thereby becomes operationally total without deriving a uniquely correct selection. Boundary applies when the act fixes the index under which the other claims are evaluated and therefore cannot be eliminated by the same quotient. The resulting interface theorem is derived case by case over the ledger. Record minimality is quotiented under the resource-unbounded demand. Joint interaction, contextual and corpus-native recoverability consequences, semantic equivalence, refinement, and unrestricted composition are walled from the interface. Convergence and declaration are fenced through recorded selection. Co-declaration remains a constitutive boundary. The principal dependency on the unresolved algebra is joint omission: because separate role-wise omission certificates need not compose, D₀ permits only role-wise omission and requires a joint certificate for any artifact omitting multiple roles. The result is conservative only where observational equivalence is established; the paper does not relabel restrictions, declarations, or constitutive choices as conservativity. It concludes that a runtime conforming to D₀ need not await corpus-native closure of the specification algebra. It does not conclude that the algebra is closed, that any implementation conforms, or that the result extends beyond the declared demand class.

Modern AI systems possess increasingly sophisticated generation pipelines but little shared computational infrastructure for determining when generated proposals become consequential commitments. Evaluation remains largely implicit—distributed across application logic, organizational review, institutional practice, and post hoc reconstruction—so verification, governance, and dispute resolution often require reconstructing how a decision was made rather than independently recomputing it. This paper presents Runtime Regime Infrastructure (RRI), a derived architecture for the missing computational phase between probabilistic generation and consequential action. Under a declared regime, identity and admissibility are established before evaluation; deterministic evaluation produces typed structural verdicts; policy composes over those verdicts without redefining them; portable decision artifacts preserve the dependencies required to support the result; and independent evaluators recompute the recorded result under a declared replay mode. The architecture follows from a constraint on bounded evaluators: a generated output cannot itself determine the conditions under which it acquires consequential authority. Operational terms such as same, safe, good, and allowed become reproducible only relative to a regime specifying the identity-bearing object, admissible transformations, invariants, evidence boundary, governing authority, and available verdict vocabulary. RRI therefore separates generation from commitment: a proposal becomes actionable only through evaluation under conditions declared independently of that proposal and preserved in an independently recomputable decision artifact. The paper introduces no new formal mathematics. It assembles previously established results from the Identity–Persistence Program—including sufficient regime specification, identity forcing, admissibility, deterministic evaluation, enforcement coding, bounded verification, and decision-artifact obligations—into a single runtime architecture. Each imported result retains its original scope and claim status; the architecture itself is explicitly a DERIVED / ARCHITECTURAL synthesis. The contribution is architectural rather than theorem-level: a layered runtime, typed computational contracts, authority boundaries, refusal semantics, verification model, governance interfaces, and explicit dependency structure for independently reproducible consequential computation under declared regimes.