Defines the neutral constraints under which systems remain explainable, inspectable, and contestable without embedding interpretation.

Structural Explainability (SE) defines a neutral structural substrate for recording identity, structure, transformation, and change without embedding interpretation, authority, causality, or judgment.

Structural Explainability is not anti-interpretation; it is anti-implicit interpretation. It separates structural facts from downstream claims so interpretation and disagreement can remain explicit, attributable, and contestable over time.

Record systems commit to identity semantics. When a rule is refined, an agent redeployed, a jurisdiction subdivided, or two records merged, the system decides whether the thing is still the same thing. That decision is load-bearing for citation, liability, and audit, but not always declared.

SE makes the commitment declarable and the failure detectable.

• Identity & Persistence: Persistent tracking and provenance depend on inspectable identity conditions. Identity is regime-relative: a single structural change might preserve identity under one regime, break it under another, or be identity-neutral under a third.
• Declared Identity Semantics: A system may induce a rule of sameness through an audited implementation surface based on one registered artifact or on a joint mechanism over several registered artifacts. SE makes divergence from the declared rule detectable as a finite witness and separates detection from sibling diagnosis and regime substitution.
• Plural Systems: SE is built for independent architectures that share data without requiring a centralized authority, uniform naming, or a single forced ontology.

Within the core substrate, structure, transformation, persistence, and regime behavior may be recorded. Causal explanation, normative evaluation, authority, legitimacy, obligation, and enforcement are not asserted as substrate-level commitments; they may be attached through constrained downstream mechanisms.

Downstream domains (e.g., law, infrastructure, agentic AI systems) consume the core substrate without redefining it. SE records the structural commitments of a system without deciding their ultimate meaning, enabling continuous cross-institutional coordination through persistent disagreement.

Some information systems make decisions that affect people, attribute claims to sources, record contested facts, or coordinate across institutions that disagree. Those systems can hold a compliance conclusion, a legal holding, or a causal claim while every party continues to refer to the same underlying artifacts.

They fail in a specific way. Under change, the system must decide what stays the same: whether a refined rule is the same rule, whether a redeployed agent is the same agent, whether a subdivided scope is the same scope. The implementation answers through fields, predicates, workflow states, configuration values, or joint mechanisms over several registered artifacts. Identity-relevant uses of those mechanisms may induce an operational identity partition that differs from the declared identity partition.

That divergence may remain invisible until the system is challenged in court, in audit, in appeal, in public review, or in cross-institutional reconciliation. By then, the records may already have committed to an identity model not explicitly declared.

Existing infrastructure may not catch it. Provenance models, content credentials, software bills of materials, version control, and audit logs record what happened. They do not necessarily require a system to declare its rule of sameness or compare that declaration against the identity relation induced by its implementation.

Six of the nine identity regimes sit in pairs on three sibling axes: LOC/OBJ, SCOPE-E/SCOPE-S, RULE-C/RULE-S. Each pair offers two admissible identity bases for the same kind of referent. When a sibling basis is available and also splits an examined declared class, the divergence can be classified against it; otherwise, the divergence is unpositioned.

A divergence has a specific, finite witness: a pair of records the declared partition places together and an audited surface separates. That pair is the witness. The witness alone does not establish the divergence's sibling classification or which identity basis the surface carries. A surface implements a hidden identity regime only when the sibling differs from the declared regime on the examined domain and the operational partition equals the sibling partition there.

Operational Identity defines what a system must declare for this to be checked, and gives the procedure.

SE does not replace domain vocabularies, standards, ontologies, or existing data systems. It constrains how systems implement them so that disagreement remains visible, attributable, and useful over time.

The formal core of Structural Explainability is developed in three papers. These papers are the normative specification of the framework.

SE-100 establishes the neutrality-by-design constraint: a substrate is neutral when its foundational layer is restricted to referential commitments and permitted attribution propositions.

SE-200 derives the referential-regime structure that neutral substrates require, yielding nine core identity regimes: three unpaired regimes, OBL, OCC, and REC, and six regimes paired across three sibling axes: LOC/OBJ, SCOPE-E/SCOPE-S, and RULE-C/RULE-S.

SE-210 defines the operational identity partition, the rule of sameness induced by an examined implementation surface, and a finite audit comparing it against the declared identity partition. The audit detects divergence, returns finite witnesses, and separates divergence detection from sibling positioning and regime-substitution claims.

SE-220 studies equivalence relations over distinct interpreted carriers. A finite probe family and an interpretation function connect a carrier of environments with a carrier of interpreted outputs. An output equivalence induces an interpretive kernel over environments. A declared environment regime that refines that kernel is coherent.

The companion construction also computes the least output equivalence required to make a declared environment regime coherent. The resulting adjunction transports equivalence requirements between the two carriers and returns finite witnesses when a proposed identification is not supported by interpretation.

The SE-210 verification repository contains two independent implementations of the finite operational identity audit defined by the paper.

The literal oracle constructs the relevant relations explicitly and applies the paper definitions directly. The optimized checker uses union-find closures, operational-signature grouping, and declared-block comparisons.

The implementations are compared across paper-derived regression cases, an exhaustive deterministic sweep over small instances, and randomized instances.

The SE-220 implementation work follows a related but distinct structure. The Python and Lean repositories are independent realizations; disagreement indicates at least one definition, proof, implementation, or test obligation must be repaired.

• se-interpretive-kernel implements the finite partition algebra and kernels and remains domain-agnostic.
• se-theory-interpretive-kernel formalizes the same mathematical contract in Lean.

The papers are normative. Companion repositories test the internal consistency, computability, algorithmic claims, and finite consequences of paper definitions.

Independent Lean 4 formalizations of the SE layers. Each is a machine-checked realization of the corresponding normative paper. All namespaces are under SE.*.

SE repositories use manifests to describe the repository's role, scope, dependencies, provided artifacts, validation expectations, governance, and traceability.

The SE manifest complements external metadata standards. Citation metadata belongs in CITATION.cff; software supply-chain metadata may use SPDX; software discovery metadata may use CodeMeta; research-object packaging may use RO-Crate.

flowchart TD
  subgraph SE["Structural Explainability Core"]
    NS["SE-100<br/>Neutral Substrates"]
    RR["SE-200<br/>Referential Regimes"]
    OI["SE-210<br/>Operational Identity"]
    IK["SE-220<br/>Interpretive Kernel"]

    NS --> RR
    RR --> OI
    OI --> IK
  end

  subgraph IKR["Interpretive-Kernel Realizations"]
    IKPY["se-interpretive-kernel<br/>Python construction"]
    IKLEAN["se-theory-interpretive-kernel<br/>Lean formalization"]

    IK <--> IKPY
    IK <--> IKLEAN
  end

• To understand the theory, read the three papers.
• To compare a record system's declared and implemented rules of sameness, apply the Operational Identity audit from SE-210.