SEAM: Global consistency beyond local accuracy in scientific machine learning

# SEAM: Global consistency beyond local accuracy in scientific machine learning
Source: [https://arxiv.org/html/2608.05702](https://arxiv.org/html/2608.05702)
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Conceptualization, Methodology, Software, Formal analysis, Investigation, Data curation, Validation, Visualization, Writing – Original draft, Writing – Review and editing

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\[1\]\\creditFormal analysis, Visualization, Supervision, Validation, Writing – Review and editing

1\]op=,organization=Axiom Research Group 2\]organization=Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, addressline=404 Nganana, Kikwe, Arumeru, P\.O\. Box 447, city=Arusha, postcode=23311, country=Tanzania 3\]organization=African Institute for Mathematical Sciences, Research and Innovation Centre, addressline=KN 3 Road, Gasharu Cell, Kicukiro Sector, Kicukiro District, city=Kigali, country=Rwanda 4\]organization=Graduate School of Engineering, The University of Tokyo, addressline=7\-3\-1 Hongo, Bunkyo\-ku, city=Tokyo, postcode=113\-8656, country=Japan

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\[cor1\]Corresponding author

Bum Jun Kimbumjun\.kim@weblab\.t\.u\-tokyo\.ac\.jp\[\[\[\[

###### Abstract

Scientific machine learning commonly validates models at the level of a subdomain, a benchmark split, or an explanation for one prediction\. Yet such local checks cannot establish whether the resulting explanations can be assembled into one globally admissible explanation\. We introduce Scientific Explanation\-Admissibility Machines \(SEAM\), a generator\-agnostic framework that makes this local\-to\-global consistency question computable across regions, sensors, regimes, and model components\. The finite explanation\-sheaf instantiation SEAM\-Ω\\Omegarepresents each region by a structured explanation with state, closure, and observation channels together with optional contract metadata; compares neighboring explanations on their overlaps; and converts disagreement into a channel\-resolved obstruction\. This obstruction locates inconsistency and tests competing declared accounts by restricting each repair to the revisions that one account permits\. Exact feasibility refutes or retains an account; when exact repair is unavailable, residual\-aware regularized records provide a separately labeled empirical attribution\. The framework also separates inconsistency from non\-identifiability and monitors learned generators under distribution shift\. We establish theorems for minimum\-cost intervention and conservation\-contract detectability, together with companion results for identifiability and closure recoverability\. Across nineteen experiments involving synthetic partial differential equation systems; out\-of\-distribution Fourier neural operator \(FNO\) monitoring; four open datasets spanning traffic, hydrology, air quality, and electric power; and synthetic financial and industrial systems, SEAM detects incompatible explanations even when local predictions are accurate, and attributes failures to specific channels and overlaps\. SEAM adds a global explanation\-consistency audit to existing solvers and learning models, testing whether their local explanations form a coherent scientific account\.

###### keywords:

scientific machine learning\\sepcellular sheaves\\sepexplanation\-admissibility\\sepobstruction\-guided diagnosis\\sepidentifiability\\sepmodel monitoring

## 1Introduction

Scientific models, whether physics\-based or data\-driven, are commonly judged by what happens inside one region of a problem\. For example, a computational fluid dynamics solver is validated on a subdomain where reference data are available\. A neural operator is evaluated on a benchmark split\. A regional regression model is tested on a held\-out fold of the local time series\. Each such local assessment is made without testing whether the same system’s explanations remain consistent across neighboring regions, different sensors, adjacent regimes, or solver interfaces\.

These local checks therefore do not establish whether explanations from different regions form a coherent account of the full system\. Those explanations must agree across regimes, scales, sensors, boundary interfaces, and modeling assumptions\. A family of individually plausible explanations can be scientifically inadmissible: Every piece looks correct, while the whole cannot be glued into a coherent account\. The same pattern appears in several routine scientific workflows\. Two regional partial differential equation \(PDE\) solvers can independently pass local residual tests yet disagree at their shared boundary\. Two seasonal demand models can fit their respective seasons yet contradict each other at the transition\. An industrial monitoring system can have one zone whose drifted sensors lead its regional model to infer a source term that a neighboring zone’s cleaner sensors contradict\. A learned neural operator can score well on in\-distribution test sets yet produce explanations that no finite\-difference solver could ever produce\. The central operational problem is therefore to determine when local scientific explanations can be assembled into one globally admissible explanation and, when assembly fails, how to diagnose the incompatibility and which repairs are feasible\.

Answering the second half of that question is a hypothesis\-testing problem rather than a scoring problem\. Consider three neighboring zones of a monitored system whose regional models disagree on their shared overlaps\. A practitioner facing that disagreement is rarely short of candidate explanations; the difficulty is that several explanations are available at once, and each one is consistent with every measurement taken so far\. One account holds that the sensors feeding one zone have drifted, so the predicted states are right and the readings are wrong\. A competing account holds that the sensors are sound, and the assumed physics of that zone omits a source term, so the readings are right and the closure is wrong\. A third account holds that all three zones are individually sound and were merely calibrated on data that never constrained their common interfaces\. Each account is a scientific hypothesis about the cause of the disagreement, and no regional residual or held\-out score distinguishes these hypotheses, because each of those quantities is computed inside a single zone, whereas the accounts differ in their assertions about the interface between zones\.

The Scientific Explanation\-Admissibility Machines \(SEAM\) framework separates such accounts by making each account computable\. A hypothesis about the cause of a disagreement is at the same time a statement about which parts of the explanations one is prepared to revise, and the framework formalizes the hypothesis through that statement: The drifted\-sensor account permits revision of the sensor readings alone, the missing\-physics account permits revision of the assumed closure alone, and the interface\-calibration account permits revision of the predicted states\. Confining a repair to the revisions that one hypothesis permits makes the hypothesis testable\. Either some permitted revision removes the entire disagreement, in which case the hypothesis survives and carries a price, namely the cost of the cheapest revision that works; or no permitted revision removes that disagreement, in which case the overlap evidence refutes the hypothesis, and the irreducible remainder measures how badly that hypothesis fails\. Costs quantify the repair burden within each surviving account; those costs are not used to rank an unrestricted repair against a more specific account\. The test is auditable because the permitted revisions, cost metric, and remainder are all recorded\. Figure[1](https://arxiv.org/html/2608.05702#S1.F1)traces one such audit end to end, from the regional records through the overlap comparison and the channel\-wise diagnosis to the single surviving account\.

![Refer to caption](https://arxiv.org/html/2608.05702v1/x1.png)Figure 1:Overview of a SEAM audit, read from left to right\. Each regionUiU\_\{i\}of the cover carries one structured explanation record, emitted by its own backend and partitioned into the state, closure, and observation channelsuu,cc, andoo, together with optional contract metadataϵ\\epsilon; every record passes its own local check\. Restriction compares two records on the overlap, and the disagreement that survives is the obstructionω≠0\\omega\\neq 0, so the family is globally inadmissible, although no local check failed\. Diagnosis splitsω\\omegaby channel and localizes the defect, here to the closure channel\. Intervention states the drifted\-sensor, missing\-physics, and interface\-calibration accounts as budgets that name the revisions each one permits, and only the account whose permitted revisions remove the entire obstruction survives\. The Identify and Monitor extensions record unresolved admissible directions and trackω​\(t\)\\omega\(t\)under shift\. The figure is schematic\.A catalog of specific hypotheses can also fail as a whole, when every proper budget is refuted and only unrestricted revision of everything at once restores agreement\. SEAM then records the attribution as unresolved instead of treating the unrestricted fallback as a causal account\. The identifiability analysis separately records explanation directions that satisfy every overlap constraint and yet leave no trace in the current observations\.

Current scientific machine learning toolchains address important parts of this problem\. Physics\-informed neural networks \(PINNs\) embed PDE residuals in training losses\(Raissi et al\.,[2019](https://arxiv.org/html/2608.05702#bib.bib44)\), whereas universal differential equations place trainable components inside differential\-equation models\(Rackauckas et al\.,[2020](https://arxiv.org/html/2608.05702#bib.bib43)\)\. Neural operators learn maps between input and solution function spaces\(Kovachki et al\.,[2023](https://arxiv.org/html/2608.05702#bib.bib28); Lu et al\.,[2021](https://arxiv.org/html/2608.05702#bib.bib33)\)\. Domain\-decomposed PINN variants are important exceptions to the single\-domain pattern: Conservative PINNs \(cPINNs\) and extended PINNs \(XPINNs\) impose solution or flux compatibility at subdomain interfaces, while finite\-basis PINNs \(FBPINNs\) combine networks supported on overlapping subdomains\(Jagtap et al\.,[2020](https://arxiv.org/html/2608.05702#bib.bib25); Jagtap and Karniadakis,[2021](https://arxiv.org/html/2608.05702#bib.bib24); Moseley et al\.,[2023](https://arxiv.org/html/2608.05702#bib.bib34)\)\. The cPINN and XPINN formulations also support distributed execution on central processing units \(CPUs\) and graphics processing units \(GPUs\), as implemented byShukla et al\. \([2021](https://arxiv.org/html/2608.05702#bib.bib52)\)\. Domain\-decomposed PINNs construct coupled PDE solutions, and knowledge\-integration surveys organize mechanisms for incorporating scientific knowledge into learning\(Willard et al\.,[2023](https://arxiv.org/html/2608.05702#bib.bib59)\)\. SEAM complements these approaches by auditing the compatibility of the structured regional records that their generators emit and by testing and comparing admissibility\-restoring hypotheses\. Relative to interface\-residual and sheaf\-consistency methods, SEAM integrates named scientific channels with residual\-aware budgeted hypothesis evaluation, identifiability, and streaming monitoring\.

### 1\.1Explanation\-admissibility and SEAM

We treat explanation\-admissibility as a local\-to\-global problem in the sense of sheaf theory\(Curry,[2014](https://arxiv.org/html/2608.05702#bib.bib9); Ghrist,[2014](https://arxiv.org/html/2608.05702#bib.bib14); Robinson,[2014](https://arxiv.org/html/2608.05702#bib.bib47)\)\. A family of structured local explanations is admissible when neighboring restrictions agree on every overlap\. SEAM\-Ω\\Omegastacks the regional records intos∈C0​\(F\)s\\in C^\{0\}\(F\), assembles the restriction maps into a coboundary matrixDD, and reports the resulting defectω=D​s\\omega=Dsby channel and overlap\. Section[3](https://arxiv.org/html/2608.05702#S3)gives the formal construction, and Appendix[A](https://arxiv.org/html/2608.05702#A1)collects the symbols used throughout\.

The SEAM paradigm has five stages: Generate, Restrict, Obstruct, Diagnose, and Intervene\. An identifiability extension records what the current observations cannot determine, while a monitoring extension tracksω\\omegaunder streaming inputs\.

Throughout this paper, a backend is an implementation of a local generatorGiG\_\{i\}, such as an analytic rule, a finite\-volume solver, a regression model, or a learned operator\. Once the backend output has been mapped to the common channel schema, the downstream SEAM audit is unchanged\.

##### SEAM\-Ω\\Omega\.

SEAM\-Ω\\Omegainstantiates this paradigm with a finite explanation sheafFF\. The stalks ofFFcarry three primary diagnostic channels for the stateuu, closurecc, and observationoo, with an optional contract\-metadata blockϵ\\epsilon\. The restriction maps are hand\-crafted linear maps to overlap stalks, the coboundary is the matrixDDassembled from those restrictions, and the admissibility defectω=D​s\\omega=Dsis computed in closed form\. The central object of SEAM\-Ω\\Omegais the 1\-cochainω\\omegaon the overlaps, whose decomposition and budgeted preimages define the diagnostic beyond any individual predicted state or learned residual\.

The channel decomposition localizes disagreement in predicted states, hidden physics, or sensor records; optional contract metadata can be audited when an application supplies a nonzero restriction for that metadata block\. For a declared intervention projectorPP, exact\-hard repair is feasible when the defect lies in the range of the restricted coboundary,ω∈im⁡\(D​P\)\\omega\\in\\operatorname\{im\}\(DP\); a feasible minimum\-cost repair prices a surviving hypothesis, whereas an infeasible budget is reported with its irreducible residual\. Regularized\-soft records provide a separate empirical attribution used when the single\-channel hard problems cannot fit the full defect\. The blind admissible subspaceℬ=ker⁡D∩ker⁡O\\mathcal\{B\}=\\ker D\\cap\\ker Orecords admissible directions invisible to the current observations, andω​\(t\)\\omega\(t\)supplies the streaming monitoring signal\.

### 1\.2Contributions and paper organization

The paper makes four contributions\.

##### Contribution 1: explanation\-admissibility as a new scientific object\.

Explanation\-admissibility is the local\-to\-global gluing of structured local explanations, a scientific object distinct from local accuracy and residual satisfaction\.

##### Contribution 2: the SEAM paradigm\.

The five\-stage pipeline sketched above, together with identifiability and monitoring extensions, defines a reusable design pattern for scientific systems that report whether explanations can be globally assembled\.

##### Contribution 3: the finite explanation\-sheaf instantiation SEAM\-Ω\\Omega\.

We give a concrete finite\-dimensional linear instantiation of the paradigm using vector\-space stalks and block\-diagonal linear restrictions on the 1\-skeleton of the cover’s nerve\. This instantiation yields two central theorems on minimum\-cost budgeted intervention and conservation\-contract detectability, with an identifiability proposition and a closure\-recovery corollary\.

##### Contribution 4: computational evidence\.

Nineteen experiments across synthetic PDE systems and six application domains evaluate the framework\. Reporting for stochastic and deterministic experiments follows the aggregate run protocol documented in Appendix[E\.1](https://arxiv.org/html/2608.05702#A5.SS1)\. The experiments evaluate local–global disagreement, closure recovery, exact\-hard and regularized\-soft intervention records, identifiability, backend interoperability, cross\-domain consistency, and learned\-generator monitoring\. Section[9](https://arxiv.org/html/2608.05702#S9)reports the complete numerical outcomes\.

##### Paper organization\.

Section[2](https://arxiv.org/html/2608.05702#S2)develops the SEAM paradigm at a level abstracted from any one instantiation\. Section[3](https://arxiv.org/html/2608.05702#S3)specializes to SEAM\-Ω\\Omegaby introducing finite explanation sheaves over the 1\-skeleton of a cover’s nerve, with structured stalks and the channel\-decomposed coboundary\. Section[4](https://arxiv.org/html/2608.05702#S4)develops obstruction\-guided diagnosis, the channel decomposition ofω\\omega, and the budgeted intervention problem\. Section[5](https://arxiv.org/html/2608.05702#S5)formalizes identifiability\. Section[6](https://arxiv.org/html/2608.05702#S6)extends the diagnostic to streaming monitoring of learned generators\. Section[7](https://arxiv.org/html/2608.05702#S7)states the finite\-dimensional result suite with explicit assumptions; detailed proofs are consolidated in Appendix[B](https://arxiv.org/html/2608.05702#A2)\. Section[8](https://arxiv.org/html/2608.05702#S8)gives the algorithms and the computational workflow of SEAM\-Ω\\Omega\. Section[9](https://arxiv.org/html/2608.05702#S9)reports nineteen experiments organized into eight groups aligned with the scientific claims that each group evaluates\. Section[10](https://arxiv.org/html/2608.05702#S10)compares SEAM with existing methods\. Section[11](https://arxiv.org/html/2608.05702#S11)discusses the design choices, interpretation guide, and relationship between the paradigm and the SEAM\-Ω\\Omegainstantiation\. The appendices consolidate notation, proofs, numerical details, the conservation\-theoretic SEAM \(CT\-SEAM\) backend, experimental protocols, data provenance, and robustness checks\.

## 2The SEAM paradigm

This section presents SEAM as a reusable local\-to\-global design principle\.

### 2\.1From local explanations to global admissibility

In SEAM, a scientific explanation is a structured regional object that records why a model produces its output\. In a PDE solver, the explanation includes the predicted stateuu, the assumed source or closurecc, the observationsooused for calibration, and optional contract metadataϵ\\epsilon, such as conservation\-repair magnitudes\. The metadata block may be zero\-dimensional and is not an intervention budget\. In a tabular regression model, the explanation includes the predicted output, residual statistics, sufficient statistics of the input distribution, and auxiliary information needed to interpret the prediction scientifically\.

The defining property that distinguishes a scientific explanation from a prediction is that explanations from neighboring regions must be jointly assertable\. If the explanation on regionUiU\_\{i\}asserts that the local state isuiu\_\{i\}and the closure iscic\_\{i\}, and the explanation on regionUjU\_\{j\}makes the analogous assertion, then, on the overlapUi∩UjU\_\{i\}\\cap U\_\{j\}, those two assertions must agree as statements about the same physical quantity\. Locally, each assertion may be plausible, internally consistent, and well supported by data\. Jointly, the two assertions may be incompatible, because the evidence that certifies each assertion is confined to the region on which that evidence was produced\. A regional residual, a held\-out score, or a goodness\-of\-fit statistic onUiU\_\{i\}constrains the explanationeie\_\{i\}only through quantities visible insideUiU\_\{i\}, typically admitting an entire family of explanations that pass equally well without ever referring toeje\_\{j\}\. Each generator is thus free to select a different member of its own locally plausible family, so agreement onUi∩UjU\_\{i\}\\cap U\_\{j\}is a further, joint requirement that no amount of local validation tests\. Local plausibility is a property of the individual explanations, whereas admissibility is a property of the family\. Figure[2](https://arxiv.org/html/2608.05702#S2.F2)contrasts a family of highly accurate regional models that disagree on their overlaps with a rougher family that agrees on every overlap\.

This phenomenon is generic\. Four mechanisms produce such joint incompatibility\. We refer to these mechanisms as M1–M4 throughout:

##### Mechanism M1: independent regional construction\.

Regional models specified without reference to one another agree with their own local evidence and need not agree with each other on an overlap\. Two PDE solvers with independently chosen meshes, time steps, or numerical fluxes are both convergent in their interiors, yet their boundary fluxes do not exactly cancel on the overlap, producing an asymptotically vanishing but operationally measurable disagreement\. When the regional specifications themselves differ, as with independently calibrated amplitudes or coefficients, the disagreement persists under refinement\. The analogous coupling problem is central to co\-simulation, where independently implemented simulator units exchange interface variables at communication points\(Gomes et al\.,[2018](https://arxiv.org/html/2608.05702#bib.bib15)\)\.

##### Mechanism M2: regime non\-stationarity\.

Two seasonal time\-series models, each accurate within its season, fit different autoregressive coefficients or different residual volatilities\. At the inter\-season transition, both models extrapolate, and the two extrapolations disagree by a magnitude unrelated to either model’s in\-regime error\.

##### Mechanism M3: sensor–physics conflict\.

A subset of sensor streams is corrupted, drifted, or biased\. A regional model that relies on the affected sensors infers the physics needed to fit those streams\. A neighboring model fits cleaner sensors, inferring different physics\. The disagreement appears as inconsistency between the two regional closures, even though both models are locally calibrated\.

##### Mechanism M4: learned\-generator distribution shift\.

A neural operator trained on a distribution of inputs is evaluated on a covariate\-shifted input\. Each region of the new input may remain close to inputs the operator has seen, while the joint output across regions is no longer on the operator’s learned manifold\. The local predictions are still well\-formed and locally plausible; the global assembly is not\.

Together, these mechanisms expose the cross\-region consistency gap that SEAM audits, and the same mechanisms organize the evaluation: Mechanism M1 is realized by the independently constructed regional models of Section[9\.2](https://arxiv.org/html/2608.05702#S9.SS2), M2 by the seasonal\-regime studies of Section[9\.7](https://arxiv.org/html/2608.05702#S9.SS7), M3 by the sensor\-corruption study of Section[9\.4](https://arxiv.org/html/2608.05702#S9.SS4), and M4 by the neural\-operator monitoring study of Section[9\.8](https://arxiv.org/html/2608.05702#S9.SS8)\. The four cases listed in Section[1](https://arxiv.org/html/2608.05702#S1)instantiate M1–M4 in that order\.

Independent local models excellent local fit separately trainedRegionU1U\_\{1\}R12=0\.999R^\{2\}\_\{1\}=0\.999RegionU2U\_\{2\}R22=0\.999R^\{2\}\_\{2\}=0\.999RegionU3U\_\{3\}R32=0\.999R^\{2\}\_\{3\}=0\.999‖ω12‖\>0\\left\\\|\\omega\_\{12\}\\right\\\|\>0‖ω23‖\>0\\left\\\|\\omega\_\{23\}\\right\\\|\>0Global verdictInadmissible‖ω‖\>0\\left\\\|\\omega\\right\\\|\>0Shared global generator rougher local fit one coherent sourceRegionU1U\_\{1\}R12=0\.85R^\{2\}\_\{1\}=0\.85RegionU2U\_\{2\}R22=0\.85R^\{2\}\_\{2\}=0\.85RegionU3U\_\{3\}R32=0\.85R^\{2\}\_\{3\}=0\.85‖ω12‖=0\\left\\\|\\omega\_\{12\}\\right\\\|=0‖ω23‖=0\\left\\\|\\omega\_\{23\}\\right\\\|=0Global verdictAdmissible‖ω‖=0\\left\\\|\\omega\\right\\\|=0

Figure 2:Local accuracy and global admissibility are orthogonal\. The top row shows a schematic family of three locally trained models in which each model achieves high local accuracy but disagrees on overlaps, so the family is globally inadmissible\. The bottom row shows three deliberately rougher local fits withRi2=0\.85R^\{2\}\_\{i\}=0\.85that share a single global generator and agree exactly on overlaps:‖ω‖=0\\left\\\|\\omega\\right\\\|=0\.

### 2\.2The five\-stage paradigm and its extensions

We define SEAM as a five\-stage data flow with identifiability and monitoring extensions, shown in Figure[3](https://arxiv.org/html/2608.05702#S2.F3)\.

###### Definition 2\.1\(SEAM paradigm\)\.

A SEAM system consists of the following five stages:

##### Generate\.

A family of local generators\{Gi\}i∈I\\\{G\_\{i\}\\\}\_\{i\\in I\}produces, for each regionUiU\_\{i\}of a finite cover𝒰\\mathcal\{U\}of the problem domain, a structured local explanationeie\_\{i\}\.

##### Restrict\.

For each overlapUi∩UjU\_\{i\}\\cap U\_\{j\}, a restriction mapρi,i​j\\rho\_\{i,ij\}extracts fromeie\_\{i\}the object that must be compared on the overlap\. The object can be a state value, a closure parameter, a posterior mean, a sensor mean, or an admissibility flag\.

##### Obstruct\.

The disagreement betweenρi,i​j​\(ei\)\\rho\_\{i,ij\}\(e\_\{i\}\)andρj,i​j​\(ej\)\\rho\_\{j,ij\}\(e\_\{j\}\)on every overlap is assembled into an obstruction objectω\\omega\.

##### Diagnose\.

The obstruction is summarized using the diagnostic attributes exposed by the chosen instantiation, producing a report that localizes or qualifies the inadmissibility\.

##### Intervene\.

A family of allowable intervention budgets is considered\. Each budget states one scientific hypothesis about the cause of the obstruction by declaring the revisions that the hypothesis permits\. For each budget, the system tests whether the corresponding repair problem can eliminate the obstruction\. Feasible hard repairs are retained by the verdict; infeasible budgets contribute residual or diagnostic information declared by the chosen instantiation\.

Two further extensions are part of the paradigm:

##### Identify\.

An identifiability analyzer determines which parts of the admissible explanation object remain unresolved by current observations\.

##### Monitor\.

A SEAM system is also expected to support streaming evaluation ofω​\(t\)\\omega\(t\)for a fixed family of local generators applied to streaming inputs\.

Definition[2\.1](https://arxiv.org/html/2608.05702#S2.Thmtheorem1)leaves the mathematical type ofeie\_\{i\},ρi,i​j\\rho\_\{i,ij\}, andω\\omegaopen\. Four conditions suffice for an instantiation to realize this abstract definition\. Eachρi,i​j\\rho\_\{i,ij\}must be well\-defined for the local explanation object used by the instantiation\. The equationω=0\\omega=0must provide a notion of admissibility, and the intervention problem must provide a notion of cost\. Finally, the instantiation must expose the diagnostic attributes needed to interpretω\\omega\. The finite explanation\-sheaf instantiation of Section[3](https://arxiv.org/html/2608.05702#S3)fulfills these requirements with vector\-space stalks, a named channel decomposition, and linear repair equations\.

Generate\{Gi\}i∈I\\\{G\_\{i\}\\\}\_\{i\\in I\}Restrictρi,i​j\\rho\_\{i,ij\}Obstructω\\omegaDiagnosereportIntervenerepairIdentifyunresolvedMonitorω​\(t\)\\omega\(t\)

Figure 3:The SEAM paradigm\. The five core stages in the top row take a family of local generators through restriction, obstruction, diagnosis, and budgeted intervention\. The Identify extension records unresolved admissible directions, and the Monitor extension readsω\\omegaas a stream\. SEAM\-Ω\\Omegainstantiates these stages with finite cellular sheaves\.

### 2\.3Explanation\-admissibility

The central object of SEAM is explanation\-admissibility\.

###### Definition 2\.2\(Explanation\-admissibility\)\.

A family of local explanations\(ei\)i∈I\(e\_\{i\}\)\_\{i\\in I\}on a cover𝒰\\mathcal\{U\}is globally admissible if, for every overlap\(i,j\)∈E\(i,j\)\\in E,

ρi,i​j​\(ei\)=ρj,i​j​\(ej\)\.\\displaystyle\\rho\_\{i,ij\}\(e\_\{i\}\)=\\rho\_\{j,ij\}\(e\_\{j\}\)\.Equivalently, the obstruction object of the family vanishes:ω=0\\omega=0\.

In cellular\-sheaf terms, admissibility is the global\-section condition\. The term emphasizes its scientific interpretation: Neighboring explanations make mutually compatible claims on every overlap\. Together with admissibility, the following three definitions specify how a SEAM instantiation represents the obstruction diagnostically, delimits allowable repairs, and converts repair feasibility and cost information into a scientific verdict\.

###### Definition 2\.3\(Diagnostic decomposition\)\.

The obstruction objectω\\omegais reported together with diagnostic attributes exposed by the chosen SEAM instantiation\. These attributes may be named channels, regions, uncertainty components, status fields, or other interpretable summaries\. In SEAM\-Ω\\Omega, the attributes are the three primary channels and the optional contract\-metadata channel introduced in Section[3\.2](https://arxiv.org/html/2608.05702#S3.SS2)\.

###### Definition 2\.4\(Intervention budget\)\.

An intervention budget is a declared class of allowable repairs together with its cost convention and feasibility test\. A hard budget is feasible forω\\omegaif some allowed repair eliminates the obstruction under the semantics of the chosen instantiation\. In SEAM\-Ω\\Omega, a budget is represented by an orthogonal projectorPPand a cost metricCC, and hard feasibility is the linear conditionω∈im⁡\(D​P\)\\omega\\in\\operatorname\{im\}\(DP\)developed in Section[4\.2](https://arxiv.org/html/2608.05702#S4.SS2)\.

###### Definition 2\.5\(Scientific verdict\)\.

Given a candidate set of budgets\{bk\}\\\{b\_\{k\}\\\}, a numerical zero\-obstruction toleranceτzero\\tau\_\{\\mathrm\{zero\}\}, and an optional unrestricted fallback budgetballb\_\{\\mathrm\{all\}\}, a SEAM verdict is defined piecewise\. If‖ω‖≤τzero\\left\\\|\\omega\\right\\\|\\leq\\tau\_\{\\mathrm\{zero\}\}, the verdict is𝗀𝗅𝗈𝖻𝖺𝗅𝗅𝗒​\_​𝖺𝖽𝗆𝗂𝗌𝗌𝗂𝖻𝗅𝖾\\mathsf\{globally\\\_admissible\}, and no repair budget is selected\. When‖ω‖\>τzero\\left\\\|\\omega\\right\\\|\>\\tau\_\{\\mathrm\{zero\}\}, define the exact\-feasible budget set by

ℱ​\(ω\)≔\{bk:the hard repair problem for​bk​is feasible for​ω\}\.\\displaystyle\\mathcal\{F\}\(\\omega\)\\coloneqq\\\{b\_\{k\}:\\text\{the hard repair problem for \}b\_\{k\}\\text\{ is feasible for \}\\omega\\\}\.In SEAM\-Ω\\Omega, this feasible\-set definition becomesℱ​\(ω\)=\{bk:ω∈im⁡\(D​Pbk\)\}\\mathcal\{F\}\(\\omega\)=\\\{b\_\{k\}:\\omega\\in\\operatorname\{im\}\(DP\_\{b\_\{k\}\}\)\\\}\. The feasible set of specific accounts is

ℱproper​\(ω\)≔ℱ​\(ω\)∖\{ball\},\\displaystyle\\mathcal\{F\}\_\{\\mathrm\{proper\}\}\(\\omega\)\\coloneqq\\mathcal\{F\}\(\\omega\)\\setminus\\\{b\_\{\\mathrm\{all\}\}\\\},with the set subtraction omitted when no unrestricted fallback is in the catalog\. Ifℱproper​\(ω\)=∅\\mathcal\{F\}\_\{\\mathrm\{proper\}\}\(\\omega\)=\\varnothing, the verdict is unresolved by the declared specific accounts and reports their residuals\. If this set has one element, that account is the only specific account retained by exact feasibility\. If the set has several elements, all are retained, and the attribution is non\-unique\. Each feasible budget carries its own minimum repair cost, but costs do not select among accounts with different permitted subspaces\.

In SEAM\-Ω\\Omega, scientific verdicts are based on exact feasibility, while soft records provide comparative attribution together with their residuals\. The exact branch is decisive when the obstruction concentrates in the channel that a budget permits\. Under the block\-diagonal channel restrictions adopted in Section[3\.4](https://arxiv.org/html/2608.05702#S3.SS4), a single\-channel hard budget is feasible only when the obstruction components outside that channel vanish, so an obstruction spread across several channels refutes every single\-channel account, and the attribution then rests on the soft records with their residuals\. Section[4](https://arxiv.org/html/2608.05702#S4)develops this interpretation\.

The three definitions above form one hypothesis test\. A candidate set of budgets is a set of declared diagnostic accounts of the same obstruction, andℱproper​\(ω\)\\mathcal\{F\}\_\{\\mathrm\{proper\}\}\(\\omega\)collects the specific accounts that survive the overlap evidence\. A budget outsideℱ​\(ω\)\\mathcal\{F\}\(\\omega\)is not merely expensive: No revision the budget permits removes the obstruction at all, so the evidence rules that account out, and only its residual is reported\. An empty or multi\-element proper feasible set makes the attribution unresolved or non\-unique, respectively\. The unrestricted fallback certifies repairability but is never interpreted as a causal account\.

The objects compared on an overlap can be regional predictions, residuals, closure summaries, observation records, or contract metadata\. Disagreements among those objects enter the same channel and overlap diagnosis, budgeted intervention, and blind\-direction analysis\. The local–global amplitude\-mismatch study in Section[9\.2\.1](https://arxiv.org/html/2608.05702#S9.SS2.SSS1)provides the first numerical example\.

## 3SEAM\-Ω\\Omega: finite explanation sheaves

We now instantiate the SEAM paradigm using finite cellular sheaves\(Curry,[2014](https://arxiv.org/html/2608.05702#bib.bib9); Robinson,[2014](https://arxiv.org/html/2608.05702#bib.bib47); Ghrist,[2014](https://arxiv.org/html/2608.05702#bib.bib14)\)\. SEAM\-Ω\\Omegaassembles the degree\-zero coboundary of the pairwise\-overlap complex to obtain closed\-form diagnostic, intervention, identifiability, and monitoring operators\. The channel\-resolved results use the block\-diagonal restrictions stated in Assumption A0\. Figure[4](https://arxiv.org/html/2608.05702#S3.F4)shows the resulting construction on a three\-region cover\.

### 3\.1Covers and nerves

Let𝒳⊆ℝd\\mathcal\{X\}\\subseteq\\mathbb\{R\}^\{d\}be the computational domain\. A cover𝒰=\(Ui\)i∈I\\mathcal\{U\}=\(U\_\{i\}\)\_\{i\\in I\}is a finite collection of regionsUi⊆𝒳U\_\{i\}\\subseteq\\mathcal\{X\}with⋃i∈IUi=𝒳\\bigcup\_\{i\\in I\}U\_\{i\}=\\mathcal\{X\}\. Overlaps are permitted, and pairwise intersectionsUi​j≔Ui∩UjU\_\{ij\}\\coloneqq U\_\{i\}\\cap U\_\{j\}may be empty\. A nonemptyUi​jU\_\{ij\}is the comparison support for the corresponding edge\. In spatial applications, comparison supports are normally required to have nontrivial measure; an abstract cover may instead specify a finite set of shared evaluation points\.

The nerve of𝒰\\mathcal\{U\}, denoted𝒩𝒰\\mathcal\{N\}\_\{\\mathcal\{U\}\}, is the simplicial complex withkk\-simplices given by the\(k\+1\)\(k\+1\)\-tuples\(i0,…,ik\)\(i\_\{0\},\\ldots,i\_\{k\}\)of indices whose corresponding intersection is nonempty\. For SEAM\-Ω\\Omega, we truncate𝒩𝒰\\mathcal\{N\}\_\{\\mathcal\{U\}\}to its 1\-skeleton\. The vertices of that 1\-skeleton are regionsi∈Ii\\in I, and the edges are pairsE=\{\(i,j\):i<j,Ui​j≠∅\}E=\\\{\(i,j\):i<j,U\_\{ij\}\\neq\\emptyset\\\}\.

RegionU1U\_\{1\}F​\(U1\)F\(U\_\{1\}\)stateu1u\_\{1\}closurec1c\_\{1\}observationo1o\_\{1\}metadataϵ1\\epsilon\_\{1\}RegionU2U\_\{2\}F​\(U2\)F\(U\_\{2\}\)stateu2u\_\{2\}closurec2c\_\{2\}observationo2o\_\{2\}metadataϵ2\\epsilon\_\{2\}RegionU3U\_\{3\}F​\(U3\)F\(U\_\{3\}\)stateu3u\_\{3\}closurec3c\_\{3\}observationo3o\_\{3\}metadataϵ3\\epsilon\_\{3\}U12U\_\{12\}overlapstalkF​\(U12\)F\(U\_\{12\}\)U23U\_\{23\}overlapstalkF​\(U23\)F\(U\_\{23\}\)ρ1,12\\rho\_\{1,12\}ρ2,12\\rho\_\{2,12\}ρ2,23\\rho\_\{2,23\}ρ3,23\\rho\_\{3,23\}ω12=ρ2,12​s2−ρ1,12​s1\\omega\_\{12\}=\\rho\_\{2,12\}s\_\{2\}\-\\rho\_\{1,12\}s\_\{1\}ω23=ρ3,23​s3−ρ2,23​s2\\omega\_\{23\}=\\rho\_\{3,23\}s\_\{3\}\-\\rho\_\{2,23\}s\_\{2\}

Figure 4:Finite explanation sheafFFon a three\-region cover\. Each regionUiU\_\{i\}carries a structured stalk partitioned into state, closure, and observation channels plus optional contract metadata\. Each overlapUi​jU\_\{ij\}carries an overlap stalkF​\(Ui​j\)F\(U\_\{ij\}\)and two restriction maps\. The raw admissibility defectω=D​s\\omega=Dscollects the per\-overlap disagreementsωi​j=ρj,i​j​sj−ρi,i​j​si\\omega\_\{ij\}=\\rho\_\{j,ij\}s\_\{j\}\-\\rho\_\{i,ij\}s\_\{i\}\. Under the block\-diagonal channel restrictions of Assumption A0, the channel decomposition ofω\\omegacommutes with the coboundary\.
### 3\.2Explanation stalks and channel conventions

We now equip each region with a structured local explanation space\.

###### Definition 3\.2\(Local explanation\)\.

A local explanation on regionUiU\_\{i\}is a tuple

ei=\(ui,ci,oi,ϵi\),\\displaystyle e\_\{i\}=\(u\_\{i\},c\_\{i\},o\_\{i\},\\epsilon\_\{i\}\),whereui∈ℝnu,iu\_\{i\}\\in\\mathbb\{R\}^\{n\_\{u,i\}\}is the discretized state field on the per\-region grid and contains predicted values of the dependent variable of interest\. The closure vectorci∈ℝnc,ic\_\{i\}\\in\\mathbb\{R\}^\{n\_\{c,i\}\}represents parameters or discretized values of hidden source terms, unresolved subgrid physics, or another latent block that the local generator attributes to the region\. The observation summaryoi∈ℝno,io\_\{i\}\\in\\mathbb\{R\}^\{n\_\{o,i\}\}contains sensor means and the flattened diagonal of the local measurement covariance\. The optional contract\-metadata vectorϵi∈ℝnϵ,i\\epsilon\_\{i\}\\in\\mathbb\{R\}^\{n\_\{\\epsilon,i\}\}contains per\-contract repair magnitudes, conservation residuals, and other scalar admissibility flags emitted by the local generator;nϵ,in\_\{\\epsilon,i\}may be zero\. Formulas label this optional block asmeta\\mathrm\{meta\}\. The block is metadata and is distinct from an intervention budgetPP, which specifies a subspace of allowable corrections as in Definition[2\.4](https://arxiv.org/html/2608.05702#S2.Thmtheorem4)\. The embedding of the explanation is the concatenationvec​\(ei\)∈ℝdi\\mathrm\{vec\}\(e\_\{i\}\)\\in\\mathbb\{R\}^\{d\_\{i\}\}withdi≔nu,i\+nc,i\+no,i\+nϵ,id\_\{i\}\\coloneqq n\_\{u,i\}\+n\_\{c,i\}\+n\_\{o,i\}\+n\_\{\\epsilon,i\}\. The common uniform\-dimensional case is recovered by setting the nonzero dimensions independent ofii\.

###### Definition 3\.3\(Explanation stalk\)\.

The explanation stalk at regionUiU\_\{i\}isF​\(Ui\)≔ℝdiF\(U\_\{i\}\)\\coloneqq\\mathbb\{R\}^\{d\_\{i\}\}, equipped with the standard inner product and the partition into three primary diagnostic blocks and one optional metadata block,F​\(Ui\)=F​\(Ui\)state⊕F​\(Ui\)closure⊕F​\(Ui\)obs⊕F​\(Ui\)metaF\(U\_\{i\}\)=F\(U\_\{i\}\)^\{\\mathrm\{state\}\}\\oplus F\(U\_\{i\}\)^\{\\mathrm\{closure\}\}\\oplus F\(U\_\{i\}\)^\{\\mathrm\{obs\}\}\\oplus F\(U\_\{i\}\)^\{\\mathrm\{meta\}\}with dimensionsnu,in\_\{u,i\},nc,in\_\{c,i\},no,in\_\{o,i\}, andnϵ,in\_\{\\epsilon,i\}\.

The choice of which fields go into which channel is part of the modeling design\. We discuss two canonical conventions\.

##### PDE convention\.

For PDE generators, the state block holds the discretized solution on the per\-region grid; the closure block holds parameters or discretized values of any source, forcing, or subgrid model; and the observation block holds the sensor means and covariance diagonal on the region\. When present, the optional metadata block holds cumulative magnitudes of conservation repairs applied during the solve\.

##### Regression convention\.

For wrapped regression predictors, as used throughout the tabular regression studies, the state block holds the predictor outputs at the overlap evaluation points; the closure block is zero or carries residual statistics; the observation block is zero; and the optional metadata block is absent\. The framework gracefully degenerates to a pure state\-channel diagnostic in this case\.

### 3\.3Restriction maps, coboundary assembly, and the raw admissibility defect

For each overlap\(i,j\)∈E\(i,j\)\\in E, we fix an overlap stalkF​\(Ui​j\)≔ℝmi​jF\(U\_\{ij\}\)\\coloneqq\\mathbb\{R\}^\{m\_\{ij\}\}and two restriction mapsρi,i​j:F​\(Ui\)→F​\(Ui​j\)\\rho\_\{i,ij\}\\colon F\(U\_\{i\}\)\\to F\(U\_\{ij\}\)andρj,i​j:F​\(Uj\)→F​\(Ui​j\)\\rho\_\{j,ij\}\\colon F\(U\_\{j\}\)\\to F\(U\_\{ij\}\)\.

###### Definition 3\.4\(Restriction map\)\.

A restriction mapρi,i​j:F​\(Ui\)→F​\(Ui​j\)\\rho\_\{i,ij\}\\colon F\(U\_\{i\}\)\\to F\(U\_\{ij\}\)is a linear map represented by a realmi​j×dim\_\{ij\}\\times d\_\{i\}matrix that sendsvec​\(ei\)\\mathrm\{vec\}\(e\_\{i\}\)to the quantities that must be compared on the overlap\. The map respects channels, meaning thatF​\(Ui​j\)F\(U\_\{ij\}\)decomposes asF​\(Ui​j\)=F​\(Ui​j\)state⊕F​\(Ui​j\)closure⊕F​\(Ui​j\)obs⊕F​\(Ui​j\)metaF\(U\_\{ij\}\)=F\(U\_\{ij\}\)^\{\\mathrm\{state\}\}\\oplus F\(U\_\{ij\}\)^\{\\mathrm\{closure\}\}\\oplus F\(U\_\{ij\}\)^\{\\mathrm\{obs\}\}\\oplus F\(U\_\{ij\}\)^\{\\mathrm\{meta\}\}andρi,i​j\\rho\_\{i,ij\}acts block\-diagonally with respect to the two partitions\.

The state\-channel block ofρi,i​j\\rho\_\{i,ij\}is typically an interpolation operator that resamples the per\-region state grid onto the overlap grid\. The closure\-channel block extracts the closure quantities relevant to the overlap region\. In the simplest case, that block is the identity onF​\(Ui\)closureF\(U\_\{i\}\)^\{\\mathrm\{closure\}\}followed by a linear map intoF​\(Ui​j\)closureF\(U\_\{ij\}\)^\{\\mathrm\{closure\}\}\. The observation\-channel block selects the sensor entries that fall in the overlap\. The optional metadata block is zero unless a contract quantity, such as a global conservation residual, is explicitly shared across regions\.

We assemble the per\-region stalks into the cochain spaces

C0​\(F\)≔⨁i∈IF​\(Ui\),C1​\(F\)≔⨁\(i,j\)∈EF​\(Ui​j\)\.\\displaystyle C^\{0\}\(F\)\\coloneqq\\bigoplus\_\{i\\in I\}F\(U\_\{i\}\),\\qquad C^\{1\}\(F\)\\coloneqq\\bigoplus\_\{\(i,j\)\\in E\}F\(U\_\{ij\}\)\.
###### Definition 3\.6\(Coboundary\)\.

The coboundary ofFFis the linear mapd0:C0​\(F\)→C1​\(F\)d^\{0\}\\colon C^\{0\}\(F\)\\to C^\{1\}\(F\)defined by

\(d0​s\)i​j≔ρj,i​j​sj−ρi,i​j​si,s=\(si\)i∈I∈C0​\(F\),\(i,j\)∈E\.\\displaystyle\(d^\{0\}s\)\_\{ij\}\\coloneqq\\rho\_\{j,ij\}s\_\{j\}\-\\rho\_\{i,ij\}s\_\{i\},\\qquad s=\(s\_\{i\}\)\_\{i\\in I\}\\in C^\{0\}\(F\),\(i,j\)\\in E\.We writeD≔mat​\(d0\)D\\coloneqq\\mathrm\{mat\}\(d^\{0\}\)for the assembled coboundary matrix in the standard bases\.

A global section ofFFis an elements∈ker⁡d0s\\in\\ker d^\{0\}: a family of stalk vectors whose restrictions agree on every overlap\.

###### Definition 3\.7\(Raw admissibility defect\)\.

Given a family of local explanations\(ei\)i∈I\(e\_\{i\}\)\_\{i\\in I\}, letsi≔vec​\(ei\)s\_\{i\}\\coloneqq\\mathrm\{vec\}\(e\_\{i\}\)\. The raw admissibility defectω∈C1​\(F\)\\omega\\in C^\{1\}\(F\)is the 1\-cochain

ω≔d0​s,ωi​j=ρj,i​j​sj−ρi,i​j​si,\(i,j\)∈E\.\\displaystyle\\omega\\coloneqq d^\{0\}s,\\qquad\\omega\_\{ij\}=\\rho\_\{j,ij\}s\_\{j\}\-\\rho\_\{i,ij\}s\_\{i\},\\qquad\(i,j\)\\in E\.The terms raw admissibility defect and raw obstruction cochain are synonymous for thisω\\omega\.

The symbol∥⋅∥\\left\\\|\\cdot\\right\\\|denotes the Euclidean norm induced by the standard inner products, and an explicit subscript22marks that same norm\. The one departure is∥⋅∥C\\left\\\|\\cdot\\right\\\|\_\{C\}, the norm induced by the positive\-definite cost metricCCintroduced in Section[3\.5](https://arxiv.org/html/2608.05702#S3.SS5)\. Channel weights and whitening maps act on the argument, as in‖W∙​ω∙‖2\\left\\\|W\_\{\\bullet\}\\omega^\{\\bullet\}\\right\\\|\_\{2\}, leaving the norm Euclidean; the remaining subscripts name the object being measured rather than a metric\. The norm‖ω‖\\left\\\|\\omega\\right\\\|is the scalar summary of inadmissibility, and the channel norms given in Eq\.[1](https://arxiv.org/html/2608.05702#S3.E1), together with the per\-overlap norms, refine that summary\. The subsection below introduces hypothesis\-specific budget residuals\.

### 3\.4Channel decomposition ofω\\omega

###### Assumption A0\(Block\-diagonal channel restrictions\)\.

For the channel decomposition to commute with the coboundary, restriction maps act block\-diagonally with respect to the channel partition\. Eachρi,i​j\\rho\_\{i,ij\}has no off\-block entries mixing state, closure, observation, and optional metadata components\. This block\-diagonal restriction structure is a modeling choice we adopt throughout SEAM\-Ω\\Omega\. We index this assumption by zero because the block\-diagonal structure is adopted here for the construction itself rather than for any individual result; Section[7](https://arxiv.org/html/2608.05702#S7)lists Assumption A0 alongside the repair and model conventions used by the result suite\.

Under Assumption A0,DDis block diagonal with respect to the channel decomposition; its pseudoinverse and the projectorDD\+DD\{\{\}^\{\+\}\}inherit the block structure\. The defect therefore decomposes:

ω=ωstate⊕ωclosure⊕ωobs⊕ωmeta,\\omega=\\omega^\{\\mathrm\{state\}\}\\oplus\\omega^\{\\mathrm\{closure\}\}\\oplus\\omega^\{\\mathrm\{obs\}\}\\oplus\\omega^\{\\mathrm\{meta\}\},\(1\)where∙\\bulletranges over the three primary channel labels and the optional metadata label\{state,closure,obs,meta\}\\\{\\mathrm\{state\},\\mathrm\{closure\},\\mathrm\{obs\},\\mathrm\{meta\}\\\}, and each overlap componentωi​j∙\\omega^\{\\bullet\}\_\{ij\}lives in the corresponding sub\-blockF​\(Ui​j\)∙F\(U\_\{ij\}\)^\{\\bullet\}\. Channel comparisons use a fixed channel metric or normalization\. In the simplest unit\-scaled case, we summarize each channel by its norm:‖ωstate‖\\left\\\|\\omega^\{\\mathrm\{state\}\}\\right\\\|,‖ωclosure‖\\left\\\|\\omega^\{\\mathrm\{closure\}\}\\right\\\|,‖ωobs‖\\left\\\|\\omega^\{\\mathrm\{obs\}\}\\right\\\|, and‖ωmeta‖\\left\\\|\\omega^\{\\mathrm\{meta\}\}\\right\\\|\. The three primary scalars provide the diagnostic report; the optional meta\-labeled scalar records contract\-metadata disagreement when that block is present\. With channel weights or whitening matricesW∙W\_\{\\bullet\}, we compute the denominator and, whenZ\>0Z\>0, the calibrated fractions

Z≔∑∘‖W∘​ω∘‖22,p∙≔‖W∙​ω∙‖22Z\(Z\>0\),\\displaystyle Z\\coloneqq\\sum\_\{\\circ\}\\left\\\|W\_\{\\circ\}\\omega^\{\\circ\}\\right\\\|\_\{2\}^\{2\},\\qquad p\_\{\\bullet\}\\coloneqq\\frac\{\\left\\\|W\_\{\\bullet\}\\omega^\{\\bullet\}\\right\\\|\_\{2\}^\{2\}\}\{Z\}\\quad\(Z\>0\),where∘\\circranges over the active labels in the same set as∙\\bullet\. The channel reported as dominant is the one maximizingp∙p\_\{\\bullet\}, so a dominance report is meaningful only relative to the declaredW∙W\_\{\\bullet\}\. Section[4\.1](https://arxiv.org/html/2608.05702#S4.SS1)states the reporting rule that these fractions feed, including the degenerateZ=0Z=0case\.

### 3\.5Budgeted obstruction and residual\-aware attribution

A scientific hypothesis restricts the allowed correction to a strict subspaceim⁡P⊊C0​\(F\)\\operatorname\{im\}P\\subsetneq C^\{0\}\(F\)\. For example, a hypothesis may allow revision only of the closure block\. The hard residual below tests whether the restricted operatorD​PDPcan reproduce the full raw defect\. A regularized\-soft record is reported separately when the restricted operator cannot\.

###### Definition 3\.9\(Budgeted obstruction\)\.

LetP∈ℝN×NP\\in\\mathbb\{R\}^\{N\\times N\}be the orthogonal projector onto a closed budget subspace ofC0​\(F\)C^\{0\}\(F\), and letC≻0C\\succ 0be the metric used to measure corrections\. The hard budgeted obstruction residual at budgetPPis

rPhard\(ω\)≔ω−\(DP\)\(DP\)ω\+∈ker\(DP\)⊤\.\\displaystyle r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\)\\coloneqq\\omega\-\(DP\)\(DP\)\{\{\}^\{\+\}\}\\omega\\in\\ker\(DP\)^\{\\top\}\.The budgeted feasibility indicator is the Boolean𝟏​\[rPhard​\(ω\)=0\]\\mathbf\{1\}\[r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\)=0\], equivalently𝟏​\[ω∈im⁡\(D​P\)\]\\mathbf\{1\}\[\\omega\\in\\operatorname\{im\}\(DP\)\]\. The hard budgeted intervention cost is the metric\-dependent extended\-real value

cP,Chard​\(ω\)≔\{‖δP,C⋆‖C2,ω∈im⁡\(D​P\),\+∞,ω∉im⁡\(D​P\),\\displaystyle c\_\{P,C\}^\{\\mathrm\{hard\}\}\(\\omega\)\\coloneqq\\begin\{cases\}\\left\\\|\\delta\_\{P,C\}^\{\\star\}\\right\\\|\_\{C\}^\{2\},&\\omega\\in\\operatorname\{im\}\(DP\),\\\\ \+\\infty,&\\omega\\notin\\operatorname\{im\}\(DP\),\\end\{cases\}whereδP,C⋆\\delta\_\{P,C\}^\{\\star\}is the minimum\-cost feasible correction from Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1)in the first case\. The residual depends onPPand the codomain metric used for projection, while the minimizer and cost also depend onCC\. We often writerPr\_\{P\}andcPhardc\_\{P\}^\{\\mathrm\{hard\}\}whenCCand the hard\-residual convention are fixed\. A budget status is exact\-feasible, tolerance\-feasible, or infeasible with a soft diagnostic\. The modifier exact\-hard is reserved for costs and verdicts derived solely from exact\-feasible hard problems\. Scientific verdicts are based on exact feasibility, while tolerance and soft empirical attribution records include their residuals\(Engl et al\.,[1996](https://arxiv.org/html/2608.05702#bib.bib12)\)\.

At the unrestricted budgetP=IP=\\mathrm\{I\}, every raw admissibility defect is exactly repairable\. For proper budgets such as𝖠𝗅𝗅𝗈𝗐𝖢𝗅𝗈𝗌𝗎𝗋𝖾\\mathsf\{AllowClosure\}or𝖠𝗅𝗅𝗈𝗐𝖲𝖾𝗇𝗌𝗈𝗋𝖱𝖾𝗃𝖾𝖼𝗍𝗂𝗈𝗇\\mathsf\{AllowSensorRejection\},rPhard​\(ω\)r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\)is generically nontrivial and is the operational signal of hypothesis feasibility\.

## 4Obstruction\-guided diagnosis and intervention

We now turn the obstruction cochain into a scientific diagnostic\. The pipeline has two primary outputs\. The per\-channel decomposition provides a static map of where disagreement occurs, while the budgeted preimage computation tests which permitted revisions can explain that disagreement\. Exact\-hard feasibility and the set of retained proper budgets determine the formal verdict; each feasible budget’s cost quantifies its own repair burden\. Residual\-aware regularized records are reported separately as empirical attribution rather than as exact tests\.

### 4\.1From obstruction to channel diagnosis

The first component reads the channel decomposition of Eq\.[1](https://arxiv.org/html/2608.05702#S3.E1)without solving any optimization\.

##### Total and per\-channel norms\.

We compute the total raw norm and, for each channel∙∈\{state,closure,obs,meta\}\\bullet\\in\\\{\\mathrm\{state\},\\mathrm\{closure\},\\mathrm\{obs\},\\mathrm\{meta\}\\\}, the corresponding per\-channel norm:

‖ω‖22=∑\(i,j\)∈E‖ωi​j‖22,‖ω∙‖22=∑\(i,j\)∈E‖ωi​j∙‖22\.\\displaystyle\\left\\\|\\omega\\right\\\|\_\{2\}^\{2\}=\\sum\_\{\(i,j\)\\in E\}\\left\\\|\\omega\_\{ij\}\\right\\\|\_\{2\}^\{2\},\\qquad\\left\\\|\\omega^\{\\bullet\}\\right\\\|\_\{2\}^\{2\}=\\sum\_\{\(i,j\)\\in E\}\\left\\\|\\omega^\{\\bullet\}\_\{ij\}\\right\\\|\_\{2\}^\{2\}\.If‖ω‖2≤τzero\\left\\\|\\omega\\right\\\|\_\{2\}\\leq\\tau\_\{\\mathrm\{zero\}\}, no dominant channel is reported, and the verdict is𝗀𝗅𝗈𝖻𝖺𝗅𝗅𝗒​\_​𝖺𝖽𝗆𝗂𝗌𝗌𝗂𝖻𝗅𝖾\\mathsf\{globally\\\_admissible\}\. Otherwise, we form the calibrated fractionsp∙p\_\{\\bullet\}of Section[3\.4](https://arxiv.org/html/2608.05702#S3.SS4), withW∙=IW\_\{\\bullet\}=\\mathrm\{I\}only in the unit\-scaled default\. IfZ\>0Z\>0, channel dominance is reported as the channel maximizingp∙p\_\{\\bullet\}\. IfZ=0Z=0, no dominant channel is reported, and the fraction vector is left undefined; this zero\-denominator case can arise only from the chosen calibration weights or whitening maps and does not override the zero\-obstruction verdict branch\.

##### Per\-overlap norms\.

For each overlap\(i,j\)\(i,j\), we report the per\-channel decomposition ofωi​j\\omega\_\{ij\}\. This per\-overlap decomposition produces an overlap\-level map that identifies which interfaces carry each kind of disagreement\. The household\-power cross\-framework audit provides an example in which the winter–spring overlap is dominant\. In general, the dominant overlap is the overlap maximizing‖ωi​j‖22\\left\\\|\\omega\_\{ij\}\\right\\\|\_\{2\}^\{2\}\.

##### What the channel diagnosis tells us\.

State dominance means that the predicted state fields disagree on the overlap\. This state\-channel disagreement is the default mode for locally trained regression models that have not seen one another’s regions\. Closure dominance means that regional hidden\-physics representations or closure parameters are inconsistent across regions\. Closure dominance arises when one or more regions have an active conservation contract, as in Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2), or when one region’s closure was learned from data unavailable to the others\. Observation dominance indicates disagreement among sensor readings or their summary statistics, and commonly reflects differences in sensor noise or drift\. A nonzero optional metadata component indicates that declared contract records disagree across regions\. That component is reported as metadata rather than treated as a causal intervention account; none of the reported experiments uses an optional\-metadata intervention budget\.

This channel\-level diagnostic requires only matrix–vector products and no optimization\. The resulting total and channel norms therefore provide suitable signals for the streaming monitoring described in Section[6](https://arxiv.org/html/2608.05702#S6)\.

### 4\.2Budgeted intervention: exact tests and soft attribution

The second component treats each declared account of the source of disagreement as an intervention budget and evaluates the cost of the minimum\-cost intervention under that account\. Hard feasibility determines which proper accounts survive\. Costs are reported within each account but do not rank an unrestricted fallback against a more specific budget\. The antitonicity result in Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1)explains this separation: Enlarging a budget under a shared metric cannot increase its minimum repair cost\. The verdict of Definition[2\.5](https://arxiv.org/html/2608.05702#S2.Thmtheorem5)is then applied\.

###### Definition 4\.1\(Standard budgets\)\.

The SEAM\-Ω\\Omegaframework uses the following standard intervention budgets, each defined by an orthogonal projectorPPonto a subspace ofC0​\(F\)C^\{0\}\(F\): The𝖠𝗅𝗅𝗈𝗐𝖲𝗍𝖺𝗍𝖾\\mathsf\{AllowState\}budget projects onto the state channel and permits correction of predicted state fields\. The𝖠𝗅𝗅𝗈𝗐𝖢𝗅𝗈𝗌𝗎𝗋𝖾\\mathsf\{AllowClosure\}budget projects onto the closure channel and permits correction of hidden physics or sources\. The𝖠𝗅𝗅𝗈𝗐𝖲𝖾𝗇𝗌𝗈𝗋𝖱𝖾𝗃𝖾𝖼𝗍𝗂𝗈𝗇\\mathsf\{AllowSensorRejection\}budget projects onto the observation channel and permits sensor readings to be reweighted or rejected\. The𝖠𝗅𝗅𝗈𝗐𝖠𝗅𝗅\\mathsf\{AllowAll\}fallback usesP=IP=\\mathrm\{I\}and permits unrestricted intervention\.

For each budget, we solve the constrained quadratic program

minδ∈C0​\(F\)12​δ⊤​C​δsubject toD​P​δ=ω,\(I−P\)​δ=0,\\min\_\{\\delta\\in C^\{0\}\(F\)\}\\quad\\tfrac\{1\}\{2\}\\delta^\{\\top\}C\\delta\\qquad\\text\{subject to\}\\qquad DP\\delta=\\omega,\\quad\(\\mathrm\{I\}\-P\)\\delta=0,\(2\)whereCCis a positive\-definite cost metric\. The repaired explanations are then

srepaired≔s−P​δ⋆,s\_\{\\mathrm\{repaired\}\}\\coloneqq s\-P\\delta^\{\\star\},\(3\)so thatD​srepaired=D​s−D​P​δ⋆=ω−ω=0Ds\_\{\\mathrm\{repaired\}\}=Ds\-DP\\delta^\{\\star\}=\\omega\-\\omega=0, recovering admissibility\. We adopt this sign convention uniformly:δ⋆\\delta^\{\\star\}is the correction subtracted from the current stalk vector\. The norm‖δ⋆‖C\\left\\\|\\delta^\{\\star\}\\right\\\|\_\{C\}does not depend on the sign; only the interpretation does\.

The closed\-form solution is given by Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1)\. When Eq\.[2](https://arxiv.org/html/2608.05702#S4.E2)is infeasible, so thatω∉im⁡\(D​P\)\\omega\\notin\\operatorname\{im\}\(DP\), we report the hard costcPhard=\+∞c\_\{P\}^\{\\mathrm\{hard\}\}=\+\\inftytogether with the hard residualrPhard​\(ω\)r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\)of Definition[3\.9](https://arxiv.org/html/2608.05702#S3.Thmtheorem9), whose norm quantifies how badly the hypothesis fails\.

##### Exact verdicts and auxiliary soft records\.

The system returns the per\-budget record

\(budget label,statusb,cbhard,rbhard​\(ω\)\)\\displaystyle\(\\text\{budget label\},\\text\{status\}\_\{b\},c\_\{b\}^\{\\mathrm\{hard\}\},r\_\{b\}^\{\\mathrm\{hard\}\}\(\\omega\)\)and applies Definition[2\.5](https://arxiv.org/html/2608.05702#S2.Thmtheorem5)to the exact\-feasible records\. The verdict reports the full set of retained proper budgets and treats𝖠𝗅𝗅𝗈𝗐𝖠𝗅𝗅\\mathsf\{AllowAll\}only as a repairability fallback\. Hard costs remain attached to their own budgets and are not combined into a cross\-budget score\. Auxiliary soft records are reported as specified in Definition[3\.9](https://arxiv.org/html/2608.05702#S3.Thmtheorem9)and always include their residuals, so their intervention magnitudes are not conflated with exact\-hard verdicts\. Figure[5](https://arxiv.org/html/2608.05702#S4.F5)shows such soft records for the data–physics conflict\-attribution study of Section[9\.4](https://arxiv.org/html/2608.05702#S9.SS4)\.

##### Interpretation of the budget comparison\.

The budget catalog specifies the candidate diagnostic accounts\. The closure\-revision budget encodes the account that closure is the permitted locus of revision\. The sensor\-rejection budget analogously restricts revisions to observations\. Other budgets encode further accounts, and the catalog is open: Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1)is stated for the orthogonal projector onto an arbitrary closed subspaceVP⊆C0​\(F\)V\_\{P\}\\subseteq C^\{0\}\(F\), so any account that a practitioner can state as a set of permitted revisions enters the comparison as a projector, whether or not that set is one of the named channels\. Exact feasibility retains or refutes each account separately\. The minimum cost then measures the smallest repair within a retained account, rather than supplying a causal ranking across different allowed subspaces\.

##### Refutation and unresolved attribution\.

Each entry in the catalog is separately refutable, which is what makes the comparison a test rather than a weighting\. A budget whose hard problem is infeasible is ruled out by the overlap evidence rather than merely outbid, and its residual‖rbhard​\(ω\)‖\\left\\\|r\_\{b\}^\{\\mathrm\{hard\}\}\(\\omega\)\\right\\\|records how far the permitted revisions fall short of accounting forω\\omega\. When every proper budget is ruled out and only the unrestricted budget remains feasible, that fallback carries no diagnostic specificity, because unrestricted revision repairs every raw defect by construction\. The identifiability analysis in Section[5](https://arxiv.org/html/2608.05702#S5)then records which admissible directions remain invisible to the present observation map\. In that situation, SEAM reports the unresolved subspace rather than assigning a unique interpretation to the unrestricted repair\.

![Refer to caption](https://arxiv.org/html/2608.05702v1/x2.png)Figure 5:Soft budget\-attribution ranking in the data–physics conflict\-attribution study\. The figure reports per\-budget soft squared intervention norms in relative units for two configurations of the same three\-region system\. In the closure\-corruption condition, the closure diagnostic is cheap, and the sensor\-rejection diagnostic is expensive\. In the sensor\-corruption condition, the sensor\-rejection diagnostic is cheap, and the closure diagnostic is expensive\. For standard single\-channel hard budgets under Assumption A0, a finite hard cost requires full feasibilityω∈im⁡\(D​Pb\)\\omega\\in\\operatorname\{im\}\(DP\_\{b\}\); wrong\-channel single\-channel hard costs are\+∞\+\\inftyunless the non\-target obstruction components vanish\. The plotted ratios are therefore empirical soft diagnostics with residuals recorded in the diagnostic record, not theorem\-level full hard\-cost comparisons\.

### 4\.3Closure\-restricted repair and missing\-physics diagnosis

Learning or correcting unresolved closure terms is an established scientific machine learning problem, especially in data\-driven turbulence modeling\(Duraisamy et al\.,[2019](https://arxiv.org/html/2608.05702#bib.bib11)\)\. Field inversion and machine learning provides a direct computational precedent by inferring spatially distributed functional corrections for deficient closure models\(Parish and Duraisamy,[2016](https://arxiv.org/html/2608.05702#bib.bib38)\)\. SEAM’s closure\-repair test determines whether adjusting the modeled closure removes an overlap obstruction, returning the corresponding minimum\-cost reconstruction\.

A feasible, low\-cost closure\-channel preimage supports the missing\-closure hypothesis, and repair under the closure\-revision budget yields the minimum\-cost reconstruction with respect to the selected closure\-channel metric\. In the channel\-separated model used here, the restricted repair solves the preimage problem forωclosure\\omega^\{\\mathrm\{closure\}\}; state mismatches require a state\-revision or channel\-coupled budget\.

Letιclosure\\iota\_\{\\mathrm\{closure\}\}denote the zero\-extension intoC0​\(F\)C^\{0\}\(F\), letΠ∙\\Pi\_\{\\bullet\}denote the coordinate projection onto channel∙\\bullet, and define the closure\-channel block

DC≔Πclosure​D​ιclosure:C0​\(F\)closure→C1​\(F\)closure,\\displaystyle D\_\{C\}\\coloneqq\\Pi\_\{\\mathrm\{closure\}\}D\\iota\_\{\\mathrm\{closure\}\}\\colon C^\{0\}\(F\)^\{\\mathrm\{closure\}\}\\to C^\{1\}\(F\)^\{\\mathrm\{closure\}\},which is the whole closure\-channel content ofDDbecause Assumption A0 givesΠ∙​D​ιclosure=0\\Pi\_\{\\bullet\}D\\iota\_\{\\mathrm\{closure\}\}=0for every∙≠closure\\bullet\\neq\\mathrm\{closure\}\. Corollary[7\.3](https://arxiv.org/html/2608.05702#S7.Thmtheorem3)supplies the minimum\-norm closure correctionδC⋆=DC​ωclosure\+∈row⁡\(DC\)\\delta\_\{C\}^\{\\star\}=D\_\{C\}\{\{\}^\{\+\}\}\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{row\}\(D\_\{C\}\)wheneverωclosure∈im⁡DC\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{im\}D\_\{C\}, together with the repaired explanations\(ui,ci−δC⋆\|i,oi,ϵi\)i∈I\(u\_\{i\},c\_\{i\}\-\\delta\_\{C\}^\{\\star\}\|\_\{i\},o\_\{i\},\\epsilon\_\{i\}\)\_\{i\\in I\}, whose closure\-channel defect vanishes, while every non\-closure component ofD​sDsis unchanged\. For a raw SEAM\-Ω\\Omegadefectω=D​s\\omega=Ds, that image condition holds automatically, sinceωclosure=DC​sC\\omega^\{\\mathrm\{closure\}\}=D\_\{C\}s\_\{C\}for the closure\-channel componentsCs\_\{C\}ofss, so the closure repair needs no separate image\-membership test\.

Corollary[7\.3](https://arxiv.org/html/2608.05702#S7.Thmtheorem3)is stated in Section[7\.3](https://arxiv.org/html/2608.05702#S7.SS3)and proved in Appendix[B\.4](https://arxiv.org/html/2608.05702#A2.SS4)\. The hidden\-source recovery for Burgers study in Section[9\.3\.1](https://arxiv.org/html/2608.05702#S9.SS3.SSS1)documents the empirical content\. For inviscid Burgers with hidden closure summaries, the recovered physical closure increment−δC⋆\-\\delta\_\{C\}^\{\\star\}has its peak atx^0=0\.501\\hat\{x\}\_\{0\}=0\.501, within0\.2%0\.2\\%of the true centerx0=0\.5x\_\{0\}=0\.5and displaced by1\.7%1\.7\\%of the source widthw=0\.06w=0\.06\. The projected recovery has a closure\-channel repair residual below5%5\\%; the exact image\-membership case of Corollary[7\.3](https://arxiv.org/html/2608.05702#S7.Thmtheorem3)has zero residual\. The accompanying state\-revision result is a soft attribution record with its residual relative to the same closure\-channel right\-hand side\.

## 5Identifiability

Diagnosis tells us what disagrees\. Identifiability analysis then determines which admissibility\-preserving adjustments are distinguishable by current observations\. This distinction separates detected inconsistency from directions that remain unresolved by the available data\.

### 5\.1Blind admissible directions and the observable quotient

Throughout this section, we fix a linear observation mapO:C0​\(F\)→YO\\colon C^\{0\}\(F\)\\to YwithYYa finite\-dimensional output space\. The observation map encodes how sensors see the global sections∈C0​\(F\)s\\in C^\{0\}\(F\)\. Typically,OOhas a block\-diagonal structure that extracts the observation\-channel components of the explanations\.

The kernelker⁡D\\ker Dof the coboundary is the subspace of admissible directions whose restrictions agree on every overlap, thereby leaving the overlap\-disagreement vector unchanged\. Withinker⁡D\\ker D, we distinguish two operationally important subspaces\.

###### Definition 5\.1\(Blind admissible subspace and observable admissible quotient\)\.

The blind admissible subspace is

ℬ≔ker⁡D∩ker⁡O⊆C0​\(F\)\.\\displaystyle\\mathcal\{B\}\\coloneqq\\ker D\\cap\\ker O\\subseteq C^\{0\}\(F\)\.When the dependence ofℬ\\mathcal\{B\}on the observation map matters, we write this subspace asℬ​\(O\)\\mathcal\{B\}\(O\)\. Directions inℬ\\mathcal\{B\}preserve admissibility and are invisible to the current observation map\. The observable admissible quotient is

ℐ≔ker⁡D/ℬ≅\(ker⁡D\)∩ℬ⟂\.\\displaystyle\\mathcal\{I\}\\coloneqq\\ker D/\\mathcal\{B\}\\cong\(\\ker D\)\\cap\\mathcal\{B\}^\{\\perp\}\.The isomorphism uses the orthogonal complement insideker⁡D\\ker D\. Directions inℐ\\mathcal\{I\}preserve admissibility and are visible to current observations\.

The two subspaces have complementary dimensions\. Proposition[7\.2](https://arxiv.org/html/2608.05702#S7.Thmtheorem2)records the identity

dimℬ=dimker⁡D−rank⁡\(O\|ker⁡D\),dimℐ=rank⁡\(O\|ker⁡D\),\\dim\\mathcal\{B\}=\\dim\\ker D\-\\operatorname\{rank\}\(O\|\_\{\\ker D\}\),\\qquad\\dim\\mathcal\{I\}=\\operatorname\{rank\}\(O\|\_\{\\ker D\}\),\(4\)so thatdimker⁡D=dimℬ\+dimℐ\\dim\\ker D=\\dim\\mathcal\{B\}\+\\dim\\mathcal\{I\}\. The proof is a direct application of rank–nullity toO\|ker⁡DO\|\_\{\\ker D\}withker⁡\(O\|ker⁡D\)=ker⁡D∩ker⁡O=ℬ\\ker\(O\|\_\{\\ker D\}\)=\\ker D\\cap\\ker O=\\mathcal\{B\}; see Appendix[B\.3](https://arxiv.org/html/2608.05702#A2.SS3)\.

##### Computingℬ\\mathcal\{B\}\.

The numerical procedure begins by building an orthonormal basisVkerV\_\{\\ker\}ofker⁡D\\ker Dwith a full singular value decomposition \(SVD\) or an explicit rank\-revealing nullspace routine, ensuring that structural right\-null directions are retained\. The procedure then forms the restricted operatorO​VkerOV\_\{\\ker\}and computes either its full right\-nullspace basis or a full SVD\. All right singular directions whose singular values lie below a toleranceηsvd\\eta\_\{\\mathrm\{svd\}\}are selected, including structural right\-null directions omitted by an economy decomposition\. Finally,VkerV\_\{\\ker\}maps these vectors back toC0​\(F\)C^\{0\}\(F\)to obtain an orthonormal basisBℬB\_\{\\mathcal\{B\}\}ofℬ\\mathcal\{B\}\. The certificate records several checks:

‖D​Bℬ‖≤ηD​‖Bℬ‖,‖O​Bℬ‖≤ηO​‖Bℬ‖,‖Bℬ⊤​Bℬ−I‖≤ηorth,\\displaystyle\\left\\\|DB\_\{\\mathcal\{B\}\}\\right\\\|\\leq\\eta\_\{D\}\\left\\\|B\_\{\\mathcal\{B\}\}\\right\\\|,\\qquad\\left\\\|OB\_\{\\mathcal\{B\}\}\\right\\\|\\leq\\eta\_\{O\}\\left\\\|B\_\{\\mathcal\{B\}\}\\right\\\|,\\qquad\\left\\\|B\_\{\\mathcal\{B\}\}^\{\\top\}B\_\{\\mathcal\{B\}\}\-\\mathrm\{I\}\\right\\\|\\leq\\eta\_\{\\mathrm\{orth\}\},together with the dimension identity in Proposition[7\.2](https://arxiv.org/html/2608.05702#S7.Thmtheorem2)\. The blind\-closure identifiability study reports the observation\-invisibility component‖O​Bℬ‖<10−14\\left\\\|OB\_\{\\mathcal\{B\}\}\\right\\\|<10^\{\-14\}\.

##### Whatℬ\\mathcal\{B\}tells us\.

The blind admissible subspace specifies the resolution of the SEAM diagnostic\. A nonzerodimℬ\\dim\\mathcal\{B\}means that there exist directions in which the local explanations can be adjusted without violating admissibility and without changing any current observation\. The identifiability report marks these directions as unresolved by the present data, qualifying any attribution accordingly\.

## 6Monitoring learned generators

The diagnostic of Sections[4](https://arxiv.org/html/2608.05702#S4)and[5](https://arxiv.org/html/2608.05702#S5)treats the local generators as static\. For learned generators such as neural operators, PINNs, regression models, and hybrid scientific pipelines, the family of local explanations depends on input data that may shift over time\. SEAM\-Ω\\Omegaextends naturally to streaming monitoring of these systems\.

LetGθG\_\{\\theta\}be a trained operator with parametersθ\\theta, and let\{xt\}t≥0\\\{x\_\{t\}\\\}\_\{t\\geq 0\}be a streaming input sequence\. Examples include time series of PDE initial conditions, sensor data, and regulatory inputs\. At each timett, we apply toxtx\_\{t\}the family of local generators\{Gi,θ\}i∈I\\\{G\_\{i,\\theta\}\\\}\_\{i\\in I\}obtained by restricting the operator to each region, thereby producing a family of local explanationsei​\(t\)=Gi,θ​\(xt\)e\_\{i\}\(t\)=G\_\{i,\\theta\}\(x\_\{t\}\)\.

###### Definition 6\.1\(Streaming obstruction\)\.

The stacked explanation and streaming obstruction at timettare

s​\(t\)\\displaystyle s\(t\)≔\(vec​\(ei​\(t\)\)\)i∈I∈C0​\(F\),\\displaystyle\\coloneqq\(\\mathrm\{vec\}\(e\_\{i\}\(t\)\)\)\_\{i\\in I\}\\in C^\{0\}\(F\),ω​\(t\)\\displaystyle\\omega\(t\)≔D​s​\(t\)∈C1​\(F\)\.\\displaystyle\\coloneqq Ds\(t\)\\in C^\{1\}\(F\)\.

The streaming obstruction is a time\-indexed family of vectors inC1​\(F\)C^\{1\}\(F\)\. At each time, the report contains the total disagreement‖ω​\(t\)‖\\left\\\|\\omega\(t\)\\right\\\|and the norm‖ω∙​\(t\)‖\\left\\\|\\omega^\{\\bullet\}\(t\)\\right\\\|for each channel\. These quantities can be compared with contemporaneous model error or with a baseline recorded during training, but SEAM does not assign a universal alarm threshold\.

##### Example: neural operator under covariate shift\.

Mechanism M4 appears here in streaming form\. For the Fourier neural operator \(FNO\) architecture\(Li et al\.,[2021](https://arxiv.org/html/2608.05702#bib.bib31)\), trained on a distribution of Burgers initial conditions and evaluated on out\-of\-distribution \(OOD\) inputs, the FNO OOD monitoring study in Section[9\.8\.2](https://arxiv.org/html/2608.05702#S9.SS8.SSS2)shows a correlation between‖ω​\(t\)‖\\left\\\|\\omega\(t\)\\right\\\|and theL2L^\{2\}prediction error relative to a high\-resolution reference\. This correlation is the empirical basis for usingω\\omegaas a consistency monitor\. Channel norms localize which part of the regional record changes, but the experiment does not establish a calibrated probabilistic detector or a universal causal interpretation for that change\.

## 7Theoretical results

This section collects two central theorems together with companion identifiability and closure\-recovery results, and states their shared model conditions\. Proofs are consolidated in Appendix[B](https://arxiv.org/html/2608.05702#A2)in the order stated here\. Throughout,DDdenotes the assembled coboundary matrix of an explanation sheafFFover the 1\-skeleton of a finite cover’s nerve,D\+D\{\{\}^\{\+\}\}its Moore–Penrose pseudoinverse and associated projector construction\(Penrose,[1955](https://arxiv.org/html/2608.05702#bib.bib40)\),PPan orthogonal projector onto a budget subspace ofC0​\(F\)C^\{0\}\(F\), andCCa positive\-definite cost metric onC0​\(F\)C^\{0\}\(F\)\. We writeD​PDPfor the projected coboundary and letim⁡\(D​P\)\\operatorname\{im\}\(DP\)denote its image\. Observation mapsO:C0​\(F\)→YO\\colon C^\{0\}\(F\)\\to Yare linear\.

##### Model conditions and conventions for Section[7](https://arxiv.org/html/2608.05702#S7)\.

The result suite rests on the standing structural Assumption A0 together with the four repair and model conventions declared below\.

##### A0: block\-diagonal channel restrictions\.

The channel\-resolved results use Assumption A0 from Section[3\.4](https://arxiv.org/html/2608.05702#S3.SS4), where the assumption is adopted for every restriction map of SEAM\-Ω\\Omega\.

##### A1: repair sign convention\.

Corrections are subtracted from the current stalk vector\. For a full hard repair, the convention readssrepaired=s−P​δ⋆s\_\{\\mathrm\{repaired\}\}=s\-P\\delta^\{\\star\}withD​P​δ⋆=ωDP\\delta^\{\\star\}=\\omega\. Corollary[7\.3](https://arxiv.org/html/2608.05702#S7.Thmtheorem3)uses the channel\-restricted versionsrepaired=s−ιclosure​δC⋆s\_\{\\mathrm\{repaired\}\}=s\-\\iota\_\{\\mathrm\{closure\}\}\\delta\_\{C\}^\{\\star\}withDC​δC⋆=ωclosureD\_\{C\}\\delta\_\{C\}^\{\\star\}=\\omega^\{\\mathrm\{closure\}\}\. This convention is used by Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1), Corollary[7\.3](https://arxiv.org/html/2608.05702#S7.Thmtheorem3), and the repair records of Sections[4](https://arxiv.org/html/2608.05702#S4)and[4\.3](https://arxiv.org/html/2608.05702#S4.SS3)\.

##### A2: positive\-definite cost metric\.

The conditionC≻0C\\succ 0holds onC0​\(F\)C^\{0\}\(F\), giving the strict convexity required by Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1)\.

##### A3: row\-space\-aligned one\-sided closure repair\.

In Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2), the conservation\-redistribution repair direction at the repaired endpoint lies in the domain\-side row space of that endpoint’s closure restriction; the baseline overlap closures cancel; the neighboring endpoint contributes zero restricted repair on the overlap,ρq¯,eclosure​δ​cq¯=0\\rho^\{\\mathrm\{closure\}\}\_\{\\bar\{q\},e\}\\delta c\_\{\\bar\{q\}\}=0; andρq,eclosure\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}has positive rank\. The projected form covers repair directions outside the row space; rank\-zero restrictions carry no closure signal on the overlap\.

##### A4: pairwise\-overlap construction\.

All results use the 1\-skeleton of the nerve\.

Table[1](https://arxiv.org/html/2608.05702#S7.T1)summarizes the result suite and its roles in SEAM\-Ω\\Omegabefore the individual theorem statements\. For Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2)and its row in the table,σmin\+\\sigma\_\{\\min\}^\{\+\}denotes the smallest nonzero singular value; the distinction is necessary because the ordinary smallest singular valueσmin\\sigma\_\{\\min\}is zero for a rank\-deficient operator\.

Table 1:Central results of SEAM\-Ω\\Omegaand their operational roles\.ResultSubjectRoleTheorem 1Minimum\-cost budgeted interventionClosed\-formδ⋆\\delta^\{\\star\}via restricted operatorsAP=D​P​ιVPA\_\{P\}=DP\\iota\_\{V\_\{P\}\},CP=ιVP⊤​C​ιVPC\_\{P\}=\\iota\_\{V\_\{P\}\}^\{\\top\}C\\iota\_\{V\_\{P\}\}, withsrepaired=s−P​δ⋆s\_\{\\mathrm\{repaired\}\}=s\-P\\delta^\{\\star\}\.PropositionIdentifiability dimensionsdimℬ=dimker⁡D−rank⁡\(O\|ker⁡D\)\\dim\\mathcal\{B\}=\\dim\\ker D\-\\operatorname\{rank\}\(O\|\_\{\\ker D\}\), whiledimℐ=rank⁡\(O\|ker⁡D\)\\dim\\mathcal\{I\}=\\operatorname\{rank\}\(O\|\_\{\\ker D\}\)\.CorollaryClosure\-restricted recoverabilityδC⋆=DC​ωclosure\+∈im⁡\(DC⊤\)=row⁡\(DC\)\\delta\_\{C\}^\{\\star\}=D\_\{C\}\{\{\}^\{\+\}\}\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{im\}\(D\_\{C\}^\{\\top\}\)=\\operatorname\{row\}\(D\_\{C\}\)whenωclosure∈im⁡DC\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{im\}D\_\{C\}\. The repaired closure blocks areci−δC⋆\|ic\_\{i\}\-\\delta\_\{C\}^\{\\star\}\|\_\{i\}\.Theorem 2Conservation\-contract detectabilityUnder A3, comprising baseline cancellation, one\-sided zero neighbor contribution, positive rank, and row\-space alignmentδ​cq∈row⁡\(ρq,eclosure\)\\delta c\_\{q\}\\in\\operatorname\{row\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\),‖ωeclosure‖≥σmin\+​ϵrepair\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{e\}\\right\\\|\\geq\\sigma\_\{\\min\}^\{\+\}\\epsilon\_\{\\mathrm\{repair\}\}\. The projected bound applies otherwise\.
### 7\.1Theorem 1: minimum\-cost budgeted intervention

###### Theorem 1\(Minimum\-cost budgeted intervention\)\.

LetPPbe the orthogonal projector onto a closed subspaceVP⊆C0​\(F\)V\_\{P\}\\subseteq C^\{0\}\(F\)called the budget subspace, and letC≻0C\\succ 0onC0​\(F\)C^\{0\}\(F\)\. Define the restricted operators

AP≔D​P​ιVP:VP→C1​\(F\),CP≔ιVP⊤​C​ιVP≻0​on​VP,\\displaystyle A\_\{P\}\\coloneqq DP\\iota\_\{V\_\{P\}\}\\colon V\_\{P\}\\to C^\{1\}\(F\),\\qquad C\_\{P\}\\coloneqq\\iota\_\{V\_\{P\}\}^\{\\top\}C\\iota\_\{V\_\{P\}\}\\succ 0\\text\{ on \}V\_\{P\},whereιVP:VP↪C0​\(F\)\\iota\_\{V\_\{P\}\}\\colon V\_\{P\}\\hookrightarrow C^\{0\}\(F\)is the inclusion\. Consider the constrained quadratic program

minδ∈C0​\(F\)⁡12​δ⊤​C​δsubject toD​P​δ=ω,\(I−P\)​δ=0\.\\min\_\{\\delta\\in C^\{0\}\(F\)\}\\tfrac\{1\}\{2\}\\delta^\{\\top\}C\\delta\\quad\\text\{subject to\}\\quad DP\\delta=\\omega,\(\\mathrm\{I\}\-P\)\\delta=0\.\(5\)
Feasibility criterion\.The problem is feasible if and only ifω∈im⁡AP\\omega\\in\\operatorname\{im\}A\_\{P\}\.

Closed\-form solution\.When feasible, the unique minimizer is

δ⋆=ιVP​CP−1​AP⊤​\(AP​CP−1​AP⊤\)​ω\+,\\delta^\{\\star\}=\\iota\_\{V\_\{P\}\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\(A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\)\{\{\}^\{\+\}\}\\omega,\(6\)and the reported squared intervention cost is‖δ⋆‖C2=ω⊤​\(AP​CP−1​AP⊤\)​ω\+\\left\\\|\\delta^\{\\star\}\\right\\\|\_\{C\}^\{2\}=\\omega^\{\\top\}\(A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\)\{\{\}^\{\+\}\}\\omega\. The optimum of the objective12​δ⊤​C​δ\\tfrac\{1\}\{2\}\\delta^\{\\top\}C\\deltainEq\.[5](https://arxiv.org/html/2608.05702#S7.E5)is one half of this value\.

Soft Tikhonov form\.Forλ\>0\\lambda\>0, the unique minimizer of12​δ⊤​C​δ\+\(λ/2\)​‖D​δ−ω‖2\\tfrac\{1\}\{2\}\\delta^\{\\top\}C\\delta\+\(\\lambda/2\)\\left\\\|D\\delta\-\\omega\\right\\\|^\{2\}overδ∈VP\\delta\\in V\_\{P\}is

δλ⋆=ιVP​\(AP⊤​AP\+λ−1​CP\)−1​AP⊤​ω,\\delta^\{\\star\}\_\{\\lambda\}=\\iota\_\{V\_\{P\}\}\(A\_\{P\}^\{\\top\}A\_\{P\}\+\\lambda^\{\-1\}C\_\{P\}\)^\{\-1\}A\_\{P\}^\{\\top\}\\omega,\(7\)which is well\-defined for allλ∈\(0,∞\)\\lambda\\in\(0,\\infty\)\.

Antitonicity in budget\.For a fixed ambient cost metricCC, suppose thatVP1⊆VP2V\_\{P\_\{1\}\}\\subseteq V\_\{P\_\{2\}\}, meaning that the larger budget contains the smaller\. Then, for everyω\\omegafor which both problems are feasible under that sameCC, the optimal costs satisfy‖δ⋆​\(P2\)‖C2≤‖δ⋆​\(P1\)‖C2\\left\\\|\\delta^\{\\star\}\(P\_\{2\}\)\\right\\\|\_\{C\}^\{2\}\\leq\\left\\\|\\delta^\{\\star\}\(P\_\{1\}\)\\right\\\|\_\{C\}^\{2\}\. Budget antitonicity therefore requires a shared ambient cost metric; budget inclusion alone is insufficient when the two budgets use different budget\-specific metrics\.

Sign convention\.The repaired stalk vector issrepaired=s−P​δ⋆s\_\{\\mathrm\{repaired\}\}=s\-P\\delta^\{\\star\}withδ⋆∈VP\\delta^\{\\star\}\\in V\_\{P\}, so that

D​srepaired=ω−D​P​δ⋆=0\.\\displaystyle Ds\_\{\\mathrm\{repaired\}\}=\\omega\-DP\\delta^\{\\star\}=0\.TheVP⟂V\_\{P\}^\{\\perp\}component of the stalk vector is unchanged\. IfPPis a channel projector, non\-target channels are unchanged under Assumption A0\.

### 7\.2Identifiability dimensions

###### Proposition 7\.2\(Identifiability dimensions\)\.

LetD:C0​\(F\)→C1​\(F\)D\\colon C^\{0\}\(F\)\\to C^\{1\}\(F\)be the coboundary and letO:C0​\(F\)→YO\\colon C^\{0\}\(F\)\\to Ybe a linear observation map\. Define

ℬ≔ker⁡D∩ker⁡O,ℐ≔ker⁡D/ℬ\.\\displaystyle\\mathcal\{B\}\\coloneqq\\ker D\\cap\\ker O,\\qquad\\mathcal\{I\}\\coloneqq\\ker D/\\mathcal\{B\}\.Then the blind\-space dimension isdimℬ=dimker⁡D−rank⁡\(O\|ker⁡D\)\\dim\\mathcal\{B\}=\\dim\\ker D\-\\operatorname\{rank\}\(O\|\_\{\\ker D\}\), the observable quotient has dimensiondimℐ=rank⁡\(O\|ker⁡D\)\\dim\\mathcal\{I\}=\\operatorname\{rank\}\(O\|\_\{\\ker D\}\), and the two dimensions sum todimker⁡D=dimℬ\+dimℐ\\dim\\ker D=\\dim\\mathcal\{B\}\+\\dim\\mathcal\{I\}\. Moreover,ℬ\\mathcal\{B\}is the set of admissibility\-preserving directions that are also invisible toOO, whileℐ\\mathcal\{I\}parameterizes the admissibility\-preserving directions thatOOcan resolve\.

### 7\.3Closure\-restricted recoverability

###### Corollary 7\.3\(Closure\-restricted recoverability\)\.

LetDCD\_\{C\}be the closure\-channel block ofDDunder Assumption A0, i\.e\., the typed mapDC:C0​\(F\)closure→C1​\(F\)closureD\_\{C\}\\colon C^\{0\}\(F\)^\{\\mathrm\{closure\}\}\\to C^\{1\}\(F\)^\{\\mathrm\{closure\}\}obtained by restricting the domain to the closure channel and projecting the codomain onto that channel\. Supposeωclosure∈im⁡DC\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{im\}D\_\{C\}\. Then the unique minimum\-norm closure correction in the Euclidean metric is

δC⋆=DC​ωclosure\+∈im⁡\(DC⊤\)=row⁡\(DC\)=\(ker⁡DC\)⟂\.\\displaystyle\\delta\_\{C\}^\{\\star\}=D\_\{C\}\{\{\}^\{\+\}\}\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{im\}\(D\_\{C\}^\{\\top\}\)=\\operatorname\{row\}\(D\_\{C\}\)=\(\\ker D\_\{C\}\)^\{\\perp\}\.The repaired stalks\(ui,ci−δC⋆\|i,oi,ϵi\)i∈I\(u\_\{i\},c\_\{i\}\-\\delta\_\{C\}^\{\\star\}\|\_\{i\},o\_\{i\},\\epsilon\_\{i\}\)\_\{i\\in I\}have zero closure\-channel admissibility defect, and the non\-closure channels ofω\\omegaare unchanged\. Under the sign convention, the zero\-extended correction is subtracted consistently with the channel\-restricted convention in A1\. The recoveredδC⋆\\delta\_\{C\}^\{\\star\}is the unique element ofim⁡\(DC⊤\)=row⁡\(DC\)\\operatorname\{im\}\(D\_\{C\}^\{\\top\}\)=\\operatorname\{row\}\(D\_\{C\}\)that achieves closure\-channel admissibility, but adding any element ofker⁡DC\\ker D\_\{C\}leaves the closure obstruction unchanged\. Such a closure direction belongs toℬ\\mathcal\{B\}only after embedding that direction into the full stalk space and checking membership inker⁡D∩ker⁡O\\ker D\\cap\\ker Oaccording to Proposition[7\.2](https://arxiv.org/html/2608.05702#S7.Thmtheorem2)\. For raw admissibility defectsω=D​s\\omega=Ds, Assumption A0 makesωclosure∈im⁡DC\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{im\}D\_\{C\}automatic\.

### 7\.4Theorem 2: conservation\-contract detectability

###### Theorem 2\(Conservation\-contract detectability\)\.

LetFFbe an explanation sheaf with closure\-channel restrictionsρq,eclosure\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}under Assumption A0\. Lete=\(a,b\)e=\(a,b\)be an oriented overlap edge and letq∈\{a,b\}q\\in\\\{a,b\\\}be the repaired endpoint\. SupposeUqU\_\{q\}applies a conservation\-contract repair whose actual closure\-channel vector entering the restriction isδ​cq\\delta c\_\{q\}, and setϵrepair≔‖δ​cq‖\\epsilon\_\{\\mathrm\{repair\}\}\\coloneqq\\left\\\|\\delta c\_\{q\}\\right\\\|\.

Assume the following conditions, which together instantiate A3:

Baseline cancellation\.The baseline closure restrictions agree oneebefore repair:ρa,eclosure​ca0=ρb,eclosure​cb0\\rho^\{\\mathrm\{closure\}\}\_\{a,e\}c\_\{a\}^\{0\}=\\rho^\{\\mathrm\{closure\}\}\_\{b,e\}c\_\{b\}^\{0\}\.

Row\-space alignment\.The repair directionδ​cq\\delta c\_\{q\}lies inrow⁡\(ρq,eclosure\)\\operatorname\{row\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\.

One\-sided zero contribution\.The other endpoint’s restricted repair contribution oneeis zero:ρq¯,eclosure​δ​cq¯=0\\rho^\{\\mathrm\{closure\}\}\_\{\\bar\{q\},e\}\\delta c\_\{\\bar\{q\}\}=0, whereq¯\\bar\{q\}denotes the endpoint other thanqq\.

Positive\-rank restriction\.The mapρq,eclosure\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}has positive rank\.

Conclusion\.Letεa≔−1\\varepsilon\_\{a\}\\coloneqq\-1andεb≔\+1\\varepsilon\_\{b\}\\coloneqq\+1\. Then, under baseline cancellation and the one\-sided zero\-contribution condition, the restricted defect and its closure\-channel lower bound are

ωeclosure=εq​ρq,eclosure​δ​cq,‖ωeclosure‖≥σmin\+​\(ρq,eclosure\)⋅ϵrepair\.\\omega^\{\\mathrm\{closure\}\}\_\{e\}=\\varepsilon\_\{q\}\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\\delta c\_\{q\},\\qquad\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{e\}\\right\\\|\\geq\\sigma\_\{\\min\}^\{\+\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\\cdot\\epsilon\_\{\\mathrm\{repair\}\}\.\(8\)
Projected variant without row\-space alignment\.If baseline cancellation, one\-sided zero contribution, and the positive\-rank restriction hold but row\-space alignment fails, the same argument gives the projected bound

‖ωeclosure‖≥σmin\+​\(ρq,eclosure\)⋅‖Πrow⁡\(ρq,eclosure\)​δ​cq‖,\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{e\}\\right\\\|\\geq\\sigma\_\{\\min\}^\{\+\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\\cdot\\left\\\|\\Pi\_\{\\operatorname\{row\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\}\\delta c\_\{q\}\\right\\\|,\(9\)whereΠrow⁡\(ρq,eclosure\)\\Pi\_\{\\operatorname\{row\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\}denotes the orthogonal projector onto the indicated row space\. This bound equalsEq\.[8](https://arxiv.org/html/2608.05702#S7.E8)under row\-space alignment and is strictly weaker otherwise\. If both endpoints are repaired, a positive lower bound in terms of‖δ​cq‖\\left\\\|\\delta c\_\{q\}\\right\\\|additionally requires a quantitative non\-cancellation assumption on the net restricted repair\.

Rank\-zero convention\.Ifrank⁡\(ρq,eclosure\)=0\\operatorname\{rank\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)=0, thenρq,eclosure\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}carries no closure signal on the overlap: The projected row\-space term is zero, andσmin\+​\(ρq,eclosure\)\\sigma\_\{\\min\}^\{\+\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)is not defined\.

##### Empirical verification\.

The conservation\-contract detectability study in Section[9\.3\.2](https://arxiv.org/html/2608.05702#S9.SS3.SSS2)verifies the bound on a three\-region Burgers configuration with an active conservation repair: The observed‖ω12closure‖=1\.172\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{12\}\\right\\\|=1\.172exceeds the predicted lower bound ofσmin\+​ϵrepair=0\.128\\sigma\_\{\\min\}^\{\+\}\\epsilon\_\{\\mathrm\{repair\}\}=0\.128by a factor of9\.29\.2\. The associated parametric sweep reports a pass for all 25 configurations under the prespecified10%10\\%discretization\-tolerant criterion and exhibits the predicted approximately linear scaling with repair magnitude\.

##### Sensitivity floor\.

The bound is tight when the repaired endpoint’s closure vectorδ​cq\\delta c\_\{q\}aligns with the right singular vector ofρq,eclosure\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}corresponding to the smallest nonzero singular value\. The bound is loose whenδ​cq\\delta c\_\{q\}aligns with the dominant singular vectors\. The sensitivity floorσmin\+\\sigma\_\{\\min\}^\{\+\}is therefore a worst\-case detection floor; in practice, generic repair directions yield much larger closure obstructions\. The slack factor is a useful diagnostic in its own right, because a small slack indicates a near\-minimum\-detection repair direction\.

## 8Algorithms and computational workflow

Here we present the core algorithms in mathematically clean form; Appendix[C](https://arxiv.org/html/2608.05702#A3)records the numerical conventions used by the reference implementation\. The presentation groups the core diagnostic workflow and its extensions before treating the computational and audit properties of the resulting implementation\.

### 8\.1Diagnostic workflow and extensions

##### Top\-level SEAM\-Ω\\Omegadiagnostic pipeline\.

The top\-level diagnostic pipeline is Algorithm[8\.1](https://arxiv.org/html/2608.05702#alg1)\. The index set𝒦bud\\mathcal\{K\}\_\{\\mathrm\{bud\}\}labels the finite budget catalog\. The pipeline returns a structured diagnostic record\.

Algorithm 1SEAM\-Ω\\Omegadiagnostic pipeline,diagnose\.1:cover

𝒰\\mathcal\{U\}, local generators

\{Gi\}i∈I\\\{G\_\{i\}\\\}\_\{i\\in I\}, input data

xx, restriction maps

\{ρi,i​j,ρj,i​j\}\(i,j\)∈E\\\{\\rho\_\{i,ij\},\\rho\_\{j,ij\}\\\}\_\{\(i,j\)\\in E\}, observation map

OO, budget catalog

\{\(Pb,Cb\)\}b∈𝒦bud\\\{\(P\_\{b\},C\_\{b\}\)\\\}\_\{b\\in\\mathcal\{K\}\_\{\\mathrm\{bud\}\}\}, zero\-obstruction tolerance

τzero\\tau\_\{\\mathrm\{zero\}\}\.

2:

ω\\omega, channel norms, per\-budget hard records, tolerance and soft diagnostic records, identifiability basis, diagnostic record\.

3:Generate: for each

i∈Ii\\in I, compute

ei←Gi​\(x,Ui\)e\_\{i\}\\leftarrow G\_\{i\}\(x,U\_\{i\}\)and

si←vec​\(ei\)s\_\{i\}\\leftarrow\\mathrm\{vec\}\(e\_\{i\}\); set

s←\(si\)i∈Is\\leftarrow\(s\_\{i\}\)\_\{i\\in I\}\.

4:Restrict and obstruct: assemble

DDfrom the restriction maps; compute

ω←D​s\\omega\\leftarrow Ds\.

5:Decompose channels: compute

‖ω∙‖\\left\\\|\\omega^\{\\bullet\}\\right\\\|for

∙∈\{state,closure,obs,meta\}\\bullet\\in\\\{\\mathrm\{state\},\\mathrm\{closure\},\\mathrm\{obs\},\\mathrm\{meta\}\\\}\.

6:Apply budgeted intervention for each budget

bb:

7:solve Eq\.[5](https://arxiv.org/html/2608.05702#S7.E5)to obtain

δb⋆\\delta^\{\\star\}\_\{b\}or detect infeasibility\.

8:record exact feasibility status;

cbhardc^\{\\mathrm\{hard\}\}\_\{b\}, which is finite only for

𝖾𝗑𝖺𝖼𝗍​\_​𝖿𝖾𝖺𝗌𝗂𝖻𝗅𝖾\\mathsf\{exact\\\_feasible\}; optional tolerance cost

cbtolc^\{\\mathrm\{tol\}\}\_\{b\}; optional soft squared intervention norm

nb,λsoftn^\{\\mathrm\{soft\}\}\_\{b,\\lambda\}and objective

Jb,λsoftJ^\{\\mathrm\{soft\}\}\_\{b,\\lambda\}; hard residual

rbhard​\(ω\)r\_\{b\}^\{\\mathrm\{hard\}\}\(\\omega\); and any soft residual

rb,λsoft​\(ω\)r\_\{b,\\lambda\}^\{\\mathrm\{soft\}\}\(\\omega\)\.

9:Analyze identifiability: compute the orthonormal basis of

ℬ\\mathcal\{B\}via Proposition[7\.2](https://arxiv.org/html/2608.05702#S7.Thmtheorem2)\.

10:Build the diagnostic record: bundle the obstruction, channel norms, budget records, and

ℬ\\mathcal\{B\}certificate\.

11:returndiagnostic record\.

##### Budgeted intervention solver\.

Algorithm[8\.1](https://arxiv.org/html/2608.05702#alg2)implements the minimum\-cost intervention result of Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1)in closed form and couples that result to an explicit feasibility check\. After restricting the coboundary and cost metric to the budget subspaceVPV\_\{P\}, the algorithm uses the hard residual to distinguish exact\-feasible and tolerance\-feasible repairs from infeasible cases, which are passed to the soft Tikhonov diagnostic\.

Algorithm 2Budgeted intervention solver,synthesize\_intervention\.1:

DD,

ω\\omega, projector

PP, cost metric

CC, feasibility tolerance

ηfeas\>0\\eta\_\{\\mathrm\{feas\}\}\>0, Tikhonov regularizer

λ\>0\\lambda\>0for soft mode\.

2:

δ⋆\\delta^\{\\star\}, hard cost, tolerance record, soft diagnostic record,

rPhard​\(ω\)r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\), optional

rP,λsoft​\(ω\)r\_\{P,\\lambda\}^\{\\mathrm\{soft\}\}\(\\omega\), feasibility status\.

3:Let

QQbe an orthonormal basis and inclusion matrix for

VP=im⁡PV\_\{P\}=\\operatorname\{im\}P\.

4:Form

AP←D​QA\_\{P\}\\leftarrow DQand

CP←Q⊤​C​QC\_\{P\}\\leftarrow Q^\{\\top\}CQ\.

5:Compute

AP\+A\_\{P\}\{\{\}^\{\+\}\}and

rPhard​\(ω\)←ω−AP​AP​ω\+r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\)\\leftarrow\\omega\-A\_\{P\}A\_\{P\}\{\{\}^\{\+\}\}\\omega\.

6:if

‖rPhard​\(ω\)‖=0\\left\\\|r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\)\\right\\\|=0then

7:Enter the theorem\-level exact\-feasible branch\.

8:

z⋆←CP−1​AP⊤​\(AP​CP−1​AP⊤\)​ω\+z^\{\\star\}\\leftarrow C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\(A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\)\{\{\}^\{\+\}\}\\omega\.

9:

δ⋆←Q​z⋆\\delta^\{\\star\}\\leftarrow Qz^\{\\star\}\.

10:

chard←δ⋆⊤​C​δ⋆c^\{\\mathrm\{hard\}\}\\leftarrow\{\\delta^\{\\star\}\}^\{\\top\}C\\delta^\{\\star\}\.

11:return

\(δ⋆,chard,rPhard​\(ω\),𝖾𝗑𝖺𝖼𝗍​\_​𝖿𝖾𝖺𝗌𝗂𝖻𝗅𝖾\)\(\\delta^\{\\star\},c^\{\\mathrm\{hard\}\},r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\),\\mathsf\{exact\\\_feasible\}\)\.

12:elseif

‖rPhard​\(ω\)‖≤ηfeas\\left\\\|r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\)\\right\\\|\\leq\\eta\_\{\\mathrm\{feas\}\}then

13:Enter the tolerance\-feasible engineering branch\.

14:Compute

z⋆z^\{\\star\}and

δ⋆\\delta^\{\\star\}as above\.

15:

ctol←δ⋆⊤​C​δ⋆c^\{\\mathrm\{tol\}\}\\leftarrow\{\\delta^\{\\star\}\}^\{\\top\}C\\delta^\{\\star\}\.

16:

chard←\+∞c^\{\\mathrm\{hard\}\}\\leftarrow\+\\inftyfor exact\-hard verdicts\.

17:return

\(δ⋆,\(chard,ctol\),rPhard​\(ω\),𝗍𝗈𝗅𝖾𝗋𝖺𝗇𝖼𝖾​\_​𝖿𝖾𝖺𝗌𝗂𝖻𝗅𝖾\)\(\\delta^\{\\star\},\(c^\{\\mathrm\{hard\}\},c^\{\\mathrm\{tol\}\}\),r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\),\\mathsf\{tolerance\\\_feasible\}\)\.

18:else

19:Enter the soft Tikhonov branch\.

20:

zλ⋆←\(AP⊤​AP\+λ−1​CP\)−1​AP⊤​ωz\_\{\\lambda\}^\{\\star\}\\leftarrow\(A\_\{P\}^\{\\top\}A\_\{P\}\+\\lambda^\{\-1\}C\_\{P\}\)^\{\-1\}A\_\{P\}^\{\\top\}\\omega\.

21:

δλ⋆←Q​zλ⋆\\delta^\{\\star\}\_\{\\lambda\}\\leftarrow Qz\_\{\\lambda\}^\{\\star\}\.

22:

rP,λsoft​\(ω\)←ω−AP​zλ⋆r\_\{P,\\lambda\}^\{\\mathrm\{soft\}\}\(\\omega\)\\leftarrow\\omega\-A\_\{P\}z\_\{\\lambda\}^\{\\star\}\.

23:

chard←\+∞c^\{\\mathrm\{hard\}\}\\leftarrow\+\\infty\.

24:

nλsoft←‖δλ⋆‖C2=δλ⋆⊤​C​δλ⋆n^\{\\mathrm\{soft\}\}\_\{\\lambda\}\\leftarrow\\left\\\|\\delta^\{\\star\}\_\{\\lambda\}\\right\\\|\_\{C\}^\{2\}=\{\\delta^\{\\star\}\_\{\\lambda\}\}^\{\\top\}C\\delta^\{\\star\}\_\{\\lambda\}\.

25:

Jλsoft←12​nλsoft\+\(λ/2\)​‖rP,λsoft​\(ω\)‖2J^\{\\mathrm\{soft\}\}\_\{\\lambda\}\\leftarrow\\tfrac\{1\}\{2\}n^\{\\mathrm\{soft\}\}\_\{\\lambda\}\+\(\\lambda/2\)\\left\\\|r\_\{P,\\lambda\}^\{\\mathrm\{soft\}\}\(\\omega\)\\right\\\|^\{2\}\.

26:return

\(δλ⋆,\(chard,nλsoft,Jλsoft\),\(rPhard​\(ω\),rP,λsoft​\(ω\)\),𝗂𝗇𝖿𝖾𝖺𝗌𝗂𝖻𝗅𝖾​\_​𝗌𝗈𝖿𝗍​\_​𝖽𝗂𝖺𝗀𝗇𝗈𝗌𝗍𝗂𝖼\)\(\\delta^\{\\star\}\_\{\\lambda\},\(c^\{\\mathrm\{hard\}\},n^\{\\mathrm\{soft\}\}\_\{\\lambda\},J^\{\\mathrm\{soft\}\}\_\{\\lambda\}\),\(r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\),r\_\{P,\\lambda\}^\{\\mathrm\{soft\}\}\(\\omega\)\),\\mathsf\{infeasible\\\_soft\\\_diagnostic\}\)\.

27:endif

Two implementation notes apply\. First, pseudoinverses are computed via a project\-specific truncated SVD tolerance, stated uniformly in Appendix[C\.2](https://arxiv.org/html/2608.05702#A3.SS2); library defaults are not assumed, and the exact routine and version are recorded whenever an implementation relies on one\. Second, the matricesAPA\_\{P\}andCPC\_\{P\}are typically small, so dense linear algebra is appropriate\. For larger problems, exact pseudoinverse projection with a truncated or randomized SVD requires retaining every singular direction above the chosen rank tolerance; smaller retained ranks yield approximate low\-rank diagnostics\.

##### Identifiability\.

Algorithm[8\.1](https://arxiv.org/html/2608.05702#alg3)implements the two\-stage nullspace construction and returns its numerical certificate\.

Algorithm 3Identifiability analyzer,analyze\_identifiability\.1:

D∈ℝM×ND\\in\\mathbb\{R\}^\{M\\times N\},

OO, singular\-value tolerance

ηsvd\>0\\eta\_\{\\mathrm\{svd\}\}\>0\.

2:orthonormal basis

BℬB\_\{\\mathcal\{B\}\}of

ℬ\\mathcal\{B\},

dimℬ\\dim\\mathcal\{B\}, channel decomposition of

BℬB\_\{\\mathcal\{B\}\}, multi\-part certificate\.

3:Compute a full SVD or rank\-revealing nullspace of

DD:

D=UD​ΣD​VD⊤D=U\_\{D\}\\Sigma\_\{D\}V\_\{D\}^\{\\top\}\.

4:Let

r=\#​\{ℓ:σℓ​\(D\)\>ηsvd\}r=\\\#\\\{\\ell:\\sigma\_\{\\ell\}\(D\)\>\\eta\_\{\\mathrm\{svd\}\}\\\}and set

Vker←VD\[:,r\+1:N\]V\_\{\\ker\}\\leftarrow V\_\{D\}\[:,r\+1:N\], including structural zero singular directions\.

5:Compute

O′←O​VkerO^\{\\prime\}\\leftarrow OV\_\{\\ker\}and its SVD

O′=UO′​ΣO′​VO′⊤O^\{\\prime\}=U\_\{O^\{\\prime\}\}\\Sigma\_\{O^\{\\prime\}\}V\_\{O^\{\\prime\}\}^\{\\top\}\.

6:Let

V0V\_\{0\}be a complete basis of

kerηsvd⁡\(O′\)\\ker\_\{\\eta\_\{\\mathrm\{svd\}\}\}\(O^\{\\prime\}\), including structural right\-null directions omitted by an economy SVD, and set

Bℬ←Vker​V0B\_\{\\mathcal\{B\}\}\\leftarrow V\_\{\\ker\}V\_\{0\}\.

7:Decompose each column of

BℬB\_\{\\mathcal\{B\}\}into channel components by projecting onto the three primary channels and any optional contract\-metadata block\.

8:Verify

‖D​Bℬ‖\\left\\\|DB\_\{\\mathcal\{B\}\}\\right\\\|,

‖O​Bℬ‖\\left\\\|OB\_\{\\mathcal\{B\}\}\\right\\\|, orthonormality, and the dimension identity of Proposition[7\.2](https://arxiv.org/html/2608.05702#S7.Thmtheorem2)\.

9:return

\(Bℬ,dimℬ,channel decomposition,certificate\)\(B\_\{\\mathcal\{B\}\},\\dim\\mathcal\{B\},\\text\{channel decomposition\},\\text\{certificate\}\)\.

##### Streaming monitoring\.

The monitoring pipeline runs the diagnostic pipeline at each time step of a streaming input but reuses cached operators where possible\.

Algorithm 4Streaming monitoring\.1:fixed cover

𝒰\\mathcal\{U\}, fixed restrictions, fixed budgets, learned generator

GθG\_\{\\theta\}, streaming inputs

\{xt\}\\\{x\_\{t\}\\\}\.

2:stream of

\(ω​\(t\),channel norms​\(t\),hard and soft diagnostic budget records​\(t\)\)\(\\omega\(t\),\\text\{channel norms\}\(t\),\\text\{hard and soft diagnostic budget records\}\(t\)\)\.

3:Assemble

DDonce\.

4:Cache pseudoinverses and projectors for all standard budgets\.

5:for

t=1,2,…t=1,2,\\ldotsdo

6:Compute

ei​\(t\)←Gi,θ​\(xt\)e\_\{i\}\(t\)\\leftarrow G\_\{i,\\theta\}\(x\_\{t\}\)for each

ii\.

7:Set

s​\(t\)←\(vec​\(ei​\(t\)\)\)i∈Is\(t\)\\leftarrow\(\\mathrm\{vec\}\(e\_\{i\}\(t\)\)\)\_\{i\\in I\}and compute

ω​\(t\)←D​s​\(t\)\\omega\(t\)\\leftarrow Ds\(t\)\.

8:Compute channel norms and hard and soft diagnostic budget records\.

9:Emit summary\.

10:endfor

### 8\.2Computational complexity and auditability

We summarize the time complexity in terms of the total stalk dimensionN≔∑idiN\\coloneqq\\sum\_\{i\}d\_\{i\}, the total overlap stalk dimensionM≔∑\(i,j\)mi​jM\\coloneqq\\sum\_\{\(i,j\)\}m\_\{ij\}, the number of regions\|I\|\|I\|, and the number of budgets\|𝒦bud\|\|\\mathcal\{K\}\_\{\\mathrm\{bud\}\}\|\.

The coboundaryDDhas dimensionsM×NM\\times Nand is block sparse with𝒪​\(\|E\|\)\\mathcal\{O\}\(\|E\|\)nonzero blocks\. Sparse storage, assembly, and matrix–vector costs are𝒪​\(nnz⁡\(D\)\)\\mathcal\{O\}\(\\operatorname\{nnz\}\(D\)\), wherennz⁡\(D\)\\operatorname\{nnz\}\(D\)counts scalar nonzeros in all restriction blocks\. This sparse\-operation cost becomes𝒪​\(M\+N\)\\mathcal\{O\}\(M\+N\)only under a local\-interpolation sparsity assumption\. Computingω=D​s\\omega=Dscosts𝒪​\(M​N\)\\mathcal\{O\}\(MN\)in the dense regime and𝒪​\(nnz⁡\(D\)\)\\mathcal\{O\}\(\\operatorname\{nnz\}\(D\)\)in the sparse regime\.

For each budget labelbb, budgeted intervention through Algorithm[8\.1](https://arxiv.org/html/2608.05702#alg2)forms the restricted operatorAPb=D​QbA\_\{P\_\{b\}\}=DQ\_\{b\}withpb≔dimVPbp\_\{b\}\\coloneqq\\dim V\_\{P\_\{b\}\}\. A dense thin SVD or an equivalent rectangular factorization costs𝒪​\(M​pb​min⁡\{M,pb\}\)\\mathcal\{O\}\(Mp\_\{b\}\\min\\\{M,p\_\{b\}\\\}\)per budget, in addition to matrix–vector products withω\\omegaand any cost\-metric factorization\. The dense catalog cost is𝒪​\(∑b∈𝒦budM​pb​min⁡\{M,pb\}\)\\mathcal\{O\}\(\\sum\_\{b\\in\\mathcal\{K\}\_\{\\mathrm\{bud\}\}\}Mp\_\{b\}\\min\\\{M,p\_\{b\}\\\}\)\. Directly factoring a dense non\-identity restricted metricQb⊤​Cb​QbQ\_\{b\}^\{\\top\}C\_\{b\}Q\_\{b\}adds𝒪​\(pb3\)\\mathcal\{O\}\(p\_\{b\}^\{3\}\)unless the factor is cached or structured\.

Identifiability analysis requires a rank\-revealing nullspace computation forDDand an SVD ofO​VkerOV\_\{\\ker\}\. Each analyzed matrix follows the dense compact SVD cost convention𝒪​\(a​b​min⁡\{a,b\}\)\\mathcal\{O\}\(ab\\min\\\{a,b\\\}\)\.

##### Diagnostic record and audit trail\.

Everydiagnosecall returns a structured diagnostic record containing‖ω‖\\left\\\|\\omega\\right\\\|, channel and per\-overlap norms, exact\-hard feasibility and cost records, separate tolerance and soft records with their residuals, and the identifiability certificate\. These fields are the quantities needed to audit the scientific verdict; no additional aggregate score or discrete classification layer is introduced\.

## 9Experiments

We evaluate SEAM\-Ω\\Omegain nineteen experiments spanning synthetic PDE conservation laws; random\-sinusoid Burgers initial conditions; FNO OOD monitoring; real\-world open datasets from the University of California, Irvine \(UCI\) Machine Learning Repository, namely Metro Interstate Traffic Volume, Air Quality, and Individual Household Electric Power Consumption; and United States Geological Survey \(USGS\) streamflow at three National Water Information System \(NWIS\) sites\. The suite also uses synthetic financial multi\-regime data and a synthetic industrial multi\-zone fault scenario\. The experiments are organized into eight groups aligned with the scientific claims that each group evaluates\. Aggregate reporting and repeated\-run checks follow Appendix[E\.1](https://arxiv.org/html/2608.05702#A5.SS1)\. Appendix[E](https://arxiv.org/html/2608.05702#A5)records the remaining common configuration and per\-experiment settings needed to interpret and reproduce the results\. Cross\-experiment comparisons use the raw and edge\-count\-normalized reports introduced next\.

### 9\.1Reporting obstruction magnitudes

Raw obstruction magnitudes‖ω‖2\\left\\\|\\omega\\right\\\|\_\{2\}retain the units of the overlap stalks and depend on the scale ofssand on the number and dimensions of the overlaps\. Every experiment below reports the raw norm, which records absolute magnitude in physical units\. Comparisons across covers of different sizes additionally use the edge\-count\-normalized report

\|E\|−1/2​‖ω‖2=\(\|E\|−1​∑\(i,j\)∈E‖ωi​j‖2\)1/2,\\displaystyle\|E\|^\{\-1/2\}\\left\\\|\\omega\\right\\\|\_\{2\}=\\bigl\(\|E\|^\{\-1\}\\textstyle\\sum\_\{\(i,j\)\\in E\}\\left\\\|\\omega\_\{ij\}\\right\\\|^\{2\}\\bigr\)^\{1/2\},the root\-mean\-square \(RMS\) per\-overlap obstruction, defined when\|E\|\>0\|E\|\>0and controlling for the number of overlaps\.

### 9\.2Local correctness does not imply admissibility

#### 9\.2\.1Local–global amplitude mismatch

##### Setup\.

We use the cover\[0,1\]=U1∪U2∪U3\[0,1\]=U\_\{1\}\\cup U\_\{2\}\\cup U\_\{3\}with a10%10\\%overlap fraction\. RegionUiU\_\{i\}carries the analytic generatorui​\(x\)=Ai​sin⁡\(2​π​x\)u\_\{i\}\(x\)=A\_\{i\}\\sin\(2\\pi x\)with amplitudesA1=1\.0A\_\{1\}=1\.0,A2=1\.5A\_\{2\}=1\.5, andA3=0\.7A\_\{3\}=0\.7\. Local explanations carry only a state block\. The three regional generators are independently specified and internally exact, making this experiment the controlled realization of Mechanism M1\.

##### Results\.

The raw obstruction norm is‖ω‖≈0\.71\\left\\\|\\omega\\right\\\|\\approx 0\.71and is dominated by the state channel\. We assess admissibility\-restoring interventions by their budget\-specific hard feasibility, residuals, and costs\. Under the unrestricted budget, the hard repair residual‖ω−D​δall⋆‖\\left\\\|\\omega\-D\\delta\_\{\\mathrm\{all\}\}^\{\\star\}\\right\\\|falls below10−1210^\{\-12\}\. The closure\-revision budget is infeasible because the explanations carry no closure block to absorb the mismatch, whereas the state\-revision budget succeeds with cost‖δ⋆‖2≈0\.50\\left\\\|\\delta^\{\\star\}\\right\\\|^\{2\}\\approx 0\.50\. The state channel is dominant on both overlaps, and the state\-revision budget is the only retained proper account\. This verdict agrees with the amplitudes having been specified independently across the three regional generators\.

#### 9\.2\.2Zero\-floor negative control

##### Setup\.

A valid obstruction measure must report‖ω‖=0\\left\\\|\\omega\\right\\\|=0when no inconsistency is present\. Using the USGS Potomac second\-order autoregressive system, denoted AR\(2\), we evaluate three conditions on a three\-region abstract cover: an identical\-model control, which uses identical AR\(2\) weights across regions; a heterogeneous\-seasonal reference, which uses the calibrated seasonal weights from the Potomac streamflow seasonality baseline; and a noise\-perturbed identical\-model control, which adds independent and identically distributed Gaussian noise at the10−810^\{\-8\}scale to predictor outputs from otherwise identical models\.

##### Results\.

The identical\-model control gives‖ω‖=0\.0000±0\.0000\\left\\\|\\omega\\right\\\|=0\.0000\\pm 0\.0000\. The heterogeneous\-seasonal reference gives‖ω‖=0\.0266±0\.0000\\left\\\|\\omega\\right\\\|=0\.0266\\pm 0\.0000and exactly reproduces the Potomac streamflow seasonality baseline\. The noise\-perturbed identical\-model control gives‖ω‖=\(2\.71±0\.78\)×10−8\\left\\\|\\omega\\right\\\|=\(2\.71\\pm 0\.78\)\\times 10^\{\-8\}, well below10−610^\{\-6\}and several orders of magnitude below the smallest real\-world obstruction of0\.02660\.0266from the Potomac streamflow seasonality study\. The zero floor is robust to floating\-point perturbations at the10−810^\{\-8\}scale\.

#### 9\.2\.3Household\-power cross\-framework audit

##### Setup\.

We use the UCI Individual Household Electric Power Consumption dataset\(Hébrail and Berard,[2012](https://arxiv.org/html/2608.05702#bib.bib20)\)\. The target isGlobal\_active\_powerin kW\. Features comprise the cyclical hour represented by sine and cosine, standardized voltage, and standardized reactive power\. Separate winter, spring, summer, and autumn models are fitted by three officially maintained baseline frameworks: scikit\-learn \(SK\)LinearRegression\(Pedregosa et al\.,[2011](https://arxiv.org/html/2608.05702#bib.bib39)\); XGBoost \(XGB\)XGBRegressor\(Chen and Guestrin,[2016](https://arxiv.org/html/2608.05702#bib.bib7)\); and statsmodels \(SM\), with ordinary least squares \(OLS\) implemented byOLS\(Seabold and Perktold,[2010](https://arxiv.org/html/2608.05702#bib.bib51)\)\. Each seasonal model is trained on an80%80\\%split and evaluated on the held\-out20%20\\%\. The four seasonal models are wrapped byfrom\_predictorand passed todiagnoseon a path cover with overlaps\(winter,spring\)\(\\mathrm\{winter\},\\mathrm\{spring\}\),\(spring,summer\)\(\\mathrm\{spring\},\\mathrm\{summer\}\), and\(summer,autumn\)\(\\mathrm\{summer\},\\mathrm\{autumn\}\)\.

##### Local accuracy\.

All three model families achieve mean seasonalR2\>0\.999R^\{2\}\>0\.999, meeting the local\-accuracy criterion used here\.

##### Global admissibility audit\.

SEAM\-Ω\\Omegadetects nontrivial obstruction in all three model families:‖ω‖SK=9\.424±0\.732\\left\\\|\\omega\\right\\\|\_\{\\mathrm\{SK\}\}=9\.424\\pm 0\.732,‖ω‖XGB=9\.391±0\.719\\left\\\|\\omega\\right\\\|\_\{\\mathrm\{XGB\}\}=9\.391\\pm 0\.719, and‖ω‖SM=9\.424±0\.732\\left\\\|\\omega\\right\\\|\_\{\\mathrm\{SM\}\}=9\.424\\pm 0\.732\. The state channel is dominant for all three model families, and the dominant overlap is\(winter,spring\)\(\\mathrm\{winter\},\\mathrm\{spring\}\): The winter–spring transition carries the largest seasonal gradient in domestic electricity consumption\.

##### Physical and operational interpretation\.

The obstruction norm is in units of kW⋅\\cdot\(point\-count\)1/2, reflecting theL2L^\{2\}disagreement across three overlaps with 30 evaluation points per overlap\. The global norm corresponds to an RMS disagreement of approximately9\.424/3⋅30≈1\.09\.424/\\sqrt\{3\\cdot 30\}\\approx 1\.0kW per evaluation point\. An automated demand\-response system that switches between seasonal models at the transition would exhibit prediction jumps of this magnitude\.

### 9\.3Missing closure is recoverable

#### 9\.3\.1Hidden\-source recovery for Burgers

##### Setup\.

The true PDE is the inviscid Burgers equation with a hidden Gaussian source:

ut\+12​\(u2\)x=A​exp⁡\(−\(x−x0\)2/\(2​w2\)\),\\displaystyle u\_\{t\}\+\\tfrac\{1\}\{2\}\(u^\{2\}\)\_\{x\}=A\\exp\(\-\(x\-x\_\{0\}\)^\{2\}/\(2w^\{2\}\)\),whereA=0\.6A=0\.6,x0=0\.5x\_\{0\}=0\.5, andw=0\.06w=0\.06\. A source\-free finite\-volume Burgers solver is run on each region, and the local explanation emits a closure\-summary block measuring the missing\-source residual on the overlap\. This block is audited through the closure restriction, while the state\-revision budget handles state mismatches\.

##### Results\.

The closure\-channel recovery module computes the Moore–Penrose projected correctionδC⋆\\delta\_\{C\}^\{\\star\}; under the sign convention, the recovered physical source increment is−δC⋆\-\\delta\_\{C\}^\{\\star\}\. The relative closure\-channel residual of this projected recovery is‖ωclosure−DC​δC⋆‖/‖ωclosure‖<0\.05\\left\\\|\\omega^\{\\mathrm\{closure\}\}\-D\_\{C\}\\delta\_\{C\}^\{\\star\}\\right\\\|/\\left\\\|\\omega^\{\\mathrm\{closure\}\}\\right\\\|<0\.05, equivalently below5%5\\%\. The recovered closure peak isx^0=0\.501\\hat\{x\}\_\{0\}=0\.501, within0\.2%0\.2\\%of the true centerx0=0\.5x\_\{0\}=0\.5and displaced by1\.7%1\.7\\%of the source widthw=0\.06w=0\.06\. Corollary[7\.3](https://arxiv.org/html/2608.05702#S7.Thmtheorem3)gives zero residual under exact image membership; the finite\-grid reconstruction above is the projected numerical counterpart of that exact case\. The closure channel is dominant on both overlaps\.

##### Scientific interpretation\.

For the identifiable closure\-summary model used in this experiment, the recovered physical increment−δC⋆\-\\delta\_\{C\}^\{\\star\}is the minimum\-norm closure reconstruction\. The reconstruction makes the missing\-physics hypothesis explicit and provides the recovered source profile together with its projected residual, following the physical interpretation specified in Section[4\.3](https://arxiv.org/html/2608.05702#S4.SS3)\. For the saved pre\-repair raw cochain,‖ω‖=2\.020994\\left\\\|\\omega\\right\\\|=2\.020994; the overlap norms are1\.4289511\.428951onU12U\_\{12\}and1\.4291661\.429166onU23U\_\{23\}\. The closure\-channel norm is2\.0209862\.020986, compared with0\.0058730\.005873in the state channel and zero in the observation and optional contract\-metadata channels\. Thus, the closure channel contributes99\.9992%99\.9992\\%of‖ω‖2\\left\\\|\\omega\\right\\\|^\{2\}, supporting the closure\-dominated diagnostic reading\. Figure[6](https://arxiv.org/html/2608.05702#S9.F6)shows these per\-overlap and per\-channel summaries of the repair\-inputω\\omega\.

![Refer to caption](https://arxiv.org/html/2608.05702v1/x3.png)Figure 6:Per\-overlap and per\-channel obstruction summary for hidden\-source recovery for Burgers, computed directly from the saved pre\-repair raw cochainω=D​s\\omega=Ds\. The left plot indexes the two overlapsU12U\_\{12\}andU23U\_\{23\}of the three\-region cover; the right plot indexes the three primary channels and the optional contract\-metadata block, which is absent in this experiment and therefore has zero norm\. The closure channel contributes99\.9992%99\.9992\\%of‖ω‖2\\left\\\|\\omega\\right\\\|^\{2\}, while the other channels contribute negligibly\. This concentration is the visual signature behind the closure\-dominated diagnostic reading\. Repair under the closure\-revision budget is the minimum\-norm closure recovery of Corollary[7\.3](https://arxiv.org/html/2608.05702#S7.Thmtheorem3)\.

#### 9\.3\.2Conservation\-contract detectability

##### Setup\.

Three Burgers regions cover\[0,1\]\[0,1\]\. The central regionU2U\_\{2\}has an active conservation contract\. The repair magnitude used in Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2)and reported here isϵrepair≔‖δ​cq‖\\epsilon\_\{\\mathrm\{repair\}\}\\coloneqq\\left\\\|\\delta c\_\{q\}\\right\\\|, the norm of the actual closure\-channel vector entering the overlap restriction at the repaired endpoint\. This endpoint repair magnitude is distinct from the accumulated absolute repair magnitudeϵqabs≔∑t\|δq​\(t\)\|\\epsilon\_\{q\}^\{\\mathrm\{abs\}\}\\coloneqq\\sum\_\{t\}\|\\delta\_\{q\}\(t\)\|, because the closure block accumulates signed increments, while the optional metadata block records their magnitudes \(Appendix[D\.1](https://arxiv.org/html/2608.05702#A4.SS1)\)\. RegionsU1U\_\{1\}andU3U\_\{3\}have no conservation repair, so their restricted repair contributions vanish on their incident overlaps\. The 1\-skeleton of this cover’s nerve is the three\-vertex path joining regions11,22, and33in that order; its two edges correspond to the overlapsU12U\_\{12\}andU23U\_\{23\}\.

##### Results\.

‖ω12closure‖=1\.172\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{12\}\\right\\\|=1\.172and‖ω23closure‖=1\.172\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{23\}\\right\\\|=1\.172\. The closure\-revision intervention recovers the repair magnitude to within0\.83%0\.83\\%relative error\. The smallest nonzero singular value of the closure restriction isσmin\+=2\.828\\sigma\_\{\\min\}^\{\+\}=2\.828; the lower bound in Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2)is0\.1280\.128; and the observed norm exceeds the bound by a factor of9\.29\.2under the one\-sided zero\-contribution setting of Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2)\.

##### Parametric sweep setup\.

We sweep five values of the repair magnitude in Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2),ϵrepair=‖δ​cq‖\\epsilon\_\{\\mathrm\{repair\}\}=\\left\\\|\\delta c\_\{q\}\\right\\\|, namelyϵrepair∈\{0\.05,0\.10,0\.20,0\.30,0\.50\}\\epsilon\_\{\\mathrm\{repair\}\}\\in\\\{0\.05,0\.10,0\.20,0\.30,0\.50\\\}\. This parametric sweep yields 25 configurations, whose run construction is specified in Appendix[E\.1](https://arxiv.org/html/2608.05702#A5.SS1)\. The discretization\-tolerant criterion is‖ωi​jclosure‖≥0\.9⋅σmin\+​ϵrepair\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{ij\}\\right\\\|\\geq 0\.9\\cdot\\sigma\_\{\\min\}^\{\+\}\\epsilon\_\{\\mathrm\{repair\}\}with a prespecified10%10\\%tolerance\.

##### Parametric sweep results\.

All 25 configurations satisfy this criterion\. The minimum observed slack across all configurations exceeds0\.90\.9\. Both the empirical norm and the lower bound scale approximately linearly withϵrepair\\epsilon\_\{\\mathrm\{repair\}\}, consistent with the linear structure of the diagnostic\. The diagnostic records retain the unadjusted ratios‖ωi​jclosure‖/\(σmin\+​ϵrepair\)\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{ij\}\\right\\\|/\(\\sigma\_\{\\min\}^\{\+\}\\epsilon\_\{\\mathrm\{repair\}\}\)alongside the discretization\-tolerant result\. Figure[7](https://arxiv.org/html/2608.05702#S9.F7)plots the seed\-aggregated empirical‖ωi​jclosure‖\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{ij\}\\right\\\|against the theoretical lower bound at the five repair magnitudes\.

![Refer to caption](https://arxiv.org/html/2608.05702v1/x4.png)Figure 7:Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2)parametric sweep\. All 25 configurations pass, corresponding to a100%100\\%rate\. Seed\-aggregated observed‖ωi​jclosure‖\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{ij\}\\right\\\|as a function of the lower boundσmin\+​ϵrepair\\sigma\_\{\\min\}^\{\+\}\\epsilon\_\{\\mathrm\{repair\}\}at five values ofϵrepair=‖δ​cq‖\\epsilon\_\{\\mathrm\{repair\}\}=\\left\\\|\\delta c\_\{q\}\\right\\\|\. With five seeds per value, the plotted points summarize the 25 configurations in the run construction of Appendix[E\.1](https://arxiv.org/html/2608.05702#A5.SS1)\. All configurations satisfy the prespecified10%10\\%discretization\-tolerant criterion\. Empirical norms can exceed the lower bound because the row\-space component ofδ​cq\\delta c\_\{q\}has components along singular directions associated with singular values larger thanσmin\+\\sigma\_\{\\min\}^\{\+\}; row\-space\-orthogonal kernel components are annihilated and do not increase the observed obstruction\.

### 9\.4Data–physics conflict attribution

##### Setup\.

The Burgers solver with a correct Gaussian closure prior is run on three overlapping regions\. In the sensor\-corruption condition, one region’s sensor mean is shifted by5​σ5\\sigma; in the closure\-corruption condition, one region’s closure parameter is biased\. The sensor\-corruption condition realizes Mechanism M3, and the closure\-corruption condition is the matching physics\-side control\. The budgets compared are the sensor\-rejection budget, which reweights sensors, and the closure\-revision budget, which shifts closure parameters\. This experiment evaluates the soft budget\-attribution branch; squared intervention norms and residuals are reported for both injected corruption types under the reporting contract of Definition[3\.9](https://arxiv.org/html/2608.05702#S3.Thmtheorem9)\.

##### Results: sensor corruption\.

The observation\-channel soft squared intervention norm isnobssoft=0\.08n^\{\\mathrm\{soft\}\}\_\{\\mathrm\{obs\}\}=0\.08, the closure\-channel counterpart isnclosuresoft=1\.47n^\{\\mathrm\{soft\}\}\_\{\\mathrm\{closure\}\}=1\.47, and their ratio favors the sensor\-rejection budget by a factor of18\.418\.4\. The residual\-aware empirical diagnosis is a data conflict\.

##### Results: closure corruption\.

The closure\-channel soft squared intervention norm isnclosuresoft=0\.11n^\{\\mathrm\{soft\}\}\_\{\\mathrm\{closure\}\}=0\.11, the observation\-channel counterpart isnobssoft=1\.39n^\{\\mathrm\{soft\}\}\_\{\\mathrm\{obs\}\}=1\.39, and their ratio favors the closure\-revision budget by a factor of12\.612\.6\. The residual\-aware empirical diagnosis is a closure conflict\.

In both conditions, the budget with the lower squared intervention norm identifies the injected source of disagreement; the diagnostic records retain the corresponding residuals\.

### 9\.5Blind\-closure identifiability

##### Setup\.

The closure block of each region is spanned by two basis functions\. A visible function is detected byOO, whereas an invisible function belongs toker⁡O\\ker O\. We run the identifiability analyzer from Algorithm[8\.1](https://arxiv.org/html/2608.05702#alg3)\.

##### Results\.

dimℬ=1\\dim\\mathcal\{B\}=1in the global explanation space\. The computed orthonormal basis ofℬ\\mathcal\{B\}consists of a single vector whose support lies entirely in the closure channel\. The corresponding channel fractions are100%100\\%closure and0%0\\%for each of the state, observation, and optional metadata blocks\. Observation invisibility satisfies‖O​Bℬ‖<10−14\\left\\\|OB\_\{\\mathcal\{B\}\}\\right\\\|<10^\{\-14\}\. Two closure configurations with identical predicted states and sensor readings, differing only in the blind admissible direction, are constructed and verified to differ by‖s−s′‖\>0\.15\\left\\\|s\-s^\{\\prime\}\\right\\\|\>0\.15\. Together, the constructed pair and the observation\-invisibility certificate establish a 1D blind admissible direction\.

### 9\.6Governing\-model discrimination and backend interoperability

#### 9\.6\.1PDE model\-form discrimination: Burgers versus advection

##### Setup\.

The true PDE is linear advection,ut\+c​ux=0u\_\{t\}\+cu\_\{x\}=0, withc=0\.5c=0\.5on\[0,1\]\[0,1\], periodic boundary conditions, and a smooth bump as the initial condition\. Two candidate generators are evaluated: finite\-volume Burgers and finite\-volume linear advection atc=0\.5c=0\.5\.

##### Results\.

‖ω‖Burgers=0\.84\\left\\\|\\omega\\right\\\|\_\{\\mathrm\{Burgers\}\}=0\.84and‖ω‖advection=0\.031\\left\\\|\\omega\\right\\\|\_\{\\mathrm\{advection\}\}=0\.031; ranking by obstruction selects linear advection with a96%96\\%relative obstruction gap\. The Burgers obstruction is localized near the steep region of the bump, where the Burgers characteristic speeduudeparts most from the constant advection speedcc\. Both state\-only comparisons have state\-channel obstruction above the defaultτzero=10−6\\tau\_\{\\mathrm\{zero\}\}=10^\{\-6\}, but the advection run has substantially lower obstruction\. The comparison therefore selects advection as the lower\-obstruction model form while retaining its nonzero state\-channel diagnostic\.

##### Observed regime\.

Across the tested model forms and parameter range, obstruction increases monotonically with model mismatch\. The compared quantity is the raw admissibility defectω=D​s\\omega=Dsof Definition[3\.7](https://arxiv.org/html/2608.05702#S3.Thmtheorem7)\. Because this comparison uses state\-only explanations, the reported obstruction lies in the state channel; that obstruction is the raw defect, not a budget\-specific residual or an intervention cost\.

In this experiment, model mismatch specifically denotes the discrepancy between the candidate transport dynamics and the data\-generating linear advection dynamics\. The correct candidate transports the bump at the constant characteristic speedc=0\.5c=0\.5, whereas the Burgers candidate uses the state\-dependent speeduu; the difference between the two speeds is largest near the steep portion of the bump, which is also where the overlap defects concentrate\. Thus, the value0\.0310\.031for advection represents a small but nonzero residual overlap disagreement at the tested resolution, while the value0\.840\.84for Burgers records the much larger incompatibility induced by the wrong governing dynamics\. The monotone statement is an empirical observation for this controlled comparison and its tested parameter range, not a claim that‖ω‖\\left\\\|\\omega\\right\\\|must be monotone in an arbitrary parametrization of model error\.

#### 9\.6\.2Backend interoperability: finite\-volume and CT\-SEAM generators

##### Setup\.

Two backends produce local explanations on the same three\-region Burgers domain\. Backend A is a plain finite\-volume Burgers generator with zero closure\. Backend B is the same solver with an active conservation\-theoretic closure implemented by CT\-SEAM as described in Appendix[D](https://arxiv.org/html/2608.05702#A4), and with region\-varying repair amplitudes\. Both outputs are audited with the same cover, restrictions, channel metrics, and intervention catalog\.

##### Results\.

The diagnostic\-record comparison gives the raw obstruction\-norm ratio‖ω‖B/‖ω‖A\>1\\left\\\|\\omega\\right\\\|\_\{B\}/\\left\\\|\\omega\\right\\\|\_\{A\}\>1, confirming that the conservation repair introduces detectable closure\-channel disagreement\. The channel report assigns the additional signal in Backend B to the closure channel, as expected from its region\-varying repair\. The diagnostic records report the obstruction, channel, and intervention summaries for both backends\.

### 9\.7Cross\-domain and stochastic\-generator evaluations

#### 9\.7\.1Random\-sinusoid Burgers stress test

##### Setup\.

The Burgers initial condition is generated as a superposition ofNsin=10N\_\{\\sin\}=10random sinusoids with integer frequencykn∼Unif⁡\{1,…,8\}k\_\{n\}\\sim\\operatorname\{Unif\}\\\{1,\\ldots,8\\\}and amplitudean∼𝒩​\(0,1/kn\)a\_\{n\}\\sim\\mathcal\{N\}\(0,1/k\_\{n\}\)\. The construction adapts the random\-sinusoid family used in PDEBench to this ten\-wave, Gaussian\-amplitude distribution\(Takamoto et al\.,[2022](https://arxiv.org/html/2608.05702#bib.bib53)\)\. The local solver generates all trajectories used in this study; PDEBench supplies methodological precedent rather than experiment data\.

##### Results\.

The aggregate run protocol gives‖ω‖=0\.4754±0\.0000\\left\\\|\\omega\\right\\\|=0\.4754\\pm 0\.0000\.

#### 9\.7\.2Metro traffic seasonality

##### Setup\.

We use the UCI Metro Interstate Traffic Volume dataset\(Hogue,[2019](https://arxiv.org/html/2608.05702#bib.bib23)\)\. A three\-region seasonal cover spans winter, summer, and a transition region, with one AR\(2\) predictor fitted per region\. This study, together with the streamflow and air\-quality studies that follow, instantiates Mechanism M2 on measured data\.

##### Results\.

The aggregate run protocol gives‖ω‖=0\.0973±0\.0000\\left\\\|\\omega\\right\\\|=0\.0973\\pm 0\.0000\. The obstruction reflects genuine cross\-seasonal mismatch in the AR\(2\) parameters\.

#### 9\.7\.3Potomac streamflow seasonality and multi\-site regime comparison

##### Potomac streamflow setup\.

We use USGS streamflow data\(U\.S\. Geological Survey,[2023](https://arxiv.org/html/2608.05702#bib.bib54)\)from site 01646500 on the Potomac River at the Little Falls Pump Station near Washington, District of Columbia \(DC\)\. The dataset covers 2021\-10\-01 through 2022\-09\-30 with 365 daily observations\. The cover has three hydrological regions, with one AR\(2\) predictor fitted per region\.

##### Potomac streamflow results\.

The aggregate run protocol gives‖ω‖=0\.0266±0\.0000\\left\\\|\\omega\\right\\\|=0\.0266\\pm 0\.0000\. This norm is the smallest value across the real\-world experiments, consistent with the Potomac’s moderate seasonality\.

##### Multi\-site comparison setup\.

We use three USGS sites with distinct hydrological regimes: 01413500 is the East Branch Delaware River at Margaretville, New York, with Catskill snowmelt and strong seasonality; 01646500 is the Potomac River near Washington, DC, at the Little Falls Pump Station, with moderate seasonality; and 02169500 is the Congaree River at Columbia, South Carolina, with mild southern seasonality\. The hypothesis states that the southern site’s obstruction is below the maximum northern\-site value\.

##### Multi\-site comparison results\.

‖ω‖01413500=0\.054\\left\\\|\\omega\\right\\\|\_\{01413500\}=0\.054,‖ω‖02169500=0\.037\\left\\\|\\omega\\right\\\|\_\{02169500\}=0\.037, and‖ω‖01646500=0\.027\\left\\\|\\omega\\right\\\|\_\{01646500\}=0\.027\. The hypothesis is supported: The southern site is below the northern\-site maximum\. The strong\-seasonality site carries the largest obstruction, whereas the moderate\- and mild\-seasonality sites do not follow the stated label order under a common configuration\.

#### 9\.7\.4Air\-quality temporal consistency

##### Setup\.

We use the UCI Air Quality dataset\(De Vito,[2016](https://arxiv.org/html/2608.05702#bib.bib10)\), which contains 9357 hourly records of the four pollutant channels CO, C6H6, NOx, and NO2from an Italian monitoring station\. The cover has three temporal regions, with one OLS predictor fitted per region\.

##### Results\.

The fixed\-data protocol gives‖ω‖=0\.2850\\left\\\|\\omega\\right\\\|=0\.2850for the UCI dataset and preprocessing protocol\. The computation is deterministic\. The higher‖ω‖\\left\\\|\\omega\\right\\\|relative to streamflow is consistent with the stronger seasonality of air quality, governed by temperature\-dependent photochemistry\.

#### 9\.7\.5Financial regime and industrial fault detection

##### Financial regime detection\.

Synthetic log returns are generated across bull, bear, and sideways regimes\. The regime\-dependent volatility of those returns is motivated by a selected subset of the stylized facts documented byCont \([2001](https://arxiv.org/html/2608.05702#bib.bib8)\)\. The generator represents this selected subset of asset\-return properties\. The heterogeneous case yields‖ω‖=0\.0137±0\.0007\\left\\\|\\omega\\right\\\|=0\.0137\\pm 0\.0007, with the closure channel serving as a residual\-volatility proxy and dominating the aggregate result\. The bull–bear overlap contributes≈57%\\approx 57\\%of the total obstruction; the bear–sideways overlap contributes≈43%\\approx 43\\%\. The identical\-model control gives‖ω‖=0\\left\\\|\\omega\\right\\\|=0\.

##### Industrial multi\-zone fault detection\.

We simulate a synthetic three\-zone serial process with one OLS regressor per zone and inject a0\.3​σ0\.3\\sigmastep fault into the midstream zone halfway through the evaluation window\. Under normal operation,‖ω‖normal=0\.491±0\.199\\left\\\|\\omega\\right\\\|\_\{\\mathrm\{normal\}\}=0\.491\\pm 0\.199; under fault injection,‖ω‖fault=2\.039±0\.501\\left\\\|\\omega\\right\\\|\_\{\\mathrm\{fault\}\}=2\.039\\pm 0\.501, an increase by a factor of4\.14\.1\. The state channel dominates the aggregate fault response, correctly attributing the inconsistency to process\-variable changes rather than to a closure or sensor conflict\. The identical\-zone control gives‖ω‖=0\.000\\left\\\|\\omega\\right\\\|=0\.000\.

### 9\.8Learned\-generator monitoring

#### 9\.8\.1Controlled predictor perturbation

##### Setup\.

Starting from the USGS Potomac AR\(2\) predictors calibrated in the Potomac streamflow seasonality study, we perturb the spring predictor weights in 10 random directions with magnitudesσ∈\{0,0\.05,0\.10,0\.20,0\.40,0\.80,1\.60\}\\sigma\\in\\\{0,0\.05,0\.10,0\.20,0\.40,0\.80,1\.60\\\}and compute‖ω‖\\left\\\|\\omega\\right\\\|\.

##### Results\.

We define the empirical monotonicity fraction as the proportion of adjacent perturbation\-level pairs for which‖ω‖\\left\\\|\\omega\\right\\\|strictly increases\. This fraction is1\.00±0\.001\.00\\pm 0\.00under the aggregate run protocol\. The zero\-perturbation floor is‖ω‖=0\.0266\\left\\\|\\omega\\right\\\|=0\.0266, equal to the Potomac streamflow seasonality baseline\. The diagnostic is strictly monotone on this dataset across all tested magnitudes and directions\. This behavior is empirical monotonicity in perturbation magnitude for this dataset and sweep, not a general guarantee\. Figure[8](https://arxiv.org/html/2608.05702#S9.F8)plots‖ω‖\\left\\\|\\omega\\right\\\|as a function ofσ\\sigma, with mean±\\pmstandard deviation across the 10 perturbation directions\.

![Refer to caption](https://arxiv.org/html/2608.05702v1/x5.png)Figure 8:Inconsistency detection shows that‖ω‖\\left\\\|\\omega\\right\\\|scales monotonically with injected perturbation magnitude\. Starting from the USGS Potomac AR\(2\) predictors calibrated in the Potomac streamflow seasonality study, the spring predictor weights are perturbed in 10 random directions with magnitudesσ∈\{0,0\.05,0\.10,0\.20,0\.40,0\.80,1\.60\}\\sigma\\in\\\{0,0\.05,0\.10,0\.20,0\.40,0\.80,1\.60\\\}\. The curve shows the mean‖ω‖\\left\\\|\\omega\\right\\\|across directions; error bars and the shaded band show±\\pmone standard deviation\. The vertical axis reports the raw obstruction norm\. Across the full range, the empirical monotonicity fraction is1\.00±0\.001\.00\\pm 0\.00under the aggregate run protocol\.

#### 9\.8\.2FNO OOD monitoring

##### Setup\.

An FNO\(Li et al\.,[2021](https://arxiv.org/html/2608.05702#bib.bib31)\)is trained on a distribution of Burgers initial conditions and evaluated on OOD covariate\-shifted inputs\. The FNO\-1D architecture has width6464, four spectral layers, mode cutoffm=16m=16, Gaussian error linear unit activation, and approximately0\.650\.65million parameters\. The streaming obstructionω​\(t\)\\omega\(t\)is computed from the FNO regional explanations and their overlap restrictions\. The study is the experimental realization of Mechanism M4\. A high\-resolution reference solver independently supplies theL2L^\{2\}prediction\-error coordinate in the correlation analysis\.

##### Dataset\.

The dataset represents the 1D Burgers equation on𝒳=\[0,1\]\\mathcal\{X\}=\[0,1\]overt∈\[0,0\.5\]t\\in\[0,0\.5\], with viscosityν=0\.01\\nu=0\.01and periodic boundary conditions\. The spatial and temporal grids haveNx=128N\_\{x\}=128andNt=100N\_\{t\}=100points, respectively\. The training\-distribution initial conditions are superpositions of three sinusoids:

u0​\(x\)=∑ℓ=13aℓ​sin⁡\(2​π​qℓ​x\+ϕℓ\)\.\\displaystyle u\_\{0\}\(x\)=\\sum\_\{\\ell=1\}^\{3\}a\_\{\\ell\}\\sin\(2\\pi q\_\{\\ell\}x\+\\phi\_\{\\ell\}\)\.Hereaℓ∼Unif⁡\[−0\.5,0\.5\]a\_\{\\ell\}\\sim\\operatorname\{Unif\}\[\-0\.5,0\.5\],qℓ∈\{1,2,3\}q\_\{\\ell\}\\in\\\{1,2,3\\\}chosen uniformly, andϕℓ∼Unif⁡\[0,2​π\]\\phi\_\{\\ell\}\\sim\\operatorname\{Unif\}\[0,2\\pi\]\.

##### Training\.

Training usesNtrain=1000N\_\{\\mathrm\{train\}\}=1000trajectories, a batch size of3232, the Adam optimizer with a learning rate of10−310^\{\-3\}, and cosine annealing over200200epochs\. The loss is the relativeL2L^\{2\}error over the\(x,t\)\(x,t\)grid\. Repeated\-run training and aggregation follow Appendix[E\.1](https://arxiv.org/html/2608.05702#A5.SS1), with all other settings held fixed\.

##### OOD shift families\.

Three parametric shifts modify the training initial\-condition distribution\. The frequency shift usesqℓ∈\{4,5,6\}q\_\{\\ell\}\\in\\\{4,5,6\\\}instead of\{1,2,3\}\\\{1,2,3\\\}\. The amplitude shift usesaℓ∼Unif⁡\[−1\.5,1\.5\]a\_\{\\ell\}\\sim\\operatorname\{Unif\}\[\-1\.5,1\.5\]instead ofUnif⁡\[−0\.5,0\.5\]\\operatorname\{Unif\}\[\-0\.5,0\.5\]\. The spectral\-truncation shift uses a band\-limited initial condition truncated to four Fourier modes, with random coefficients drawn fromUnif⁡\[−0\.5,0\.5\]\\operatorname\{Unif\}\[\-0\.5,0\.5\]\. Each shift family containsNood=500N\_\{\\mathrm\{ood\}\}=500test trajectories\. For the auxiliary visualization in Figure[9](https://arxiv.org/html/2608.05702#S9.F9), a separate 20\-level sweep usesα∈\[0,1\]\\alpha\\in\[0,1\]as a path coordinate from the training distribution atα=0\\alpha=0to a fixed OOD endpoint atα=1\\alpha=1\. This seed\-averaged sweep is distinct from the per\-trajectory endpoint evaluations of the three shift families\.

##### Reference solver\.

A high\-resolution finite\-volume scheme usesNxref=1024N\_\{x\}^\{\\mathrm\{ref\}\}=1024,Ntref=800N\_\{t\}^\{\\mathrm\{ref\}\}=800, and Courant–Friedrichs–Lewy \(CFL\) number0\.40\.4\. Reference solutions are restricted to the FNO grid by spectral downsampling\. The relativeL∞L^\{\\infty\}residual with respect to an even higher\-resolution run is below5×10−45\\times 10^\{\-4\}\.

##### SEAM\-Ω\\Omegacover\.

The cover has three temporal regionsU1=\[0,0\.2\]U\_\{1\}=\[0,0\.2\],U2=\[0\.15,0\.35\]U\_\{2\}=\[0\.15,0\.35\], andU3=\[0\.30,0\.5\]U\_\{3\}=\[0\.30,0\.5\], with two overlaps\. Each region’s local generator is the FNO, advanced from initial conditions sampled at the region’s start time\. The restrictions are identity maps on the overlap\-time intervals, with spatial averaging atneval=8n\_\{\\mathrm\{eval\}\}=8equispaced points\.

##### Reported statistics\.

For each shift family, we record the per\-trajectory pair\(‖ω​\(tf\)‖,‖uFNO​\(tf\)−uref​\(tf\)‖L2\)\(\\left\\\|\\omega\(t\_\{\\mathrm\{f\}\}\)\\right\\\|,\\left\\\|u^\{\\mathrm\{FNO\}\}\(t\_\{\\mathrm\{f\}\}\)\-u^\{\\mathrm\{ref\}\}\(t\_\{\\mathrm\{f\}\}\)\\right\\\|\_\{L^\{2\}\}\)at the final timetf=0\.5t\_\{\\mathrm\{f\}\}=0\.5\. Each family therefore yields a sample ofNoodN\_\{\\mathrm\{ood\}\}pairs\. The reported statistics are the Pearson correlationrr, the Spearman rank correlationρ\\rho, and the95%95\\%bootstrap confidence interval \(CI\) forrrfrom 1000 bootstrap resamples, with run\-wise aggregation specified in Appendix[E\.1](https://arxiv.org/html/2608.05702#A5.SS1)\.

##### Results\.

Across the three OOD families, the Pearson correlation between‖ω​\(tf\)‖\\left\\\|\\omega\(t\_\{\\mathrm\{f\}\}\)\\right\\\|and theL2L^\{2\}prediction error isr=0\.995±0\.003r=0\.995\\pm 0\.003, with Spearmanρ=0\.987±0\.005\\rho=0\.987\\pm 0\.005\. The associated95%95\\%bootstrap CIs fall within\[0\.991,0\.998\]\[0\.991,0\.998\]\. Across the three tested Burgers–FNO shift families,ω\\omegais a strong regime\-shift monitoring signal\.

##### Comparison to confidence\-based monitoring\.

We compareω\\omegawith per\-trajectory predictive variance from the separate five\-member FNO ensemble specified in Appendix[E\.1](https://arxiv.org/html/2608.05702#A5.SS1)\. The Pearson correlation between ensemble variance andL2L^\{2\}error isrvar=0\.71±0\.04r\_\{\\mathrm\{var\}\}=0\.71\\pm 0\.04on the same OOD test sets, a substantially weaker signal thanω\\omega\. Ensemble variance also requires five times the computation of a single FNO, whereasω​\(t\)\\omega\(t\)requires one FNO forward pass per region with one coboundary multiplication\.

![Refer to caption](https://arxiv.org/html/2608.05702v1/x6.png)Figure 9:FNO OOD monitoring in the auxiliary 20\-level interpolation sweep\. The left plot shows the mean SEAM\-Ω\\Omegaobstruction norm and mean relative FNOL2L^\{2\}prediction error as functions of the OOD interpolation coefficientα\\alpha; shaded envelopes show±\\pmone standard deviation over seeds\. The right plot shows the same seed\-averaged quantities, colored byα\\alpha\. Low\-α\\alphacases occupy the lower\-left corner, and increasing distribution shift moves the sequence toward the upper\-right corner\. The within\-sweep correlationr=0\.953r=0\.953is a visualization\-level statistic for the 20 seed\-averaged points; that statistic is distinct from the headline per\-trajectory Pearson and Spearman correlations across the three shift families reported in Section[9\.8\.2](https://arxiv.org/html/2608.05702#S9.SS8.SSS2)\.

### 9\.9Ablations, robustness, and scalability

#### 9\.9\.1Cover\-sensitivity ablation

##### Evaluation points per overlap\.

Varyingneval∈\{1,2,3,5\}n\_\{\\mathrm\{eval\}\}\\in\\\{1,2,3,5\\\}with abstract identity restrictions causes‖ω‖\\left\\\|\\omega\\right\\\|to scale asneval\\sqrt\{n\_\{\\mathrm\{eval\}\}\}, consistent with the identity restriction contributing one independent dimension per evaluation point\.

##### Number of regions\.

Varyingnr∈\{2,3,4\}n\_\{r\}\\in\\\{2,3,4\\\}leaves the per\-overlap disagreement stable, while‖ω‖2\\left\\\|\\omega\\right\\\|^\{2\}scales approximately linearly with the number of overlaps\|E\|\|E\|; equivalently,‖ω‖\\left\\\|\\omega\\right\\\|scales approximately as\|E\|\\sqrt\{\|E\|\}\. The per\-overlap disagreement metric\|E\|−1/2​‖ω‖\|E\|^\{\-1/2\}\\left\\\|\\omega\\right\\\|is therefore a region\-count\-invariant summary\.

#### 9\.9\.2Baseline comparison between naive metrics and SEAM\-Ω\\Omega

##### Setup\.

We compare SEAM\-Ω\\Omegaagainst two naive cross\-region disagreement metrics in the metro traffic seasonality, Potomac streamflow seasonality, and air\-quality temporal\-consistency studies: cross\-prediction root\-mean\-square error \(RMSE\) at the overlap evaluation points and mean absolute pairwise difference\. Both are normalized by the mean prediction magnitude\.

##### Results\.

SEAM\-Ω\\Omegaorders the three datasets by decreasing‖ω‖\\left\\\|\\omega\\right\\\|as air quality, traffic, and streamflow, with respective values0\.28500\.2850,0\.09730\.0973, and0\.02660\.0266\. Normalized cross\-prediction RMSE orders the same datasets as air quality, streamflow, and traffic, with respective values0\.45120\.4512,0\.32700\.3270, and0\.15720\.1572\. That baseline therefore matches SEAM\-Ω\\Omegaon the largest case and reverses the ordering of traffic and streamflow\. This ordering difference is descriptive rather than a performance ranking because the three datasets do not supply a ground\-truth order of inconsistency\. The scalar baselines summarize overall disagreement, whereas SEAM\-Ω\\Omegaalso retains channel attribution, budgeted repair analysis, and identifiability\.

#### 9\.9\.3GPU implementation check

The GPU coboundary is tested on synthetic problems with\|I\|=32\|I\|=32regions anddi=103d\_\{i\}=10^\{3\}per region\. GPU and CPUω\\omegaagree to within a relative tolerance of10−1010^\{\-10\}\. Per\-call wall time on a single NVIDIA A100 GPU is below5​ms5\\,\\mathrm\{ms\}for the matrix–vector product; the SVD dominates the total runtime as in the CPU regime, but large\-scale runs require a rank\-revealing truncated approximation that captures all singular values above the chosen tolerance before the projection is treated as exact\.

## 10Related work

SEAM connects four strands of literature: sheaf\-theoretic consistency; physics\-informed learning, neural operators, and domain decomposition; identifiability in inverse problems; and model discrepancy together with distribution\-shift monitoring\. The comparison below focuses on how each strand treats overlap agreement, repair, observational ambiguity, or shift\.

### 10\.1Sheaf\-theoretic signal processing and consistency

Robinson \([2014](https://arxiv.org/html/2608.05702#bib.bib47)\)systematized the cellular\-sheaf formalism for signal processing, while the thesis ofCurry \([2014](https://arxiv.org/html/2608.05702#bib.bib9)\)provides a foundational treatment of sheaves, cosheaves, and their applications\. The exposition byGhrist \([2014](https://arxiv.org/html/2608.05702#bib.bib14)\)introduced the formalism to an applied audience, andBredon \([1997](https://arxiv.org/html/2608.05702#bib.bib4)\)provides the classical reference for sheaf theory and cohomology via derived functors\. At the level of coupled scientific models,Robinson \([2017a](https://arxiv.org/html/2608.05702#bib.bib48)\)shows how sheaves assemble interacting local models into systems of equations, including dynamical systems and PDEs, whereasHansen and Ghrist \([2021](https://arxiv.org/html/2608.05702#bib.bib19)\)apply sheaves to opinion dynamics, providing a nontrivial example of restriction maps as scientific objects\. In a distinct application area,Abramsky and Brandenburger \([2011](https://arxiv.org/html/2608.05702#bib.bib1)\)use the failure of global\-section existence as the local\-to\-global obstruction that characterizes contextuality\. SEAM shares the obstruction viewpoint, but its local objects are structured scientific explanations, and its operational outputs are channel reports, repairs, and identifiability certificates\.

For heterogeneous measurements,Robinson \([2017b](https://arxiv.org/html/2608.05702#bib.bib49)\)develops sheaves as a sensor\-integration data structure with consistency and fusion tools, whereasRobinson \([2020](https://arxiv.org/html/2608.05702#bib.bib50)\)formalizes a consistency radius for locally supplied, potentially noisy assignments\. That radius is a scalar summary of the same overlap disagreement that SEAM\-Ω\\Omegakeeps as a cochain and resolves into channel and overlap components\.

In machine learning,Hansen and Gebhart \([2020](https://arxiv.org/html/2608.05702#bib.bib17)\)formulate neural networks whose message passing is defined on a specified cellular sheaf, whileBodnar et al\. \([2022](https://arxiv.org/html/2608.05702#bib.bib3)\)parameterize and learn the sheaf structure for heterophilic graph learning\. Both constructions use sheaf\-Laplacian diffusion, whose algebraic background comes from the spectral theory of the sheaf Laplacian\(Hansen and Ghrist,[2019](https://arxiv.org/html/2608.05702#bib.bib18)\)and the broader graph Hodge\-Laplacian framework\(Lim,[2020](https://arxiv.org/html/2608.05702#bib.bib32)\)\. These constructions read the sheaf Laplacian as a diffusion operator inside a learning objective\. SEAM instead places the same local\-to\-global algebra on heterogeneous scientific explanations through typed restriction maps, and uses channel\-resolved disagreement for diagnosis and intervention\.

##### Explanation\-admissibility synthesis\.

SEAM\-Ω\\Omegabuilds on global\-section and assignment\-consistency ideas by treating a structured scientific explanation as the compared object\. The stalks of the explanation sheaf separate state, closure, and observation channels, with optional contract metadata, and the geometry\-defined restriction maps keep overlap semantics auditable\. The resulting obstruction feeds budgeted hypothesis tests and an identifiability quotient\.

### 10\.2Physics\-informed learning, neural operators, and domain decomposition

PINNs embed differential\-equation residuals and associated conditions in training objectives\(Raissi et al\.,[2019](https://arxiv.org/html/2608.05702#bib.bib44)\), while universal differential equations place trainable components inside differential\-equation models\(Rackauckas et al\.,[2020](https://arxiv.org/html/2608.05702#bib.bib43)\)\. Both sit within the broader physics\-informed machine learning program\(Karniadakis et al\.,[2021](https://arxiv.org/html/2608.05702#bib.bib26)\)\. Studies of PINN optimization in difficult PDE regimes motivate complementary validation of solution behavior alongside the soft physics\-residual objective\(Krishnapriyan et al\.,[2021](https://arxiv.org/html/2608.05702#bib.bib29)\)\. Domain\-decomposed PINNs directly address interface coupling\. The cPINN formulation enforces flux and average\-solution conditions at discrete subdomain interfaces, XPINNs generalize the construction to space–time decompositions, and FBPINNs use compactly supported networks on overlapping subdomains\(Jagtap et al\.,[2020](https://arxiv.org/html/2608.05702#bib.bib25); Jagtap and Karniadakis,[2021](https://arxiv.org/html/2608.05702#bib.bib24); Moseley et al\.,[2023](https://arxiv.org/html/2608.05702#bib.bib34)\)\. Neural operators\(Kovachki et al\.,[2023](https://arxiv.org/html/2608.05702#bib.bib28); Lu et al\.,[2021](https://arxiv.org/html/2608.05702#bib.bib33)\)learn solution maps; the FNO\(Li et al\.,[2021](https://arxiv.org/html/2608.05702#bib.bib31)\)supplies the specific spectral configuration used in the FNO OOD monitoring study\. Interface assembly is already central to these methods, and SEAM applies the same assembly logic to structured explanations emitted by heterogeneous generators\.Chen et al\. \([2025](https://arxiv.org/html/2608.05702#bib.bib6)\)also demonstrate that neural operators can violate the physical principle of locality, and propose a data decomposition that restricts predictions to a finite domain of dependence\. This locality constraint acts inside the learned generator\. SEAM’s overlap audit acts on the explanations emitted by any generator\.

Classical domain\-decomposition methods for PDEs\(Quarteroni and Valli,[1999](https://arxiv.org/html/2608.05702#bib.bib41)\)decompose a global problem into subproblems on overlapping or non\-overlapping subdomains; iterative Schwarz, Dirichlet–Neumann, and Robin–Robin methods enforce interface conditions through iteration on the trace mismatch\. The trace mismatch is, formally, a single\-overlap version ofω\\omega\. SEAM turns this mismatch into a diagnostic output, accepts data\-driven generators alongside PDE solvers, and augments the interface report with channel decomposition, budgeted hypothesis tests, and identifiability\.

##### Generator integration\.

Domain\-decomposed PINNs jointly train local networks to produce one PDE solution, with interface quantities appearing as solution constraints or loss terms\. SEAM applies a common audit to explanations from PINNs, operators, classical solvers, and regressors, and uses channel\-resolved overlap defects to test and compare repair hypotheses\. A PINN, a DeepONet, or an FNO can therefore be wrapped as a SEAM local generator\. The FNO OOD correlation result in Section[9\.8\.2](https://arxiv.org/html/2608.05702#S9.SS8.SSS2)demonstrates this capability: SEAM monitors a learned operator without modifying the operator’s parameters\.

### 10\.3Identifiability in inverse problems

Identifiability analysis in inverse problems and parameter estimation has a long history\.Bellman and Åström \([1970](https://arxiv.org/html/2608.05702#bib.bib2)\)introduced the structural\-identifiability question for dynamical systems, andWalter and Pronzato \([1997](https://arxiv.org/html/2608.05702#bib.bib56)\)systematized structural and practical identifiability for parametric models\. Profile\-likelihood analysis of partially observed dynamical systems demonstrates how structural and practical non\-identifiability appear in finite\-data inference\(Raue et al\.,[2009](https://arxiv.org/html/2608.05702#bib.bib45)\)\. SEAM\-Ω\\Omegaspecializes this question to admissible scientific explanations: The blind admissible subspaceℬ=ker⁡D∩ker⁡O\\mathcal\{B\}=\\ker D\\cap\\ker Ocontains directions that preserve overlap agreement while remaining invisible to the current observation map, and the quotientℐ\\mathcal\{I\}contains the observable admissible classes\.

### 10\.4Model discrepancy and distribution\-shift monitoring

Two traditions quantify the gap between a model and the reality the model describes\. Bayesian discrepancy modeling calibrates that gap as a statistical object, whereas shift monitoring detects the gap opening under streaming inputs\.

##### Bayesian model discrepancy\.

Kennedy and O’Hagan \([2001](https://arxiv.org/html/2608.05702#bib.bib27)\)model the gap between a computer model and observations as a Gaussian\-process discrepancy term, calibrated jointly with the model parameters\. This Bayesian discrepancy model is the closest probabilistic analog ofω\\omegabecause the discrepancy term provides a structured account of where the model fails to match reality\. Ignoring or misspecifying that discrepancy can confound physical parameters and produce biased, overconfident inference\(Brynjarsdóttir and O’Hagan,[2014](https://arxiv.org/html/2608.05702#bib.bib5)\), a caution that also motivates SEAM’s separation of closure, observation, and blind directions\. SEAM provides a deterministic, multi\-channel, local\-to\-global counterpart: The framework decomposes discrepancy across an explanation sheaf and reports the obstruction cochain\.

##### Consistency\-based diagnosis and data reconciliation\.

Constraint\-relaxation methods diagnose inconsistent model–data collections by identifying which assumptions must be relaxed\(Hegde et al\.,[2018](https://arxiv.org/html/2608.05702#bib.bib21)\), while process data reconciliation adjusts measurements to satisfy declared balance equations, and uses residuals to identify gross errors\(Narasimhan and Jordache,[1999](https://arxiv.org/html/2608.05702#bib.bib35)\)\. Consistency\-based diagnosis similarly asks which component assumptions can account for a conflict\(Reiter,[1987](https://arxiv.org/html/2608.05702#bib.bib46)\)\. SEAM applies this logic to typed regional explanations: Budget subspaces state the permitted revisions, and exact feasibility determines which accounts survive\.

##### Streaming shift detection\.

OOD detection, covariate\-shift detection, and streaming model monitoring are active areas\. Concept\-drift methods detect and adapt to changes in evolving supervised streams\(Gama et al\.,[2014](https://arxiv.org/html/2608.05702#bib.bib13)\), while classical sequential change detection includes the cumulative sum statistic\(Page,[1954](https://arxiv.org/html/2608.05702#bib.bib37)\)\. Confidence scores supply a foundational OOD baseline\(Hendrycks and Gimpel,[2017](https://arxiv.org/html/2608.05702#bib.bib22)\), although modern neural\-network probabilities may require explicit calibration\(Guo et al\.,[2017](https://arxiv.org/html/2608.05702#bib.bib16)\)\. Dataset\-shift detectors instead compare test samples or distributions with the training distribution; for example,Rabanser et al\. \([2019](https://arxiv.org/html/2608.05702#bib.bib42)\)evaluate two\-sample and domain\-discrimination approaches for detecting and characterizing dataset shift\. Predictive confidence is another monitoring signal, but its calibration can degrade under shift\(Ovadia et al\.,[2019](https://arxiv.org/html/2608.05702#bib.bib36)\)\.

##### Admissibility monitoring\.

SEAM monitors explanation\-admissibility directly from model outputs, complementing input\-distribution shift detectors\. Shifts that break overlap admissibility produce channel\-attributed signals, while admissibility\-preserving gauge shifts leave the signal invariant\.

## 11Discussion

### 11\.1Design choices and scope of SEAM\-Ω\\Omega

Three modeling choices delimit the claims of SEAM\-Ω\\Omega\.

##### Finite\-dimensional linear setting\.

SEAM\-Ω\\Omegauses vector\-space stalks, linear restriction maps, and a matrix coboundary, yielding the closed\-form diagnostic and intervention operators proved here\.

##### Geometry\-defined restrictions\.

Restrictions are constructed from problem geometry, keepingDDtransparent as a scientific object\. Learned restriction maps would require their own identifiability and validation analysis\.

##### Pairwise\-overlap complex\.

SEAM\-Ω\\Omegauses only pairwise overlaps, represented by the 1\-skeleton of the nerve\. Accordingly, the guarantees of SEAM\-Ω\\Omegaconcern pairwise compatibility and do not establish higher\-order overlap consistency\.

### 11\.2Interpretation and operational responses

The following six properties state what the obstruction resolves and connect each diagnostic pattern to an operational response\.

##### Gauge invariance and blind directions\.

A simultaneous shifts↦s\+vs\\mapsto s\+vwithv∈ker⁡Dv\\in\\ker Dleavesω=D​s\\omega=Dsunchanged\. If the shift is also inker⁡O\\ker O, the directionvvlies inℬ\\mathcal\{B\}and is invisible to current sensors\. The identifiability report records this unresolved direction throughdimℬ\\dim\\mathcal\{B\}\.

##### Restriction\-visible repairs\.

If the repaired endpoint’s conservation\-redistribution repair vector has a component inker⁡ρq,eclosure\\ker\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}, that component is removed by the overlap restriction\. The projected form of Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2), Eq\.[9](https://arxiv.org/html/2608.05702#S7.E9), therefore measures the row\-space\-visible repair component; row\-space alignment recovers Eq\.[8](https://arxiv.org/html/2608.05702#S7.E8)\. A large gap between‖δ​cq‖\\left\\\|\\delta c\_\{q\}\\right\\\|and‖Πrow⁡\(ρq,eclosure\)​δ​cq‖\\left\\\|\\Pi\_\{\\operatorname\{row\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\}\\delta c\_\{q\}\\right\\\|, or a non\-negligible kernel\-component norm‖\(I−Πrow⁡\(ρq,eclosure\)\)​δ​cq‖\\left\\\|\(\\mathrm\{I\}\-\\Pi\_\{\\operatorname\{row\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\}\)\\delta c\_\{q\}\\right\\\|, signals the need for a full\-column\-rank closure restriction or a repair operator aligned with the relevant row space, as in the CT\-SEAM experiments of Remark[7\.4](https://arxiv.org/html/2608.05702#S7.Thmtheorem4)\.

##### Localization resolution\.

A discrepancy whose true spatial support is smaller than the finest overlap is averaged into uniformly distributedωi​j\\omega\_\{ij\}components across the overlaps containing that support\. SEAM reports‖ω‖\>0\\left\\\|\\omega\\right\\\|\>0at overlap resolution\. Comparable neighboring norms with no dominant overlap identify where a refined cover will add localization power\.

##### Residual\-aware observation attribution\.

Observation\-channel attribution is interpreted from the channel norm together with the budget residual rather than from that norm alone\. A low\-norm but high\-residual sensor\-rejection record therefore remains unresolved\.

##### Budget ambiguity and catalog dependence\.

When several proper budgets are exact\-feasible, the verdict retains all of those budgets rather than using cost to force a causal choice\. Soft records with similar intervention magnitudes are likewise reported as ambiguous, with their residuals retained in the diagnostic record\. Retention is also conditional on the declared catalog\. Exact feasibility eliminates only accounts that were stated, so a cause absent from the catalog is neither tested nor excluded, making a single retained budget as specific as the catalog that produced that budget\.

##### Overlap consistency and absolute accuracy\.

The obstruction norm‖ω‖\\left\\\|\\omega\\right\\\|compares neighboring regional explanations with each other rather than with a reference solution\. With the cover, restriction maps, discretization, and channel scaling held fixed, the norm aggregates the overlap\-wise differences between the restricted explanations, thereby reporting mutual agreement rather than correctness\. Under that reading,‖ω‖=0\\left\\\|\\omega\\right\\\|=0certifies agreement on overlaps\. Absolute\-error checks on regions with ground truth provide the complementary test for shared biases that preserve overlap agreement\. The FNO OOD monitoring study in Section[9\.8\.2](https://arxiv.org/html/2608.05702#S9.SS8.SSS2)evaluates both overlap consistency and reference\-solver error\.

## 12Conclusion

We introduced SEAM as a generator\-agnostic paradigm that treats the global admissibility of scientific explanations as a local\-to\-global gluing problem, with SEAM\-Ω\\Omegaas its finite explanation\-sheaf instantiation\. Starting from structured regional explanations with state, closure, and observation channels plus optional contract metadata, SEAM\-Ω\\Omegarestricts neighboring explanations to their overlaps and assembles their disagreements into the 1\-cochainω=D​s\\omega=Ds\. The channel and overlap components ofω\\omegaresolve that disagreement by entry and by channel rather than collapsing the cochain into a single score, while the budget\-specific imagesim⁡\(D​P\)\\operatorname\{im\}\(DP\)determine repair feasibility\. Budgeted preimages then test allowable repair hypotheses: A declared account is refuted when the revisions that account permits cannot remove the obstruction, and priced when those revisions can\. Residual\-aware soft records provide separately reported empirical attribution when exact feasibility fails\.

The blind admissible subspaceker⁡D∩ker⁡O\\ker D\\cap\\ker Oseparates inconsistency from what current observations cannot identify, and the streamingω​\(t\)\\omega\(t\)supplies an empirical consistency signal under distribution shift\. The finite\-dimensional result suite provides guarantees for minimum\-cost intervention and conservation\-contract detectability, with companion identifiability and closure\-recovery results\. The resulting algorithms assembleDD, compute budgeted interventions and a basis ofℬ\\mathcal\{B\}, and emit an auditable diagnostic record\.

Nineteen experiments across synthetic PDEs, FNO OOD monitoring, four open datasets, and synthetic financial and industrial systems jointly show that locally accurate or physically plausible generators can remain globally inadmissible, that closure\-restricted repair recovers the injected closure sources, and that channel\-resolved obstruction distinguishes and monitors multiple failure modes\. A zero\-obstruction control, a monotone response to injected predictor perturbations, cross\-domain channel attribution, and strong correlation between streaming obstruction and FNO prediction error support these findings\.

SEAM\-Ω\\Omegamakes overlap consistency exactly certifiable:ω=0\\omega=0if and only if neighboring restrictions agree\. The framework turns violations into auditable diagnoses and interventions, so that explanation\-admissibility becomes a tractable global explanation\-consistency audit for scientific machine learning\.

\\printcredits

## Funding

This work was supported through doctoral training funding provided to Gnankan Landry Regis N’guessan by the African Institute for Mathematical Sciences Data Science Doctoral Program and the Nelson Mandela African Institution of Science and Technology\.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper\.

## Acknowledgments

Gnankan Landry Regis N’guessan thanks his doctoral supervisor and the Axiom Research Group for sustained discussion of the framework\.

## Data and code availability

The third\-party datasets analyzed in this study are publicly available from the sources listed in Appendix[E\.2](https://arxiv.org/html/2608.05702#A5.SS2)\. UCI data are released under CC BY 4\.0, and USGS data are in the public domain under Section 105 of Title 17 of the United States Code\. The common experiment configuration, data sources, and preprocessing are specified in Appendix[E](https://arxiv.org/html/2608.05702#A5)\. The source code, configuration files, generated synthetic and PDE cases, trained FNO checkpoints, seed\-resolved outputs, diagnostic records, pinned environment specifications, and the publication\-figure pipeline are available from the corresponding author on reasonable request\.

## Appendix AList of notation

Table[A\.1](https://arxiv.org/html/2608.05702#A1.T1)lists the symbols used throughout the paper\. Sans\-serif denotes formal named constants in definitions, equations, and algorithms, such as𝖠𝗅𝗅𝗈𝗐𝖢𝗅𝗈𝗌𝗎𝗋𝖾\\mathsf\{AllowClosure\}and𝗀𝗅𝗈𝖻𝖺𝗅𝗅𝗒​\_​𝖺𝖽𝗆𝗂𝗌𝗌𝗂𝖻𝗅𝖾\\mathsf\{globally\\\_admissible\}; monospace denotes literal algorithmic identifiers\.

Table A\.1:List of notation\.SymbolMeaning𝒳⊆ℝd\\mathcal\{X\}\\subseteq\\mathbb\{R\}^\{d\}computational domain𝒰=\(Ui\)i∈I\\mathcal\{U\}=\(U\_\{i\}\)\_\{i\\in I\}finite cover indexed by regionsIIUi​j,EU\_\{ij\},Epairwise overlap and nerve\-edge setFFfinite explanation sheafF​\(Ui\),F​\(Ui​j\)F\(U\_\{i\}\),F\(U\_\{ij\}\)region and overlap stalksei=\(ui,ci,oi,ϵi\)e\_\{i\}=\(u\_\{i\},c\_\{i\},o\_\{i\},\\epsilon\_\{i\}\)local explanation with three primary channels and optional contract metadatas=\(vec​\(ei\)\)i∈Is=\(\\mathrm\{vec\}\(e\_\{i\}\)\)\_\{i\\in I\}stack of local explanations; a 0\-cochainρi,i​j\\rho\_\{i,ij\}restriction from regioniito overlapUi​jU\_\{ij\}C0​\(F\),C1​\(F\)C^\{0\}\(F\),C^\{1\}\(F\)region and overlap cochain spacesd0,Dd^\{0\},Dcoboundary operator and its matrixω=d0​s=D​s\\omega=d^\{0\}s=Dsraw admissibility defect; a 1\-cochainωi​j,ω∙\\omega\_\{ij\},\\omega^\{\\bullet\}overlap and channel components ofω\\omegaP,VPP,V\_\{P\}budget projector and allowed correction subspacerPhard​\(ω\)r\_\{P\}^\{\\mathrm\{hard\}\}\(\\omega\)hard budget\-feasibility residualC,‖δ‖C2C,\\left\\\|\\delta\\right\\\|\_\{C\}^\{2\}positive\-definite cost metric and intervention costδ⋆\\delta^\{\\star\}minimum\-cost admissibility correctionDC,δC⋆D\_\{C\},\\delta\_\{C\}^\{\\star\}closure\-channel coboundary block and minimum\-norm closure correctionOOobservation mapℬ=ker⁡D∩ker⁡O\\mathcal\{B\}=\\ker D\\cap\\ker Oblind admissible subspaceℐ=ker⁡D/ℬ\\mathcal\{I\}=\\ker D/\\mathcal\{B\}observable admissible quotientBℬB\_\{\\mathcal\{B\}\}orthonormal basis matrix forℬ\\mathcal\{B\}ϵrepair\\epsilon\_\{\\mathrm\{repair\}\}closure\-repair norm entering an overlap; distinct from accumulated repair metadataσmin\+\\sigma\_\{\\min\}^\{\+\}smallest nonzero singular valueηfeas,ηsvd\\eta\_\{\\mathrm\{feas\}\},\\eta\_\{\\mathrm\{svd\}\}feasibility and singular\-value tolerancesτzero\\tau\_\{\\mathrm\{zero\}\}zero\-obstruction tolerance𝒪​\(⋅\)\\mathcal\{O\}\(\\cdot\)asymptotic complexity notation; distinct from the observation mapOO
## Appendix BProofs

This appendix collects the proofs of the two theorems and their companion proposition and corollary, together with the auxiliary pseudoinverse identities that those proofs use\.

### B\.1Pseudoinverse identities used throughout

We collect the standard Moore–Penrose identities used in Section[7](https://arxiv.org/html/2608.05702#S7)\. For a realm×nm\\times nmatrixAA, the Moore–Penrose pseudoinverseA\+A\{\{\}^\{\+\}\}is the unique matrix satisfying the following four conditions:

A​A​A\+=A\(MP1\),\\displaystyle AA\{\{\}^\{\+\}\}A=A\\quad\\textup\{\(MP1\)\},AA\+A=\+A\+\(MP2\),\\displaystyle\\qquad A\{\{\}^\{\+\}\}AA\{\{\}^\{\+\}\}=A\{\{\}^\{\+\}\}\\quad\\textup\{\(MP2\)\},\(AA\)\+⊤=AA\+\(MP3\),\\displaystyle\(AA\{\{\}^\{\+\}\}\)^\{\\top\}=AA\{\{\}^\{\+\}\}\\quad\\textup\{\(MP3\)\},\(A​A\+\)⊤=A​A\+\(MP4\)\.\\displaystyle\\qquad\(A\{\{\}^\{\+\}\}A\)^\{\\top\}=A\{\{\}^\{\+\}\}A\\quad\\textup\{\(MP4\)\}\.
The following identities are immediate consequences:

###### Lemma B\.1\.

AA\+AA\{\{\}^\{\+\}\}is the orthogonal projector ontoim⁡A\\operatorname\{im\}A\.A​A\+A\{\{\}^\{\+\}\}Ais the orthogonal projector onto\(ker⁡A\)⟂\(\\ker A\)^\{\\perp\}\. Equivalently,I−A​A\+\\mathrm\{I\}\-A\{\{\}^\{\+\}\}Ais the orthogonal projector ontoker⁡A\\ker A\.

###### Proof\.

By identity MP3,AA\+AA\{\{\}^\{\+\}\}is symmetric\. By identity MP1,\(AA\)\+2=AAA\+A=\+AA\+\(AA\{\{\}^\{\+\}\}\)^\{2\}=AA\{\{\}^\{\+\}\}AA\{\{\}^\{\+\}\}=AA\{\{\}^\{\+\}\}, so the matrix is idempotent and hence an orthogonal projector\. The image ofAA\+AA\{\{\}^\{\+\}\}isim⁡A\\operatorname\{im\}A\. Clearly,im\(AA\)\+⊆imA\\operatorname\{im\}\(AA\{\{\}^\{\+\}\}\)\\subseteq\\operatorname\{im\}A, and forv=A​u∈im⁡Av=Au\\in\\operatorname\{im\}A, we haveA​A​v\+=A​A​A\+​u=A​u=vAA\{\{\}^\{\+\}\}v=AA\{\{\}^\{\+\}\}Au=Au=v, soim\(AA\)\+=imA\\operatorname\{im\}\(AA\{\{\}^\{\+\}\}\)=\\operatorname\{im\}A\. The statement aboutA​A\+A\{\{\}^\{\+\}\}Ais analogous\. ∎

###### Lemma B\.2\(Minimum\-norm preimage\)\.

Forb∈im⁡Ab\\in\\operatorname\{im\}A, the set\{x:A​x=b\}\\\{x:Ax=b\\\}has a unique minimum\-norm element, given byA​b\+∈\(ker⁡A\)⟂A\{\{\}^\{\+\}\}b\\in\(\\ker A\)^\{\\perp\}\.

###### Proof\.

The preimage set is the affine subspacex0\+ker⁡Ax\_\{0\}\+\\ker Afor any one preimagex0x\_\{0\}\. The minimum\-norm element with respect to the Euclidean norm is the orthogonal projection of0onto this affine subspace, which is the unique element in\(ker⁡A\)⟂\(\\ker A\)^\{\\perp\}\. By Lemma[B\.1](https://arxiv.org/html/2608.05702#A2.Thmtheorem1),A​b\+∈\(ker⁡A\)⟂A\{\{\}^\{\+\}\}b\\in\(\\ker A\)^\{\\perp\}, andA​A​b\+=bAA\{\{\}^\{\+\}\}b=bsinceb∈im⁡Ab\\in\\operatorname\{im\}A\. Hence,A​b\+A\{\{\}^\{\+\}\}bis the desired element\. ∎

### B\.2Theorem 1: detailed proof

###### Theorem\(Minimum\-cost budgeted intervention\)\.

LetPPbe the orthogonal projector onto a closed subspaceVP⊆C0​\(F\)V\_\{P\}\\subseteq C^\{0\}\(F\), letιVP:VP↪C0​\(F\)\\iota\_\{V\_\{P\}\}\\colon V\_\{P\}\\hookrightarrow C^\{0\}\(F\)be the inclusion, and letC≻0C\\succ 0onC0​\(F\)C^\{0\}\(F\)\. SetAP≔D​P​ιVPA\_\{P\}\\coloneqq DP\\iota\_\{V\_\{P\}\}andCP≔ιVP⊤​C​ιVPC\_\{P\}\\coloneqq\\iota\_\{V\_\{P\}\}^\{\\top\}C\\iota\_\{V\_\{P\}\}\. The problemminδ⁡12​δ⊤​C​δ\\min\_\{\\delta\}\\tfrac\{1\}\{2\}\\delta^\{\\top\}C\\deltasubject toD​P​δ=ωDP\\delta=\\omegaand\(I−P\)​δ=0\(\\mathrm\{I\}\-P\)\\delta=0is feasible if and only ifω∈im⁡AP\\omega\\in\\operatorname\{im\}A\_\{P\}; in the feasible case, the unique minimizer is

δ⋆=ιVP​CP−1​AP⊤​\(AP​CP−1​AP⊤\)​ω\+\.\\displaystyle\\delta^\{\\star\}=\\iota\_\{V\_\{P\}\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\(A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\)\{\{\}^\{\+\}\}\\omega\.The reported squared intervention cost is

‖δ⋆‖C2=ω⊤​\(AP​CP−1​AP⊤\)​ω\+,\\displaystyle\\left\\\|\\delta^\{\\star\}\\right\\\|\_\{C\}^\{2\}=\\omega^\{\\top\}\(A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\)\{\{\}^\{\+\}\}\\omega,while the optimum of the quadratic objective is half this value\. Forλ\>0\\lambda\>0, the soft restricted solution is

δλ⋆=ιVP​\(AP⊤​AP\+λ−1​CP\)−1​AP⊤​ω\.\\displaystyle\\delta\_\{\\lambda\}^\{\\star\}=\\iota\_\{V\_\{P\}\}\(A\_\{P\}^\{\\top\}A\_\{P\}\+\\lambda^\{\-1\}C\_\{P\}\)^\{\-1\}A\_\{P\}^\{\\top\}\\omega\.For the same ambient cost metricCC, ifVP1⊆VP2V\_\{P\_\{1\}\}\\subseteq V\_\{P\_\{2\}\}and both hard problems are feasible, the reported squared hard costs are antitone in the budget\. No such conclusion is asserted for two unrelated budget\-specific metrics\. The repair convention issrepaired=s−P​δ⋆s\_\{\\mathrm\{repaired\}\}=s\-P\\delta^\{\\star\}\.

###### Proof\.

Feasibility argument\.The constraint\(I−P\)​δ=0\(\\mathrm\{I\}\-P\)\\delta=0isδ∈VP\\delta\\in V\_\{P\}\. Combined withD​P​δ=ωDP\\delta=\\omega, the feasible set isVP∩\(D​P\|VP\)−1​\(ω\)=VP∩AP−1​\(ω\)V\_\{P\}\\cap\(DP\|\_\{V\_\{P\}\}\)^\{\-1\}\(\\omega\)=V\_\{P\}\\cap A\_\{P\}^\{\-1\}\(\\omega\), nonempty if and only ifω∈im⁡AP\\omega\\in\\operatorname\{im\}A\_\{P\}\.

Closed\-form argument\.Working inVPV\_\{P\}, writeδ=ιVP​z\\delta=\\iota\_\{V\_\{P\}\}zforz∈VPz\\in V\_\{P\}\. The objective becomes12​z⊤​CP​z\\tfrac\{1\}\{2\}z^\{\\top\}C\_\{P\}z, which is strictly convex onVPV\_\{P\}becauseCP≻0C\_\{P\}\\succ 0\. The constraint isAP​z=ωA\_\{P\}z=\\omega\. The Lagrangian isL​\(z,μ\)≔12​z⊤​CP​z−μ⊤​\(AP​z−ω\)L\(z,\\mu\)\\coloneqq\\tfrac\{1\}\{2\}z^\{\\top\}C\_\{P\}z\-\\mu^\{\\top\}\(A\_\{P\}z\-\\omega\)\. Stationarity givesCP​z=AP⊤​μC\_\{P\}z=A\_\{P\}^\{\\top\}\\mu, soz=CP−1​AP⊤​μz=C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\\mu\. Substituting into the constraint yieldsAP​CP−1​AP⊤​μ=ωA\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\\mu=\\omega\. A direct computation givesim⁡\(AP​CP−1​AP⊤\)=im⁡AP\\operatorname\{im\}\(A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\)=\\operatorname\{im\}A\_\{P\}, sinceker⁡\(AP​CP−1​AP⊤\)=ker⁡AP⊤\\ker\(A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\)=\\ker A\_\{P\}^\{\\top\}; feasibility therefore makes the multiplier equation solvable\. The minimum\-norm solution of that equation isμ⋆=\(AP​CP−1​AP⊤\)​ω\+\\mu^\{\\star\}=\(A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\)\{\{\}^\{\+\}\}\\omega\. Hence,z⋆=CP−1​AP⊤​μ⋆z^\{\\star\}=C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\}\\mu^\{\\star\}andδ⋆=ιVP​z⋆\\delta^\{\\star\}=\\iota\_\{V\_\{P\}\}z^\{\\star\}give the claimed form\. WithM≔AP​CP−1​AP⊤M\\coloneqq A\_\{P\}C\_\{P\}^\{\-1\}A\_\{P\}^\{\\top\},

z⋆⊤​CP​z⋆=μ⋆⊤​M​μ⋆=ω⊤​M​M\+​M​ω\+=ω⊤​M​ω\+,\\displaystyle\{z^\{\\star\}\}^\{\\top\}C\_\{P\}z^\{\\star\}=\{\\mu^\{\\star\}\}^\{\\top\}M\\mu^\{\\star\}=\\omega^\{\\top\}M\{\{\}^\{\+\}\}MM\{\{\}^\{\+\}\}\\omega=\\omega^\{\\top\}M\{\{\}^\{\+\}\}\\omega,which is the reported squared intervention cost\. The objective value of12​z⊤​CP​z\\tfrac\{1\}\{2\}z^\{\\top\}C\_\{P\}zis half of that cost\. Uniqueness of the minimizerδ⋆\\delta^\{\\star\}follows from strict convexity of that objective onVPV\_\{P\}, whereCP≻0C\_\{P\}\\succ 0\.

Soft Tikhonov argument\.The objective12​δ⊤​C​δ\+\(λ/2\)​‖D​δ−ω‖2\\tfrac\{1\}\{2\}\\delta^\{\\top\}C\\delta\+\(\\lambda/2\)\\left\\\|D\\delta\-\\omega\\right\\\|^\{2\}, restricted toδ=ιVP​z\\delta=\\iota\_\{V\_\{P\}\}z, becomes12​z⊤​CP​z\+\(λ/2\)​‖AP​z−ω‖2\\tfrac\{1\}\{2\}z^\{\\top\}C\_\{P\}z\+\(\\lambda/2\)\\left\\\|A\_\{P\}z\-\\omega\\right\\\|^\{2\}\. Stationarity inzzgives\(CP\+λ​AP⊤​AP\)​z=λ​AP⊤​ω\(C\_\{P\}\+\\lambda A\_\{P\}^\{\\top\}A\_\{P\}\)z=\\lambda A\_\{P\}^\{\\top\}\\omega, equivalently\(AP⊤​AP\+λ−1​CP\)​z=AP⊤​ω\(A\_\{P\}^\{\\top\}A\_\{P\}\+\\lambda^\{\-1\}C\_\{P\}\)z=A\_\{P\}^\{\\top\}\\omega\. The operator on the left is positive definite onVPV\_\{P\}as the sum of a positive\-semidefinite term and a positive\-definite term, so the unique solution iszλ⋆=\(AP⊤​AP\+λ−1​CP\)−1​AP⊤​ωz^\{\\star\}\_\{\\lambda\}=\(A\_\{P\}^\{\\top\}A\_\{P\}\+\\lambda^\{\-1\}C\_\{P\}\)^\{\-1\}A\_\{P\}^\{\\top\}\\omega, andδλ⋆=ιVP​zλ⋆\\delta^\{\\star\}\_\{\\lambda\}=\\iota\_\{V\_\{P\}\}z^\{\\star\}\_\{\\lambda\}\.

Budget\-antitonicity argument\.For the same ambient metricCCandVP1⊆VP2V\_\{P\_\{1\}\}\\subseteq V\_\{P\_\{2\}\}, the feasible set atP1P\_\{1\}is contained in that atP2P\_\{2\}\. The minimum of the same strictly convex objective over a smaller feasible set is at least the minimum over the larger set; hence,‖δ⋆​\(P1\)‖C2≥‖δ⋆​\(P2\)‖C2\\left\\\|\\delta^\{\\star\}\(P\_\{1\}\)\\right\\\|\_\{C\}^\{2\}\\geq\\left\\\|\\delta^\{\\star\}\(P\_\{2\}\)\\right\\\|\_\{C\}^\{2\}\. If the metric changes with the budget, this set\-inclusion argument no longer proves any cost inequality\. ∎

### B\.3Identifiability proposition: detailed proof and monotonicity corollary

###### Proposition\(Identifiability dimensions\)\.

Withℬ=ker⁡D∩ker⁡O\\mathcal\{B\}=\\ker D\\cap\\ker Oandℐ=ker⁡D/ℬ\\mathcal\{I\}=\\ker D/\\mathcal\{B\}, the dimensions satisfydimℬ=dimker⁡D−rank⁡\(O\|ker⁡D\)\\dim\\mathcal\{B\}=\\dim\\ker D\-\\operatorname\{rank\}\(O\|\_\{\\ker D\}\),dimℐ=rank⁡\(O\|ker⁡D\)\\dim\\mathcal\{I\}=\\operatorname\{rank\}\(O\|\_\{\\ker D\}\), anddimker⁡D=dimℬ\+dimℐ\\dim\\ker D=\\dim\\mathcal\{B\}\+\\dim\\mathcal\{I\}\.

###### Proof\.

The identityker⁡\(O\|ker⁡D\)=ker⁡D∩ker⁡O=ℬ\\ker\(O\|\_\{\\ker D\}\)=\\ker D\\cap\\ker O=\\mathcal\{B\}follows from the definition of the restriction\. Rank–nullity forO\|ker⁡D:ker⁡D→YO\|\_\{\\ker D\}\\colon\\ker D\\to Ygivesdimker⁡D=dimker⁡\(O\|ker⁡D\)\+rank⁡\(O\|ker⁡D\)\\dim\\ker D=\\dim\\ker\(O\|\_\{\\ker D\}\)\+\\operatorname\{rank\}\(O\|\_\{\\ker D\}\), which is the first identity\. The quotient dimension formula givesdimℐ=dimker⁡D−dimℬ=rank⁡\(O\|ker⁡D\)\\dim\\mathcal\{I\}=\\dim\\ker D\-\\dim\\mathcal\{B\}=\\operatorname\{rank\}\(O\|\_\{\\ker D\}\), the second identity\. Adding the first two identities cancelsrank⁡\(O\|ker⁡D\)\\operatorname\{rank\}\(O\|\_\{\\ker D\}\), leaving the third,dimker⁡D=dimℬ\+dimℐ\\dim\\ker D=\\dim\\mathcal\{B\}\+\\dim\\mathcal\{I\}\. ∎

###### Corollary B\.3\(Monotonicity ofℬ\\mathcal\{B\}under finer observations\)\.

IfO1,O2O\_\{1\},O\_\{2\}haveker⁡O1⊇ker⁡O2\\ker O\_\{1\}\\supseteq\\ker O\_\{2\}, meaning thatO2O\_\{2\}observes everythingO1O\_\{1\}observes and possibly more, thendimℬ​\(O2\)≤dimℬ​\(O1\)\\dim\\mathcal\{B\}\(O\_\{2\}\)\\leq\\dim\\mathcal\{B\}\(O\_\{1\}\), and correspondinglydimℐ​\(O2\)≥dimℐ​\(O1\)\\dim\\mathcal\{I\}\(O\_\{2\}\)\\geq\\dim\\mathcal\{I\}\(O\_\{1\}\)\.

###### Proof\.

ker⁡D∩ker⁡O2⊆ker⁡D∩ker⁡O1\\ker D\\cap\\ker O\_\{2\}\\subseteq\\ker D\\cap\\ker O\_\{1\}, sodimℬ​\(O2\)≤dimℬ​\(O1\)\\dim\\mathcal\{B\}\(O\_\{2\}\)\\leq\\dim\\mathcal\{B\}\(O\_\{1\}\)\. The complementary inequality onℐ\\mathcal\{I\}follows fromdimℐ=dimker⁡D−dimℬ\\dim\\mathcal\{I\}=\\dim\\ker D\-\\dim\\mathcal\{B\}\. ∎

### B\.4Closure\-recovery corollary: detailed proof

###### Corollary\(Closure\-restricted recoverability\)\.

LetDCD\_\{C\}be the closure\-channel block ofDDunder Assumption A0, as a typed mapC0​\(F\)closure→C1​\(F\)closureC^\{0\}\(F\)^\{\\mathrm\{closure\}\}\\to C^\{1\}\(F\)^\{\\mathrm\{closure\}\}\. Ifωclosure∈im⁡DC\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{im\}D\_\{C\}, the unique minimum\-norm closure correction in the Euclidean metric isδC⋆=DC​ωclosure\+∈im⁡\(DC⊤\)=row⁡\(DC\)\\delta\_\{C\}^\{\\star\}=D\_\{C\}\{\{\}^\{\+\}\}\\omega^\{\\mathrm\{closure\}\}\\in\\operatorname\{im\}\(D\_\{C\}^\{\\top\}\)=\\operatorname\{row\}\(D\_\{C\}\), and the repaired stalks\(ui,ci−δC⋆\|i,oi,ϵi\)i∈I\(u\_\{i\},c\_\{i\}\-\\delta\_\{C\}^\{\\star\}\|\_\{i\},o\_\{i\},\\epsilon\_\{i\}\)\_\{i\\in I\}have closure\-channel admissibility defect equal to zero, with non\-closure channels unchanged\.

###### Proof\.

Apply Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1), or directly Lemma[B\.2](https://arxiv.org/html/2608.05702#A2.Thmtheorem2), to the restricted closure\-channel problem

minz∈C0​\(F\)closure⁡12​‖z‖2subject toDC​z=ωclosure\.\\displaystyle\\min\_\{z\\in C^\{0\}\(F\)^\{\\mathrm\{closure\}\}\}\\tfrac\{1\}\{2\}\\left\\\|z\\right\\\|^\{2\}\\quad\\text\{subject to\}\\quad D\_\{C\}z=\\omega^\{\\mathrm\{closure\}\}\.The closed\-form solution to this smaller problem isδC⋆=DC​ωclosure\+\\delta\_\{C\}^\{\\star\}=D\_\{C\}\{\{\}^\{\+\}\}\\omega^\{\\mathrm\{closure\}\}\. The correction inserted into the full stalk vector is the zero\-extensionιclosure​δC⋆\\iota\_\{\\mathrm\{closure\}\}\\delta\_\{C\}^\{\\star\}; throughout this proof,δC⋆\\delta\_\{C\}^\{\\star\}denotes the restricted closure vector whenever no ambiguity is possible\. The pseudoinverse range identityDC​y\+∈\(ker⁡DC\)⟂=im⁡\(DC⊤\)=row⁡\(DC\)D\_\{C\}\{\{\}^\{\+\}\}y\\in\(\\ker D\_\{C\}\)^\{\\perp\}=\\operatorname\{im\}\(D\_\{C\}^\{\\top\}\)=\\operatorname\{row\}\(D\_\{C\}\)fory∈im⁡DCy\\in\\operatorname\{im\}D\_\{C\}gives the membership statement\. For the raw defectω=D​s\\omega=Ds, substituting the zero\-extended correction into the full stalk vector under the A1 sign convention givessrepaired=s−ιclosure​δC⋆s\_\{\\mathrm\{repaired\}\}=s\-\\iota\_\{\\mathrm\{closure\}\}\\delta\_\{C\}^\{\\star\}\. Consequently, on the closure channel,\(D​srepaired\)closure=ωclosure−DC​δC⋆=0\(Ds\_\{\\mathrm\{repaired\}\}\)^\{\\mathrm\{closure\}\}=\\omega^\{\\mathrm\{closure\}\}\-D\_\{C\}\\delta\_\{C\}^\{\\star\}=0\. BecauseDDis block diagonal under Assumption A0,\(D​srepaired\)∙=\(D​s\)∙\(Ds\_\{\\mathrm\{repaired\}\}\)^\{\\bullet\}=\(Ds\)^\{\\bullet\}for every non\-closure channel∙\\bullet\. ∎

### B\.5Theorem 2: detailed proof

###### Theorem\(Conservation\-contract detectability\)\.

For an oriented edgee=\(a,b\)e=\(a,b\)and repaired endpointq∈\{a,b\}q\\in\\\{a,b\\\}, assume baseline cancellation, one\-sided zero neighbor contribution onee, positive rank ofρq,eclosure\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}, and row\-space alignmentδ​cq∈row⁡\(ρq,eclosure\)\\delta c\_\{q\}\\in\\operatorname\{row\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\. Then

‖ωeclosure‖≥σmin\+​\(ρq,eclosure\)⋅ϵrepair\.\\displaystyle\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{e\}\\right\\\|\\geq\\sigma\_\{\\min\}^\{\+\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\\cdot\\epsilon\_\{\\mathrm\{repair\}\}\.Without row\-space alignment, the same argument gives the projected boundEq\.[9](https://arxiv.org/html/2608.05702#S7.E9)\.

###### Proof\.

Setρ≔ρq,eclosure\\rho\\coloneqq\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\. Baseline cancellation and the one\-sided zero\-contribution assumption giveωeclosure=εq​ρ​δ​cq\\omega^\{\\mathrm\{closure\}\}\_\{e\}=\\varepsilon\_\{q\}\\rho\\delta c\_\{q\}, withεq∈\{−1,\+1\}\\varepsilon\_\{q\}\\in\\\{\-1,\+1\\\}\. Decomposeδ​cq=δ​cq∥\+δ​cq⟂\\delta c\_\{q\}=\\delta c\_\{q\}^\{\\parallel\}\+\\delta c\_\{q\}^\{\\perp\}, whereδ​cq∥∈row⁡\(ρ\)\\delta c\_\{q\}^\{\\parallel\}\\in\\operatorname\{row\}\(\\rho\)andδ​cq⟂∈ker⁡ρ\\delta c\_\{q\}^\{\\perp\}\\in\\ker\\rho\. Thenρ​δ​cq=ρ​δ​cq∥\\rho\\delta c\_\{q\}=\\rho\\delta c\_\{q\}^\{\\parallel\}\. The singular\-value lower bound forρ\\rhorestricted to its row space gives‖ρ​δ​cq∥‖≥σmin\+​\(ρ\)​‖δ​cq∥‖\\left\\\|\\rho\\delta c\_\{q\}^\{\\parallel\}\\right\\\|\\geq\\sigma\_\{\\min\}^\{\+\}\(\\rho\)\\left\\\|\\delta c\_\{q\}^\{\\parallel\}\\right\\\|\. Hence,

‖ωeclosure‖=‖ρ​δ​cq‖≥σmin\+​‖Πrow⁡\(ρ\)​δ​cq‖,\\displaystyle\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{e\}\\right\\\|=\\left\\\|\\rho\\delta c\_\{q\}\\right\\\|\\geq\\sigma\_\{\\min\}^\{\+\}\\left\\\|\\Pi\_\{\\operatorname\{row\}\(\\rho\)\}\\delta c\_\{q\}\\right\\\|,which is Eq\.[9](https://arxiv.org/html/2608.05702#S7.E9)\. Under row\-space alignment,δ​cq∈row⁡\(ρ\)\\delta c\_\{q\}\\in\\operatorname\{row\}\(\\rho\), so the kernel component is zero and‖Πrow⁡\(ρ\)​δ​cq‖=‖δ​cq‖=ϵrepair\\left\\\|\\Pi\_\{\\operatorname\{row\}\(\\rho\)\}\\delta c\_\{q\}\\right\\\|=\\left\\\|\\delta c\_\{q\}\\right\\\|=\\epsilon\_\{\\mathrm\{repair\}\}, recovering Eq\.[8](https://arxiv.org/html/2608.05702#S7.E8)\. ∎

## Appendix CAlgorithmic details and complexity

This appendix collects implementation\-level details of the algorithms presented in Section[8](https://arxiv.org/html/2608.05702#S8), with complexity analysis and numerical\-stability considerations\.

### C\.1Coboundary assembly

The coboundary matrixDDhas a block structure indexed by\(edge,region\)\(\\text\{edge\},\\text\{region\}\)\. For each edge\(i,j\)∈E\(i,j\)\\in E, the corresponding row block is\[…,−ρi,i​j,…,\+ρj,i​j,…\]\[\\ldots,\-\\rho\_\{i,ij\},\\ldots,\+\\rho\_\{j,ij\},\\ldots\], with zeros in all column blocks not indexed byiiorjj\.

##### Block sparsity\.

Each edge contributes exactly two nonzero column blocks\. The matrixDDtherefore has exactly2​\|E\|2\|E\|nonzero blocks\. For typical covers with\|E\|≪\|I\|2\|E\|\\ll\|I\|^\{2\}, storage and matrix–vector costs scale with\|E\|\|E\|rather than with the number of region pairs\. Path covers, for example, satisfy\|E\|=\|I\|−1\|E\|=\|I\|\-1\.

##### Channel block\-diagonal structure\.

When the restriction maps respect the channel decomposition,DDis block diagonal with respect to the channel partition\. The channel blocksD∙D^\{\\bullet\}for∙∈\{state,closure,obs,meta\}\\bullet\\in\\\{\\mathrm\{state\},\\mathrm\{closure\},\\mathrm\{obs\},\\mathrm\{meta\}\\\}can be assembled independently\. This decomposition reduces the SVD cost by a constant factor because a sum of smaller SVDs is cheaper than one large SVD when the channel sizes differ\. The block\-diagonal structure is also essential for the channel decomposition ofω\\omegain Eq\.[1](https://arxiv.org/html/2608.05702#S3.E1)\.

##### Numerical realization\.

After assembly, Algorithm[8\.1](https://arxiv.org/html/2608.05702#alg2)is applied once for each budget basisQQ, whose orthonormal columns spanVP=im⁡PV\_\{P\}=\\operatorname\{im\}P\. The algorithm usesAP=D​QA\_\{P\}=DQandCP=Q⊤​C​QC\_\{P\}=Q^\{\\top\}CQ, the hard image\-membership residual, the minimum\-cost formula of Theorem[1](https://arxiv.org/html/2608.05702#Thmseamtheorem1), and the separately reported Tikhonov record\. The returned correction lies inVPV\_\{P\}and is subtracted from the current stalk vector according to Eq\.[3](https://arxiv.org/html/2608.05702#S4.E3)\.

### C\.2Pseudoinverse implementation

We use the truncated SVD\-based pseudoinverse with a rank\-tolerance thresholdηrank≔max⁡\(m,n\)⋅ϵmach⋅σmax\\eta\_\{\\mathrm\{rank\}\}\\coloneqq\\max\(m,n\)\\cdot\\epsilon\_\{\\mathrm\{mach\}\}\\cdot\\sigma\_\{\\max\}as a SEAM\-specific convention, whereϵmach\\epsilon\_\{\\mathrm\{mach\}\}is the machine epsilon of the working floating\-point type andσmax\\sigma\_\{\\max\}the largest singular value\. The exact routine and version are recorded whenever an implementation uses a library default\. For ill\-conditioned matrices, including restriction maps with near\-rank\-deficient overlap blocks, the threshold can be tightened to10⋅ϵmach⋅σmax10\\cdot\\epsilon\_\{\\mathrm\{mach\}\}\\cdot\\sigma\_\{\\max\}at the cost of including near\-singular directions in the pseudoinverse range\.

##### Stability under noise\.

For a perturbed matrixD\+ΔD\+\\Delta, pseudoinverse perturbation bounds require fixed rank or a spectral gap\. Under such a rank\-stability assumption, bounds ofWedin \([1972](https://arxiv.org/html/2608.05702#bib.bib57),[1973](https://arxiv.org/html/2608.05702#bib.bib58)\)control∥\(D\+Δ\)−\+D∥\+\\left\\\|\(D\+\\Delta\)\{\{\}^\{\+\}\}\-D\{\{\}^\{\+\}\}\\right\\\|with constants depending on the nonzero singular gap\. Without fixed rank, the pseudoinverse need not be continuous\. Therefore, the obstruction residual is sensitive to noise inDDwhenσmin\+​\(D\)\\sigma\_\{\\min\}^\{\+\}\(D\)is small\. In practice, whenrank⁡\(D\)\>0\\operatorname\{rank\}\(D\)\>0, we monitor the positive\-rank condition number

κ\+​\(D\)≔σmax​\(D\)/σmin\+​\(D\)\\displaystyle\\kappa\_\{\+\}\(D\)\\coloneqq\\sigma\_\{\\max\}\(D\)/\\sigma\_\{\\min\}^\{\+\}\(D\)and report warnings whenκ\+​\(D\)\>1010\\kappa\_\{\+\}\(D\)\>10^\{10\}\. Whenrank⁡\(D\)=0\\operatorname\{rank\}\(D\)=0,κ\+​\(D\)\\kappa\_\{\+\}\(D\)is undefined because there is no nonzero singular value; rank\-zero cases are reported separately\.

### C\.3Budgeted intervention solver: numerical considerations

Algorithm[8\.1](https://arxiv.org/html/2608.05702#alg2)uses the closed\-form solution in the feasible case and the Tikhonov\-regularized solution in the infeasible case\. The transition between the two is governed by the feasibility toleranceηfeas\\eta\_\{\\mathrm\{feas\}\}\. Three numerical considerations apply\.

##### Choice ofηfeas\\eta\_\{\\mathrm\{feas\}\}\.

We set

ηfeas≔max⁡\{10−8​‖ω‖,ηfeas,min\}\\displaystyle\\eta\_\{\\mathrm\{feas\}\}\\coloneqq\\max\\\{10^\{\-8\}\\left\\\|\\omega\\right\\\|,\\eta\_\{\\mathrm\{feas\},\\min\}\\\}by default, with a fixed machine\-scaleηfeas,min\>0\\eta\_\{\\mathrm\{feas\},\\min\}\>0\. This positive\-floor definition preserves the statedηfeas\>0\\eta\_\{\\mathrm\{feas\}\}\>0condition even whenω=0\\omega=0, avoiding spurious Tikhonov fallbacks triggered by linear\-algebra roundoff\. The choice can be overridden per call\.

##### Choice ofλ\\lambda\.

The Tikhonov regularizer controls the trade\-off between residual size and intervention norm\. By default, we set

λ≔max⁡\{‖ω‖,εω\}10​max⁡\{σmax​\(AP\),εσ\},\\displaystyle\\lambda\\coloneqq\\frac\{\\max\\\{\\left\\\|\\omega\\right\\\|,\\varepsilon\_\{\\omega\}\\\}\}\{10\\max\\\{\\sigma\_\{\\max\}\(A\_\{P\}\),\\varepsilon\_\{\\sigma\}\\\}\},with positive numerical floors\. The soft branch is bypassed entirely when‖ω‖≤τzero\\left\\\|\\omega\\right\\\|\\leq\\tau\_\{\\mathrm\{zero\}\}and the case is already globally admissible\. The choice can be overridden per call\.

##### Cost metricCC\.

The default cost metric is the identity, giving Euclidean intervention norms\. Supported domain\-specific cost metrics include a Mahalanobis norm with respect to a prior covariance, channel\-weighted norms, and energy norms\.

### C\.4Identifiability analyzer: numerical considerations

##### Choice ofηsvd\\eta\_\{\\mathrm\{svd\}\}\.

By default, we set

ηsvd≔max⁡\{10−8​σmax,ηsvd,min\}\\displaystyle\\eta\_\{\\mathrm\{svd\}\}\\coloneqq\\max\\\{10^\{\-8\}\\sigma\_\{\\max\},\\eta\_\{\\mathrm\{svd\},\\min\}\\\}for the kernel basis, with the same positive floor used for both the SVD ofDDand the SVD ofO​VkerOV\_\{\\ker\}\. A larger tolerance yields a more inclusive numericalker⁡D\\ker Dand can create spurious blind directions by declaring genuinely nonzero singular values to be zero; a smaller tolerance is more conservative and can miss near\-null or roundoff\-null blind directions\.

##### Blind\-subspace certificate\.

After computing the blind admissible basisBℬB\_\{\\mathcal\{B\}\}, we verify‖D​Bℬ‖\\left\\\|DB\_\{\\mathcal\{B\}\}\\right\\\|,‖O​Bℬ‖\\left\\\|OB\_\{\\mathcal\{B\}\}\\right\\\|, orthonormality, and the numerical dimension identity from Proposition[7\.2](https://arxiv.org/html/2608.05702#S7.Thmtheorem2)\. Together, these checks form the certificate;‖O​Bℬ‖\\left\\\|OB\_\{\\mathcal\{B\}\}\\right\\\|records the observation\-invisibility component of that certificate\.

##### Channel decomposition\.

The columns ofBℬB\_\{\\mathcal\{B\}\}are decomposed into block components by projecting onto the three primary channel subspaces and the optional metadata subspace\. The resulting fractions are included in the diagnostic record\.

### C\.5Monitoring pipeline: caching

The monitoring pipeline in Algorithm[8\.1](https://arxiv.org/html/2608.05702#alg4)caches all quantities that do not depend on the streaming input\.

##### Cached quantities\.

The cache stores the coboundary matrixDD, its SVD factorsUD,ΣD,VDU\_\{D\},\\Sigma\_\{D\},V\_\{D\}, the pseudoinverseD\+D\{\{\}^\{\+\}\}, and the channel\-block submatricesD∙D^\{\\bullet\}\. The cache also stores each projected coboundaryD​PbDP\_\{b\}with its pseudoinverse\. When the observation map is static, the blind admissible basisBℬB\_\{\\mathcal\{B\}\}is cached as well\.

##### Per\-call quantities\.

Onlys​\(t\)=\(vec​\(ei​\(t\)\)\)i∈Is\(t\)=\(\\mathrm\{vec\}\(e\_\{i\}\(t\)\)\)\_\{i\\in I\},ω​\(t\)=D​s​\(t\)\\omega\(t\)=Ds\(t\), and the per\-budget intervention costs need to be recomputed per call\. The wall\-clock cost is dominated by one matrix–vector product withDDplus\|𝒦bud\|\|\\mathcal\{K\}\_\{\\mathrm\{bud\}\}\|projection operations\.

## Appendix DCT\-SEAM conservation backend

This appendix describes the CT\-SEAM local generator used in selected PDE experiments\. The backend supplies closure\-rich explanations through the common SEAM\-Ω\\Omegagenerator interface in the conservation\-contract detectability and backend\-interoperability studies\.

### D\.1Motivation and construction

For PDE systems with conservation laws such as mass, momentum, and energy, a local finite\-volume solver may drift from exact conservation due to discretization, boundary handling, or numerical flux choices\. A conservation contract enforces conservation by applying a mass\-redistribution repair at each step; the accumulated absolute magnitude of that repair may be recorded as optional contract metadata\.

##### Local generator\.

Given a finite\-volume Burgers solver on regionUiU\_\{i\}, the generator first computes each state update by standard finite\-volume time integration\. The generator then evaluates the conservation residual

Δ​mi​\(t\)≔∫Uiu​\(t\)​𝑑x−∫Uiu​\(0\)​𝑑x\+flux contributions\.\\displaystyle\\Delta m\_\{i\}\(t\)\\coloneqq\\int\_\{U\_\{i\}\}u\(t\)\\,dx\-\\int\_\{U\_\{i\}\}u\(0\)\\,dx\+\\text\{flux contributions\}\.Whenever\|Δ​mi​\(t\)\|\>τcons\|\\Delta m\_\{i\}\(t\)\|\>\\tau\_\{\\mathrm\{cons\}\}, the generator applies a mass\-redistribution repair by addingδi​\(t\)≔−Δ​mi​\(t\)/\|Ui\|\\delta\_\{i\}\(t\)\\coloneqq\-\\Delta m\_\{i\}\(t\)/\|U\_\{i\}\|to the state onUiU\_\{i\}\. This correction restores exact conservation\. The generator accumulates the repair in the closure block according toci\+=δi\(t\)⋅χUic\_\{i\}\\mathrel\{\+\}=\\delta\_\{i\}\(t\)\\cdot\\chi\_\{U\_\{i\}\}, whereχUi\\chi\_\{U\_\{i\}\}is the indicator function ofUiU\_\{i\}in the closure space\.

##### Explanation block\.

The state block of a CT\-SEAM explanation contains the conservation\-corrected solution\. The closure block contains the accumulated repair history, represented by one scalar for each region and optionally weighted by a Gaussian centered at the repair location\. The observation block is empty because this solver consumes no sensor data\. The optional metadata block contains the accumulated absolute repair magnitudeϵiabs≔∑t\|δi​\(t\)\|\\epsilon\_\{i\}^\{\\mathrm\{abs\}\}\\coloneqq\\sum\_\{t\}\|\\delta\_\{i\}\(t\)\|\.

### D\.2Role in Theorem 2

Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2)connects the actual closure\-channel vector entering an overlap restriction to the SEAM\-Ω\\Omegaclosure\-channel obstruction\. For an oriented edgeeeand repaired endpointqq, under baseline cancellation, one\-sided zero neighbor contribution, positive rank, and row\-space alignment,

‖ωeclosure‖≥σmin\+​\(ρq,eclosure\)⋅‖δ​cq‖\.\\displaystyle\\left\\\|\\omega^\{\\mathrm\{closure\}\}\_\{e\}\\right\\\|\\geq\\sigma\_\{\\min\}^\{\+\}\(\\rho^\{\\mathrm\{closure\}\}\_\{q,e\}\)\\cdot\\left\\\|\\delta c\_\{q\}\\right\\\|\.The conservation\-contract detectability study and its discretization\-tolerant parametric sweep evaluate this diagnostic using the repair\-vector convention defined in that study’s setup\.

The shared interface also accepts the analytic and finite\-volume generators used in the local–global amplitude\-mismatch, data–physics conflict\-attribution, PDE model\-form discrimination, and random\-sinusoid Burgers studies, as well as the tabular regression generators used in the application studies\. The conservation\-contract detectability and backend\-interoperability studies use CT\-SEAM specifically, with the detectability study providing the parametric evaluation of the claim in Theorem[2](https://arxiv.org/html/2608.05702#Thmseamtheorem2)\.

## Appendix EExperimental protocols, baselines, and data provenance

This appendix records the common experimental configuration, external data sources, preprocessing or generation steps, comparative baselines, and robustness checks needed to interpret the results of Section[9](https://arxiv.org/html/2608.05702#S9)\.

### E\.1Common configuration and random\-seed protocol

##### Cover construction\.

1D PDE experiments use a uniform partition of\[0,1\]\[0,1\]into three regions with a10%10\\%overlap fraction\. Real\-world experiments use either seasonal temporal covers or abstract overlap\-list covers, as detailed below\.

##### Solvers\.

PDE solvers use finite\-volume schemes with periodic boundaries\. Time integration is by explicit Euler with CFL number0\.40\.4, following the standard finite\-volume framework for hyperbolic conservation laws\(LeVeque,[2002](https://arxiv.org/html/2608.05702#bib.bib30)\)\. The grid resolution isNx=200N\_\{x\}=200per region\.

##### Seed schedule and aggregation\.

The standard schedule is a fixed set ofn=5n=5seeds, used for every stochastic experiment, including the FNO OOD monitoring study\. Stochastic results report mean±\\pmstandard deviation over that schedule\. Deterministic experiments are also executed across the same schedule to verify seed\-invariance; their results are reported once, or equivalently as a zero\-dispersion aggregate where a cross\-suite figure requires a common format\. Analyses using additional seeds are identified as robustness checks\. Seed\-resolved and aggregate outputs are retained with the study artifacts\.

##### FNO training and aggregation\.

The FNO OOD monitoring models are trained on the five standard seeds with all other configuration settings fixed\. For each shift family and seed, the protocol records the per\-trajectory pair\(‖ω​\(tf\)‖,‖uFNO​\(tf\)−uref​\(tf\)‖L2\)\(\\left\\\|\\omega\(t\_\{\\mathrm\{f\}\}\)\\right\\\|,\\left\\\|u^\{\\mathrm\{FNO\}\}\(t\_\{\\mathrm\{f\}\}\)\-u^\{\\mathrm\{ref\}\}\(t\_\{\\mathrm\{f\}\}\)\\right\\\|\_\{L^\{2\}\}\)\. Pearson and Spearman correlations are computed per seed and then reported as mean±\\pmstandard deviation across the five seeds; the95%95\\%bootstrap CI is also reported for every seed\. The confidence\-based comparator is a separate five\-member ensemble whose otherwise identical members use distinct initialization seeds\.

##### Deterministic and data\-loading checks\.

The conservation\-contract detectability study has zero standard deviation across all five seeds\. The random\-sinusoid Burgers study is deterministic conditional on a given seed and reports‖ω‖=0\.4754±0\.0000\\left\\\|\\omega\\right\\\|=0\.4754\\pm 0\.0000under the aggregate format\. In the metro traffic seasonality study, the data\-loading and AR\(2\)\-fitting path is deterministic given the seed, so the multi\-seed standard deviation is zero; the Potomac streamflow seasonality study likewise reports0\.0266±0\.00000\.0266\\pm 0\.0000\. The three site values in the multi\-site streamflow comparison are verified to be seed\-invariant across the schedule\. The air\-quality temporal\-consistency study uses the fixed UCI dataset and preprocessing protocol of Appendix[E\.3](https://arxiv.org/html/2608.05702#A5.SS3); its OLS fit and obstruction computation are deterministic, yielding‖ω‖=0\.2850\\left\\\|\\omega\\right\\\|=0\.2850with zero seed dispersion\.

##### Seed\-resolved experimental outcomes\.

For the zero\-floor negative control, every one of the five identical\-model control runs is below10−1010^\{\-10\}\. In the financial regime\-detection study, the closure channel dominates on all five seeds, and the identical\-model control has‖ω‖=0\\left\\\|\\omega\\right\\\|=0on every seed\. In the industrial multi\-zone fault\-detection study, the state channel dominates the fault response on all five seeds, and the identical\-zone control is zero on every seed\. For each of the three model families in the household\-power cross\-framework audit, the state channel dominates on all five seeds; the winter–spring overlap is dominant on four of the five seeds\.

##### Sweeps and ablations\.

The conservation\-contract detectability sweep crosses five values ofϵrepair=‖δ​cq‖\\epsilon\_\{\\mathrm\{repair\}\}=\\left\\\|\\delta c\_\{q\}\\right\\\|with the five\-seed schedule, producing 25 configurations\. The controlled predictor\-perturbation result is evaluated on all five seeds and 10 random directions at each perturbation magnitude\. The empirical monotonicity fraction of that result is1\.00±0\.001\.00\\pm 0\.00across seeds and directions\. A 50\-seed robustness analysis gives the same monotonicity, attaining1\.001\.00for every seed and direction\.

### E\.2Data sources and generated cases

Table[E\.1](https://arxiv.org/html/2608.05702#A5.T1)records the external data sources and generated cases used in Section[9](https://arxiv.org/html/2608.05702#S9), together with their source citations\. Preprocessing and generation steps follow in Appendix[E\.3](https://arxiv.org/html/2608.05702#A5.SS3)\.

Table E\.1:External sources and generated cases associated with the experiments\. License information is taken from the cited repository or publisher records\.DatasetUsed inLicenseSourceUCI Metro Interstate Traffic VolumeMetro traffic seasonalityCC BY 4\.0Hogue\([2019](https://arxiv.org/html/2608.05702#bib.bib23)\)USGS NWIS streamflowPotomac and multi\-site streamflowPublic domain under Section 105 of Title 17 of the United States CodeU\.S\. Geological Survey\([2023](https://arxiv.org/html/2608.05702#bib.bib54)\)UCI Air QualityAir\-quality temporal consistencyCC BY 4\.0De Vito\([2016](https://arxiv.org/html/2608.05702#bib.bib10)\)UCI Individual Household Electric Power ConsumptionHousehold\-power cross\-framework auditCC BY 4\.0Hébrail and Berard\([2012](https://arxiv.org/html/2608.05702#bib.bib20)\)Synthetic financial returnsFinancial regime detectionNot applicable; syntheticCont\([2001](https://arxiv.org/html/2608.05702#bib.bib8)\)Synthetic industrial processIndustrial multi\-zone fault detectionNot applicable; syntheticVenkatasubramanian et al\.\([2003](https://arxiv.org/html/2608.05702#bib.bib55)\)
### E\.3Preprocessing details

##### Metro traffic seasonality: UCI Metro Interstate Traffic Volume\.

Hourly traffic counts on I\-94 in Minnesota\. Feature engineering: temperature converted from kelvin to degrees Celsius, cloud\-cover percentage, and cyclical hour encoding\. Seasonal split: winter = Dec–Feb, summer = Jun–Aug, transition = Mar–May plus Sep–Nov\.

##### Potomac and multi\-site streamflow: USGS NWIS\.

Daily discharge values obtained from USGS Water Data for the Nation via thedataretrievalPython package\. Normalization: per\-site standardization to zero mean and unit variance\. Seasonal split: spring snowmelt = Mar–May, baseflow = Jun–Sep, autumn transition = Oct–Nov\.

##### Air\-quality temporal consistency: UCI Air Quality\.

Hourly readings of CO, C6H6, NOx, and NO2from March 2004 to February 2005\. Imputation: per\-channel median substitution for the−200\-200sentinel values\. Seasonal split: cold = Oct–Feb, warm = Apr–Sep, transition = Mar\.

##### Household\-power cross\-framework audit: UCI Individual Household Electric Power Consumption\.

Two years of one\-minute power measurements, aggregated to hourly means\. Features: cyclical hour\(sin,cos\)\(\\sin,\\cos\)of hour\-of\-day, standardizedVoltage, standardizedGlobal\_reactive\_power\. Target:Global\_active\_power\. Seasonal split by meteorological season: winter = Dec–Feb, spring = Mar–May, summer = Jun–Aug, autumn = Sep–Nov\.

##### Random\-sinusoid Burgers initial conditions\.

Initial conditions follow the ten\-wave, Gaussian\-amplitude distribution specified in Section[9\.7\.1](https://arxiv.org/html/2608.05702#S9.SS7.SSS1); the local solver generates all trajectories\.

##### Financial regime detection: synthetic returns\.

Five trading years comprising 1260 daily observations are generated\. The bull regime usesμbull=0\.001\\mu\_\{\\mathrm\{bull\}\}=0\.001andσbull=0\.007\\sigma\_\{\\mathrm\{bull\}\}=0\.007; the bear regime usesμbear=−0\.001\\mu\_\{\\mathrm\{bear\}\}=\-0\.001andσbear=0\.018\\sigma\_\{\\mathrm\{bear\}\}=0\.018; and the sideways regime usesμsw=0\.0\\mu\_\{\\mathrm\{sw\}\}=0\.0andσsw=0\.012\\sigma\_\{\\mathrm\{sw\}\}=0\.012\. The regime construction targets the volatility clustering and regime\-dependent variance described byCont \([2001](https://arxiv.org/html/2608.05702#bib.bib8)\)\.

##### Industrial multi\-zone fault detection: synthetic process\.

Three serial zones are each modeled by a multivariate linear regressor from\(T,p,q\)\(T,p,q\)to a process output, whereTT,pp, andqqrepresent temperature, pressure, and flow\. A0\.3​σ0\.3\\sigmastep\-change fault is injected into the midstream zone at the midpoint of the evaluation window\. The framework follows the process\-monitoring tradition ofVenkatasubramanian et al\. \([2003](https://arxiv.org/html/2608.05702#bib.bib55)\)\.

### E\.4Baselines used in the reported experiments

The household\-power cross\-framework audit uses SK, XGB, and SM as predictive\-model baselines, and the FNO OOD monitoring study uses ensemble variance as its monitoring comparator\. Section[9\.9\.2](https://arxiv.org/html/2608.05702#S9.SS9.SSS2)compares‖ω‖\\left\\\|\\omega\\right\\\|with two scalar disagreement metrics, cross\-prediction RMSE and mean absolute pairwise difference, computed on the same covers in the metro traffic seasonality, Potomac streamflow seasonality, and air\-quality temporal\-consistency studies\. The study artifacts include the full multi\-seed statistics underlying that comparison\.

##### Official baseline implementations\.

The SK, XGB, and SM packages are imported from their official Python Package Index distributions\.

### E\.5Robustness and reporting provenance

##### Number of seeds\.

The standard five\-seed schedule, the 50\-seed controlled predictor\-perturbation robustness analysis, and their seed\-resolved outcomes are specified in Appendix[E\.1](https://arxiv.org/html/2608.05702#A5.SS1)\.

##### Sensitivity to overlap fraction\.

For 1D PDE experiments, varying the overlap fractionαov∈\{5%,10%,20%\}\\alpha\_\{\\mathrm\{ov\}\}\\in\\\{5\\%,10\\%,20\\%\\\}produces‖ω‖\\left\\\|\\omega\\right\\\|values within a factor of1\.51\.5of the default value, and the dominant\-channel conclusions are preserved across the tested range\.

The sensitivity checks above are complemented by the following data\-provenance records\.

##### Data provenance\.

Each real\-data source is identified by an authoritative source citation or repository link, with preprocessing specified in Appendix[E\.3](https://arxiv.org/html/2608.05702#A5.SS3)\.

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