Elimination Geometry

Mian Huang; Xueqin Wang

Source-linked AI summary

Elimination Geometry

Mian Huang, Xueqin Wang

arXiv:2608.17646v1Aug 18, 2026cs.LG

TL;DR

Shared deployment rules can impose losses distinct from local approximation, finite-sample, and implementation error. The monograph develops an audit-oriented elimination geometry framework and reports formal, application-specific certificates while limiting claims about universal architecture choice or guaranteed repair. These distinctions guide whether structural nonrealizability should be addressed through deployment-contract changes rather than better local models or solvers.

Problem
The central question is whether loss caused by a shared deployment contract can be separated from local approximation, generalization, and implementation error in the native objective scale.
Method
Elimination geometry types assumptions and conclusions across established mathematical traditions through a common native-loss interface and audit workflow.
Results
The monograph closes architecture-specific certificates for sparse inference and a separate sealed diabetes decision, without claiming general architecture-choice theory or external replication.
Takeaways & Limitations
Structural nonrealizability calls for changing the deployment contract, whereas misspecification, finite-sample ambiguity, and implementation error require different interventions.
Takeaways & Limitations
The framework does not establish that a native defect implies architecture obstruction, that an obstruction implies successful repair, or that structural repair guarantees held-out improvement.

Abstract

from arXiv · show

This monograph develops elimination geometry (EG), a typed, native-loss, audit-oriented framework for studying when locally optimal objects can be realized by a shared deployment rule. Elimination and compression may erase distinctions required by prediction, inference, control, or representation. EG asks which distinctions are lost, whether the induced defect is visible to the declared task, and whether changing information, architecture, action space, or deployment domain can repair it. EG separates local solvability, global realizability, and finite-sample certifiability. It derives native defects from the original objective and distinguishes architecture obstruction from model approximation, generalization, and implementation error. The monograph synthesizes tools from geometry, optimization, information theory, statistics, and machine learning into interfaces for integrability, representation admissibility, resource constraints, observational overlap, and common deployment. Formal results address regular, coordination, singular, compositional, and resource-limited mechanisms with explicit antecedents and claim boundaries. Applications include sparse model selection, distribution-free prediction, observational treatment policies, routed expert and retrieval systems, and learned score fields. Obstruction-Aware Learning and Inference links structural diagnosis to finite-data authorization, mechanism-matched intervention, and independent validation. Reproducible synthetic and real-data studies illustrate how certificates can guide architecture repair while recording failed gates and unresolved cases. The framework requires the deployment contract, native endpoint, competing explanations, information and compute budgets, and validation rule to be fixed before a persistent performance floor is attributed to architecture.

Notation and Standing Conventions

The notation fixes typed objects for elimination geometry, including lifted and eliminated objectives, oracle sets, native defects, deployable architectures, and population obstruction. It also defines symbols for information loss, resource-limited distortion, certified worlds, deployment exactness, and recursive quotient stability.

Core elimination and architecture: The core notation distinguishes the lifted objective H_x(a), eliminated objective J(x), oracle set O(x), native defect D_x(a), deployable architecture class 𝔄, deployed field A, and population architecture obstruction 𝔒𝔓(𝔄).The native defect is defined as H_x(a) − J(x), while the eliminated objective is inf_{a∈A_x} H_x(a) and the oracle set is arg min_{a∈A_x} H_x(a).
Information, resources, and coordination: The conventions define conditional dual-oracle signatures, carrier information loss, quotient-reduced lift frontiers, resource grammars, native distortion, coordination residuals, and declared downstream tasks.These symbols organize decoder construction, visible signature cardinality, resource constraints, and local coordination costs.
Certified learning and intervention: Certified learning uses population-world, fixed-record confidence, and anytime confidence-world sequences whose declared coverage events contain the true world at protected records or times.The notation also includes identified images, typed certificates, certificate margins, certificate truth or architecture-action colors, and alternative parameter sets.
Deployment and recursive stability: Deployment notation separates the carrier H_dep and action decoder d_dep, with exactness requiring a★ = d_dep ◦ H_dep, and tracks retained evidence, recursive carriers, joint colors, and stable quotient class counts.The world-independent channel T_ev maps raw observations or transcripts to retained evidence, while φ_rec preserves declared colors and observation-labeled successor laws.

Orientation of asymmetric divergences

Asymmetric divergences place the evaluated or trial object first and the tangent, reference, or oracle object second. This order is preserved under conditioning and pushforward, including for KL divergence.

Orientation of asymmetric divergences: The first argument is the evaluation point, while the second is the tangent or reference point.
Orientation of asymmetric divergences: For KL divergence, the first argument is the integration law and the second is the reference law, with KL(Q∥P) = +∞ when Q is not absolutely continuous with respect to P.
Orientation of asymmetric divergences: Conditioning and pushforward preserve this order, so native defects place the trial or deployed object first and the oracle second.

Standing distinctions

The framework preserves distinctions among attainment, risk contracts, error sources, output semantics, impossibility claims, and statistical sampling units. These distinctions prevent stronger conclusions or substitutions than the stated contract supports.

Standing distinctions: A zero infimum does not imply attainment of an exact minimizer or global section.
Standing distinctions: Uniform, average, and task-weighted risks are distinct contracts and cannot be interchanged without an explicit theorem.
Standing distinctions: Local oracle, architecture, generalization, and implementation errors remain separate unless an exact identity combines them.
Standing distinctions: Point-, set-, quotient-, projector-, and distribution-valued outputs represent different semantic contracts.
Standing distinctions: Deterministic obstruction, statistical impossibility, posterior credibility, computational intractability, and sampling-unit declarations are distinct claims.The independent statistical unit is the declared sampling unit, not automatically an individual cell, token, edge, or repeated measurement.

Infima and measurability

The framework takes infima over nonempty declared classes, states attainment conditions locally, and uses ε-optimal selections when approximation rather than attainment is required.

Infima and measurability: Infima range over nonempty declared classes; measurability and attainment assumptions are imposed locally, while approximation uses infima and ε-optimal selections.Relevant local conditions include lower semicontinuity, compactness, coercivity, and closure.

The Structural Realizability Problem

Modern learning systems reuse representations, parameterizations, memories, and output interfaces across local problems, making shared simultaneous realization by one declared deployment system a central question beyond local optimality.

Learning systems reuse representations, parameterizations, memories, and output interfaces across many local problems.
The existence of local optima does not by itself establish that they can be realized simultaneously.
The structural realizability problem asks whether one declared deployment system can realize the local optima simultaneously.

Structural Realizability under Shared Deployment

Structural realizability asks whether one deployment rule can realize a family of locally optimal objects under shared representation, memory, regularity, and resource constraints. The framework separates local solvability, global realizability, and finite-sample certifiability, assigning different remedies to structural, statistical, modeling, and implementation failures.

Core distinction: Local solvability does not ensure global realizability when one contract must deploy across many local problems.The mismatch persists after local oracles are fixed and optimization within the declared deployment class is complete.
Canonical example: A constant deployment rule incurs an irreducible loss of 1/4 for exact two-point quadratic local optima.Reparameterization cannot remove this floor; allowing deployment to depend on x removes the obstruction.
Three questions: The framework separates local solvability, global realizability, and finite-sample certifiability because these questions require different assumptions and remedies.A population obstruction may be undetectable at realistic sample sizes, while unresolved certification does not establish absence; local misspecification can also invalidate an otherwise realizable deployment.
Intervention: Interventions must match the failure: richer local models address misspecification, better data address statistical ambiguity, better solvers address implementation error, and contract changes address structural nonrealizability.Contract changes can include quotient representations, atlases, additional memory, different sharing, or larger resource budgets.
Position relative to neighboring theory: The framework positions its contribution as a general structural diagnosis rather than a priority claim about sharing-induced suboptimality.Related work distinguishes amortization and variational-family approximation [Cremer et al., 2018], characterizes attainable amortized optima, and studies shared representations.

4 CHAPTER 1. STRUCTURAL REALIZABILITY UNDER SHARED DEPLOYMENT

This chapter develops an audit language for structural nonrealizability under shared deployment, measuring unavoidable native loss when locally optimal objects cannot be coherently realized. It separates architecture obstruction from approximation, optimization, statistical, and implementation explanations through explicit contracts, certificates, interventions, and validation.

Sparse-model mechanism: A sparse KKT construction exposes incompatible oracle Jacobian rows, converts the conflict into a quantitative native-loss floor for a one-pass class, and computes sufficient proximal repair depth.The exact priority of this result remains unresolved and is recorded in Appendix G.
Native defects and population floors: Structural obstruction is an exact excess objective in the native loss scale, yielding a positive population floor when the deployment class cannot realize the fixed local oracle family.The framework treats architecture obstruction independently from generic approximation error only after the local oracle, defect, representation, deployment class, statistical unit, and implementation error are declared separately.
Audit workflow: The recommended audit workflow fixes the native loss, validates defect and representation descriptions, identifies deployment obstructions, assesses finite-data observability, and tests minimal interventions against held-out risk.The five questions cover native loss scale, structural validity, regular, coordination, singular, or resource obstruction, finite-data certification and task observability, and intervention-based validation.
Falsifiable structural evidence: A performance plateau alone does not establish architecture obstruction; stronger evidence requires a certified nonzero floor, persistence under contract-preserving capacity or compute increases, and a mechanism-matched intervention.Misspecification, optimization failure, data scarcity, regularization, and metric ceilings can produce the same curve, so structural explanations must remain falsifiable.
Scope and positioning: The framework complements approximation, generalization, optimization, information theory, and representation learning by asking what shared systems can realize and when changing deployment is justified.In settings with a unique oracle and norm-equivalent defect, the obstruction may reduce to ordinary approximation error, but the local target, loss scale, contract, and population quantifier remain explicit.

Exercises

The exercises apply elimination geometry by constructing architecture obstructions, analyzing projector-based realizability, and separating local solvability, global realizability, and finite-sample certifiability.

Exercise 1.1: A finite instance space should exhibit existing local optima but positive architecture obstruction under constant deployment, then be minimally enlarged to eliminate that obstruction.This exercise targets the relationship between deployment-class expressivity and architecture obstruction.
Exercise 1.2: The projector representation preserves the eigenspace while changing the realizability problem because P(0) = P(2π) but v(2π) = −v(0).The exercise asks why identical projectors can coexist with sign-reversed eigenvectors.
Exercise 1.3: A modern learning system should be analyzed through separate questions about local solvability, global realizability, and finite-sample certifiability, alongside a negative control for structural explanations.The exercise requires proposing one negative control capable of falsifying a structural account.

Certified Elimination Systems

Certified elimination defines native residuals from an exact objective identity, while certified learning systems add deployment, representation, task, and statistical components that determine realizability and certifiability. The framework distinguishes native defects from architecture changes and requires geometry that respects observational equivalence.

Certified elimination: Certified elimination uses a triple (H, J, D) with nonempty auxiliary fibers and proper, finitely bounded-below lifted objectives, making the residual well defined without requiring attainment.The residual remains defined even when a trial state has infinite lifted cost.
Certified elimination: A distance to a selected optimizer is not itself a certificate; certification requires a proved exchange inequality linking distance to the native objective gap H_x(a) − J(x).
Certified elimination: In nonidentifiable mixtures, positive parameter distance can coexist with zero native defect, so the correct geometry is a quotient or unordered output space.Permutation invariance makes the oracle fiber an orbit rather than a point.
Certified learning systems: A certified learning system packages measurable sections, a representation and resource grammar, and a statistical experiment around the objective, with each component governing a distinct certification role.The objective determines the defect; deployment determines the architecture class; the task contract determines operational visibility; and the statistical experiment determines what can be certified.
Certified learning systems: Changing architecture or representation while fixing the objective changes the obstruction but not the native defect, whereas changing the lifted objective can change the statistical target.
Certified learning systems: Under finite-population assumptions, every deployed field satisfies an exact architecture decomposition, provided the architecture class contains a finite-defect field and the deployed field itself has finite defect.

8 CHAPTER 2. CERTIFIED ELIMINATION SYSTEMS

Certified elimination systems distinguish irreducible class-level floors from optimization failure and decompose obstruction across representation levels. The framework also separates uniform from average risk and requires attainment conditions before interpreting a zero obstruction as exact realizability.

Uniform risk can retain a fixed topological seam tax even when the seam has arbitrarily small probability mass under average risk.This distinguishes a persistent worst-case obstruction from concentrated failure under the data distribution.
Oracle families may be regular, finitely branched with monodromy, singular, or naturally set- or distribution-valued, changing whether pointwise global realization is appropriate.Singular families require defects that price branch collisions or multiplicity changes, while point-valued encodings can create artificial obstructions for set-valued orbits, projectors, sets, and laws.
Exact elimination decomposes total loss into coarse-state error and the vertical cost of realizing that state within its fine fiber.For restricted fine architectures, the resulting fine obstruction is expressed through an infimal fiber-realization tax.
The obstruction tower recursively separates coarse-state selection from the architectural cost of realizing its canonical fine fiber.An objective-faithful coarse representation need not preserve obstruction structure.
A zero obstruction means exact oracle containment only when the infimum is attained; an unattained zero indicates arbitrarily good approximation, while zero average obstruction can coexist with positive uniform obstruction.Attainment requires conditions such as compactness, lower semicontinuity, coercivity, or finite-dimensional closure.

Exercises

The exercises test three structural phenomena: attainment of population architecture obstruction, separation between population and infinite-sample defects, and label-dependent realization costs.

Exercises: For finite instance spaces and auxiliary fibers, the population architecture obstruction is attained by every nonempty architecture class.This exercise asks for a proof of universal attainment under finiteness conditions.
Exercises: A continuous selector on a circle with a shrinking-measure seam can yield 𝔒𝑃(𝔄) = 0 but 𝔒∞(𝔄) > 0.The exercise asks for an explicit construction separating the population and infinite-sample defects.
Exercises: Under a label permutation in a mixture model, a coarse oracle can have zero coarse defect while a labeled fine architecture pays a positive realization tax.The exercise asks for an example showing that label resolution can create a realization cost absent from the coarse representation.

A Four-Component Decomposition of Population Risk

The section decomposes excess risk into model, architecture, generalization, and implementation components with distinct reference objects and intervention implications. It frames the identity and certified bound as diagnostic tools whose structural floors persist unless the local model or deployment contract changes.

Decomposition: The four components separate local model inadequacy, shared-deployment obstruction, finite-sample uncertainty, and within-class empirical suboptimality.Δmodel compares the best local oracle family with the scientific target; Δarch measures loss forced by the deployment contract; generalization compares population and empirical risks; Δopt_n measures empirical suboptimality within the declared class.
Reference objects: The algebraic identity fixes reference objects rather than providing a probabilistic bound, keeping architecture obstruction distinct from local model misspecification.All quantitatively added terms must share a risk scale or be connected by an explicit exchange theorem.
Certified learning bound: The certified learning bound places learned excess risk around a structural floor, while data and optimization reduce only generalization and implementation terms.Under a fixed deployment class, uniform deviation, approximate empirical minimization, and R★ ≤ Roracle ≤ inf_A∈𝔄 R(A), positive Δmodel + Δarch requires changing the local model or deployment contract.
Interactions and limitations: The components interact: richer local families can increase architecture obstruction, data-driven atlases randomize deployment classes, and test-time computation can alter both implementation and class size.Nonconvex optimization may also reach only a subset of a nominal architecture class.
Diagnostic use: The decomposition should diagnose the dominant mechanism before intervention, because performance plateaus alone do not establish structural failure.Structural interpretation requires separately controlling local model inadequacy, finite-sample uncertainty, and implementation error; different loss scales remain distinct without a conversion theorem.

Exercises … Proof of the oracle-variation transport bound

The merged material develops elimination geometry as a contract-relative framework separating native defects, architecture obstruction, optimization, representation, and certification. It establishes exact identities, repair rules, quotient and integrability criteria, and an oracle-variation transport bound with explicit validity boundaries.

Proofs and exercises: The four-component identity decomposes excess risk into model, architecture, optimization, generalization, and implementation terms, with residuals vanishing under the corresponding attainment conditions.The certified upper bound requires only uniform generalization and empirical optimization control, while population attainment is unnecessary.
Representation, validity, and structural laboratories: Representation repair is admissible only when it preserves oracle-relevant distinctions: projection can restore probability validity, quotienting can remove gauge, and coarse carriers incur native conditional or fiber taxes when distinctions are lost.A valid repair may preserve statistical rates without becoming the constrained optimizer in another metric, and architecture enlargement is not always necessary.
Shared deployment and resource refinement: Shared deployment can create coordination, memory, and resource defects that width alone cannot remove; rectangularization, memory refinement, quotienting, or computation-budget changes repair different obstructions.The one-pass quadratic example changes the deployment contract from 𝔄0 to 𝔄≤𝑘, while its positive obstruction must be separately established rather than assumed.
Native defects and elimination calculus: Elimination-generated defects are native directed divergences whose orientation is fixed by the original objective, while towers and the P/G/X/V/C calculus track where transformations change or propagate costs.Numerical equality does not imply operational equality, and contract changes must remain distinct from the five transformation modes.
Exactification and optimization certificates: Exactification restores the target jet by subtracting the native defect, and conversely every target-jet-preserving scalar correction has the same defect-jet subtraction up to a k-flat term.This is a local normal form rather than a claim of global optimizer equivalence; numerical tolerances become objective and gradient certificates.
Integrability and global defect consistency: Global defect consistency requires vanishing local curvature plus zero periods on domains with holes, whereas transport or sampling contracts may legitimately admit nonconservative fields.Local square circulation yields falsification certificates, but operational curvature depends on whether the declared task can observe the discrepancy.
Lift admissibility and visible reduction: Conic lift admissibility is governed by slack-operator factorization and quotient-faithful extraction, while finite-complexity target-calling constructions show that exact lifts can otherwise add dummy coherence without meaningful deployment content.Visible capacity and cone size are distinct units, and target-calling no-go and extraction claims are identified as program-specific.
Regular obstruction transfer and flow duality: The oracle-variation transport theorem converts oracle variation beyond an L-Lipschitz deployment budget into a quantitative architecture lower bound, but the certificate may be zero when variation remains within budget.Its proof yields an architecture certificate rather than a universal impossibility result; related quadratic graph certificates follow by convex duality.

Rectangularity, Coordination Tax, and Memory … Exercises

The supplied sections develop elimination geometry as a framework for diagnosing coordination, representation, topology, resource, and deployment obstructions in native objective scales. They distinguish structural impossibility from decoder, task-visibility, statistical, and implementation limitations, while characterizing targeted repairs and honest certificates.

Rectangularity, Coordination Tax, and Memory: Rectangularization removes cross-history coupling, but the exact coordination tax measures the native cost of forcing one shared parameter, memory state, or network across histories.The tax can be localized to histories where sharing is most expensive; rectangularization is an analytical relaxation rather than the deployed architecture.
Rectangularity, Coordination Tax, and Memory: Memory compression can preserve coarse objective values while failing to preserve canonical fine conditional kernels, so universal equality requires canonical-fiber saturation rather than objective preservation alone.Approximate saturation yields an additive upper bound, making saturation an architecture-level condition rather than sufficiency for one fixed oracle.
Singular Fibers, Monodromy, and Catastrophe Taxes: Topological incompatibility creates native loss floors: winding, monodromy, degree, and radical branch obstructions can prevent continuous global selections, with costs controlled by clearance from singularities.The exact radical result requires 𝑘 divides ℓ for continuous zero-defect selection; the tax can vanish as the target approaches the discriminant, and representation artifacts need not reflect irregularity of the underlying object.
Atlases, Quotients, and Randomized Repair: Atlas architectures remove some global point-selection obstructions by routing inputs to local oracle sections, but the theorem establishes zero uniform obstruction without claiming minimal chart counts or gating generalization.For approximate repair, the inverse chart-budget/loss frontier is preferred because minimum chart counts can jump while achievable loss is more stable.
Resource-Constrained Architecture Rate-Distortion: Architecture rate–distortion separates carrier information loss from resource-limited decoder nonsaturation, and zero native distortion requires both task-relative sufficiency and a capable canonical decoder.Raw bit, dimension, or label counts become meaningful only through an extraction theorem; operational visibility still requires transmission and exposure gates.
Operational Semantics and Contextual Observability: Task envelopes and contextual closure identify exactly which native differences legal contexts and tasks can observe, so a positive internal defect may be clipped, smoothed, or remain operationally invisible.The contextual quotient is task- and grammar-relative, and contextual observability is a gate or transform rather than an additional scalar tax.
Composition, Base Change, and Dequantization: Eliminating conditional dependence can preserve visible marginals while destroying encoder semantics, because productization may make one carrier pair compatible with several input symbols; optimization within the restricted class cannot remove that architecture term.Candidate failures require the final architecture-level gate, and the result is presented as classical information theory plus rigorous numerics rather than a consequence of EG terminology.
Confidence Worlds and Statistical Elimination Geometry: Statistical certification must report identified images, margins, and unresolved regions, because pointwise feasibility does not guarantee a common deployable witness and repairs may require changing outputs, actions, data, or deployment grammar.On the coverage event, the standard certificate is honest and maximally decisive; common-deployment conflicts can persist even when every compatible world is individually feasible.

A Falsifiable Research Program for Structural Learning … D.1 Uniform concentration

The research program seeks to determine when shared deployment contracts realize local optima, quantify native defects when they do not, certify conclusions from finite data, and validate minimal structural repairs. It narrows its scope through explicit success criteria, failure rules, task-relative semantics, and technical tools for computation-dependent, singular, quantum, topological, convex, graph, and statistical settings.

27.1 Data-dependent and computation-dependent deployment: Computation-dependent architecture theory must distinguish reduced implementation error, enlarged test-time function classes, and optimizer- or implicit-bias restrictions on reachable functions.Data-dependent repaired classes likewise require complexity control for random atlases, quotients, memory refinements, or routing structures without discarding their certificate-generated structure.
27.2 Population effects of singularity and composition: Singular and coordination theories must relate deployment regularity, population mass, induced state distributions, and function approximation to average defects and continuous-state coordination limits.Task-relative semantics require retaining distinctions necessary for local oracles and downstream decisions while quotienting distinctions irrelevant to both.
27.3 Scientific AI and foundation-model contracts: Scientific applications count only when structural objects change a scientific or decision endpoint, including in multi-site, multi-batch, drifting-sensor, shared-latent-state, routing, retrieval, memory, and structured-output systems.The hypotheses must remain tied to native objectives and falsifiable deployment contracts rather than treating an atlas, quotient, or certificate as sufficient by itself.
27.4 Success criteria and stopping rules: Success requires computable architecture-obstruction certificates, sharp saturation theorems, reproducible repaired deployments, statistically valid unresolved certificates, and resource–risk frontiers.The program should be narrowed when certificates are vacuous or uncomputable, generic baselines explain gains, realistic samples leave cases unresolved, or structural distinctions do not change independent decisions.
27.4 Success criteria and stopping rules: The stopping rules require abandoning a structural explanation when its defect lacks downstream-task relevance, affects negligible population mass, is explained by matched generic baselines, or is too unstable for structural decisions.Operational examples include memory blindness to opposite resource-adjusted choices and sparse-inference certificates that pair native-loss floors with proximal repair depth.
27.4 Success criteria and stopping rules: The program’s mission is to test whether shared deployment contracts realize local optima, quantify unavoidable native risk, certify it from finite data, and identify the smallest independently validated structural repair.No theorem currently achieves this in complete generality; the proved results close specific interfaces across convex, graph, conditional, singular, finite-sample, and quantum settings.
Quantum antecedents and the rigidity specialization: The quantum boundary distinguishes established information-theoretic antecedents from a narrower rigidity specialization: a field-independent Gibbs base-change law for all Hermitian external fields implies an additive bipartite Hamiltonian without genuine interaction.The chapter presents this as a transparent specialization under sharp noncommutative quantifiers, while measurement generally yields lower certificates rather than an exact classical defect ledger.
D.1 Uniform concentration: The technical appendices supply reusable foundations: convex and Bregman geometry, graph and partial-map operators, covering and singularity tools, and uniform concentration over the actual deployment class.For finite loss classes Hoeffding applies, while infinite or calibration-dependent classes require Rademacher, covering-number, stability, or PAC-Bayes analyses.

D.2 Sample splitting … E.2 Validity spine

The statistical and validation spine specifies honest certification through independent splitting, uncertainty sets, sequential guarantees, lower bounds, and hierarchy-aware sampling. It then orders structural audits from native defects and integrability through admissible lifts, target-visible reduction, repair, and independent validation without treating audit transitions as mathematical implications.

D.2 Sample splitting: Independent scientific-unit splits make adaptive candidates fixed for test analysis, providing the simplest route to honest OALI validation.Calibration, training, tuning, and test samples are separated before test analysis.
D.3 Confidence sets and three-way decisions: Confidence sets support valid shared declarations, unresolved outputs when compatible worlds disagree, and explicit model conflict when the set is empty.Coverage at probability 1 − α guarantees validity for declarations shared by all worlds in the confidence set.
D.5 Testing lower bounds: Le Cam, Fano, and Assouad arguments yield risk lower bounds from indistinguishable alternatives, while certificate resolution restricts attention to opposite-certificate worlds.Only alternatives that can change the certificate are binding for certificate-resolution analysis.
D.6 Clustered data: For clustered observations, the effective independent test size is the number of biological units, so resampling and confidence intervals must respect the hierarchy.Cells, time points, or repeated observations nested within a unit do not provide independent unit-level replicates.
Appendix E; Validation and Dependency Map: The audit workflow delays structural lower-bound interpretation until native defects, defect-system integrability, and every nonnative carrier’s admissibility are established.The book’s validation spine is an audit workflow rather than a chain of theorem implications; each gate may require a new theorem, modeling assumption, or independent experiment.
E.1 Constructive spine: The constructive spine organizes the framework around conjugate defect identities, P/G/X/V/C distinctions, exactification and approximate-jet certificates, and graph-CDF validity repair with CRPS risk decomposition.These components define the main constructive sequence from defect specification to certified repair and risk accounting.
E.2 Validity spine; Validation and Dependency Map: The validity spine assembles local defect reports, uses Hodge and period terms to diagnose nonintegrability, excludes dummy or target-calling carriers, and links admissible lifts to deployment capacity through proved model-specific gates.Target-visible reduction and quotient-faithful extraction connect admissible representations to deployment only when the relevant gate is established.

E.3 Architecture spine … Source Manuscript Crosswalk

The monograph’s architecture spine separates obstruction mechanisms, resource and semantic effects, statistical authorization, and intervention repairs, while explicit non-circularity statements delimit key results. Its glossary and source crosswalk fix terminology, boundaries, and the locations of the underlying constructions.

E.3 Architecture spine: Architecture obstruction is treated as a second elimination, with EOT yielding metric and flow lower bounds, COT yielding rectangularity and coordination decompositions, and singular EG yielding exact catastrophe taxes.Atlas, quotient, set-valued, and randomized repairs alter the declared deployment contract.
E.4 Resources, semantics, and composition: Resource rate–distortion separates carrier information loss from decoder nonsaturation, while operational semantics determines which internal differences are contextually visible.Foundations and Part V govern composition, base change, and limits.
E.5 Statistical and intervention spine: Simultaneous defect and grammar envelopes bracket architecture frontiers, persistent atlas inverse frontiers remain stable, and three-way decisions encode unresolvedness rather than forced conclusions.Typed realization joins finite-information authorization, recursive closure, and one common deployment witness without identifying their different units.
E.6 No circularity in the graph-CRPS result; E.7 No circularity in exactification: The graph-CRPS validity repair and exact risk decomposition are independent of the graph-universal probability-validity classification, while the sharp path minimax theorem is summarized rather than fully reproved.The constructive exactification theorem separately requires a verified defect-remainder or acceptance condition before target descent; its converse establishes the defect-jet form modulo a flat term.
Controlled Glossary of Structural Terms: The controlled glossary fixes working definitions and boundaries, distinguishing native defects, certified systems, common deployment, operational visibility, resource units, and model conflict from nearby concepts.It emphasizes that obstruction claims are conditional on the declared package and that risk components remain non-interchangeable without compatible reference identities.
Obstruction-aware learning and inference: Obstruction-aware learning and inference uses contract-relative visibility, partial and set-valued representations, resource frontiers, task envelopes, and three-way certificates to distinguish realizability, nonrealizability, and unresolvedness.Target-calling lifts can manufacture fake coherence and are excluded by intrinsic lift admissibility; randomized repairs require a declared law-valued or sampled-action loss.
Source Manuscript Crosswalk: The source crosswalk reorganizes the manuscripts by concept, locating native defect, exactification, integrability, representation limits, common deployment, typed realization, certificate statistics, risk theory, OALI, transport recovery, and validation across the cited chapters and packages.The crosswalk identifies the source locations rather than presenting a new empirical or theoretical result.

G.1 Principal-result audit ledger … Thematic Index

The audit ledger inventories 37 completely proved formal-result families while distinguishing classical imports, specializations, synthesis interfaces, narrow program-specific increments, and unresolved priority candidates. The monograph’s claimed increment is its terminology, typed interfaces, audit order, and cross-domain interpretation rather than 37 uniformly original mechanisms, with the thematic index organizing the resulting concepts.

G.1 Principal-result audit ledger: The ledger records 37 formal-result families with complete proofs, but this is a proof-coverage count rather than a count of independent contributions.The rows include classical identities, workflow lemmas, book-level interfaces, and narrow priority candidates; treating all 37 as principal results would overstate original mathematics.
G.1 Principal-result audit ledger: The audit codes distinguish imported results, direct specializations, possible narrow increments, unresolved priority-audit candidates, and book-level synthesis interfaces.Book-level synthesis should not be cited as one wholly new mathematical mechanism, and possible increments are explicitly bounded by close antecedents.
E.7. NO CIRCULARITY IN EXACTIFICATION: The result-level audit repeatedly attributes mathematical cores to established identities and methods while limiting novelty claims to typed specializations, native-defect formulations, or interface-level organization.Examples include the conjugate defect identity from Fenchel–Young/Bregman theory [Bregman, 1967, Rockafellar, 1970], rectangularity from robust control [Epstein and Schneider, 2003, Iyengar, 2005, Nilim and El Ghaoui, 2005], and conic lifting from Gouveia et al. [2013] extending Yannakakis [1991].
E.7. NO CIRCULARITY IN EXACTIFICATION: Several retained results are explicitly framed as narrow or organizational increments, including frozen-lift jet exactification, task-envelope realization, common-deployment quantification, and typed no-compensation.The source claims a normal form within the declared frozen-lift class, a task-restricted two-sided min-plus realization, separation of pointwise feasibility from one common witness, and logical independence of multiple impossibility gates.
E.7. NO CIRCULARITY IN EXACTIFICATION: The ledger rejects the summary that four results are original and nine are program-specific, retaining elementary or classical consequences for their fit with the audit framework rather than as established new mechanisms.The finite target-calling construction, radical value, and evidence-slack criterion are specifically identified as elementary or short consequences, while the universal Gibbs base-change rigidity receives a dedicated audit.
G.2 Editorial principle: The editorial principle is that every inventoried result has a complete chapter-local appendix proof, while model-specific variants and advanced results outside the book remain in source manuscripts or are marked and excluded.The monograph claims terminology, typed interfaces, audit order, and cross-domain interpretation as its contribution.
Thematic Index: The thematic index maps the audit framework’s recurring vocabulary across architecture obstruction, native loss, shared deployment, resource limits, certification, observational overlap, and obstruction-aware learning and inference.It also indexes specific interfaces and mechanisms such as the deployment contract, common witness, rate–distortion, recursive quotient, architecture repair, and finite-sample certifiability.