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Nonperturbative Definition and Completion Criteria

A nonperturbative definition must determine observables beyond every formal expansion, not merely organize perturbative coefficients. At minimum it must specify degrees of freedom, states or an observable algebra, dynamics, regulator removal, all relevant sectors, and a rule that gives unique answers throughout its claimed domain.

Required background. Holographic Duality: Claims, Dictionaries, and Regimes fixes the claim type; Exact Statements, Saddle Expansions, and Conditional Derivations separates exact data from expansions; Dictionary Completeness and Global Data supplies the global sectors that a completion must address.

Helpful background. EFT Truncation Errors and Breakdown Diagnostics identifies what a low-energy expansion leaves undetermined. The Quantum-Gravity Handoff marks the point where semiclassical variables cease to be sufficient, while Duality Checks, Evidence Independence, Status, and Failure Modes supplies the broader duality comparison.

Criteria for a definition

Section titled “Criteria for a definition”

A proposal should answer the following questions within a declared domain.

  1. Objects: What are the fundamental degrees of freedom or observables?
  2. States: What is the Hilbert space, algebraic state space, or replacement?
  3. Dynamics: What exact Hamiltonian, path integral, algebraic rule, or transition amplitude defines evolution?
  4. Regulation: How is the construction regulated, and does regulator removal exist?
  5. Sectors: Are branes, topology, flux, charge, and nonperturbative exponential effects included?
  6. Uniqueness: Do the stated data select one theory rather than a family of completions sharing an asymptotic expansion?
  7. Recovery: Which controlled limits reproduce perturbative strings, gravity, and QFT observables?
  8. Computability: Is there a principled way, even if difficult, to determine the claimed observables?

Failure of one item need not make a proposal useless. It limits what may be called defined.

Three proposals under one test

Section titled “Three proposals under one test”
ProposalClearly specified dataPrincipal completion question
BFSS matrix quantum mechanicsFinite-NN matrices, Hamiltonian, gauge constraintRecovery and completeness of uncompactified M-theory in the required large-NN limit
Covariant string field theoryString fields, BRST structure, interaction vertices around a backgroundBackground scope, nonperturbative sectors, and globally complete state space
AdS/CFT as definitionA nonperturbatively defined boundary theory and boundary observablesCompleteness and uniqueness of the bulk map, especially beyond perturbative code subspaces

BFSS can be an exact quantum-mechanical model at finite NN while the M-theory claim remains a large-NN inference Banks et al. 1997. Covariant closed-string field theory supplies a quantum action with a Batalin–Vilkovisky structure around specified backgrounds Zwiebach 1993, without thereby supplying every background or sector. AdS/CFT can define bulk quantities when the boundary theory and map are complete enough; it cannot define an observable that the dictionary has not specified merely by invoking duality.

The out-of-sector test

Section titled “The out-of-sector test”

Ask each proposal for an unprotected observable outside the regime used to motivate it: a finite-NN amplitude, a topology-changing quantity, a sector with new charge, or a background not included in the construction.

There are three scientifically distinct outcomes:

  • the rules determine a unique answer;
  • the observable is demonstrably absent from the theory’s domain;
  • the proposal supplies no defined answer.

Only the first two are complete responses. The third does not refute the controlled sector, but it forbids describing the proposal as a completion for that observable.

What this checklist establishes

Section titled “What this checklist establishes”

Passing the criteria is evidence that a construction is operationally complete for a stated domain. It is not a proof of uniqueness among all formulations or of empirical correctness. Chapter 5 applies the test to string- and M-theory proposals; Chapters 18 and 20 treat low-dimensional and ensemble completions; Volume XVI owns mathematical existence standards.

Evidence cutoff. Proposal-status statements are fixed to 25 July 2026 and require a Research update if the literature changes.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

Section titled “References”