Nonperturbative Definition and Completion Criteria

A nonperturbative definition supplies exact rules for a declared set of questions, not merely a formal series that approximates their answers. A nonperturbative completion adds the data needed to turn an incomplete perturbative or asymptotic description into such rules. Neither phrase means that every observable is easy—or even algorithmically possible—to compute. The central issue is determinacy: after the physical inputs and the accepted notion of equivalence have been fixed, does the proposal select a well-defined answer in its stated 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.

Reading path. Begin with the seven claim types, use the operational criteria, then compare the three proposals and their adversarial queries. The final worked application explains when boundary data define a bulk theory.

Seven claims that should not be conflated

The word nonperturbative says what a construction does not rely on; it does not by itself say what has been defined. The following claims require different evidence and have different ceilings.

Distinct kinds of exact, nonperturbative, and completion claims

Claim	Data that make it meaningful	What it does not establish by itself
Exact finite or regulated model	A Hilbert space or algebra, constraints, dynamics, boundary data, and observable rule at fixed volume, cutoff, or matrix size	A continuum, infinite-volume, large-N, or target-spacetime limit
Constructive continuum definition	A regulated family, tuned bare data, declared order of limits, convergence of a separating observable set, and the relevant positivity or reconstruction conditions	A UV completion of every low-energy effective theory, or sectors outside the constructed continuum model
UV completion of an effective theory	Exact high-energy degrees of freedom and dynamics, together with renormalization-group flow or matching that recovers the specified low-energy theory	Constructive existence, or backgrounds and global sectors outside the high-energy theory's declared domain
Asymptotic or resurgent completion	An analyticity and summability domain, a summation direction or contour, and, when needed, additional sectors, derived Stokes data, and free transseries parameters	A unique theory when necessary parameters remain free, or observables outside the completed sector
Sector-limited definition	Exact rules at fixed asymptotics, background, charge, flux, topology, or superselection label	An extension to sectors that were explicitly excluded
Axiomatic or bootstrap constraints	Consistency equations, positivity or unitarity conditions, and a specified space of candidate data	Existence or uniqueness unless a reconstruction theorem or constructive solution is supplied
Conditional dual definition	An independently exact nongravitational theory and a complete-enough dictionary for the targeted gravitational observables	The duality itself or any dictionary entries not supplied; these remain hypotheses unless separately established

A sector boundary is not a defect when it is stated honestly. Conversely, calling a model “UV complete” does not automatically include every compactification, boundary condition, charge sector, or topology. The claim must name both its domain and its equivalence relation: gauge-related descriptions, unitarily isomorphic Hilbert-space presentations, and genuinely different completions are not the same notion of uniqueness.

Operational criteria for a declared domain

The applicable conditions below are necessary obligations for a definition claim. A proposal must address each field as applicable and explain why an inapplicable field is absent. These conditions are not a theorem that a proposed theory exists, is consistent, is unique among all formulations, or describes nature.

Operational criteria for a nonperturbative definition or completion

Obligation	Record explicitly	Passing evidence	Failure signal or claim ceiling
Domain and equivalence	Asymptotics, backgrounds, couplings, boundaries, charges, volumes, and which presentations count as the same theory	Every target observable falls inside a declared domain and is invariant under the named equivalences	An observable changes when an allegedly redundant presentation changes, or its domain is unstated
Objects and states	Degrees of freedom or observable algebra, state space, constraints, and physical inner product or positivity condition	The construction identifies physical states and gauge-invariant observables without double counting	Only kinematic variables or gauge-variant quantities are specified
Dynamics or correlation rule	A self-adjoint Hamiltonian and its domain, an algebraic time-evolution automorphism, a transfer matrix, or a measure or cycle with gauge quotient, counterterms, boundary conditions, and reconstruction data	The rule produces well-defined amplitudes, correlators, expectation values, or algebraic evolution in the stated domain	An action, formal integrand, or correlator list is given without the data needed for integration, evolution, or reconstruction
Regulation and limits, when claimed	Every cutoff and finite-volume parameter, tuned bare data, order of limits, convergence mode, and any claimed regulator-independence test	The claimed continuum or large-system observables converge; if universality or regulator independence is asserted, matched observables agree across the tested regulators	A required limit is missing, order-dependent but unordered, or defined only by extrapolation from a protected regime; an exact finite model need not claim such a limit
Sectors and global data	Which topology, brane, charge, flux, defect, and superselection sectors are fixed, summed, or excluded, together with their weights	The rule either determines the included sectors or explicitly excludes them from a narrower claim	A claimed sector has no state space, measure, weight, or observable map
Completion and uniqueness	Analyticity and summability hypotheses, contours, boundary conditions, spectral data, and, when present, derived Stokes constants or free transseries parameters invisible to the starting expansion	Fixed inputs select one answer up to the declared equivalence	Inequivalent answers share all data the proposal actually supplied
Controlled recovery	The target observables, small parameters, limiting path, error terms, and comparison data	Independent calculations reproduce the claimed perturbative string, gravitational, or QFT regime with controlled errors	Only qualitative resemblance or checks that reuse the same assumptions are available
Determinacy and validation	A principled observable rule plus consistency, falsification, and cross-check conditions	The observable is mathematically fixed even when its evaluation is difficult	The answer depends on an unstated choice; practical difficulty alone is not a failure

For Hamiltonian data, writing $H$ is not enough. A schematic exact specification is

$$\bigl(\mathcal H,\mathcal D(H),H=H^\dagger,\mathcal A, \text{constraints and boundary data}\bigr), \qquad U(t)=e^{-itH}.$$

For a path integral, the analogous list includes the configuration space, measure or integration cycle, gauge treatment, renormalization prescription, and boundary conditions. Different admissible cycles can give different answers and can jump across Stokes lines Witten 2011, §§ 2.1 and 3.1. An action alone is therefore not an exact definition.

Exact specification is also weaker than universal solvability. There are explicit families of local two-dimensional quantum spin Hamiltonians for which deciding the thermodynamic spectral gap is algorithmically undecidable Cubitt, Pérez-García, and Wolf 2022, § 1.2, Theorem 3, pp. 5–6. The Hamiltonians are defined; one property has no general decision algorithm.

Why perturbation theory can miss a completion

Suppose $F_0(g)$ has a power-series asymptotic expansion as $g\to0^+$. For any $A>0$ and constant $C$, define

$$F_C(g)=F_0(g)+C e^{-A/g}.$$

For every nonnegative integer $n$,

$$\lim_{g\to0^+}\frac{e^{-A/g}}{g^n}=0.$$

Every $F_C$ therefore has the same power-series asymptotics as $F_0$, although different values of $C$ give different exact functions. Perturbative coefficients alone cannot determine $C$. A physical construction may fix it through an integration cycle, boundary condition, spectral requirement, Stokes cancellation, or another exact principle—but that extra datum is part of the completion. This example proves logical underdetermination, not the existence of several consistent quantum-gravity theories.

A Borel-summable series in a nonsingular direction may need no extra transseries sector or free parameter once the required analyticity and growth conditions hold Sokal 1980, pp. 261–263. When singular directions and exponential sectors are present, lateral or median resummation, Stokes data, and any genuinely free transseries parameters must instead be distinguished Aniceto and Schiappa 2015, § 2, pp. 5–10. The Semiclassics, Resurgence, and Transseries guide develops these cases.

Regulators and ordered limits

A finite regulated theory can already be exact in its own domain. A continuum claim is a second statement. For a compact gauge group $G$, for example, a finite lattice gauge theory begins with an integral such as

$$Z_{a,L}=\int\prod_{\ell}dU_\ell\,e^{-S_a[U]},$$

where each $dU_\ell$ is normalized Haar measure. A continuum, infinite-volume claim then requires a tuned limit of a separating set of renormalized observables. Schematically,

$$G_R^{(n)}(\{x_i\};\mu) = \underset{\substack{a\to0,\;La\to\infty\\\lambda_0=\lambda_0(a)}}{\operatorname{Lim}} G_{R,a}^{(n)}(\{x_i^{(a)}\};\mu), \qquad x_i^{(a)}\to x_i.$$

Here lattice points $x_i^{(a)}$ approach fixed physical insertion points $x_i$, the renormalization scale $\mu$ is held fixed, $\lambda_0(a)$ denotes every relevant tuned bare parameter, and $G_{R,a}^{(n)}$ includes the required operator mixing and additive or multiplicative subtractions. The construction must state whether the limits are sequential or joint and what convergence means. A proved limit in one regulator can define a continuum theory when the relevant reconstruction hypotheses hold. Cross-regulator agreement is an additional test only when universality or regulator independence is claimed, and then requires matching renormalized parameters, schemes, observables, and universality class. Wilson’s lattice formulation supplies exact gauge-invariant regulated data Wilson 1974, § III.A–B, pp. 2450–2452; recovering a relativistic Lorentzian theory from Euclidean functions requires additional reconstruction hypotheses Osterwalder and Schrader 1975, § IV, pp. 287–305.

Order is physical data whenever limits do not commute. The toy family

$$X_{N,\Lambda}=\frac{N}{N+\Lambda}$$

obeys

$$\lim_{\Lambda\to\infty}\lim_{N\to\infty}X_{N,\Lambda}=1, \qquad \lim_{N\to\infty}\lim_{\Lambda\to\infty}X_{N,\Lambda}=0.$$

This formula is not a model of matrix theory. It isolates the logical point: “take the cutoff away and then take large $N$” is different input from the reverse order.

Three proposals under one test

The comparison below applies the same fields to three proposals without assuming that their targets coincide. “Exact input” refers only to what is independently specified; “target claim” records the additional identification being tested.

Three proposals and their strongest licensed conclusions

Proposal	Independently supplied object	Additional target claim	Strongest licensed conclusion
BFSS	Fixed-N supersymmetric matrix quantum mechanics	An appropriate large-N regime gives uncompactified eleven-dimensional M-theory	Exact matrix model plus a conditional target-theory limit
Covariant string field theory	Gauge-consistent string perturbation theory around chosen backgrounds	One quantum prescription covers all couplings, backgrounds, and sectors	All-orders perturbative definition with controlled sectoral applications, not a generic nonperturbative completion
AdS/CFT as a definition	An independently exact finite-N boundary theory	An exact duality and dictionary define a specified asymptotically AdS bulk	Conditional bulk definition for mapped observables and declared sectors

BFSS matrix quantum mechanics

• Exact input and conventions. The physics formulation supplies gauge-constrained maximally supersymmetric $U(N)$ matrix quantum mechanics at fixed $N$, including its Hamiltonian and Gauss-law constraint. Treating it as mathematically exact input additionally requires the relevant self-adjoint operator domains.
• Observable and domain. Matrix-model observables are questions in that quantum mechanics. The additional target is uncompactified eleven-dimensional M-theory scattering in the appropriate large-$N$ regime.
• Control data. In the original spacelike-circle convention one records $P_{11}=N/R_{11}$ and the uncompactified infinite-momentum or large-$N$ limit. The separate finite-$N$ DLCQ proposal uses $x^\pm=(x^0\pm x^{11})/\sqrt2$, a null circle $x^-\sim x^-+2\pi R_-$, and $P^+=N/R_-$. Each scattering or infrared limit and its order must also be stated.
• Logical status and evidence class. The matrix model is the exact input; the target identification is a conjecture supported by supersymmetry and spectrum checks and by selected low-velocity, large-separation terms, including restricted or protected $v^4$ and $v^6$ structures—not by arbitrary unprotected or multiparticle amplitudes.
• Uncertainty and falsifier. Existence and completeness of the target limit, emergent locality, and Lorentz invariance are not proved. A failure of the prescribed large-$N$ limit or a target observable that disagrees after all limits would defeat the broad identification.
• Licensed conclusion. BFSS is an exact matrix model with a conditional M-theory interpretation, not by itself a completed definition of arbitrary uncompactified M-theory observables.

The original paper states the momentum convention and large-$N$ conjecture and records unresolved questions about the existence of that limit, Lorentz invariance, and locality Banks et al. 1997, §§ 2, 4, 8, and 11. Interpreting finite $N$ as discrete light-cone quantization is a further claim, developed by Susskind 1997, § 1, pp. 1–2 and Seiberg 1997, § 1, pp. 1–2; it should not be silently merged with the fixed-$N$ matrix model. For a current account of the restricted derivative and kinematic regimes in amplitude comparisons, see Lin 2026, §§ 4.1–4.2, pp. 18–22.

Covariant string field theory

• Exact input and conventions. Choose a conformal background, BRST complex, string fields, BV action and vertices, and the integration prescriptions needed order by order.
• Observable and domain. The construction supplies off-shell data and on-shell amplitudes in perturbative string theory around that background.
• Control data. The expansion is organized by string coupling or genus and by a decomposition of moduli space. A universal all-coupling contour or transseries datum does not follow automatically.
• Logical status and evidence class. This is a gauge-consistent definition of string perturbation theory and supports important sectoral nonperturbative applications; here it is the perturbative control case.
• Uncertainty and falsifier. Convergence, arbitrary background change, and a globally complete sector sum remain outside the general construction. A claimed broad completion fails if fixed perturbative input admits inequivalent nonperturbative answers.
• Licensed conclusion. Retain the all-orders perturbative definition and explicitly constructed sectoral results, not a universal nonperturbative string definition.

Zwiebach’s quantum closed-string field theory constructs a BV master action perturbatively around a chosen conformal background and explicitly leaves convergence and nonperturbative completion open Zwiebach 1993, § 1, pp. 1–5, and § 4.4. Modern string field theory gives gauge-invariant formulations of perturbative string theories and supports nonperturbative classical solutions and other controlled applications Sen and Zwiebach 2024, § 1, pp. 2–3. Those achievements do not by themselves provide one quantum prescription across all backgrounds and couplings.

AdS/CFT as a definition

• Exact input and conventions. Start from an independently exact finite-$N$ boundary theory, including its global form, state space or algebra, couplings, boundary conditions, and sectors.
• Observable and domain. Boundary observables may be exact. Bulk relational, asymptotic, or otherwise gauge-invariant observables are defined only where the dictionary identifies them.
• Control data. The defining bridge is an exact duality and dictionary. Bulk $1/N$ and inverse-coupling expansions are recovery tests, not defining input.
• Logical status and evidence class. This is a conditional nonperturbative definition of a specified asymptotically AdS bulk; evidence is pair- and observable-dependent.
• Uncertainty and falsifier. An incomplete known dictionary is an epistemic limitation, not proof that no map exists. Two inequivalent maps that survive the same fixed input but predict different target observables would defeat uniqueness.
• Licensed conclusion. The exact claim covers the boundary theory; the conditional bulk claim covers only mapped observables and declared sectors.

Maldacena introduced the large-$N$ gravity/gauge-theory relation as a conjecture Maldacena 1998, abstract and § 6, pp. 16–17. Witten then made the source-to-observable ansatz precise and explained its Hamiltonian interpretation Witten 1998, §§ 2.3 and 3.3, pp. 8–11 and 34–36. A fully specified boundary theory can be exact even when no closed-form calculation is available; using it as an exact bulk definition remains conditional on the duality and on a dictionary for the bulk question being asked.

Adversarial queries

An adversarial query should probe the edge of the claimed domain. If the observable was explicitly excluded from the outset, the test confirms honest scope but says nothing about a broader completion. If the proposal initially claims to include it, three outcomes must be distinguished:

1. the fixed rules predict one answer up to the declared equivalence;
2. the analysis proves that the observable is absent or belongs outside a newly narrowed domain;
3. the claim includes the observable, but the supplied rules do not determine it.

The first outcome passes the original inclusion claim. The second is a complete scope diagnosis only after the claim is revised; it is not a completion for the excluded observable. The third forces additional completion data or the same kind of scope correction.

Adversarial tests and the exact downgrade each failure requires

Proposal	Challenge	Response from the supplied rules	Classification and downgrade
BFSS finite-N DLCQ claim	For a specified null-circle sector with p⁺ = N/R₋, request an amplitude between states explicitly mapped to matrix-model asymptotic states	The matrix dynamics can address the mapped quantity only after the DLCQ dictionary, operator or state map, and scattering limit are supplied	Conditional but well posed when all maps are present; a failed map or disagreement defeats the finite-N DLCQ identification, not the matrix model
BFSS uncompactified claim	For a specified eleven-dimensional amplitude, execute the declared large-N, radius, and scattering scaling and test the limit	A missing or nonexistent limit leaves no target answer; a convergent answer can be compared with the independently defined target observable	Undefined without the limit prescription; nonexistence or disagreement falsifies the broad identification for that observable
Covariant string field theory	Request a nonperturbative exponential in the string coupling, or a transition to a background not included in the starting conformal field theory	The perturbative BV construction does not select the missing exponential datum or a new-background quantum prescription	Undefined for the broad nonperturbative claim; retain the all-orders perturbative and explicitly constructed sectoral claims
Conditional AdS/CFT	Given exact boundary data but a partial dictionary, request a topology-changing or charged bulk observable with no specified boundary image	Named boundary observables remain exact; the bulk label has no unique operational meaning until the dictionary and sector rule are supplied	Undefined as a bulk question in the present dictionary; retain the exact boundary theory and the mapped portion of the conditional duality

The AdS/CFT result is deliberately epistemic: failure to exhibit a bulk map does not prove that the exact dual lacks one. It limits what the present formulation lets a reader calculate or claim.

When boundary data define a bulk theory

Consider the most ambitious common use of AdS/CFT as a nonperturbative definition. Two logically different readings must be kept separate: the exact boundary theory may define mapped bulk observables, or an independently specified bulk path integral may be conjectured to reproduce the boundary answers. The first reading does not presuppose the second.

1. Boundary theory. Fix a particular finite-$N$ theory, global form, couplings or conformal data, Hilbert spaces or observable algebras, boundary conditions, defects, and superselection sectors independently of bulk perturbation theory.
2. Target observable. Choose an exact boundary quantity—such as a spectrum, correlator, partition function, or state amplitude—and give its normalization and, when a regulator is used, its regulator-removal prescription.
3. Dictionary entry. Map boundary sources and operators to gauge-invariant, relational, or asymptotic bulk observables. A local coordinate label in gravity is not enough.
4. Definition by duality. If an exact duality is assumed, define each mapped bulk answer by its boundary image and state the equivalence under which that image is unique. Unknown bulk saddle weights are then reconstruction data, not missing input to the boundary definition.
5. Independent bulk representation, if claimed. Only when a separate bulk path integral is asserted must its asymptotic boundary conditions, allowed fillings and topologies, field sectors, contour or Lorentzian prescription, counterterms, and relative weights be independently fixed.
6. Logical status. Say whether the dictionary entry is a conditional definition, an exact identity established in a restricted setting, or a conjecture tested only in a large-$N$ or strong-coupling expansion.
7. Independent checks. Test symmetries, Ward identities, anomalies, spectra, unitarity or reflection positivity, factorization, and controlled bulk limits without counting consequences of one protected identity as independent evidence.

Under the boundary-definition reading, the generating-functional statement is a conditional definition,

$$Z_{\mathrm{QG}}^{(\mathrm{def})} \!\left[\partial X=M,\ \Phi|_{\partial X}\sim J\right] :=Z_{\mathrm{CFT}}[M,J].$$

This assignment is conditional on the exact duality and dictionary, but it is not circular: the independently defined CFT supplies the answer. Any bulk representation must reproduce that total answer, although its decomposition into topologies, sectors, contours, or saddles need not be uniquely reconstructible from the CFT quantity. Witten’s generating-functional ansatz motivates this assignment Witten 1998, § 2.3, pp. 8–11, eqs. (2.10)–(2.13).

A stronger, separate representation claim writes an independently constructed $Z_{\mathrm{QG}}$ and tests whether it equals $Z_{\mathrm{CFT}}$. In that case the bulk symbol is not defined by notation: its field space, sectors, topology rule, contour, boundary terms, and normalization must be fixed; Witten’s discussion of summing bulk fillings illustrates the extra choice Witten 1998, § 3.2, pp. 28–29, eqs. (3.2)–(3.3). Selected semiclassical saddles may approximate either formulation, but the saddle expansion is not the exact definition. If two inequivalent dictionaries assign different boundary observables to the same proposed bulk quantity, uniqueness has not yet been shown. If no image is supplied, the bulk question is undefined within the stated dictionary even though the boundary theory remains exact.

Claim ceilings and handoffs

Satisfying the operational fields above shows that a proposal is sufficiently specified to be tested in a stated domain. It does not replace a construction or proof of existence, consistency, convergence, uniqueness, dual equivalence, or empirical correctness. Those conclusions require evidence appropriate to the claim type.

Evidence cutoff. 27 August 2026. The foundational claim boundaries are fixed by the cited primary sources, the 2024 string-field-theory review, and the current matrix-theory review cited above; broader literature-sensitive assessments follow the Research guide below. Reassess this page if a controlled construction establishes a presently missing large-$N$ limit, all-coupling string-field prescription, exact bulk dictionary, or contrary observable test.

The Nonperturbative String- and M-Theory Definition Proposals chapter applies these distinctions in depth to BFSS, string field theory, and conditional AdS/CFT. The low-dimensional cases require additional care about averages and nonunique completions: see Fixed-Theory, Disorder-Average, and Ensemble Distinctions and Fixed-Theory Factorization and Nonperturbative Completion Tests. Existence, Uniqueness, and Equivalence Claims develops the corresponding mathematical standards.

For literature-sensitive assessments, use the Holography and Quantum Gravity Research guide, which states its own evidence cutoff and reassessment triggers. Foundational papers establish the original constructions and their logical status; they do not certify a current field-wide verdict. Continue with Evidence Programs and Duality Tests to design independent checks.

Common pitfalls

Exact model versus exact target theory. An exactly specified finite-$N$ Hamiltonian does not by itself prove that a particular large-$N$ limit exists or equals uncompactified M-theory. State the model and the target identification as separate claims.

A formula versus a definition. A Lagrangian, Euclidean action, or formal path integral omits operator domains, contours, gauge quotients, boundary conditions, and renormalization data that can change the answer.

All perturbative orders versus an exact answer. Terms smaller than every power of the expansion parameter can remain invisible to the entire perturbative series. Record the prescription that fixes them.

Difficulty versus indefiniteness. A well-defined observable may be analytically intractable or even undecidable in a general family. The failure relevant here is dependence on an unstated input, not merely the absence of a fast algorithm.

A scoped exclusion versus a universal completion. Demonstrating that an observable lies outside a declared sector is a complete scope diagnosis. It does not extend the definition to that observable.

Exercises

1. Invisible completion data

Let $A>0$ and $F_C(g)=F_0(g)+C e^{-A/g}$ for $g>0$. Show that the coefficient $C$ cannot be recovered from the power-series asymptotic expansion at $g=0^+$. Name two kinds of additional data that could fix it.

Solution

For every integer $n\ge0$, set $x=A/g$. Then $e^{-A/g}/g^n=A^{-n}x^n e^{-x}\to0$ as $x\to\infty$. The added term is therefore smaller than every power of $g$, so changing $C$ changes no perturbative coefficient. An integration contour, boundary condition, exact spectral condition, Stokes-cancellation rule, or independently defined microscopic theory could fix $C$.

2. Which limit was defined?

For $X_{N,\Lambda}=N/(N+\Lambda)$, compute both iterated limits as $N,\Lambda\to\infty$. What must a proposal record if these variables represent matrix size and a regulator?

Solution

At fixed $\Lambda$, $X_{N,\Lambda}\to1$ as $N\to\infty$, so taking $\Lambda\to\infty$ afterward leaves $1$. At fixed $N$, $X_{N,\Lambda}\to0$ as $\Lambda\to\infty$, so the reverse order gives $0$. The proposal must state the order or joint scaling path, the observables held fixed, and evidence that the chosen limit exists and is regulator-independent in the claimed sense.

3. Exact, conditional, or undefined?

Classify each request: (a) a gauge-invariant expectation value of the fixed-$N$ BFSS Hamiltonian; (b) an uncompactified M-theory amplitude obtained from BFSS without a stated large-$N$ prescription; (c) a boundary-CFT correlator in an independently defined finite-$N$ theory; (d) a proposed local bulk field with neither gravitational dressing nor a boundary image.

Solution

(a) is an exact matrix-model question once the state and operator are specified. (b) is undefined from the stated input because the target limit is missing. (c) is an exact boundary-theory question once its regulator, state, sources, and normalization are fixed. (d) is not yet a gauge-invariant bulk observable and has no dictionary entry, so the bulk question is undefined; this does not prove that no suitable relational or boundary representation exists.

4. Does crossing define a CFT?

Suppose a proposed conformal field theory supplies crossing equations, positive squared OPE coefficients, and numerical bounds but no spectrum and OPE data that solve the equations. Which claim type applies, and which operational obligations remain unmet?

Solution

This is an axiomatic or bootstrap constraint system, not yet an existence result. At minimum, Objects and states remains unmet because no spectrum and OPE data have been supplied, while Determinacy and validation remains unmet because the constraints have not selected and reconstructed a theory. A constructive or reconstruction step must provide consistent correlation functions or operator data. Even an isolated numerical island also needs an explicit equivalence criterion before it establishes uniqueness.

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

• Aniceto, Inês, and Ricardo Schiappa. 2015. “Nonperturbative Ambiguities and the Reality of Resurgent Transseries.” Communications in Mathematical Physics 335: 183–245. DOI. Open PDF.
• Banks, Tom, Willy Fischler, Stephen H. Shenker, and Leonard Susskind. 1997. “M Theory as a Matrix Model: A Conjecture.” Physical Review D 55: 5112–5128. DOI. Open PDF.
• Cubitt, Toby, David Pérez-García, and Michael M. Wolf. 2022. “Undecidability of the Spectral Gap.” Forum of Mathematics, Pi 10: e14. DOI. Open PDF.
• Lin, Henry W. 2026. “TASI Lectures on Matrix Theory from a Modern Viewpoint.” Preprint arXiv:2508.20970v3. Open PDF.
• Maldacena, Juan M. 1998. “The Large N Limit of Superconformal Field Theories and Supergravity.” Advances in Theoretical and Mathematical Physics 2: 231–252. DOI. Open PDF.
• Osterwalder, Konrad, and Robert Schrader. 1975. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42: 281–305. DOI. Open PDF.
• Seiberg, Nathan. 1997. “Why Is the Matrix Model Correct?” Physical Review Letters 79: 3577–3580. DOI. Open PDF.
• Sen, Ashoke, and Barton Zwiebach. 2024. “String Field Theory: A Review.” In Handbook of Quantum Gravity, edited by Cosimo Bambi, Leonardo Modesto, and Ilya Shapiro, 2385–2600. Singapore: Springer Nature. DOI. Open PDF.
• Sokal, Alan D. 1980. “An Improvement of Watson’s Theorem on Borel Summability.” Journal of Mathematical Physics 21: 261–263. DOI.
• Susskind, Leonard. 1997. “Another Conjecture about M(atrix) Theory.” Preprint arXiv:hep-th/9704080. Open PDF.
• Wilson, Kenneth G. 1974. “Confinement of Quarks.” Physical Review D 10: 2445–2459. DOI.
• Witten, Edward. 1998. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2: 253–291. DOI. Open PDF.
• Witten, Edward. 2011. “Analytic Continuation of Chern–Simons Theory.” In Chern–Simons Gauge Theory: 20 Years After, 347–446. AMS/IP Studies in Advanced Mathematics 50. Providence, RI: American Mathematical Society. Open PDF.
• Zwiebach, Barton. 1993. “Closed String Field Theory: Quantum Action and the Batalin–Vilkovisky Master Equation.” Nuclear Physics B 390: 33–152. DOI. Open PDF.