Falsifiers, Negative Results, and Counterexamples

A failed approximation is not automatically a failed duality, and a successful approximation is not automatically evidence for a complete one. A useful negative result identifies the proposition being tested: the dictionary itself, the existence of a regime, a quantitative error bound, a proposed sufficient criterion, or a computational method.

Required background. Holographic Duality: Claims, Dictionaries, and Regimes identifies the claim, while Evidence Programs for Holographic Duality identifies the test and its assumptions.

Helpful background. Duality Checks, Evidence Independence, Status, and Failure Modes gives the general duality setting. EFT Truncation Errors and Breakdown Diagnostics distinguishes expected breakdown from contradiction.

Five kinds of negative result

Failure	What is challenged	What may survive
Dictionary contradiction	Two mapped observables disagree in the same controlled domain	Other, narrower sectoral maps
Missing regime	Conditions required for a proposed bulk description cannot be met	The boundary theory and perhaps a non-Einstein bulk
Error-bound violation	The remainder exceeds its declared bound	The leading mechanism without claimed precision
Counterexample to sufficiency	Proposed criteria hold but conclusion fails	Criteria as necessary or heuristic indicators
Method failure	A solver, saddle selection, or extrapolation is unreliable	The physical claim tested by another method

A genuine falsifier fixes all nuisance choices that could otherwise move the target: normalization, state, ensemble, sector, contour, regulator, and limit order.

Finite gap, exact consistency, and numerical error

Consider a large-$N$ CFT with a finite higher-spin gap. Three negative observations differ.

1. Higher-derivative corrections become order one at the energy of interest. This refutes an Einstein-EFT approximation at that precision, not CFT consistency.
2. Crossing or unitarity fails for the proposed CFT data. This challenges the boundary data themselves and hence any dictionary using them.
3. A numerical large-$N$ extrapolation changes under grid, cutoff, or fit-window refinement. This invalidates that numerical conclusion until convergence is restored; it need not say anything about the exact theory.

Causality constraints on higher-derivative graviton couplings give a concrete version of the first distinction: certain corrections require new higher-spin states near the associated scale Camanho et al. 2016. The result bounds the low-energy description rather than disproving the ultraviolet theory that supplies those states.

Perturb until the approximation fails

Suppose an observable is represented as

$$F=F_0+\epsilon F_1+R_2, \qquad \lvert R_2\rvert\le K\epsilon^2$$

for $\epsilon<\epsilon_*$. Increase $\epsilon$ through $\epsilon_*$. Disagreement beyond that point is expected and does not falsify the underlying theory. Disagreement inside the domain, after independent verification of $K$, challenges the stated bound or the calculation.

This adversarial continuation should be performed before assigning a negative result. It records the strongest surviving statement: perhaps a leading scaling law, a non-Einstein holographic regime, or only the exact boundary observable.

Evidence ceiling

This classification does not decide current controversies. Individual chapters own domain-specific counterexamples; Volume XVI owns mathematical countermodels; Research records new negative results and responses. The stable lesson is to attach every “failure” to one explicit proposition.

Evidence cutoff. Literature examples are fixed to 25 July 2026.

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

• Camanho, Xian O., Jose D. Edelstein, Juan Maldacena, and Alexander Zhiboedov. 2016. “Causality Constraints on Corrections to the Graviton Three-Point Coupling,” Journal of High Energy Physics 02, 020.
• Heemskerk, Idse, João Penedones, Joseph Polchinski, and James Sully. 2009. “Holography from Conformal Field Theory,” Journal of High Energy Physics 10, 079.
• Maldacena, Juan M. 1998. “The Large N Limit of Superconformal Field Theories and Supergravity,” Advances in Theoretical and Mathematical Physics 2, 231–252.