Certification in the Numerical Conformal Bootstrap

Numerical conformal bootstrap results range from reproducible high-precision computations to formal, machine-checkable proofs, and those categories should not be conflated. A dual functional can certify infeasibility of a finite semidefinite program; an end-to-end certificate for an infinite-dimensional CFT bound must additionally control conformal-block approximation, spin and derivative truncations, rounding, solver residuals, and any scan used to define an island.

Evidence cutoff. 11 August 2026.

Required background. Solver certificates and independent verification defines primal and dual evidence; precision, convergence, and error budgets supplies truncation tests. Helpful background. Benchmark reproduction and provenance supplies clean-room checks, block approximations and semidefinite programs supplies the finite reduction, and optimization from crossing fixes the logical meaning of exclusion.

Certificates for bounds, islands, and spectra

Normative question. Which bounds, islands, and inferred spectra have solver-independent certificates and reproducible control of truncation and precision?

The scope is numerical bootstrap based on linear or semidefinite optimization. “Solver-independent” means that a small verifier, separate from the search solver, can check a preserved witness against a frozen problem specification. Re-running the same executable at higher precision is valuable convergence evidence but is not, by itself, an independent formal certificate.

Four layers of a defensible claim

Layer	Required artifact	What it certifies
Crossing specification	Correlators, representations, normalizations, gaps, block conventions, derivative basis, spin treatment	The physical assumptions and finite optimization problem
Analytic reduction	Error bounds for block rational approximations, pole truncation, asymptotic spin tail, and domain	That the finite problem safely represents the declared infinite constraints
Optimization witness	Dual functional or primal feasible data with sufficient precision and residual margins	Exclusion or feasibility of the finite problem
Independent verification	Versioned input, witness, exact or interval evaluation, hashes, and a separate verifier	Reproducibility without trusting the original solver’s control flow

SDPB made arbitrary-precision primal–dual semidefinite programming practical and records residuals and feasibility diagnostics (Simmons-Duffin 2015); its later distributed implementation scales the same class of problems across many nodes (Landry and Simmons-Duffin 2019). Its output can be a strong witness. Floating-point “dual feasible” status still depends on margins relative to rounding and on the correctness of the generated polynomial matrices. A certificate that checks only the solver’s final matrices proves the discretized optimization result, not the conformal-block tail used to create those matrices.

For a one-sided bound, a verified dual functional with strict positivity supplies a natural certificate. An island is more demanding: the complement of a region must be covered by exclusions, the search mesh must not leave holes, and nonlinear assumptions such as operator uniqueness or OPE-angle parametrization must be included. The high-precision three-dimensional Ising island of Kos and collaborators is supported by systematic increases of derivative order and agreement with other methods (Kos et al. 2016). Its published uncertainty is a carefully tested numerical inference, not an interval-certified theorem about the untruncated crossing problem.

Spectrum inference has a lower claim ceiling. Extremal functionals and navigator minima can reconstruct candidate dimensions and OPE coefficients near a boundary, but near-degenerate operators, Karateev et al. 2019, fake-primary effects, finite derivative order, and movement of the extremal solution can produce apparently stable levels. Navigator functions improve continuity and optimization diagnostics (Reehorst et al. 2021); they do not turn identification of a known model into a theorem of existence or uniqueness.

Failure modes and independence

Material failure modes include insufficient precision, loss of strict dual feasibility, block poles or spin tails outside the certified domain, inconsistent normalization, hidden warm-start state, an incomplete island scan, and repository drift. Two runs using the same block generator and solver share the most consequential code path. Stronger independence comes from a second block implementation, a minimal witness verifier, exact rational or directed-rounding arithmetic, and a clean-room reproduction from a frozen specification.

Assessment. The field has strong reproducible numerical evidence and partial formal certification. Many landmark bounds and islands show stable convergence and can be reproduced from public tools. End-to-end, solver-independent certificates that cover the analytic reduction as well as optimization remain uncommon, and inferred spectra generally do not support the same claim strength as exclusion bounds.

What would count as certification?

For a bound, publish the exact assumptions, generator and solver versions, input hashes, a strict dual witness, and a compact verifier using exact or interval arithmetic; prove positivity between sampled points and beyond the finite spin or pole cutoff. For an island, additionally publish a verified cover of its complement. For spectrum claims, provide stability under independent correlator systems and held-out crossing equations, and distinguish a candidate spectrum from a proved unique CFT. A reproducible failure of any witness margin under directed rounding would retract the formal-certificate claim while leaving ordinary numerical evidence to be reassessed.

Related routes are conformal field theory and bootstrap, analytic and numerical conformal bootstrap, the three-dimensional Ising benchmark, and from crossing solutions to actual CFTs.

Evidence cutoff and source selection

The finite set includes SDPB, a landmark mixed-correlator island, navigator methods, a documented block pathology, and reproducibility standards. Targeted arXiv, journal, software, and citation-chain searches covered public evidence through 11 August 2026. “Rigorous” in a paper’s numerical terminology was not upgraded to an end-to-end formal certificate without a separately checkable witness and approximation bounds.

References

• Karateev, Denis, Petr Kravchuk, Marco Serone, and Alessandro Vichi. “Fermion Conformal Bootstrap in 4D.” Journal of High Energy Physics 2019, no. 6 (2019): 088. arXiv.
• Kos, Filip, David Poland, David Simmons-Duffin, and Alessandro Vichi. “Precision Islands in the Ising and $O(N)$ Models.” Journal of High Energy Physics 2016, no. 8 (2016): 036. DOI.
• Landry, Walter, and David Simmons-Duffin. “Scaling the Semidefinite Program Solver SDPB.” arXiv:1909.09745 (2019). arXiv.
• Reehorst, Marten, Slava Rychkov, David Simmons-Duffin, Benoit Sirois, Ning Su, and Balt van Rees. “Navigator Function for the Conformal Bootstrap.” SciPost Physics 11 (2021): 072. DOI.
• Simmons-Duffin, David. “A Semidefinite Program Solver for the Conformal Bootstrap.” Journal of High Energy Physics 2015, no. 6 (2015): 174. arXiv.