Extremal Functionals, Navigators, and Spectrum Reconstruction

Extremal functionals, primal solutions, and navigator objectives turn exclusion boundaries into candidate CFT data and efficient searches. They are inference tools, not existence theorems: reconstructed dimensions and OPE coefficients depend on extremality, degeneracies, basis choices, finite cutoffs, and the selected objective.

Required background. Mixed-Correlator Islands supply the conditional allowed region. Solver Certificates and Independent Verification supply primal-dual checks. Helpful background. Linear Functionals and Positivity supply the functional zeros used below.

Evidence cutoff: 2026-08-09. Navigator implementations and reconstructed spectra are research-sensitive. This page states method-level checks and exact synthetic fixtures; it does not assert a current spectrum estimate for a named interacting CFT.

Extremal zeros and primal data

At a smooth boundary point, an extremal functional $\alpha_*$ often obeys

$$\alpha_*[\mathbf F_{\Delta_i,\ell_i}]=0$$

on operators that enter a candidate boundary solution. Interior zeros in a continuous dimension sector are typically tangent zeros, so the derivative with respect to $\Delta$ also vanishes when the zero is nondegenerate. Endpoint zeros obey different conditions. The zero pattern depends on functional normalization and finite basis.

Given candidate locations, solve the truncated primal equation

$$\mathbf F_{\mathbf1} +\sum_{i=1}^{n}p_i\mathbf F_{\Delta_i,\ell_i} =\mathbf r, \qquad p_i\geq0,$$

and report the residual $\mathbf r$ in the original unscaled crossing basis. Mixed systems reconstruct PSD OPE matrices or vectors up to signs and rotations in degenerate subspaces. A small residual with an unstable operator list indicates overfitting to the finite basis rather than a stable spectrum.

The extremal functional method relates boundary zeros to candidate low-lying data and must be tested as cutoffs increase El-Showk and Paulos 2013, §§2–4.

Navigator functions

A navigator replaces a binary feasibility query by a continuous objective $\mathcal N(p)$ on parameter space. In one common convention,

$$\mathcal N(p)<0\ \text{in the allowed region}, \qquad \mathcal N(p)=0\ \text{on its boundary}, \qquad \mathcal N(p)>0\ \text{in the excluded region}.$$

The sign and normalization belong to the chosen deformation of crossing; record them. $\mathcal N$ is not a metric distance to the set of exact CFTs. Gradients can guide optimization and trace boundaries, but local minima, nonsmooth changes of active constraints, and objective dependence require multiple starts and direct certificate checks Reehorst et al. 2021, §§2–4.

Stability protocol

Track each candidate operator across derivative order, block order, spin coverage, precision, and nearby parameter points. Match operators by quantum numbers and a continuity criterion declared before looking at the result. Report zero multiplicity, singular values of the truncated primal system, positivity margins, and rotations within near-degenerate subspaces. Disappearing high operators are expected; a claimed low-lying datum must be stable within a stated envelope. The relationship among extremal, primal, and navigator methods is reviewed in Rychkov and Su 2024, §§III.C–III.D.

For a generalized-free-field fixture, exact exchanged dimensions and OPE coefficients provide a target. Deliberately truncate the tower, reconstruct low states, and verify that the residual decreases with the declared tail treatment. This tests reconstruction mechanics without asserting an interacting model.

The shared figure places navigator searches, extremal reconstructions, and benchmark reproductions in separate evidence classes. Compare their supported-claim boxes with the common boundary against model identity, existence, and uniqueness.

Schematic evidence classification for navigator and extremal methods. A navigator locates points relative to a chosen deformed feasibility problem; extremal and primal data reconstruct a cutoff-dependent candidate spectrum; benchmark reproduction checks a frozen target and environment. None proves uniqueness, model identity, or existence without further checks.

The seven visible columns have this semantic mapping:

Computed or reproduced object	Strongest supported claim	Not established by that object alone
Certified excluded point	The represented problem is infeasible at that point under the stated assumptions	Nonexistence of an exact CFT outside the controlled representation
Certificate-backed conditional bound	A finite upper or lower boundary in the declared represented problem	Realization of a CFT at the boundary
Kink or feature	A stable geometric feature after the stated refinements	Identification with a particular theory
Scan-delimited conditional island	A not-excluded component bounded by tested certificates and brackets	Realization of every interior point or uniqueness
Navigator objective and search	Position relative to the declared finite deformed objective in the searched domain	A physical distance to theory space or a unique model
Extremal reconstruction	Cutoff-dependent candidate dimensions and OPE data	An exact full spectrum or model identity
Benchmark reproduction	A frozen observable or certificate is reproduced within the declared tolerance	Correctness of every method or identification of a theory

The structured claim taxonomy is:

Claim class	Rigorous output	Interpretation	Assumptions	Convergence	Falsifier	Evidence requirement	Prohibited wording
Navigator sign	sign of a declared finite objective	location relative to its deformed feasible set	deformation and normalization fixed	repeated precision and cutoff settings	direct feasibility contradicts the sign	objective definition and saved solver records	geometric distance to an exact CFT
Navigator minimum	local minimum found by the stated search; global only with a separate guarantee	candidate region for further study	objective, domain, starts, optimizer	multiple starts and stable gradient	lower value or missed component	dated search record and direct feasibility checks	unique theory
Extremal zeros	zeros of a finite functional	candidate exchanged dimensions	extremal boundary and sector convention	stable positions and multiplicities	zeros drift or lose positivity	saved functional and cutoff sequence	exact operator spectrum
Primal reconstruction	nonnegative truncated solution with residual	approximate low spectrum and OPE data	selected zero set and degeneracy treatment	residual and low data stable across cutoffs	negative weights or ill-conditioned instability	primal-dual record and independent residual	proof of full crossing solution
Candidate model data	no theorem beyond finite feasibility and compatibility	possible identification	convention dictionary and identification hypothesis	several observables and methods agree	conflicting symmetry or operator data	current external evidence and dated reproduction	existence or uniqueness theorem

Failure tests

Multiplicity test. Treat a tangency zero as two distinct operators. The primal system should expose rank loss or unstable coefficients.

Objective test. Change the navigator deformation while keeping the physical feasible set. A moving minimum must not be called physical data without qualification.

Degeneracy test. Rotate a near-degenerate OPE subspace. Only invariant combinations should remain stable.

Finish with Benchmark Reproduction and Data Provenance before treating any reconstructed numbers as reproducible evidence.

References

• El-Showk, Sheer, and Miguel F. Paulos. “Bootstrapping Conformal Field Theories with the Extremal Functional Method.” Physical Review Letters 111 (2013): 241601. DOI. Open PDF
• 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. Open PDF
• Rychkov, Slava, and Ning Su. “New Developments in the Numerical Conformal Bootstrap.” Reviews of Modern Physics 96 (2024): 045004. DOI. Open PDF