From Crossing Equations to Convex Optimization

A numerical exclusion begins with a logical gate: the crossing equation must be expressible as a positive combination of known vectors or matrices. When that gate is open, convex separation converts a spectral hypothesis into a falsifiable feasibility problem. When coefficients are not positive—because the theory is nonunitary, the channel is not reflection-positive, or products of distinct OPE data occur without a PSD organization—the same optimization may be exploratory, but it is not a positive-cone certificate.

Required background. Crossing equations and positivity fix the CFT sign conditions. Convex cones, separation, and conic duality supply the separation theorem. Helpful background. Conformal Hamiltonian truncation illustrates a different finite approximation whose errors must not be conflated with functional truncation.

Crossing as cone membership

Fix the external data, identity normalization, symmetry sectors, and a trial spectral hypothesis $H$. Write

$$-\mathbf F_{\mathbf1} =\sum_{a\in H}p_a\mathbf F_a, \qquad p_a\geq0.$$

The right side is the cone $C_H$ generated by the allowed block vectors. If a real linear functional $\alpha$ satisfies

$$\alpha(\mathbf F_{\mathbf1})=1, \qquad \alpha(\mathbf F_a)\geq0\quad\text{for every }a\in H,$$

then applying it to the original crossing equation gives $1+\sum_a p_a\alpha(\mathbf F_a)=0$, a contradiction. This proves that no exact positive solution satisfying $H$ exists—provided the functional and its positivity on the full declared continuum have genuinely been established. The normalization may instead be $-1$ if every sign is reversed consistently.

Common hypotheses include a scalar gap, spin-dependent unitarity thresholds, global-symmetry representations, and assumptions that certain low operators are unique. These define the cone; they are not consequences of the computed bound.

Finite projection and its logical scope

Choose functionals $e_i$, for example derivatives at the crossing-symmetric point, and project each exact vector to

$$\mathbf f_a=(e_1[\mathbf F_a],\ldots,e_M[\mathbf F_a])\in\mathbb R^M.$$

Separation in $\mathbb R^M$ is exact for the projected vectors if their values and positivity are certified. It is generally weaker than separation in the full function space: increasing $M$ can exclude more trial spectra, but failure to exclude at finite $M$ is not evidence that an exact CFT exists. A numerical implementation adds block, spin-tail, polynomial, and roundoff approximations, each requiring its own check Poland, Rychkov, and Vichi 2019, §§VI.B–VI.C. Polynomial-matrix positivity and its conic formulation are made explicit in Simmons-Duffin 2015, §2.

Exact finite-cone check

Let $v(x)=(1,x,x^2)$ at $x=0,1,2$. The target $t_{\rm in}=v(1)$ is feasible with the exact nonnegative coefficient vector $(0,1,0)$. The target $t_{\rm out}=(1,1,0)$ is separated by

$$\alpha(y_0,y_1,y_2)=y_0-2y_1+y_2,$$

because $\alpha(v(x))=(x-1)^2\geq0$ at all three generators while $\alpha(t_{\rm out})=-1$. This rational example tests sign, normalization, and serialization before any conformal blocks enter.

Primal, dual, and boundary cases

The primal question asks for nonnegative spectral weights reproducing $-\mathbf F_{\mathbf1}$. The dual asks for a separating functional. Strict feasibility gives the cleanest alternative, but bootstrap optima often lie on cone boundaries where neither side has a generous interior. Solver labels such as “primal feasible” or “dual feasible” therefore need residuals, scaling information, and an independently evaluated certificate; near-boundary termination alone is not a theorem.

If a proposed decomposition contains signed coefficients, complex data without a real PSD reformulation, or nonlinear constraints on unknown dimensions and OPE coefficients, stop the cone argument. One may discretize and solve algebraic equations, minimize crossing residuals, or search a nonconvex objective, but the conclusion is then a finite exploratory solution, not a separation-based exclusion.

The diagram below shows which information must survive from crossing to the final claim. Inspect the branch where independent verification can reject a solver output even when termination was reported.

Schematic certification path from an exact positive crossing equation to a finite conic problem, solver output, independent residual and positivity checks, and a conditional physics statement. Every transformation carries its normalization, truncations, precision, and hashes; a solver status can terminate without producing a claim.

The structured equivalent is:

Stage	Mathematical object	Required declaration	Independent check	Maximum conclusion
Exact crossing	Infinite block cone	External data, sectors, gaps, positivity	Algebraic crossing and sign check	Well-posed hypothesis
Finite projection	Vectors or polynomial matrices	Functional basis and all truncations	Regenerate entries and bound approximation	Finite problem defined
Solver	Primal/dual conic variables	Scaling, precision, tolerances, version	Recompute residuals	Candidate certificate
Verification	Serialized certificate	Hashes and continuum domain	Interval/tail positivity and objective gap	Finite exclusion
Interpretation	Conditional CFT statement	Assumptions and convergence study	Remove assumptions or refine cutoffs	Bounded claim only

Failure tests

Sign test. Flip the identity normalization while leaving the functional normalization unchanged. The supposed contradiction disappears or reverses.

Continuum test. Verify positivity between interpolation nodes and beyond the largest explicitly treated spin or dimension. Sampled positivity alone leaves gaps through which a negative region can pass.

Nonunitary test. Replace the nonnegative coefficients by signed ones. The separating functional no longer excludes an algebraic crossing solution.

Continue to Linear Functionals and Positivity for functional construction and to Solver Certificates and Independent Verification for certificate semantics.

References

• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI. Open PDF
• Simmons-Duffin, David. “A Semidefinite Program Solver for the Conformal Bootstrap.” Journal of High Energy Physics 06 (2015): 174. DOI. Open PDF