Automated Crossing-System Generation

Automating a crossing system is safe only when the generator emits more than solver input. It must also emit a canonical, human-readable specification from which representations, tensor structures, permutation signs, positive-semidefinite sectors, and every numerical table can be checked independently.

Required background. Mixed correlators and symmetry sectors supply the coupled crossing structure. Block Approximations and Semidefinite Programs supply the numerical target format. Helpful background. Spinning operators and blocks explain basis and statistics data beyond scalar systems.

Evidence cutoff: 2026-08-09. Software examples and supported formats were checked through this date against the method survey in Rychkov and Su 2024, §II.C. The durable requirements below are format-independent; a later generator release must be revalidated rather than assumed compatible.

A declarative crossing specification

The source specification should contain, at minimum:

• spacetime dimension and Euclidean/Lorentzian convention;
• external primaries, global-symmetry representations, spins, parity, reality conditions, and two-point normalizations;
• an ordered basis of three- and four-point tensor structures;
• permutation matrices, statistics phases, and crossing orbits;
• exchanged representation sectors, spin parities, unitarity thresholds, gaps, and uniqueness assumptions;
• the order of OPE-vector components in every matrix-positive sector; and
• block normalization, evaluation coordinates, derivative basis, and approximation settings.

For scalar external operators $\phi_i$, a generated mixed system has the form

$$\sum_{\mathcal O\in R} \boldsymbol\lambda_{\mathcal O}^{\mathsf T} \mathbf F^{R}_{\Delta,\ell} \boldsymbol\lambda_{\mathcal O}=0,$$

where the vector order must be fixed globally. A permutation of external labels acts on both the tensor basis and OPE vector; transforming only one creates a crossing system that can look dimensionally correct while being false.

Algebraic checks before block evaluation

1. Projectors are idempotent, mutually orthogonal where appropriate, and sum to the identity on the declared product representation.
2. Permutation matrices obey the group relations, including fermionic signs.
3. Applying the crossing map around a closed permutation cycle returns the original basis.
4. Bose or Fermi symmetry removes exactly the expected structures and spins.
5. Conserved structures have the rank predicted after Ward identities and dimension-specific identities.
6. Every scalar or PSD contribution has the sign implied by reflection positivity in the chosen two-point basis.

The autoboot construction demonstrates how global-symmetry representations can generate mixed-scalar equations, but any concrete output remains tied to its supported input class and release Go and Tachikawa 2019, §§2–4. A generator does not remove the need for an independent small-system derivation.

Canonical serialization and hashes

Canonical output sorts sectors by a declared stable key, writes exact rational or symbolic group-theory coefficients without locale-dependent formatting, fixes matrix row and column order, and records the generator version plus source revision. Compute hashes separately for:

• the human-readable physical specification;
• symbolic projectors and crossing maps;
• block-generation inputs;
• numerical block tables;
• conic solver input; and
• any certificate or reconstructed spectrum.

A hash identifies bytes, not correctness. It becomes useful only after the hashed object has passed the relevant mathematical checks. Generated timestamps, absolute paths, and nondeterministic map ordering should not enter canonical scientific content.

The figure below shows how exact symbolic data and numerical approximations join. Inspect the independent-verification branch: it reads frozen artifacts rather than mutable in-memory generator objects.

Schematic route from representation and tensor-basis input to exact crossing, finite block data, a conic problem, and a bounded claim. Canonical ordering, versions, hashes, precision, and truncations accompany each edge; independent algebraic and numerical evaluators can stop the route when they disagree.

The same route in structured form is:

Generated object	Canonical content	Independent identity	Failure signal
Representation file	Irreps, product rules, projectors	Projector completeness and character count	Missing or duplicated sector
Tensor-basis file	Ordered structures and parity	Rank and permutation closure	Singular or incomplete basis
Crossing file	Exact matrices and statistics signs	Closed-cycle permutation test	Wrong sign after a cycle
Positivity file	OPE-vector order and PSD blocks	Reflection-positive quadratic form	Signed term placed in a PSD block
Block input	Convention, coordinates, derivatives, poles	Direct low-order block comparison	Normalization drift
Solver input	Scaled matrices and objective	Regeneration with matching hash	Nondeterministic bytes or sector order

Failure tests

Closure test. Remove one correlator from a permutation orbit. The generated equations must report that the system is not closed.

Serialization test. Randomize dictionary insertion order. Canonical output and hashes must remain unchanged.

Sign test. Exchange two identical fermions. A generator that omits the statistics phase must fail an exact cycle check.

Continue to Precision, Convergence, and Numerical Error Budgets and Solver Certificates and Independent Verification.

References

• Go, Mocho, and Yuji Tachikawa. “autoboot: A Generator of Bootstrap Equations with Global Symmetry.” Journal of High Energy Physics 06 (2019): 084. 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