Precision, Convergence, and Numerical Error Budgets

A numerical-bootstrap error budget is a set of controlled approximation axes tied to the reported observable, not a single solver tolerance. Derivative order, spin coverage, block approximation, arithmetic precision, conic residuals, parameter-search resolution, and basis conditioning can move a bound in different directions and must be varied independently.

Required background. Block Approximations and Semidefinite Programs identify the finite representations being refined. Helpful background. Complete lattice error budgets provide useful general principles for separating systematic and numerical effects.

Define the observable and acceptance rule first

Let $B(\Lambda,\ell_{\max},n_{\rm pole},p,\tau)$ denote a bound extracted at derivative order $\Lambda$, explicit spin cutoff $\ell_{\max}$, block or pole order $n_{\rm pole}$, arithmetic precision $p$, and solver tolerance settings $\tau$. A convergence study chooses a preregistered grid and an acceptance rule such as

$$\max_{c\in\mathcal C_{\rm refined}} \lvert B(c)-B(c_*)\rvert\leq\varepsilon_B,$$

where $c_*$ is the reference configuration and $\mathcal C_{\rm refined}$ changes at least two independent axes. This is an empirical envelope for the tested family, not a proof of the infinite-cutoff limit.

Nonmonotonic sequences are normal near basis changes, topology changes in an island, or marginal feasibility. Do not discard them to force a smooth extrapolation. Report the sequence, explain the chosen envelope, and test whether changing the fit window changes the conclusion. The distinct cutoff, precision, conditioning, and solver controls used in modern computations are reviewed in Rychkov and Su 2024, §§II.B–II.D.

Separate the error sources

Source	Diagnostic	Control	What it does not control
Functional truncation	repeat at several $\Lambda$	stable exclusion or bound sequence	block-table accuracy
Spin truncation	enlarge explicit spins and alter tail threshold	stability plus tail positivity	finite-$\Delta$ interpolation
Block approximation	increase radial/pole/polynomial order	direct-block residual on a fixed domain	solver roundoff
Arithmetic	use a precision ladder	stable residuals and objective	wrong algebraic input
Solver termination	tighten primal/dual criteria	independently recomputed residuals	continuum positivity
Search mesh	refine and shift parameter samples	stable boundary interpolation and topology	certificate at unsolved points
Conditioning	change scaled bases	invariant unscaled result	omitted physical sector

An output uncertainty should be expressed in the units of the observable. A primal residual of $10^{-30}$ is not an uncertainty of $10^{-30}$ in a scaling dimension. The map from residuals to a bound displacement can be ill-conditioned and should be measured by controlled perturbations or bracketed feasibility searches. The underlying polynomial-matrix residuals and solver tolerances are defined in Simmons-Duffin 2015, §§2.4–2.5.

A synthetic refinement design

For the fixed generalized-free-boson fixture, a prospective study can use derivative orders $4,8,12,16$, arithmetic precisions $64,128,256$ bits, two block orders, and two spectrum-positivity representations. The exact correlator and OPE data provide an independent target. No result is asserted until all configurations, certificates, and a direct residual evaluator exist.

The acceptance sequence should include adversarial injections: a stale block table, one sign-flipped coefficient, insufficient printed precision, and a tail cutoff that omits a known positive contribution. A test suite that only reruns successful settings measures repeatability, not robustness.

Numerical certificate and provenance table

The following semantic record states what can be established now and what must remain conditional.

Problem	Approximation	Solver build	Precision	Certificate	Residual	Error budget	Input hash	Output hash	Independent rerun	Evidence cutoff
Rational cone $v(x)$ at $x=0,1,2$	None; exact finite generators	Exact rational evaluator	Exact	$\alpha=(1,-2,1)$, $\alpha(t_{\rm out})=-1$	Algebraically zero on serialized equalities; minimum generator action $0$	No truncation	Required on canonical rational input	Required on evaluation record	Must reproduce exact signs	Stable mathematical fixture
One-dimensional generalized-free boson	Proposed $\Lambda=4,8,12,16$ and two block/tail representations	Version and source revision required	$64,128,256$ bits proposed	Not yet produced	Not yet evaluated	Must separate functional, block, tail, precision, and solver effects	Required	Required	Required before any numerical claim	No executed result as of 2026-08-09
Dated phenomenological benchmark	Absent without a named source-specific dossier	Not selected	Not selected	None	None	Cannot be assigned	None	None	None	No reproduction claimed as of 2026-08-09

The first row is an exact algebraic check. The other rows are acceptance specifications, not completed computations.

Stopping and reporting

A defensible report includes the full refinement table, not just its finest row; unscaled primal and dual residuals; feasibility brackets rather than a single boundary point; the interval or tail positivity method; search-grid topology; and the criterion that ended the study. If the conclusion changes under a refinement, the claim remains provisional even when the solver converges at every setting.

Failure tests

One-axis test. Increase precision while keeping an inadequate spin tail. Apparent digit stability does not establish convergence.

Shared-code test. Compare two block tables produced by the same erroneous recurrence. Add a direct hypergeometric or radial-series evaluator.

Boundary test. Move a parameter by the stated error bar and rerun feasibility. If the classification is unstable, widen the envelope.

Continue to Solver Certificates and Independent Verification for the residual semantics.

References

• Rychkov, Slava, and Ning Su. “New Developments in the Numerical Conformal Bootstrap.” Reviews of Modern Physics 96 (2024): 045004. 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