Solver Certificates and Independent Verification

A solver termination message is evidence about an algorithmic run, not by itself a mathematical certificate. Certification begins when primal or dual variables are exported with their convention, precision, scaling, and exact input hash, then evaluated by a separate implementation on the unscaled problem.

Required background. Precision, Convergence, and Numerical Error Budgets fix the refinement and tolerance plan. Automated Crossing-System Generation fixes canonical input and sector order. Helpful background. Forms, adjoints, and isometries clarify adjoints under basis changes.

Evidence cutoff: 2026-08-09. Solver formats and implementations are mutable; the release-aware numerical-bootstrap software landscape is reviewed in Rychkov and Su 2024, §II.C. The equations below define the durable verification contract; any concrete version must be named, hashed, and checked against its own documentation and serialized output.

Fix the conic convention

Use the primal form

$$A x=b, \qquad x\in K, \qquad \text{minimize }c^{\mathsf T}x,$$

and the dual form

$$A^{\mathsf T}y+s=c, \qquad s\in K^*, \qquad \text{maximize }b^{\mathsf T}y.$$

Then weak duality gives $c^{\mathsf T}x-b^{\mathsf T}y=\langle s,x\rangle\geq0$. Record whether the actual solver negates the objective or exchanges primal and dual labels. Report normalized unscaled residuals, for example

$$r_p=\frac{\lVert Ax-b\rVert_2}{1+\lVert b\rVert_2}, \qquad r_d=\frac{\lVert A^{\mathsf T}y+s-c\rVert_2}{1+\lVert c\rVert_2},$$

together with cone violations, objective gap, and complementarity. A small $r_p$ cannot compensate for a negative eigenvalue of $x$.

For pure feasibility $Ax=b$, $x\in K$, a dual vector satisfying

$$A^{\mathsf T}y\in K^*, \qquad b^{\mathsf T}y=-1$$

is an infeasibility certificate: any feasible $x$ would give $-1=\langle A^{\mathsf T}y,x\rangle\geq0$. This normalization makes the sign test exact. The polynomial-matrix primal and dual conventions used by SDPB, including residual and termination data, are derived in Simmons-Duffin 2015, §§2.2–2.5.

Independent evaluation

The verifier should not import solver memory or reuse its matrix assembly routine. It reads the canonical physical specification, conic input, certificate, scaling map, and hashes, then:

1. parses every number at higher precision than the serialized certificate;
2. reconstructs unscaled $A,b,c$ and checks input hashes;
3. recomputes $r_p,r_d$, cone eigenvalue bounds, objective gap, and complementarity;
4. evaluates bootstrap functional normalization and signs directly from frozen blocks;
5. proves or bounds positivity between samples and in the dimension/spin tail; and
6. records its own version, precision, rounding mode, and output hash.

Interval arithmetic can enclose scalar polynomial minima or matrix eigenvalues on compact intervals. Beyond a finite threshold, an analytic asymptotic bound or a rigorously enclosed recurrence must control the tail. Dense sampling is a useful adversarial test but does not close a continuum.

Strict, weak, and ambiguous termination

Strict primal or dual feasibility with margins much larger than verified residuals is robust. At a boundary, weak feasibility can make both the condition number and interpretation delicate. An “unknown,” stalled, or mutually inconsistent primal/dual result supports no exclusion. A certificate that passes the finite conic equations but fails the independently evaluated conformal-block positivity supports only the conic statement, not the CFT claim.

The figure below separates these layers. Inspect the final branch: only the verified normalization, residual, PSD, interval, and tail conditions reach the conditional exclusion.

Schematic solver-to-certificate path for a finite bootstrap problem. Versions, precision, basis scaling, sector order, and hashes are fixed inputs. Failure of any independently recomputed equality, PSD condition, interval bound, or tail estimate terminates the physics claim.

The structured equivalent is:

Check	Serialized input	Verified quantity	Passing condition	Failure consequence
Identity	physical and conic hashes	byte-for-byte correspondence	all named hashes match	wrong problem
Equalities	$A,b,c,x,y,s$	$r_p,r_d$ after unscaling	below declared tolerances with margin	no finite certificate
Cone	primal/dual blocks	enclosed minimum eigenvalues	nonnegative within outward-rounded bound	feasibility not established
Gap	objectives and scaling	duality gap/complementarity	consistent with claim and tolerance	optimum or boundary uncertain
Functional	frozen blocks and coefficients	normalization and sector actions	exact sign convention holds	no bootstrap exclusion
Continuum	intervals and tail data	global positivity	all intervals and tails enclosed	sampled result only

A reproducible calculation should test the exact rational cone before a generalized-free crossing fixture at several derivative orders and precisions. This page claims no calculation execution or tolerance result.

Failure tests

Scaling test. Verify residuals before undoing the solver’s row and column scaling. The result must be rejected as incomparable to the physical problem.

Serialization test. Round one functional coefficient to ordinary double precision. The higher-precision verifier must expose any lost sign margin.

Tail test. Pass every sampled point but insert a negative polynomial interval above the last sample. Continuum certification must fail.

The verified finite statement can now feed Single-Correlator Bounds or Numerical Boundary and Defect Bootstrap, with their additional physical assumptions.

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