Numerical bootstrap bounds and truncation control

Fix spacetime dimension, external operators, internal symmetries and correlation functions. State any assumed gaps, parity sectors, conserved currents and uniqueness conditions. These assumptions define the allowed exchanged operators and their nonnegative coefficient structure. For a single identical real scalar, crossing is a vector equation after selecting a set of derivatives or other linear measurements. Mixed-correlator problems generally replace scalar nonnegative coefficients by positive semidefinite matrices and require corresponding matrix constraints.

A linear functional acts on the crossing vectors. If it is positive on the fixed identity contribution and nonnegative on every allowed exchanged contribution, applying it to the crossing sum gives a strictly positive number that cannot equal zero. This separates the proposed spectrum from the crossing cone. The sign convention can be reversed consistently, but identity and exchange signs must be checked together. A functional verified only at a few sampled dimensions does not prove positivity between those samples.

Using finitely many derivatives restricts the space of functionals one searches. For exact blocks and certified positivity over the full allowed spectrum, a functional in that smaller space can still establish a valid exclusion; failing to find one is weaker evidence. Increasing derivative order can strengthen ideal exact bounds. Actual computations also approximate blocks, use finite precision and select spin treatments, so observed monotonic convergence must be checked rather than assumed from the ideal argument.

A finite spin list without a valid remaining-spin treatment leaves a gap in a claimed universal exclusion. Polynomial or rational block approximations used in semidefinite programs also need accuracy control on the relevant dimension domain. Primal and dual residuals, positivity margins, repeated precision settings and stable changes of derivative order are practical diagnostics. They do not automatically make an output mathematically certified unless approximation errors and all omitted sectors are rigorously bounded.

A narrow allowed island can strongly constrain a known theory when multiple correlators and justified gaps are included. Its shape remains conditional on those assumptions and on the numerical treatment. An extremal functional can suggest approximate operator dimensions, but extracting them is an additional inference rather than an existence theorem. A complete local theory requires compatible correlators and operator products beyond a finite collection of necessary constraints, even if those constraints are solved to impressive precision.

The exact arithmetic of a small cone example makes the exclusion logic visible without pretending to run a CFT computation. By contrast, a finite feasible toy system illustrates only that selected equations have a solution. The optimization lab solves gradient descent for one quadratic objective; convergence there says nothing about conformal blocks, dual certificates or a physical theory. Neither a smooth displayed trajectory nor a tiny objective value can replace verification of the actual infinite positivity conditions.