Does positivity give only bounds, or does it sometimes give values?

The tension. The bounding-mechanism table says positivity gives inequalities and never values. But the Tsirelson run produced exactly 2√2 — a value. Either the table is wrong or "bound" and "value" coincide under some condition, and the condition was not stated.

The test. Delsarte's linear-programming bound for kissing numbers: optimise over Gegenbauer coefficients subject to a sign condition on the forbidden arc. Finite-dimensional, and the true answer is known in several dimensions, so correctness is checkable.

Resolution. Positivity does not produce values. It produces bounds that sometimes coincide with values, and the coincidence is a property of the dimension, not of the method. Tight at d = 2 and d = 8 — and, in the literature, at d = 24 — loose by 6–16% everywhere between. The same exceptional pair where the sphere-packing proofs go through.

And two of this project's mechanisms meet on one number. 240 is reachable as a count (dim M₄(SL₂(ℤ)) = 1 forcing the theta series) and as an attained positivity bound (an LP over Gegenbauer coefficients). Two independent instruments, one answer, in the one dimension where both work. First instance of that convergence.

Two methodological notes. The LP was written backwards first — maximising where it should minimise — and returned inf rather than a plausible wrong number, so the error announced itself. And SCS ran five minutes without converging on a problem HiGHS solved in under a second: it is a linear program, and it was handed to a conic solver.