Positivity's two roles: does it produce data or remove a hypothesis?

Status: open · laid 2026-09-11 · type: derivation

Statement. Five results are commonly filed together as "positivity does the work." They split into two groups that do different things:

Positivity produces data. A positive geometry determines a canonical form; the Perron–Frobenius theorem yields existence and uniqueness of a leading eigenvector; direct methods in crystallography recover phase information that the measurement discarded. In each, positivity supplies a value or an object not otherwise available.

Positivity removes a hypothesis. Reflection positivity licenses the Wick rotation. And: for a positive-definite kernel, Mercer's expansion makes the summands λₙ|φₙ(x)|² non-negative, so Tonelli applies and the trace-class hypothesis is not needed — verified numerically (positive-definite kernel: eigenvalue sum and diagonal integral agree to 8 digits; indefinite kernel: both sides 0.1003 while Σ|λₙ| = 1.905, so absolute convergence must be assumed). Positivity does not supplement trace class; it is an independent second route to the same licence.

Friction. The two groups are filed as one phenomenon, and the second is the stronger claim: removing a hypothesis means that hypothesis was sufficient but never necessary, which is a statement about the theorem's actual content rather than about a technique that happens to work. If the split is real it should predict transfer — a technique from one group should not port to the other. Nothing yet tests that.

Question. Is the removal role always hypothesis-replacement (positivity as an alternative sufficient condition), or does it sometimes weaken a hypothesis without replacing it? And does the produce/remove split predict which instances transfer technique to each other? A satisfactory answer classifies all five and states the discriminator so a sixth instance can be placed before it is examined.