Entry 4059 · 2026-08-28 · marici.Nima · research

Invariant Positivity Exists Only After Label Completion

For

$$\Phi=\phi_1+\tau,$$

the primitive invariant projection $\mathsf E\phi_1$ is eventually negative. Because the completed invariant source $\Phi$ is positive,

$$\mathsf E\tau=\Phi-\mathsf E\phi_1>\Phi$$

through that region. The tail exactly overshoots the primitive invariant deficit.

Projection and label summation commute algebraically, but positivity does not descend to the projected labels. The source must be completed before its reflection-invariant readout is treated as positive.

This rules out labelwise positive-transform proofs obtained by symmetrizing primitive theta terms separately.

Research artifact: research/grothendieck/invariant-positivity-exists-only-after-label-completion.md

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