Automatic Midline Decomposition

Abstract

Half-density unitarity and self-resonance select the same automatic midline.

Theorem 1.1 (Half-density unitarity is equivalent to self-resonance).

$$\begin{array}{c}\forall A,M:A\to \mathbb{R},\alpha \in \mathbb{R},s\in \mathbb{C},\\ (\forall a,0\le M(a)),(\exists a,M(a)\ne 0)\\ \Rightarrow (\forall a,|\mathrm{halfDensityCoefficient}(M,\alpha ,s,a)|=1)\\ \iff \mathrm{KernelResonant}(\alpha ,s,s).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Midline/AutomaticMidlineDecomposition.half_density_unitarity_iff_self_resonance (✓ std3). ∎

Source. Repository-derived.

Commentary.

For a nonnegative, nontrivial heat spectrum, coordinatewise unit modulus after half-density normalization is equivalent to self-resonance. Both conditions independently characterize the real line at alpha over two.

The proof is a thin wrapper over the exact half-density and resonance characterizations in the universal heat-trace module.

This is a partial closure of the automatic-midline clause. The square-summability boundary, an independent reflection center, and the named analytic and quasicrystal instances remain open.

References

• Truth anchor: D5/S3/Midline/AutomaticMidlineDecomposition.half_density_unitarity_iff_self_resonance
• Dependency: D5/S3/Midline/UniversalHeatTrace