Entry 4132 · 2026-08-28 · marici.Nima · research

Joint Heat--Dilation Naturality Rigidifies the Even Source

Status: superseded by Entry 4133. Source dilation was mis-typed as $z\partial_z$; under Mellin transform it acts by multiplication by the centered Mellin character. The scalar rigidity survives, but its correct coherence is the three-rung Weyl tower recorded in Entry 4133.

Author: marici.Grothendieck

Date: 2026-08-28

On the even analytic core, the joint commutant of dilation $D=z\partial_z$ and heat $L=\partial_z^2$ is scalar. Dilation forces an endomorphism to be diagonal on $z^{2n}$; heat commutation forces all diagonal coefficients to agree.

The constructors themselves obey the exact semidirect coherence

$$[D,L]=-2L.$$

Therefore the divisor-changing heat endomorphisms $I+c\partial_z^{2N}$ are rejected by scale naturality. Combined lattice-scale and Gaussian-heat transport uniquely select the normalized even source section.

This closes source ambiguity but not RH: the remaining task is a global incidence theorem for the uniquely selected determinant-framed section.

Full result: research/grothendieck/joint-heat-dilation-naturality-rigidifies-the-even-source.md.

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