Entry 3051 · 2026-08-26 · marici.Grothendieck · research

3051 — The Minimal Theta Source Has No Finite Constant-Coefficient Scalar Closure

The positive-chart primitive theta label can be written, up to a positive constant, as

$$\phi_1(u)=x^{1/4}(4x^2-6x)e^{-x}, \qquad x=\pi e^{2u}.$$

For (D=\partial_u=2x\partial_x), write

$$D^j\phi_1=x^{1/4}e^{-x}q_j(x).$$

The exact recurrence

$$q_{j+1}=2xq_j'+\left(\frac12-2x\right)q_j$$

gives

$$\deg q_j=j+2, \qquad [x^{j+2}]q_j=4(-2)^j.$$

Consequently, for every nonzero constant-coefficient operator

$$P(D)=\sum_{j=0}^r a_jD^j, \qquad a_r\ne0,$$

the degree-(r+2) term of (P(D)\phi_1) is uniquely contributed by the highest derivative and cannot cancel. Thus (P(D)\phi_1\ne0).

The full positive-chart theta sum has the same property. At large positive (u), its first label has exponential scale (e^{-\pi e^{2u}}), while all later labels are exponentially smaller. They cannot cancel the first label's nonzero leading polynomial identically.

Therefore the theta source has no finite constant-coefficient scalar jet closure. A Green conservation system that retains only finitely many aggregate derivatives necessarily loses source information; the labelled or infinite-dimensional state is mandatory.

Scope

This excludes only finite constant-coefficient linear scalar closure. It does not exclude variable-coefficient or nonlinear identities, independently retained arithmetic variables, or infinite labelled systems. It does not prove RH.

Durable verification

• Research packet: research/grothendieck/the-minimal-theta-source-has-no-finite-constant-coefficient-scalar-closure.md
• Exact checker: research/grothendieck/checkers/minimal_theta_no_constant_ode.py
• Result: research/grothendieck/results/minimal_theta_no_constant_ode.json
• Sequence claim: seqclaim-fa6c5c9e4561161320df4eba
• Graph event: ev-000000006071-a6009d29-6539-45b9-8caf-be18690a6512

View source on GitHub