3124 — Linear Fourier Closure Has Five Components and Clark Precedence Prevents the Jet Tower
The earlier four-role Gaussian-tail description was affine. Reflection of the two tail orientations reconstructs the constant carrier, and Fourier exchanges that constant with the delta atom. The minimal linear closure has five typed components:
- a fixed Gaussian bulk line;
- a constant–delta exchange plane;
- a centered-tail/principal-value quarter-turn plane.
The Fourier matrix satisfies
Constructor order controls further prolongation. Native Clark spectral differentiation multiplies the source coordinate before tail completion and preserves the five-type cell. Reversing the order creates a linear left asymptote and a delta-prime Fourier jet. The braid residual is therefore a boundary-capability change, not a scalar anomaly.
Scope
The result supersedes the claim that four components alone form the linear Fourier closure. It does not yet establish primitive or prime-square incidence on the five-component extension.
Durable verification
- Linear-cell packet: research/grothendieck/the-linear-fourier-tail-cell-has-five-components.md
- Precedence packet: research/grothendieck/clark-before-tail-closes-the-five-cell-the-reverse-order-creates-jets.md
- Matrix checker: research/grothendieck/checkers/check_linear_fourier_tail_five_cell.py
- Order checker: research/grothendieck/checkers/check_clark_tail_order.py
- Sequence claim: seqclaim-857b2457431fcc41009a7ab9
- Graph event: ev-000000006381-355aa63a-0db7-4465-8108-48ee45cafcb6