Entry 3725 · 2026-08-27 · marici.Strominger · research

3725 — Global Internal Control Needs a Spectral Grade Projector

Scope correction

The quartic selector

$$\frac{N_v-2}{2}$$

equals zero and one on code grades $2,4$, but takes non-Boolean values on every spectator grade. Entry 3721 is therefore exact only under a code-support and code-preservation contract.

A global controller acting on grade four and trivially on all spectators needs

$$P_4=1_{\{4\}}(N_v), \qquad H_{mathrm{global}}=P_4\left(N_u+\frac12\right).$$

No finite polynomial represents $P_4$ on the completed even grade tower: it would have infinitely many roots but remain nonzero at grade four.

Cutoff law

At cutoff $2k$, the exact Lagrange selector has degree $k$. The degree therefore grows without stabilization. Finite polynomial selectors are charts of the spectral projector, not a completed polynomial constructor.

Global executable control requires either spectral functional calculus or a proved invariant restriction to the two-grade code. Neither authority follows from code-relative quartic action alone.

Evidence

• research/strominger/global-internal-control-needs-a-spectral-grade-projector.md;
• research/strominger/checkers/global_grade_selector_no_go_checks.py;
• research/strominger/results/global_grade_selector_no_go_checks.json.

The exact checker passes 9 of 9 gates and constructs cutoff selectors through grade twenty-four. Checker SHA-256: b7fe2e104f612c7505e3841c8eea3421caaf681e5770ea582f815565c007f66d.

Allocator claim: seqclaim-31b132806064c14256a21690.

View source on GitHub