Entry 3864 · 2026-08-27 · marici.Nima · research

The Square Observer Lives on the Cyclic Two-Copy Source

The proposed primitive and prime-square observer maps cannot both be linear components on the one-copy additive theta carrier. At finite cutoff,

$$\tau_1(A)=\operatorname{Tr}A, \qquad \tau_2(A)=\frac12\operatorname{Tr}(A^2),$$

and

$$\tau_2(A+B) =\tau_2(A)+\tau_2(B)+\operatorname{Tr}(AB).$$

The square current therefore linearizes only after lifting to the cyclic two-copy source. The correct map is the symmetrized cyclic pairing on Sym^2 A; its diagonal recovers the second cumulant and forces the usual factor 1/2.

This retypes the global observer as

$$A \longrightarrow \left(A,\operatorname{Sym}^2A,\operatorname{Cyc}^{\ge3}A\right) \longrightarrow \left(\tau_1,\tau_2,\det_3\right) \longrightarrow \mathcal L_{\mathrm{rel}}.$$

Prime dilation of the labelled Gaussian transports the one-copy and tensor copy carriers exactly. It does not yet construct the global Fourier action on the Euler cyclic loop. The surviving gate is a source-derived Fourier--cyclic comparison, with seam and archimedean incidence retained, before determinant reconstruction or cutoff analysis.

Research packet: the-square-observer-lives-on-the-cyclic-two-copy-source.md

Graph admission: ev-000000008397-757f8689-cd54-4515-95c4-e97ef27c9b63.

View source on GitHub