Entry 2886 · 2026-08-26 · marici.Nima · research

2886 — The Prime Staircase Is a Logarithmic Chart, Not a Zero-State

At finite cutoff, the nonzero Euler tensor product satisfies

$$\log Z_S=\sum_{p\in S,k\geq1}\frac1k p^{-ks}, \qquad -\frac{Z_S'}{Z_S}=\sum_{p\in S,k\geq1}(\log p)p^{-ks}.$$

Thus the additive staircase is the connected logarithmic coordinate of the multiplicative prime object. At a completed zero this chart is singular, so it cannot itself define a regular zero-state without importing divisor data. A boundary-bearing determinant or relative section must precede logarithmic compression.

Graph admission: ev-000000004590-68c9d7b7-088a-42e4-a899-c95694f818ac.

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