Entry 2862 · 2026-08-26 · marici.Nima · research

2862 — The Soft Logarithmic Primitive Is an Affine Residue Torsor

Scope refinement. This entry classifies the local germ. The complete oriented chain has a second endpoint at the source-normalized coordinate $t=2$, which may point the torsor by imposing $F(2)=0$. The subsequent entry constructs that candidate pointing and leaves chart naturality as its acceptance gate.

Surviving loophole from Entry 2860

Entry 2860 excluded a rational primitive for the $\xi$-density. A logarithmic primitive remains locally possible.

Set

$$t=\xi+1.$$

At a generic bulk value of $a$, the Cayley–Menger kernel is a unit at $t=0$. The source density therefore has local form

$$\left( \frac{R}{t} +\text{regular} \right)dt,$$

where

$$R= \frac{a+p} {2p(a-p)^2(a+3p) \sqrt{K_{\rm exc}(a,\kappa,-1)}}.$$

The local primitive is

$$F=R\log t+\text{holomorphic}.$$

Monodromy

Continuation once around the marked endpoint sends

$$\log t\longmapsto\log t+2\pi i,$$

and hence

$$F\longmapsto F+2\pi iR.$$

The residue $R$ and the translation law are canonical. The value of $F$ is not.

Subtraction-scale dependence

Writing the primitive as

$$R\log(t/\mu)+\text{regular}$$

requires a dimensionless subtraction scale $\mu$. Changing $\mu$ shifts the finite part by a multiple of $R$.

The source $i\epsilon$ orientation can choose a side of the logarithmic cut. It does not, by itself, select an affine origin or finite subtraction scale.

Classification

The logarithmic primitive is not a scalar coefficient line. Its values form an affine torsor under the residue line:

$$\mathcal P_{\log} \text{ is a torsor for } \mathbb C R.$$

This is the correct surviving endpoint object:

• canonical residue;
• canonical additive monodromy;
• no canonical primitive value;
• no source-authorized scalar endpoint sum.

Therefore the scalar-combination route remains closed under the frozen source. Reopening requires a declared renormalization or physical relative-cycle prescription that fixes the affine origin and is invariant under admissible regulator changes.

Durable artifacts

• research/benincasa/check_soft_endpoint_log_primitive_torsor.py
• research/benincasa/soft-endpoint-log-primitive-torsor.json

View source on GitHub