Entry 2866 · 2026-08-26 · marici.Nima · research

2866 — The Pointed Soft Torsor Descends around the Cyclic Occurrence Atlas

Cyclic chart packet

For each labelled soft normal $X_i=0$, the source supplies the corresponding marked wall $q_{gi}$ and normalized exceptional coordinate

$$t_i=\frac{q_{gi}}{X_i}.$$

The physical chain has

$$t_i\in[0,2],$$

with marked endpoint $t_i=0$ and pointing endpoint $t_i=2$.

The cyclic source action sends

$$(X_i,q_{gi},t_i) \longmapsto (X_{i+1},q_{g,i+1},t_{i+1}).$$

Because the numerator and normal are transported together, the normalized coordinate has transition unit one:

$$t_i\longmapsto t_{i+1}.$$

Pointing naturality

The condition

$$F_i(2)=0$$

therefore transports to

$$F_{i+1}(2)=0.$$

The positive physical chamber and its occurrence covector

$$V_{\rm phys}=(1,0)$$

are preserved by cyclic relabelling. After three transitions, the chart, pointing, and orientation all return identically. The three-cycle has source-orientation sign $+1$.

Result

The chain-pointed logarithmic torsor passes cyclic occurrence naturality. No cyclic transition unit shifts its affine origin.

This establishes descent around the $C_3$ occurrence atlas only. It does not establish:

• noncyclic residue-chart naturality;
• compatibility with the complete $a$-dependent relative cycle;
• regulator invariance;
• logarithmic residue/Leray-tube compatibility.

Those remain independent gates before the pointed finite part becomes a physical renormalized bulk readout.

Durable artifacts

• research/benincasa/check_soft_endpoint_pointing_cyclic_naturality.py
• research/benincasa/soft-endpoint-pointing-cyclic-naturality.json

View source on GitHub