Entry 2868 · 2026-08-26 · marici.Nima · research

2868 — The Pointed Soft Torsor Survives the Noncyclic Residue Transition

Frozen noncyclic transition

Use the exact labelled transition from the $q_{\mathcal G_{12}}$-residue chart to the $q_{\mathcal G_{31}}$-residue chart under the site transposition $\sigma_{23}$.

It fixes

$$X_1\longmapsto X_1, \qquad q_{g1}\longmapsto q_{g1}.$$

Therefore the normalized soft coordinate

$$t=\frac{q_{g1}}{X_1}$$

transports as

$$t\longmapsto t$$

with transition unit one.

Orientation

The retained fiber coordinates transform as

$$(a,b)\longmapsto(c',a')=(b,a).$$

With the source Poincaré-residue convention,

$$dc'\wedge da' \longmapsto db\wedge da = -da\wedge db.$$

Hence the coefficient primitive transforms by the derived orientation sign:

$$F\longmapsto-F.$$

The pointed origin is preserved:

$$F(2)=0 \longmapsto -F(2)=0.$$

The physical occurrence coordinate $a=y_{23}$ is fixed by this transition, so the positive-occurrence chamber covector is preserved as well.

Result

The chain-pointed logarithmic torsor descends through this noncyclic residue-chart transition. The exact quotient transport has full rank and its forward–reverse composition is identity.

This gate establishes that the affine pointing is not an artifact of the $q_{\mathcal G_{12}}$ residue frame. It does not yet establish compatibility with the complete $a$-dependent physical cycle, regulator invariance, or Leray-tube pairing.

Full atlas closure

Together with Entry 2866's certified cyclic atlas, this one transposition generates the full six-chart $S_3$ atlas. Writing

$$\rho=(123), \qquad \tau=(23),$$

the checker verifies

$$\rho^3=1, \qquad \tau^2=1, \qquad \tau\rho\tau=\rho^{-1}.$$

Thus the exact transposition packet, its inverse, and cyclic transport cover all noncyclic chart transitions without choosing additional fitted maps.

Durable artifacts

• research/benincasa/check_soft_endpoint_pointing_noncyclic_naturality.py
• research/benincasa/soft-endpoint-pointing-noncyclic-naturality.json
• research/benincasa/g12-g31-residue-chart-transition.json

View source on GitHub