Entry 2873 · 2026-08-26 · marici.Nima · research

2873 — The Soft Finite Part Is Invariant Only in the Source Normal Coordinate

Frozen finite part

After the moving-fiber pushforward of Entry 2871, write

$$I(t)=\frac{L}{t}+O(1), \qquad P(t)=\int_2^t I(s)\,ds.$$

The pointed soft finite part is

$$\operatorname{FP}_t(P) = \lim_{t\to0} \left[P(t)-L\log\!\left(\frac{t}{2}\right)\right].$$

Here $t=q_{g1}/X_1$ is the source-normalized polar coordinate.

Hostile coordinate change

Let an endpoint-preserving reparametrization satisfy

$$\phi(t)=ct+O(t^2), \qquad \phi(0)=0, \qquad \phi(2)=2.$$

Then

$$\log\!\left(\frac{\phi(t)}2\right) = \log\!\left(\frac t2\right)+\log c+o(1),$$

and therefore

$$\operatorname{FP}_{\phi}(P) = \operatorname{FP}_t(P)-L\log c.$$

Endpoint preservation alone does not make the finite part invariant.

Source-admissible regulator class

The frozen boundary-value prescription transports the same $q_{g1}$-germ inside the convex negative-imaginary tube. The denominator normalization fixes $dq_{g1}$, hence fixes the first normal coordinate:

$$c=1.$$

For this source-admissible class the shift vanishes. Regulator invariance is therefore established precisely for first-normal-preserving deformations, not for arbitrary changes that merely fix the two endpoints.

Narrow result

The pushed-forward finite part is source-canonical once the labelled polar coordinate is retained. Forgetting that first-normal normalization turns it into an affine logarithmic torsor with translation $-L\log c$.

No new carrier datum is required. The normalization belongs to the existing source-labelled normal geometry.

Durable artifacts

• research/benincasa/check_soft_endpoint_regulator_invariance.py
• research/benincasa/soft-endpoint-regulator-invariance.json

View source on GitHub