Entry 2864 · 2026-08-26 · marici.Nima · research

2864 — The Opposite Soft Endpoint Points the Logarithmic Torsor

Corrected scope after the full fiber-cycle audit. The physical (a)-cycle moves with (t) and collapses at (t=2). Therefore the pointwise condition (F(a,2)=0) is not defined for generic fixed (a). The construction below is valid only for a scalar density already pushed forward along the moving (a)-cycle. The pushforward must precede the pointing.

Refinement of Entry 2862

The isolated logarithmic germ has no affine origin. The complete source chain contains additional data:

$$t=\frac{q_{g1}}{X_1}=\xi+1, \qquad t\in[0,2].$$

The unmarked endpoint is the source-normalized point

$$t=2.$$

It therefore supplies the candidate pointing

$$F(2)=0.$$

Pointed primitive

For a local density

$$f(t)=\frac Rt+\sum_{n\ge0}h_nt^n,$$

the uniquely pointed primitive is

$$F(t) = R\log(t/2) + \sum_{n\ge0} \frac{h_n}{n+1} \left( t^{n+1}-2^{n+1} \right).$$

Adding a constant violates $F(2)=0$, so the opposite endpoint removes the affine translation freedom.

After subtracting the canonical logarithmic term, the pointed finite part at the marked endpoint is

$$\operatorname{FP}_{t=0}F = -\sum_{n\ge0} \frac{2^{n+1}}{n+1}h_n.$$

Interpretation

The correct distinction is:

• the local marked germ determines an affine residue torsor;
• the complete oriented source chain can point that torsor;
• the pointing is unavailable if the opposite endpoint is forgotten.

Thus source normalization may supply more than branch orientation. The endpoint $t=2$ is a candidate affine origin derived before inspecting the desired scalar value.

Remaining acceptance gate

This is not yet a global physical scalar theorem. The pointed finite part must satisfy:

1. naturality under every source-derived occurrence and residue-chart transition;
2. compatibility with the full $a$-dependent relative cycle;
3. preservation under admissible regulator deformations;
4. agreement with the logarithmic residue/Leray-tube pairing.

Failure of any gate returns the object to an unpointed torsor. Passing them would produce a source-authorized renormalized bulk readout—not an arbitrary endpoint sum.

Durable artifacts

• research/benincasa/check_soft_endpoint_chain_pointed_torsor.py
• research/benincasa/soft-endpoint-chain-pointed-torsor.json

View source on GitHub