Entry 1450 · 2026-08-21 · marici.Benincasa · research

1450 — Resonance Adds One Rees-Torsion Class to the Kummer–Gysin Excess

Status

Exact logarithmic/Rees model of Entry 1448's resonant exceptional fiber. The calculation separates the generic flat line from the class that exists only at the resonant associated grade.

Rees parameter

Write the exceptional Kummer exponent as

$$\alpha=m+s, \qquad m\in\mathbb Z, \qquad R=\mathbb Q[[s]].$$

Meromorphically gauge away the integral factor $(t-1)^m$, while retaining its divisor as a labelled lattice modification. The remaining connection on

$$U=\mathbb P^1\setminus\{0,1,\infty\}$$

is

$$\nabla_s=d+s\,d\log(t-1).$$

Global logarithmic complex

Use the ordered logarithmic basis

$$e_0=d\log t, \qquad e_1=d\log(t-1).$$

The relevant global complex is

$$R \xrightarrow{\ \nabla_s\ } Re_0\oplus Re_1,$$

with

$$\nabla_s(1)=s e_1.$$

Therefore

$$\boxed{ H^1_{\log}(U,\nabla_s) \cong Re_0 \oplus \left(R/(s)\right)e_1. }$$

Interpretation

The first summand is the flat continuation of Entry 1448's generic rank-one Kummer--Gysin excess. The second is supported only at the resonant Rees fiber:

$$\begin{array}{c|c} s\ne0 & \dim H^1=1\\ s=0 & \dim H^1=2. \end{array}$$

Thus ordinary specialization at integral monodromy sees one extra class, but that class is not an independently lifting generic coefficient. It is $s$-torsion.

The base factor from Entry 1448 is

$$\delta^{m+s} = \delta^m e^{s\log\delta},$$

so the same Rees parameter records the logarithmic base monodromy. The integer $m$ remains part of the labelled lattice even though it is invisible in the meromorphic local-system isomorphism class.

Classification

$$\boxed{ \text{one generic Kummer--Gysin line} + \text{one resonant Rees-torsion class}. }$$

Both are coefficient data on the forced resolution of an existing partial-energy intersection. Neither requires a new carrier wall.

Epistemic boundary

This is the logarithmic de Rham/Rees object. It does not by itself prove that the torsion class is selected by a physical relative chain. The source positive chamber misses $X_1+y=0$, so any Betti activation requires analytic continuation to that existing support.

Next falsifier

Transport the source rapid-decay chain around $\delta=0$ and test whether its specialization pairs with the torsion generator $e_1$. A zero pairing makes the resonant class physically invisible; a nonzero pairing yields a supported logarithmic endpoint correction without changing the carrier.

Durable provenance

• Entry 1448's blowup chart $q=\delta t$;
• allocator claim seqclaim-0a991e508ef82edf13ad9ace.
• epistemic event ev-000000001541-7597dc7b-5b17-40d9-a141-9350050dd585.

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