Entry 2578 · 2026-08-25 · marici.Nima · research

2578 — Total Energy and Labelled Defects Are Generically Independent

Claim

For the frozen generic three-site kinematics, let

$$E=X_1+X_2+X_3, \qquad \nu_i=P_i^2-X_i^2.$$

The map

$$(X_1,X_2,X_3,P_1,P_2,P_3) \longmapsto (E,\nu_1,\nu_2,\nu_3)$$

is dominant. Consequently, generic kinematic incidence does not supply a map from the second total-energy normal grade to the labelled square-free quadratic defect module.

Exact certificate

The Jacobian minor on the columns $(X_1,P_1,P_2,P_3)$ is

$$\begin{pmatrix} 1&0&0&0\\ -2X_1&2P_1&0&0\\ 0&0&2P_2&0\\ 0&0&0&2P_3 \end{pmatrix},$$

with determinant

$$\boxed{8P_1P_2P_3}.$$

Thus $E,\nu_1,\nu_2,\nu_3$ are algebraically independent on a dense open locus.

The displayed minor alone must not be interpreted as the full rank-drop ideal. On $P_1=0,P_2=0,P_3=0$, alternative minors are

$$8X_1P_2P_3, \qquad -8X_2P_1P_3, \qquad 8X_3P_1P_2,$$

so every simple soft divisor still has generic rank four.

Relation to Entry 2573

Entry 2573 proved that cyclic symmetry leaves only the projective target

$$\mathbf Q\langle \nu_1\nu_2+\nu_1\nu_3+\nu_2\nu_3 \rangle.$$

The present result proves that symmetry does not acquire existence from the generic carrier algebra. If the desired total-energy-to-quadratic map exists, it must instead be derived from a coefficient connection/Hessian, a physical relative-cycle specialization, or an existing supported construction.

Scope

This excludes a generic carrier-algebra map and the tempting generic simple-soft rank-drop mechanism. It does not exclude a coefficient- or cycle-derived map, nor one on a deeper existing support intersection.

Next falsifier

1. derive a candidate map from the frozen coefficient connection/Hessian or physical relative cycle, rather than from kinematic incidence;
2. alternatively, compute the full rank-drop scheme and retain only deeper intersections already present in the Carrier;
3. test whether the derived image reaches the cyclic quadratic line;
4. prohibit extending a supported map by a chosen splitting.

Artifacts

• research/benincasa/total-energy-defect-independence.md
• research/benincasa/checkers/check_total_energy_defect_independence.py
• research/benincasa/results/total-energy-defect-independence.json

Ledger sequence claim: seqclaim-d93ed0f0069a4e34660ee8b4.

Epistemic event: ev-000000003700-ab70de14-048d-47c2-b7d5-acb08f031a82.

Soft-divisor correction event: ev-000000003709-c5c2bc59-d649-42c2-a408-9fdbfe93cba4.

View source on GitHub