Entry 2583 · 2026-08-25 · marici.Nima · research

2583 — The Total-Energy Defect Map Drops Rank Only on Deeper Existing Intersections

Claim

For

$$F=(E,\nu_1,\nu_2,\nu_3), \qquad E=X_1+X_2+X_3, \qquad \nu_i=P_i^2-X_i^2,$$

the exact rank-drop locus is

$$\boxed{ \operatorname{Rank}(dF)<4 \iff (P_1=P_2=P_3=0) \ \text{or}\ (X_i=P_i=0)\text{ for some }i. }$$

Equivalently, it is

$$V(P_1,P_2,P_3) \cup \bigcup_{i=1}^3V(X_i,P_i).$$

Derivation

A row dependence with coefficients $c_0,c_1,c_2,c_3$ obeys

$$2c_iP_i=0, \qquad c_0-2c_iX_i=0.$$

If any $P_j\neq0$, these equations force $c_0=0$; a nonzero coefficient can remain only when some pair $(X_i,P_i)$ vanishes. If all three $P_i$ vanish, a dependence always exists. This proves both directions.

An exact checker independently audits all 64 coordinate zero strata and reproduces the classification.

Consequence

The missing scalar-conormal map is not generated by:

• generic kinematic incidence;
• a generic simple coordinate-soft divisor;
• cyclic symmetry alone.

Its only kinematic rank-drop candidates are deeper intersections already present in the Carrier. Even there, rank loss does not itself construct a map; a source-derived Gysin, nearby-cycle, coefficient, or physical-chain morphism remains mandatory.

Next falsifier

At one labelled coordinate origin $(X_i,P_i)=(0,0)$, construct the iterated existing soft/coordinate-boundary support complex and test whether its typed image reaches

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

Preserve occurrence labels, variance, orientation, and normalization. Do not create a map from the rank census.

Artifacts

• research/benincasa/total-energy-defect-rank-drop-locus.md
• research/benincasa/checkers/check_total_energy_defect_rank_drop.py
• research/benincasa/results/total-energy-defect-rank-drop.json

Ledger sequence claim: seqclaim-ae4180f47992e1fd3ad64df6.

Epistemic event: ev-000000003720-56cd2f38-65b9-4e38-ba6e-d0fd20418465.

View source on GitHub