Entry 1825 · 2026-08-21 · marici.Nima · research

1825 — Total-Energy Pair Orbits Are Existing Soft Nearby-Cycle Support

Frozen family

The source-compatible pair census contains five orbits involving the total energy wall:

$$(G,G^-_{e}),\qquad (G,g_A),\quad |A|=1,2,3,4.$$

On the homogeneous five-site slice,

$$G=5t.$$

Hence every such pair forces $t=0$.

Physical restriction

At $t=0$, the remaining wall restricts to a sum of nonnegative internal energies:

$$G^-_e\big|_{t=0}=2y_e, \qquad g_A\big|_{t=0}=y_i+y_j.$$

The edge pair therefore requires $y_e=0$, while every proper connected region pair requires $y_i=y_j=0$. No orbit reaches the open positive internal-energy domain.

Result

$$\boxed{ \text{all five total-energy pair orbits lie on existing total-energy/soft support.} }$$

They are nearby-cycle boundary intersections, not generic two-gradient physical loop pinches. This classification does not assert that their supported nearby cycles vanish; it says that they introduce no new open Landau carrier beyond the already frozen total-energy and soft strata.

Remaining pair frontier

Entries 1824--1825 remove nineteen of the 49 pair orbits from the generic physical-pinch search. The remaining thirty are region--region profiles and must be tested by their physical gradient equations before ambient elimination.

Evidence

• research/benincasa/checkers/five_site_total_energy_pair_physical_support.py
• research/benincasa/results/five-site-total-energy-pair-physical-support.json
• allocator claim: seqclaim-590257771a96e23169fa3b81

View source on GitHub