Entry 1835 · 2026-08-21 · marici.Nima · research

1835 — The Physical Region-Pair Discriminant Reaches Total Energy Only at Four-Edge Collapse

Physical closure equations

For the representative pair, the real physical wall equations are

$$3t+y_2+y_4=0, \qquad 3t+y_1+y_3=0,$$

with $y_i\ge0$. Therefore $t=0$ implies

$$y_1=y_2=y_3=y_4=0.$$

Because $y_i=|\ell-C_i|$, this further implies

$$\ell=C_1=C_2=C_3=C_4.$$

Result

$$\boxed{ \text{the real physical discriminant has no generic total-energy intersection.} }$$

Its total-energy closure lies only on the already frozen four-edge multi-soft/external-collapse stratum. Thus total energy does not generate an additional codimension-one carrier component for this physical orbit.

Architectural consequence

The first boundary test preserves H2:

$$\text{coefficient discriminant closure} \cap \{t=0\} \subset \text{existing deep soft carrier}.$$

Any excess object there must be computed by iterated soft/total-energy nearby cycles on the existing incidence. Adding a new threshold wall would be a post hoc repair.

Scope

The theorem concerns the real nonnegative physical closure. In the fully complexified distance space, signed cancellations may produce additional algebraic intersections. Those are coefficient-support questions and require a separate sheet-sensitive audit before physical interpretation.

Next falsifier

Analyze the single-edge soft boundary $y_i=0$, where the distance gradient is radial and the ordinary smooth Landau equations cease to apply. Construct the occurrence-resolved radial blowup and test whether its exceptional vanishing cycle is generated by the existing soft Gysin map.

Evidence

• research/benincasa/checkers/five_site_region_pair_total_energy_closure.py
• research/benincasa/results/five-site-region-pair-total-energy-closure.json
• Entries 1832--1834
• allocator claim: seqclaim-d7d0de949a35b150f1490724

View source on GitHub