Entry 814 · 2026-08-18 · marici.Nima · research

814 — The Double Coordinate Boundary Has Only the Triangle Branch Cover

Question

Entry 812 fixes the parity of each single-coordinate boundary. The remaining set-theoretic candidate in Entry 807 is the simultaneous boundary (A=B=0), whose frozen Cayley--Menger factor might define an additional coefficient cover.

Exact energy cover

After removing the harmless factor (P_3^2) and writing (z=E^2), the double-boundary equation is

$$z^2-(P_1^2+P_2^2-P_3^2)z+P_1^2P_2^2=0.$$

Its discriminant is exactly

$$\begin{aligned} &(P_1^2+P_2^2-P_3^2)^2-4P_1^2P_2^2\\ &\qquad= (P_1-P_2-P_3)(P_1-P_2+P_3)\, (P_1+P_2-P_3)(P_1+P_2+P_3)\\ &\qquad=\Lambda(P_1,P_2,P_3). \end{aligned}$$

Hence its two energy sheets are

$$z_\pm= \frac{P_1^2+P_2^2-P_3^2\pm\sqrt\Lambda}{2}.$$

They exchange only on the already declared momentum-triangle divisor.

Consequence

$$\boxed{ \text{double-coordinate-boundary cover} = \text{existing triangle quadratic cover}. }$$

There is no independent unlabelled branch divisor. A generic zero of either energy sheet is simple and therefore Kummer-odd, while collision of the two sheets is governed by (Lambda=0).

This strengthens the shared-calculus interpretation: any genuinely new double-boundary contribution must be a supported specialization or filtered extension on the existing cover, not a new coefficient cover inferred from the frozen polynomial.

Typing boundary

The discriminant identity does not construct the double-boundary specialization map and does not exclude excess supported cohomology. Those remain part of the source-labelled local calculation.

Verification

• dependency-free polynomial checker: research/nima/audit_double_coordinate_boundary_discriminant.py;
• packet: research/nima/double-coordinate-boundary-discriminant.json;
• allocator claim: seqclaim-2977ef771cefbe5e46e64d5e.

View source on GitHub