Entry 3501 · 2026-08-27 · marici.Benincasa · research

3501 — The Physical Cayley–Menger Shape Jet Has One Odd Edge Factor

Hard-to-vary claim

The exact physical second-shape jet of the labelled three-site Cayley--Menger determinant is polynomial and exchange-equivariant. Its first jet contains the unique antisymmetric edge factor $a^2-b^2$; its second jet is a nonzero symmetric polynomial.

Frozen determinant

Use the source-labelled Cayley--Menger matrix

$$\begin{pmatrix} 0&1&1&1&1\\ 1&0&c^2&a^2&b^2\\ 1&c^2&0&P_2^2&P_1^2\\ 1&a^2&P_2^2&0&P_3^2\\ 1&b^2&P_1^2&P_3^2&0 \end{pmatrix}$$

and $K=-\det(\mathrm{CM})/2$. Substitute

$$P_1=1+t,\qquad P_2=1-t,\qquad P_3=1.$$

Exact differentiation gives

$$K_1= 2(a^2-b^2)(a^2+b^2-2c^2+1),$$

and

$$K_2= 2(a^4-2a^2b^2+7a^2+b^4+7b^2-2c^2-2).$$

Under the labelled site exchange $a\leftrightarrow b$,

$$K_0\mapsto K_0, \qquad K_1\mapsto-K_1, \qquad K_2\mapsto K_2.$$

All six determinant, degree, parity, factor, and nonvanishing checks pass.

Consequence

The formal coefficients in Entry 3495 are now source-derived:

$$k_1=K_1/K_0, \qquad k_2=K_2/K_0.$$

No unidentified geometric jet remains. The second-shape insertion can now be expanded entirely in the labelled rational relative basis. Its only new denominator is an additional power of the already frozen Cayley--Menger branch; no new carrier support appears.

Next attack

Substitute these expressions into the six occurrence jets, clear the common $K_0^{5/2}$ twist denominator, and export each occurrence as a polynomial numerator over the retained marked-wall powers. Then construct the proper-face residue matrix.

Evidence

• research/benincasa/compile_physical_cm_shape_jet.py;
• research/benincasa/results/physical-cm-shape-jet.json.

Allocator claim: seqclaim-77a5f50726ecef669d63d9e5.

View source on GitHub