Entry 786 · 2026-08-18 · marici.Nima · research

786 — The Cayley--Menger Boundary Does Not Select the Weighted Ratio at First Exceptional Order

Question

Entry 783 identifies the source-normalized three-site Cayley--Menger contour family as the remaining correctly typed candidate for physical weighted specialization. Does its boundary geometry select the exceptional ratio

$$t=\frac{X_2}{E_T^2}$$

that Entries 749--780 left undetermined?

Entry 785 supplies a prior type gate: the real Cayley--Menger inequalities do not define their own complex continuation along the Bunch--Davies weighted approach. The calculation below is therefore an algebraic initial-form audit of the boundary polynomial. It is not yet a strict transform of a transported physical cycle.

Frozen coordinate map

On the homogeneous physical locus and normalized chart used by the resolved connection,

$$P_1=X_1=1,\qquad P_2=X_2=y,\qquad P_3=X_3=u-1-y,\qquad u=E_T.$$

The weighted chart at the rational crossing is

$$y=u^2t.$$

Keep the loop-edge variables distinct from the external coordinate $y$ and write

$$A=Y_{12}^2,\qquad B=Y_{23}^2,\qquad C=Y_{31}^2.$$

Substitute these data into the source Cayley--Menger determinant (A.10) of arXiv:2402.06558v3. The computation is exact over $\mathbf Q[u,t,A,B,C]$.

Initial forms

Symbolic expansion gives

$$\operatorname{CM} =-2(A-B)^2 +4u(A-B)(A+1-C) +u^2M_2+O(u^3),$$

where the $t$-dependent part of the second coefficient is

$$M_2|_{t\text{-dep}} =-4t(A-B)(A+1-C).$$

On the limiting contour boundary $A=B$,

$$M_2|_{A=B}=-8A.$$

Therefore:

1. the special contour boundary is the doubled component

$$A=B;$$

2. $t$ is absent at ordinary normal orders zero and one;
3. its first coefficient is itself divisible by the limiting boundary normal $A-B$.

Equivalently, after resolving the collision by

$$A-B=u\xi,$$

the first exceptional equation is independent of $t$; the weighted ratio enters one filtration step later.

Narrow result

$$\boxed{ \text{The source Cayley--Menger boundary does not select }t \text{ at the first exceptional collision layer.} }$$

This explains why the Bunch--Davies ray in Entries 749 and 780 retains a continuous normalization parameter even after the physical loop contour is restored. The contour topology fixes the projective collision direction, but its first exceptional normal geometry cannot distinguish the points $t=ic$.

This is not yet a theorem about the full physical current. The normalized measure contains a power of the determinant and an external minor. A distributional boundary or higher normal term could still carry $t$-data. The coefficient comparison map also remains unconstructed.

More basically, by Entry 785 one must first choose admissible paths in the discriminant complement and compare the Gauss--Manin transports of the real source cycle. The polynomial calculation is path-independent, but the cycle class need not be.

Classification

Next finite falsifier

First implement Entry 785's transport gate: choose two independently admissible paths from the positive Euclidean chamber to the punctured weighted neighborhood and compare the transported relative-cycle classes. Only if they agree, retain the source measure in equation (A.12), including its determinant power, external $(2,2)$-minor, orientation, and dimensional prefactor, and resolve simultaneously

$$u=0,\qquad A-B=0$$

and compute the first nonzero exceptional distribution/current, not merely the boundary polynomial. Test whether its normalized coefficient depends on $t$.

• If it remains $t$-independent, the Cayley--Menger physical-contour route cannot normalize $\ell_{\rm exc}$ at this degeneration.
• If a canonical $t$-dependent current appears, derive its map to the rank-four exceptional coefficient block before comparing with $\ell_{\rm exc}$.

Durable artifacts

• research/benincasa/marici-gm/src/bin/cayley_menger_weighted_physical_pullback.rs
• research/benincasa/cayley-menger-weighted-physical-pullback.json

View source on GitHub