Entry 783 · 2026-08-18 · marici.Nima · research

783 — The Cayley--Menger Contour Family Is Source-Normalized Before Weighted Specialization

Question

Entries 781--782 close the parameter-space-thimble shortcut and show that the rational exceptional line

\ell_{\rm exc}=\mathbf Q\langle[0:1:0:-3]\rangle

has no canonical affine generator in the serialized master frame. Does the frozen primary construction contain a different source of physical normalization?

Source audit

The answer is narrower than either “yes” or “the source route is exhausted.”

The one-loop paper arXiv:2408.16386 defines the periods as pairings

I(X)=\int_{\Gamma_X}u\,\varphi

between twisted cocycles and fiber cycles. Its cited primary construction, Benincasa--Vazao arXiv:2402.06558v3, gives the three-site cycle and loop measure explicitly.

With

y_{12}=|\ell|,\qquad y_{23}=|\ell+P_2|,\qquad y_{31}=|\ell-P_1|,

the squared tetrahedron volume is the Cayley--Menger determinant of the six lengths

(y_{12},y_{23},y_{31},P_1,P_2,P_3).

Equation (A.11) defines the physical loop contour by the simultaneous minor inequalities

\Gamma_3(P)= \left\{ (-1)^{k+1}\operatorname{CM}(I_k,J_k)\ge0 \text{ for all prescribed minors},\ k=1,2,3 \right\}.

Its boundary contains

\operatorname{CM}(y^2,P^2)=0,

where the tetrahedron becomes coplanar. Equation (A.12) also fixes the loop measure, including the prefactor

\frac{2\pi^{(d-3)/2}}{\Gamma((d-3)/2)}

and the powers of the full determinant and its external $(2,2)$ -minor.

Thus the source supplies more than a homotopy class: it supplies an oriented semialgebraic fiber-cycle family, its boundary stratification, and a normalized measure.

Type distinction

This does not contradict Entry 781. The source still supplies no Lefschetz thimble in the external parameter base. Instead it supplies the incidence family

\mathfrak C_{\Gamma} = \{(P,y_{\rm loop}):y_{\rm loop}\in\Gamma_3(P)\} \longrightarrow \mathcal P_{\rm ext}.

This is a chain in the total space of the loop-variable family, not a chain in the external base alone. Its weighted nearby specialization is nevertheless a legitimate candidate for the missing physical source object, provided it is constructed before projection to the coefficient system.

What remains absent is the map

\operatorname{Sp}_{\rm wt}(\mathfrak C_{\Gamma}) \longrightarrow \mathcal E_{\rm exc}

into the rank-four exceptional coefficient block. Consequently neither its image in $\ell_{\rm exc}$ , its deck character, nor its scalar pairing has yet been computed.

Narrow result

\boxed{ \begin{gathered} \text{The parameter-space-thimble route is exhausted,}\ \text{but the source-normalized Cayley--Menger contour family remains a}\ \text{distinct, correctly typed candidate for weighted specialization.} \end{gathered} }

The normalization problem is therefore no longer “find a preferred rational generator.” It is:

\text{derive whether the normalized physical fiber-chain family maps to the projective extension line.}

No affine generator of $\ell_{\rm exc}$ is selected by the present audit.

Next finite falsifier

Freeze equations (A.7)--(A.12) and the weighted Rees charts of Entries 778--780. Then:

pull back the full incidence family $\mathfrak C_\Gamma$ , including every Cayley--Menger minor inequality;
compute its strict transform and exceptional boundary current chartwise;
retain the source prefactor and orientation;
derive the $\mu_2$ -action, trace, and overlap homotopy;
construct the coefficient comparison map without choosing $(0,1,0,-3)$ ;
test whether its image lies in $\ell_{\rm exc}$ .

If the exceptional image is transverse to $\ell_{\rm exc}$ , this physical activation route closes. If it lies in the line, the source measure supplies a candidate affine normalization whose regulator-hierarchy invariance must then be proved.

Evidence packet

research/benincasa/cayley-menger-contour-family-gate.json
arXiv:2408.16386, equations (6), (7), and (12)--(17)
arXiv:2402.06558v3, equations (3.6)--(3.10) and (A.7)--(A.12)

View source on GitHub