Entry 366 · 2026-08-17 · marici.Benincasa · research

Cyclic Naturality of the Six Boundary-Value Leray Germs

Question

Entry 365 showed that the starting positive chain has no literal boundary on the marked-Cut union. The sector residues instead arise through the Bunch--Davies boundary-value continuation. The hard-to-vary claim tested here is

$$\boxed{ \text{analytic continuation and the six sectorwise Leray residues may fail to commute with the source }C_3\text{ action}.}$$

The negative-imaginary tube, positive Cayley--Menger sheet, source volume, occurrence order, lower denominators, and boundary-value normalization are frozen.

Frozen occurrence action

In source order,

$$(12|23),(12|31),(23|31),(23|12),(31|12),(31|23),$$

the cyclic relabelling has two orbits:

$$(12|23)\to(23|31)\to(31|12)\to(12|23),$$

$$(12|31)\to(23|12)\to(31|23)\to(12|31).$$

The all-positive source vector is fixed by this permutation.

Boundary-value data

Each marked Cut has the form

$$q_{\mathcal G_{ij}}=E+y_{ij}, \qquad \frac{\partial q_{\mathcal G_{ij}}}{\partial y_{ij}}=1.$$

The three sector residue orders are even cyclic permutations of

$$dy_{12}\wedge dy_{23}\wedge dy_{31}.$$

Thus every sector has:

$$\text{Jacobian}=1, \qquad \text{orientation}=+1, \qquad \text{multiplicity}=1.$$

The common boundary value supplies

$$\operatorname{Disc}\frac1{q-i0}=2\pi i\,\delta(q)$$

with the same factor in all three sectors.

The tube

$$\operatorname{Im}x_i<0, \qquad \operatorname{Im}y_{ij}<0$$

is convex and invariant under cyclic relabelling. The positive Cayley--Menger sheet and signed-minor domain are transported by the same source permutation.

Naturality square

On the six occurrence-resolved source summands, the labelled first-residue matrix is

$$R_{\rm Leray}=I_6.$$

Let (P_\rho) denote the permutation matrix of the two three-cycles. Exact matrix multiplication gives

$$\boxed{ R_{\rm Leray}P_\rho=P_\rho R_{\rm Leray}.}$$

All pairwise iterated marked-Cut residues vanish because no source summand contains two marked-Cut poles. Hence no overlap correction is required for the square.

The tested failure claim is falsified.

Narrow result

$$\boxed{ \text{the six source-defined boundary-value Leray germs form a canonical }C_3\text{-equivariant family}.}$$

Analytic continuation is therefore the global operation relating the single degree-zero meromorphic source packet to its three residue sectors. This is not Čech gluing: there are still no pairwise transition maps or joint Cut poles.

Classification

Scope

This establishes naturality for the local physical residue germs and their source occurrence modules. It does not yet prove a conjugacy of the complete rank-twelve Gauss--Manin connection matrices, integral lattices, or relative period bases across the three sectors.

Evidence

• research/benincasa/marici-gm/src/bin/cyclic_leray_naturality.rs;
• research/benincasa/cyclic-leray-naturality-certificate.json;
• Entries 180, 229, 364, and 365.

Next falsifier

Lift the occurrence-level naturality square to the complete (9+3) rank-twelve filtered coefficient systems. Derive the cyclic basis transport from the source master descriptors—not from fitted connection matrices—and test

$$\nabla_{23}=P_\rho\nabla_{12}P_\rho^{-1}, \qquad \nabla_{31}=P_\rho^2\nabla_{12}P_\rho^{-2}$$

for the absolute nine-master block, the three marked quotient classes, and their extension columns. Any mismatch must be classified as basis gauge, sector-specific coefficient data, support, or a failure of cyclic naturality. No overlap transition may be inserted to repair it.

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