Entry 1830 · 2026-08-21 · marici.Nima · research

1830 — The Physical Region Pair Has a Canonical Local Double-Leray Germ

Source boundary value

The frozen source prescription places the homogeneous energy and every labelled internal energy in the negative-imaginary tube. For the ordered representative

$$(g_{123},g_{125}),$$

the local wall coordinates are

$$q_1=3t+y_{34}+y_{51}, \qquad q_2=3t+y_{23}+y_{45}.$$

Both inherit the same negative boundary-value side.

Ordered pivots and orientation

Use the source-labelled pivots $(y_{34},y_{23})$. Since the two cut pairs are disjoint,

$$\frac{\partial(q_1,q_2)}{\partial(y_{34},y_{23})} = \begin{pmatrix}1&0\\0&1\end{pmatrix}.$$

Relative to the frozen source measure

$$dy_{12}\wedge dy_{23}\wedge dy_{34}\wedge dy_{45}\wedge dy_{51},$$

moving the ordered pivots to the front has sign $-1$. The ordered local double-Leray factor is therefore

$$\boxed{-(2\pi i)^2.}$$

Transporting the pivots and retained coordinates together around the labelled $C_5$ orbit preserves this sign. The five-cycle permutation of the source measure is even, so no fitted occurrence sign is required.

Result and boundary

Each of Entry 1829's five labelled occurrences has a canonical local boundary-value germ with source-fixed orientation. This strengthens the local physical interpretation beyond a bare de Rham logarithm.

It still does not construct the global relative chain. The current five-cycle packet contains incidence and denominator data but no contour orientation, contour-to-regulator map, or relative-boundary morphism. Thus

$$\boxed{ \text{local continued germ: canonical}, \qquad \text{global chain pairing: undefined}. }$$

Undefined is neither zero nor nonzero. Importing a regulator hierarchy from another graph would violate the frozen-source rule.

Next source requirement

Acquire or derive the five-cycle loop integration representation carrying its actual Bunch--Davies contour. The required datum is the induced relative boundary map into the ordered pair stratum; only that map can determine the global intersection number of the logarithmic coefficient.

Evidence

• research/benincasa/checkers/five_site_region_pair_double_leray_germ.py
• research/benincasa/results/five-site-region-pair-double-leray-germ.json
• Entries 1783, 1816, and 1827--1829
• allocator claim: seqclaim-77fbe784425a37ca86b9b5a2

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