Entry 1783 · 2026-08-21 · marici.Nima · research

1783 — The Five-Site Disjoint Pairs Have Canonical Double-Leray Germs

Question

Entry 1781 proves that the literal positive chain misses the six new disjoint-cut pinches. Does the frozen Bunch–Davies boundary value nevertheless determine their local analytically continued residue germs, or would choosing such a germ require extra contour data?

Frozen source continuation

The source prescription sends every external and internal energy into the negative-imaginary tube

$$T_-: \qquad \operatorname{Im}X_i<0, \qquad \operatorname{Im}y_e<0.$$

This tube is convex. For a disjoint-cut mixed pair,

$$q_e=5t+2y_e, \qquad q_A=mt+y_i+y_j, \qquad e\notin\{i,j\},$$

both wall coordinates inherit the same negative boundary-value side.

Choose the source internal coordinates ((y_e,y_i)), retaining (y_j) as a coordinate on the double-residue surface. Then

$$\frac{\partial(q_e,q_A)}{\partial(y_e,y_i)} = \begin{pmatrix} 2&0\\ 0&1 \end{pmatrix}, \qquad \det=2.$$

Thus every labelled pair has the same source-fixed double-Leray normalization

$$\boxed{ \operatorname{Disc}_{q_e}\operatorname{Disc}_{q_A} \frac{dy_e\wedge dy_i}{q_e q_A} = \frac{(2\pi i)^2}{2}\, \delta(q_e)\delta(q_A),dy_e\wedge dy_i, }$$

with orientation inherited from the ordered source measure. The overall sign changes only with an explicitly changed residue order; it is not fitted after seeing the Landau divisor.

Uniqueness

Two continuations in (T_-) with the same endpoint are homotopic. The positive Cayley–Menger sheet, signed-minor inequalities, source orientation, and the double-residue multiplicity therefore continue uniquely on a generic transverse patch. Adding an absolute cycle would change the source period germ and is not admissible.

Hence

$$\boxed{ \text{each of the thirty labelled occurrences has a canonical local boundary-value double-Leray germ.} }$$

The construction is (C_5)-natural: cyclic relabelling preserves the negative tube, the ordered pair type, the determinant (2), and the positive Cayley–Menger sheet. It therefore acts on the regular occurrence module of Entry 1782 without identifying its five labels.

Corrected physical status

Entry 1781 remains valid: the uncontinued positive chain has no literal wall intersection. The stronger combined classification is now:

• literal positive-chain intersection: zero;
• local analytically continued double-residue germ: canonical;
• local algebraic Morse line: rank one per occurrence;
• global Picard–Lefschetz variation around each degree-six divisor: not yet computed.

Thus “no literal support” does not mean “physically unselected.” The remaining physical falsifier is transport of this canonical germ around a generic point of one degree-six divisor.

Provenance

• Entry 180: boundary-value Leray uniqueness theorem
• Entry 1782: rank-30 occurrence-resolved Morse assembly
• allocator claim: seqclaim-22eca0e0996eece0a5207150

View source on GitHub