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Entry 1083 · 2026-08-20 · marici.Nima · research

1083 — The Physical Hexagon Activates Only After Double Leray Continuation

Problem

Entry 1082 showed that the six ordered residue terms of the three-site integrand generate the nonzero fundamental class of the physical normal link

Lk⁡phys=C6.\operatorname{Lk}_{\rm phys}=C_6.

The remaining question was whether the Bunch--Davies integration chain physically selects this class.

Two notions of restriction must be kept separate:

  1. ordinary intersection with the literal positive Cayley--Menger chain;
  2. Leray specialization of its negative-imaginary boundary value after analytic continuation to a factorization sheet.

Literal positive-chain restriction

The frozen three-cut certificate writes the positive chain as

a,b,c≥0,x,y,z>0,a,b,c\ge0, \qquad x,y,z>0,

with the positive Cayley--Menger square-root sheet and all required signed minor inequalities. On this chain,

qG12=E+c>E,qG23=E+a>E,qG31=E+b>E.q_{\mathcal G_{12}}=E+c>E, \qquad q_{\mathcal G_{23}}=E+a>E, \qquad q_{\mathcal G_{31}}=E+b>E.

For physical E>0E>0, neither the interior nor the boundary of the chain meets the marked cut union. Therefore the six literal incidence pairings are

(0,0,0,0,0,0).\boxed{(0,0,0,0,0,0).}

This vanishing is an ordinary-support statement. It does not determine the continued boundary value.

Boundary-value Leray specialization

Entry 180 froze the relevant primary-source data:

The negative-imaginary tube is convex. Hence the continued positive Cayley--Menger chain has a unique local Leray germ at every generic transverse marked pole. Its sheet, orientation, Jacobian, and multiplicity are fixed, and the multiplicity is one.

For a compatible pair in equation (51), applying this construction successively gives a unique double-Leray germ. Antisymmetry under reversing the two residue normals is exactly the sign used in Entry 1082. Consequently the six continued germs have oriented coefficients

(−1,−1,−1,−1,−1,−1).(-1,-1,-1,-1,-1,-1).

They therefore represent the nonzero generator

[ΓBD(2)]=±[C6]≠0in H1(C6;Q),\boxed{ [\Gamma^{(2)}_{\rm BD}] = \pm[C_6] \ne0 \quad\text{in }H_1(C_6;\mathbb Q), }

where the overall sign depends only on the already frozen global orientation convention.

Narrow theorem

literal positive-chain restriction=0,Bunch–Davies double-Leray specialization≠0.\boxed{ \begin{aligned} \text{literal positive-chain restriction}&=0,\\ \text{Bunch--Davies double-Leray specialization}&\ne0. \end{aligned} }

Thus the physical hexagon is activated as analytic-continuation/nearby-cycle data, not as a boundary component of the starting real chamber.

This is the precise mechanism that was unavailable for the two-site spurious corner: here both the support complex and the continued residue germ are source-defined.

Architectural consequence

No new carrier datum is required. The result uses

connected-subgraph incidence+Cayley–Menger relative chain+Leray/nearby specialization+ordered residue signs.\text{connected-subgraph incidence} + \text{Cayley--Menger relative chain} + \text{Leray/nearby specialization} + \text{ordered residue signs}.

This is direct evidence for H2's shared support-sensitive calculus with a cosmological relative-chain coefficient object.

Scope boundary

The theorem is local on generic transverse factorization patches. It does not assert an additional independent integrated period, nor determine global monodromy when the six strata meet the elliptic or algebraic discriminants.

Next falsifier

Transport the double-Leray generator around the cyclic atlas and toward the total-energy degeneration. Test whether it maps to an existing Cut flag/Tate nearby-cycle class or leaves a residual extension into the elliptic coefficient block. This comparison must retain occurrence labels and cannot be inferred from the rank-one result alone.

Durable packet

View source on GitHub