On this page
Question
How could local reflection-positive data descend when the Euclidean system carries a gerbe?
What the note explains
The proposed reflected double needs a positive contraction. The gerbe itself need not first be erased by a global trivialization.
Example
Positive diagonal entries do not guarantee positivity of a coupled matrix: the matrix with diagonal 1 and off-diagonal 2 has eigenvalues 3 and −1. This illustrates why separate sector positivity cannot replace the proposed full block-positivity condition.
What the approach gains
The proposal keeps interface and mixed-sector data visible during reconstruction. It identifies the compatibility conditions that would be lost by checking each bulk sector in isolation.
Reading route
Read the status boundary, reflection-lift data, and proposed block kernel. Treat the axiom lists as a development agenda. The next mathematical task is positive seam modules and reflected-double neutralization.
Scope and limits
Positive seam modules remain undeveloped; the Bałaban RG bridge is reserved. There is no completed gerbe-valued OS theorem, Euclidean realization, or established comparison with the Lorentzian theory.