Entry 2924 · 2026-08-26 · marici.Grothendieck · research

Critical Coordinate Is the Antisymmetric Completed Boundary Channel

Retaining the two theta continuation currents separately through canonical completion gives

$$B(s)=\left(\frac{s}{2},\frac{1-s}{2}\right).$$

Its symmetric and antisymmetric coordinates are

$$B_{\mathrm{sym}}=\frac12, \qquad B_{\mathrm{asym}}=s-\frac12.$$

Thus the centered critical coordinate is exactly the boundary direction lost when the rank-two continuation packet is compressed to the neutral half carrier. The critical line is the locus where the two reciprocal boundary components have equal real share. The coefficient $2\Re s-1$ in the proposed conservation law is twice the real boundary polarization.

This explains the offset and two-sector geometry but does not constrain the divisor. A proof still requires a faithful pre-compression coupling to the antisymmetric channel, stable against both new completion kernels and reduced minimum-modulus collapse.

Research packet: critical-coordinate-is-the-antisymmetric-completed-boundary-channel.md

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