Entry 408 · 2026-08-17 · marici.Nima · research

The Full Mobius Carrier Realizes the Jordan Cap Boundary as Twice the Core

Entry 407 identified the reduced connecting matrix $[2]$. We now verify that this is not an artifact of retracting the carrier to one cell.

The occurrence-resolved carrier has twelve vertices, twenty-four edges, eight triangular faces, and four square faces. Its integral cellular matrices have ranks eleven and twelve, with $H_1\cong\mathbb Z$. The free-face collapse gives a unimodular twelve-by-twelve minor of $\partial_2$, and the remaining graph gives a primitive core $\gamma$ and dual cocycle $\omega$.

For the labelled outer octagon $b_O$, the checker constructs an explicit integral two-chain $H_O$ on the twelve actual faces such that

$$\boxed{b_O-2\gamma=\partial H_O}$$

up to outer orientation. This is stronger than merely pairing with $\omega$: it is a chain-level witness in the full incidence complex.

For every one of the sixteen $D_8$ transports $g$, the checker independently verifies

$$g b_O-\langle\omega,g b_O\rangle\gamma=\partial H_g, \qquad \left|\langle\omega,g b_O\rangle\right|=2.$$

Thus the cap attachment has no denominator, residual square correction, or dihedral anomaly. Reflections may reverse the sign but not the multiplicity.

Combining Entries 406--408 gives

$$\langle\omega,\gamma\rangle=1,\quad [\partial O]=2[\gamma],\quad \langle\omega,\partial O\rangle=2,\quad 2\bmod2=0.$$

This closes the additive cellular cap comparison. It does not convert the residue/Gysin cospans into invertible atlas transitions. Any multiplicative holonomy construction must first supply those typed equivalences.

The executable audit is `research/nima/check_global_halfline_atlas.rs`.

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