Introduction and logical scope

Changing a conical deficit requires more than changing a label. The sector opening, seam identification, realized slices, adapted charts, and matching witnesses must change coherently. This paper supplies such a constructor on an explicit compact interval and proves that a specified particle-link event persists throughout it.

The argument has three independent stages:

1. construct and certify the geometry for every scalar deficit in the interval;

2. substitute the expectation of a fixed named Hermitian matrix into that scalar construction;

3. implement a finite algebraic restriction on classical codes carrying a compatible, externally supplied source token.

The geometric theorem does not depend on the source law. The finite implementation does not acquire the expectation and does not turn the continuum semantic rule into a quantum channel.

The affine matching interface of Landmark VIII-A CheungVIIIA is restated here. Its fixed-presentation uniqueness and two rejected updates are proved locally for variable deficit. The accessible-current and residual distinctions reported in VII-E CheungVIIE, and the affinity distinction reported in VII-G CheungVIIG, are also proved in the present notation. None of these companion results is a formal premise.

The physical setting is motivated by classical $2+1$-dimensional gravity with zero cosmological constant: source-free regions are locally flat DJH1984, and polygon descriptions can be reconstructed from first-order gravity Kadar2005. These references provide context. We import neither a mass normalization nor a Hamiltonian or constraint-algebra theorem.

Faces in the present construction may be nonconvex, and regular vertices need not be trivalent. The results concern the explicit matching contract below, not every more restrictive polygon gauge.

and conventions. Matter states are $\varrho$; the sector radial coordinate is $\rho$. Developed spatial rays are $e_i\in {\mathbb{R}}^2$, whereas $\mathbf{t}=(1,0,0{)}^{\mathsf{T}}$ is timelike. The second oblique coordinate is $\zeta$, distinct from the Minkowski metric $\eta$. All matrix operator norms are Euclidean spectral norms, including norms of homogeneous affine matrices; $\Vert \cdot {\Vert }_1$ is the trace norm. Report precision, clock precision, and permission width are $p_{\mathrm{rep}}$, $p_{\mathrm{clk}}$, and $w_{\mathrm{src}}$. Lengths and the slicing parameter are expressed in fixed normalized units.