The sharp bevel is a global map from sites to points #2H7DL8

LUFT's refounded chamfer has an unusually strong regularity. A canonical edge or vertex site s, its occupancy star, and the width w determine one point

An oriented face does not invent that point. It merely includes the same G(s) as every other face incident to the site. The CPU classifies four edge stars and four vertex stars into one 32-bit shape word; the vertex shader decodes those decisions and realizes sixteen points on the same 4 by 4 patch for every face. Fixed positive and negative index templates draw eighteen triangles. #RUAWR5 explains the displacement rule and its closure tests.

This is correct by construction in a specific sense: incident faces agree, the surface is watertight without stitching, cubical transforms preserve the rule, and CPU decisions have one continuous realization at the live width. It is also inherently a sharp-miter system. If several chamfer strips meet at one lattice vertex, they all have exactly one G(s) at which to terminate.

That limitation is honest and local. The ordinary convex and concave bevel geometry keeps its coherent structure; only a mixed junction asks for a codomain richer than one point.

Mentioned in: Every vertex is an offset-line intersection

A miter arc carries the band; it does not decorate a singularity #DJK8HW

Three user-supplied screenshots preserve the visual argument. The first is the original LUFT junction, deliberately zoomed and framed on the problem: the green chamfer strip loses its width and collapses into the one shared point.

The second is the first attempted repair. Extra points do not carry the strip around the corner; they enlarge the old singularity into a conspicuous pinwheel of skinny triangles.

The third is Blender's Bevel modifier with Outer Miter set to Arc. It is an independent visual oracle, not a requirement to copy Blender's mesh representation. Broad bevel bands turn through the mixed junction without a central star, a sliver fan, or a strip narrowing to a point. Blender's bevel documentation describes the Arc choice as two vertices near the meeting joined by a curved arc.

The transferable requirements are relational: approximately constant band width, an ordered turn, and a crease graph which follows the band. A broad architectural plate such as #Z5NDTA supplies context, but it cannot replace the user-framed closeup as evidence.

Mentioned in: The miter family portrait keeps the awkward stars in context, The minimal-arc transition crossed the architecture halfway, Every vertex is an offset-line intersection, Turn, don't die: sharp miters leave a (3,1,1) fingerprint, Blender's Arc bevel is an executable oracle, Implement the site-stream LUFT renderer backbone

The minimal-arc transition crossed the architecture halfway #RJ0RLN

Commit 4724734 tried to move from the sharp junction to an Arc-like one while retaining a fixed face patch. It made two changes which need to be judged separately.

The useful representational observation was that one compact corner direction cannot describe several directed endpoints around the same vertex. The commit therefore replaced the CPU-authored edge and corner conclusions in the shape word with four complete eight-cell stars. It expanded every face from sixteen to twenty-four implicit points by adding two half-edge points at each corner, and from eighteen to twenty-six triangles.

But this also partly reversed the refoundation's CPU/GPU boundary. Before the commit the CPU classified occupancy and the shader decoded the answer. After it, the shader again counted occupancy, computed minority moments, projected half-edge stars, recognized miter configurations, and selected point formulas. The CPU still extracted a bounded neighborhood into a 16-byte record, so this was not a return to arbitrary shader-side world reads, but discrete bevel meaning had moved back into every vertex invocation.

Most importantly, the topology did not make the same transition as the point data. Directional endpoints were inserted into the old face-owned patch and triangulated back to its shared centre. The representation wanted a coherent vertex junction; the indices still described a centre-oriented face fan. The visible pinwheel in #DJK8HW is that mismatch, not a small tuning error.

LUFT has therefore returned to the sharp baseline. The failed transition is kept as evidence rather than retained as latent complexity in every ordinary face.

Mentioned in: Turn, don't die: sharp miters leave a (3,1,1) fingerprint, Faces, bands, and fans: three instance streams

Structural proofs implemented the rejected specification faithfully #7LEM72

The minimal-arc experiment remained watertight, preserved shared incident-face positions, transformed over the cubical symmetry group, produced consistently wound nondegenerate triangles, and agreed between independent CPU and shader arithmetic to floating-point tolerance. Those checks established that the program implemented its specification.

They did not establish that the specification was an Arc bevel. None asked whether a strip retained its width, whether the crease graph contained an unexpected high-valence centre, or whether the triangulation formed slivers. The rendered closeup rejected the result immediately.

This separates three claims which future work must not conflate:

1. the records and triangles are combinatorially valid;
2. CPU and GPU realize the same formulas; and
3. the formulas and connectivity express the intended bevel.

The first two remain necessary engineering evidence. The third needs an independent geometric oracle and human inspection. Exhausting all 256 stars cannot rescue a predicate which never states the visible requirement.

Mentioned in: Every vertex is an offset-line intersection, Blender's Arc bevel is an executable oracle

The sharp baseline remains inspectable at the problem scale #L7N4MO

The miter study of #Z5NDTA retains the exact #xCD wall termination alongside five-, six-, and seven-cell junctions. Two named captures use the production renderer, the normal 0.11-cell chamfer width, one camera, and construction ink as the only changed variable. They are diagnostic fixtures, not approved goldens.

Mentioned in: Implement the site-stream LUFT renderer backbone

The sharp junction with its construction edges #78WA2W

The same sharp junction without diagnostic ink #6X0WRV

A larger local object may preserve the old system's regularity #GBCLEV

The known constraint is small: one point cannot represent a band entering and leaving a mixed vertex at distinct locations. It does not yet determine the right replacement. Several questions remain open:

• Is the geometric value at a vertex a cyclic boundary, a filled patch, or a small topology identifier whose realization remains fixed and implicit?
• Can ordinary faces keep the sixteen-point sharp path while only exceptional vertex stars publish junction records, avoiding a universal per-face cost?
• Should the CPU publish explicit miter decisions, as the refoundation intended, or is some bounded star interpretation genuinely better in the shader?
• Are outer and inner miters complements of one rule, or distinct policies as Blender's vocabulary suggests?
• What independent construction defines constant width and angular order over the relevant cubical stars before triangulation is chosen?
• Can all required junctions be selected from the finite 256-star alphabet without turning the table itself into an unexplained collection of pictures?

A promising design would generalize the gluing invariant rather than abandon it: every incident face derives the same ordered boundary from the same canonical vertex star, and the junction owns whatever fills that boundary. That is a consequence of the failure, not yet a selected implementation.