Kernel rig: two chains that now build graded by their exact volumes

With the tangent chain on, two cases round more than the edge they pick, and their old
expectations were the single edge's:
- 1207-fillet-box-end-edge-into-unblended-round: the r1 chain is the end face's top edge, the r3
  round's end arc and the back upright; 34 A + (pi/2) A (3 - c) by Pappus on the arc:
  V = 7953.139216 (it built 7953.139216, the old 7957.505 passed only inside 1e-3), 10 faces.
- 851-fillet-top-front-then-front-left-upright: the upright's chain runs up the first round's
  end arc on x = 0 and along the left top edge, so the corner becomes the R2 ball's eighth:
  V0 less the first round's spandrel to the wall (35.344727) less 27 (4 - pi) + 2 pi/3:
  V = 9530.777979 (built 9530.777987), 10 faces.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01DtL5M41G8Nua7HvuFPZkmJ

tests/kernel-cases/fillets/1207-fillet-box-end-edge-into-unblended-round/expected.json

@@ -2,13 +2,14 @@
  "sources": [
   "tangency-labs/solidcraft#1207"
  ],
- "what": "an edge that runs on tangentially into a round left out of the blend (tangent chain off) is rounded on past the round's end arc until it leaves the material",
+ "what": "an edge that runs on tangentially into a round's end arc, its tangent chain rounded with it: the r1 round follows the end face's outline over the r3 round and down the back edge",
  "source": "analytic",
- "method": "cube 20 (V0 = 8000), its top back edge (y=20, z=20) rounded r3: removes 20 (9 - 9 pi/4) = 38.628316. The end face's top edge (x=0, z=20, y 0..17) then rounded r1 alone, tangent_chain false: over y 0..17 it removes 17 (1 - pi/4) = 3.648231; at y=17 the edge goes on tangentially into the r3 round's end arc, and the r1 round's cylinder (axis x=1, z=19) runs on along y through the r3 round, cutting the part of its section under the round's surface z = 17 + sqrt(9 - (y-17)^2) until that surface drops below z=19 at y = 17 + sqrt(5): with t = sqrt(9 - (y-17)^2) - 2, the section's area under it is t - (t sqrt(1-t^2) + asin t)/2, integrated over y 17..17+sqrt(5) (Simpson, 200000 steps) = 0.218236. V = 8000 - 38.628316 - 3.648231 - 0.218236 = 7957.505202",
+ "method": "cube 20 (V0 = 8000), its top back edge (y=20, z=20) rounded r3: removes 20 (9 - 9 pi/4) = 38.628331. The end face's top edge (x=0, z=20, y 0..17) then rounded r1 with its tangent chain: at y = 17 it runs on tangentially into the r3 round's end arc on x = 0 (convex, a quarter circle about y = z = 17), which runs on at (0, 20, 17) into the back upright (x=0, y=20, z 0..17), so the chain is those three edges; it ends at the front face y = 0 and the bottom z = 0, each square to its edge, so each straight round ends in a plane section. With A = 1 - pi/4 the r1 spandrel and c = r (10 - 3 pi)/(3 (4 - pi)) = 0.223368 its centroid's distance from either face: the top edge and the upright remove 17 A each; along the arc the spandrel, in the half plane through the r3 axis, lies at radius 3 - u from the axis (u its distance from the round's surface), and is turned through the arc's quarter turn: (pi/2) A (3 - c) = 0.935991 (Pappus). V = 8000 - 38.628331 - 34 A - 0.935991 = 7953.139216. Faces: the cube's 6, the r3 round, and the r1 chain's three (cylinder, torus, cylinder): 10.",
  "outcome": "solid",
  "body_count": 1,
- "volume_mm3": 7957.505202,
+ "volume_mm3": 7953.139216025938,
  "continuity": {
   "order": "g1"
- }
+ },
+ "faces": 10
 }

tests/kernel-cases/fillets/851-fillet-top-front-then-front-left-upright/expected.json

@@ -3,9 +3,9 @@
   "tangency-labs/solidcraft#851",
   "tangency-labs/solidcraft#854"
  ],
- "what": "on #851's knife-edge part (tessellation/1263): the top front edge R2, then the front-left upright R2 as a fillet of its own, ending in the first round",
- "source": "invariants",
- "method": "the rolling ball of R2 fits on every one of these edges and where they meet; the corner between rounds that end in each other has no closed form here: one closed valid solid whose display mesh closes. Each round meets the faces it rolls on G1: graded at its seams' middles (the top front round's at z = 8 on the front and y = -19 on the top; the arc's at rho 23 on the top and rho 21, z = 8 on the wall; the uprights' at their contacts, z = 5). Where a round ends on a face or on another round the seam is a crease by construction and is not graded.",
+ "what": "on #851's knife-edge part (tessellation/1263): the top front edge R2, then the front-left upright R2 with its tangent chain: up the first round's end arc and along the left top edge",
+ "source": "analytic",
+ "method": "plate x 0..41.1667, y -21..0 and the corner x 41.1667..62.1667 less the cylinder r 21 about (62.1667, -21), 10 high (tessellation/1263): V0 = 10 (62.1667 x 21 - pi 21^2/4) = 9591.394099. The top front edge rounded R2 first: no ball of R2 inside the part (centre y >= -19, z <= 8) reaches its spandrel (y in [-21,-19], z in [8,10], outside the circle about (-19, 8)), so it removes the spandrel's material, from x = 0 to the wall x_w(y) = 62.1667 - sqrt(441 - (y+21)^2): Int over the spandrel of x_w(y) = 35.344727 (Simpson in y = -19 - 2 sin t, 200000 steps). Then the front-left upright (x=0, y=-21, z 0..8) with its tangent chain: at z = 8 it runs on tangentially into the first round's end arc on x = 0 (a quarter circle about y = -19, z = 8, convex), which runs on at (0, -19, 10) into the left top edge (x=0, z=10, y -19..0), ending on the back face y = 0, square to it. The three are the corner's other two edges and the arc between: the result is the part with all three edges at (0, -21, 10) rolled by the R2 ball, its corner the ball's eighth about (2, -19, 8) (the arc's round is the torus of tube centre radius 0 about the first round's axis, the ball itself). With A = (4 - pi) the R2 spandrel: the upright removes 8 A and the left top edge 19 A; in the corner cube [0,2] x [-21,-19] x [8,10] the first round left the quarter cylinder 2 pi and the ball leaves its eighth 4 pi/3: 2 pi/3. V = 9591.394099 - 35.344727 - 27 A - 2 pi/3 = 9530.777979. Faces: the part's 6, the first round, and the chain's three (cylinder, sphere, cylinder): 10. Each round meets the faces it rolls on G1: graded at its seams' middles (the top front round's at z = 8 on the front and y = -19 on the top; the uprights' at their contacts, z = 5).",
  "body_count": 1,
  "watertight": true,
  "continuity": {
@@ -32,5 +32,7 @@
     5
    ]
   ]
- }
+ },
+ "volume_mm3": 9530.777979288185,
+ "faces": 10
 }