LIDO Stage 2, proposal P01, asset tet bodies

Tet Null Space Plan

2026-10-14 Worktree lido-p01-tetmesh, diagnosis of 2026-10-13, Gate D record of 2026-10-14

This page explains why the asset tet bodies of P01 collapse, what the two fixes do, what the probes and the trained arms show, and what we decide next. Teal marks the tet mesh and its energy. Gray marks the hex carrier grid. Orange marks contact and constraints. Green marks a kept state, red a defect. Every number traces to a file listed in the footer. Where a figure is a sketch and not data, the caption says so.

1. The problem in one picture

An asset tet body rides on a hex carrier grid. Each tet vertex sits at fixed trilinear coordinates inside one cell. Its position is the weighted sum of that cell's eight corners, x_v = J x. The corners are the only degrees of freedom. The tet energy reads J x and nothing else. When the grid has more corners than the mesh has vertices, there are corner motions that move no vertex. These motions form the null space of J. The energy cannot see them.

Figure 1. The null space in one picture, in 2D. A vertex position is the weighted sum of its cell's corners, x_v = J x. Panel B moves the corners along a direction d with J d = 0, computed for this very drawing. The corners move by up to 0.20 cells. The vertices do not move. The tet energy sees only the vertices, so it sees no difference between A and B. The real bodies are 3D with eight corners per cell; the mechanism is the same.

Only the inertia term touches the corners directly, and inertia resists only a change of velocity. Once a null-space velocity exists, nothing slows it down. Three sources feed it: the initial deformation velocity field, the contact forces of the voxel path, which act on the hex face samples and not on J x, and the network's per-query proposals, which no energy loss corrects in ker J. These three sources are the diagnosis report's, section "Summary".

2. Why assets and not lattice bodies

A lattice body puts a tet vertex on every grid corner. Its J is the identity. Nothing can move unseen. An asset mesh fills the cells with tets whose vertices sit inside the cells, so the grid has far more corners than the mesh has vertices.

Figure 2. Why asset bodies and not lattice bodies. A lattice body puts a vertex on every corner, so J is the identity and nothing can move unseen. The asset plate 353 has 4675 hex corners for 2847 tet vertices, the compact body 145 has 4256 for 2462. The rank of J is 2846 and 2462, so dim ker J is 5487 and 5382. The same plate occupancy filled with lattice tets has 4675 vertices on 4675 corners and no null space. Counts from report.md section 2 and nullspace_check.md.

Measured for the two bodies of the soft well: the plate 353 has 2268 cells, 8090 tets, 2847 vertices and 4675 hex corners. The compact body 145 has 1994 cells, 7730 tets, 2462 vertices and 4256 corners. The rank of J is 2846 of 2847 rows for the plate and 2462 of 2462 for the compact body. So dim ker J is 5487 of 14025 corner degrees of freedom for the plate and 5382 of 12768 for the compact body. Across the library the surplus of corners over vertices is 1700 to 1800 per body for the three failing meshes 353, 145 and 485. The lattice arm has no surplus.

3. What happens in a rollout

The diagnosis probe rolls the T1 epoch 20 checkpoint on the demo scene voxel_well_soft exactly as the renderer does: 8 queries per step, no noise, 450 steps. It records the carrier's lattice edge ratio and the tet edge ratio per body and per step. Both are the current edge length over the rest edge length, maximum over the body.

Figure 3. The rollout of T1 epoch 20 on the soft well, 450 steps, 8 queries per step, no noise. Gray: the hex carrier's maximal edge ratio. Teal: the tet edge ratio. Solid: plate 353. Dashed: compact body 145. The carrier drifts from step 0 with no contact pair until step 22. Hex cells invert from step 40. The tets stay at 1.00 to 1.08 until step 60 and below 1.4 until step 250. A few body-body pairs appear from step 257. The scrambled hex surfaces of the two bodies meet in force around step 290, and the first tet inverts at step 294, when the plate owns 174 body pairs. Data: runs/T1_soft_baseline.json.

The order of events, all from the per-step record. The carrier drifts from step 0. After the 8 queries of step 0 the plate's longest hex edge is already 1.4 percent long and the compact body's 6 percent. No contact pair exists before step 22 in the per-step record; the report rounds this to the first 25 steps. The lattice edge ratio reaches 4.6 for the plate and 7.7 for the compact body by step 60. In free flight alone, with no plane and no walls, the drift rate is 0.16 to 0.19 edge-length units per step, which is 0.020 to 0.024 cells per query, about one step cap of 0.01625 cells per query. The first hex cell inverts at step 40 for the compact body and step 44 for the plate. The tet edge ratio stays at 1.00 to 1.08 until step 60. The scrambled hex surfaces produce spurious contacts and energy: the plate's kinetic energy is 13 J at step 275 and 637 J at step 300. The two hex surfaces meet around step 290. The first tet inverts at step 294, when the plate owns 174 body pairs. At step 449 the plate has 4063 inverted tets and the compact body 3874.

step	plate carrier edge max	compact carrier edge max	plate tet edge max	compact tet edge max	plate inverted hex cells	compact inverted hex cells	plate inverted tets	compact inverted tets	plate body pairs	compact body pairs
0	1.01	1.06	1.00	1.03	0	0	0	0	0	0
10	1.13	1.87	1.02	1.04	0	0	0	0	0	0
20	1.38	2.89	1.02	1.06	0	0	0	0	0	0
30	1.87	3.95	1.02	1.05	0	0	0	0	0	0
40	2.57	4.82	1.03	1.06	0	1	0	0	0	0
45	2.94	5.56	1.04	1.06	2	1	0	0	0	0
60	4.58	7.74	1.05	1.08	6	2	0	0	0	0
100	10.38	11.45	1.17	1.30	20	88	0	0	0	0
200	17.00	33.53	1.38	1.87	265	216	0	0	0	0
275	48.81	86.15	1.35	1.36	430	377	0	0	0	5
290	70.87	135.69	1.92	1.54	448	407	0	0	36	5
295	80.91	156.63	2.86	2.81	497	416	18	3	198	18
300	90.27	179.80	4.37	5.10	557	413	132	8	440	27
350	280.45	494.16	26.09	71.83	890	569	1610	1033	1672	1000
449	1548.17	1936.33	285.66	446.10	1306	1215	4063	3874	1982	888

The training runs show the same collapse. The cheap validation of one physical step cannot see a drift that needs advance to turn a displacement into a velocity, so it reported 8 of 8 survivors for T1. The full-horizon validation of 8 queries by 128 steps on 4 held-out scenes reports 0 of 4 survivors for T1, T2 and T2_hand at every epoch from epoch 4 on, all collapsed with hex penetration above 3 r and 592 to 1327 inverted hex cells. The training guard replaced 45 to 78 scenes per epoch from epoch 6 on. T0, the lattice arm, keeps 4 of 4 and replaces 0 to 3 scenes per epoch.

Figure 4. Training evidence from the Gate D record. The validation rolls 4 held-out scenes for 128 steps of 8 queries; the count sums the last 4 epochs. Every lattice arm with its contact path survives 16 of 16. Every asset arm without a carrier fix survives 0 of 16. The same record notes that T1 has 0 of 4 at every epoch from epoch 4 on. Data: p01_gateD_20261014/README.md and validation_summary.json.

4. The controls that rule out other causes

Each control changes one thing on the same scene seed. The first inverted tet moves by a few steps at most. Only the lattice representation removes the failure. The table lists the step of the first inverted tet, the step of the first inverted hex cell, and the inverted tets at the end of 450 steps. Each cell reads plate / compact body.

Figure 5. The controls in one view. Each box reads plate / compact body. Only the two representation controls, lattice tets on the same occupancy and Newton VBD on the bare tet mesh, remove the collapse. Numbers from report.md sections 3 and 4.

control	first inverted tet	first inverted hex cell	inverted tets at the end	reading
T1 baseline, 8 queries	294 / 294	44 / 40	4063 / 3874	the failure
T0 checkpoint, lattice bodies of the same seed, 7 bodies	never / never	never	0	max edge ratio 1.70; one body has 5 inverted tets at step 193 under contact, 0 at the end
kappa raised to 191.15	301 / 289	61 / 52	4104 / 3819	contact stiffness changes nothing of substance
inversion factor 0	295 / 295	46 / 37	3354 / 3859	the inversion rule is not the cause
inversion factor 30	260 / 286	45 / 36	3212 / 3876	same
zero body velocity and zero drift	261 / 250	41 / 40	4006 / 3919	velocity is not the cause
at rest on the floor, no deformation field	216 / 236	25 / 5	3966 / 3904	fails earlier: contact on the hex surface drives the carrier too
1 query per step instead of 8	348 / 124	80 / 29	156 / 1578	slower damage, same drift
plate alone, free flight, no plane, no walls	341 / -	51 / -	3160 / -	no contact at all, same drift and collapse
compact body alone, free flight	- / 305	- / 40	- / 3805	same
same occupancies with lattice tets	never / never	never	0 / 0	max edge ratio 2.59 and 1.84; the problem is gone
Newton SolverVBD on the bare tet mesh, 10 iterations, dt 1/300 s, 450 substeps	4 at substep 150 / never	-	0 / 0	plate: transient at the floor impact, final tet edge ratio max 1.12; the mesh and the material are fine

Thinness and sliver content do not separate the failing bodies from the rest: 82 to 84 percent of the library has a median voxel thickness of 2 cells or less in every split, and all four inspected meshes have a minimum dihedral angle of 7 to 9 degrees. What the failing bodies share and the lattice bodies lack is the surplus of corners over vertices.

5. Every candidate, one picture each

Six ways to close the null space are on the table. Each panel uses the toy of Figure 1: three grey cells, teal tet vertices, dark corner dots. The purple arrows are the null-space corner motion of Figure 1, the same vector d with J d = 0. Orange marks the element that pins that motion. The chip gives the status and the measured number when one exists. Sections 6 to 8 give the details of the first three.

Figure 6. Every candidate, one picture each, on the toy of Figure 1. Purple: the same null vector d as in Figure 1, J d = 0. Orange: the element that pins that motion. Panels 1 and 2 are implemented; their numbers are the Gate D record's and the probes'. Panel 3 is the ADR 0003 fallback. Panel 4's numbers are the asset-library builder trials a to d on the same 24 source meshes, totals computed from the per-entry seconds and the rejection reasons in their index.json: edge 1.0 accepts 14 of 24 in 17.6 min at a median 4.8 tets per cell; edge 0.7 accepts 14 of 24 in 31.6 min at 9.8 tets per cell; edge 0.5 accepts 9 of 24 in 74.9 min at 28.7 tets per cell, 4.3 times the time of edge 1.0, and still rejects 2 meshes for corner coverage; a second edge 1.0 run, trial d, accepts 15 of 24 in 10.3 min. Panel 5 is the follow-up suggestion recorded in p01_carrier.md section 4.3, not measured. Panel 6 is out of scope because ADR 0003 keeps the corners as the only degrees of freedom and leaves the network and the fusion unchanged.

6. Fix 1: the carrier weight

Switch carrier_hex_weight = w. With w > 0 the potential of a tet batch is

Phi(x) = E_tet_elastic(J x) + E_tet_damping(J x) + w E_hex_elastic(x) + inertia + contact

E_hex_elastic is the hex Neo-Hookean elastic energy of the carrier cells: eight Gauss points per cell, the body's own lam and mu_scale, the inversion rule of stable_neo_hookean, no hex damping. In code it is physics.carrier_hex_elastic: scale * (hc.weights * stable_neo_hookean(F, lam)).sum(-1) per cell, with F the Gauss-point deformation gradient of the cell's eight corners, then w times the per-object sum. The term is zero at the rest lattice and depends on x itself, so a ker J displacement costs energy and gets a restoring force in every direction. The price is a bias: the body as a whole is stiffer by about w times the material's stiffness on the filled part of each cell, because a deformation of J x deforms the carrier too. On lattice tets the hex and tet energies agree on affine deformations, so the term would only stiffen the body uniformly; the lattice arms stay at w = 0 and are bitwise unchanged.

Figure 7. Fix 1, the carrier weight. The hex Neo-Hookean elastic energy of the carrier cells is added at weight w. It depends on x itself, so a ker J displacement costs energy and gets a restoring force. Right: the inference probe on the T1 checkpoint, soft well, 120 steps. Weight 0 reproduces the drift. Every weight from 0.02 on removes it. At 0.02 the compact body still inverts one hex cell at step 106. From 0.05 on no hex cell and no tet inverts. Data: p01_carrier_20261013/runs/T1_soft_w*.json.

What the probe measures at step 120 of the soft well, plate and compact body: carrier edge max 12.18 and 13.63 at weight 0, 1.60 and 4.02 at 0.02, 1.28 and 3.37 at 0.05, 1.28 and 2.81 at 0.1, 1.30 and 2.22 at 0.2. The 60-step drift slope falls from 0.057 and 0.129 per step at weight 0 to at most 0.004 per step from 0.05 on. The maximal penetration falls from 1.28 and 2.24 r to 0.04 to 0.09 r. The tet edge max is 1.15 to 1.19 for the plate and 1.30 to 1.33 for the compact body at every weight, so the material is almost untouched at the probe. The network was trained without the term, so the probe is a stabilisation test, not an accuracy test.

Figure 8. Training outcome of the weight arms, Gate D record, 20 epochs on 4 GPUs each, seed 73. On the voxel collision surface the weight 0.2 arm survives 14 of 16 and the weight 0.05 arm 3 of 16. On the tet collision surface with division planes or the coupled constraint both weights survive 16 of 16. The pairs law arm C1_c05 survives 0 of 16, as its lattice twin C1L does. Data: p01_gateD_20261014/README.md.

Inference from these numbers, not a measurement: on the voxel surface the contact forces act on the hex face samples, so the carrier has to carry the contact, and a weight of 0.05 is not enough there. On the tet surface the carrier carries only the network's features and frames, and both weights suffice. The accuracy cost of the bias is not measured yet.

7. Fix 2: the minimal-norm reset

Switch carrier_reset. It adds no energy. At the end of every committed physical step, in Step.advance after the DAT truncation and before the velocity update, every asset body's corners are rebuilt. From the docstring of carrier_reset.py:

v        = J x_committed                           the vertex positions the physics produced
x_rigid  = R (x_rest - rc) + vc                    the rest corners under the rigid transform that best fits
                                                   the vertex motion: vertex-mass weighted Procrustes fit
x_new    = x_rigid + J^T (J J^T)^-1 (v - J x_rigid) the minimum-norm corner correction that reproduces v

J x_new equals v to rounding. The material is unchanged. The carrier loses every accumulated component in ker J. V = X_new - X_prev sees the reset state. Pinned corners keep their positions and the least-squares runs on the free corners only. The system J_F J_F^T is fixed per mesh and pin set, assembled once, and solved per batch layout with one factorisation: dense Cholesky up to 1024 vertices per block, cuDSS above it, both in float64. Lattice bodies are skipped. The reset is not differentiable and never runs inside a query.

Figure 9. Fix 2, the minimal-norm reset, computed on the 2D toy of Figure 1. Panel 1: the committed state carries a scrambled carrier, a rigid motion and a real stretch of the material. Panel 2: the rest corners under the rigid transform that best fits the vertex motion, a vertex-mass weighted Procrustes fit. Panel 3: the smallest corner correction that reproduces the vertices exactly. The ker J component relative to the rigid pose drops from 0.37 cells to 2e-16 cells in this drawing. The formula is the one in carrier_reset.py. The reset runs on the committed state after each step, never inside a query, and adds no energy.

What the probe measures with the reset on, same scene, same checkpoint, weight 0. No tet inverts in 120 steps on either body. The compact body's tet edge max is 1.08, the best of all rows, against 1.30 at weight 0 and 1.32 at weight 0.05. Its elastic plus damping energy at step 120 is 0.39 J against 22.8 J at weight 0. But the plate's carrier edge ratio is still 10.7 at step 120 with 127 inverted hex cells, and the 60-step slopes fall only from 0.057 and 0.129 to 0.021 and 0.043 per step. The reason is in the per-corner dumps: the plate bends. Its vertices leave the rigid pose by 2.50 cells RMS and 8.90 cells max at step 119 at a tet edge ratio of 1.21. The minimum-norm reconstruction of that bend is not smooth.

weight	reset	body	carrier edge max at 120	inverted hex cells at 120	tet edge max at 120	max penetration, r
0	off	plate 353	12.18	32	1.15	1.28
0	off	compact 145	13.63	117	1.30	2.24
0.05	off	plate 353	1.28	0	1.19	0.04
0.05	off	compact 145	3.37	0	1.32	0.06
0	on	plate 353	10.71	127	1.21	0.38
0	on	compact 145	3.52	0	1.08	1.57

Figure 10. Why the reset leaves the plate's carrier distorted. This is a sketch of the mechanism read from dump_analysis.md, not a drawing of data. The 1.2 m plate bends by up to 8.9 cells while its tet edges stay at 1.21. The correction J^T (J J^T)^-1 r lands on each corner in proportion to its weights. Heavy corners follow the material, light corners stay near the rigid pose, and the carrier edges between them stretch: median incident edge ratio 1.91 for corners at 1 to 10 percent of the median mass against 1.36 for the heavy ones at step 119.

Cost, measured with CUDA events: 2.94 ms per step on the two-body probe scene with 5309 vertices. On a D1 scene with 34135 cells, 22 bodies, 18 of them assets with 36002 vertices in total, the cuDSS reset costs 2.74 ms per step on average, median 2.62 ms, with 42 MB of device memory for the factor; the dense alternative costs 25.9 ms and 962 MB. One query of such a scene costs tens of milliseconds and the reset runs once per 8 queries.

Open problem of the reset path. The three cluster jobs with the reset, T1_p, D1_p and D1_pc05, failed twice each on 4 GPUs with an NCCL collective timeout at the first training update. One rank hangs. The copied logs of one attempt each show it: ranks 0, 2 and 3 report a watchdog timeout of the all-reduce with sequence number 101 after 10800000 ms, and rank 1 reports its last enqueued work as 100. progress.json of each job stops at epoch 1 with 1 completed update. The cause is not yet known. Two local smoke runs with the reset switch finished today without a hang: a two-rank run of 6000 cells completed 2 epochs and 31 updates, and a 32000-cell run completed 1 epoch and 6 updates. So the hang is not reproduced locally yet. Until the cause is found the reset has no training outcome.

8. Fix 3 in reserve: Option B of ADR 0003

ADR 0003 chose one deformation gradient per tet from the embedded vertices; the spec and the plan page call this Option A. The ADR kept one alternative as the fallback, in its words "the hex deformation gradient evaluated at tet quadrature points"; the spec calls it Option B. For linear tets the quadrature point is the tet centroid and its weight is the tet volume. The ADR says of it: it keeps the energy a function of the cell modes and cannot leave corners without weight. So there is no null space in practice. It is not Newton VBD's tet potential, so the lattice-tet fidelity gate would lose its exact reference. Spec section 12 records that Option B stays the fallback behind the two switches.

Figure 11. The three approaches side by side. Option B, the spec's name for the ADR's fallback, evaluates the hex deformation gradient at the tet quadrature points, so the energy stays a function of the cell modes and cannot leave corners without weight. It is not Newton VBD's tet potential, so the lattice-tet fidelity gate would lose its exact reference. Source: ADR 0003 and spec section 12. The "price" and "status" lines are my summary, not quotes.

9. The plan

Figure 12. The plan. The survival numbers are the Gate D record's. The beam error against VBD on asset meshes is not measured yet; it is the next measurement. The hang of the reset arms is the open problem of the reset path.

Runs now. The weight arms are done: T1_c05 3 of 16, T1_c20 14 of 16, D1_c05 16 of 16, D1_c20 16 of 16, K1_c05 16 of 16. The reset arms T1_p, D1_p and D1_pc05 are blocked by the hang at the first update.
Measure next. The beam error of the weight arms against Newton VBD on asset meshes, which is the number that prices the stiffness bias. Renders of the asset arms. The two-rank reproduction of the hang, then the reset arms again.
Decision rule. Pick the fix whose asset arms match the lattice arms' survival, 16 of 16, and do not lose beam accuracy against VBD. A combination is allowed; D1_pc05 is that arm. If neither fix passes, build Option B.
Question for Anka. Weight, reset or Option B? Which bias do you accept: a stiffness of about w times the material on the filled part of each cell, a per-step projection that distorts the carrier of a bent sheet, or a hex energy that is not VBD's tet potential?