Exact certification of the 95.0668% incumbent

The older numerical result now has a verified exact realization with the same 45 physical groups and 95.06675512568251% coverage. The remaining 4.93324487431749% is still unsolved. This is an exact partial construction, not a complete dissection and not a fabrication/assembly certificate.

The literal saved floating-point mesh is not exact. We recovered consistent algebraic parameters and trimmed microscopic boundary protrusions and overlaps. The original numerical files are unchanged. The certificate describes the repaired geometry; it does not pretend the old float coordinates passed exact equality checks.

Artifacts

The certificate preserves all 45 group IDs. It has 331 convex computational cells, one more than the numerical mesh after overlap trimming. Those cells remain joined into the same 45 face-connected physical groups.

What changed and why

The saved rotations are all exact proper axis permutations: 14 distinct rotation matrices across 24 distinct poses. The difficulties were the translations and shared plane parameters, not irrational rotations.

Treating the stored IEEE-754 floats as exact binary rationals, 201 of 330 cell pairs fail a necessary exact congruence test under their saved poses. The largest coordinate-extremum discrepancy is 1/18014398509481984, approximately 5.5511e-17. This proves that the literal saved coordinates cannot simply be relabeled as an exact certificate. It is not evidence of a large geometric error.

The new recovery producer associates approximately shared planes, then solves their constraints exactly in the existing number field. For paired source/target supporting planes, a rigid pose imposes

n_target R = k n_source
 d_target = k d_source - n_target · t.

Root-support planes anchor the system to the actual algebraic TI and exact unit cube. The recovered system has 611 shared planes, 24 poses, 2,257 equations, rank 613, and 70 free scalar parameters. Only unconstrained parameters use decimal hints. Exact substitution checks every equation. The association tolerance proposes a system; it is not used to accept geometric correctness.

Producer diagnostics estimated a maximum normalized plane-offset change of 4.8983e-13 and translation change of 1.3811e-13. These are floating diagnostic estimates, not a certified Hausdorff-distance claim. The separate comparison report passed exact inequalities tying the final certificate to the old records. All 330 original cells remain represented, and all 45 physical groups are preserved. Computed exact maxima (displayed here approximately) are:

The comparison checks pose and halfspace bounds against 1e-8, and per-cell reported-volume changes against 1e-9, using exact arithmetic. These bounds are comparison criteria only; they do not weaken the exact geometric verifier. They are not a two-sided Hausdorff-distance certificate. The matched exact volume also passes the rational interval test for rounding to 95.0668%.

Some faces omitted by the original tolerance-based geometry left microscopic target protrusions. The producer intersects each candidate with both exact roots. It also subtracts earlier occupied material at both endpoints, correcting four positive-overlap encounters. No overlapping material is counted twice; any loss remains in the explicitly defined residual. All cuts remain exact and poses remain rigid.

Why the direct certificate is valid

Rebuilding 330 cells through one global cut tree introduced many unnecessary subdivisions. The new scissors-convex-partition-v1 format instead gives exact source halfspaces and a pose for every convex cell. The independent verifier:

  1. Reconstructs the whitelisted exact roots and clips each source cell from the source root.
  2. Checks every pose for exact orthogonality and determinant +1.
  3. Checks target containment and pairwise disjoint interiors at both endpoints, including cells belonging to the same group.
  4. Checks that every physical group has one pose and is connected through positive-area faces.
  5. Adds exact cell volumes and reports the exact residual volume.

Containment and disjoint interiors establish a valid partial union. The residual is defined as the regular-closed set difference between the root and the matched union. It is not silently discarded. For a complete claim, the verifier additionally requires exactly zero residual volume. With these finite closed polyhedral pieces, containment plus disjoint interiors plus full exact volume proves coverage; volume equality alone would not.

The acceptance check uses only Python's standard library, the exact algebraic field, and convex-geometry routines. NumPy is used in the untrusted association producer only. No geometric acceptance decision uses an epsilon. The first full verification took about 66.5 seconds on this laptop. The implementation remains trusted Python, not Lean-verified.

Reproduce

Run from the project root:

# Independent verification of the saved realization; no SciPy required.
uv run python -S -m experiments.certification verify \
  docs/research/certificates/incumbent_95_exact.json --expect-ti --max-pieces 45

# Recreate the shared-plane proposal from the original numerical geometry.
uv run --with scipy python -m experiments.certification recover \
  docs/research/free_dissection_search_policy_15min_continuation.geometry.json \
  --output .research-cache/certification/incumbent-recovered.proposals.json

# Repair exact overlaps and verify before saving the accepted certificate.
uv run python -m experiments.certification realize \
  .research-cache/certification/incumbent-recovered.proposals.json \
  --output .research-cache/certification/incumbent-exact.json --seconds 300

The bound comparison can also be reproduced with:

uv run python -S -m experiments.certification compare \
  docs/research/certificates/incumbent_95_exact.json \
  docs/research/free_dissection_search_policy_15min_continuation.geometry.json \
  --output .research-cache/certification/incumbent-comparison.json

realize keeps a .candidate.json for diagnostics; that file is unverified until the independent checker succeeds. The accepted output and .verification.json are saved only after success. Runtime budgets are cooperative and exclude indivisible final verification work. --require-complete must reject this partial certificate.

What this unlocks—and what remains

We can now use an exact 95.0668% matched union as the starting mathematical state. Further completion can reopen any of its groups; preserving this union is not mandatory. We no longer need to reason from the older 87.8523% exact prefix.

The current joint-completion importer accepts cut trees, whereas this certificate defines residuals by exact set difference. Before that search can consume this file, it needs either explicit residual-cell materialization or direct convex-partition support. The importer rejects this format with an explanatory message rather than silently treating it as a cut tree.

Exact fabrication clearance, minimum neck thickness, weight and an executable assembly sequence remain separate requirements. Certification of the matched union does not establish that the remaining 4.9332% can be finished within the 99-piece budget.

Validation

All 108 Python tests pass. Added cases cover sparse exact plane recovery, inconsistent equations, literal binary-rational mismatch witnesses, target escape, source/target overlaps below 1e-30, residual slivers, disconnected groups, exact overlap repair, rigid-motion rejection, and stdlib-only CLI verification. A separate stdlib-only process reverified the full 45-piece certificate and its comparison bounds. git diff --check passes.