Can any model be fabricated? Inverse operation based planning for hybrid additive-subtractive manufacturing

Charlie C.L. Wang; Kun Qian; Tao Liu; Tianyu Zhang; Weiming Wang; Yongxue Chen; Yuming Huang; Zikang Shi

REVIEW 2 major objections 4 minor 2 references

Can any model be fabricated? Inverse operation based planning for hybrid additive-subtractive manufacturing

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that any stable voxel model can be fabricated exactly by interleaved additive and subtractive operations, by planning the reversed dismantling sequence.

desk verdict Clever inverse-planning algorithm with strong demos, but the 'any model' completeness theorem is a sketch that doesn't handle 3D tool access inside cavities. read the letter →

arxiv 2509.10599 v1 pith:BVDJJ3OK submitted 2025-09-12 cs.GR

Yongxue Chen , Tao Liu , Yuming Huang , Weiming Wang , Tianyu Zhang , Kun Qian , Zikang Shi , Charlie C.L. Wang This is my paper · ORCID

classification cs.GR

keywords hybridmanufacturingadditive-subtractiveinverseoperationplanningnullificationalgorithmvoxelmodelsprocesssupportstructuresfabricationcompleteness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tackles a fundamental question in hybrid manufacturing: given a model that cannot be printed in one pass or milled in one pass, can it still be made exactly by interleaving additive (printing) and subtractive (machining) steps? The authors answer yes for any stable voxel model, and prove it by planning the whole sequence backward. Instead of asking how to build the target from nothing, they ask how to dismantle the target to nothing using two inverse operations: erosion (removing material, the reverse of printing) and accretion (adding material, the reverse of machining). The reversed dismantling sequence is a valid fabrication plan that produces the target exactly, with temporary supports printed and later machined away. The completeness proof, the local stability check, and physical prototypes support the claim that hybrid machines can fabricate arbitrary voxel shapes, including parts from unconstrained topology optimization.

What carries the argument

The central machinery is the nullification algorithm built from two inverse operators: erosion, the inverse of an additive-manufacturing step, which removes a voxel, and accretion, the inverse of a subtractive-manufacturing step, which adds a voxel. The algorithm repeatedly takes the current top layer, classifies voxels there as erosion-feasible or not, and for non-feasible boundary voxels grows a connected support collar ring by ring below that layer until erosion becomes feasible, then erodes the layer. The completeness argument rests on the topmost-layer rule (erosion above, accretion below) and on the ring-wise accretion claim that a boundary voxel can always be made erosion-feasible with a subtractive tool of length $\bar{L}\ge 2$; scalability comes from replacing the global stability flood-fill with a conservative localized check on a $\Delta$-neighborhood.

What would settle it

Construct a stable voxel model with a deep internal cavity whose only opening is narrower than the subtractive tool, and run the algorithm to empty; if it stops before empty or produces a reversed plan that leaves material where it should not be, the completeness claim fails.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a completeness theorem for hybrid manufacturing planning: every stable voxel model can be reduced to empty by a finite sequence of feasible erosion and accretion operations, and the reverse of that sequence is a valid additive-manufacturing/subtractive-manufacturing fabrication plan that reproduces the target exactly, with every intermediate state stable and no over- or under-cuts. The enabling observation is that inverse planning can create temporary support material for free: accretion adds voxels that will later be eroded, which in the forward direction means supports that are printed and then machined away. The theorem is argued by restricting erosion to the topmost layer and accretion to below it, together with a ring-wise growing argument for boundary voxels; the authors report a voxel implementation that handles models up to roughly a million solid voxels, succeeds on 100 random benchmark models, and produces physical prototypes. The paper also reports that a topology-optimized MBB beam made by its hybrid plan is 30.51% stiffer than the self-support-constrained version.

Load-bearing premise

The theorem presumes that any boundary voxel on the current topmost layer can always be made removable by adding a ring of support beneath it with a subtractive tool at least two voxels long, without requiring a full 3D accessibility analysis for enclosed pockets.

Editorial extensions

If this is right

Topology-optimized parts no longer need to satisfy self-support constraints before fabrication, so designers can keep the mechanically optimal layout; the paper's MBB beam test shows a 30.51% stiffness improvement over a self-support-constrained version.
A hybrid machine can be given a fully automatic, exact plan for complex high-genus voxel models such as a TPMS structure with genus 72, including temporary supports that are removed by later subtractive steps.
Because the fabrication plan is the reverse of a verified nullification sequence, the final geometry is exact by construction, avoiding the over-cuts and under-cuts that forward-search planners can produce.
The algorithm remains usable at engineering scales: planning completed for models with up to roughly 0.98 million solid voxels, and the planned sequences were validated by physical fabrication on a five-axis hybrid machine.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the completeness theorem is stated for voxel models with isotropic material, so a natural next question is whether the same inverse-planning scheme extends to multi-resolution representations or to anisotropic materials; the paper itself flags anisotropy as future work.
Editorial extension: the implemented algorithm uses a finite $\Delta$-neighborhood stability check instead of the global check used in the proof, so the code is not formally covered by the completeness theorem; a search for a pathological voxel configuration with small $\Delta$ would show whether the gap is real.
Editorial extension: the proof's localization to the top layer makes enclosed cavities with narrow openings the likely stress case for the claim; testing deep, tortuous voids would test whether exact fabrication of any model includes models with internal sealed chambers.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here is genuinely nice: run the planner backwards from the target voxel model, using erosion as the inverse of deposition and accretion as the inverse of machining, so temporary supports that would be dead-ends in forward search become legal reverse steps. The implementation is also serious: local stability checks, a pre-processing heuristic, validation on 100 Thingi10K models, and actual physical parts machined on a hybrid system with a stiffness test. The code is public. That is real engineering and deserves credit.

The soft spot is the headline claim. The abstract says “any model can be fabricated exactly,” and that claim rests on the completeness proof in Sec. 3.3. As written, that proof is a 2D picture and a few sentences. It does not rigorously establish that the ring-wise accretion step is always feasible for a boundary voxel on the topmost layer when the needed support voxels lie inside an enclosed cavity. The accretion operation is only legal if the corresponding forward SM removal is collision-free, and for an interior cavity with a narrow opening that is not obviously true (indeed, with the paper's own conservative tool shapes, it can be false). The induction over layers is also sketched, not proven. On top of that, the implemented algorithm uses a localized stability check and a pre-processing step, and the paper never shows these preserve the completeness theorem; it even admits the local check can misclassify stable states as unstable, which is fine for soundness but leaves the implementation's completeness claim unsupported.

I want to be fair: this is not a paper that is wrong from the ground up. The inverse formulation is new, the scalability results are convincing, and the physical demonstration is far beyond what most papers in this area do. The empirical success on a varied dataset suggests the method works well in practice. But the title's question—“Can any model be fabricated?”—is answered with a proof that does not cover the hard cases. The paper would be honest and still strong if it either fixed the proof under explicit accessibility assumptions or stated the theorem with those caveats.

A serious editor should send this to peer review. The right reviewers will push on the geometry of accessibility, and the authors have enough material to respond. I would ask for a revised version that states precisely what is proven and what is heuristic, and that either strengthens Sec. 3.3 or changes the wording of the completeness claim.

Referee Report

2 major / 4 minor

Summary. The paper proposes a voxel-based planner for hybrid additive–subtractive manufacturing. Instead of searching forward from the empty state to the target, it searches backward by applying inverse operations: erosion (the inverse of additive deposition) and accretion (the inverse of subtractive removal). The algorithm repeatedly identifies erosion-feasible voxels on the topmost layer, uses accretion below that layer to make remaining boundary voxels erosion-feasible, and thereby reduces the model to the empty set; the reversed operation sequence is claimed to be a valid hybrid fabrication plan. The central theoretical claim is completeness: every stable voxel model can be nullified, and hence exactly fabricated, given an SM tool of length L≥2. The paper also reports a scalable implementation with a localized stability check, a pre-processing heuristic to reduce temporary supports, computational experiments on several benchmark models, and physical fabrication on a custom hybrid machine.

Significance. If the completeness theorem were correct, the inverse-operation approach would be a notable contribution: it would give a constructive, universal feasibility guarantee for hybrid AM/SM process planning, with practical scalability to hundreds of thousands of voxels and physical validation. The paper is clearly written, the inverse operation formulation is elegant, the implementation and data are shared (github link), and the physical experiments support the practical utility of the method for the tested geometries. However, the theoretical claim is the main advertised contribution, and it is not established as stated. The proof in Sec. 3.3 is a sketch that does not handle 3D tool accessibility in interior cavities, and there is a concrete stable model (a box with a fully enclosed internal void) for which the algorithm, as described, cannot generate a nullification sequence. Since the central claim is false in the current form, the significance of the paper is substantially reduced; the practical results remain interesting but do not compensate for the unsupported universality claim.

major comments (2)

[Sec. 3.3] The completeness theorem is not valid for the stated problem domain. The proof relies on the assertion that any boundary voxel on the topmost layer can be made erosion-feasible by accretion with an SM tool of length L≥2, but this is not shown for voxels adjacent to fully enclosed interior voids. Consider the stable 5×5×5 voxel box with a 3×3×3 central void (all solid voxels connected to the base). The center voxel (3,3,5) of the top plate has no solid neighbor at height 4 (the void extends underneath), so it is not self-supported and hence not erosion-feasible. To accrete at (3,3,4) below this voxel, the corresponding inverse SM operation would require a collision-free tool path. A vertical SM tool collides with the solid voxel (3,3,5) itself; horizontal tools, with widths implied by Def. 2.4, collide with the solid side walls at x=1,5 or y=1,5. No feasible accretion exists, so the algorithm cannot erode the top layer and the nullification process gets stuck. This is not a mere gap in presentation: it is a counterexample to the claimed theorem, which states that every stable voxel model can be nullified. The authors need to either add an assumption (e.g., cavity accessibility) or prove that the algorithm can create an opening in the top layer by eroding supported perimeter voxels first and then erode the center voxel without support below; the latter contradicts Def. 3.1. As written, the central claim cannot be repaired by a local proof tightening.
[Sec. 4.1] The completeness guarantee is proven for the abstract algorithm with a global stability check, but the implemented algorithm replaces that check with a conservative local Δ-neighborhood test. The paper claims (Sec. 4.1) that conservative misclassifications 'do not compromise manufacturability,' yet no proof is given that the local-check version inherits the completeness theorem. Since the local check can reject erosion and accretion operations that are feasible under the global criterion, the algorithm may fail to find a nullification sequence even when one exists for the abstract algorithm. The empirical verification on 100 Thingi10K models (Sec. 6.1.5) is not a formal argument. To support the claim that the implemented planner is complete, the authors must either prove that the local check preserves completeness (for example by showing that any rejected operation can be replaced by an equivalent sequence) or explicitly restrict the completeness statement to the global-check version and describe the local version as heuristic.

minor comments (4)

[Sec. 3.2, Step 4] The algorithm description in Step 4 is ambiguous about the order of checking erosion-feasibility with respect to the growing set Λ. In particular, the footnote says 'we cannot determine the order of following erosion operators in advance,' but the pseudocode in the text is not sufficiently formal to make the dependency clear. A precise pseudocode or a formal invariant for Λ would help reproduce the method.
[Sec. 3.3 and Fig. 7] Fig. 7 is a 2D illustration and the proof text refers to '46 configurations' without specifying whether peach voxels are at layer k−1 or at the same layer k. This makes the self-support condition in the proof difficult to verify. A 3D statement of the lemma and a careful distinction between support from below and same-layer neighbors is needed.
[Sec. 2.2, Def. 2.4] The occupancy set definitions of the SM tool are given in half-space form, but the physical interpretation of the tool length L̄ and the subtraction of the tool axis are not described in words. For reproducibility, the paper should state explicitly what material volumes these sets represent (e.g., the tool shank, the cutter body, and the depth of cut).
[Sec. 6.1.1, Table 1] The column 'Support Voxels' reports both pre-processing and total support counts, but the operational meaning of '# Operations' (e.g., whether it counts each voxel-level AM/SM action, not toolpaths) could be clarified in the caption or text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the completeness theorem is a constructive claim about the paper's own operators, with independent empirical validation.

full rationale

The central claim—that any stable voxel model can be nullified by the algorithm and the reversed sequence is a valid hybrid AM/SM plan—is a mathematical theorem about the paper's own accretion and erosion operators (Defs. 3.1-3.2), not a quantity fitted to data and then reproduced as a prediction. The feasibility of each operation is checked against explicit tool-occupancy and stability conditions (Defs. 2.3-2.6, 3.1-3.2), and the completeness argument in Sec. 3.3 is attempted from local boundary configurations rather than assumed. The only self-citations appear in related-work and toolpath-optimization contexts and are not load-bearing for the completeness proof. The proof sketch in Sec. 3.3 does contain a rigor gap: the assertion that a top-layer boundary voxel can always be made erosion-feasible by ring-wise accretion is illustrated mainly in 2D (Fig. 7) and does not fully establish 3D tool accessibility through interior cavities. Additionally, Sec. 6.3 explicitly concedes that chip removal in deep or tortuous internal voids is not guaranteed, which narrows the practical universality of the claim. These are scope and rigor limitations, not circular reasoning. The localized stability check with parameter Δ is a heuristic that affects scalability and solution quality but is not used to derive the theorem, and validation on 100 Thingi10K models plus physical fabrication provides external evidence. No step reduces by construction to its own inputs.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The central claim rests on a specific voxel-based process model: flat-head vertical AM, five-axis straight-tool SM with length >=2, a topological connectivity notion of stability, and an assumption that the input model is stable and the material is stiff and isotropic. The paper states the last two limitations explicitly in Sec. 6.3. No new physical entities are introduced; the accretion and erosion operators are algorithmic constructs, not new forces or particles.

free parameters (4)

Self-support angle = pi/4 (45 degrees)
Assumed threshold for AM self-support in Def. 2.5 and in the pre-processing cone search; chosen from material assumptions, not fitted to data from this paper.
SM tool length L̄ = 10 voxels (12 mm) in most tests; range 2-100 studied
Hardware parameter that strongly affects the number of operations and support voxels; user-selected, not fitted. The completeness proof requires L̄>=2.
Local stability search range Δ = 10 or 20 voxels
Chosen by hand in Sec. 6.1.3 to balance computation time vs. support voxel count; affects result quality and is not derived from the completeness proof.
Minimum printable feature size M = 10 voxels
Optional heuristic in Sec. 5.2 to enforce filament-based AM feature size; selected for the test examples.

assumptions (8)

domain assumption Input model is stable (all solid voxels connected to base layer)
Sec. 3.2 states 'Without loss of generality, we assume the input model is stable.' This excludes disconnected or floating-component targets from the 'any model' claim.
domain assumption Stability is equivalent to face/edge connectivity to the base layer
Def. 2.1 and 2.2 define stability topologically, ignoring gravity-induced deformation and material stiffness. The paper acknowledges this in Sec. 6.3.
domain assumption AM tool deposits only vertically downward with a flat head, occupancy set T_A = {gamma > k}
Prerequisite 2.1 and Def. 2.3 restrict AM geometry; the completeness proof relies on this flat-head model.
domain assumption SM tool operates along five axes with a straight tool of length L̄ and the occupancy sets in Def. 2.4
Prerequisite 2.2. The completeness proof requires L̄>=2 (Sec. 3.3), and horizontal accessibility is assumed for boundary voxels.
domain assumption Voxelized target exactly represents the model to be fabricated
The algorithm operates on voxel sets; 'exact fabrication' means exact reproduction of the voxelization, not arbitrary continuous geometry.
domain assumption Local stability check never misclassifies an unstable configuration as stable
Claimed in Sec. 4.1, based on Lemmas 4.1 and 4.2; the proof of Lemma 4.1's sufficiency is described as 'intuitive' and the finite search range Δ can introduce conservative errors.
domain assumption Material is isotropic and sufficiently stiff that geometric connectivity suffices for structural stability
Sec. 6.3 states deformation by gravity is not accounted for, and only isotropic materials are considered.
domain assumption SM chips inside closed cavities can be removed or are ignored
Sec. 6.3 acknowledges that trapped chips remain a practical concern; the planning model does not address it.

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Cite this review

Pith. "Pith review of Can any model be fabricated? Inverse operation based planning for hybrid additive-subtractive manufacturing." pith.science (2026). https://pith.science/paper/BVDJJ3OK

@misc{pith2026250910599,
  author       = {Pith},
  title        = {Pith review of: Can any model be fabricated? Inverse operation based planning for hybrid additive-subtractive manufacturing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BVDJJ3OK}},
  note         = {Machine review of arXiv:2509.10599}
}

read the original abstract

This paper presents a method for computing interleaved additive and subtractive manufacturing operations to fabricate models of arbitrary shapes. We solve the manufacturing planning problem by searching a sequence of inverse operations that progressively transform a target model into a null shape. Each inverse operation corresponds to either an additive or a subtractive step, ensuring both manufacturability and structural stability of intermediate shapes throughout the process. We theoretically prove that any model can be fabricated exactly using a sequence generated by our approach. To demonstrate the effectiveness of this method, we adopt a voxel-based implementation and develop a scalable algorithm that works on models represented by a large number of voxels. Our approach has been tested across a range of digital models and further validated through physical fabrication on a hybrid manufacturing system with automatic tool switching.

Figures

Figures reproduced from arXiv: 2509.10599 by the authors.

**Figure 1.** Existing manufacturing approaches often fail to fabricate complex geometries using a single modality, as illustrated by the unsuccessful 3D printing of [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** Topology optimization of the Messerschmitt-Bölkow-Blohm (MBB) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 4.** 2D illustrations of (a) an AM tool and (b) SM tools, shown with tool [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (14 more)

**Figure 5.** Figure 5: Illustration of the nullification process in the 2D case, where the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: For enhancing the erosion-feasibility of a voxel [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: All 46 configurations – excluding symmetric cases – of existing voxels (peach) around a boundary voxel [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 8.** Figure 8: Four 2D example cases illustrating the stability of a model after [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: Pre-processing to add temporary supports. (a) As illustrated in a [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 12.** Figure 12: The result of a TPMS structure with complex topology (genus: 72) [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]

**Figure 11.** Figure 11: The result of the hybrid manufacturing sequence determined by [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

**Figure 13.** Figure 13: Results using different tool lengths on the Fertility model (voxel [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]

**Figure 14.** Figure 14: Results using different tool lengths on the GE-Bracket model (voxel [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]

**Figure 15.** Figure 15: Study for the influence of the search range [PITH_FULL_IMAGE:figures/full_fig_p010_15.png]

**Figure 19.** Figure 19: Collision occurs (with interference regions highlighted in red) when [PITH_FULL_IMAGE:figures/full_fig_p011_19.png]

**Figure 18.** Figure 18: Quantitative comparison of the Fertility model, enhanced with [PITH_FULL_IMAGE:figures/full_fig_p011_18.png]

**Figure 20.** Figure 20: Three-point bending tests (a) taken on the physical specimens of [PITH_FULL_IMAGE:figures/full_fig_p012_20.png]

**Figure 21.** Figure 21: Results of our method on 100 models randomly selected from the Thingi10K dataset. Support voxels added during the nullification step are shown [PITH_FULL_IMAGE:figures/full_fig_p013_21.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

2 extracted references · 1 canonical work pages

[2021]

ACM Transactions on Graphics40, 4, Article 180 (2021)

Geometry and Tool Motion Planning for Curvature Adapted CNC Machining. ACM Transactions on Graphics40, 4, Article 180 (2021). Morad Behandish, Saigopal Nelaturi, and Johan de Kleer. 2018. Automated Process Planning for Hybrid Manufacturing.Computer-Aided Design102 (2018), 115 – 127. Li Chen, Tak Yu Lau, and Kai Tang. 2020. Manufacturability Analysis and P...

work page 2021
[2024]

Matthijs Langelaar

Learning Based Toolpath Planner on Diverse Graphs for 3D Printing.ACM Transactions on Graphics43, 6, Article 229 (2024). Matthijs Langelaar. 2016. Topology Optimization of 3D Self-Supporting Structures for Additive Manufacturing.Additive Manufacturing12 (2016), 60 – 70. Hak Yong Lee, Mu Zhu, and James K. Guest. 2022. Topology Optimization Considering Mult...

arXiv 2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [2021]

ACM Transactions on Graphics40, 4, Article 180 (2021)

Geometry and Tool Motion Planning for Curvature Adapted CNC Machining. ACM Transactions on Graphics40, 4, Article 180 (2021). Morad Behandish, Saigopal Nelaturi, and Johan de Kleer. 2018. Automated Process Planning for Hybrid Manufacturing.Computer-Aided Design102 (2018), 115 – 127. Li Chen, Tak Yu Lau, and Kai Tang. 2020. Manufacturability Analysis and P...

work page 2021

[2] [2024]

Matthijs Langelaar

Learning Based Toolpath Planner on Diverse Graphs for 3D Printing.ACM Transactions on Graphics43, 6, Article 229 (2024). Matthijs Langelaar. 2016. Topology Optimization of 3D Self-Supporting Structures for Additive Manufacturing.Additive Manufacturing12 (2016), 60 – 70. Hak Yong Lee, Mu Zhu, and James K. Guest. 2022. Topology Optimization Considering Mult...

arXiv 2024