# Verification report — OWR-13678-010 (C^1 splines on bipyramid cells, Colvin–DiMatteo–Sorokina conjecture)

Verification date: 2026-09-30. Revised on 2026-09-30, after the second independent verification run.

**Verdict.** The paper proves an exact formula for dim S^1_d(Δ) on every generic bipyramid cell. With n base vertices and m slopes:
- dim S^1_d = 1, 4, 11 for d = 0, 1, 2;
- for d ≥ 3 it is 2n·C(d,3) + 6d − 1 + (4 − m)_+ if m ≥ 3, and 8·C(d,3) + 8d − 3 − (4 − d)_+ if m = 2 (then n = 4).

For d ≥ 3 these are exactly the upper bounds printed in the Oberwolfach abstract. Hence the conjecture of Colvin, DiMatteo and Sorokina (CDS) holds in every remaining generic case except m = 3, d = 2. There dim S^1_2 = 11, while the printed bounds are [10, 12]. This happens exactly for the generic cells with m = 3, which have 3 ≤ n ≤ 6.

The bounds used are those printed in Oberwolfach Report 21/2015. The journal version (CAGD 45, 2016) could not be accessed. If its bounds for d ≥ 3 are the printed ones and it words the case m = 3, d = 2 differently, the conjecture holds as stated there. For any wording, the exact formula answers the question, since it gives the dimension itself.

**Credit.** The dimension is the sum of the dimensions dim H^1_k of the homogeneous components.
- For k ≥ 3r + 2 = 5 these dimensions also follow from the formula of Alfeld, Neamtu and Schumaker (1996, Theorem 3), evaluated with the number of distinct planes around each interior edge.
- dim H^1_2 = 7 is elementary (Proposition 1.3).
- The new content is therefore the determination of H^1_3 and H^1_4, together with the gluing step and the resulting uniform formula with its exception.
- The CDS conjecture concerns only the homogeneous degrees 2–4.

The note is unrefereed.

## Statement checked
- **Primary source.** T. Sorokina (joint work with J. Colvin and D. DiMatteo), "Dimension of trivariate C^1 splines on bipyramid cells", Oberwolfach Reports 12 (2015), no. 2, Report No. 21/2015, pp. 1184–1186, doi:10.4171/OWR/2015/21.
  - The PDF of the report was read. The copy used has SHA-256 3c2f206268cfe2ef9eb59c9c4bb6f04f29806192d6381767438703ed8ada5561. Both independent verification runs obtained the same file.
  - The page with Theorems 1–4 (p. 1186) was also checked on a rendered image, because text extraction flattens the binomial coefficients.
  - Theorem 4 (generic case), as printed:
    - for m ≥ 3 and d ≥ 2: 2n·C(d,3) + 6d − 2 ≤ dim ≤ 2n·C(d,3) + 6d − 1 + (4 − m)_+;
    - for m = 2: dim S^1_2 = 11, and for d ≥ 3: 8·C(d,3) + 8d − 5 ≤ dim ≤ 8·C(d,3) + 8d − 3 − (4 − d)_+.
  - The conjecture, for the remaining cases where these bounds differ: "we conjecture that the dimension coincides with the upper bound".
- **Journal version.** J. Colvin, D. DiMatteo, T. Sorokina, Comput. Aided Geom. Design 45 (2016) 140–150, doi:10.1016/j.cagd.2015.12.001. Its full text was not accessible:
  - the publisher page returned HTTP 403 and the text-mining API returned 400;
  - Crossref has no abstract;
  - the second verification run also found no abstract in OpenAlex, Semantic Scholar or zbMATH.

  Its wording of Theorem 4 is therefore unknown.
- **Corpus record.** ulamai/UnsolvedMath, OWR-13678-010 (status `open`, title "Exact Dimensions of Trivariate Spline Spaces"). Statement: "In the remaining cases where the upper and lower dimension bounds differ, does the dimension equal the upper bound?"

## Readings
| Reading | Answer | Witness |
|---|---|---|
| CDS conjecture with the bounds as printed in OWR 21/2015 (dim = UB whenever LB < UB, generic case) | holds in every case except (m, d) = (3, 2) | Thm 1.1, Cor. 1.2; triangular cell of Example 1.4: dim S^1_2 = 11 < UB = 12 |
| The same, restricted to d ≥ 3 | true | Thm 1.1 |
| The same, restricted to m ≠ 3 | true | Thm 1.1 |
| The journal version, if its bounds for d ≥ 3 are the printed ones and it restricts or corrects the m = 3 bound at d = 2 | true | Thm 1.1 |
| "What is the dimension in the remaining cases?" | exact formula for all n, m, d | Thm 1.1, Prop. 4.1 |

## Results in the paper
- **Theorem 1.1.** The exact formula above, for every generic cell and every d. Proposition 4.1 gives the homogeneous components:
  - dim H^1_k = 1, 3, 7 for k ≤ 2;
  - dim H^1_k = 2n·C(k−1,2) + 6 + ε_k for k ≥ 3, where ε_3 = (4−m)_+ + (3−m)_+, ε_4 = (5−2m)_+ + 2(3−m)_+, and ε_k = 2(3−m)_+ for k ≥ 5.

  The components with k ≥ 5 were known (see the next bullet). What is new is the formula in degrees k = 3, 4, and the complete statement for all n, m and d.
- **Components of degree k ≥ 5 (known).** For k ≥ 3r + 2 = 5, Proposition 4.1 also follows from Alfeld–Neamtu–Schumaker, SIAM J. Math. Anal. 27 (1996), Theorem 3, eq. (15). So do the homogeneous values in the proof of Theorem 4.2. The paper evaluates the formula in Section 6 ("The formula of Alfeld, Neamtu and Schumaker"). The numbers of distinct planes around the interior edges are:
  - generic case: 3 around each base edge and m around each apex edge;
  - coplanar cases: 2 around each base edge in P;
  - collinear case: 2 around every base edge.

  For k = 3, 4 the theorem does not apply. On generic cells with m ≤ 3 its expression is smaller than the true dimension: by 1 at k = 3 if m = 3, and by 1 at k = 3 and at k = 4 if m = 2. For m ≥ 4 it happens to give the true values at k = 3, 4 as well.
- **Corollary 1.2.** When LB < UB, dim = UB, except at (m, d) = (3, 2), where 10 < 11 < 12. The upper bound is always valid. The lower bound is never attained when the bounds differ.
- **Proposition 1.3** (elementary, degree 2). S^1_2(Δ) consists of the polynomials of degree ≤ 2 plus multiples of (ℓ_B)_+^2, for every generic cell and every n, m. Proof: the cofactors around each base edge satisfy a relation among the squares of three pairwise independent linear forms, so they vanish. For n = m = 5 this value was also obtained by DiPasquale–Villamizar 2021, via Mourrain–Villamizar 2014.
- **Example 1.4.** v0 = 0, v1 = (1,0,0), v2 = (0,1,0), v3 = (−1,−1,0), v4 = (0,0,1), v5 = (1,−2,−1); here n = m = 3 and dim S^1_d = 1, 4, 11, 24, 48 for d ≤ 4.
- **Standard ingredients (attributed).**
  - The homogeneous splitting: Alfeld–Neamtu–Schumaker 1996; DiPasquale–Villamizar 2021, eq. (3).
  - The dimension of H^r_k for k ≥ 3r + 2 on any cell: Alfeld–Neamtu–Schumaker 1996, Theorem 3; see also DiPasquale–Villamizar 2021, Remark 4.6 and Section 6.2. The paper's proof does not use it.
  - The algebraic C^r criterion: Billera 1988; Billera–Rose 1991, Prop. 1.2.
  - Lemma 2.1: the classical dimension formula for splines on a planar fan (Schumaker 1979; Lai–Schumaker 2007), proved by apolarity as in Geramita–Schenck 1998.
  - Lemma 2.3: the homogeneous form of the orange projection formula, Sirvent–Sorokina–Villamizar–Yuan 2024, Thm 1.1 (see also Alfeld–Schumaker–Sirvent 1992).
- **New steps.**
  - Proposition 3.1 (gluing): dim H^1_k = dim W_k + σ_{k−1} + 2·Σ_{j≤k−2} σ_j.
  - With it, the determination of H^1_3 and H^1_4 for every generic cell (Proposition 4.1 for k = 3, 4). The theorem of Alfeld–Neamtu–Schumaker does not cover these degrees. Together with k = 2, they are the only degrees on which the answer to the conjecture depends.
  - Lemma 3.2 (trace smoothness): in the generic case W_k = H^2_k(F). The paper relates this supersmoothness to Sorokina 2010 and to the additional smoothness that CDS used for their upper bound. The paper does not claim that Lemma 3.2 is new.
- **Theorem 4.2.** A uniform re-derivation of CDS Theorems 1–3 (collinear, coplanar I, coplanar II) from the same reduction.
- **Not claimed.** Nothing for C^r with r ≥ 2 (Section 6 explains why the gluing step becomes a coupled system).

## Computations (exact; scripts and outputs in reproducibility/)
- **Lead** (`lead/check_bipyramid.py`, standard library only, one to two minutes). It was written for the paper and imports nothing from the finder's code. It imposes that the jump and its normal derivatives vanish on each face plane (no cofactor unknowns) and computes ranks exactly over Q. It checks:
  - the triangular cell: dim S^1_d = 1, 4, 11, 24, 48 (d ≤ 4) without the homogeneous splitting, and the explicit basis of S^1_2;
  - Prop. 4.1 on 48 generic cells (all admissible (n, m), 3 ≤ n ≤ 8);
  - Thm 4.2 on 38 collinear and coplanar cells (2 ≤ d ≤ 5);
  - DiPasquale–Villamizar Table 1 ("symdim", n = m = 5), r = 1..4;
  - Cor. 1.2 for n ≤ 60, d ≤ 40;
  - Section 6, added in the revision: the formula of Alfeld–Neamtu–Schumaker for k ≥ 5.
    - The numbers of distinct planes are counted on each computed cell.
    - The formula is compared with the exact dim H^1_k: 115 comparisons.
    - It is compared with Prop. 4.1 and with the homogeneous values in the proof of Thm 4.2 (n ≤ 60, 5 ≤ k ≤ 40).
    - Those homogeneous values are also checked against the exact dimensions for 3 ≤ k ≤ 5: 114 comparisons.
    - It prints the differences at k = 3, 4.

  All checks pass. In the revision the section titles were aligned with the numbering of the paper (for example "vs Proposition 4.1" instead of "vs Theorem 1.1"), and Section 6 was added. Sections 1–5 print the same values as before.
- **Finder** (`claimant/`). The linear systems use cofactor unknowns, with exact ranks over Q. The programs check:
  - Lemma 2.1 in 1674 cases;
  - 185 cells of all four cases (n = 3..10, k ≤ 7), with a second implementation on a subset;
  - the closed formulas against CDS Theorems 1–4 for n ≤ 40, d ≤ 30;
  - non-homogeneous computations for d ≤ 4;
  - generic cells with n = 11..14;
  - DiPasquale–Villamizar Table 1 for r = 1..4;
  - the Euler-characteristic observation of Section 6.

  No mismatch occurred. Both independent runs re-ran these scripts on 2026-09-30, and the outputs were identical to the recorded ones apart from timing lines.
- **First independent verification run** (AI-assisted; `verifier/README.md`). Its own non-homogeneous code used ranks modulo two 31-bit primes, plus an exact rank over Q for the triangular cell. It reported:
  - 303 comparisons on random cells of all four cases (n ≤ 8, d ≤ 5);
  - 160 homogeneous comparisons (k ≤ 8, n ≤ 10);
  - no mismatch.

  Its code was kept in a temporary directory that was cleared before packaging, so it is not included.
- **Second independent verification run** (AI-assisted; `independent_run_2/`, code and outputs included). Its programs were written before it read the author's programs, and they import nothing from them. They use two formulations:
  - the Bernstein–Bézier form of the full space S^1_d, on cells moved by random integer affine maps;
  - homogeneous cofactor equations.

  All ranks are exact over Q. It found:
  - Example 1.4: dim S^1_d = 1, 4, 11, 24, 48, 90, 156, 252 for d ≤ 7;
  - 126 random generic cells with n = 3..12: 639 comparisons of dim S^1_d (d ≤ 6) and 1004 of dim H^1_k (k ≤ 10);
  - 13 generic cells in special position;
  - 384 comparisons on collinear and coplanar cells;
  - DiPasquale–Villamizar Table 1 for r = 1..4 on two pentagonal cells;
  - the Euler characteristics of Section 6;
  - Lemma 2.1 in 540 cases;
  - the expression of Alfeld–Neamtu–Schumaker against Prop. 4.1 for k ≥ 5.

  No mismatch occurred. The files were renamed for the release, and labels that used a different numbering were aligned with the paper. The outputs were regenerated and agree with the run's originals apart from timing lines and those labels (see `independent_run_2/README.md`).
- **External confirmation.** DiPasquale–Villamizar (SIAM J. Appl. Algebra Geom. 5 (2021); arXiv:2005.13043, Section 6.2, Table 1) computed dim H^r_d with Macaulay2 for a pentagonal cell with n = m = 5. This cell is generic in the sense of CDS; DiPasquale–Villamizar call it a "non-generic bipyramid".
  - For r = 1 their values 7, 16, 36, 66, 106, 156, 216, 286 (d = 2..9) are those of Prop. 4.1.
  - Of these, the values for d ≥ 5 also follow from Alfeld–Neamtu–Schumaker, and d = 2 follows from Mourrain–Villamizar. The values for d = 3, 4 were obtained there by computation.
  - The exact computations also reproduce their values for r = 2, 3, 4.

## Independent verification
### First run (2026-09-30)
The first independent verification run reported:

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (OWR PDF read independently; Theorem 4 bounds and the conjecture as quoted) |
| Proofs | CONFIRMED (every step checked; no gap) |
| Computations | CONFIRMED (independent code; claimant scripts reproduced) |
| Answer as posed | RESOLVED: exact formula; the conjecture holds except at (m, d) = (3, 2) |
| Novelty | none found in print (no priority claim) |
| Presentation | fixes required |

The required fixes and how they were applied:
1. **Lead with the exact formula, not with a disproof.** Done: the title, abstract, Theorem 1.1 and Corollary 1.2 state the formula first. The (3, 2) case is described as the single exception to the bound as printed.
2. **Keep the source caveat.** Done: the abstract, the introduction, the text after Corollary 1.2 and the Scope paragraph all state:
   - that the bounds are those printed in OWR 21/2015;
   - that CAGD 2016 was inaccessible;
   - that the conjecture holds as stated there if that version words the (3, 2) case differently.
3. **Attribute the standard ingredients precisely.**
   - Done: Lemma 2.1 is marked classical (Schumaker; Lai–Schumaker; apolarity as in Geramita–Schenck), Lemma 2.3 is SSVY24 Thm 1.1, and the homogeneous splitting is cited (ANS96; DV21).
   - Novelty was claimed only for the gluing step, Theorem 1.1, the exception, and the uniform re-derivation of CDS Theorems 1–3. The novelty claim for Theorem 1.1 was narrowed further after the second run (below).
   - For Lemma 3.2 the paper points to the supersmoothness used by CDS and to Sorokina 2010.
4. **Include the elementary d = 2 argument, the counterexample, and the DiPasquale–Villamizar comparison.** Done: Proposition 1.3 with its proof in Section 4.3, Example 1.4, and Section 5 (with a table for r = 1 and the r = 2..4 agreement).
5. **C^r (optional).** The paper makes no C^r claim. Section 6 explains the coupled gluing system for r ≥ 2.

### Second run (2026-09-30)
The second independent verification run, also AI-assisted, reported:

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED. The OWR PDF was downloaded anonymously and read on rendered pages. The live corpus row was checked. The question answered is the one asked. |
| Proofs | CORRECT. Every step was checked by hand, and no gap was found. |
| Computations | CONFIRMED. Its own code (exact over Q) found no mismatch, and all release scripts were re-run with identical outputs apart from timing lines. |
| Answer as posed | Exact answer: yes for every d ≥ 3 and for d = 2 with m ≠ 3; no for m = 3, d = 2 (dim 11, LB 10, UB 12). |
| Novelty and credit | NEW RESULT, with one credit omission. The components k ≥ 5 follow from Alfeld–Neamtu–Schumaker 1996, Theorem 3, as recalled by DiPasquale–Villamizar for this configuration. The omission is not fatal. |
| Presentation | GOOD. House style is followed; there were minor points. |

Required fixes and how they were applied:
1. **Credit the known high-degree components.**
   - *Hypotheses checked.* Theorem 3 of Alfeld–Neamtu–Schumaker was checked in a preprint version posted on the third author's homepage (https://math.vanderbilt.edu/schumake/hdim.pdf, 18 pages, SHA-256 63c3a3e70848fe821c6e346fa37d2115a81dd55bc4bfcd1c37ec88aec39a6335). It was fetched anonymously. Its numbering (Theorem 3, eqs. (15) and (16)) agrees with the citations of DiPasquale–Villamizar. The statement is as follows:
     - hypotheses: r ≥ 0, d ≥ 3r + 2, and any trihedral decomposition;
     - for a total decomposition, which is a closed vertex star after the interior vertex is moved to 0, dim H^r_d = (d−r)(d−2r)V − 2d² + 6dr − 3r² + 3r + 2 + σ;
     - here σ = Σ_v Σ_{j=1}^{d−r} (r + j + 1 − j·e_v)_+, and e_v is the number of distinct planes containing the faces that meet at the ray v;
     - no genericity is assumed.

     Its results in lower degree are not used. Theorem 16 treats d = 3r + 1 without degenerate faces, and Theorem 18 treats r = 1, d ∈ {2, 3, 4} under genericity with respect to perturbations. The faces U_i and L_i of a bipyramid cell are degenerate at v_i.
   - *Method and relation to earlier work.* A new paragraph covers the following:
     - it states the theorem for k ≥ 3r + 2 = 5, with the numbers of distinct planes, citing DV21 Remark 4.6 and Section 6.2;
     - it notes that the theorem reproduces Proposition 4.1 for k ≥ 5 and the homogeneous values used for Theorem 4.2;
     - the new content of Theorem 1.1 and Proposition 4.1 is the determination of H^1_3 and H^1_4, with H^1_2 elementary (Proposition 1.3);
     - given ANS96, the CDS conjecture concerns only the homogeneous degrees 2–4.
   - *DV21 paragraph.* It now says that DV21 obtain d ≥ 3r + 2 from ANS96, so for their cell and r = 1 only degrees 3 and 4 were left to Macaulay2. The sentence "We found no earlier statement of … Theorem 1.1" is qualified accordingly.
   - *Scope and priority.* This paragraph is qualified in the same way.
   - *Abstract.* It now states that the dimensions of the homogeneous components of degree at least 5 also follow from a general formula of Alfeld, Neamtu and Schumaker, and that the new part is the determination of the components of degrees 3 and 4.
   - *New paragraph in Section 6.* It evaluates the formula explicitly for bipyramid cells in all four cases and records where it differs from the true dimensions at k = 3, 4.
   - *Checks.* The lead script's new Section 6 checks all of this against exact computations, and the second run's `run2_ans_check.py` checks it at formula level.
2. **Propagate to the release.** Done:
   - this report (Verdict, Credit, Results, New steps, Closest work, Priority);
   - RESULT.md (top note);
   - `paper.pdf`, `main.tex`, `references.bib` (the ANS96 entry notes the preprint that was checked);
   - `source.zip`, the Zenodo copies and `ZENODO_METADATA.md`, whose description is the new abstract.

Optional suggestions:
- **CAGD caveat made precise:** applied. The abstract, the text after Table 1 and the Scope paragraph now say that the conclusion transfers to the journal version if its bounds for d ≥ 3 agree with the printed ones. In any case the exact formula answers the question for any version of the bounds.
- **m = 3 needs 3 ≤ n ≤ 6:** applied, in the introduction and after Table 1. The exceptional cells are exactly the generic cells with m = 3, in degree 2.
- **DV21 call their configuration of Section 6.2 a "non-generic bipyramid":** applied, in the introduction and in Section 5. Their Section 6.1 treats generic vertex positions.
- **Connect the Euler characteristic in "The lower bound" to the ANS value:** not applied as a separate statement. It would add a further computational observation. The new paragraph of Section 6 records where the ANS expression differs from dim H^1_k at k = 3, 4.
- **Include a reproducible AI-assisted run:** applied. The programs and outputs are in `reproducibility/independent_run_2/` and are described as item 4 of the Verification paragraph.
- **Align the naming in the lead script:** applied. Section 2 is now "vs Proposition 4.1", Section 1 speaks of the exception instead of a counterexample, the faces are labelled U_i, L_i, B_i as in the paper, and Section 6 was added. The output was regenerated.
- **Minor tidying:** partly applied. Section 6 is now mentioned in "Organisation". The title was not changed (no subtitle) and no further references were added; neither is needed for the required fixes.

## Relation to the literature, novelty and scope
- **Searches (September 2026).**
  - Sources: arXiv, Crossref, OpenAlex, zbMATH, Semantic Scholar, and three web searches (one in the finding stage and one in each independent verification run). All requests were anonymous.
  - Queries covered bipyramid cells, trivariate spline dimension, vertex stars, supersmoothness and homogeneous splines, and the works of Sorokina, Colvin, DiMatteo, DiPasquale and Villamizar.
  - OpenAlex and Semantic Scholar list only DiPasquale–Villamizar 2021 as citing CAGD 2016.
  - Crossref reference lists of the 2024 Springer volume on splines and algebra were also checked; none of its chapters cites CAGD 2016.
  - For the revision, open copies of Alfeld–Neamtu–Schumaker 1996 were sought.
    - OpenAlex and Semantic Scholar mark it closed.
    - The CiteSeerX copies listed by OpenAlex were unavailable.
    - The preprint on L. L. Schumaker's homepage was read.

    No web search was used in the revision.
- **Closest work.**
  - Alfeld–Neamtu–Schumaker 1996, Theorem 3, gives dim H^r_k on any closed vertex star for k ≥ 3r + 2. For bipyramid cells and r = 1 this is Proposition 4.1 for k ≥ 5, and it also gives the homogeneous values behind Theorem 4.2 for k ≥ 5. It does not cover k = 3, 4.
  - DiPasquale–Villamizar 2021, Section 6.2, treats one pentagonal cell (n = m = 5). They call it a "non-generic bipyramid"; it is generic in the sense of CDS.
    - They note the spline z_+^{r+1}.
    - They show via Mourrain–Villamizar 2014 that in low degree the dimension equals that of the span of the global polynomials and the multiples of z_+^{r+1}.
    - They obtain d ≥ 3r + 2 from Alfeld–Neamtu–Schumaker and compute the dimensions with Macaulay2. So for r = 1 only degrees 3 and 4 rest on their computation alone.
    - They give no general formula and do not discuss the conjecture.
  - Sirvent–Sorokina–Villamizar–Yuan 2024 supplies the projection used in Lemma 2.3 but does not treat bipyramid cells.
  - Later work found (DiPasquale–Villamizar 2024; Checa et al., arXiv:2606.18298) concerns other partitions.
- **Priority.**
  - The components of degree k ≥ 5 of Theorem 1.1 follow from Alfeld–Neamtu–Schumaker 1996.
  - The component of degree 2 is elementary, and for n = m = 5 it is in DiPasquale–Villamizar 2021.
  - No earlier statement was found of the formula in degrees 3 and 4, of Proposition 3.1, or of the exception at (m, d) = (3, 2).

  This negative search is not a proof of priority.
- **Caveat.** The full text of CAGD 45 (2016) 140–150 was not seen.
- **Scope.** The result covers every generic bipyramid cell, including special positions within the generic case; the dimension depends only on (n, m, d). The collinear and coplanar cases were already known (CDS Theorems 1–3); the paper re-derives them. C^r for r ≥ 2 is not treated.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/