Computer Bounds for Kronheimer-Mrowka Foam Evaluation

David Boozer

REVIEW 1 major objections 6 minor 11 references

Computer Bounds for Kronheimer-Mrowka Foam Evaluation

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A computer program proves that the combinatorial foam invariant equals the Tait coloring count on two fullerene webs, and constrains the dodecahedral web to 58 or 60.

desk verdict The first exact computations for nonreducible webs are real, but the missing code means the central theorem is only as good as an uncheckable table. read the letter →

arxiv 1908.07133 v2 pith:62GOM4DC submitted 2019-08-20 math.GT math.CO

David Boozer This is my paper · ORCID

classification math.GTmath.CO MSC 57K1805C15

keywords foamevaluationTaitcoloringswebsandfoamsdodecahedralwebfullerenegraphsfour-colortheoremquantumdimensionSmithnormalform

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

The paper builds a computer program that computes lower bounds on the dimension and graded dimension of $J^\flat(K)$, the combinatorial foam invariant associated to a planar trivalent graph, or web. It uses these bounds to test the open question of whether $\dim J^\flat(K)$ always equals the number of Tait colorings $\mathrm{Tait}(K)$ of $K$. For the fullerene webs $W_2$ and $W_3$ the program proves the equality; for the dodecahedral web $W_1$ it proves $\dim J^\flat(W_1)\in\{58,60\}$ and, because the lower bound saturates at 58 across thousands of half-foams, suggests the true value is 58 rather than the Tait count 60. If that value stands, $W_1$ is the first nonreducible web where the combinatorial invariant and the Tait count diverge, settling the open question negatively without disturbing the gauge-theoretic route to the four-color theorem.

What carries the argument

The load-bearing object is the closed-foam evaluation formula implemented by the program. For a closed foam $F$, the formula expresses a symmetric polynomial $\langle F\rangle=\sum_{c\in\mathrm{adm}(F)}P(F,c)/Q(F,c)$ over the field of two elements, summing over admissible colorings of the foam's facets, and $J^\flat(F)$ is this polynomial evaluated at $E_1=E_2=E_3=0$. The program enumerates facets and admissible colorings, generates a finite set of half-foams with top boundary $K$, forms the matrix of pairings $(F_i,F_j)_\varphi$ in the polynomial ring $\mathbb{F}[E]$, and performs a Smith normal form decomposition over that ring. The rank of the pairing matrix on a finite subspace is a lower bound for $\dim J^\flat(K)$, while the theorem that $\langle K\rangle_\varphi$ is a free module of rank $\mathrm{Tait}(K)$ supplies the matching upper bound; when the two bounds coincide, all intermediate inequalities collapse to equalities.

What would settle it

Recompute the ranks in Tables 2 and 3 with an independent implementation of the closed-foam evaluation formula; any difference would invalidate Theorem 4.2, and for $W_1$ a single new half-foam that raises the lower bound above 58 would refute the suggested value.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a pair of computations. Theorem 4.2 states that for the webs $W_2$ and $W_3$ shown in Figure 6, $\dim J^\flat(K)=\operatorname{rank}\langle K\rangle_\varphi=\mathrm{Tait}(K)$ and $\operatorname{qdim} J^\flat(K)=\operatorname{qrank}\langle K\rangle_\varphi=\ell_q(K)$, so both the dimension and the graded dimension collapse to the Tait data. For the dodecahedral web $W_1$, the program yields $\dim J^\flat(W_1)\in\{58,60\}$; the lower bound 58 is reached after 156 half-foams and remains 58 through 6,727 of the 11,160 half-foams generated, which the paper takes as strong evidence that $\dim J^\flat(W_1)=58$. Since $\mathrm{Tait}(W_1)=60$, this would make $W_1$ the first nonreducible web for which the equality question is answered negatively.

Load-bearing premise

The exact statements for $W_2$ and $W_3$ assume that the paper's computer program correctly implements the closed-foam evaluation formula and the matrix computations; the suggested value for $W_1$ additionally assumes that the saturation of the lower bound at 58 over thousands of half-foams is a reliable sign of the true dimension.

Editorial extensions

If this is right

For $W_2$ and $W_3$, equality between $\dim J^\flat(K)$ and $\mathrm{Tait}(K)$ is now established for nonreducible webs, the first such examples, and the graded version is established as well through $\ell_q(K)$.
For $W_1$, the dimension is proven to be either 58 or 60, leaving a single binary choice; the saturation data favor 58.
If $\dim J^\flat(W_1)=58$, the open question asking whether $\dim J^\flat(K)=\mathrm{Tait}(K)$ for every web would have a negative answer, ruling out the specific strategy of identifying $J^\flat$ with the gauge-theoretic functor.
For the other tested fullerene webs $W_4$ through $W_7$, the method gives nontrivial lower bounds strictly below the Tait counts, so their true dimensions are constrained to short intervals, such as 178, 179, or 180 for $W_4$.
The inequalities $\dim J^\flat(K)\le\mathrm{Tait}(K)\le\dim J^\sharp(K)$ are tight for $W_2$ and $W_3$, so the whole chain collapses to a single number for those webs.

Reading between the lines

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

The exactness of the results for $W_2$ and $W_3$ could be made independently checkable by releasing the program's source code and a transcript of the Smith normal form computations; as presented, the tables are the only evidence, and a single bug in facet enumeration or arithmetic could change a rank.
The saturation at 58 for $W_1$ is a heuristic, not a proof; a finite certificate for $\dim J^\flat(W_1)=58$ would require either a spanning set of half-foams for the whole space or a proof that the pairing matrix rank cannot increase past 58, for example from a degree bound on generators.
The dodecahedral web is a pivot between two conjectures: the symmetry of $\operatorname{qrank}\langle K\rangle_\varphi$ under $q\mapsto q^{-1}$ would force $\dim J^\flat(W_1)=60$, so the suggested 58 would refute that symmetry as well as the equality question.
The same computational pipeline could probe other small fullerenes; for each new web the ratio $\ell(K)/\mathrm{Tait}(K)$ gives a cheap indicator of where the equality question might fail first.

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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

First thing to know: the paper gives the first exact evaluations of dim J^flat for two nonreducible fullerene webs, and narrows the dodecahedral web to 58 or 60 with a suggested 58. If the computations are right, that's a meaningful data point for Khovanov-Robert's Question 2.1, and the author is honest that a negative answer for W1 wouldn't break Kronheimer-Mrowka's four-color strategy.

What's genuinely new is the data. The lower-bound method—using finite sets of half-foams and the rank of the restricted bilinear form—is known from Kronheimer-Mrowka and Khovanov-Robert; the novelty is that the author pushed it far enough to get exact dimensions for W2 and W3. That is a real computational achievement. The proof skeleton in Section 4 is clean: Corollary 3.1 and Theorem 4.1 sandwich qdim, and if the table values are correct, equality follows. The Smith normal form over F[E] is explained carefully, and the modular/quantum rank arguments check out.

Now the soft spot, and it's the load-bearing one. Tables 2 and 3 come from a Mathematica program that is not in the arXiv record. Reference [3] is a bare author URL with no version, no checksum, no code listing. That means Theorem 4.2 rests entirely on an unverifiable computation. A bug in facet enumeration, coloring enumeration, or Smith normal form would change the results. This is not a minor issue; it is the entire difference between a proven theorem and a conjecture from a black box. The author should be required to ship the code (or a certificate) as a condition of publication.

The W1 'suggestion' of 58 is handled properly. It is labeled a suggestion based on saturation of the lower bound after many half-foams, not a theorem. The paper even notes that the alternative 60 would keep Question 2.1 open. That's honest. It doesn't need to be fixed; it just shouldn't be over-read.

Citations look fine. No parameter fitting, no circularity. The computational lower bounds are genuinely independent of the conclusion.

Who is this for? People working on foam evaluation and the Kronheimer-Mrowka program, and anyone interested in whether J^flat equals Tait for nonreducible webs. A serious referee should be assigned, because the data matter and the computation must be checked—preferably by running the code. I'd condition acceptance on code release and independent verification. Don't desk reject; do send it to review.

Referee Report

1 major / 6 minor

Summary. The paper describes a Mathematica program that computes lower bounds on the dimension and quantum dimension of the Kronheimer-Mrowka combinatorial foam-evaluation functor J^flat for planar webs, using Khovanov-Robert's closed-foam evaluation formula on finite sets of half-foams. These lower bounds are sandwiched between the known upper bounds (dim J^flat(K) <= Tait(K) and qdim J^flat(K) <= qrank<K>_phi) to obtain exact values for two fullerene webs W2 and W3 (Theorem 4.2). For the dodecahedral web W1 the computation gives a lower bound of 58 while Tait(W1)=60, and the paper suggests, without claiming a proof, that dim J^flat(W1)=58. The paper also develops structural results (Corollary 3.1, Theorems 4.1 and 4.3) that make the computational results rigorous conditional on the correctness of the program's output.

Significance. If the computations are correct, the paper achieves the first nonreducible webs for which dim J^flat equals the Tait number, and the first computation of qdim J^flat equal to qrank of the phi-module for such webs. The dodecahedral web W1 becomes a concrete candidate for a negative answer to Khovanov-Robert's Question 2.1. The mathematical framework is clean and parameter-free, and the paper is explicit in distinguishing proved theorems from the heuristic suggestion for W1. The main weakness is that the central numerical claims rest on an unversioned external program that the reader cannot inspect or verify from the manuscript.

major comments (1)

[Section 3.2, Remark 3.6, Tables 2-3] The exact entries in Tables 2 and 3, on which Theorem 4.2 and the W1 discussion rest, are outputs of a Mathematica program that is not included in the submission; reference [3] is a bare URL with no version, checksum, or file name. The manuscript describes the algorithm at a high level but provides no test cases, no output files, and no certificates for the Smith normal form computations in Eqs. (23)-(32). Because a bug in facet enumeration, coloring enumeration, closed-foam evaluation, or the Smith normal form computation would invalidate Theorem 4.2, the central claim is not independently checkable as submitted. I request that the complete program be made available in a versioned form as supplementary material, and that the authors provide verifiable certificates for the table values (for example, the matrices S, B, T for the Smith decompositions, or an independent verification script).

minor comments (6)

[Section 4, proof of Theorem 4.2] The inference from ell(K)=Tait(K) to equality of the three quantum dimensions in Eq. (33) is valid but requires the observation that ell_q(1)=ell(K)=Tait(K)=qrank<K>_phi(1); since the inequalities in Eq. (33) are coefficientwise, this forces ell_q = qdim J^flat(K) = qrank<K>_phi. Please add a sentence spelling out this argument.
[Introduction, page 1] The sentence 'dim J^flat(W1) must be either 58 and 60' should read 'either 58 or 60'.
[Section 3.1] The statement 'For the empty web K = emptyset we take S(K) = emptyset' is ambiguous: if S(emptyset) is the empty set, the recursive construction for reducible webs cannot produce any half-foams. Please clarify whether S(emptyset) is intended to be the singleton set containing the empty half-foam, and make the convention explicit.
[Reference [3]] The reference to the computer program should be a complete citation with a specific file or URL, an access date, and a version or checksum, rather than a bare homepage URL.
[Table 2] The columns N_ell, N_e, and N are defined in the text but not in the caption; please include their definitions in the caption so that the table is self-contained.
[Remark 4.2] In the sentence 'if use only IH cobordisms' the word 'we' is missing; please fix this typographical error.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a computation of finite-rank lower bounds feeding into external inequalities by Khovanov–Robert, not a repackaging of the conclusion.

full rationale

The paper's central results, Theorem 1.1 and Theorem 4.2, are derived by computing lower bounds ell(K) and ell_q(K) from finite sets of half-foams and comparing them with the independently stated Tait numbers and the Khovanov–Robert rank bound. Corollary 3.1 supplies dim J^flat(K) <= Tait(K), Theorem 4.1 supplies ell_q(K) <= qdim J^flat(K), and Theorem 3.1 supplies rank< K >_phi = Tait(K). For W2 and W3 the table shows ell(K) = Tait(K), so the inequalities collapse to equalities. No parameter is fitted to the target dimension, and no conclusion is assumed in the computation: the bilinear-form ranks are computed directly from Khovanov and Robert's closed foam evaluation formula, which is an external proved result (arXiv:1808.09662), not a result of this paper. The lower bounds are genuine lower bounds by the linear-algebra argument of Remark 2.4, and the saturation value ell(K) is reported as a lower bound, not as an input. The only caveats are empirical and reproducibility-related rather than circular: the Mathematica program is referenced via the author's website [3] without a version or hash, so Tables 2 and 3 cannot be independently checked from the arXiv record, and the 'suggestion' that dim J^flat(W1) = 58 is an extrapolation from saturation after 6,727 of 11,160 half-foams. Neither of these involves defining a quantity in terms of the target result, fitting a parameter to a prediction, or importing a load-bearing conclusion from the author's own prior unverified work. The cited Kronheimer–Mrowka and Khovanov–Robert results are external mathematical theorems whose assumptions do not include the paper's computed values. Therefore the derivation chain is self-contained conditional on the correctness of the external evaluation formula and the computation, which is a correctness/reproducibility risk, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The numerical outputs are deterministic ranks computed from a fixed evaluation formula; the only external inputs are proved theorems from the literature and finite graph enumeration data.

assumptions (5)

domain assumption Khovanov and Robert's formula gives a well-defined closed foam evaluation J^b(F) and the associated functor J^b, with rank<K>_phi = Tait(K) for every web K.
Proved in [6]; used throughout to provide upper bounds and grading. The paper does not reprove it.
standard math The universal construction turns the closed foam evaluation into a TQFT-style functor with nondegenerate bilinear form on J^b(K).
Standard construction from Blanchet-Habegger-Masbaum-Vogel, used in [7] and Section 2.
standard math F[E] is a PID, so submodules of a free module are free and Smith normal form applies to matrices over F[E].
Used in Section 4 for Smith decomposition and in Theorem 4.3 for full-rank free submodules.
domain assumption The reduction algorithm for reducible webs yields bases of half-foams whose images span J^b(K') in the reducible case.
From [6]; used to generate the finite sets S(K) for nonreducible webs via local moves.
domain assumption The listed Tait numbers and fullerene enumeration data for W1 through W7 are correct.
These finite counts are used as upper bounds; a miscount would affect Theorem 4.2 and the W1 analysis.

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Cite this review

Pith. "Pith review of Computer Bounds for Kronheimer-Mrowka Foam Evaluation." pith.science (2026). https://pith.science/paper/62GOM4DC

@misc{pith2026190807133,
  author       = {Pith},
  title        = {Pith review of: Computer Bounds for Kronheimer-Mrowka Foam Evaluation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/62GOM4DC}},
  note         = {Machine review of arXiv:1908.07133}
}

abstract

Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. Their approach is based on a functor $J^\sharp$, which they define using gauge theory, from the category of webs and foams to the category of vector spaces over the field of two elements. They also consider a possible combinatorial replacement $J^\flat$ for $J^\sharp$. Of particular interest is the relationship between the dimension of $J^\flat(K)$ for a web $K$ and the number of Tait colorings $\mathrm{Tait}(K)$ of $K$; these two numbers are known to be identical for a special class of "reducible" webs, but whether this is the case for nonreducible webs is not known. We describe a computer program that strongly constrains the possibilities for the dimension and graded dimension of $J^\flat(K)$ for a given web $K$, in some cases determining these quantities uniquely. We present results for a number of nonreducible example webs. For the dodecahedral web $W_1$ the number of Tait colorings is $\mathrm{Tait}(W_1) = 60$, but our results suggest that $\dim J^\flat(W_1) = 58$.

Figures

Figures reproduced from arXiv: 1908.07133 by the authors.

**Figure 1.** Local models for closed foams F. (a) Local model near a regular point p ∈ F. (b) Local model near a seam point p ∈ F. (c) Local model near a tetrahedral point p ∈ F. singular cobordism between two webs. More precisely, a closed foam F is a singular 2D surface embedded in R3 in which every point p ∈ F has a neighborhood that takes the form of one of three local models shown in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Local replacements K → K0 for reducible webs. The web K0 is identical to K outside the depicted region. (a) Disk elimination K → K0 1 . (b) Bigon elimination K → K0 2 . (c) Triangle elimination K → K0 3 . (d) Square elimination K → K0 4a and K → K0 4b . to the local replacements shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Elementary cobordisms for reducible webs. The cobordisms are taken to be the identity outside of the depicted region. Note that the cobordisms C˙ 1, C¨ 1, and C˙ 2 have facets that are decorated with dots. Remark 3.3. For simplicity, we do not actually use the above algorithm for reducible webs to construct the generating sets S(K0 ) for reducible webs K0 . Rather, given a reducible web K0 we apply a recursive algor… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Local replacements K → K0 for nonreducible webs. The web K0 is identical to K outside the depicted region. (a) Zip. (b) Unzip. (c) Saddle. (d) IH. and define R = F[E1, E2, E3] to be the subring of R0 consisting of symmetric polynomials in X1, X2, and X3. Given a closed…

**Figure 5.** Figure 5: Elementary cobordisms for nonreducible webs. The cobordisms are taken to be the identity outside of the depicted region. (a) Zip. (b) Unzip. (c) Saddle. (d) IH. for half-foams F1 and F2 with top boundary K. To obtain the facets of F, we enumerate the facets of F1 and F…

**Figure 6.** Figure 6: Example webs. The web W1 is the dodecahedral web. that we constructed via the algorithm described in Section 3.1. The lower bound remains 58 even after examining Ne = 6 727 of the N = 11 160 half-foams we constructed, which suggests that dim J [ (W1) = 58. If so, this …

**Figure 7.** Figure 7: Graph of `n(W1) versus n for the dodecahedral web W1. only Unzip cobordisms we obtain `(W1) = 58, if we use only Saddle cobordisms we obtain `(W1) = 58, and if use only IH cobordisms we obtain `(W1) = 58. This fact provides additional evidence that perhaps dim J [ (W1)…

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Works this paper leans on

11 extracted references · 7 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [3]

A. D. Boozer, Computer program for ﬁnding lower bounds on J ♭(K), http://www.math.ucla.edu/~davidboozer. COMPUTER BOUNDS FOR KRONHEIMER-MROWKA FOAM EV ALUATION 15

work page

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Appel and W

K. Appel and W. Haken, Every planar map is four colorable , volume 98 of Contemporary Mathematics. American Mathe- matical Society, Providence, RI, 1989

work page 1989

[3] [2]

Blanchet, N

C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological quantum ﬁeld theories derived from the Kauﬀman bracket, Topology, 34(1995) 883–927

work page 1995

[4] [4]

Brinkmann and W

G. Brinkmann and W. M. Dress, A Constructive Enumeration of Fullerenes , J. Algorithms, 23(1997) 245–358

work page 1997

[5] [5]

Gonthier, Formal Proof–The Four-Color Theorem, Notices of the American Mathematical Society, 55(2008) 1382–139

G. Gonthier, Formal Proof–The Four-Color Theorem, Notices of the American Mathematical Society, 55(2008) 1382–139

work page 2008

[6] [6]

Khovanov and L.-H

M. Khovanov and L.-H. Robert, Foam evaluation and Kronheimer–Mrowka theories , preprint, arXiv:1808.09662

arXiv

[7] [7]

P. B. Kronheimer and T. S. Mrowka, Tait colorings, and an instanton homology for webs and foams , preprint, arXiv:1508.07205

arXiv

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P. B. Kronheimer and T. S. Mrowka, Exact triangles for SO(3) instanton homology of webs , preprint, arXiv:1508.07207

work page Pith review arXiv