On the statistical nature of Betti numbers and Euler characteristic of smooth random fields

Pravabati Chingangbam

REVIEW 4 major objections 6 minor 74 references

On the statistical nature of Betti numbers and Euler characteristic of smooth random fields

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Betti numbers of smooth random fields can be read as sums of independent coin-flip-like counting variables.

desk verdict A clean bookkeeping device plus a useful observation about bsum's stability, but the Binomial independence model contradicts the paper's own measured negative covariance and the CLT is heuristic; deserves referee time, but the distributional claims should not survive review as they stand. read the letter →

arxiv 2507.06255 v1 pith:LHKNRCCQ submitted 2025-07-07 math.ST astro-ph.COmath-phmath.MPmath.PRstat.TH

Pravabati Chingangbam This is my paper

classification math.STastro-ph.COmath-phmath.MPmath.PRstat.TH MSC 60G6060F0562E20

keywords BettinumbersEulercharacteristicexcursionsetsBinomialdistributionGaussianrandomfieldstopologicalstatisticscentrallimittheoremcosmology

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 claims that the random topology of a smooth random field can be understood through a small set of counting variables. Any excursion set—the region where the field exceeds a threshold—is written as a union of basic building blocks (disks, disks with holes, and so on), and the numbers $m_j$ of each building block are modelled as independent Binomial variables. Betti numbers, the Euler characteristic, and the sum of Betti numbers are then literal sums of these $m_j$'s, which makes their probability distributions follow from simple combinatorial ones and become Gaussian when the field of view divided by the correlation length is large. If true, this gives cosmologists a principled reason for when Gaussian error bars on topological statistics are trustworthy and when they are not.

What carries the argument

The topological basis $B=\{Q_0,Q_1,Q_2,\ldots\}$, where each $Q_j$ is a connected component with exactly $j$ holes (in three dimensions, additional indices track handles and cavities), together with a colored-balls-in-a-box model that realizes each coefficient $m_j$ as the count of draws of color-$j$ balls in $N$ trials, i.e., a Binomial variable. The identities $b_0=\sum_j m_j$, $b_1=\sum_j j m_j$, $\chi=\sum_j(1-j)m_j$, and $b_{\rm sum}=\sum_j(1+j)m_j$—obtainable from the generating function $h(\alpha)=\sum_j m_j e^{-j\alpha}$—are what carry the argument, because they reduce the statistical question about topology to the classical central limit theorem for sums of independent discrete variables.

What would settle it

Simulate many realizations of a zero-mean Gaussian field on a fixed grid, compute the excursion sets at a threshold, and count the basis coefficients $m_0,m_1,\ldots$ across realizations. If the sample covariance of, say, $m_0$ and $m_1$ is systematically nonzero beyond Monte Carlo noise, the independence assumption fails and the predicted product-form PDFs do not hold. Alternatively, at a threshold where $N\simeq 1$ (e.g., $\nu>3$), the Binomial model predicts variance $Np(1-p)$; a measured variance that departs from this with the same mean signals a misspecified model.

Watch

Extended reading notes

Core claim

The paper's central claim is that the coefficients $m_j$ of the topological basis are Binomial random variables. With $b_0 = \sum_j m_j$, $b_1 = \sum_j j m_j$, $\chi = \sum_j (1-j)m_j$, and $b_{\rm sum} = \sum_j (1+j)m_j$, the probability distribution of each topological statistic is determined by the product of Binomial distributions of the $m_j$'s. Assuming independence and a growing number of building blocks, a central limit argument shows that $b_0,b_1,\chi,b_{\rm sum}$ become Gaussian at each threshold as $q\equiv (L/r_c)^d\to\infty$. The paper tests this model on Gaussian temperature maps and finds the Binomial and Gaussian descriptions fit the simulated PDFs well except at high thresholds where the effective sample size $N$ is of order one.

Load-bearing premise

The whole distributional story rests on the assumption that the coefficients $m_j$ are statistically independent Binomial variables with parameters $(p_j,N)$ set by the field; the paper does not derive this from the field's probability structure.

Editorial extensions

If this is right

Topological statistics are asymptotically Gaussian at each threshold provided $q\to\infty$, with the convergence rate depending on threshold through $j_{\max}(\nu)$.
At intermediate thresholds, the PDFs of $b_0,b_1,\chi,b_{\rm sum}$ are well approximated by Binomial or Gaussian models, while at $|\nu|>3$, where $N$ is of order one, the Gaussian approximation breaks down.
The anti-correlation between $b_0$ and $b_1$ makes $b_{\rm sum}$ have lower variance than either, which the paper notes could yield tighter cosmological parameter constraints.
The Binomial model explains the threshold dependence: at $\nu>1$, $b_0\sim\chi\sim b_{\rm sum}\sim m_0$ with $p\sim p_0$, while at $\nu<-1$, $b_1\sim -\chi\sim b_{\rm sum}\sim j_{\max}\,m_{j_{\max}}$.
The 3D generalization expresses $b_0,b_1,b_2,\chi,b_{\rm sum}$ as sums of coefficients $m_{ij_0j_1\ldots}$, with the same Binomial modeling and the same asymptotic Gaussian behavior restricted to $q\to\infty$.

Reading between the lines

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

If the Binomial model holds, the full distribution of any topological statistic is determined by its mean and variance at each threshold; this two-parameter description could let observers convert a measured value into a statistical significance without running simulations.
The independence assumption could be relaxed to block independence; a direct measurement of covariances between $m_j$'s on synthetic fields would settle whether the paper's variance formulas need correction terms.
Because the basis coefficients count components by their number of holes, the same Binomial logic may constrain the fluctuations of persistence diagrams, whose birth-death counts share a similar combinatorial structure.
The paper claims the Binomial form is general, but its numerical test is limited to Gaussian fields; checking the model on lognormal or Rayleigh random fields is a direct next test of the claim.

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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

Colleague,

The paper is easy to summarize: excursion sets are written as unions of basic pieces Q_j (disks with j holes), with nonnegative integer coefficients m_j. Betti numbers, Euler characteristic, and bsum are linear combinations of the m_j's, and the author proposes to model the m_j's as independent Binomial variables so that the topological statistics inherit their distributions as sums of Binomials. There is a real kernel of value here: the representation is a clean bookkeeping device, the generating function h(α) is neat, and the numerical study of 10^4 CMB maps is solid work. The practical observation that bsum has smaller variance than b0 or b1 because b0 and b1 are anti-correlated is useful and likely robust.

The soft spots are concentrated in the probabilistic content. The independence assumption on the m_j's is not just unproved; it is contradicted by the paper's own numbers. Under independence, Cov(b0,b1) = Σ j Var(m_j) ≥ 0, but Fig. 4 shows negative covariance. The colored-ball analogy the author invokes points the same way: counting colors from N draws gives multinomial counts, which are negatively correlated, not independent. So Eq. (44), the product-form PDF for b0, and the Binomial-sum distributions for b1, χ, and bsum rest on an assumption that the author's own data falsify. The later claim that cross-correlation corrections to means and variances are 'likely small' is hard to square with the fact that the negative covariance visibly changes σ_χ and σ_bsum (Eqs. 28–29).

The CLT section is heuristic. It invokes block-sum independence and boundedness but never verifies that the variance sums diverge or that the block sums are actually independent. The result is plausible, but it is a sketch, not a theorem, and it does not engage with the rigorous CLTs already in Owada–Thomas and Owada–Samorodnitsky (refs 56/57) beyond citing them.

The Binomial validation is partly circular in the way the reader says: N and p_j are calibrated from the ensemble mean and variance of each statistic, and the same numerical PDFs are then used for comparison. That leaves only the skewness/kurtosis as a true test, which is weaker than the paper implies. A two-parameter Binomial matching mean and variance can look reasonable even when the independence structure is wrong.

There are also small slips: the text says Binomial corresponds to drawing without replacement (it is the opposite; hypergeometric is without replacement), and there is a typo in the binomial coefficient in Eq. (47).

Who is this for? Cosmologists using Betti numbers for error bars would benefit from a correct version of this, but they should not yet trust the Binomial-sum formulas. The paper deserves a serious referee because the problem is important, the numerical work is substantial, and the representation could be the starting point for a proper treatment. I would send it to review, but with the expectation that the distributional claims need to be re-derived with dependence included (e.g., multinomial) or substantially weakened. As it stands, reject.

Referee Report

4 major / 6 minor

Summary. The paper proposes a representation of the excursion sets of smooth random fields as finite unions of a topological basis, with nonnegative integer coefficients m_j counting the numbers of connected components of each homological type. Betti numbers b0 and b1, the Euler characteristic χ, and the sum of Betti numbers bsum are then expressed as linear combinations of these coefficients. The paper assumes that the m_j are statistically independent and models each as a Binomial variable, and it uses this to derive that the topological statistics are asymptotically Gaussian when the packing parameter q=(L/r_c)^d tends to infinity. The claims are tested numerically on Gaussian CMB maps at several smoothing scales, and the paper identifies threshold regimes where Gaussian approximations hold. The central assertion is that the Binomial nature of the coefficients controls the statistical nature of the topological statistics.

Significance. If the distributional claims were sound, the paper would provide a useful framework for attaching error bars to Betti numbers and Euler characteristics in cosmological analyses, and the observation that bsum has smaller variance than b0 or b1 separately is potentially valuable. The topological-basis representation itself is a transparent bookkeeping device, and the numerical experiments are extensive and clearly described. However, the paper provides no derivation of the Binomial model, its own numerical results contradict the independence assumption on which the model rests, and the numerical validation is circular because the model parameters are calibrated from the very quantities used for comparison. The central probabilistic content is therefore not established, and the asymptotic Gaussian conclusion is unsupported.

major comments (4)

[Sec. IV.B and Sec. IV.B.1, Eqs. (19), (30), (34)] The independence assumption stated in Sec. IV.B is contradicted by the paper's own numerical results. Since b0 = Σ m_j and b1 = Σ j m_j, if the m_j are independent then Cov(b0,b1) = Σ_{j≥1} j Var(m_j) ≥ 0. The paper reports Cov(b0,b1) < 0 from the simulations (Fig. 4, Eq. (34)). This makes the factorization in Eq. (19) and the product formula in Eq. (44) internally inconsistent with the model used to interpret the numerical data. The discussion following Eq. (34) attempts to reinterpret the inequality through the variances, but it does not repair the sign contradiction with independence.
[Sec. IV.B.3 and Sec. IV.B.4] The colored-ball model of Sec. IV.B.3 describes drawing N balls from a box with distinct colors, which yields multinomial counts with Cov(m_j,m_k) = -N p_j p_k < 0 for j ≠ k. The coefficients are therefore negatively dependent, not independent, under the very analogy used to justify the Binomial assumption. Moreover, the note in Sec. IV.B.4 that Binomial modeling corresponds to 'Bernoulli trials without replacement' is erroneous: Binomial counts arise from independent trials (sampling with replacement, or infinite population), while sampling without replacement gives hypergeometric counts. The statistical model is thus internally inconsistent.
[Sec. IV.B.4, Eqs. (55)-(56), (61)-(62)] The numerical validation is circular. The values of N(ν) and p_j(ν) are fixed by equating the Binomial formulas for the mean and variance to the analytic mean of χ and the numerically computed standard deviation σ_χ (or, in the intermediate regime, to the ensemble mean and standard deviation of each statistic). The resulting Binomial PDFs are then compared to the numerical PDFs obtained from the same ensemble. A two-parameter family will generally match the first two moments of a given distribution; this procedure does not test whether the m_j are Binomial, does not test independence, and does not test the product structure of Eq. (44). The agreement displayed in Fig. 5 is therefore not evidence for the central model.
[Sec. IV.B.2] The central limit theorem argument is not a proof. It invokes block-sum independence without specifying the dependence structure or verifying a mixing condition for the m_j's, and it does not verify that the variance of the sum diverges. Boundedness of the summands is not sufficient for asymptotic normality; a Lindeberg-type condition is required. More fundamentally, since the m_j are dependent for the Gaussian fields studied in the paper (as shown by the negative covariance in Sec. IV.B.1), the classical CLT does not apply to b0, b1, χ, or bsum. The conclusion that these statistics 'tend to Gaussian random variables provided q → ∞' is therefore unsupported.

minor comments (6)

[Sec. III] There is a typo: 'distinguishuable' should be 'distinguishable'.
[Sec. IV.B.3] The notation 'N C_k p^k (1-p)^{N-k}' is nonstandard; the binomial coefficient is usually written '\binom{N}{k}' or 'C(N,k)'.
[Sec. IV.B.4] The statement that 'b0(ν) = b1(−ν)' as a reflection of Alexander duality is an approximate identity for finite maps and should be qualified as approximate; it is not an exact equality for a single finite-volume realization.
[References] Reference [2] is listed with a title and URL identical to Reference [1]; the correct reference is R. J. Adler and J. E. Taylor, 'Random Fields and Geometry' (Springer, 2007).
[Sec. IV.B.4] The phrase 'we are assuming Bernoulli trials without replacement' is not only incorrect but also unconceptual; consider rewriting the modeling discussion to distinguish independent trials (Binominal) from finite-population sampling (hypergeometric).
[Fig. 5 caption] The caption states that 'the values of N, p shown are obtained using the ensemble expectation and standard deviation of each statistic, at each ν, as inputs in the Binomial distribution.' This explicitly confirms the circular calibration procedure and should be flagged as a limitation of the validation.

Circularity Check

2 steps flagged · score 6.0 of 10

The Binomial 'validation' fits N and p to the very numerical moments of the statistics whose PDFs are then compared, so the agreement is a two-moment calibration rather than an independent prediction.

fitted input called prediction [Sec. IV.B.4, Eqs. (51)-(56) and Fig. 5 (large positive ν)]
"Using the right hand side of Eq. (31) and equating it to ⟨b0(ν)⟩ ∼ p0(ν)N(ν), we get p0(ν) = A2/N(ν) νe^{−ν^2/2}. ... Then, using numerically computed σχ and equating with the right hand side of Eq. (56), we can obtain N(ν). ... The solid orange lines represent the corresponding Binomial distributions, which fit the numerical PDFs well."

In this regime the paper itself sets b0 ∼ m0 ∼ χ ∼ bsum (Eqs. 51-54), so the plotted Binomial curve for b0, χ and bsum is built with N and p0 chosen to reproduce the analytic mean ⟨χ⟩ and the numerically computed variance σχ². Since b0=χ=bsum when b1≈0, the comparison is between the empirical PDF of a statistic and a two-parameter Binomial whose mean and variance were fixed using that same statistic (or its exact proxy). The first two moments match by construction; calling the resulting agreement a validation of the Binomial model for the m_j is a fitted-input-called-prediction step, not an independent test.
fitted input called prediction [Sec. IV.B.4, Fig. 5 caption and intermediate-regime discussion (−2 < ν < 2)]
"Nevertheless, we assume b0 and b1 to be single Binomial variables and estimate N and p (shown in the panels) using their numerically computed ensemble means and standard deviations as inputs. The solid magenta lines represent the corresponding Binomial distributions ... At |ν| = 1, the magenta and black lines agree well with the numerical PDFs."

Here N and p for each statistic are fit to that same statistic's numerical mean and variance, and the resulting Binomial PDF is then compared with the numerical PDF of the same statistic. Any two-parameter distribution matched to the empirical mean and variance will reproduce the location and width of the empirical PDF by construction, so the agreement is a check of the assumed Binomial shape, not a prediction from the m_j product structure of Eq. (44). The Binomial model's product form and independence assumption are not independently tested by this in-sample calibration.

full rationale

The paper's asymptotic Gaussianity claims rest on a standard CLT argument applied to sums of independent, bounded discrete variables, and are not themselves circular: given the stated block-sum independence and boundedness assumptions, the conclusion follows from a textbook theorem. The main circularity is confined to the Binomial modeling section. There, the 'validity test' for the Binomial model calibrates the free parameters N and p from the analytic mean of χ and the numerically computed standard deviation of χ (or, in the intermediate regime, from the numerical mean and standard deviation of each statistic individually), then compares the resulting Binomial PDFs with the same numerical PDFs. In the high-|ν| regimes, the statistics being compared are effectively identical to χ or −χ, so the Binomial curve is fitted to the target statistic and then presented as agreement. This is a fitted input labelled as validation rather than an independent prediction. No load-bearing self-citation or uniqueness-imported-from-authors pattern appears: self-citations are contextual, and the central CLT step does not depend on them. The separate issue that the paper's own reported negative Cov(b0,b1) is inconsistent with the independence assumption on which Eq. (44) rests is a correctness concern, not a circularity of the derivation chain.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation is a bookkeeping identity plus a Binomial ansatz. The identities are exact, but all probabilistic content flows from unverified assumptions about the coefficients.

free parameters (2)

N(ν) = fitted numerically via Eqs. (55)-(56)/(61)-(62) using σχ
The total number of Bernoulli trials in the Binomial model; not derived from the field, set to match the measured standard deviation of χ at each threshold.
p_j(ν) = p0 and pjmax computed from analytic ⟨χ⟩ and fitted N via Eqs. (55),(61)
Success probabilities for the Binomial model of each topological basis element; determined semi-empirically, varying with threshold and smoothing scale.

assumptions (4)

ad hoc to paper Statistical independence of the coefficients m_j at each threshold
Assumed in Sec. IV.B to factor the joint PDF; explicitly acknowledged as a simplifying assumption, not derived from the field.
ad hoc to paper Binomial distribution of each m_j
Postulated in Sec. IV.B.3 via a balls-in-a-box analogy; no derivation from the random field's probability structure.
ad hoc to paper Block-sum independence and divergence of variance sums for the CLT
The CLT proof in Sec. IV.B.2 requires m_j to be block-sum independent with s_n → ∞; these conditions are stated but not proven for excursion set coefficients.
domain assumption Finite jmax bounded by packing fraction q
Assumed in Sec. IV.A that jmax is finite and set by the parameter q = (L/r_c)^d, excluding configurations with unbounded isoperimetric ratios.

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Cite this review

Pith. "Pith review of On the statistical nature of Betti numbers and Euler characteristic of smooth random fields." pith.science (2026). https://pith.science/paper/LHKNRCCQ

@misc{pith2026250706255,
  author       = {Pith},
  title        = {Pith review of: On the statistical nature of Betti numbers and Euler characteristic of smooth random fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHKNRCCQ}},
  note         = {Machine review of arXiv:2507.06255}
}

read the original abstract

We represent excursion sets of smooth random fields as unions of a topological basis consisting of a sequence of simply and multiply connected compact subsets of the underlying manifold. The associated coefficients, which are non-negative discrete random variables, reflect the randomness of the field. Betti numbers of the excursion sets can be expressed as summations over the coefficients, and the Euler characteristic and the sum of Betti numbers can also be expressed as their (alternating) sum. This enables understanding their statistical properties as sums (or differences) of discrete random variables. We examine the conditions under which each topological statistic can be asymptotically Gaussian as the size of the manifold and the resolution increase. The coefficients of the basis elements are then modeled as Binomial variables, and the statistical natures of Betti numbers, Euler character and sum of Betti numbers follow from this fundamental property. We test the validity of the modeling using numerical calculations, and identify threshold regimes where the topological statistics can be approximated as Gaussian variables. The new representation of excursion sets thus maps the properties of topological statistics to combinatorial structures, thereby providing mathematical clarity on their use for physical inference, particularly in cosmology.

Figures

Figures reproduced from arXiv: 2507.06255 by the authors.

**Figure 2.** FIG. 2. Examples of basic topological spaces in three di [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Examples of excursion sets (yellow regions) at three [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Ensemble expectations (left) and standard deviations (middle) of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. PDFs of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

An example is a power law form with spectral index α

P (k) does not contain any intrinsic physical cut- off scale other than L and Rs. An example is a power law form with spectral index α. This case is of interest in cosmology because cosmological fields are assumed to have evolved from primordial den- sity fluctuations with nearly scale invariant power spectrum within the inflationary paradigm. In this cas...

work page

[2] [2]

Such exam- ples arise when some physical process erases small and/or lager scale fluctuations

P (k) has a form that contains intrinsic small scale cut-off at some k = ks > R−1 s , or large scale cut- off at some k = kL < L−1, or both. Such exam- ples arise when some physical process erases small and/or lager scale fluctuations. In this case we have rc = rc(k−1 L , k−1 p ), and it is independent of L, Rs. Next, let us define a ‘packing fraction’ pa...

work page

[3] [3]

For this purpose, we simulate 104 Gaussian CMB temperature fluctuation maps whose pixel resolution is set by the Healpix parame- ter Nside = 512

Comparison of standard deviations of the topological statistics for Gaussian fields Before using the topological basis further we examine numerically computed ensemble expectations and stan- dard deviations of b0, b1, χ and bsum. For this purpose, we simulate 104 Gaussian CMB temperature fluctuation maps whose pixel resolution is set by the Healpix parame...

work page

[4] [4]

For this purpose let us reca- pitulate the central limit theorem (CLT) for statistically independent discrete random variables [63, 64]

Central limit theorem for topological statistics We are interested in the behaviour of b0, b1, χ and bsum in the limit of large jmax. For this purpose let us reca- pitulate the central limit theorem (CLT) for statistically independent discrete random variables [63, 64]. LetXi be n independent discrete random variables that are not nec- essarily identicall...

work page

[5] [5]

In this case, for resolution q, the variable having the largest (finite) support is mjmax , and it satisfies the bound mjmax ≤ jmax

Then X = b1. In this case, for resolution q, the variable having the largest (finite) support is mjmax , and it satisfies the bound mjmax ≤ jmax. Therefore, b1 must also be asymptotically Gaussian for jmax → ∞. CLT for χ and bsum: For χ, let m′ j = (1 − j)mj. Then χ can be written as χ = jmaxX j=0 m′ j. (39) In this case also |m′ j| are bounded, with the ...

work page

[6] [6]

We now take a different approach and focus on the nature of mj’s

Modeling mj’s as Binomial distributed variables In practical applications the limit q → ∞discussed in the previous subsection is not of much use since observed data covers finite spatial extent and finite resolution. We now take a different approach and focus on the nature of mj’s. Consider a box containing balls of distinct colours. Let each element of t...

work page

[7] [7]

Specifically we focus on Gaussian random fields, for which the closed form analytic formula for ⟨χ⟩ is available

Testing the Binomial modeling for the case of Gaussian fields Since pj(ν)’s and N (ν) are not known in general, we now adopt a semi-numerical approach. Specifically we focus on Gaussian random fields, for which the closed form analytic formula for ⟨χ⟩ is available. This allows us to test the validity of the Binomial model in relevant threshold regimes. As...

work page

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I. Gott, J. Richard, A. L. Melott, and M. Dickinson, ApJ 306, 341 (1986)

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