Pith. sign in

REVIEW 3 major objections 4 minor 3 cited by

Stable Real-Space Invariants and Topology Beyond Symmetry Indicators

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper introduces stable real-space invariants that fully classify when atomic insulators are stably equivalent, determine all symmetry indicators, and diagnose almost every split elementary band representation.

desk verdict Real-space invariants finally classify stable equivalence of atomic insulators, but the one-to-one map to symmetry data hinges on an explicitly unproven identity. read the letter →

arxiv 2505.09697 v2 pith:66QTTF5L submitted 2025-05-14 cond-mat.mes-hall cond-mat.mtrl-sci

Yoonseok Hwang , Vaibhav Gupta , Frank Schindler , Luis Elcoro , Zhida Song , B. Andrei Bernevig , Barry Bradlyn This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords topologicalquantumchemistrystablereal-spaceinvariantssymmetryindicatorselementarybandrepresentationsWyckoffpositionsWannierfunctionssplitEBRsadiabaticequivalence
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 introduces stable real-space invariants (SRSIs): integer and modulo-$n$ linear combinations of Wannier-orbital counts at the symmetry-distinct points of a crystal, computed from the Smith normal form of a matrix that lists all symmetry-preserving orbital moves. It argues that these invariants completely classify the stable equivalence of atomic insulators: two atomic insulators with matching SRSIs can always be adiabatically deformed into one another after adding the same auxiliary trivial bands, and conversely. It further claims that the integer SRSIs are in one-to-one correspondence with momentum-space symmetry data, so they determine every symmetry indicator, while the $\mathbb{Z}_2$- and $\mathbb{Z}_4$-valued SRSIs carry information momentum space cannot see. Applied to all known split elementary band representations, the $\mathbb{Z}_n$ SRSIs certify band topology in 203 of 211 cases across 51 space groups, the 8 remaining exceptions all occurring with spin-orbit coupling. If correct, this gives a real-space classification that contains the symmetry-indicator framework and detects topological gaps that momentum-space data miss.

What carries the argument

The load-bearing object is the adiabatic-process matrix $q$, whose columns are the elementary symmetry-preserving moves of Wannier orbitals between connected Wyckoff positions, together with its Smith decomposition $q=L\Lambda R$. The SRSIs are rows of $L^{-1}$: rows paired with zero elementary divisors give $\mathbb{Z}$-valued invariants, and rows paired with divisors $n=2,4$ give $\mathbb{Z}_n$-valued invariants modulo $n$. Because every adiabatic process changes the irrep-multiplicity vector $p$ by an integer combination of columns of $q$, these combinations are unchanged by construction. The same matrix equation, combined with the band-representation matrix $BR$ mapping real-space multiplicities to momentum-space little-group irreps, is what lets the paper prove the one-to-one ZSRSI-to-symmetry-data correspondence and derive the split-elementary-band-representation criterion.

What would settle it

Search the kernel of the band-representation matrix in any of the 230 space groups for a vector $p_0$ with $\Theta^{(0)} p_0 \neq 0$; if one exists, two atomic insulators with identical momentum-space symmetry data but different integer SRSIs can be constructed, and the claimed one-to-one mapping fails. On the split-EBR side, compute the Wilson loops of the 8 exceptional cases; finding a trivial, non-winding Wilson spectrum for both the valence and conduction bands without any large-gauge transformation would show that matching SRSIs do not always certify triviality, while finding winding would confirm the paper's diagnosis.

Watch

Extended reading notes

Core claim

The paper's central claim is that stable equivalence of band representations—equivalence up to adding the same set of trivial atomic bands—is fully diagnosed by the SRSIs. For any atomic insulator, assemble a vector $p$ of site-symmetry-irrep multiplicities at all Wyckoff positions, and collect all adiabatic deformations into a matrix $q$; the Smith decomposition $q=L\Lambda R$ produces invariant combinations $\theta=(L^{-1}p)$ reduced mod the elementary divisors. The authors prove that two atomic insulators have matching SRSIs if and only if they are adiabatically deformable into each other in the presence of auxiliary trivial bands. They then establish, by exhaustive computation for all 230 nonmagnetic space groups with and without spin-orbit coupling, that the $\mathbb{Z}$-valued SRSIs are in one-to-one correspondence with momentum-space symmetry data, hence determine the symmetry indicators; the $\mathbb{Z}_n$-valued SRSIs are not fixed by momentum data and serve as sufficient criteria for non-symmetry-indicated topology. This diagnoses all but eight of 211 split elementary band representations in 51 space groups, with the eight exceptions being cases where the split representation is stably equivalent to the proposed sum of elementary band representations.

Load-bearing premise

The load-bearing premise is that no 'hidden' orbital configuration exists: the authors verify numerically, in every space group and spin-orbit setting, that any configuration with zero momentum-space symmetry data also has zero integer SRSI, but they leave an analytic proof of this to future work, and the whole classification further assumes the listed elementary orbital moves generate every symmetry-preserving adiabatic process.

Editorial extensions

If this is right

  • Two atomic insulators with equal SRSI values are guaranteed to be connected by an adiabatic path once the same auxiliary trivial bands are added on both sides, so the SRSIs give a complete stable-equivalence classification in real space.
  • Because the $\mathbb{Z}$-SRSIs determine the symmetry-data vector, every symmetry indicator can be written as a linear function of the $\mathbb{Z}$-SRSIs; fractional $\mathbb{Z}$-SRSIs and nonzero symmetry indicators become the same statement.
  • A mismatch in $\mathbb{Z}_n$ SRSIs between a split band representation and its proposed decomposition is a sufficient criterion for topology, so band splittings that look trivial in momentum space can still be certified topological.
  • The method resolves almost all known split-elementary-band-representation cases (203 of 211), and the eight exceptions are precisely the cases where the split representation is stably equivalent to the sum.
  • The framework subsumes earlier local and composite real-space invariants and matches the structure of the $E^2_{0,0}$ page of the real-space Atiyah–Hirzebruch spectral sequence, giving an explicit construction of those invariants.

Reading between the lines

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

  • If the one-to-one mapping holds, symmetry-indicator-based materials databases could in principle be re-expressed in terms of occupancy counts at Wyckoff positions, which would make high-throughput searches for 'topological-only' candidates a real-space calculation once automated from Wannier functions.
  • The same Smith-decomposition construction should extend to magnetic space groups beyond the paper's illustrative example; K-theory work cited by the authors suggests magnetic groups could harbor adiabatic equivalences that produce new $\mathbb{Z}_n$ invariants, possibly with $n>4$.
  • The eight exceptional split-EBR cases are natural targets for fragile topology or large-gauge transformations; if no large-gauge equivalence exists in those cases, then SRSIs are not quite complete as a diagnostic and an additional invariant is needed to close the gap.
  • A practical testable extension would be to compute SRSIs from first-principles Wannier functions for the obstructed atomic insulator the paper analyzes and check whether the predicted real-space mismatch appears in the realistic material's occupied subspace.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper introduces stable real-space invariants (SRSIs), obtained from the Smith normal form of a matrix q that encodes adiabatic processes between site-symmetry irreps at Wyckoff positions. The authors prove that two atomic insulators are stably equivalent if and only if their SRSIs match, generalize earlier local and composite RSIs, and enumerate SRSIs for all 230 nonmagnetic space groups with and without spin-orbit coupling. They further claim that the Z-valued SRSIs are in one-to-one correspondence with momentum-space symmetry-data vectors, hence determine all symmetry indicators, while the Zn-valued SRSIs provide information not captured by momentum-space data. The framework is applied to 211 split elementary band representations in 51 space groups, diagnosing band topology in all but 8 cases, and is illustrated with tight-binding models in SG P 41′.

Significance. If the central claims hold, this work provides a comprehensive real-space classification that goes beyond symmetry indicators, gives an explicit construction of the E2_{0,0} page of the real-space Atiyah–Hirzebruch spectral sequence, and offers practical diagnostics for non-symmetry-indicated topology. The Smith-decomposition derivation in Sec. III and the stable-equivalence theorem in Sec. IV are clean and well presented, and the exhaustive tables in the Supplemental Material are a valuable resource. The split-EBR analysis is timely and the tight-binding examples are convincing. However, the one-to-one mapping between ZSRSIs and symmetry-data vectors rests on an identity that the authors state is only numerically verified, not proven; this gap is load-bearing for the paper's headline claim that ZSRSIs determine all symmetry indicators.

major comments (3)
  1. [SM SIII B 2; Eq. (S86)] The identity Θ^(0)·p0 = 0 for every p0 in ker BR is explicitly stated to be only numerically checked, with an analytic proof deferred (SM SIII B 2). This identity is load-bearing: it underlies the rank relation r_BR = Nρ_UC − rank(q) in Eq. (S87) and, via Eqs. (S88)–(S89) and Sec. V, the one-to-one correspondence between ZSRSIs and symmetry-data vectors, and hence the claim that ZSRSIs determine all symmetry indicators. Without an analytic proof or machine-checkable certificates covering all 230 space groups, the central claim is not fully established.
  2. [Sec. III; SM SIII A (definition of q)] The adiabatic-process matrix q is introduced as a finite basis that generates all adiabatic processes, but the completeness of this basis is assumed rather than proved. The stable-equivalence theorem in Sec. IV and SM SIII B 4 relies on the assertion that every adiabatic deformation corresponds to an integer linear combination of columns of q; a missing adiabatic process would break the 'if' direction of the matching-SRSI criterion. Please state precisely why the listed processes generate the full lattice of adiabatic deformations, or prove completeness from Wyckoff-position connectivity.
  3. [SM SVI tables] The exhaustive enumeration results are presented in large tables without accompanying code or machine-readable certificates. Because the Θ^(0)·p0 = 0 identity is only numerically verified, a single arithmetic error in a large space-group computation could invalidate the central conclusions; releasing the verification code and the generated data would make the exhaustive claims reproducible and would substantially strengthen the paper.
minor comments (4)
  1. [SM SIII B 1] The text contains an editorial artifact, '[YH: changed display of Eq. (S75)]', which should be removed from the published version.
  2. [Sec. IX, Fig. 3] The caption of Fig. 3 refers to 'double winding' of the Wilson loop without explaining what this implies for the fragile topology; a brief clarifying sentence in the caption or text would help the reader.
  3. [SM SIII C 1] The sentence 'In Sec. SII D, we apply Eqs. (S103) and (S108)...' appears to refer to the wrong section; Sec. SII D is about local RSIs, whereas the discussion of SRSIs relating to local RSIs is in Sec. SIII C.
  4. [References] Reference [66] is cited as 'In preparation'; the authors should either provide a public preprint or clarify the status of this work in the bibliography.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: SRSIs are constructed from the Smith normal form of q; the only load-bearing gap is an explicitly unproven numerical identity, which is a proof gap rather than a circular step.

full rationale

The paper's central construction is self-contained: SRSIs are defined as the invariants of the integer lattice generated by the adiabatic-process matrix q via its Smith normal form (Eqs. (2)-(3)), and the stable-equivalence theorem (SM SIII B 4) is a proof that two atomic insulators have matching SRSIs if and only if their symmetry-representation vectors differ by an integral combination of q columns in the presence of auxiliary bands. This is a mathematical characterization of the quotient lattice, not a renaming of the conclusion or a fit masquerading as a prediction. The one-to-one mapping between ZSRSIs and symmetry-data vectors (Sec. V, SM SIII B 3) rests on the identity Theta^(0)*p0 = 0 on ker BR; the paper explicitly states in SM SIII B 2: "We numerically checked that Theta^(0)*p0 = 0 holds for all SG with and without SOC. We leave a proof of this result analytically as a future research." That is an honest missing-proof flag, and it is a correctness risk for the mapping claim, but it is not circular: the identity is a property of two independently defined integer matrices, and no parameter is fitted from the data being predicted. The split-EBR diagnoses use ZnSRSI additivity as a derived sufficient criterion, and the 211-case enumeration is an independent algebraic computation rather than a restatement of the paper's inputs. Self-citations to prior TQC and RSI papers supply context and earlier theorems, but the new SNF-based invariants and the split-EBR tables do not reduce to those citations. A score of 2 reflects the presence of a load-bearing but unproven numerical check and heavy same-author contextual citation, not any circular reduction of the derivation.

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

No free parameters are fitted. The paper introduces no new physical entities; SRSIs are invariants derived from the Smith normal form, not postulated particles or forces. The load-bearing assumptions are the standard Wannier-locality premise, the completeness of the adiabatic-process matrix, and the numerically checked kernel-containment identity.

assumptions (4)
  • domain assumption An atomic insulator admits exponentially localized symmetric Wannier functions, so it is described by a non-negative integer site-symmetry representation vector p.
    This is the standard TQC premise (SM SI A); if false for some band structure, real-space multiplicity invariants do not apply.
  • domain assumption Every adiabatic process that preserves symmetries and the gap changes p by an integer vector in the column lattice of the adiabatic-process matrix q.
    The entire SRSI construction in Sec. III and SM SIII A depends on q generating all adiabatic processes; completeness is asserted, not proven.
  • ad hoc to paper The identity Theta^(0) * p0 = 0 holds for all nonmagnetic space groups with and without spin-orbit coupling.
    Used in Sec. V and SM SIII B 2 to prove the one-to-one ZSRSI/symmetry-data mapping; only numerically verified, with the analytic proof deferred.
  • standard math Smith normal form and Hermite normal form of integer matrices give complete invariants of the quotient lattice Z^N / im(q).
    Background used in Eqs. (2)-(3) and SM SIII A.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stable Real-Space Invariants and Topology Beyond Symmetry Indicators." pith.science (2026). https://pith.science/paper/66QTTF5L

@misc{pith2026250509697,
  author       = {Pith},
  title        = {Pith review of: Stable Real-Space Invariants and Topology Beyond Symmetry Indicators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66QTTF5L}},
  note         = {Machine review of arXiv:2505.09697}
}
abstract

We introduce stable real-space invariants (SRSIs), topological invariants defined from adiabatic deformations between Wannier states, generalizing previously discovered local and composite real-space invariants. SRSIs are $\mathbb{Z}$- and $\mathbb{Z}_n$-valued ($n=2,4$) linear combinations of Wannier state multiplicities characterizing the stable equivalence of atomic insulators. We enumerate all SRSIs in nonmagnetic space groups with and without spin-orbit coupling. $\mathbb{Z}$SRSIs are in one-to-one correspondence with momentum-space symmetry data and thus determine symmetry indicators of topology (SIs). $\mathbb{Z}_n$SRSIs capture real-space information beyond momentum-space symmetry data and SIs. Applying SRSIs to split elementary band representations (EBRs) whose symmetry data decomposes into positive sums of other EBR symmetry data, we diagnose the topology of all 211 cases across 51 space groups except for 8 exceptions in 5 space groups. Our results solidify Topological Quantum Chemistry beyond SIs and momentum-space symmetry data. Finally, we use SRSIs to diagnose an obstructed atomic insulator in a realistic material.

Figures

Figures reproduced from arXiv: 2505.09697 by the authors.

Figure 1
Figure 1. FIG. 1. RSIs in the SG [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Stable equivalence in SG [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Tight-binding models in SG [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Higher-Order Topological States with Cleavage-Dependent Dirac Mass

    cond-mat.mes-hall 2026-08 conditional novelty 6.0 of 10

    Corner states with e/2 charges appear on half the corners of a cleaved obstructed atomic insulator only when the dangling bonds at those corners are tilted, not aligned with the cut.

  2. Fragile Topology is Unstable Under Translation Refinement

    cond-mat.mes-hall 2026-07 conditional novelty 6.0 of 10

    Every symmetry-indicated fragile phase in any 2D wallpaper group is adiabatically connected to an atomic insulator in some finite symmetry-compatible supercell, with the minimal supercell index tabulated for every sym...

  3. Topological Dislocation Response in Elementary Semiconductors

    cond-mat.mes-hall 2026-02 conditional novelty 6.0 of 10

    Edge dislocations in silicon, diamond, germanium, and black phosphorene are predicted to bind mid-gap polarization bands protected by a filling anomaly, while screw dislocations are trivial.

Reference graph

Works this paper leans on

36 extracted references · 24 canonical work pages · cited by 3 Pith papers

  1. [1]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Fel ser, M. I. Aroyo, and B. A. Bernevig, Nature 547, 298 (2017)

    work page 2017

  2. [2]

    The SIs without SOC are listed in Tab. V. For each space group, we list its number in the International Tables for Crystallography, Vol. A , to- gether with the corresponding list of SIs that are expressed in terms of the SRSIs of Table IV. SG# SIs with SOC 2 z2w,1 = − 2θ1 + 2θ2 + 2θ5 mod 2 z2w,2 = − 2θ1 + 2θ2 + 2θ4 mod 2 z2w,3 = − 2θ1 + 2θ2 + 2θ3 mod 2 z...

    work page

  3. [3]

    Zak, Phys

    J. Zak, Phys. Rev. Lett. 45, 1025 (1980)

    1980

  4. [4]

    H. C. Po, A. Vishwanath, and H. Watanabe, Nature Communications 8, 50 (2017)

    2017

  5. [5]

    Kruthoff, J

    J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Sla ger, Phys. Rev. X 7, 041069 (2017)

    work page 2017

  6. [6]

    Zak, Phys

    J. Zak, Phys. Rev. B 23, 2824 (1981)

    1981

  7. [7]

    Aroyo, J

    M. Aroyo, J. Perez-Mato, D. Orobengoa, E. Tasci, G. De La Flor, an d A. Kirov, Bulgarian Chemical Communications 43, 183 (2011)

    work page 2011

  8. [8]

    M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wo ndratschek, Bilbao crystallographic server, http://www.cryst.ehu.eus

    work page

Show all 36 references
  1. [9]

    Z.-D. Song, L. Elcoro, Y.-F. Xu, N. Regnault, and B. A. Bernev ig, Phys. Rev. X 10, 031001 (2020)

    2020

  2. [10]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, M. G. Vergniory, J. Cano, Z. Wang, C. Fel ser, M. I. Aroyo, and B. A. Bernevig, Phys. Rev. B 97, 035138 (2018)

    2018

  3. [11]

    H. C. Po, H. Watanabe, and A. Vishwanath, Phys. Rev. Lett. 121, 126402 (2018)

    2018

  4. [12]

    Z.-D. Song, L. Elcoro, and B. A. Bernevig, Science 367, 794 (2020)

    2020

  5. [13]

    Peri, Z.-D

    V. Peri, Z.-D. Song, M. Serra-Garcia, P. Engeler, R. Queiroz, X. Huang, W. Deng, Z. Liu, B. A. Bernevig, and S. D. Huber, Science 367, 797 (2020)

    2020

  6. [14]

    Cano and B

    J. Cano and B. Bradlyn, Annual Review of Condensed Matter Physics 12, 225 (2021)

    2021

  7. [15]

    J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. G. Vergniory, C. Fe lser, M. I. Aroyo, and B. A. Bernevig, Phys. Rev. Lett. 120, 266401 (2018)

    2018

  8. [16]

    Y. Xu, L. Elcoro, G. Li, Z.-D. Song, N. Regnault, Q. Yang, Y. Sun, S. Parkin, C. Felser, and B. A. Bernevig, arXiv preprint arXiv:2111.02433 (2021)

    2021 arXiv

  9. [17]

    Altmann and P

    S. Altmann and P. Herzig, Point-group Theory Tables , Oxford science publications (Clarendon Press, 1994)

    1994

  10. [18]

    Elcoro, B

    L. Elcoro, B. Bradlyn, Z. Wang, M. G. Vergniory, J. Cano, C. Fe lser, B. A. Bernevig, D. Orobengoa, G. de la Flor, and M. I. Aroyo, Journal of Applied Crystallography 50, 1457 (2017)

    2017

  11. [19]

    Cohen, A course in computational algebraic number theory , Vol

    H. Cohen, A course in computational algebraic number theory , Vol. 138 (Springer Science & Business Media, 2013)

    2013

  12. [20]

    C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press Oxford, Oxford, United Kingdom, 1972)

    1972

  13. [21]

    J. Cano, L. Elcoro, M. I. Aroyo, B. A. Bernevig, and B. Bradlyn , Phys. Rev. B 105, 125115 (2022)

    2022

  14. [22]

    Elcoro, Z

    L. Elcoro, Z. Song, and B. A. Bernevig, Physical Review B 102, 035110 (2020)

    2020

  15. [23]

    For each space group, we list its number in the International Tables for Crystallography, Vol

    The SIs with SOC are listed in Table VI. For each space group, we list its number in the International Tables for Crystallography, Vol. A , together with the corresponding list of SIs that are expressed in terms of the SRSIs of Table III. SG# SIs without SOC 2 z2,1 = − 2θ1 + 2...

  16. [24]

    Belov, N

    N. Belov, N. Neronova, and T. Smirnova, Sov. Phys. Crystall ogr. 2, 311 (1957)

    1957

  17. [25]

    Z. Song, T. Zhang, Z. Fang, and C. Fang, Nature Communications 9, 3530 (2018)

    2018

  18. [26]

    Z.-D. Song, L. Elcoro, Y.-F. Xu, N. Regnault, and B. A. Berne vig, Phys. Rev. X 10, 031001 (2020)

    2020

  19. [27]

    Hwang, J

    Y. Hwang, J. Ahn, and B.-J. Yang, Physical Review B 100, 10.1103/PhysRevB.100.205126 (2019)

    2019 doi

  20. [28]

    Schindler and B

    F. Schindler and B. A. Bernevig, Phys. Rev. B 104, L201114 (2021)

    2021

  21. [29]

    J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. G. Vergniory, C. Fe lser, M. I. Aroyo, and B. A. Bernevig, Phys. Rev. B 97, 035139 (2018)

    2018

  22. [30]

    Bouhon, A

    A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager, Phys. Rev. B 100, 195135 (2019)

    2019

  23. [31]

    Bradlyn, Z

    B. Bradlyn, Z. Wang, J. Cano, and B. A. Bernevig, Phys. Rev. B 99, 045140 (2019)

    2019

  24. [32]

    Shiozaki and M

    K. Shiozaki and M. Sato, Physical Review B 90, 10.1103/PhysRevB.90.165114 (2014)

    2014 doi

  25. [33]

    J. Ahn, S. Park, and B.-J. Yang, Phys. Rev. X 9, 021013 (2019)

    2019

  26. [34]

    R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, Phys. Rev. B 84, 075119 (2011)

    2011

  27. [35]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005)

    2005

  28. [36]

    B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006)

    2006

Pith tools

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