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Exact g-function without strings

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Pith's one-line read This paper derives the exact g-function of sine-Gordon theory with integrable boundaries as a ratio of Fredholm determinants evaluated on the solution of a single nonlinear integral equation, bypassing the thermodynamic Bethe ansatz and…

desk verdict A genuinely new formula for the sine-Gordon g-function with real consistency checks, but the boundary-energy subtraction has an admitted ambiguity that needs a proof before I would call it exact. read the letter →

arxiv 2412.12869 v1 pith:VNW2VCIY submitted 2024-12-17 hep-th cond-mat.stat-mechnlin.SI

Yi-Jun He , Yunfeng Jiang This is my paper · ORCID
classification hep-thcond-mat.stat-mechnlin.SI
keywords g-functionboundaryentropysine-GordonmodelintegrableconditionsnonlinearintegralequationFredholmdeterminantBetheansatzlatticeregularization
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 exact g-function, or boundary entropy, of sine-Gordon theory with integrable boundaries can be computed without invoking the thermodynamic Bethe ansatz or the magnetic string hypothesis. The route is an integrable lattice regularization: the g-function is read off the finite part of the overlap between the ground state and an integrable boundary state on a finite cylinder. In the continuum limit the overlap splits into a boundary-energy term proportional to the cylinder radius and a finite remainder, and that remainder equals $\ln g = \ln g_{\mathrm{pref}} + \ln g_{\mathrm{det}}$, with both parts expressed through the counting function $Z(u)$ that solves a single nonlinear integral equation. If correct, this supplies the first exact finite g-function for a non-diagonal scattering integrable quantum field theory, a quantity relevant to boundary renormalization-group flows and worldsheet correlation functions.

What carries the argument

The engine is the light-cone six-vertex lattice regularization of sine-Gordon theory on a cylinder with integrable boundaries. The closed-channel partition function is written as an exact overlap of two integrable boundary states with a Bethe state of the double-row transfer matrix; the overlap is nonzero only for paired Bethe roots and is given by an exact formula containing a product over roots and a ratio of Gaudin-like determinants. In the continuum limit the anti-ferromagnetic vacuum dominates, and the root distribution is encoded in a counting function $Z_N(u)$ whose continuum limit satisfies a single nonlinear integral equation of NLIE type. All g-function data are extracted from this one counting function: the prefactor comes from the product of root-dependent boundary factors $f(u_j)$, the determinant part comes from the continuum limit of the Gaudin determinants, and the extensive piece proportional to the cylinder radius is identified with and subtracted as the boundary energy. This machinery avoids the infinitely many Y-functions and magnetic-string complications of the thermodynamic Bethe ansatz route.

What would settle it

Compute the boundary energy independently, for instance from the exact lattice overlap at small system size or from a separate thermodynamic calculation, and check whether the finite part of $\ln g$ extracted from equations (27)-(29) is invariant under the alternative pole prescriptions described in Appendix D; any shift of the finite part would falsify the claim. A second direct check is to evaluate the lattice boundary-state overlap exactly for small $N$ and compare its continuum extrapolation with the NLIE-based formula at generic values of $\gamma$, $a$, and $b$.

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Extended reading notes

Core claim

The central claim is that the exact g-function of the sine-Gordon theory with identical integrable boundaries is given by equations (27) and (29): $\ln g$ decomposes into a boundary-parameter-dependent prefactor $\ln g_{\mathrm{pref}}$ and a universal ratio of Fredholm determinants, $\ln g_{\mathrm{det}} = \frac{1}{2} \ln\left[\det(1-\hat H^+)/\det(1-\hat H^-)\right]$, both evaluated on the counting function $Z(u)$ that solves the nonlinear integral equation (24). The prefactor contains an integral of $\ln(1+e^{iZ})$ against a kernel built from boundary data plus discrete pole and zero terms, while the determinant part depends only on the bulk theory. The paper verifies the formula at the free-fermion point, where it agrees with the thermodynamic Bethe ansatz up to a recovered constant $-\frac12\ln2$, and in the UV and IR limits; numerically it yields finite g-functions that flow from $2\ln|g|\to 0$ in the UV to a $\gamma$-dependent IR constant.

Load-bearing premise

The result assumes that the boundary-energy subtraction is unambiguous: the pole choice in the contour integral that defines it is admitted to be "not completely clear", and if that choice shifts the finite part, or if the assumed Wick rotation selects a different state, the claimed g-function changes.

Editorial extensions

If this is right

  • For the first time, a finite exact g-function is obtained for a non-diagonal scattering integrable quantum field theory, namely sine-Gordon theory with integrable boundaries.
  • Computing the g-function reduces to solving one nonlinear integral equation for $Z(u)$ and evaluating two Fredholm determinants, a substantial simplification compared with TBA's system of Y-functions.
  • The universal determinant factor separates from the boundary-dependent data, so changing boundary conditions changes only the prefactor and discrete terms while the determinant ratio is shared.
  • The free-fermion limit reproduces the TBA result with the missing $-\tfrac12\ln2$ recovered, and the UV and IR limits give $\ln|g|\to0$ and $\ln|g|^2\to\tfrac12\ln(\tfrac12-\tfrac{\gamma}{2\pi})$, consistent with boundary-flow expectations.

Reading between the lines

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

  • Beyond the paper: the same lattice-overlap strategy should yield finite worldsheet g-functions in planar $\mathcal N=4$ super-Yang-Mills theory and ABJM theory, where the scattering is non-diagonal and TBA treatments lead to divergent individual g-functions.
  • Beyond the paper: because the determinant ratio is universal, it may be possible to rewrite it purely in terms of Y-functions after relating the counting function to TBA; such a rewriting would indicate how the TBA prescription must be regularized.
  • Beyond the paper: a direct lattice check at small system size, comparing the exact overlap with the continuum extrapolation of equations (27)-(29), would settle whether the boundary-energy pole-choice ambiguity noted in Appendix D leaves the finite part truly untouched.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a new approach to computing the exact g-function for integrable quantum field theories with non-diagonal scattering, exemplified by sine-Gordon theory with integrable boundaries. The idea is to use the light-cone six-vertex lattice regularization: the cylinder partition function in the closed channel is written as a sum over Bethe states (19), the integrable boundary-state overlap with the ground (AFV) state is computed exactly by the Pozsgay–Rákos overlap formula [37], and the continuum limit is taken via the Destri–de Vega scaling (23). After subtracting the extensive boundary-energy contribution, the authors obtain the main result, Eqs. (27) and (29): ln g = ln g_pref + ln g_det, where ln g_pref is a boundary-parameter-dependent integral over the counting function Z(u) (the solution of the NLIE (24)) plus discrete terms, and ln g_det is a universal ratio of Fredholm determinants. The paper reports consistency checks — the free fermion point where the NLIE result agrees with TBA up to the expected log-2 shift, the IR limit (32), the UV limit ln g → 0, and numerical boundary-flow monotonicity consistent with the g-theorem — and gives a numerical algorithm in App. G.

Significance. If correct, this is the first exact, finite g-function for a non-diagonal scattering IQFT, resolving a long-standing obstacle: TBA computations for such theories diverge [27]. The approach is conceptually simpler than TBA (closed-channel overlap, one NLIE instead of a system of Y-functions, no string hypothesis), and the derivation is not circular: the overlap uses the independent formula of [37], the NLIE is the established equation of [32], and the boundary energy is checked against [33]. The paper ships reproducible numerics (App. G) and explicit falsifiable predictions: the free-fermion log-2 shift, the IR value (32), the UV limit ln g → 0, and monotone boundary flow. These strengths, together with the absence of ad hoc free parameters, make the claim credible if the two admitted gaps are closed.

major comments (3)
  1. [Appendix D.2, Eqs. (D14)–(D25) and SM (C20)] The boundary-energy subtraction is load-bearing and is admitted to be ambiguous. The text states that 'how to deform the contour from the real axis to pick up precisely these two poles is not completely clear' and that keeping only one of the two poles at ±iπ/γ changes εa+εb relative to (D25); it then asserts, without demonstration, that 'they do not affect our result for the g-function.' Because εa is the coefficient of R in the overlap exponent (Eqs. (4) and (26)), an O(m) change in εa from the one-pole choice alters the subtracted extensive term by O(mR), which is O(1) at fixed r = mR and hence shifts ln g(r) directly; this cannot be absorbed into UV counterterms. The derivation of (27) in SM C.2 explicitly relies on this identification, since after (C20) the first term is declared extensive 'as explained in appendix D 2.' The match of (D25) with [33] for a>γ/2 corroborates the two-pole choice only in that regime. The paper should either prove that the finite remainder is independent of the contour deformation, or determine εa independently over the parameter range used in Figs. 2–3, or restrict the claim accordingly. The same f(u) contour integrals also control the discrete terms in (27), since the pole/zero sets {wa, za} in (C26) are contour-dependent, so the prefactor is exposed to the same ambiguity.
  2. [Footnote 2, Secs. III.B.d and III.C] The Wick rotation from the Minkowskian lattice partition function (10) to the Euclidean cylinder partition function (2)–(3) is asserted but not performed. This step is load-bearing for the central claim because it justifies both the dominance of the AFV state in the large-L limit and the identification of the overlap W in (19) with |⟨B|0⟩|², from which the g-function in (4) is extracted. The NLIE (24) is inherited from the standard light-cone framework, but the identification of the overlap is a new element specific to this paper, and the footnote explicitly flags the missing rotation. Please provide the analytic-continuation argument, or state precisely why the standard continuation applies to the overlap (and commutes with the large-L limit), with a reference where this is established.
  3. [Appendix F.2, Sec. IV.b] The UV result ln g → 0, presented in Sec. IV.b as an analytical result, rests on the statement 'Numerical evidence shows that in the UV limit, the solution of the non-linear integral equation is a(z±iξ)=1.' This matters because the additive normalization of ln g (the boundary-state normalization freedom acknowledged in Sec. IV.a) is pinned down by this UV condition. Please either provide a proof of the UV behaviour of the counting function, or state explicitly in the main text that the UV limit is a numerically supported check rather than a derivation.
minor comments (5)
  1. [Sec. III.C, Eq. (27)] The main-text formula (27) is incomplete as printed: the 'discrete terms' are not defined there, and the validity conditions on ξ (0<ξ<γ/2, and the additional conditions ξ<a, 2ξ<γ stated in SM C.2) appear only in the SM; since (27)–(29) is the headline result, the discrete terms and their domain of validity should be displayed in the main text.
  2. [Sec. IV.a, Eq. (31)] The status of the free-fermion comparison (31) should be clarified: the paper first states that a different boundary-state normalization can shift ln|g| by a constant, and then presents −(1/2)ln2 as recovered 'naturally'; please spell out that (31) is a check after fixing the normalization, and note the role of the UV condition (and hence of the numerical evidence in App. F.2) in fixing it.
  3. [Figs. 2–3, Sec. IV.c–d] The captions of Figs. 2 and 3 do not state which values of γ (and of a, b where varied) correspond to which curves; without this information the reader cannot verify the claimed plateaus against (32), (F13) and (F14), nor the stated monotonicity.
  4. [Apps. D, E and Sec. III.C] The parameter-regime restrictions are scattered: the boundary-energy match to [33] holds for a>γ/2 (App. D), the free-fermion comparison for π/4<a<3π/4 (App. E), and the generic formula (C26) depends on a pole/zero-counting convention; the main text should state explicitly the range of (γ, a, b) over which (27),(29) is claimed.
  5. [Abstract and Sec. I] Please state explicitly the sense in which the result is 'exact': the g-function is given exactly in terms of the solution Z(u) of the NLIE (24), but Z(u) itself is obtained numerically for generic r (App. G); the paper would benefit from one sentence making this precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the g-function is obtained from an independently derived Bethe-ansatz overlap and external NLIE/lattice inputs; the boundary-energy pole ambiguity is a correctness risk, not a by-construction reduction.

full rationale

The derivation chain is: (i) define the g-function by the non-extensive overlap (4); (ii) compute the lattice partition function and the overlap W from the independently derived transfer-matrix and overlap results (10),(15)-(19) of [34,37]; (iii) take the continuum limit using the NLIE (24) obtained by standard Destri-de Vega contour deformation [32]; (iv) subtract the boundary energy, whose coefficient is computed in Appendix D and matches the independent result [33]. At no point is Eq. (27) or (29) used as an input to define Z or the overlap; the NLIE solution Z comes from the Bethe equations for the AFV state, not from a fit to the target g-function. The self-citation [34] is a published, independently derived lattice result and is also sketched in Supplement B; the other self-citations are motivational, not load-bearing. The only serious caveats are the admitted pole-selection ambiguity in Appendix D.2 and the unperformed Wick rotation in footnote 2. These are correctness and rigor gaps: if a different pole choice changed the finite remainder, the numerical value of ln g would shift, but that would be an error in a non-circular derivation, not an equivalence of the result to its own inputs by construction. The boundary energy is not chosen as 'whatever makes the remainder finite'; it is identified with the known boundary energy of [33]. Hence no step reduces to itself or to a fitted parameter, and the paper is self-contained against external benchmarks.

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

The central claim rests on standard integrability results (BAE, overlap formulas, NLIE) taken from the literature rather than on new free parameters. No data are fitted; all parameters are physical parameters of the bulk and boundary theory. The main input assumptions are the validity of the lattice regularization and the AFV state as the vacuum, plus two unresolved technical steps (Wick rotation, contour pole selection) that the paper itself flags.

assumptions (5)
  • domain assumption The light-cone six-vertex model with K-matrices gives an integrable lattice regularization of sine-Gordon, with the mapping ν=π/γ and Eq. (9).
    Invoked in Sec. II.C; relations from [24,33]. Standard but assumed.
  • domain assumption The AFV Bethe state represents the sine-Gordon ground state in the continuum limit with R=mNΔ fixed.
    Used in Sec. III.C to select the dominant state in the large-L limit; from Destri-de Vega [32].
  • domain assumption The overlap formula (16)-(18) for integrable boundary states applies to this lattice model and to the paired Bethe roots.
    Taken from [37] in Sec. III.B, not re-derived here.
  • ad hoc to paper A Wick rotation connects the Minkowskian lattice partition function (10) to the Euclidean cylinder partition function without changing the overlap.
    Footnote 2 states this is needed but supplies no rotation; the large-L AFV dominance depends on it.
  • ad hoc to paper In the UV limit, the NLIE solution satisfies a(z±iξ)=1, so the prefactor integral vanishes and ln g tends to 0.
    App. F.2 uses 'numerical evidence' rather than a proof; this fixes the normalization of the g-function.

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Pith. "Pith review of Exact g-function without strings." pith.science (2026). https://pith.science/paper/VNW2VCIY

@misc{pith2026241212869,
  author       = {Pith},
  title        = {Pith review of: Exact g-function without strings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNW2VCIY}},
  note         = {Machine review of arXiv:2412.12869}
}
abstract

We propose a new approach to compute exact $g$-function for integrable quantum field theories with non-diagonal scattering S-matrices. The approach is based on an integrable lattice regularization of the quantum field theory. The exact $g$-function is encoded in the overlap of the integrable boundary state and the ground state on the lattice, which can be computed exactly by Bethe ansatz. In the continuum limit, after subtracting the contribution proportional to the volume of the closed channel, we obtain the exact $g$-function, given in terms of the counting function which is the solution of a nonlinear integral equation. The resulting $g$-function contains two parts, the scalar part, which depends on the boundary parameters and the ratio of Fredholm determinants, which is universal. This approach bypasses the difficulties of dealing with magnetic excitations for non-diagonal scattering theories in the framework of thermodynamic Bethe ansatz. We obtain numerical and analytical results of the exact $g$-function for the prototypical sine-Gordon theory with various integrable boundary conditions.

Figures

Figures reproduced from arXiv: 2412.12869 by the authors.

Figure 1
Figure 1. FIG. 1. The light cone six-vertex model with integrable [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Plot of 2 ln [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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    Bulk S-matrix The scattering between (anti-)solitons is described by the following S-matrix [29]: S++ ++ (θ) = S−− −− (θ) = S0(θ), S +− +− (θ) = S−+ −+ (θ) = ST (θ)S0(θ), S −+ +− (θ) = S+− −+ (θ) = SR(θ)S0(θ), ST (θ) = sinh (λθ) sinh (λ(iπ − θ)) , S R(θ) = i sin (λπ) sinh (λ(i...

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    Boundary S-matrix The (anti-)soliton boundary scattering amplitudes can be expressed by the matrix [28, 30] Rs(u) = P + 0 (ζ, ϑ, u) Q0(u) Q0(u) P − 0 (ζ, ϑ, u) R0(u) σ(ζ, u) cos(ζ) σ(iϑ, u) cosh(ϑ) , (A6) P ± 0 (ζ, ϑ, u) = cos(λu) cos(ζ) cosh(ϑ) ∓ sin(λu) sin(ζ) sinh(ϑ) , (A7)...

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    Eigenvalue of the transfer matrix The transfer matrix TR(u) can be diagonalized by algebraic Bethe ansatz. Using the crossing symmetry ofR-matrix, it can be shown that τ (˜u; {θ}) = bτ (u; {θ}). (B10) We define the monodromy matrix T (2N ) a (u; {θj}) in the usual way T (2N ) ...

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    Non-linear Integral Equation For later convenience, we introduce the function ϕν(u) := i ln sinh(iγν + u) sinh(iγν − u) , which is analytic in the strip | Im u| ≤min(νγ, π− νγ ) for real γ. We define the counting function ZN (u) ≡ N ϕ1/2(u − Θ) + ϕ1/2(u + Θ) − NX j=1 ϕ1(u − uj...

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    The g-F unction On the lattice, the squared g-function |g|2 comes from the following overlap: ⟨Φ+ 0 (−Θ − iγ 2 )|U †|u⟩⟨u|Φ− 0 (−Θ − iγ 2 )⟩ ⟨u|u⟩ = sinhN (2Θ − iγ) sinh2N (2Θ + iγ) (C11) × NY k=1 sinh uk − Θ + iγ 2 sinh uk − Θ − iγ 2 N/2Y j=1 F (u+ j ) det G+ jk det G− jk . O...

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    Bulk Energy The bulk energy of the ground state comes from the eigenvalue of the operatorbτ (Θ; {±Θ})τ (Θ; {±Θ}) corresponding to the AFV state, given by τR(−2Θ − iγ) = [sinh(2Θ + iγ) sinh(iγ)]2N NY k=1   sinh uk − Θ + iγ 2 sinh uk − Θ − iγ 2 sinh uk + Θ − iγ 2 sinh uk + Θ +...

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    Thermodynamic Bethe Ansatz Approach tog-function At the free fermion point, we have solitons and anti-solitons which scatter freely in the bulk. The boundary scattering process of solitons and anti-solitons can be described by the following Faddeev-Zamolodchikov (FZ) algebra [...

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    We take γ = π 2 in the NLIE and the expressions (C25) for the g-function

    NLIE Approach to g-function Now we compute the g-function using the lattice regularization approach. We take γ = π 2 in the NLIE and the expressions (C25) for the g-function . In this case, G(u) = 0 and the NLIE simplifies to Z(u) = mR sinh πu γ = mR sinh(2u). (E14) As a resul...

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    IR Limit We first consider the IR limit mR ≫ 1. Following the procedure in [32], we define the following quantities, ϵ(θ) ≡ −iZ( γ π θ + iγ 2 ), (F1) ¯ϵ(θ) ≡ iZ( γ π θ − i γ 2 ), (F2) G0(θ) ≡ γ π G( γ π θ) = Z +∞ −∞ dk 4π eikθ sinh ( π2 2γ − π)k sinh ( π2 2γ − π 2 )k cosh π 2 ...

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    We will argue that the logarithm of g-function should be zero in this limit

    UV Limit Now we consider the UV limit mR → 0. We will argue that the logarithm of g-function should be zero in this limit. First, the logarithm of the Fredholm determinant part should be zero in the UV limit. Take the integral contour in (C22) to be {R + iξ} ∪ {R − iξ}, with 0...

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    Solving the Non-linear Integral Equation The starting point is the following non-linear integral equation for vacuum: Z(u) = mR sinh πu γ + 2 Im Z +∞ −∞ dvG(u − v − iξ) ln 1 + eiZ(v+iξ) . (G1) 23 This equation can be analytically continued and rewritten as ϵ(θ, ξ) = −imR sinh(...

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    Give a cutoff L to functions in the equation and divide the interval [ −L, L] into N part with length 2L N

    Make the grid. Give a cutoff L to functions in the equation and divide the interval [ −L, L] into N part with length 2L N

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    Sample the points in the grid on the variable θ of source function −imR sinh(θ + iξ)

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    This can be done in the following way

    Calculate the convolution term using the convolution theorem. This can be done in the following way. Define the function L(θ, ξ) ≡ ln 1 + e−ϵ(θ,ξ) , then upon discretization, the convolution term can be written as Z +∞ −∞ G0(u − v)L(v)dv ∼δ N/2−1X m=−N/2 G0(nδ − mδ)L(mδ), (G6)...

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    Putting the source function −imβ sinh(θ + iξ) to be the initial function for iteration

    Sum all terms on the right hand side of the equation (G2) and set-up the iteration to solve the non-linear integral equation. Putting the source function −imβ sinh(θ + iξ) to be the initial function for iteration. After sufficient times of iteration, we can obtain the numerica...

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    (G13) To use the numerical solution, one should first rescale the integral variable and the imaginary shift ξ by γ π

    Computation of F redholm Determinant Recall the expression for the logarithm of g-function − Im Z +∞ −∞ dv dx 2π (ln f )′(−x − iξ) (δ − G)(x − v) ln 1 + eiZ(v+iξ) + 1 2 ln det(1 − ˆH +) det(1 − ˆH −) + 1 2 Z +∞ −∞ dvG(−v − iξ) ln 1 + eiZ(v+iξ) − 1 2 ln 1 + eiZ(0) , (G12) where...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.