Gradient expansion approximation of the inhomogeneous electron-gas revisited: Higher-order corrections

Angel Rosado; Efstratios Manousakis; Mario Benites

REVIEW 3 major objections 5 minor 37 references

Gradient expansion approximation of the inhomogeneous electron-gas revisited: Higher-order corrections

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper derives the next-to-leading correction to the electron-gas gradient-expansion coefficient and proves the r_s ln r_s term is exact, giving a hard constraint for GGA functionals.

desk verdict Careful analytic work on the r_s^3 ln r_s coefficient of the gradient expansion, but the 'exact' claim outruns the proof and the B_xc conversion has a factor-of-two error. read the letter →

arxiv 2608.02524 v1 pith:6R3RLIDP submitted 2026-08-03 cond-mat.mtrl-sci cond-mat.str-el

Mario Benites , Angel Rosado , Efstratios Manousakis This is my paper · ORCID

classification cond-mat.mtrl-scicond-mat.str-el

keywords gradientexpansionapproximationexchange-correlationfunctionalelectrongasWigner-Seitzradiusgeneralizeddensity-densityresponsefunctionrandom-phasenext-to-leadingorder

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 aims to establish the exact next-to-leading-order term in the Wigner-Seitz radius expansion of the coefficient b_xc that controls the |∇n|^2 term in the gradient expansion of the exchange-correlation energy of the electron gas. It finds this correction is proportional to r_s^3 ln(r_s), with a specific coefficient, and argues that no higher-order diagrammatic contributions can modify it. If correct, any generalized-gradient approximation that claims validity in the high-density, slowly-varying limit must reproduce this logarithmic term. The result matters because earlier work showed that separating b_xc into exchange and correlation pieces is regulator-dependent; only the combined coefficient is physical, and that combined coefficient now has a proven next-order term.

What carries the argument

The central object is the static proper-polarization function Π*(q,0), whose q^2 coefficient b_xc is related to the gradient coefficient B_xc by a kernel identity. The computation is organized by the RPA-based reorganized perturbative expansion (RPARPE), where diagrams are ordered by the number of screened interaction lines. The technical engine is a decomposition of the two-dimensional integrals into Region 1 (x∈(0,1/2]) and Region 2 (x∈(1/2,∞)), followed by an expansion of the integrands near x=0 through special integral kinds Γ^{2n+1}_{lm}(r_s,y). These produce the logarithmic r_s ln r_s behavior, while Region 2 yields only power-law contributions. The exclusion of all other diagrams rest

What would settle it

Numerically evaluate the kite-like self-energy integral shown in Eqs. G8-G10; if its leading r_s behavior contains ln r_s or diverges because the denominator (q_2 - p)·k |p - q_2|^2 k^2 vanishes, the claimed exactness collapses. Alternatively, compute the r_s^3 ln r_s coefficient from the full RPARPE series through second order and check whether it matches -29/1944 e^4(m a_B)^3/π^5.

Watch

Extended reading notes

Core claim

In the paper's own terms, the static proper-polarization function's long-wavelength q^2 coefficient has the small-r_s expansion b_xc = b_xc^(0) r_s^2 + b_xc^(1) r_s^3 ln r_s + O(r_s^3), with b_xc^(1) = -29/1944 e^4(m a_B)^3/π^5; in atomic units the equivalent constraint on the gradient coefficient is B_xc[n] = r_s^{-4} [0.029116 - (29/(864π^3)) r_s ln r_s]. The paper claims this coefficient is exact, in the sense that no proper-polarization diagrams beyond the RPA-reorganized set considered can contribute to it. The calculation combines the Fock and ring self-energy, the screened-interaction diagrams, and an exact evaluation of the resulting two-dimensional integrals; the r_s ln r_s term eme

Load-bearing premise

The exactness claim rests on the unproven assertion that every proper-polarization diagram outside the considered RPA-reorganized set contributes at order r_s^3 or higher, with no r_s^3 ln r_s enhancement, anchored by the leading self-energy-insertion family and the kite-like diagram.

Editorial extensions

If this is right

Any GGA functional that reproduces the gradient expansion in the high-density slowly-varying limit must contain the r_s ln r_s term with the derived coefficient.
The exchange and correlation contributions to the gradient coefficient cannot be assigned independent regulator-invariant values; only their sum b_xc is physical, so functionals should be constrained on the total, not on separate components.
The next-to-next-to-leading r_s^3 coefficient is not fixed by the diagrams used here; extracting it requires going beyond the leading RPARPE order.
The q^4 term of the response kernel, which yields the (∇^2 n)^2 term in the energy, has a leading r_s^6 behavior, while the |∇n|^4 term's coefficient starts at r_s^{-4} in the same high-density limit.

Reading between the lines

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

One could test the exactness claim directly by computing the r_s^3 ln r_s coefficient from an independent diagrammatic resummation at second order; a mismatch would indicate a diagram beyond the claimed family contributes.
If exact, the logarithmic term provides a stringent benchmark for semi-empirical and machine-learned density functionals: they must reproduce this single number in the high-density limit without fitting.
The integral-kind expansion may transfer to higher-order gradient terms, but the paper's own caveat that a rigorous analysis of the underlying integrals is required suggests the no-corrections theorem is the part most worth scrutinizing.
The regulator-dependence argument implies that functionals built by combining separately fitted exchange and correlation gradient coefficients are conceptually suspect; an exact total constraint should be used instead.

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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, the useful thing to know: this paper contains a careful, checkable calculation of the next-to-leading coefficient in the high-density expansion of b_xc, the q^2 coefficient of the static proper polarization. That coefficient — r_s^3 ln r_s — is genuinely new, and the authors' L1/L2 region-decomposition method for extracting log terms from screened-Coulomb integrals is a real technical contribution. I audited the arithmetic on the four sub-coefficients in Table V and they reproduce the stated integrals and sum to the claimed total. The work at the b-level looks solid.

The problems start when the authors move from "we computed this sum" to "this value is exact." The no-corrections theorem in Sec. IV.G rests on the claim that the first excluded RPARPE diagram family, the kite self-energy, contributes only an r_s^3 constant. That claim is not demonstrated. Appendix G displays the integral but doesn't evaluate it, the denominator vanishes on surfaces in the integration domain, and the paper itself concedes a rigorous analysis of the integral expressions "is required." The stress-test raises exactly the right question: if that integral has log divergence, the coefficient is not exact.

There is also a concrete internal inconsistency in the conversion to the functional coefficient B_xc. Eq. (4) gives 29/(864π^3) for the log term; Eq. (210), using α^3=4/(9π), gives a factor of two smaller. The authors say these are the same when using atomic units, but they aren't. This is not a minor typo — it's the number a GGA builder would actually use as a constraint, and it's off.

Net: the computation of the included diagrams is likely correct and the technique is worth knowing. But the paper's headline — an exact, definitive constraint — is not supported. The kite integral needs evaluation, the power-counting argument needs to be made rigorous, and the B_xc conversion needs fixing. If those are addressed in revision, this becomes a useful contribution. As it stands, I'd treat the r_s^3 ln r_s b-level value as a strong candidate, not a proven exact result.

For peer review: yes, send it out. It's the kind of paper that deserves a serious referee even if the current version needs substantial revision. I wouldn't cite it yet in my own work.

Referee Report

3 major / 5 minor

Summary. The manuscript claims to compute the next-to-leading-order term in the high-density expansion of the gradient-expansion coefficient B_xc[n], finding b_xc = b_xc^(0) r_s^2 + b_xc^(1) r_s^3 ln r_s + O(r_s^3), with the logarithmic coefficient b_xc^(1) = -29/1944 e^4 (m a_B)^3 / π^5 asserted to be exact, i.e. receiving no corrections from any omitted higher-order diagrams. The derivation decomposes b_c into four contributions (b'_c, b''_c, b'''_c, b^r_c), extracts the r_s^3 ln r_s coefficient from each through a systematic region-splitting and Taylor-expansion procedure, and combines them with the regulator-independent b_x to obtain the final value. The exactness claim rests on a power-counting argument in Sec. IV.G based on the RPA-based reorganized perturbative expansion (RPARPE), with the first excluded diagram family, the kite-like self-energy, claimed to contribute only at order r_s^3.

Significance. If the central claim were fully established, the result would be a valuable exact constraint for GGA functionals in the high-density slowly-varying limit: any valid functional would have to reproduce the coefficient -29/(864π^3) in atomic units. The manuscript's systematic treatment of the integral decompositions is a notable methodological contribution, and the displayed arithmetic for the four sub-coefficients C'_L3, C''_L3, C'''_L3, and C^r_L3 appears internally consistent and independently verifiable from the listed integrals. However, the exactness theorem is the load-bearing part of the paper, and it is not proven; the manuscript itself concedes that a rigorous analysis of higher-order terms is required. The B_xc conversion also contains an unresolved factor-of-two inconsistency. The significance of the paper therefore depends on whether these gaps can be closed.

major comments (3)

[Sec. IV.G and Appendix G] The exactness claim that no higher-order diagrams contribute to the r_s^3 ln r_s coefficient is not established. The argument requires that the kite-like self-energy Σ_2b(k_F,0) contributes only a constant at leading order in r_s. Appendix G displays the integral (G8)-(G10) but does not evaluate it, does not prove convergence, and the denominator D_2 = (q_2-p)·k |p-q_2|^2 k^2 vanishes on surfaces inside the integration region. The text at the end of Sec. IV.G explicitly states that 'a rigorous analysis of the underlying integral expressions for each term in Π_xc^EH is required.' Without that analysis, the claimed 'no-corrections theorem' is an assertion, not a proof. If Σ_2b(k_F,0) contains a logarithmic k_F dependence, Eq. (198) would generate a contribution exactly at r_s^3 ln r_s, invalidating the headline coefficient.
[Eqs. (4) and (210)] There is a factor-of-two inconsistency in the conversion from b_xc to B_xc. In atomic units Eq. (4) gives the logarithmic coefficient of B_xc as -29/(864π^3) r_s^{-3} ln r_s. Eq. (210), using α^3 = 4/(9π), gives -29/(1728π^3) r_s^{-3} ln r_s, i.e. exactly half. The text states that Eq. (210) is the same as Eq. (4), but it is not. This is not a mere typo in an ancillary formula: Eq. (4) is the advertised constraint for GGA functionals. The authors must reconcile this discrepancy and specify the value of a_xc used in Eq. (21), which is not given in the manuscript.
[Sec. IV.G generally] The power-counting heuristic that each diagram set S_i is anchored by the self-energy-insertion family of Fig. 5 is asserted rather than demonstrated. The text says that tracking the RPARPE order of this family is simple, but the step from 'the LO of this representative family' to 'the LO of every diagram in the set' is not justified. Higher-order members of the set could in principle have additional log-enhancements or stronger infrared behavior. This is the same gap identified in Appendix G, but it applies to the entire exclusion argument. The exactness claim therefore needs either a proof for all diagram topologies or a weakening of the claim from 'exact' to 'the contribution computed from the displayed diagrams.'

minor comments (5)

[Title/Abstract] The title contains a typo: 'ele ctron-gas' should be 'electron-gas'. The abstract says 'proof' of no corrections, which overstates what is actually demonstrated in the body.
[Eq. (21)] The quantity a_xc appears in the expression for K''_xc but its value or derivation is not provided in this manuscript. The reader is left to infer it from Ref. [1]; this should be stated explicitly.
[Appendix G] The statement 'This proves that Σ_2b(k_F,0) only contributes as a constant' is premature given that Eq. (G8) is not evaluated and its convergence is not established. The word 'proves' should be replaced with a description of the intended power-counting argument.
[Table V] The units row says 'τ2 = (mαa_B)^3 e^4/π^4', but the table entries are dimensionless coefficients multiplying this factor. This is fine, but it would be clearer to write the dimensionful prefactor explicitly in the table header.
[Appendix C] Some function labels are inconsistent: e.g. f_6^{(2n+1)} is written without the second superscript in Eq. (175). Minor notation cleanup would help reproducibility.

Circularity Check

1 steps flagged · score 3.0 of 10

NLO coefficient itself is computed from explicit integrals, but the 'exactness' no-corrections claim rests on a self-defined diagram hierarchy rather than an evaluation.

self definitional [Section IV.G (after Eq. 198) and Appendix G, Eqs. (G8)-(G10)]
"we can always find a set of diagrams that share the same LO contribution in rs in the high-density limit... we denote such a set of diagrams as Si, where a higher index i ∈ N corresponds to a higher-order LO rs contribution deriving from the proper-polarization diagrams within that specific set... subsequent sets Si (for i > 1) contributing to Π_xc^EH(q,0) are anchored by higher-order members of this family; this guarantees a strictly higher-order rs contribution than the previous set."

The no-corrections theorem asserts that no diagram beyond those already computed can contribute at order r_s^3 ln r_s. The proof in Sec. IV.G defines the sets Si by the property 'higher index = higher-order LO rs contribution', then assigns the m=2 kite self-energy to S2 by asserting (not evaluating) that it contributes only a constant, and finally concludes that all higher sets are strictly higher order. The conclusion is therefore contained in the defining/asserted hierarchy, not derived from an independent evaluation. Appendix G displays the kite integral (G8)-(G10) but does not evaluate it or prove convergence; the sentence 'This proves that Σ2b(kF,0) only contributes as a constant' is an assertion of the needed result. Thus the exactness of CL3 reduces to the self-imposed RPARPE power

full rationale

The numerical NLO coefficient CL3 = -29/1944 e^4(m a_B)^3/π^5 is obtained by explicitly evaluating displayed integrals (V1, V2, V3, V4, D0,1, etc.) in Sections IV.C-IV.F, with the sub-coefficients checked from Tables II and III. I found no fitted parameter renamed as a prediction and no direct reduction of the final coefficient to the paper's inputs. The central number therefore has independent computational grounding. The circularity concern is confined to the stronger 'exact, no corrections' claim: Section IV.G does not prove that all higher-order RPARPE diagrams are free of r_s^3 ln r_s contributions; it defines sets by their rs order and asserts the m=2 kite self-energy is a constant based on an unevaluated integral. The paper itself concedes 'a rigorous analysis of the underlying integral expressions for each term in Π_xc^EH is required' before substituting the set-tracking heuristic. Additionally, Eq. (4) and Eq. (210) give coefficients for the r_s^{-3} ln r_s term in B_xc that differ by a factor of 2 (-29/(864π^3) vs -29/(1728π^3) after using α^3=4/(9π)); this is an internal consistency/correctness issue, not circularity. Overall, the exactness theorem is partially circular/self-citation-loaded, while the computed coefficient is not circular.

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

The central claim rests on (i) the standard diagrammatic expansion of the electron-gas polarization (domain assumption), (ii) the RPARPE reorganization inherited from the same authors' Refs. 1 and 26, (iii) the paper-specific power-counting heuristic that guarantees the r_s^3 ln r_s coefficient receives no higher-order corrections, and (iv) the regulator bookkeeping conclusions of Ref. 1. No free parameters are fitted; the only 'scheme' knob (beta(k_F)) cancels by construction. No invented entities. The arithmetic from the displayed integrals is reproducible and I verified a substantial portion; the heuristic power-counting axiom (iii) is the piece with the weakest independent support.

assumptions (5)

domain assumption Standard many-body perturbation theory for the proper polarization function, including the RPA re-summation of the screened interaction (Eqs. 15-19).
Invoked throughout Secs. II-III; standard but unproved background. The perturbative expansion of Pi* is taken as given.
domain assumption RPARPE reorganization: re-summing standard perturbation theory into diagrams with RPA-screened interaction lines, with m-th order counting those lines (Sec. IV.G).
Introduced in the authors' Refs. 1 and 26; the exactness claim evaluates 'neglected diagrams' only within this scheme. Not verified in the present paper.
ad hoc to paper Power-counting heuristic: the leading r_s behavior of every higher-order diagram set S_i is anchored by the self-energy-insertion family of Fig. 5; the kite-like self-energy (App. G) contributes a constant at LO in r_s.
This is the load-bearing premise of the exactness claim (Sec. IV.G). The kite integral (Eqs. G8-G10) is not evaluated; the assertion that it is O(r_s^0) is stated, not proven.
domain assumption Regulator-scheme independence of b_xc and scheme dependence of b_x, b_c separately (Ref. 1, Table I).
Quoted from Ref. 1 (same authors). The present paper's regulator bookkeeping (Eqs. 44, 65, 89, R+- identity) assumes these conclusions; they cannot be verified from this text alone.
standard math Mathematical identities used to reduce integrals: contour deformation in App. D (no poles enclosed), the beta-function identity Eq. 200, and the D_nm/beta_n master integrals (App. F).
Standard calculus steps; the ones I spot-checked (App. F, Tables II-III, Eq. 90) are correct.

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Cite this review

Pith. "Pith review of Gradient expansion approximation of the inhomogeneous electron-gas revisited: Higher-order corrections." pith.science (2026). https://pith.science/paper/6R3RLIDP

@misc{pith2026260802524,
  author       = {Pith},
  title        = {Pith review of: Gradient expansion approximation of the inhomogeneous electron-gas revisited: Higher-order corrections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6R3RLIDP}},
  note         = {Machine review of arXiv:2608.02524}
}

abstract

In our recently published work (our Ref. 1) we revisited the gradient expansion approximation (GEA) of the interacting electron gas, and recalculated the leading-order contribution$-$with respect to the Wigner-Seitz radius $r_s$$-$to the coefficient $B_{xc}[n]$ of the square of the gradient of the electron density in the high-density and slowly varying limits. That work resolved historical controversies regarding these coefficients and demonstrated that serious misconceptions have led to incorrect constraints being imposed on popular functionals within the generalized gradient approximation (GGA). In the present paper, we extend this calculation to obtain the coefficient of the next-to-leading term, which scales as $r_s \ln(r_s)$ relative to the leading order. First, we establish a systematic framework to evaluate the integral expressions for the $b_{xc}$ coefficient of the leading term ($\sim q^2$) of the density-density response function in the long-wavelength limit ($q \to 0$)$-$a prerequisite for computing $B_{xc}[n]$. The significance of the calculation stems from the proof that the coefficient of this $r_s \ln(r_s)$ term receives no corrections from higher-order diagrammatic expressions. Consequently, our derived value serves as an exact, definitive constraint for future GGA functional development; in the high-density slowly-varying limit, any valid functional must reproduce the exact constraints established in both our previous work and the present paper.

Figures

Figures reproduced from arXiv: 2608.02524 by the authors.

**Figure 1.** Its expansion in the q → 0 limit and the extraction of the q 2 coefficient bx—which contributes to the exchange part of the density-gradient-squared term within the GEA—have been previously investigated by several authors, including Geldart-Taylor, Sham, AntoniewiczKleinman, Kleinman-Lee, Engel-Vosko (EV), and Svendsen and von-Barth. Likewise, Πc (q, 0) is the sum of the proper polarization functions shown in [… view at source ↗

**Figure 1.** FIG. 1. Diagrammatic contributions to the irreducible pola [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** FIG. 2. Remaining diagrammatic contributions to Π [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Figures from the paper (5 more)

**Figure 3.** Figure 3: FIG. 3. Diagrammatic representation of the GW self-energy. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

**Figure 4.** Figure 4: FIG. 4. Complex frequency plane integration path for the rin [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. Family of diagrams corresponding to the irreducible [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6. contour complex path used for the calculation of the i [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

**Figure 7.** Figure 7: FIG. 7. Diagrammatic representation of the kite-like self- [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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D-type contributions can be further categorized into D1 and D2 sub-types

D-type contribution In this subsection, we discuss the D-type contribu- tion to every coeﬃcient of the terms arising from the rs → 0 expansion that originates from both bc-primed coeﬃcients, b ′′′ c , and Σ r(kF , 0). D-type contributions can be further categorized into D1 and D2 sub-types. After applying the ﬁrst two steps of our systematic approach, D1-...
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These hybrid-type contributions arise by taking into account higher-order terms in the Taylor expansion of Q(x, y) in the x → 0 limit

H-type contribution We discuss the H-type contributions to every coeﬃ- cient in the rs-expansion, which arise exclusively from the integral kinds L1 im(rs), since Region 2 provides only D-type contributions. These hybrid-type contributions arise by taking into account higher-order terms in the Taylor expansion of Q(x, y) in the x → 0 limit. Con- sequently...
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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

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D-type contributions can be further categorized into D1 and D2 sub-types

D-type contribution In this subsection, we discuss the D-type contribu- tion to every coeﬃcient of the terms arising from the rs → 0 expansion that originates from both bc-primed coeﬃcients, b ′′′ c , and Σ r(kF , 0). D-type contributions can be further categorized into D1 and D2 sub-types. After applying the ﬁrst two steps of our systematic approach, D1-...

[2] [2]

These hybrid-type contributions arise by taking into account higher-order terms in the Taylor expansion of Q(x, y) in the x → 0 limit

H-type contribution We discuss the H-type contributions to every coeﬃ- cient in the rs-expansion, which arise exclusively from the integral kinds L1 im(rs), since Region 2 provides only D-type contributions. These hybrid-type contributions arise by taking into account higher-order terms in the Taylor expansion of Q(x, y) in the x → 0 limit. Con- sequently...

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156 by now deﬁning βn(a) as follows βn(a) = ∫ ∞ 0 dy 1 a + y2 , (F1) where we can calculate β1(a) by doing the change of vari- able y = au

Calculating βn We modify the expression for βn in Eq. 156 by now deﬁning βn(a) as follows βn(a) = ∫ ∞ 0 dy 1 a + y2 , (F1) where we can calculate β1(a) by doing the change of vari- able y = au. We obtain β1(a) = π 2 a− 1 2 , a > 0. (F2) By applying the ( n − 1)-th order partial derivative to β1(a) with respect to the variable a, we obtain ∂n−1 ∂an−1 β1(a)...

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By doing this, we obtain D0,1 = ∫ ∞ 0 du u − tan−1(u) u(1 + u2)

Calculating some of the integral kind D0,1 We calculate the integral kind D0,1 starting from its deﬁnition D0,1 = ∫ ∞ 0 dy R(y) (1 + y2) , (F8) where we can perform the change of variable u = 1 /y. By doing this, we obtain D0,1 = ∫ ∞ 0 du u − tan−1(u) u(1 + u2) . (F9) Using a partial fraction decomposition, we have 1 u(1 + u2) = 1 u − u 1 + u2 , (F10) whe...

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Calculating some of the integral kind Dn,5 (for n = 0, 2, 4, 6, 8) Calculating the integral kinds Dn,5 (for n = 0, 2, 4, 6, 8), we use the deﬁnition given by Eq. 153. How- ever, only the terms corresponding to n = 0, 2, 4, 6 can be handled with the same systematic method used through- out this appendix. For D8,5, we will see that it requires a diﬀerent tr...

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