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REVIEW 4 major objections 6 minor 39 references

The $\mathcal{L}$-Calculus for Causal Variational Principles: An Exterior Differential Calculus on Non-Smooth Spaces

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that the support of a critical causal variational principle carries an exterior calculus with de Rham cohomology, Stokes and Gauß theorems.

desk verdict A genuinely new but scope-limited exterior calculus: the math is checkable for smooth Lagrangians, but the physical Lagrangian isn't smooth, so the abstract overclaims and the Poincaré lemma is near-tautological. read the letter →

arxiv 2608.08811 v1 pith:YT7BKXWY submitted 2026-08-09 math-ph math.ATmath.DGmath.MP

Felix Finster , Niky Kamran , Frank van der Top This is my paper · ORCID
classification math-phmath.ATmath.DGmath.MP MSC 58A1058A1249Q20
keywords causalvariationalprinciplesfermionsystemsnon-smoothspacesexteriorcalculusdeRhamcohomologyStokestheoremsurfacelayerintegralsosculatingvacua
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 sets out to build an exterior differential calculus on the support of a causal variational principle, a closed subset of a smooth manifold that need not be a submanifold and may even be discrete. It defines vector fields, weak directional derivatives, differential forms, an exterior derivative, de Rham cohomology, restriction and extension constructions, a Mayer-Vietoris sequence, a Künneth formula, and a Poincaré lemma, together with versions of Stokes' theorem and the Gauß divergence theorem. The point is to bring the standard tools of manifold topology to the singular or discrete spacetimes that arise in causal variational principles, where ordinary differential topology does not apply.

What carries the argument

The central object is the osculating vacuum $M_p$: a smooth vector-space submanifold of the ambient manifold attached to each point $p$, chosen so that it approximates the non-smooth support near $p$ and plays the role of a tangent space. The Lagrangian $L$ itself serves as a smoothing kernel, so weak derivatives are defined by mollified evaluation, as in $\int L(x,y)\,D^\kappa f(y)\,d\tilde\rho(y)=(-1)^{|\kappa|}\int D^\kappa_2 L(x,y)f(y)\,d\tilde\rho(y)$. Parallel frames built from the L-induced connection $\nabla^L_{q,p}$ allow these derivatives to be expressed in local coordinates. This combination of osculating vacua, Lagrangian mollification, and parallel frames carries the whole exterior calculus.

What would settle it

Evaluate the cohomology of the discrete circle of Section 10.4 with the Lagrangian (10.17) at $c=1/2$, $a=0$; the paper predicts $H^0 \cong H^1 \cong \mathbb{R}^2$. A direct computation of the complex (10.21) giving any other dimensions would falsify the claimed cohomology definition.

Watch

Extended reading notes

Core claim

The paper claims that the support $\tilde M$ of a critical measure of a causal variational principle carries an exterior calculus once a smooth Lagrangian and continuously chosen osculating vacua are given. The exterior derivative $d$ is defined on equivalence classes of alternating tensor sections via totally anti-symmetrized weak derivatives, and satisfies $d^2=0$, giving de Rham cohomology groups $H^r(\tilde M)$ and Betti numbers. Versions of Stokes' theorem and the Gauß divergence theorem hold, with boundary integrals replaced by surface layer double integrals and with the Euler-Lagrange equations providing the cancellations. Under additional conditions, the paper proves the Mayer-Vietoris sequence, a Künneth formula for compactly supported forms on countable discrete spaces, and a Poincaré-type lemma for L-star-shaped regions.

Load-bearing premise

The framework assumes the Lagrangian is smooth, while the causal Lagrangians of the physical applications are only locally Hölder continuous; it also assumes osculating vacua can be chosen continuously at every point, a construction deferred to a separate paper.

Editorial extensions

If this is right

  • The support of a minimizer of a causal variational principle carries Betti numbers, so a singular or discrete spacetime can be assigned the usual cohomological invariants of a manifold.
  • Stokes' theorem holds only for critical measures: boundary integrals become surface layer double integrals, and the Euler-Lagrange equations are what make the boundary terms cancel.
  • The Mayer-Vietoris sequence applies whenever the two pieces $U\setminus V$ and $V\setminus U$ L-separate the forms, which is automatic when their L-neighborhoods are disjoint.
  • On countable discrete products, compactly supported cohomology satisfies a Künneth isomorphism; the paper shows this fails without compact support, as on $\mathbb{Z}^2$ the function $\delta_{n,m}$ is not in the algebraic tensor product.
  • The Poincaré lemma holds only in L-star-shaped regions with a spectral-gap condition on the Lagrangian convolution operator, making exactness a quantitative, scale-dependent statement.

Reading between the lines

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

  • If the calculus is correct, the de Rham cohomology of the support of a minimizing measure is an invariant of the causal variational principle itself, not merely of the underlying topological space; two Lagrangians on the same support could yield different Betti numbers, giving a new classification tool.
  • The L-star-shaped condition suggests a notion of cohomology at resolution $\epsilon$: as the lattice spacing tends to zero and the spectral gap closes, the Poincaré lemma may fail, producing a discretized de Rham complex whose cohomology tracks the continuum limit.
  • The smoothness idealization might be testable: if a Hölder-continuous causal Lagrangian is mollified while preserving the Euler-Lagrange equations, the same theorems should hold with quantitative error estimates, extending the calculus to the physical case.
  • The failure of Künneth for non-compact forms indicates that a norm or topology on the form spaces, which the paper deliberately omits, is not cosmetic but essential for analytic tensor products; endowing forms with such structure is the natural next step.
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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

4 major / 6 minor

Summary. The paper develops an exterior differential calculus on the support M̃ of a minimizing measure ρ̃ for a causal variational principle in the smooth setting L ∈ C∞(F × F). The main constructions are: directional derivatives defined by testing against the Lagrangian (Section 4), a divergence and Gauß theorem (Section 5), L-induced charts and a parallel transport connection (Section 6), weak derivatives and an exterior derivative on spaces of equivalence classes of tensor sections (Section 7.1–7.2), de Rham cohomology (Section 7.3), a Stokes theorem for top forms using surface layer integrals (Section 7.4), restriction and extension lemmas with a Mayer-Vietoris sequence (Sections 7.5–7.7), a Künneth formula for compactly supported forms on countable discrete spaces (Section 7.8), a Poincaré-type lemma under an L-star-shaped condition (Section 7.9), a higher-codimension Stokes theorem for softened surface layer integrals (Section 8), and a general tensor calculus (Section 9). The last section computes cohomology for one-point, two-point, discrete line, discrete circle, lattice, and L-star-shaped lattice examples. The paper positions itself as a generalization of the de Rham calculus to non-smooth spaces that arise as the support of critical measures.

Significance. If the results are correct, this is a substantial contribution: it provides a concrete, computable exterior calculus on spaces that are not manifolds, including discrete spaces, and it proves analogues of Stokes's theorem, the Gauß divergence theorem, Mayer-Vietoris sequences, and a Künneth formula. The paper is careful in many places: the proofs of Theorems 5.2, 7.5 and 8.2 are direct and transparent, the cancellations using the Euler-Lagrange equations are explicit, and the examples in Section 10 give verifiable computations of cohomology groups, including the subtle lattice examples where the Fourier method is used. The authors also honestly indicate several limitations: the smoothness of the Lagrangian is an idealization, the Poincaré lemma requires a contraction condition, and the Künneth formula is restricted to compactly supported forms on discrete countable spaces. However, these limitations are not merely cosmetic: they affect the scope of the central claim and the completeness of some proofs, so the paper needs revision before it can be accepted.

major comments (4)
  1. [Section 2.1 and Abstract] The smoothness assumption L ∈ C∞(F × F) in Eq. (2.1) is load-bearing for every subsequent construction. The directional derivative (4.3), the divergence in Definition 5.1, the weak derivatives (7.4), the exterior derivative (7.15), and the proofs of Theorems 5.2, 7.5 and 8.2 all differentiate the Lagrangian and use the Euler-Lagrange equations (2.3) to cancel terms. The paper itself states at the end of Section 2.1 that the causal Lagrangian in the physical applications is only locally Hölder continuous and that mollifying it may destroy the EL equations. Consequently, the abstract's claim of a calculus "for causal variational principles" is too broad: the theorems as stated apply only to the smooth-Lagrangian class, not to the causal action principle that motivates the theory. Please either restrict the claims explicitly to smooth causal variational principles or provide a genuine extension to the Hölder case, for example via the expedient differential calculus mentioned in Section 2.1.
  2. [Section 3, Eq. (3.2)] The existence of a continuous choice of osculating vacua Φ_p satisfying (3.2) is assumed without proof and deferred to the in-preparation reference [24]. This is not a peripheral technicality: the L-induced charts in (6.2), the connection ∇L in Section 6, the parallel frames in (7.6)–(7.7), and therefore the entire exterior calculus depend on this choice. Moreover, the non-transitive case in Section 3 is described only informally via "approximately equal" and a sketch of a variational principle. Since [24] is not available to the reader, the paper should either prove existence of a suitable Φ_p under transparent hypotheses on (F, L, ρ) or state this existence as a theorem-level hypothesis in the Introduction and in the statements of the main theorems.
  3. [Section 7.9, Definition 7.24 and Lemma 7.25] The Poincaré-type lemma is a central topological claim, but its proof as written has a gap. The L-star-shaped condition (7.53) uses an unspecified norm ‖·‖_{\hat U} and an infimum over representatives, and the proof of Lemma 7.25 concludes that the series ν = Σ_k Pω^(k) converges from the estimate ‖ω^(k)‖ ≤ c^k‖ω‖ with c < 1. This requires a completeness statement for the space of representatives (or an explicit convergence argument), which is not provided. In addition, (7.53) is a strong contraction condition that already encodes the exactness of all closed forms, so the lemma would be substantially more informative if the condition were verified for the lattice example in Section 10.6 rather than assumed as part of Definition 7.24. Please state the completeness assumption and clarify the role of (7.53) relative to the claimed exactness.
  4. [Section 7.8, Lemma 7.20 and Theorem 7.21] The Künneth formula is stated only for compactly supported differential forms on countable discrete spaces, and the restriction is acknowledged in the discussion after Theorem 7.21. Nevertheless, the proof of Lemma 7.20 has a gap: the surjectivity argument writes a compactly supported form ω on M̃ × Ñ in terms of a parallel frame (e_i)_{i=1}^N, but the corresponding parallel frame and equivalence relations for the product causal variational principle are not constructed. Since the whole point of Lemma 7.20 is to identify Ω_c^*(M̃ × Ñ) with Ω_c^*(M̃) ⊗ Ω_c^*(Ñ), the product frame and the behavior of the equivalence relations under the product Lagrangian need to be stated explicitly. Please also state precisely the hypotheses on the factor Lagrangians under which the identification holds.
minor comments (6)
  1. [Title page] The running title contains spacing artifacts: "V ARIA TIONAL" should be "VARIATIONAL".
  2. [Section 7.1, Eq. (7.4)] The displayed formula (7.4) is missing an integral sign on the left-hand side: the expression should read ∫̶_˜M L(x,y)·f(y) dρ̃(y) := ... .
  3. [Section 7.2, Eq. (7.15)] In Eq. (7.15) the index m is used both for the order of the weak derivative and for a tensor rank; this makes the formula hard to parse. Please distinguish the two uses, for example by writing r for the tensor rank.
  4. [Section 10.3] In the paragraph after Eq. (10.8), the cross-reference "eq:1-vanish" appears to refer to Eq. (10.8) but the label is not defined; please correct the reference.
  5. [Section 10.2] The sentence "In the case that some of the off-diagonal matrix elements in (10.6), we cannot use the Mayer-Vietoris sequence" is grammatically incomplete; it should read "In the case that some of the off-diagonal matrix elements in (10.6) are non-zero, ...".
  6. [Section 7.9] The notation ˇω in Definition 7.24 is introduced only implicitly; please define it explicitly as a representative of the equivalence class ω before using it in (7.53).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the integral theorems reduce to the externally given Euler–Lagrange equations, and the glueing and Poincaré statements carry explicit hypotheses that are stronger than their conclusions.

full rationale

None of the paper's claimed derivations reduces to its own inputs by construction. The Gauss and Stokes theorems are direct consequences of the independent Euler–Lagrange equations: Theorem 5.2's proof says 'The term (5.1) vanishes in view of the EL equations (2.3) for ρ̃', and Theorem 7.5's proof uses the identical cancellation at (7.26). The EL equations are an external input from the causal variational principle, not a renamed version of the conclusions, so relying on them is not circular. The glueing statements are explicitly conditional: Lemma 7.11 proves that the extension property is equivalent to the L-separating condition of Definition 7.10, and Corollary 7.14 then derives the long exact sequence once that condition is assumed. This is an identified sufficient/necessary hypothesis, not a hidden identification of hypothesis and conclusion. The Poincaré-type lemma is a contraction-mapping argument: Definition 7.24 requires the quantitative inequality (7.53) for all closed forms, which is stronger than exactness, and Lemma 7.25 obtains exactness by a Neumann series. No equation of the hypothesis literally is the theorem. The Künneth formula uses the classical algebraic Künneth theorem (Lemma 7.22, cited to [29]) after establishing an isomorphism of cochain complexes; the target result is not imported into the proof. The paper is also explicit that the smoothness assumption (2.1) is 'a mathematical idealization which does not quite hold in the physical applications', and that mollifying the causal Lagrangian may destroy the EL equations. This is a scope limitation and a correctness risk for physical applications, not a circular step. Finally, the existence of continuously chosen osculating vacua is assumed in (3.2) and its detailed construction is deferred to [24]; because it is stated as an assumption rather than derived from the paper's own conclusions, it does not create a circular chain.

Assumptions & free parameters 3 free parameters · 8 assumptions · 3 invented entities

The framework rests on the Lagrangian as a differentiation and smoothing device, so the smoothness axiom and the minimization/EL conditions are the external inputs; the vacuum dimension k and the example parameters (alpha, beta; a, b, c; matrix entries) are hand-chosen and directly control the computed cohomology. The L-separating and L-star-shaped conditions are introduced specifically to make the Mayer-Vietoris and Poincaré statements true. The osculating vacua, the L-induced connection, and the surface layer integrals are new framework entities without independent empirical or formal support; per-application verification is required for each.

free parameters (3)
  • vacuum dimension k = dim M
    The cohomology groups depend on the chosen dimension of the osculating vacuum M (Section 10.1: the one-point space has H^0 = R for k = 0 but H^1 = R for k = 1). This is a modeling choice, not derived from the data.
  • Lagrangian parameters alpha, beta (discrete line) = alpha, beta in R
    Section 10.3: the cohomology of the discrete line depends on these parameters; e.g., H^0(Z) = R^2 if beta != 0, and H^0(Z) = Omega^{0,0,1}(Z) if beta = 0 (Prop 10.1). Chosen by hand to illustrate the calculus.
  • Lagrangian matrix entries a0, bp, ap (two points) and a, b, c (discrete circle) = case-dependent values
    Sections 10.2 and 10.4: the cohomology depends on these entries (Table 1 of Section 10.4); values are chosen per case to produce the tabulated Betti numbers.
assumptions (8)
  • domain assumption L is C^infinity with compact range (2.1) and strictly positive on the diagonal
    Stated in Section 2.1 as assumptions (i) to (iii). All weak derivatives in Section 7 test against derivatives of L; flagged by the authors as an idealization since the physical causal Lagrangian is only locally Holder continuous.
  • domain assumption The measures rho (vacuum) and rho-tilde (interacting) satisfy the EL equations (2.3), and rho-tilde is a minimizer of the causal action
    In Section 2.1; required for the vanishing of term (5.1) in Theorem 5.2 and the analogous cancellations in the proofs of Theorems 7.5 and 8.2. Without criticality, Gauss and Stokes fail.
  • ad hoc to paper Existence of a continuous choice of osculating vacua (3.2) with a symmetry group G acting transitively or approximately
    Section 3 assumes symmetry transformations Phi_p with Phi_p(0) approximately p and continuity (3.2), deferring the construction to [24]; this is the paper's substitute for a tangent space.
  • ad hoc to paper L-regularity blanket assumption: the parallel transport connection is bijective on every L-ball
    Stated after Definition 7.2; required to define parallel frames (7.6) to (7.7) used throughout the form calculus. No existence theorem is given.
  • ad hoc to paper L-separating condition (Definition 7.10)
    Required for the extension lemma and Mayer-Vietoris exactness (Lemma 7.11, Corollary 7.14); shown necessary and sufficient, so it is the content of the exactness statement.
  • ad hoc to paper L-star-shaped condition with contraction inequality (7.53)
    Definition 7.24; the Poincare-type lemma 7.25 holds by iterating this inequality. The assumption is close in form to the conclusion.
  • domain assumption For the Kunneth formula: both component measures satisfy (7.43), and the spaces are countable and discrete with compactly supported forms
    Section 7.8; Theorem 7.21 restricts to compactly supported forms on countable discrete spaces; Example 7.23 shows the unrestricted statement is false.
  • standard math Standard algebraic topology background (Mayer-Vietoris proof follows Bott-Tu, algebraic Kunneth theorem of Hilton-Stammbach)
    Used in Corollary 7.14 and Lemma 7.22; external classical results.
invented entities (3)
  • Osculating vacuum M_p and associated measure rho_p
    purpose: Per-point smooth, flat tangent-like space attached to p in M-tilde, used to define directional derivatives, frames, and forms
    Section 3; existence and choice assumed via symmetry transformations, no external experimental handle; it is framework-internal machinery.
  • L-induced chart phi_p and connection nabla^L
    purpose: Chart map from M-tilde to M_p and parallel-transport connection between osculating vacua, used to define global frames and L-balls
    Section 6; defined via the Lagrangian, geometric significance deferred to the in-preparation reference [24]; no independent verification outside the same research program.
  • Surface layer integrals of co-dimension q
    purpose: Replacement for boundary integrals in the non-smooth setting; double integrals over subsets near the boundary
    Sections 2.2, 7.4, 8.1; built from the Lagrangian and characteristic functions; an internal notion with no external benchmark.

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Pith. "Pith review of The $\mathcal{L}$-Calculus for Causal Variational Principles: An Exterior Differential Calculus on Non-Smooth Spaces." pith.science (2026). https://pith.science/paper/YT7BKXWY

@misc{pith2026260808811,
  author       = {Pith},
  title        = {Pith review of: The $\mathcalL$-Calculus for Causal Variational Principles: An Exterior Differential Calculus on Non-Smooth Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YT7BKXWY}},
  note         = {Machine review of arXiv:2608.08811}
}
read the original abstract

A differential calculus for causal variational principles is developed, which generalizes the exterior calculus of differential forms and some of the associated differential topological structures to non-smooth spaces. Our calculus includes the exterior derivative, de Rham cohomology, glueing constructions (restrictions and extensions of differential forms, Mayer-Vietoris sequence), a K\"unneth formula and Poincar{\'e}'s lemma. Moreover, we prove versions of Stokes' theorem and the Gau{\ss} divergence theorem. The constructions and results are illustrated by several examples.

Figures

Figures reproduced from arXiv: 2608.08811 by the authors.

Figure 1
Figure 1. The space [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 1
Figure 1. A discrete space M˜ embedded in F. Therefore, the interesting structures come from the embedding of M˜ in F, as we now explain step-by-step. First, on F we are given the Lagrangian L : F × F → R + 0 . In the present paper, we always assume that L is smooth in both arguments (the smooth setting; for details see Section 2.1). Similar to a convolution kernel, the Lagrangian can be used as a smoothing kernel for mollify… view at source ↗
Figure 2
Figure 2. Osculating vacua of a discrete space. derivative D2,vL(x, y) (where the index two means that the derivative acts on the second argument of the Lagrangian). Considering for simplicity again a compactly supported function f : M˜ → R, we interpret the expression − ˆ Mp D2,vL(x, y) f(y) dρp(y) (1.2) as the weak derivative of f in the direction v. In this way, one can generalize certain aspects of the standard differenti… view at source ↗
Figures from the paper (4 more)
Figure 3
Figure 3. Figure 3: A surface layer integral. We finally note that all the constructions and results in this paper are functorial, if for a morphism between causal variational principles we take a smooth and closed mapping from F to F ′ which preserves the Lagrangian. A measure ρ on F is …
Figure 4
Figure 4. Figure 4: A surface layer integral of co-dimension two. Again assuming that the Lagrangian vanishes unless x and y are close together, we only get contributions to this surface layer integral if both x and y are close to the two-dimensional surface S (see [PITH_FULL_IMAGE:figur…
Figure 5
Figure 5. Figure 5: Two sets which satisfy the sufficient condition of Lemma 7.12. 7.8. A K¨unneth Formula. There are several classical computational methods for the de Rham cohomology, a particularly useful being the K¨unneth formula. In the classical case, this formula relates the cohom…
Figure 6
Figure 6. Figure 6: Glueing Construction. 7.10. Computing the Cohomology, Independence of the Osculation. By com￾bining the above constructions, one can follow the usual path for computing the de Rham cohomology. But of course, the fact that we are in the non-smooth setting leads to subtl…

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Reference graph

Works this paper leans on

39 extracted references · 24 canonical work pages

  1. [24]

    A Geometric Derivation of the Einstein Equations from the Causal Action Principle

    F. Finster and C. Krpoun, A geometric derivation of the Einstein equations from the ca usal action principle, arXiv:2607.13871 [math-ph] (2026). THE L-CALCULUS FOR CAUSAL V ARIATIONAL PRINCIPLES 57

    work page Pith review arXiv 2026

  2. [39]

    van der Top, An Exterior Differntial Calculus on Non-Smooth Spaces , Masterarbeit Mathe- matik, Universit¨ at Regensburg, in preparation (2026)

    F. van der Top, An Exterior Differntial Calculus on Non-Smooth Spaces , Masterarbeit Mathe- matik, Universit¨ at Regensburg, in preparation (2026)

    work page 2026

  3. [1]

    Link to web platform on causal fermion systems: www.causal- fermion-system.com

    work page

  4. [2]

    Arnold, Finite element exterior calculus , CBMS-NSF Regional Conference Series in Applied Mathematics, vol

    D.N. Arnold, Finite element exterior calculus , CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 93, Society for Industrial and Applied Ma thematics (SIAM), Philadelphia, PA, 2018

    work page 2018

  5. [3]

    Complexes from complexes

    D.N. Arnold and K. Hu, Complexes from complexes, arXiv:2005.12437 [math.NA], Found. Comput. Math. 21 (2021), no. 6, 1739–1774

    work page Pith review arXiv 2021

  6. [4]

    Bott and L.W

    R. Bott and L.W. Tu, Differential Forms in Algebraic Topology , Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982

    work page 1982

  7. [5]

    Casali, P

    M.R. Casali, P. T, S. Dartois, and L. Grasselli, Topology in colored tensor models via crystallization theory, arXiv:1704.02800 [math-ph], J. Geom. Phys. 129 (2018), 142–167

    arXiv 2018

  8. [6]

    Cheeger, On the spectral geometry of spaces with cone-like singulari ties, Proc

    J. Cheeger, On the spectral geometry of spaces with cone-like singulari ties, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106

    work page 1979

Show all 39 references
  1. [7]

    Cohen, M

    D.C. Cohen, M. Goresky, and L. Ji, On the K¨ unneth formula for intersection cohomology , Trans. Amer. Math. Soc. 333 (1992), no. 1, 63–69

    1992

  2. [8]

    Curiel, F

    E. Curiel, F. Finster, and J.M. Isidro, Two-dimensional area and matter flux in the theory of causal fermion systems , arXiv:1910.06161 [math-ph], Internat. J. Modern Phys. D 29 (2020), 2050098

    2020 arXiv

  3. [9]

    Dappiaggi and F

    C. Dappiaggi and F. Finster, Linearized fields for causal variational principles: Exist ence theory and causal structure , arXiv:1811.10587 [math-ph], Methods Appl. Anal. 27 (2020), no. 1, 1–56

    2020 arXiv

  4. [10]

    Desbrun, E

    M. Desbrun, E. Kanso, and Y. Tong, Discrete differential forms for computational modeling , Discrete differential geometry, Oberwolfach Semin., vol. 3 8, Birkh¨ auser, Basel, 2008, pp. 287–324

    2008

  5. [11]

    Ferri, C

    M. Ferri, C. Gagliardi, and L. Grasselli, A graph-theoretical representation of PL-manifolds—a survey on crystallizations , Aequationes Math. 31 (1986), no. 2-3, 121–141

    1986

  6. [12]

    Finster, Causal variational principles on measure spaces , arXiv:0811.2666 [math-ph], J

    F. Finster, Causal variational principles on measure spaces , arXiv:0811.2666 [math-ph], J. Reine Angew. Math. 646 (2010), 141–194. 13. , The Continuum Limit of Causal Fermion Systems , arXiv:1605.04742 [math-ph], Funda- mental Theories of Physics, vol. 186, Springer, Cham, 2016

    2010 arXiv

  7. [14]

    , Positive functionals induced by minimizers of causal varia tional principles , arXiv:1708.07817 [math-ph], Vietnam J. Math. 47 (2019), 23–37

    2019 arXiv

  8. [15]

    Finster and et al, Non-smooth generalizations of the Einstein equations for c ausal fermion systems, in preparation

    F. Finster and et al, Non-smooth generalizations of the Einstein equations for c ausal fermion systems, in preparation

  9. [16]

    Finster and A

    F. Finster and A. Grotz, A Lorentzian quantum geometry , arXiv:1107.2026 [math-ph], Adv. Theor. Math. Phys. 16 (2012), no. 4, 1197–1290

    2012 arXiv

  10. [17]

    Finster and M

    F. Finster and M. Jokel, Causal fermion systems: An elementary introduction to phys ical ideas and mathematical concepts, arXiv:1908.08451 [math-ph], Progress and Visions in Quan tum Theory in View of Gravity (F. Finster, D. Giulini, J. Kleiner, and J. To lksdorf, eds.), Birkh...

    1908 arXiv

  11. [18]

    Finster and N

    F. Finster and N. Kamran, Spinors on singular spaces and the topology of causal fermio n systems, arXiv:1403.7885 [math-ph], Mem. Amer. Math. Soc. 259 (2019), no. 1251, v+83 pp

    2019 arXiv

  12. [19]

    , Complex structures on jet spaces and bosonic Fock space dyna mics for causal variational principles, arXiv:1808.03177 [math-ph], Pure Appl. Math. Q. 17 (2021), no. 1, 55–140

    2021 arXiv

  13. [20]

    , A positive quasilocal mass for causal variational principl es, arXiv:2310.07544 [math-ph], Calc. Var. Partial Differential Equations 64 (2025), no. 3, 91

    2025 arXiv

  14. [21]

    Finster, S

    F. Finster, S. Kindermann, and J.-H. Treude, Causal Fermion Systems: An Introduction to Fun- damental Structures, Methods and Applications , arXiv:2411.06450 [math-ph], Cambridge Mono- graphs on Mathematical Physics, Cambridge University Pres s, 2025

    2025

  15. [22]

    Finster and J

    F. Finster and J. Kleiner, Noether-like theorems for causal variational principles , arXiv:1506.09076 [math-ph], Calc. Var. Partial Differenti al Equations 55:35 (2016), no. 2, 41

    2016 arXiv

  16. [23]

    , A Hamiltonian formulation of causal variational principle s, arXiv:1612.07192 [math-ph], Calc. Var. Partial Differential Equations 56:73 (2017), no. 3, 33

    2017 arXiv

  17. [25]

    Finster and C

    F. Finster and C. Langer, Causal variational principles in the σ-locally compact setting: Existence of minimizers , arXiv:2002.04412 [math-ph], Adv. Calc. Var. 15 (2022), no. 3, 551–575

    2022 arXiv

  18. [26]

    Finster and M

    F. Finster and M. Lottner, Banach manifold structure and infinite-dimensional analys is for causal fermion systems, arXiv:2101.11908 [math-ph], Ann. Global Anal. Geom. 60 (2021), no. 2, 313–354

    2021 arXiv

  19. [27]

    Finster and D

    F. Finster and D. Schiefeneder, On the support of minimizers of causal variational principl es, arXiv:1012.1589 [math-ph], Arch. Ration. Mech. Anal. 210 (2013), no. 2, 321–364

    2013 arXiv

  20. [28]

    Gurau and J.P

    R. Gurau and J.P. Ryan, Colored tensor models—a review , arXiv:1109.4812 [hep-th], SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), Paper 020, 78

    2012 arXiv

  21. [29]

    Hilton and U

    P.J. Hilton and U. Stammbach, A Course in Homological Algebra , second ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997

    1997

  22. [30]

    Hydon and E.L

    P.E. Hydon and E.L. Mansfield, A variational complex for difference equations , Found. Comput. Math. 4 (2004), no. 2, 187–217

    2004

  23. [31]

    Johnson, Atomic and nonatomic measures , Proc

    R.A. Johnson, Atomic and nonatomic measures , Proc. Amer. Math. Soc. 25 (1970), 650–655

    1970

  24. [32]

    Laugwitz, Differential and Riemannian Geometry , Academic Press, New York-London, 1965, Translated by F

    D. Laugwitz, Differential and Riemannian Geometry , Academic Press, New York-London, 1965, Translated by F. Steinhardt

    1965

  25. [33]

    Lichnerowicz, Elements of Tensor Calculus , Methuen & Co Ltd, John Wiley & Sons, London- New York, 1962, Translated by J.W

    A. Lichnerowicz, Elements of Tensor Calculus , Methuen & Co Ltd, John Wiley & Sons, London- New York, 1962, Translated by J.W. Leech

    1962

  26. [34]

    Milnor and J.D

    J.W. Milnor and J.D. Stasheff, Characteristic Classes , Princeton University Press, Princeton, N. J., 1974, Annals of Mathematics Studies, No. 76

    1974

  27. [35]

    Paganini and S

    C.F. Paganini and S. Yadav, The preserver problem for causal fermion systems , in preparation

  28. [36]

    HeeSook Park, Some remarks for K¨ unneth formula on bounded cohomology , Honam Math. J. 37 (2015), no. 1, 7–27

    2015

  29. [37]

    Steenrod, The Topology of Fibre Bundles , Princeton Landmarks in Mathematics, Princeton University Press, 1951

    N. Steenrod, The Topology of Fibre Bundles , Princeton Landmarks in Mathematics, Princeton University Press, 1951

    1951

  30. [38]

    Valette, L∞ cohomology is intersection cohomology, Adv

    G. Valette, L∞ cohomology is intersection cohomology, Adv. Math. 231 (2012), no. 3-4, 1818–1842

    2012

  31. [40]

    White and P.E

    L.C. White and P.E. Hydon, Moving frames: difference and differential-difference Lagrangi ans, arXiv:2309.09040 [math-ph], SIGMA Symmetry Integrabilit y Geom. Methods Appl. 20 (2024), Paper No. 006, 29. F akult¨at f ¨ur Mathematik, Universit ¨at Regensburg, D-93040 Regensburg, Ge...

    2024 arXiv

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