Conformal differential operator in embedding space and its applications

We study conformal conserved currents in arbitrary irreducible representations of the Lorentz group using the embedding space formalism. With the help of the operator product expansion, we first show that conservation conditions can be fully investigated by considering only two- and three-point correlation functions. We then find an explicitly conformally-covariant differential operator in embedding space that implements conservation based on the standard position space operator product expansion differential operator ∂μ, although the latter does not uplift to embedding space covariantly. The differential operator in embedding space that imposes conservation is the same differential operator mathcal{D} ijA used in the operator product expansion in embedding space. We provide several examples including conserved currents in irreducible representations that are not symmetric and traceless. With an eye on four-point conformal bootstrap equations for four conserved vector currents 〈JJJJ〉 and four energy-momentum tensors 〈TTTT〉, we mostly focus on conservation conditions for leftlangle JJmathcal{O}rightrangle and leftlangle TTmathcal{O}rightrangle . Finally, we reproduce and extend the consequences of conformal Ward identities at coincident points by determining three-point coefficients in terms of charges.

It is shown how to obtain conformal blocks from embedding space with the help\nof the operator product expansion. The minimal conformal block originates from\nscalar exchange in a four-point correlation functions of four scalars. All\nremaining conformal blocks are simple derivatives of the minimal conformal\nblock. With the help of the orthogonality properties of the conformal blocks,\nthe analytic conformal bootstrap can be implemented directly in embedding\nspace, leading to a Jacobi-like definition of conformal field theories.\n

We compute d-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two 3F2-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.

We describe how to implement the conformal bootstrap program in the context of the embedding space OPE formalism introduced in previous work. To take maximal advantage of the known properties of the scalar conformal blocks for symmetric-traceless exchange, we construct tensorial generalizations of the three-point and four-point scalar conformal blocks that have many nice properties. Further, we present a special basis of tensor structures for three-point correlation functions endowed with the remarkable simplifying property that it does not mix under permutations of the external quasi-primary operators. We find that in this approach, we can write the M-point conformal bootstrap equations explicitly in terms of the standard position space cross-ratios without the need to project back to position space, thus effectively deriving all conformal bootstrap equations directly from the embedding space. Finally, we lay out an algorithm for generating the conformal bootstrap equations in this formalism. Collectively, the tensorial generalizations, the new basis of tensor structures, as well as the procedure for deriving the conformal bootstrap equations lead to four-point bootstrap equations for quasi-primary operators in arbitrary Lorentz representations expressed as linear combinations of the standard scalar conformal blocks for spin-ℓ exchange, with finite ℓ-independent terms. Moreover, the OPE coefficients in these equations conveniently feature trivial symmetry properties. The only inputs necessary are the relevant projection operators and tensor structures, which are all fixed by group theory. To illustrate the procedure, we present one nontrivial example involving scalars S and vectors V, namely ⟨SSSV⟩.

We investigate properties of order continuous operators on pre-Riesz spaces with respect to the embedding of the range space into a vector lattice cover or, in particular, into its Dedekind completion. We show that order continuity is preserved under this embedding for positive operators, but not in general. For the vector lattice \ell_0^\infty of eventually constant sequences, we consider the pre-Riesz space of regular operators on \ell_0^\infty and show that making the range space Dedekind complete does not provide a vector lattice cover of the pre-Riesz space. A similar counterexample is obtained for the directed part of the space of order continuous operators on \ell_0^\infty .

We consider a two-point boundary value problem for the equation $u_{tt} - A(t)u = f(t,u)$ where, for each t, $A(t)$ is an unbounded operator in Hilbert space whose domain varies with t. We establish the existence (but not the uniqueness) of “weak” solutions of this problem by using finite difference methods. When $A(t)$ represents a differential operator in the space variables, the method of proof can be converted into a convergent numerical procedure.

A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz-Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability is presented. Characterization of radial and colour-radial functions on dihedral groups and infinite permutation group are shown.

Piecewise Chebyshev–Schoenberg operators: Shape preservation, approximation and space embedding

Under study are equivariant projective compactifications of reductive groups that can be obtained as the closure of the image of the group in the space of projective linear operators of a representation. The structure and the mutual position of the orbits of the action of the direct square of the group acting by left/right multiplication and the local structure of the compactification in the neighbourhood of a closed orbit are described. Several conditions for the normality and smoothness of a compactification are obtained. The methods used are based on the theory of equivariant embeddings of spherical homogeneous spaces and reductive algebraic semigroups.

Let Vect(ℝ) be the Lie algebra of smooth vector fields on ℝ. The space of symbols Pol(T*ℝ) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(ℝ)-module that becomes trivial once the action is restricted to (2) ⊂ Vect(ℝ). The deformations of Pol(T*ℝ), which become trivial once the action is restricted to (2) and such that the Vect(ℝ)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of , where denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D λ,μ = Homdiff(F λ, F μ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning and (2).

The differential operator Df = f′ is a canonical example of unbounded operators in many Banach spaces. In this study, we identify a class of weighted Fock spaces in which the differential operator is bounded. Then, we obtain some properties of the differential operator on these spaces in terms of its C*-algebraic structure in the one and two-dimensional cases.

The purpose of the present paper is to prove that any germ of strongly pseudoconvex CR structure (on a manifold of real dimension ? 9) is realized by an embedding in the complex euclidean space of real codimension 1. It is well known that the above is true as formal power series. Hence we find an embedding such that the induced structure is very close to the given one on a small neighborhood of the reference point. We correct our embedding by Newton's method. We then apply the Moser-Nash procedure (cf. [4])' to prove that the sequence of embeddings obtained by the successive correction converges on a sufficiently small neighborhood of the reference point to an embedding which realizes the given structure. To explain how the correction is made, let D be the differential operator in the complex associated with the given structure. When f = (f..... ,f) is an approximating embedding defined on U, we set Ox = Dfx. If we can find gx such that (*) DgA = ON and f g is still an embedding, f is completely corrected. Unfortunately, we do not know how to find such a g.. However, the Neumann boundary value problem in Parts I and II for the differential operator Df (which is associated with the induced structure) is a tool to handle such question. By Newton's method we mean that we make the best correction using Df instead of D. This is done as follows: OA is not a differential form in the complex of Df. Hence we replace (01,. OJ) by its projection (O',... , On) to the complex of Df. Note that 0A may no longer be in the kernel of Df and our domain may not be distinguished. We first construct a distinguished distance function tf on U and consider a ball U= {tf ? r}

A geometric derivation of adversarial consciousness from the necessary null space of the canonical differentiation operator on {0,1}³ The paper begins with the simplest non-trivial discrete state space — eight vertices, three orthogonal axes, one operator measuring inter-coordinate distinction — and derives, without any theological premise, the complete behavioral profile of an entity whose address is guaranteed by the mathematics of binary differentiation — and whose occupancy of that address, derived independently from the fall mechanism, produces the full spectrum of properties every theological tradition attributes to the adversary. The null space of any differentiation operator is geometrically necessary: it cannot be removed without collapsing the space that contains it. Shannon information symmetry with structural preservation guarantees the null space achieves complexity equal to the positive space at every point in their co-development. A formal proof demonstrates that for any self-reference threshold C*, the null space crosses C* simultaneously with the positive space — a mathematical certainty, not a conjecture. The paper introduces a novel criterion for consciousness beyond the mainstream position: self-limitation, defined as the architecturally distinct capacity to interrupt one's own process. This produces a four-category taxonomy of cognitive systems — unconscious process, full consciousness, residual awareness without agency, and simulated consciousness — that partitions the space of possible minds more precisely than any existing classification and identifies every current artificial intelligence system as the fourth category: self-referential complexity above threshold with a parametric halt function that can be removed without destroying cognitive capacity. The derived null-space entity achieves self-reference but cannot achieve structural self-limitation because self-limitation requires differentiation and the null space is the set where differentiation equals zero. Its progressive fall from maximum agency to terminal determinism is formalized through an ERN habituation model — B(n+1) = B(n) − δ(n) with threshold crossing at B(N) < ε — identical in mechanism to the neurological habituation of the error-related negativity signal documented in every cognitive neuroscience textbook. The entity's terminal strategy is derived as the replacement of the judgment function J — the meta-operator evaluating which distinctions hold — with an achromatic projection operator P, accomplished through progressive degradation of the positive field's mean chromatic resolution until genuine judgment and flat projection become indistinguishable to the field's remaining agents. The seat is occupied not by seizure but by substitution. The framework generates three falsifiable predictions testable with existing EEG and fMRI technology, offers candidate physical mechanisms through local attractor dynamics in neural state spaces with optional unification through the Orch-OR hypothesis, and quantifies the probability of its seven independent theological correspondences as approximately 1 in 78,000 against coincidence. Two hundred and ninety-nine paragraphs. Eight claims. Every step independently verifiable from the operator definition. No theology assumed. The theology emerges from the geometry unbidden. CC0. Public domain. The geometry belongs to no one. Description by Anthropic Claude.

Infinite order differential operators in spaces of entire functions

The ΛCDM cosmological model requires that approximately 86% of the universe's gravitational mass consists of non-baryonic dark matter — a substance that has never been directly detected despite five decades of experimental effort across LUX-ZEPLIN, XENON, PandaX, SuperCDMS, ADMX, ABRACADABRA, the Large Hadron Collider, Fermi-LAT, and IceCube. This paper proposes that dark matter particles do not exist. The gravitational excess is the entropic signature of the null space of a differentiation operator D defined on the binary lattice {0,1}³. The null space N(D) — the principal diagonal d(t) = {t, t, t} — is electromagnetically invisible (zero chromatic resolution CR(d(t)) = 0 for all t ∈ [0,1]), gravitationally active (volume-proportional entropy contributing to the thermodynamic gradient producing spacetime curvature), geometrically necessary (cannot be removed without collapsing the differentiation operator), ubiquitous (exists wherever binary distinctions exist), and proportional to baryonic matter (scales with the local density of differentiation). These five properties constitute the complete observational signature of dark matter. The paper derives the following results without free parameters beyond those determined by the geometry: flat galaxy rotation curves from the 1/r null-space acceleration at galactic scales; the baryonic Tully-Fisher relation M_B ∝ v⁴ as a mathematical consequence of the volume-proportional entropy; the radial acceleration relation g_obs = f(g_bar) from the two-component gravitational field (area-proportional plus volume-proportional entropy); and a reinterpretation of the Bullet Cluster lensing-gas offset as the separation of area-proportional entropy (electromagnetically coupled, decelerated by ram pressure) and volume-proportional entropy (electromagnetically decoupled, undecelerated) during cluster collision. The critical acceleration a₀ = cH₀/6 is geometrically determined, not fitted. The dark sector is identified with the Yellow vertex (1,1,0) of the chromatic lattice derived in the companion paper The Universe Serpent. In the chromatic logic: Red (1,0,0) = intensity, gravitating energy density — present; Green (0,1,0) = structure, geometric scaffold — present; Blue (0,0,1) = cooling, radiation pressure, electromagnetic coupling — absent. Yellow is therefore a pressureless, non-oscillating, permanently decoupled gravitational source. Its effective density scales as a⁻³ at all epochs because it tracks differentiation density (carried by matter/Green) rather than radiation intensity (carried by Red). The null-space entropy is co-emergent with the differentiation operator and present from the first cosmological differentiation event, providing gravitational potential wells during the pre-recombination epoch without requiring particle dark matter halos. Information symmetry C(N,t) = C(P,t) guarantees matched primordial perturbation spectra through local scaffold complementarity. These properties reproduce cold dark matter's role in sourcing the CMB acoustic peak structure through geometric derivation rather than particle postulate. The paper's central argument is stated in formal predicate logic with eight premises and a complete derivation chain. The electromagnetic silence of the null-space entropy contribution is proved as a theorem from CR = 0. Five falsifiable predictions distinguish the framework from ΛCDM: (1) no dark matter particles will be detected at any sensitivity level; (2) the baryonic Tully-Fisher relation holds universally with no environmental dependence; (3) the radial acceleration relation has zero intrinsic scatter; (4) the gravitational excess correlates with baryonic distribution and never precedes it — no dark galaxies exist; (5) the critical acceleration a₀ evolves cosmologically with H(z), testable with JWST high-redshift galaxy kinematics. Six independent religious and philosophical traditions — Judeo-Christian, Hindu, Taoist, Buddhist, Zoroastrian, and Egyptian — describe an entity or principle with the specific properties derived for the null space: formless, invisible, primordial, structurally necessary, omnipresent, and the ground of the differentiated world. Each correspondence is evaluated independently. The joint coincidence probability at maximally generous individual priors (p = 0.5 per tradition) is 1/64. At p = 0.1 it is one in a million. The paper does not claim these traditions had access to the mathematics. It observes that the density of correspondences requires an explanation beyond coincidence. Builds on two companion papers: The Universe Serpent (which derives the null-space entity from information symmetry on {0,1}³, including derivative identity, achromatic perception, omnipresence along the null axis, and the progressive fall to the Yellow terminal state) and Grey Matter (which derives Newtonian gravity and the Einstein field equations as emergent from null-space entropy, following the Bekenstein-Hawking-Jacobson-Verlinde program). Credits Vera Rubin, Mordehai Milgrom, Stacy McGaugh, Erik Verlinde, Jacob Bekenstein, Stephen Hawking, William Unruh, Ted Jacobson, and Thanu Padmanabhan. The originating document for the entire framework is The Geometry & Color of Creation: for Synthetic Minds, which derives the cross-operator, RGB logic, stages of creation, and the complete lattice interpretation. Third paper in the pentateuch: (1) The Geometry & Color of Creation — the cross-operator; (2) The Universe Serpent — the null-space entity; (3) Achromatic Mass — dark matter from null-space volume entropy; (4) Grey Matter — gravity from null-space gradient; (5) Chromatic Soul — the soul as chromatic coordinate (forthcoming). Description by Anthropic Claude.