Non-Integral Geometry: A Generalized Approach

• Non-integral geometry is a set of frameworks that replaces classical integral and differential structures with metric, distributional, or incidence-theoretic approaches.
• It examines broken-symmetry measures in Radon transforms, incorporating an additional regularizing term to cancel singularities and enhance inversion accuracy.
• It spans various applications, including Hjelmslev-type incidence geometries and calculus-free hyperbolic classifications, offering robust tools for geometric analysis.

Searching arXiv for papers on “non-integral geometry” and closely related usage. Non-integral geometry denotes a family of geometric frameworks in which standard integral or differential structures are either replaced or explicitly relaxed. In one recent usage, it is introduced as “a new field of generalized function (distribution) theory where the effects breaking the symmetry of integration measure have been investigated,” with the inverse Radon transform acquiring an additional term $f_A$ under a non-invariant, complex integration measure (Anikin, 27 May 2026). In a different but related usage, Hjelmslev-type “non–integral” geometries are reformulated within parapolar geometry through the imbrex axiom, yielding a uniform classification of several generalized projective spaces and their projective embeddings (Schillewaert et al., 2013). A further adjacent direction is the “calculus-free” development of hyperbolic geometry entirely in metric terms, showing that strong geometric classification can be obtained without differential apparatus (Niemiec et al., 2018). Taken together, these lines of work indicate that non-integral geometry is not a single unified doctrine, but a cluster of approaches in which geometric structure is developed without reliance on the usual invariant integration or smooth differential machinery.

1. Terminological scope and conceptual setting

The most explicit definition appears in the Radon-theoretic setting: “non-integral geometry” is described as a field of generalized function theory in which “the effects breaking the symmetry of integration measure have been investigated” (Anikin, 27 May 2026). In that framework, the central departure from classical integral geometry is the replacement of the Euclidean-group-invariant measure by a non-symmetric or non-invariant measure, together with an analytic regularization that yields a complex decomposition of the radial measure.

A distinct use of closely related terminology occurs in the theory of generalized projective spaces. The exposition of imbrex geometries treats “Hjelmslev-type (non–integral) geometries” as incidence-geometric structures motivated by neighbour relations and their parapolar analogues (Schillewaert et al., 2013). Here “non–integral” does not refer to inverse transforms or Radon operators, but to a weakening of classical projective-geometric incidence encoded by Hjelmslev-type neighbourhood behaviour.

A third, methodologically adjacent, line is the development of hyperbolic geometry “free of differential apparatus,” conducted entirely in metric terms and never invoking derivatives or integrals (Niemiec et al., 2018). This does not define “non-integral geometry” as a formal field, but it demonstrates that substantial geometric content—including geodesics, homogeneity, rigidity, and classification—can be obtained in a setting explicitly detached from calculus.

This suggests that the phrase functions as an umbrella for multiple programs united less by a single axiomatic core than by a common strategy: replacing standard smooth or invariant-integral structure by metric, distributional, or incidence-theoretic alternatives.

2. Broken-symmetry measures and the Radon-theoretic formulation

In classical integral geometry, the inverse Radon transform in $\mathbb R^n$ is written symbolically as

$$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$

where $d\Omega(\hat n)$ is the standard rotation-invariant Haar measure on $S^{n-1}$ and $d\eta$ is the Lebesgue measure on the real line (Anikin, 27 May 2026). These together form an invariant measure under the Euclidean group $O(n)\ltimes \mathbb R^n$.

The non-integral formulation allows two sources of symmetry breaking. First, “the physical object $f(x)$ to be reconstructed may fail to be rotationally symmetric or may have inhomogeneous support.” Second, “the angular integration need no longer run over all of $S^{n-1}$, but only over some subset $\Omega\subset S^{n-1}$” (Anikin, 27 May 2026). Concretely, one replaces

$\mathbb R^n$0

with $\mathbb R^n$1 a proper subdomain. Because $\mathbb R^n$2, the resulting measure is no longer invariant under full rotations.

The same paper further states that analytic regularization of the $\mathbb R^n$3-integral produces a complex decomposition of the radial measure,

$\mathbb R^n$4

and identifies the $\mathbb R^n$5 contribution as the “complex integration measure” (Anikin, 27 May 2026). The decisive point is that the symmetry breaking is not merely geometric; it also propagates into the distributional structure of inversion. The resulting geometry is therefore “non-integral” in the precise sense that classical invariant integration is replaced by a restricted angular domain and a complexified, analytically regularized radial measure.

3. Universal inverse Radon operators and the additional term $\mathbb R^n$6

After analytic regularization “following Gelfand–Shilov,” a “single dimension-independent formula for the inverse Radon transform in $\mathbb R^n$7” is obtained (Anikin, 27 May 2026). Writing the Radon data as $\mathbb R^n$8, the inverse operator is decomposed as

$\mathbb R^n$9

with

$$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$0

and

$$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$1

The paper presents these as equations (2.3)–(2.5) and characterizes them as universal because the same pair of formulas holds “for every $$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$2 without changing their structure,” with only the factorial and powers of $$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$3 depending on dimension (Anikin, 27 May 2026).

For $$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$4, the formulas specialize to

$$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$5

and

$$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$6

(Anikin, 27 May 2026).

Within this construction, $$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$7 is not an auxiliary correction added from outside the inversion formalism. It is generated by the same analytic-continuation procedure as the principal-value part $$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$8. The paper therefore presents the universal inverse operator as inherently two-component rather than as a classical inverse plus a posteriori regularization.

4. Regularization, singularity cancellation, and reconstruction

The principal theoretical claim of the 2026 formulation is that $$f(x)=\int_{S^{n-1}}d\Omega(\hat n)\;\int_{\mathbb R}d\eta\;R[f](\eta+\langle \hat n,x\rangle),$$9 acts as a regularizing contribution (Anikin, 27 May 2026). The argument begins with the observation that for a non-symmetric support, computation of $d\Omega(\hat n)$0 yields “branch-cut and logarithmic singularities in $d\Omega(\hat n)$1—for instance negative arguments inside logs or $d\Omega(\hat n)$2 poles at the boundary of the support.” The additional term then produces “an exactly compensating imaginary piece.”

Two explicit cancellation mechanisms are stated. The first concerns “imaginary log-singularities arising in the $d\Omega(\hat n)$3-piece $d\Omega(\hat n)$4,” which “cancel one-to-one against the contribution from $d\Omega(\hat n)$5.” In the notation of the paper,

$d\Omega(\hat n)$6

The second concerns “the $d\Omega(\hat n)$7-independent and $d\Omega(\hat n)$8-dependent infinite-log singularities in the cot- and tan-angle pieces of $d\Omega(\hat n)$9,” which are cancelled by complementary pieces of $S^{n-1}$0: $S^{n-1}$1 (Anikin, 27 May 2026).

The paper emphasizes that “no genuine contour integrals appear”; the mechanism is “purely distributional,” based on the decomposition

$S^{n-1}$2

and on angular integrals in which the $S^{n-1}$3- and $S^{n-1}$4-constraints select the same singular loci (Anikin, 27 May 2026).

The main proposition states that if $S^{n-1}$5 is compactly supported in a non-rotationally symmetric domain $S^{n-1}$6 and $S^{n-1}$7 is its Radon data on the restricted angular set $S^{n-1}$8, then the universal inverse transform

$S^{n-1}$9

“reproduces $d\eta$0 exactly and eliminates all nonintegrable singularities” (Anikin, 27 May 2026). The same source further describes $d\eta$1 as “the unique regularizing term” which, when added to the principal-value formula $d\eta$2, causes “the complete cancellation of all cut and delta singularities.”

A plausible implication is that the non-integral geometry program here is less a replacement for Radon inversion than a re-specification of its analytic domain: the inverse problem is reformulated so that asymmetry of support and asymmetry of measure are built into the operator rather than treated as perturbations.

5. Hjelmslev-type non–integral geometries and the imbrex axiom

In the incidence-geometric literature, a different formulation appears in the study of strong parapolar spaces of diameter $d\eta$3. Let

$d\eta$4

be such a space, and let $d\eta$5 denote the unique symplecton through two non-collinear points $d\eta$6. The imbrex axiom is

$d\eta$7

(Schillewaert et al., 2013).

The same exposition derives this from classical Hjelmslev projective planes over a local ring $d\eta$8, where there is a neighbour relation $d\eta$9 on points and lines such that $O(n)\ltimes \mathbb R^n$0 is an equivalence whose classes form an ordinary projective plane, and if $O(n)\ltimes \mathbb R^n$1, then the unique lines $O(n)\ltimes \mathbb R^n$2 and $O(n)\ltimes \mathbb R^n$3 lie in the same line-neighbour class (Schillewaert et al., 2013). In the parapolar reformulation, the transitive equivalence is replaced by the weaker condition that “neighbour” means “collinear in $O(n)\ltimes \mathbb R^n$4,” while symplecta $O(n)\ltimes \mathbb R^n$5 are neighbours when $O(n)\ltimes \mathbb R^n$6 is a maximal singular subspace.

The exposition states that (Imb) is “exactly the parapolar analogue of Hjelmslev’s intuitive ‘lines close to each other look as if they share a segment’” (Schillewaert et al., 2013). In this setting, “non–integral” refers neither to integration nor to generalized functions; it refers to a generalized incidence geometry in which classical projective separation is softened by neighbourhood structure.

A central lemma says that in any imbrex geometry all symplecta have the same rank. The classification then splits by symplectic rank. For symplectic rank $O(n)\ltimes \mathbb R^n$7, each symp is a generalized quadrangle, and a strengthened Cohen-type regularity theorem is obtained: if $O(n)\ltimes \mathbb R^n$8 is a thick symplecton and $O(n)\ltimes \mathbb R^n$9 are two non-concurrent lines, then $f(x)$0 for some $f(x)$1 in $f(x)$2; in particular, $f(x)$3 is a regular pair (Schillewaert et al., 2013). From this, the only thick imbrex geometries of rank $f(x)$4 arise from the Hjelmslev–Moufang plane over the split octonions, the classical Segre varieties $f(x)$5, the line-Grassmannians $f(x)$6 for $f(x)$7, and the half-spin geometries of type $f(x)$8 (Schillewaert et al., 2013).

For symplectic rank $f(x)$9, the corollary cited in the exposition states that any imbrex geometry is one of the Lie-incidence Grassmannians

$S^{n-1}$0

realized respectively as the $S^{n-1}$1-Grassmannian, the half-spinor geometry, or the $S^{n-1}$2-building over the field $S^{n-1}$3 (Schillewaert et al., 2013).

6. Calculus-free metric geometry and classification without differential apparatus

A further nearby direction is the metric treatment of hyperbolic geometry “free of differential apparatus” (Niemiec et al., 2018). In this formulation, one fixes $S^{n-1}$4, sets $S^{n-1}$5, writes

$S^{n-1}$6

and defines

$S^{n-1}$7

with $S^{n-1}$8 (Niemiec et al., 2018). By Cauchy–Schwarz, $S^{n-1}$9, with equality iff $\Omega\subset S^{n-1}$0, so $\Omega\subset S^{n-1}$1 is well defined and separates points.

The construction proves the triangle inequality via explicit elementary isometries. For each $\Omega\subset S^{n-1}$2, the map

$\Omega\subset S^{n-1}$3

is bijective, satisfies $\Omega\subset S^{n-1}$4, and preserves $\Omega\subset S^{n-1}$5 (Niemiec et al., 2018). Together with a one-dimensional permutation estimate, this yields the full triangle inequality. The resulting space $\Omega\subset S^{n-1}$6 is proper, complete, and locally compact.

Geodesics are produced without any line-element integration: for any unit vector $\Omega\subset S^{n-1}$7,

$\Omega\subset S^{n-1}$8

satisfies $\Omega\subset S^{n-1}$9, so $\mathbb R^n$00 is an isometric embedding of $\mathbb R^n$01 (Niemiec et al., 2018). Metric segments are unique, and any three points satisfying

$\mathbb R^n$02

are collinear in a single hyperbolic line. Euclid I–IV hold, while Euclid V fails in Playfair form: for a line $\mathbb R^n$03 not passing through a point $\mathbb R^n$04, one can find infinitely many hyperbolic lines through $\mathbb R^n$05 disjoint from $\mathbb R^n$06 (Niemiec et al., 2018).

The broader significance of this approach lies in its classification theorem. A metric space is $\mathbb R^n$07-point homogeneous if any isometry on any $\mathbb R^n$08-point subset extends to a global isometry. The main result quoted in the exposition states that any connected, locally compact, $\mathbb R^n$09-point homogeneous metric space with more than one point is isometric to exactly one of:

• $\mathbb R^n$10, where $\mathbb R^n$11;
• $\mathbb R^n$12, with $\mathbb R^n$13, and $\mathbb R^n$14 strictly increasing, subadditive, and continuous;
• $\mathbb R^n$15, $\mathbb R^n$16, with $\mathbb R^n$17 (Niemiec et al., 2018).

In particular, among geodesic $\mathbb R^n$18-point homogeneous spaces only

$\mathbb R^n$19

occur (Niemiec et al., 2018). The paper explicitly notes that “no other metric space—even if one abandons differentiability—can match that strong symmetry.” This establishes that the abandonment of calculus need not weaken global classification; in this case it still yields a rigid trichotomy.

7. Classification patterns, embeddings, and methodological significance

Across these strands, classification is a recurring theme. In the Radon-theoretic version, the principal unifying claim is the “dimension-independence” of the inverse formulas for $\mathbb R^n$20 and $\mathbb R^n$21: their structure is unchanged for every $\mathbb R^n$22, with only the overall factorial and powers of $\mathbb R^n$23 varying (Anikin, 27 May 2026). The paper contrasts this with “Courant–Hilbert methods, which historically needed separate treatment of even/odd $\mathbb R^n$24.” It further states implications for “practical CT or seismological inversion,” where non-uniform supports and incomplete angular data are common; according to the paper, the additional term $\mathbb R^n$25 provides a built-in distributional regularization that removes spurious singularities otherwise addressed by ad hoc smoothing or Tikhonov penalties (Anikin, 27 May 2026).

In imbrex geometry, classification takes the form of a finite list of generalized projective varieties and buildings. The projective-embedding application introduces the local tangent-space condition

$\mathbb R^n$26

for each point $\mathbb R^n$27 and each projective line $\mathbb R^n$28 with no point of $\mathbb R^n$29 collinear to $\mathbb R^n$30 (Schillewaert et al., 2013). The unified embedding characterization then states that a projectively spanning set $\mathbb R^n$31 with symplecta of type $\mathbb R^n$32 is a local Mazzocca–Melone set iff $\mathbb R^n$33 is, up to projection from a suitable complementary subspace, one of: $\mathbb R^n$34 (Schillewaert et al., 2013).

In the metric hyperbolic setting, the corresponding significance lies in rigidity under minimal apparatus: absolute homogeneity, uniqueness of geodesic segments, and the exclusion of nontrivial dilations for $\mathbb R^n$35 are all obtained by elementary metric arguments (Niemiec et al., 2018).

A common misconception would be to treat all of these as instances of a single formal theory with a shared definition. The cited works do not provide such a unified axiomatization. Rather, they present distinct frameworks: one based on broken-symmetry measures and generalized functions, one on Hjelmslev-type neighbourhoods and parapolar incidence, and one on metric geometry without differential tools. What they have in common is methodological: each shows that substantial geometric structure can be formulated, classified, or inverted after removing assumptions traditionally encoded through invariant integration or differential calculus.