arXiv subjects

Daniele Corradetti

Publications and source records attributed to Daniele Corradetti.

At least 19 recordsLinked to original sources

arXiv2026

Integral Orders for the Okubo Algebra and Idempotent Geometries of the $E_8$ Lattice

We study the Coxeter-Dickson $E_8$ order as a common integral support for the octonionic, para-octonionic and compact real Okubo products. The three algebra structures have the same additive lattice, the same positive composition norm and hence the same $240$ norm-one vectors, namely the roots of $E_8$ and the vertices of the Gosset polytope $4_{21}$. Their multiplicative structures are nevertheless different and this difference is already visible in the idempotents: among the same $240$ roots one finds respectively $1$, $57$ and $12$ nonzero integral idempotents. We show that these three counts organize the common root system in three increasingly polarized ways. The octonionic product leaves the full $E_8$ geometry unseparated; the $57$ para-octonionic idempotents reproduce the $E_7$ contact decomposition $1+56+126+56+1$; the $12$ Okubo idempotents split into four mutually orthogonal oriented $A_2$ triangles and hence determine canonically an $A_2^4$ subsystem. Choosing one triangle as the external $A_2$ then yields the $A_2+E_6$ Magic Star, while the remaining nine idempotents determine the trinification subsystem $A_2^3\subset E_6$. This provides an arithmetic interpretation of the $E_8\supset E_7$ and $E_8\supset E_6+A_2$ decompositions directly from integral idempotents.

math.RA↗

arXiv2026

Integral elements of Okubo algebra and the E8-lattice

In this work we study the interplay between the Coxeter-Dickson $E_{8}$-order, the para-octonions, and the real Okubo algebra. We first explain why a usual rotation presentation of the order-three automorphism, although algebraically correct, does not preserve the classical integral orders. We then introduce an integral order-three automorphism and prove that its Petersson isotope is the compact real Okubo algebra. The four octonionic orders with lattices $C_{8}$, $A_{2}^{4}$, $D_{4}^{2}$, and $E_{8}$ are preserved by the new realisation of the Okubo product. In particular, the same Coxeter-Dickson $E_{8}$ lattice supports octonionic, para-octonionic, and Okubo integral structures, although the resulting multiplicative systems are not isomorphic.

math.RA↗

arXiv2026

Linguistic Holonomy and Statistical Watermarks: Inner Geometry of Meaning-Preserving Transformations

Statistical watermarks for language models live in the freedom of the signifier: they choose among tokens that are nearly equivalent in meaning, and they are therefore eroded by exactly those transformations which move the form of a text while leaving its content in place. The literature measures such transformations by their endpoint, through the semantic similarity between the original and the rewritten text. We show that the endpoint is the wrong statistic. Adapting the formalism of linguistic loops, we prove that the invariant of a chain of meaning-preserving transformations factorises canonically into an endpoint part and a holonomy in the stabiliser of the initial state, the second of which the semantic deficit cannot see; the loop rotation is parallel transport on the unit sphere of the embedding space, so that the analogy with the Wilson loop becomes a theorem rather than a figure of speech. On the side of the detector we prove an exact identity: the residual statistic is proportional to the number of positions whose seeding window survived intact, from which the decay law $ρ^{h+1}$ follows as the independent-edit corollary. The identity has a disconcerting consequence, which we confirm to three decimal places: at one and the same retention rate the surviving signal may be one half of the original, one quarter of it, or exactly nothing, according only to where the edits fall.

cs.CL↗

arXiv2026

Integral Planes and Unit-Norm Polytopes

We introduce and study integral planes associated with crystallographic and non-crystallographic integral systems in real composition algebras. For an integral order $\Order$ in such an algebra we define the plane $\Order^{2}$ with quadratic form $Q(x,y)=\NN(x)+\NN(y)$, the axis shell, the balanced shell, and the corresponding unit-normalised spherical polytopes. For ten crystallographic orders we recover, in one uniform construction, the orthogonal-direct-sum root systems $2A_{1}$, $A_{2}\oplus A_{2}$, $4A_{1}$, $D_{4}\oplus D_{4}$, $16A_{1}$, and $E_{8}\oplus E_{8}$ (with classical-polytope realisations including the square, the 16-cell, the 24-cell, and the Gosset polytope $4_{21}$); for two non-crystallographic orders we obtain $H_{2}\oplus H_{2}$ (decagonal tegum) and $H_{4}\oplus H_{4}$ (600-cell tegum) over $\Z[\golden]$. We prove a rank-obstruction theorem that closes, unconditionally and by a purely Coxeter-theoretic argument, the existence question for an indecomposable rank-eight golden octonion order: no such order can exist. On the balanced shell side, we identify the genuine algebraic Hopf map $\Hopfmap_{A}(a,b)=(2a\bar b,\NN(a)-\NN(b))$ and prove that its restriction to the balanced shell is a finite principal fibration of the unit loop, valid both for the associative case and for the alternative Moufang case.

math.CO↗

arXiv2026

Non-crystallographic systems of integers over composition algebras

In this work we revisit classical systems of integers inside the real normed division algebras from the point of view of finite norm shells and root systems. Building on the icosian framework of Moody--Patera and on the integral root-system viewpoint of Chen--Moody--Patera and of Johnson, we isolate the precise axiomatic ingredients of the non-crystallographic analogue: an order over the golden ring $\Zphi$ together with a distinguished finite root shell whose Cartan coefficients lie in $\Zphi$. We show that the usual Gaussian, Eisenstein, Hamilton, Hurwitz and Coxeter--Dickson examples are recovered by separating the order, its units, and its distinguished finite shells; once the lattice requirement is replaced by a finite root-shell requirement, the golden integer ring becomes the natural coefficient ring for the non-crystallographic cases $H_2$ and $H_4$. We then construct a weak golden octonion order by Cayley--Dickson doubling of the icosian ring; the resulting free rank-$8$ $\Zphi$-order has a $240$-element finite shell of type $H_4\oplus H_4$ and its multiplication is genuinely octonionic. Finally, we prove (i) that this weak double is self-dual with respect to the polar norm pairing, hence has no strict norm-integral overorder, and (ii) that the first trace-integral discriminant tower over it contains no octonion-stable nonzero isotropic gluing.

math.RA↗

arXiv2026

Integral Shell Polytopes of Composition Algebras

Integral systems in real composition algebras give rise to finite metric configurations whose geometry is linked to both regular polytopes and root-systems. In this work we investigate, to our knowledge for the first time in this form, the shell polytopes obtained by fixing the integral norm and taking the convex hull of the corresponding integral elements. The first shells recover the familiar root-polytopal configurations attached to the classical Hurwitz systems, while the Okubo algebra gives a quite different behaviour. The Okubo integral closure does not recover the Gosset polytope directly: it selects a two-adic hierarchy whose first visible layers are a cross-polytope and a $D_8$ root polytope. We further show that the natural intermediate lattice is isometric to the rescaled cubic lattice; consequently every shell decomposes into explicit orbits of the hyperoctahedral group $W(B_8)$, and the higher Okubo shells admit a complete combinatorial description in cubic-lattice coordinates. The full $E_8$ Gosset polytope is then recovered from the intermediate lattice by maximal-isotropic gluing along $(\ZZ/2)^4$. This gives an interplay between non-unital composition, integral lattice shadows, and the geometry of $E_8$.

math.CO↗

arXiv2026

Learning Hippo: Multi-attractor Dynamics and Stability Effects in a Biologically Detailed CA3 Extension of Hopfield Networks

We present a biologically detailed extension of the classical Hopfield/Marr auto-associative memory model for CA3, implementing ten populations (two asymmetric pyramidal subtypes, eight GABAergic interneuron classes), forty-seven compartments, multi-rule plasticity (recurrent Hebb, BCM anti-saturation, mossy-fiber short-term, endocannabinoid iLTD, burst-gated Hebb), and a bimodal cholinergic encoding/consolidation cycle. Evaluated on pattern completion across auto-associative, associative, and temporal regimes, and on a controlled inhibitory-proportion manipulation at $N{=}256$, the full architecture exhibits \emph{three qualitative signatures absent from a minimal Hopfield baseline}: (i)~multi-attractor cross-seed behaviour at $K{=}5$ with biologically realistic inhibitory proportions, where two of five seeds converge to positive attractors with margin ${+}0.10{-}0.22$ (Cohen's $d{=}0.71$, one-sided $p{=}0.08$); (ii)~target-selective associative recall in paired $(A, B)$ memory at $K{\geq}5$, where the full model retrieves $B$ from a partial cue of $A$ while the minimal model echoes $A$ (Pearson margin $Δ{=}{+}0.163$ at $K{=}5$); (iii)~reduced cross-seed variance of the full model below the minimal baseline under clean upstream, with ratios $1.0{-}3.0$. These three signatures are architecture-specific: they appear consistently across independent regimes and are absent from the minimal control.

cs.NE↗

arXiv2025

Three Dixon-Rosenfeld Planes

Rosenfeld postulated ``generalized'' projective planes, which exploit a correspondence between rank-one idempotents of Jordan algebras $\mathfrak{J}_3(\mathbb{A})$ and points of projective planes $\mathbb{A}P^2$. The isometry groups of the generalized projective planes (which were later defined rigorously as homogeneous spaces) are entries of the Tits-Freudenthal magic square. Given recent interest in the Dixon algebra $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, we extend Rosenfeld's approach and present three new coset manifolds. These "Dixon-Rosenfeld planes" have isometry algebras that are obtained from Tits' magic formula and involve all tensorial components of the Dixon algebra. We show that these are the only three planes obtainable with Tits' formula that preserve the analogy with Rosenfeld's planes. These non-simple Lie algebras generalize $\mathfrak{f}_{4},\mathfrak{e}_{6},\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$ for the octonionic plane $\mathbb{O}P^{2}$ and in the octonionic Rosenfeld planes $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$, $\left(\mathbb{H}\otimes\mathbb{O}\right)P^{2}$ and $\left(\mathbb{O}\otimes\mathbb{O}\right)P^{2}$. We finally investigate the relationships between the isometry algebras of the Dixon-Rosenfeld planes and the exceptional Lie algebras.

math-ph↗

arXiv2025

RED.AI Id-Pattern: First Results of Stone Deterioration Patterns with Multi-Agent Systems

The Id-Pattern system within the RED.AI project (Reabilitação Estrutural Digital através da AI) consists of an agentic system designed to assist in the identification of stone deterioration patterns. Traditional methodologies, based on direct observation by expert teams, are accurate but costly in terms of time and resources. The system developed here introduces and evaluates a multi-agent artificial intelligence (AI) system, designed to simulate collaboration between experts and automate the diagnosis of stone pathologies from visual evidence. The approach is based on a cognitive architecture that orchestrates a team of specialized AI agents which, in this specific case, are limited to five: a lithologist, a pathologist, an environmental expert, a conservator-restorer, and a diagnostic coordinator. To evaluate the system we selected 28 difficult images involving multiple deterioration patterns. Our first results showed a huge boost on all metrics of our system compared to the foundational model.

cs.CV↗

arXiv2025

A new method for structural diagnostics with muon tomography and deep learning

This work investigates the production of high-resolution images of typical support elements in concrete structures by means of muon tomography (muography). By exploiting detailed Monte Carlo radiation-matter simulations, we demonstrate the feasibility of reconstructing 1 cm-thick iron bars inside 30 cm-deep concrete blocks, regarded as an important testbed within the structural diagnostics community. In addition, we present a new method for integrating simulated data with advanced deep learning techniques in order to improve the muon imaging of concrete structures. Through deep learning enhancement techniques, this results in a dramatic improvement in image quality and a significant reduction in data acquisition time, which are two critical limitations within the usual practice of muography for civil engineering diagnostics.

hep-ex↗

arXiv2025

Linguistic Loops and Geometric Invariants as a Way to Pre-Verbal Thought?

In this work we introduce the concepts of linguistic transformation, linguistic loop and semantic deficit. By exploiting Lie group theoretical and geometric techniques, we define invariants that capture the structural properties of a whole linguistic loop. This result introduces new line of research, employing tools from Lie theory and higher-dimensional geometry within language studies. But, even more intriguingly, our study hints to a mathematical characterization of the meta-linguistic or pre-verbal thought, namely of those cognitive structures that precede the language.

cs.CL↗

arXiv2025

Physics with non-unital algebras? An invitation to the Okubo algebra

This paper presents some preliminary discussion on the possible relevance of the Okubonions, i.e. the real Okubo algebra $\mathcal{O}$, in quantum chromodynamics (QCD). The Okubo algebra lacks a unit element and sits in the adjoint representation of its automorphism group $\text{SU}_{\mathcal{O}}$, thus being fundamentally different from the better-known octonions $\mathbb{O}$. While these latter may represent quarks (and color singlets), the Okubonions are conjectured to represent the gluons, i.e. the gauge bosons of the QCD $\text{SU}(3)$ color symmetry. However, it is shown that the $\text{SU}(3)$ groups pertaining to Okubonions and octonions are distinct and inequivalent subgroups of $Spin(8)$ that share no common $\text{SU}(2)$ subgroup. The unusual properties of Okubonions may be related to peculiar QCD phenomena like asymptotic freedom and color confinement, though the actual mechanisms remain to be investigated.

hep-th↗

arXiv2024

Synthetic Data for Discriminating Serotonergic Neurons using Convolutional Neural Networks

Serotonergic neurons in the raphe nuclei exhibit diverse electrophysiological properties and functional roles, yet conventional identification methods rely on restrictive criteria that likely overlook atypical serotonergic cells. The use of convolutional neural network (CNN) for comprehensive classification of both typical and atypical serotonergic neurons is an interesting one, but the key challenge is often given by the limited experimental data available for training. This study presents a procedure for synthetic data generation that combines smoothed spike waveforms with heterogeneous noise masks from real recordings. This approach expanded the training set while mitigating overfitting of background noise signatures. CNN models trained on the augmented dataset achieved high accuracy (96.2% true positive rate, 88.8% true negative rate) on non-homogeneous test data collected under different experimental conditions than the training, validation and testing data.

q-bio.NC↗

arXiv2024

Identification of Stone Deterioration Patterns with Large Multimodal Models

The conservation of stone-based cultural heritage sites is a critical concern for preserving cultural and historical landmarks. With the advent of Large Multimodal Models, as GPT-4omni (OpenAI), Claude 3 Opus (Anthropic) and Gemini 1.5 Pro (Google), it is becoming increasingly important to define the operational capabilities of these models. In this work, we systematically evaluate the abilities of the main foundational multimodal models to recognise and classify anomalies and deterioration patterns of the stone elements that are useful in the practice of conservation and restoration of world heritage. After defining a taxonomy of the main stone deterioration patterns and anomalies, we asked the foundational models to identify a curated selection of 354 highly representative images of stone-built heritage, offering them a careful selection of labels to choose from. The result, which varies depending on the type of pattern, allowed us to identify the strengths and weaknesses of these models in the field of heritage conservation and restoration.

cs.CV↗

arXiv2024

Deep Learning Models for Atypical Serotonergic Cells Recognition

The serotonergic system modulates brain processes via functionally distinct subpopulations of neurons with heterogeneous properties, including their electrophysiological activity. In extracellular recordings, serotonergic neurons to be investigated for their functional properties are commonly identified on the basis of "typical" features of their activity, i.e. slow regular firing and relatively long duration of action potentials. Thus, due to the lack of equally robust criteria for discriminating serotonergic neurons with "atypical" features from non-serotonergic cells, the physiological relevance of the diversity of serotonergic neuron activities results largely understudied. We propose deep learning models capable of discriminating typical and atypical serotonergic neurons from non-serotonergic cells with high accuracy. The research utilized electrophysiological in vitro recordings from serotonergic neurons identified by the expression of fluorescent proteins specific to the serotonergic system and non-serotonergic cells. These recordings formed the basis of the training, validation, and testing data for the deep learning models. The study employed convolutional neural networks (CNNs), known for their efficiency in pattern recognition, to classify neurons based on the specific characteristics of their action potentials.

q-bio.NC↗

arXiv2024

Recovering Composition Algebras from 3D Geometric Algebras

Generalized Hurwitz theorem states that there are fifteen composition algebras for any given field: seven unital, six para-unital, and two non-unital algebras. In this article we explore the recovery of such algebras from 3D Geometric Algebras. Different involutions, such as reversion, inversion, Clifford conjugation, and full grade inversion, are introduced in order to recover the norm of all composition algebras. A special attention is given to composition algebras of dimension 8, i. e. octonions, para-octonions and Okubo algebra, for which the introduction of a different product is needed.

math.RA↗

arXiv2023

Collineation groups of octonionic and split-octonionic planes

We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel and minimal construction of the same octonionic and split-octonionic planes, by exploiting two symmetric composition algebras: the Okubo algebra and the paraoctonionic algebra. Besides the intrinsic mathematical relevance of this construction of the real forms of the Cayley-Moufang plane, we expect this approach to have implications in all mathematical physics related with exceptional Lie Groups of type $G_{2},F_{4}$ and $E_{6}$.

math.RA↗

arXiv2023

All Hurwitz Algebras from 3D Geometric Algebras

Hurwitz algebras are unital composition algebras widely known in algebra and mathematical physics for their useful applications. In this paper, inspired by works of Lesenby and Hitzer, we show how to embed all seven Hurwitz algebras (division and split) in 3D geometric algebras, i.e. $\mathcal{G}\left(p,q\right)$ with $p+q=3$. This is achieved studying the even subalgebra, which is always of quaternionic or split-quaternionic type, and the algebra generated by the pseudoscalar which is always of complex or split-complex type. Reversion, inversion and Clifford conjugation correspond to biquaternionic, complex and quaternionic conjugation respectively. Octonionic algebras, i.e., octonions and split-octonions, are obtained in two ways introducing two different products, replicating a variation of the Cayley-Dickson process. Octonionic conjugation is achieved in the geometric algebra formalism through the involution defined by the full grade inversion.

math.RA↗