arXiv subjects

Joe Gildea

Publications and source records attributed to Joe Gildea.

At least 19 recordsLinked to original sources

arXiv2026

Why the Bethe Ansatz Works: A Structural Explanation via Interaction Propagation

The Bethe Ansatz provides exact solutions for certain interacting quantum many-body systems, yet its success is confined to narrow regimes and breaks down abruptly outside them. Despite extensive developments in integrable systems, a structural explanation of this phenomenon has remained elusive. In this paper we give a representation-independent account of both the existence and the failure of Bethe-type exact solvability. We identify a single governing mechanism: the behaviour of interaction propagation. For systems in which propagation terminates after finite depth without encountering structural boundaries, global interaction data factor through finitely many local components, forcing Bethe-type solvability. Conversely, once a structural boundary is encountered, irreducible interaction data arise that obstruct such finite factorization and preclude Bethe-type solutions. This yields a sharp structural dichotomy. Within this regime, exact solvability is not an analytic accident but a rigidity phenomenon, while its failure is governed by intrinsic boundary formation. In this way, the Bethe Ansatz is understood as a consequence of constrained interaction propagation rather than as a model-specific construction.

math.RA↗

arXiv2026

A Structural Criterion for the Applicability of Algebraic Phase Theory

Algebraic Phase Theory (APT) exhibits a marked structural selectivity. In certain mathematical and physical settings it gives rise to rigidity phenomena, constrained representation behaviour, and reductions in apparent degrees of freedom, while in many analytic or dynamical contexts the finite-depth APT framework does not naturally apply. This paper studies the structural origin of this asymmetry. We establish a structural criterion for the existence of a nondegenerate finite-depth Algebraic Phase Theory structure. The criterion isolates three conditions: nondegenerate phase duality, compatibility of admissible dynamics with phase interaction, and finite or terminating defect propagation. Within the framework considered here, these conditions are jointly necessary and sufficient. When they are satisfied, the resulting phase structure exhibits strong rigidity properties; when one of the conditions fails, the associated domain falls outside the intended finite-depth APT setting. As consequences, phenomena such as Fourier decomposition, Bethe-type exact solvability, rigidity of stabilizer codes, and uniqueness phenomena associated with certain canonical representations can be interpreted as structural manifestations of these conditions rather than isolated constructions. The results therefore clarify both the scope and the structural limitations of Algebraic Phase Theory within the finite-depth setting considered here.

math.RA↗

arXiv2026

Algebraic Phase Theory IV: Morphisms, Equivalences, and Categorical Rigidity

We complete the foundational architecture of Algebraic Phase Theory by developing a categorical and $2$-categorical framework for algebraic phases. Building on the structural notions introduced in Papers~I-III, we define phase morphisms, equivalence relations, and intrinsic invariants compatible with the canonical filtration and defect stratification. For finite, strongly admissible phases we establish strong rigidity theorems: phase morphisms are uniquely determined by their action on rigid cores, and under bounded defect, weak, strong, and Morita-type equivalence all coincide. In particular, finite strongly admissible phases admit no distinct models with the same filtered representation theory. We further show that structural boundaries are invariant under Morita-type equivalence and therefore constitute genuine categorical invariants. Algebraic phases, phase morphisms, and filtration-compatible natural transformations form a strict $2$-category in the strongly admissible regime. We also prove that completion defines a reflective localization of this category, with complete phases characterized as universal forced rigidifications. Together, these results elevate Algebraic Phase Theory from a collection of algebraic constructions to a categorical framework in which rigidity, equivalence collapse, boundary invariance, and completion arise as intrinsic consequences of phase interaction, finiteness, and admissibility.

math.RA↗

arXiv2026

Boundary Calculus, Rigidity Islands, and Deformation Theory in Algebraic Phase Structures

We develop a general boundary calculus for algebraic phases and use it to formulate an intrinsic structural framework for deformation and obstruction phenomena. Structural boundaries are shown to be finitely detectable and canonically stratified by failure type and depth. For each boundary we construct a canonical boundary exact sequence and identify a maximal rigid subphase, called a rigidity island, that persists beyond global boundary failure. Rigidity islands are organised by intrinsic invariants and serve as canonical base points for deformation theory. Deformation behaviour within the standing admissibility framework is governed by boundary quotients, while rigidity islands remain stable under admissible deformation. Boundary quotients act as obstruction objects whose associated strata organise higher-depth deformation behaviour. As a consequence, deformation behaviour is naturally stratified by boundary depth and failure type, while formal smoothness is associated with the vanishing of boundary data. The resulting moduli behaviour is organised by rigidity islands together with their associated obstruction patterns, without requiring intrinsic analytic or continuous deformation parameters.

math.RA↗

arXiv2026

Duality, Reconstruction, and Structural Toolkit Theorems in Algebraic Phase Theory

We study finite-depth reconstruction frameworks based on representation theory and show that non-rigid reconstruction behaviour is naturally accompanied by intrinsic structural boundaries. Within the finite-depth setting considered in this paper, reconstruction is controlled up to boundary equivalence by the associated filtered representation data together with boundary stratification. We show that algebraic phases satisfying the axioms of Algebraic Phase Theory (APT) are reconstructible up to intrinsic phase equivalence from their filtered representation categories together with their boundary structure. Reconstruction proceeds without boundary collapse on rigidity islands, while globally the remaining ambiguity is governed by intrinsic boundary phenomena. We further study a collection of structural consequences associated with the axioms of APT, including finite generation phenomena, rigidity and obstruction behaviour, finite-depth boundary detectability, and obstruction structures arising from boundary layers. These results apply across the phase models developed in the APT series. Taken together, the results of this paper further develop Algebraic Phase Theory as a structural framework for studying reconstruction, duality, rigidity, and boundary behaviour beyond rigid or semisimple settings.

math.RA↗

arXiv2026

Algebraic Phase Theory I: Radical Phase Geometry and Structural Boundaries

We develop Algebraic Phase Theory (APT), an axiomatic framework for extracting intrinsic algebraic structure from phase based analytic data. From minimal admissible phase input we prove a general phase extraction theorem that yields algebraic Phases equipped with functorial defect invariants and a uniquely determined canonical filtration. Finite termination of this filtration forces a structural boundary: any extension compatible with defect control creates new complexity strata. These mechanisms are verified in the minimal nontrivial setting of quadratic phase multiplication operators over finite rings with nontrivial Jacobson radical. In this case nilpotent interactions produce a finite filtration of quadratic depth, and no higher degree extension is compatible with the axioms. This identifies the radical quadratic Phase as the minimal example in which defect, filtration, and boundary phenomena occur intrinsically.

math.RA↗

arXiv2026

Algebraic Phase Theory II: The Frobenius Heisenberg Phase and Boundary Rigidity

We develop the representation theory intrinsic to Algebraic Phase Theory (APT) in regimes where defect and canonical filtration admit faithful algebraic realisation. This extends the framework introduced in earlier work by incorporating a representation-theoretic layer that is compatible with defect and filtration. In this setting, algebraic phases act naturally on filtered module categories rather than on isolated objects, and classical irreducibility must be replaced by a filtration-compatible notion of indecomposability forced by defect. As a central application, we analyse the Frobenius Heisenberg algebraic phase, which occupies a rigid boundary regime within the broader APT landscape, and show that it satisfies the axioms of APT in a strongly admissible form. We study the representations realising this phase and show that their non-semisimplicity and rigidity properties are consequences of the underlying algebraic structure rather than analytic or semisimple hypotheses. In particular, we establish a Stone von Neumann type rigidity theorem for Heisenberg groups associated with finite Frobenius rings. For each such ring $R$, we construct a canonical Schr\"odinger representation of the Frobenius Heisenberg group $H_R$, and show that, for a fixed nontrivial central character, every centrally faithful representation of $H_R$ is equivalent to this model. The proof is entirely algebraic and uses no topology, unitarity, Fourier analysis, or semisimplicity. Instead, rigidity emerges as a boundary phenomenon governed by defect and canonical filtration. The Frobenius hypothesis is shown to be sharp: it precisely delineates the structural boundary within APT at which Heisenberg rigidity persists, and outside the Frobenius class this rigidity necessarily fails.

math.RA↗

arXiv2026

Algebraic Phase Theory III: Structural Quantum Codes over Frobenius Rings

We develop the quantum component of Algebraic Phase Theory by showing that quantum phase, Weyl noncommutativity, and stabiliser codes arise as unavoidable algebraic consequences of Frobenius duality. Working over finite commutative Frobenius rings, we extract nondegenerate phase pairings, Weyl operator algebras, and quantum stabiliser codes directly from admissible phase data, without assuming Hilbert spaces, analytic inner products, or an externally imposed symplectic structure. Within this framework, quantum state spaces appear as minimal carriers of faithful phase action, and stabiliser codes are identified canonically with self-orthogonal submodules under the Frobenius phase pairing. CSS-type constructions arise only as a special splitting case, while general Frobenius rings admit intrinsically non-CSS stabilisers. Nilpotent and torsion structure in the base ring give rise to algebraically protected quantum layers that are invisible to admissible Weyl-type errors. These results place quantum stabiliser theory within Algebraic Phase Theory: quantisation emerges as algebraic phase induction rather than analytic completion, and quantum structure is information-complete at the level of algebraic phase relations alone. Throughout, we work over finite Frobenius rings, which are precisely the base rings for which admissible phase data become strongly admissible, and in this regime the full quantum formalism is forced by Frobenius duality.

math.RA↗

arXiv2021

Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

In this work, we define a modification of a bordered construction for self-dual codes which utilises $\lambda$-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.

cs.IT↗

arXiv2021

New binary self-dual codes of lengths 56, 62, 78, 92 and 94 from a bordered construction

In this paper, we present a new bordered construction for self-dual codes which employs $\lambda$-circulant matrices. We give the necessary conditions for our construction to produce self-dual codes over a finite commutative Frobenius ring of characteristic 2. Moreover, using our bordered construction together with the well-known building-up and neighbour methods, we construct many binary self-dual codes of lengths 56, 62, 78, 92 and 94 with parameters in their weight enumerators that were not known in the literature before.

cs.IT↗

arXiv2021

Group LCD and Group Reversible LCD Codes

In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the $n \times n$ matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element $v,$ one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element $v,$ one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters.

cs.IT↗

arXiv2021

New binary self-dual codes of lengths 80, 84 and 96 from composite matrices

In this work, we apply the idea of composite matrices arising from group rings to derive a number of different techniques for constructing self-dual codes over finite commutative Frobenius rings. By applying these techniques over different alphabets, we construct best known singly-even binary self-dual codes of lengths 80, 84 and 96 as well as doubly-even binary self-dual codes of length 96 that were not known in the literature before.

math.CO↗

arXiv2021

New binary self-dual codes of lengths 56, 58, 64, 80 and 92 from a modification of the four circulant construction

In this work, we give a new technique for constructing self-dual codes over commutative Frobenius rings using $\lambda$-circulant matrices. The new construction was derived as a modification of the well-known four circulant construction of self-dual codes. Applying this technique together with the building-up construction, we construct singly-even binary self-dual codes of lengths 56, 58, 64, 80 and 92 that were not known in the literature before. Singly-even self-dual codes of length 80 with $\beta\in\{2,4,5,6,8\}$ in their weight enumerators are constructed for the first time in the literature.

math.CO↗

arXiv2020

New Extremal Binary Self-Dual Codes from Block Circulant Matrices and Block Quadratic Residue Circulant Matrices

In this paper, we construct self-dual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2 and F2 + uF2. Using extensions, neighbours and sequences of neighbours, we construct many new self-dual codes. In particular, we construct one new self-dual code of length 66 and 51 new self-dual codes of length 68.

math.RA↗

arXiv2020

New Self-Dual Codes of length 68 from a 2 by 2 block matrix Construction and Group Rings

Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form G=(I|A), where I is the n by n identity matrix and A is the n by n matrix fully determined by the first row. In this work, we define a generator matrix in which A is a block matrix, where the blocks come from group rings and also, A is not fully determined by the elements appearing in the first row. By applying our construction over F_2+uF_2 and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual [68,34,12]-codes with the rare parameters gamma=7,8 and 9 in W_{68,2}. In particular, we find 92 new binary self-dual [68,34,12]-codes.

cs.IT↗

arXiv2020

Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes

In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We define quasi-composite G-codes and give a construction of these codes. We also study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters gamma=7,8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.

cs.IT↗

arXiv2020

New Extremal binary self-dual codes of length 68 from a novel approach to neighbors

In this work, we introduce the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the k-neighbors, we are able to construct extremal binary self-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codes of length 68 with new weight enumerators including 42 codes with gamma=8 in their W_{68,2} and 40 with gamma=9 in their W_{68,2}. These examples are the first in the literature for these gamma values. This completes the theoretical list of possible values for gamma in W_{68,2}.

math.CO↗

arXiv2020

New Self-Dual Codes from 2x2 Block Circulant Matrices, Group Rings and Neighbours of Neighbours

In this paper, we construct self-dual codes from a construction that involves 2x2 block circulant matrices, group rings and a reverse circulant matrix. We provide conditions whereby this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2, F2 + uF2 and F4 + uF4. Using extensions, neighbours and neighbours of neighbours, we construct 32 new self-dual codes of length 68.

math.RA↗