Parametric Galois Extensions and Their Spectral Invariants

Parametric Galois Extensions and Their Spectral Invariants Francesco D’Agostino October 13, 2025 Abstract In this article we develop a dynamic formulation of Galois theory in which the classical symmetry group of a polynomial is replaced by a parametrized family of groups. The construction centers on a dependency space: a parameter set Γ that determines, at each stage, which roots are admitted via a boundary condition in the base field. Each parameter γyields a restricted subfield and automorphism group, and the resulting family forms a projective system under canonical restriction maps. As the boundary varies, the system exhibits both smooth deformations and abrupt collapses of symmetry. We encode each group via its Cayley graph and use spectral invariants of the Laplacian to track structural changes: continuous eigenvalue variation signals stable phases, while jumps mark symmetry transitions. A quartic example demonstrates the framework explicitly. 1 Introduction Classical Galois theory describes the symmetries of algebraic equations in a static setting [1, 2, 3]: given a polynomial over a base field, one adjoins all its roots and studies the automorphism group of the splitting field. This yields a complete picture of the global symmetry, but offers no internal resolution. There is no way to see how parts of the field contribute, how subsets of roots interact, or how the symmetry reorganizes when the data are restricted. We introduce a dynamic extension of this framework. Instead of fixing the full splitting field, we define a parameter space that determines which roots are included at each stage. Each parameter corresponds to a boundary condition selecting a subset of roots, and hence a subfield. The automorphism groups of these subfields form a structured family—a projective system linked by restriction maps—that evolves as the boundary varies. To analyze this evolution, we represent each group by its Cayley graph and study the spectral data of the associated Laplacian. Eigenvalues provide a numerical signature of symmetry: smooth variation reflects gradual deformation, while discrete jumps mark algebraic phase transitions where the field structure reorganizes. This approach offers two advantages: first, it makes explicit the internal gradation of symmetry, showing how it builds from partial data rather than appearing all at once; second, by translating algebraic change into spectral language, it provides quantitative tools for detecting and comparing structural transitions across families of polynomials. 2 Foundations Let Kbe a field and let f(x)∈K[x] be a polynomial. The roots of f(x) need not lie in K, but they are contained in some extension field Lof K. The minimal such extension in which f(x) decomposes into linear factors is called the splitting field. Definition 1 (Splitting field).Let f(x)∈K[x]. A splitting field of f(x)over Kis the smallest field extension L/K such that f(x) = n Y i=1 (x−αi), αi∈L. (1) This field is unique up to isomorphism. The symmetries of this structure are encoded in the Galois group .¸ An automorphism of Lis a bijective map σ:L→Lthat preserves the field operations. Those automorphisms which fix Kpointwise form a group under composition. 1 Definition 2 (Galois group).If L/K is a Galois extension, the Galois group of L/K is Gal(L/K)={σ∈Aut(L):σ(k)=k∀k∈K}.(2) In the classical approach this correspondence is static: the splitting field Lis fixed, and Gal(L/K) is determined once and for all. In many contexts, however, it is natural to restrict attention to a subset of the roots, or to consider situations where roots merge under constraints. Such restrictions induce the relevant subfield and, consequently, the group of admissible automorphisms. To formalize this, we introduce the concept of a dependency space Γ. Each γ∈Γ specifies a boundary region Bγ⊆K, which determines which roots of f(x) are included. The subfield Lγis defined by adjoining to Konly those roots lying in Bγ, and the corresponding subgroup Gγ= Gal(Lγ/K) (3) encodes the symmetries of this restricted configuration. Thus the parameter space Γ acts as a control mechanism: as γvaries, the boundary regions Bγevolve, the associated subfields Lγchange, and the groups Gγvary accordingly. This passage from the static to the dynamic picture provides the foundation for the next sections, where the evolving structure will be organized via graph-theoretic representations and analyzed through spectral invariants. 3 Dynamic Galois Groups via Dependency Spaces The central object that enables the transition from a static to a dynamic Galois system is the dependency space, denoted by Γ. Its role is to index the restrictions that determine which portion of the splitting field is under consideration at a given stage. Let f(x)∈K[x] be a polynomial with splitting field Lover K, and let {α1, . . . , αn}⊆Ldenote its roots. Classically, one adjoins all these roots to Ksimultaneously, obtaining L, and the associated Galois group G= Gal(L/K) describes the full symmetry. To pass to a dynamic setting, we allow for partial selections of the root data. These selections are not arbitrary subsets of {αi}, but are governed by constraints expressed in the base field K. Concretely, we introduce a parameter space Γ, each element of which prescribes a boundary region Bγ⊆K. The interpretation is as follows: I. The boundary region Bγprovides a filter, written in the language of K, for deciding which roots of f(x) are to be included. II. Each root αi∈Lcan be assigned invariants valued in K(for instance, minimal polynomial coefficients, real/imaginary parts if K⊆R, or reductions modulo primes when K=Q). III. The rule determined by Bγis then applied to these invariants, and only those roots whose invariants fall inside Bγare retained. In this way, Bγacts not on Kitself but as a constraint mechanism: it determines a subset of the roots in Lthat survive under the given boundary. Given such a choice, we obtain a subfield Lγ=K(αi:αipasses the filter determined by Bγ),(4) and with it a subgroup Gγ= Aut(Lγ/K).(5) Thus, varying γ∈Γ produces a family of subfields {Lγ}γ∈Γand a corresponding family of symmetry groups {Gγ}γ∈Γ. Some transitions between parameters induce continuous deformations of Gγ(the relations between automorphisms persist with minor variation), while others produce abrupt changes (a root entering or leaving the admissible set, or colliding with another, may collapse the symmetry structure). The dependency space Γ is precisely the indexing device that organizes this evolving landscape. More precisely: Definition 3 (Dependency Space with Invariant Map).Let f(x)∈K[x]have splitting field L. An invariant map is a function Inv : L−→ Km(6) 2 such that Inv(α)is determined by the action of Gal(K/K)on α. A dependency space is a parameter set Γtogether with an assignment γ7−→ Bγ⊆Km.(7) For each γ∈Γ, this determines the subfield Lγ=K(αi:αi∈L, Inv(αi)∈Bγ),(8) and the corresponding dynamic Galois group Gγ= Aut(Lγ/K).(9) [8] Proposition 1 (Monotonicity of Root Selection).Let f(x)∈K[x]have splitting field Lover K, and let Γbe a dependency space with invariant map Inv : L−→ Km.(10) For γ∈Γ, define Rγ={α∈L: Inv(α)∈Bγ}, Lγ=K(Rγ).(11) Suppose γ1, γ2∈Γsatisfy Bγ1⊆Bγ2. Then Rγ1⊆Rγ2, Lγ1⊆Lγ2.(12) Proof. Let α∈Rγ1. Then by definition, Inv(α)∈Bγ1. Since Bγ1⊆Bγ2, we have Inv(α)∈Bγ2, so α∈Rγ2. Thus Rγ1⊆Rγ2. The inclusion Lγ1⊆Lγ2follows immediately, since Lγ1=K(Rγ1) and Lγ2=K(Rγ2), and adjoining a subset of elements yields a subfield. In this manner, Γ serves as the parameter space that governs which restrictions of the splitting field are under consideration, and the resulting family {Gγ}γ∈Γdescribes the evolving symmetry structure. Once the dependency space has been specified, we no longer consider the groups Gγin isolation. The assignment γ7→ Gγinduces a structured family indexed by Γ. Concretely, for each γ∈Γ one has Rγ={αi∈L: Inv(αi)∈Bγ},(13) Lγ=K(Rγ),(14) Gγ= Aut(Lγ/K).(15) This produces a map Φ:Γ−→ Grp, γ 7−→ Gγ,(16) where Grp denotes the category of groups. The structure becomes visible once we compare different parameters. Suppose γ1, γ2∈Γ with Bγ1⊆Bγ2. Then by construction Rγ1⊆Rγ2, Lγ1⊆Lγ2.(17) This nesting of subfields gives rise to a canonical restriction map ργ2→γ1:Gγ2→Gγ1, σ 7→ σ|Lγ1.(18) Each ργ2→γ1is a group homomorphism, since composition of automorphisms commutes with restriction: (σ1◦σ2)|Lγ1=σ1|Lγ1◦σ2|Lγ1.(19) Thus the collection {Gγ}γ∈Γ, together with the restriction maps {ργ2→γ1}, forms a projective system of groups. The kernel of ργ2→γ1consists of precisely those automorphisms of Lγ2that fix Lγ1pointwise: ker(ργ2→γ1) = {σ∈Gγ2:σ|Lγ1= idLγ1}.(20) The system ({Gγ},{ργ2→γ1}) is therefore the dynamic Galois system associated with f(x) and Γ. It encodes not only the individual symmetries of each restricted field, but also the natural relations between them as γvaries. To organize the family of groups {Gγ}γ∈Γ, we associate to each Gγa finite graph Gγ. The main idea is that the algebraic structure of each dynamic group is encoded in combinatorial form, so that changes in symmetry as γvaries can be analyzed by graphical and spectral means. 3 Lemma 1 (Restriction Homomorphism).Let f(x)∈K[x]have splitting field Lover K, and let Γbe a dependency space with invariant map Inv : L→Km. For each γ∈Γ, define Rγ={α∈L: Inv(α)∈Bγ}, Lγ=K(Rγ), Gγ= Aut(Lγ/K).(21) Suppose γ1, γ2∈Γsatisfy Bγ1⊆Bγ2. Then Lγ1⊆Lγ2, and there exists a canonical group homomorphism ργ2→γ1:Gγ2−→ Gγ1, ργ2→γ1(σ) = σ|Lγ1.(22) This map satisfies: (I.) ργ2→γ1is well-defined: each σ∈Gγ2restricts to a K-automorphism of Lγ1. (II.) ργ2→γ1is a group homomorphism. (III.) The kernel is ker(ργ2→γ1)={σ∈Gγ2:σ|Lγ1= idLγ1}.(23) (IV.) The map is surjective if and only if every automorphism of Lγ1/K extends to an automorphism of Lγ2/K. Proof. By Proposition 1, we have Lγ1⊆Lγ2. (i.) Let σ∈Gγ2= Aut(Lγ2/K). Since Lγ1is a subfield of Lγ2generated over K, and σis a Kautomorphism of Lγ2, we have σ(Lγ1)⊆Lγ2. We claim that σ(Lγ1) = Lγ1. Since Lγ1and σ(Lγ1) are both finite extensions of Kinside Lγ2, and σis an isomorphism, they have the same degree over K. Moreover, σ(Lγ1)⊆Lγ2is a subfield containing K. Because σis bijective on Lγ2and Lγ1⊆Lγ2is a K-subspace of the same dimension as σ(Lγ1), we must have σ(Lγ1)=Lγ1. Thus σ|Lγ1:Lγ1→Lγ1is a well-defined K-automorphism. (ii.) For σ1, σ2∈Gγ2, we compute ργ2→γ1(σ1◦σ2)=(σ1◦σ2)|Lγ1=σ1|Lγ1◦σ2|Lγ1=ργ2→γ1(σ1)◦ργ2→γ1(σ2).(24) Thus ργ2→γ1is a group homomorphism. iii. By definition, ker(ργ2→γ1)={σ∈Gγ2:σ|Lγ1= idLγ1}.(25) iv. Surjectivity of ργ2→γ1means: for every τ∈Gγ1there exists σ∈Gγ2such that σ|Lγ1=τ. Equivalently, every automorphism of Lγ1/K admits an extension to an automorphism of Lγ2/K. Theorem 1 (Dynamic Galois System as a Projective System).Let f(x)∈K[x]have splitting field L over K, and let (Γ,Inv,{Bγ}γ∈Γ)be a dependency space. For each γ∈Γ, set Rγ={α∈L: Inv(α)∈Bγ}, Lγ=K(Rγ), Gγ= Aut(Lγ/K).(26) If γ1, γ2∈Γsatisfy Bγ1⊆Bγ2, let ργ2→γ1:Gγ2→Gγ1, ργ2→γ1(σ) = σ|Lγ1.(27) Then the family {Gγ}γ∈Γ, together with the homomorphisms {ργ2→γ1}, forms a projective system of groups indexed by the poset (Γ,⊆). Proof. By Proposition 1, if Bγ1⊆Bγ2, then Lγ1⊆Lγ2. By Lemma 1, this inclusion induces a canonical homomorphism ργ2→γ1:Gγ2→Gγ1.(28) To verify the projective system axioms, we check: I. For every γ∈Γ and σ∈Gγ, we have ργ→γ(σ) = σ|Lγ=σ, (29) so ργ→γ= idGγ. 4 II. If γ1, γ2, γ3∈Γ with Bγ1⊆Bγ2⊆Bγ3, then Lγ1⊆Lγ2⊆Lγ3.(30) For any σ∈Gγ3, (ργ2→γ1◦ργ3→γ2)(σ) = ργ2→γ1(σ|Lγ2) = (σ|Lγ2)|Lγ1=σ|Lγ1=ργ3→γ1(σ).(31) Thus the maps satisfy the composition rule required of a projective system. Therefore the collection ({Gγ},{ργ2→γ1}) forms a projective system in the category of groups. Definition 4 (Cayley Graph of a Dynamic Group).Let γ∈Γand let Gγ= Aut(Lγ/K)be the corresponding dynamic group. Fix a symmetric generating set Σγ⊆Gγ, i.e. Σγgenerates Gγand satisfies Σγ= Σ−1 γ. The Cayley graph of Gγwith respect to Σγ[4, 5, 6] is the graph Gγ= Cay(Gγ,Σγ),(32) defined by V(Gγ)=Gγ, E(Gγ) = {σ, τ}⊆Gγ:σ−1τ∈Σγ.(33) By construction, Gγis undirected and |Σγ|-regular. Each vertex represents a symmetry σ∈Gγ, and two symmetries are adjacent precisely when one can be obtained from the other by multiplication with a generator. The collection of graphs {Gγ}γ∈Γmirrors the dependency system of groups. If Bγ1⊆Bγ2, then Lγ1⊆Lγ2and there is a restriction homomorphism ργ2→γ1:Gγ2−→ Gγ1.(34) This homomorphism relates the corresponding Cayley graphs. If the generating sets are chosen compatibly, in the sense that ργ2→γ1(Σγ2)⊆Σγ1,(35) then ργ2→γ1induces a graph homomorphism Gγ2−→ Gγ1.(36) In general, without such a compatibility assumption, one still obtains a natural comparison between the graphs, reflecting how the symmetry relations shrink when passing from γ2to γ1. The outcome is that the dependency space Γ indexes not only a system of groups but a corresponding system of Cayley graphs. Each Gγrepresents the symmetry relations of the restricted field Lγin combinatorial form, and these graphs provide the foundation for the spectral analysis to follow. The variation of γmanifests itself directly in the group structure and therefore in its graphical representation. As the boundary regions Bγevolve: I. the vertex set of the Cayley graph changes whenever the order of Gγchanges, so that automorphisms may appear or disappear; II. the edge set changes when generators vanish or collapse, removing certain adjacencies, or when new generators are introduced, adding new adjacencies; III. at critical values of γ, the boundary Bγmay undergo discontinuous transitions, forcing distinct automorphisms to become identified and collapsing multiple vertices into a single one. These three modes of evolution: incremental changes, discrete jumps, and collapse events; constitute the dynamic behavior of the family {Gγ}γ∈Γ. The dependency space therefore governs not only the algebraic restrictions of the splitting field but also the combinatorial evolution of the associated graphs. In the next section, we analyze this evolving structure through spectral invariants of the Cayley graphs {Gγ}γ∈Γ. 5 4 Spectral Analysis To extract numerical information from the dynamic graphs, we associate to each Cayley graph Gγa collection of canonical matrices. These matrices encode the combinatorial structure of the graph and provide the operators whose eigenvalues reflect the underlying symmetries. Definition 5 (Adjacency matrix).Let Gγbe the Cayley graph of Gγwith respect to Σγ. Label the vertices by the elements of Gγ. The adjacency matrix is the |Gγ|×|Gγ|matrix Aγ(σ, τ) = (1if {σ, τ}∈E(Gγ), 0otherwise.(37) Equivalently, Aγ(σ, τ)=1if σ−1τ∈Σγ. Definition 6 (Degree matrix).The degree matrix is the diagonal matrix Dγindexed by Gγ, defined by Dγ(σ, σ) = deg(σ)=|{τ∈Gγ:{σ, τ} ∈ E(Gγ)}|.(38) In the Cayley graph setting, deg(σ)=|Σγ|for all σ, so that Dγ=|Σγ|·I|Gγ|.(39) Definition 7 (Laplacian and normalized Laplacian).The (combinatorial) Laplacian of Gγis the matrix Lγ=Dγ−Aγ.(40) The normalized Laplacian is the matrix Lγ=I−D−1/2 γAγD−1/2 γ.(41) Since Gγis |Σγ|-regular, the normalized Laplacian simplifies to Lγ=I−1 |Σγ|Aγ.(42) These matrices form the analytic window into the dynamic system [7]. The adjacency matrix records direct connections between symmetries, the Laplacian measures diffusion of functions over the graph, and the normalized Laplacian provides a scale-invariant version suitable for comparing graphs of different size or degree. Moving onto the spectral invariants of the dynamic graphs: let Gγ= Cay(Gγ,Σγ) be the Cayley graph of the dynamic group Gγ. Suppose |Σγ|=d, so that Gγis d-regular. Denote its adjacency matrix by Aγ, degree matrix by Dγ, and Laplacian by Lγ=Dγ−Aγ. Since the graph is d-regular, we have Dγ=dI|Gγ|, and therefore Lγ=dI|Gγ|−Aγ.(43) The normalized Laplacian is given by Lγ=I−D−1/2 γAγD−1/2 γ.(44) But Dγ=dI, hence D−1/2 γ=1 √dI, and so Lγ=I−1 dAγ.(45) Thus we have explicit relations between the three matrices: Lγ=dI −Aγ,Lγ=I−1 dAγ.(46) From these identities, the spectra are directly linked. Let λ1, . . . , λ|Gγ|be the eigenvalues of Aγ. Then the eigenvalues of Lγand Lγare obtained by direct substitution: i. If Aγv=λv, then Lγv= (dI −Aγ)v= (d−λ)v, (47) so every eigenvalue µof Lγsatisfies µ=d−λfor some eigenvalue λof Aγ. 6 ii. Likewise, Lγv=I−1 dAγv=1−λ dv, (48) so each eigenvalue νof Lγsatisfies ν= 1 −λ/d. Therefore all three spectra are equivalent under affine transformations. The adjacency spectrum lies in the interval [−d, d]; the Laplacian spectrum lies in [0,2d]; the normalized Laplacian spectrum lies in [0,2]. The first key invariant is the largest adjacency eigenvalue. Because the graph is d-regular, the vector 1= (1,1,...,1)⊤is an eigenvector of Aγ: (Aγ1)(σ) = X τ∈Gγ Aγ(σ, τ)·1 = deg(σ) = d. (49) Thus Aγ1=d1, so λmax(Aγ) = d. This eigenvalue always occurs, and its multiplicity equals the number of connected components of the graph. In the connected case it has multiplicity one. The second central invariant is the algebraic connectivity [9], defined as the second smallest eigenvalue of the Laplacian. Denote the Laplacian eigenvalues by 0 = µ1≤µ2≤···≤µ|Gγ|. Then µ2is characterized by the Rayleigh quotient µ2= min f⊥1 f=0 f⊤Lγf f⊤f.(50) Writing out the quadratic form explicitly, f⊤Lγf=f⊤(dI −Aγ)f=dX σ∈Gγ f(σ)2−X σ,τ∈Gγ Aγ(σ, τ)f(σ)f(τ).(51) Equivalently, f⊤Lγf=1 2X {σ,τ}∈E(Gγ)f(σ)−f(τ)2.(52) Thus µ2is small when the graph contains a bottleneck (a sparse cut with few edges crossing), and µ2 is large when the graph is strongly connected with no narrow cuts. This makes µ2a precise measure of how tightly the group symmetries are bound together. The third invariant is the spectral gap of the adjacency matrix. Since the trivial eigenvalue dis always present, the spectral gap is defined as gap(Aγ)=d−max{λi:λi=d}.(53) Equivalently, it is the difference between the largest eigenvalue and the second largest eigenvalue. A positive gap implies strong expansion properties [10]: random walks on the graph mix rapidly, and the group symmetries are highly interwoven. A small or vanishing gap indicates fragility, allowing the symmetry structure to split or become unstable under perturbations. Finally, multiplicities of eigenvalues also carry structural information. The multiplicity of µ= 0 in the Laplacian spectrum equals the number of connected components of Gγ. The multiplicity of the trivial adjacency eigenvalue dequals the same number. Nontrivial repeated eigenvalues often signal hidden algebraic regularities, such as abelian structure or presence of normal subgroups, which cause the Cayley graph to decompose into highly symmetric pieces. In summary, the correspondences µi=d−λi, νi= 1 −λi d(54) make it transparent how adjacency, Laplacian, and normalized Laplacian spectra are linked. Tracking µ2, the adjacency spectral gap, and the multiplicities of special eigenvalues gives a precise spectral fingerprint of the dynamic symmetry encoded by Gγ. The dynamic system {Gγ}γ∈Γis not static of course: as γvaries, the underlying group Gγand its Cayley graph Gγevolve. Since the spectral invariants are defined in terms of the matrices Aγ,Lγ, and Lγ, their behavior is governed entirely by how these matrices change with γ. There are two distinct regimes to consider: continuous evolution and discrete jumps. Suppose the vertex set V(Gγ) = Gγ remains constant as γvaries in some neighborhood of γ0. In this case the group structure does not change in cardinality; rather, only the adjacency relations encoded by Σγmay evolve. Formally, fix an enumeration of the vertices by Gγ, and consider the adjacency matrix Aγ. Its entries are (Aγ)σ,τ =(1 if σ−1τ∈Σγ, 0 otherwise.(55) 7 If Σγchanges with γ(for example, by varying generators that depend continuously on a parameter inside K), then the adjacency matrix varies continuously in its entries. Because the eigenvalues of a real symmetric matrix depend continuously on its entries, it follows that γ7−→ λi(Aγ) (56) is a continuous function for each ias long as the vertex set is fixed. The same reasoning applies to the Laplacian and normalized Laplacian. Since Lγ=dI −Aγ,Lγ=I−1 dAγ,(57) the eigenvalues µi(Lγ) and νi(Lγ) vary continuously in γunder the same assumption. Thus, in the continuous regime, all spectral invariants evolve smoothly: the algebraic connectivity µ2(Lγ), the adjacency spectral gap, and multiplicities change gradually with γ. No abrupt changes occur unless the underlying group structure changes in size. If a root leaves the admissible region Bγ, then Rγshrinks, Lγdecreases, and so does Gγ. Equivalently, vertices in Gγvanish. The corresponding matrices Aγ, Lγ,Lγdrop in dimension, and the spectrum undergoes a discontinuous jump. For example, suppose |Gγ|=nin a neighborhood of γ0, and then at γ0a root exits, causing |Gγ0|=n−1. The adjacency matrix Aγis then an n×nmatrix for γ=γ0, but becomes (n−1) ×(n−1) at γ0. There is no continuous path connecting the eigenvalues before and after: one eigenvalue sequence simply disappears, reflecting the loss of a vertex. Similarly, if two roots merge at γ0, then two automorphisms collapse into one. The vertex set contracts, and two vertices in the Cayley graph are identified. In matrix terms, two rows and columns of Aγcoalesce into one, reducing dimension. The corresponding eigenvalues may jump or merge, producing discontinuous changes in the spectrum. These discontinuities signal structural phase transitions in the symmetry: the group Gγchanges order, subgroups may collapse, and the combinatorial structure of the Cayley graph alters abruptly. The spectral invariants capture these events as sudden jumps in the eigenvalue lists. In conclusion, the spectral dynamics of the system {Gγ}fall into two regimes. Within stable intervals of γ, eigenvalues vary continuously and provide smooth signatures of symmetry evolution. At singular points where the root set changes, eigenvalues jump discretely, reflecting a reorganization of the symmetry structure. These spectral transitions are the precise numerical analogues of the algebraic events in the dependency space, and they will serve as the central tool for detecting robustness and fragility of dynamic Galois symmetries. Analysis on Smooth versus Abrupt Spectral Transitions. The behavior of the spectral invariants under variation of γdepends entirely on how the admissible root set Rγevolves. When Rγremains unchanged as γvaries in a neighborhood of some point γ0, the corresponding subfield Lγand group Gγ have fixed cardinality, and the Cayley graph Gγhas a fixed vertex set of size n=|Gγ|. In this regime the adjacency matrix Aγis an n×nsymmetric matrix with entries (Aγ)σ,τ =(1 if σ−1τ∈Σγ, 0 otherwise.(58) If both the vertex set and the generating set Σγare fixed, then the adjacency matrix does not change at all, and the spectrum is constant. More generally, if Σγis altered only in a way that preserves the adjacency pattern across γ, the eigenvalues remain constant. Thus in the stable regime the spectral invariants remain constant, reflecting the absence of algebraic change in the underlying selection of roots. When the admissible root set itself changes, the situation is different. If a root is excluded from Rγ, the subfield Lγcontracts and the automorphism group Gγchanges size. The vertex set of Gγis then replaced by one of different cardinality, say from nelements to melements. The adjacency matrix Aγ, which was n×n, is replaced by a different matrix of size m×m. The two spectra, one of length nand the other of length m, cannot be continuously connected; they are simply different lists of eigenvalues. A similar discontinuity occurs when the boundary condition Bγchanges so that roots previously distinguished are now treated in the same way, collapsing Lγand thereby reducing the effective automorphism group. In each case the spectral data undergoes a discrete reorganization. The principle that emerges is interesting: as long as the dependency rule leaves Rγunchanged, the Cayley graph is fixed in size and structure, and the spectral invariants remain stable. At critical values of γwhere Rγchanges, the group Gγreorganizes and the Cayley graph is replaced by one of different size. The associated spectra are then unrelated by continuity, and the invariants jump abruptly. The smooth regime corresponds to deformation within a fixed algebraic structure, while the abrupt regime corresponds to phase transitions in the dependency space Γ. It is precisely through the spectral data that these two modes of evolution can be numerically distinguished. 8 5 Illustrative Example: The Quartic f(x) = (x2−2)(x2−3) We now turn to a concrete example in order to see the abstract machinery of Sections 2-4 in action. The aim here is not only to compute but to watch, almost step by step, how the invariant map and the boundary mechanism interact to determine which roots are admitted, how the corresponding fields appear, and how the symmetry groups evolve. By working this through carefully, the abstract formalism will become much more tangible. Consider the polynomial f(x) = (x2−2)(x2−3)=x4−5x2+ 6 ∈Q[x].(59) Its roots are easy to list: α1=√2, α2=−√2, α3=√3, α4=−√3.(60) From these we see immediately that the splitting field is L=Q(√2,√3),(61) and since √2 and √3 are independent quadratic extensions, the degree is [L:Q] = 4. The full Galois group is the Klein four-group, generated by the two independent sign changes σ2:√27→ −√2,√37→ √3, σ3:√27→ √2,√37→ −√3.(62) To connect this polynomial with the dependency-space framework, we need to decide on an invariant map. The natural choice is Inv(α) = α2,(63) which sends the two square roots of 2 to 2, and the two square roots of 3 to 3. Thus the values of the invariant cleanly separate the two quadratic factors of the polynomial. Once we have this, the boundary regions Bγcan be taken to be subsets of {2,3}, and by varying them we decide whether to include the √2-pair, the √3-pair, both, or neither. Now let us watch what happens as the boundary changes. If Bγ=∅, then no root passes the filter and we are left with Lγ=Qand the trivial group. If Bγ={2}, then both ±√2 are admitted, which forces Lγ=Q(√2). This is a quadratic extension with nontrivial automorphism √27→ −√2, so here Gγis cyclic of order two. A similar story holds when Bγ={3}: we obtain Lγ=Q(√3) with Gγ∼ =C2. Finally, if Bγ={2,3}, then all four roots are admitted, Lγis the full splitting field Q(√2,√3), and the group jumps to V4, generated by the two independent sign flips. Thus, as the boundary varies, the fields and groups change in a very controlled way. The dependency mechanism takes us from the trivial group to a cyclic one, and then further to the Klein group, depending only on which square values the boundary allows. It is the invariant map that makes this possible, since it identifies roots in a manner compatible with the algebraic structure. From the perspective of graphs, each Gγcomes with a natural Cayley graph. In the trivial case there is just a single vertex with no edges, and the spectrum is reduced to zero. In the cyclic cases we have two vertices connected by one edge; the adjacency matrix A=0 1 1 0(64) has eigenvalues {1,−1}, and the Laplacian L=1−1 −1 1 (65) has eigenvalues {0,2}. This shows that the algebraic connectivity is already positive, so the symmetry is nontrivial. Finally, in the Klein case the Cayley graph is the four-cycle, with adjacency eigenvalues {2,0,0,−2}and Laplacian eigenvalues {0,2,2,4}. One can see how the spectrum reorganizes when the group jumps from two elements to four, reflecting the new structure and the increased connectivity. By fixing a simple quartic polynomial and letting the dependency space decide which quadratic blocks are admitted, we generate a family of intermediate groups. Each choice of boundary corresponds to a clear algebraic regime, and the transition from one regime to another is visible not only in the algebra but also in the combinatorial data of the Cayley graphs and their spectra. This example makes transparent how the dependency mechanism produces a dynamic system of fields and groups, with numerical fingerprints recorded in the spectral data. 9