A Coherence-First Lattice Formulation with Spatially Varying Gauge Grammar

A Coherence-First Lattice Formulation with Spatially Varying Gauge Grammar Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] Standard lattice gauge formulations enforce consistency by postulating a fixed local symmetry group, whose exact invariance guarantees the coherent gluing of local descriptions. While sufficient, this requirement is stronger than necessary and becomes restrictive in the presence of interfaces, crossovers, or emergent gauge structure. In this work we introduce a coherence–first lattice formulation in which exact discrete consistency is enforced independently of a fixed symmetry grammar. The construction is based on a discrete coherence complex, in which local comparison data are defined on cells of increasing dimension and their failures of compositional closure are tracked by higher–level defect fields. A dynamical structure field controls the local grammar by which such defects are repaired, allowing the effective gauge structure to vary spatially across a single lattice without any change of discretization. We formulate an action that penalizes coherence defects rather than symmetry violation and show that the resulting variational dynamics preserves exact discrete closure identities, including higher Bianchi and Noether–II relations, at finite lattice spacing. Numerical implementations demonstrate stable relaxation and Langevin sampling, as well as the programmability of local gauge grammar through spatial modulation of the structure field. These results establish a consistent lattice framework in which symmetry appears as a limiting case of exact coherence, enabling the controlled modeling of interfaces and crossover regimes that are difficult to treat within symmetry–first discretizations. I. INTRODUCTION A. Motivation: limits of symmetry-first discretizations Lattice gauge theory (LGT) and related discretizations have been among the most successful nonperturbative tools in modern physics, from Wilson’s original formulation of confinement to contemporary large-scale simulations of non-Abelian gauge dynamics [ 2 – 5 ]. At their core, standard symmetry-first lattice formulations enforce consistency by fixing a local redundancy a priori: link variables are valued in a chosen gauge group G (e.g. SU ( N )), local gauge transformations act on links and matter fields, and the discretized action is engineered to be exactly invariant under this fixed symmetry [ 2 , 4 ]. This architecture is powerful because it ensures that constraint propagation and Ward identities are built in by construction, reflecting the general connection between local symmetries and identities among Euler–Lagrange operators (Noether’s second theorem) [6–8]. However, the same rigidity becomes a limitation in a range of situations in which the effective notion of “gauge structure” is not globally uniform. Interfaces and crossover regimes in condensed matter provide prominent examples: emergent gauge descriptions can appear only in the infrared, change type across a phase boundary, or hold only approximately over an intermediate window [ 13 – 15 ]. Related phenomena also arise in high-energy and quantum-gravity contexts where generalized and emergent symmetry structures play a central organizing role, but need not coincide with a fixed microscopic gauge group everywhere [ 10 – 12 ]. In such settings, standard symmetry-first discretizations are often forced into ad hoc modeling choices: one simulates separate symmetry-fixed theories and “matches” them by hand, or one adds explicit symmetry-breaking terms whose consistency control is nontransparent. This is not a criticism of LGT—it reflects the fact that symmetry-first discretizations are optimized for regimes where the local redundancy is fixed and exact. The practical message is that symmetry exactness is frequently stronger than what is operationally required. Gauge symmetry is not directly observable (only gauge-invariant operators and holonomies are), and the prohibition of spontaneous breaking of local gauge invariance (Elitzur’s theorem) highlights that local gauge symmetry is a redundancy of description rather than a directly measurable order parameter [ 9 ]. What actually matters for the existence of a well-defined global theory is that local descriptions can be glued consistently across overlaps and compositions—a notion that is geometric and categorical in origin and that persists beyond the special case of strict group-valued transition functions [ 16 , 19 , 20 ]. This observation motivates a discretization strategy that targets gluing/compatibility as the primitive consistency requirement, rather than targeting invariance under a fixed symmetry grammar as the primitive axiom. 2 B. Conceptual shift: coherence as the minimal consistency requirement We adopt the viewpoint that physics is fundamentally the problem of coherently gluing local descriptions into a global account. In differential geometry, this is expressed via transition data, cocycle conditions, and the appearance of connections and curvature as the residue of nontrivial gluing [ 17 , 18 ]. Categorically, one may view local models as objects, changes of local frame as morphisms, and relations between changes of frame as higher morphisms; coherence is then the statement that the relevant diagrams commute (strictly or up to controlled higher data) [ 16 , 19 , 20 ]. In this language, ordinary gauge theory corresponds to a first level of coherence (comparison of local frames), while higher gauge theory corresponds to enforcing coherence at the next level (comparison of comparisons) by introducing higher-form compensators [ 20 – 22 ]. The key point is that symmetry is a sufficient but not necessary condition for coherent gluing of local descriptions; the minimal requirement of consistency is the exact closure of the coherence complex. Here “coherence complex” refers to the coupled discrete structure consisting of: (i) fields assigned to cells of increasing dimension (e.g. 1-form and 2-form data on edges and plaquettes), (ii) defect maps that measure failures of compositional closure (e.g. a fake curvature FΦ and higher curvature H ), and (iii) exact identities (discrete Bianchi/Leibniz/Noether-II relations) that guarantee compatibility of these defects across cells and, crucially, across boundaries. Symmetry-first discretizations guarantee these properties by restricting the admissible local comparisons to a fixed group G and enforcing strict invariance. A coherence-first discretization instead enforces the closure identities directly at the level of the complex; symmetry then appears as a special limit in which coherence defects are strongly suppressed and the redundancy organizes into an effective group action. Key conceptual point. Symmetry is a sufficient but not necessary condition for coherent gluing of local descriptions. The minimal requirement of consistency is the exact closure of the coherence complex; symmetry arises as a limiting case in which coherence defects are dynamically suppressed. This shift becomes particularly transparent on a lattice. Discrete exterior calculus (DEC) and related geometric discretizations make the topological identities of differential forms exact at finite resolution— in particular, d2 = 0 and discrete Stokes theorems follow from incidence relations rather than from continuum limits [ 23 – 26 ]. A coherence-first lattice formulation leverages these exact identities as the backbone of consistency: defect variables are defined so that the relevant closure relations hold off-shell (independently of equations of motion), while an action functional penalizes defect magnitude and drives the dynamics toward coherent configurations. In such a setting, what is “approximate” is not the underlying closure of the complex, but rather the extent to which the resulting dynamics reproduces the familiar symmetry-imposed Ward constraints in correlation functions. The redundancy is therefore coherent (exactly compatible) even when symmetry-like behavior is only emergent. C. Contribution and scope This paper makes the above perspective concrete in a minimal executable setting. Our main contribution is a lattice formulation in which the local gauge grammar can vary spatially across a single lattice without any change of discretization, while exact discrete consistency is preserved under dynamics. Concretely: • We define a coherence-first lattice complex with 1-form and 2-form fields on edges and plaquettes and a structure field Φthat controls a local structure map (a “grammar” for how lower-level defects may be repaired by higher-level compensators). • We construct a defect-based action that penalizes coherence defects rather than imposing invariance under a fixed symmetry group, and we derive the corresponding Euler–Lagrange system and Noether-II identities in the discrete setting [6–8]. • We demonstrate numerically that exact discrete closure identities (discrete Bianchi/Leibniz relations) remain satisfied at finite lattice spacing to machine precision under both relaxation dynamics and Langevin sampling, even when Φvaries spatially (e.g. across a domain wall). • We show “programmability” of the local grammar by shifting the Φ-wall and observing that the resulting coherence-defect textures track the wall within a single fixed scheme. These results establish a practical and conceptually clean framework for modeling interfaces and crossover regimes in which the effective local redundancy is spatially dependent, without resorting to patching distinct symmetry-fixed models. 3 Throughout this paper, statements about “spatially varying symmetry” should be understood in this precise sense: it is the local grammar encoded by the structure map tΦ and the associated coherence stiffnesses that vary across the lattice, not the introduction of distinct gauge groups or independently discretized theories. Fixed-group lattice gauge theory is recovered as a limiting case in which the grammar is uniform and coherence defects are dynamically suppressed. Relation to the broader framework. The present manuscript is a focused, self-contained methods paper extracted from the broader coherence-first program developed in [ 1 ], where the continuum fieldtheoretic formulation, categorical motivation, and additional applications are presented in full. Here we restrict attention to the discrete coherence complex and its exact closure at finite lattice spacing, together with minimal numerical demonstrations of spatially varying gauge grammar. Equally important is what we do not claim here. We do not claim a new phase of matter, a confinement/deconfinement result, or any new universal critical exponent. We also restrict the present work to an abelian prototype and defer nonabelian crossed-module (2-group) implementations, as well as material-specific phenomenology, to future work [ 19 – 22 ]. The goal of this paper is methodological: to provide a compact, verifiable lattice construction showing that exact coherence closure can replace fixed symmetry as the primitive consistency principle, with symmetry recovered as a controlled limit. Coherence-First vs Symmetry-First Lattice Discretization Symmetry-first • Fixed gauge group G • Link variables ∈ G • Plaquette curvature • Exact gauge invariance • Interfaces treated ad hoc fixed grammar Coherence-first • Local states on 0-cells • Comparisons on 1-cells • Defects on 2-cells • Structure field Φ(x) • Exact coherence closure variable grammar Figure 1. Conceptual contrast between symmetry-first and coherence-first lattice discretizations. In symmetry-first formulations (left), consistency is enforced by fixing a global gauge grammar, with link variables valued in a fixed group G and exact gauge invariance imposed everywhere. In coherence-first formulations (right), local states live on vertices, comparison maps on edges, and coherence defects on plaquettes; a structure field Φ( x )modulates the local grammar by which defects are repaired, while exact closure of the coherence complex is preserved across the entire lattice. Symmetry appears as a limiting case of suppressed coherence defects. II. COHERENCE-FIRST LATTICE FORMULATION A. Lattice and cochain structure Let K be an oriented finite cell complex (e.g. a cubulation or triangulation) representing a lattice discretization of a d -dimensional manifold. We write Kp for the set of oriented p -cells (vertices, edges, plaquettes/faces, 3-cells, etc.). The basic algebraic backbone of our construction is the cochain complex C0(K)d −→ C1(K)d −→ C2(K)d −→ C3(K)d −→ ··· , d2= 0,(1) 4 where Cp ( K )denotes the space of p -cochains on K (real-valued unless stated otherwise), and d is the coboundary operator. Concretely, d is the transpose of the cellular boundary incidence map, and the identity d2= 0 is the discrete form of the topological fact that “the boundary of a boundary vanishes”. This cochain complex is not yet “physics”; it is the universal bookkeeping device for organizing localto-global compatibility. In discrete exterior calculus (DEC), one refines this by introducing a discrete Hodge star and inner products that depend on a chosen dual complex, allowing one to define adjoints d† and discrete Laplacians [ 23 – 26 ]. For the present section, we require only the algebraic identities of the cochain complex, especially d2 = 0, and we therefore keep the DEC discussion minimal. (The DEC metric structure will enter later when we define action functionals with L2 -type norms and corresponding Euler–Lagrange operators.) B. Fields and structure map We now place dynamical fields on the cells of K in a way that mirrors the hierarchical gluing logic. The minimal coherence-first prototype employs: • a 1-form field A∈C1 ( K ), assigned to oriented 1-cells (edges), encoding local comparison data between neighboring 0-cells; • a 2-form field B∈C2 ( K ), assigned to oriented 2-cells (plaquettes/faces), encoding higher comparison data (“comparison of comparisons”); • a structure field Φ ∈C0 ( K ), assigned to 0-cells (vertices), encoding the local grammar by which lower-level comparison defects may be repaired by higher-level data. In the abelian prototype developed here, the structure field induces a scalar map (a local structure morphism) tΦ≡m(Φ),(2) which acts by pointwise multiplication when evaluated on higher cochains. More explicitly, m (Φ) is a 0-cochain (vertex field); to apply it to a 2-cochain B we choose a local averaging prescription that produces a face-centered coefficient (e.g. the mean of m on the vertices of each face). We denote the resulting face-centered field by mfand write, schematically, (tΦB)(f) := mfB(f), f ∈K2.(3) Nothing essential depends on the specific averaging choice, provided it is local and fixed once and for all (and hence part of the discretization scheme). The role of tΦ is to specify, locally, how 2-level compensator data are permitted to “screen” or “repair” 1-level curvature-like defects. C. Coherence defects The central objects in a coherence-first formulation are not the raw cell fields themselves, but the defect cochains that measure failures of compositional closure in the gluing hierarchy. In the present minimal model we define: FΦ:= dA −tΦ(B)∈C2(K),(4) H:= dB ∈C3(K).(5) We refer to FΦ as the (abelian) fake curvature or first-level coherence defect. When FΦ ( f )=0on a face f , the loop composition encoded by dA is exactly repaired by the higher datum tΦ ( B )on that face; when FΦ ( f ) 6 = 0, the residual mismatch is the measurable “non-glueability” at the 1 → 2 interface. The 3-cochain H is the next-level defect: it measures failure of the 2-form data to glue consistently at the level of 3-cells. In higher gauge language, His the analogue of a 3-curvature. It is important to emphasize what these definitions do not assume. We do not begin by postulating a fixed gauge group G and then defining plaquette holonomies as group products. Instead, we begin with the cochain complex and define defects as algebraic expressions that live naturally on higher cells and quantify failures of lower-level data to close. Symmetry-first lattice gauge theory is recovered as a special case when the local grammar is fixed and exact invariance is enforced kinematically; in the coherence-first setting, FΦ and H are the primary bookkeeping variables, and symmetry emerges as the limit in which these defects are strongly suppressed. 5 D. Exact discrete coherence identities The distinguishing structural feature of coherence-first discretizations is that the gluing hierarchy closes through exact identities at finite lattice spacing. These identities hold off-shell (independent of dynamics) because they follow from the algebra of the cochain complex together with the discrete Leibniz rule for the Φ-dependent structure map. Starting from (4), apply d: dFΦ=d(dA)−d(tΦ(B)) = −d(tΦ(B)) ,(6) where d(dA)=0by d2= 0. Therefore we obtain the exact closure identity dFΦ+d(tΦ(B)) ≡0.(7) Equation (7) is the discrete coherence backbone: it is the lattice analogue of a (structure-dependent) Bianchi identity, and it remains true for any configuration of (A, B, Φ). Remark (structural vs. tautological). Although the identity (7) follows algebraically from the definitions of FΦ and the property d2 = 0, its content is not merely tautological in the lattice setting. The nontrivial aspect is that the Φ-dependent structure map tΦ and the associated discrete Leibniz rule are implemented consistently on a fixed cell complex, so that (7) – (9) remain satisfied exactly at finite lattice spacing and under numerical evolution, as verified explicitly in Fig. 2. To make explicit the role of spatially varying grammar, we expand the second term using the discrete Leibniz rule. Writing tΦ≡m (Φ), the product cochain mB is a 2-cochain whose coboundary is a 3-cochain. On a 3-cell c∈K3the discrete Leibniz rule takes the form d(mB) = m dB + (dm)∧B, (8) where dm ∈C1 ( K )is the coboundary of the 0-cochain m , and ( dm ) ∧B denotes the canonical bilinear cochain operation (cup/wedge product) that maps ( C1, C2 ) →C3 (with the usual orientation-dependent signs). The precise cup-product convention is fixed by the choice of discretization scheme (and, in DEC, by the primal/dual pairing), but once fixed it is algebraic and exact [ 23 – 25 ]. Combining (7) – (8) with H=dB gives the equivalent form dFΦ+m H + (dm)∧B≡0.(9) This identity exhibits the precise sense in which spatial variation of the local grammar acts as a structured “source term” in the closure relations: even when m (Φ) varies across the lattice (e.g. across a domain wall), the coherence complex closes exactly; the effect of d Φis not to break consistency, but to contribute explicitly through (dm)∧B. Equations (7) and (9) are purely structural; they do not rely on any assumed phase, any minimization principle, or any continuum limit. They are the precise lattice realization of the paper’s central theme: the minimal consistency requirement is exact closure of the coherence complex, while symmetry is an emergent special case obtained when the dynamics suppresses coherence defects. This exact closure identity is the discrete backbone of the coherence-first perspective developed in [ 1 ], specialized here to a minimal lattice setting suitable for computation. III. ACTION AND VARIATIONAL CONSISTENCY A. Defect-based action A coherence-first lattice formulation becomes a physical theory only once we specify a functional that selects, weights, and dynamically suppresses (or tolerates) coherence defects. The guiding principle is simple: we penalize incoherence rather than “break symmetry”. Concretely, we assign an energetic cost to the defect cochains defined in Sec. II C, namely FΦ=dA −tΦ(B)∈C2(K)and H=dB ∈C3(K). To write an action, we need an inner product on cochains. This is where the minimal DEC input enters: choose positive definite bilinear forms h·,·ip:Cp(K)×Cp(K)→R, p = 0,1,2,3,(10) 6 typically induced by a discrete Hodge star or diagonal “mass matrices” on cells [ 24 – 26 ]. We denote the induced norm by kXk2 p := hX, Xip . The adjoint (codifferential) δ : Cp ( K ) →Cp−1 ( K )is defined by the discrete integration-by-parts identity hdα, βip=hα, δβip−1, α ∈Cp−1(K), β ∈Cp(K),(11) which is the cochain analogue of Stokes’ theorem and is exact at finite resolution. With these preliminaries, we introduce the defect-based action S[A, B; Φ] = 1 2kFΦk2 2βF+βH 2kHk2 3+kB 2kBk2 2,(12) where: • kFΦk2 2βF denotes a weighted 2-cochain norm, implemented by a positive weight βF ( x ) > 0assigned to faces (or dual faces), kFΦk2 2βF:= X f∈K2 βF(f)FΦ(f)2wf,(13) with wfthe cell weight arising from the chosen inner product; •βH> 0is a stiffness for the 3-cochain defect H = dB (we keep it constant here, though it may also be Φ-dependent); •kB≥ 0is a small regularizer (or mass term) for B which ensures coercivity of the quadratic form in numerical implementations and fixes the scale of Bfluctuations. The key structural point is the dependence of βF on the structure field Φ. Since tΦ≡m (Φ) sets the local “grammar” by which Brepairs dA, it is natural to allow the defect penalty to depend on Φas well: βF(f) = βFΦloc(f),(14) where Φ|loc(f)denotes the local stencil of Φused to evaluate tΦand/or the face-averaged value of m(Φ) (cf. Sec. II B). In physical terms, Φcontrols not only how defects are repaired (through tΦ ), but also how strongly unrepaired defects are penalized. This implements, on the lattice, the notion that “symmetry” is not an axiom but a regime: when the coherence stiffness is large, defects are suppressed and the theory behaves symmetry-like; when the stiffness is moderate, defects fluctuate while the coherence complex remains exactly closed (Sec. II D). A comment on interpretation is important. In symmetry-first formulations, one often phrases physics in terms of “breaking” or “preserving” gauge invariance. In the present formulation, the primitive quantity is the defect magnitude. There is no need to interpret nonzero FΦ as a pathology: it is simply the measurable residue of non-glueability that remains after allowing the higher compensator B to contribute according to the local grammar tΦ. B. Euler–Lagrange equations as defect-reduction dynamics Because the action (12) is quadratic in the defect cochains, its Euler–Lagrange (EL) equations can be written in a compact form that makes their physical meaning transparent: they are discrete analogues of “defect-reduction” equations. For clarity, we first treat Φas a prescribed background (so that tΦ and βF are fixed functions on the lattice) and vary Aand B. We comment on Φ-variation at the end. Variation with respect to A .Since FΦ = dA −tΦ ( B ), the variation is δFΦ = d ( δA ). Using (11) we obtain δAS=hβFFΦ, d(δA)i2=hδA, δ(βFFΦ)i1,(15) hence the EL operator for Ais EA:= δ(βFFΦ)∈C1(K), EA= 0.(16) Interpretation: (16) is a discrete “divergence of defect” equation. It states that, at stationarity, the weighted defect flux has no net source on edges. When βF is large, the dynamics strongly suppresses configurations with large FΦ ; when βF is moderate, the residual defect must still satisfy the EL balance (16). 7 Variation with respect to B.Now δFΦ=−tΦ(δB)and δH =d(δB). Therefore δBS=hβFFΦ,−tΦ(δB)i2+hβHH, d(δB)i3+hkBB, δBi2.(17) Using the adjoint t† Φdefined by hX, tΦ(Y)i2=ht† Φ(X), Y i2and discrete integration by parts, we get δBS=hδB, −t† Φ(βFFΦ) + δ(βHH) + kBBi2,(18) so the EL operator for Bis EB:= −t† Φ(βFFΦ) + δ(βHH) + kBB∈C2(K), EB= 0.(19) Interpretation: (19) balances three effects: (i) the tendency of B to reduce the 1-level defect through t† Φ ( βFFΦ ), (ii) the stiffness cost of H = dB through δ ( βHH ), and (iii) the regularization scale kBB . In the regime kB→ 0and large βF , one obtains an approximate algebraic relation t† Φ ( βFFΦ ) ≈δ ( βHH ), i.e. the 2-level sector absorbs the 1-level defect subject to its own coherence constraint H. Optional variation with respect to Φ.If Φis treated as dynamical, one adds a local kinetic term and potential (both standard in lattice field theory), e.g. κΦ 2kd Φ k2 1 + Pv∈K0V (Φ( v )), and includes the Φ-dependence of tΦ and βF . The resulting Φ-equation has the universal form “(discrete wave-map / diffusion)+ ∇V =sources”, where the sources are quadratic in the defects: variations of tΦ and βF produce terms proportional to FΦ·B (and, if generalized, H· ( A, B )), matching the continuum logic that curvature defects drive the structure field toward grammars that reduce incoherence (cf. Noether-II consistency and higher-gauge kinematics [7, 8]). Defect-reduction viewpoint. Equations (16) – (19) are most intuitively read as follows: the action (12) defines a quadratic “coherence energy” in the defect variables. Stationary points are configurations in which defects are minimized subject to exact closure identities (Sec. II D). This makes precise the slogan that the theory penalizes incoherence: the dynamics does not enforce gauge invariance as an axiom; it drives the system toward regions of field space where the coherence defects become small (in the L2 sense dictated by DEC weights), and in which symmetry-like behavior emerges as a limit. C. Noether-II / Ward identities and variational consistency A central consistency test of any lattice gauge formulation is not merely that an action can be written, but that its variational structure is compatible with its exact kinematic identities. In symmetry-first formulations, this is typically ensured by exact gauge invariance: invariance of the action under local transformations yields Ward identities relating EL operators and guaranteeing constraint propagation [ 6 – 8 ]. The same logic applies here, but the relevant “gauge” transformations are best understood as gauge-of-description redundancies of the coherence complex. Gauge-of-description redundancy (abelian 1 → 2 case). In the abelian prototype, the defects (4) – (5) are invariant under the 2-level redundancy A7→ A+dα −tΦ(λ), B 7→ B+dλ, (20) with α∈C0(K)and λ∈C1(K). One verifies immediately that FΦand Hare unchanged: FΦ7→ d(A+dα −tΦ(λ)) −tΦ(B+dλ) = dA −tΦ(B) + d2α−d(tΦ(λ)) −tΦ(dλ).(21) For constant grammar, d ( tΦ ( λ )) = tΦ ( dλ )and the last bracket vanishes; for variable grammar, the discrepancy is accounted for by the exact Leibniz structure of the coherence complex (Sec. II D), and the action remains invariant because it depends only on the defect cochains and their exact closure identities. (Operationally, the discretization scheme fixes the appropriate product rule, and the redundancy (20) is understood within that fixed scheme.) Noether-II identities. Let EA and EB be the EL operators (16) – (19) . Gauge invariance of S under (20) implies the discrete Noether-II identity 0 = δS =hEA, δAi1+hEB, δBi2for arbitrary α, λ. (22) Substituting δA =dα −tΦ(λ)and δB =dλ and using (11) yields two identities: δEA≡0,(23) δEB−t† Φ(EA)≡0,(24) 8 up to the fixed Φ-dependent Leibniz terms determined by the discretization scheme. Equation (23) is the discrete analogue of a divergence identity for the 1-level EL operator; (24) is the “exchange” identity relating the 1-form and 2-form EL operators. These are the coherence-first Ward relations: they encode that what would appear as a failure of 1-level conservation is consistently absorbed by the 2-level sector through the structure map tΦ. Variational consistency (definition). We say that the lattice formulation is variationally consistent if: 1. the action S[A, B; Φ] is well-defined on the chosen discrete field spaces and inner products; 2. the variations with respect to the discrete fields exist and lead to EL operators EA, EB (and, if included, EΦ); 3. the EL operators satisfy exact Noether-II identities (Ward relations) implied by the gauge-ofdescription redundancy; 4. these Ward relations are compatible with the exact kinematic closure identities of the coherence complex, notably (7)–(9). This last point is essential: the exact discrete closure identities hold off-shell, and the variational identities hold off-shell; the two structures must mesh so that defect propagation is consistent and no contradiction can be generated by time evolution or iterative solvers. In symmetry-first formulations this meshing is often taken for granted because exact invariance is built in. In the coherence-first setting, the meshing is built into the coherence complex itself: variable grammar does not destroy consistency; it enters through explicit, exact Leibniz-type terms, which are accounted for by the same discretization scheme. Practical consequence. The combination of: (i) exact closure identities at the level of defects (Sec. II D) and (ii) exact Noether-II identities for the EL operators means that the discretization does not need to be “manually redesigned” when the local grammar varies. Interfaces and crossover regions correspond to spatial textures of Φ(hence tΦ and βF ), but the underlying scheme—cell assignment, operators, inner products, and update rules—remains unchanged. Symmetry is recovered in regimes where the dynamics suppresses defects; outside those regimes, the system remains coherent because the complex closes exactly. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 | dF + d ( mB )| on cubes 1e 16 0 100 200 300 400 500 count Exact discrete closure after dynamics (max residual 3.89e-16) Figure 2. Exact discrete closure under dynamics. Histogram of the cube-wise residual |dFΦ + d ( tΦB ) | after defect-reduction dynamics (relaxation and/or Langevin sampling), showing residuals at machine precision. This provides a direct numerical verification that variable local grammar (through tΦ ) does not spoil exact closure of the discrete coherence complex. IV. NUMERICAL DEMONSTRATIONS This section provides executable demonstrations of the central capability established in Secs. II–III: a single fixed discretization scheme supports spatially varying gauge grammar (encoded by the structure field Φand the induced map tΦ≡m (Φ)) while preserving exact discrete closure identities under dynamics. 9 We emphasize three complementary points: (i) the defect-based action drives stable defect-reduction dynamics, (ii) the same scheme accommodates spatially varying grammar without redesign, and (iii) the resulting coherence-defect textures are robust at the level of ensemble sampling. Throughout, we use the abelian 3D lattice prototype described in Sec. II. Fields A and B are stored as real-valued cochains (unwrapped variables) on a periodic cubical lattice, and tΦ is implemented as a local multiplication map obtained from a vertex field Φ( x )by a fixed stencil average to faces. The action is the quadratic (Villain-like) defect functional (12) , with a small B -regularizer kB> 0to ensure numerical coercivity. The exact closure identity (7) – (9) is verified directly by computing the cube-wise residual dFΦ+d(tΦB), which must vanish identically up to floating precision. A. Relaxation dynamics Gradient descent as defect-reduction. To exhibit that the action (12) defines meaningful dynamics and does not merely encode an abstract identity, we perform deterministic relaxation by gradient descent. Concretely, we iterate A(n+1) =A(n)−ηA∇AS[A(n), B(n); Φ], B(n+1) =B(n)−ηB∇BS[A(n), B(n); Φ],(25) with learning rates ηA, ηB> 0. The gradients are the discrete EL operators of Sec. III B (up to the standard identification between variational derivatives and gradient directions under the chosen inner products). Here and in the numerical implementations, we use the discrete inner products introduced in Sec. III A to identify variational derivatives with gradients, so that the Euler–Lagrange operators EA and EB coincide with the descent directions for the action under the chosen DEC metric structure. In practice, we employ a mild elementwise gradient clipping to prevent rare large-gradient steps from destabilizing the iterate; this does not change the stationary points, but improves numerical stability. The key property is monotonic defect reduction: for sufficiently small step sizes, S decreases along the iterates because (25) is a descent direction for the quadratic functional. Empirically, S decreases smoothly and converges to a stable low-defect configuration. This demonstrates that the coherence-first action implements an explicit energy landscape whose minimizers are coherent configurations (small FΦ and H), rather than relying on symmetry axioms. What to measure. A direct and physically interpretable diagnostic is the (face-averaged) magnitude of the fake curvature component Ffake xy , averaged over the transverse directions: h|Ffake xy |iy,z(x) := 1 |K1(y)||K1(z)|X y,z |Ffake xy (x, y, z)|.(26) This profile provides a one-dimensional view of how coherence defects distribute along x in the presence of a nontrivial structure field. Stability and exact closure under dynamics. A potential concern in variable-grammar formulations is that numerical evolution might violate the structural identities of the coherence complex. Here that does not occur: the exact closure identity is structural, and the discrete operators are fixed; therefore, the cube-wise residual dFΦ + d ( tΦB )remains at floating precision throughout the relaxation. This is precisely the sense in which the discretization targets consistency at the level of gluing (closure of the coherence complex) rather than enforcing symmetry as a primitive axiom. B. Spatially varying grammar Spatial variation of the local “gauge grammar” through a structure field Φ( x )is a central theme of the coherence-first framework [ 1 ]; in this section we demonstrate its lattice realization and numerical controllability within a single fixed discretization scheme. Domain wall in the structure field. To demonstrate the central capability—spatially varying local grammar within a single scheme—we impose a one-dimensional domain wall in Φ(x): Φ(x) = tanhx−x0 w, tΦ≡m(Φ) = mL+mR−mL 2(1 + Φ(x)) ,(27) 16 3. Relaxation solver: clipped gradient descent Relaxation uses gradient descent in the discrete field variables: A←A−ηA∇AS, B ←B−ηB∇BS, (B2) with fixed step sizes ηA, ηB . Because the action is quadratic in cochains but spatially inhomogeneous through βF (Φ), we implement mild elementwise clipping of the gradients (a standard numerical stabilization) to prevent occasional large local gradients from dominating the update. This does not change the location of stationary points; it only restricts step size locally. Convergence is assessed by: •monotonic decrease of the action density S(Fig. 3 and the auxiliary action-trace plot); •stabilization of defect norms kFΦkand kHk; •preservation of exact closure (cube-wise residual at machine precision, Fig. 2). 4. Stochastic solver: overdamped Langevin sampling Langevin sampling uses the overdamped update X←X−ηX∇XS∆t+√2T∆t ξ, X ∈ {A, B},(B3) with independent Gaussian noise fields ξ on the cochain spaces. A burn-in period is discarded, and samples are recorded with thinning to reduce autocorrelation. The principal ensemble diagnostic is the mean±std profile of h|Ffake xy |iy,z(x)(Fig. 5). We emphasize that, unlike gradient-descent relaxation, Langevin dynamics is not expected to produce a monotonic decrease of the action S . Fluctuations and occasional increases of S are intrinsic to stochastic sampling at finite effective temperature T and do not indicate instability of the scheme. Structural consistency is instead assessed by the exact closure residual dFΦ + d ( tΦB ), which remains at floating precision throughout the Langevin evolution (Fig. 2). As a structural consistency check, the cube-wise closure residual dFΦ + d ( mB )is computed at each sampling stage. Since this residual is an exact discrete identity, any nonzero value reflects numerical roundoff (or an implementation error). In all reported runs the maximum residual remains within floating precision and does not grow with time; see Fig. 2 and the additional closure histograms provided in Appendix C. 5. Exact closure verification protocol To verify the key identity dFΦ+d(mB)≡0,(B4) we compute FΦ on oriented faces, then apply the discrete coboundary d to obtain a cube-wise 3-cochain dFΦ . We separately compute the product cochain mB (with m face-centered using the same stencil used in FΦ ) and evaluate d ( mB )on cubes. The residual is then defined cube-wise as R := dFΦ + d ( mB ), and we report its histogram and its maximum absolute value. This test is performed after relaxation and after Langevin sampling and is the principal numerical evidence that the coherence complex remains exactly closed under variable grammar and dynamics. Appendix C: Additional plots and supplementary diagnostics This appendix collects additional figures that support the main narrative. 1. Structure-field and wall motion diagnostics 17 0 5 10 15 20 x 0.5 1.0 1.5 2.0 m ( ) Moving the structure-field wall shifts the local grammar center=6 center=15 Figure 6. Structure map profiles for two different wall positions. The structure field Φ( x )induces m (Φ( x )), which controls the local grammar tΦ . Shifting the wall center changes only the background grammar texture; the lattice scheme, operators, and action are unchanged. 0246810 y 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 x | Ffake xy | after relaxation, center=6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0246810 y 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 x | Ffake xy | after relaxation, center=15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 7. Side-by-side spatial slices of |Ffake xy | at fixed z = Lz/ 2for two different wall positions, after relaxation. The coherence-defect texture translates with the grammar texture, illustrating programmability of local redundancy. 2. Action traces and stochastic diagnostics 0 250 500 750 1000 1250 1500 1750 gradient steps 0.000 0.025 0.050 0.075 0.100 0.125 0.150 S (mean density) Relaxation of coherence action (clipped gradient descent) Figure 8. Action density decrease during clipped gradient-descent relaxation, demonstrating stable defect-reduction dynamics. 18 0 500 1000 1500 2000 step 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 S (mean density) Langevin sampling trace (fluctuating around relaxed minimum) Figure 9. Representative Langevin action trace showing stable stochastic sampling around low-defect configurations. (Fluctuations are expected; the key structural test is exact closure.) 3. Heatmaps and correlation-style diagnostics 0246810 y 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 x Slice: | Ffake xy | after relaxation at z = Lz /2 0.1 0.2 0.3 0.4 0.5 0.6 | Fxy | Figure 10. Spatial slice of |Ffake xy | after relaxation at fixed z = Lz/ 2. This plot is a diagnostic of where coherence defects reside under the chosen stiffness landscape and grammar texture. 4. Closure histograms beyond the main text 19 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 | dF + d ( mB )| on cubes 1e 16 0 100 200 300 400 500 count Exact discrete closure after dynamics (max residual 3.89e-16) Figure 11. Closure residual histogram after deterministic relaxation. Residuals are at machine precision, as required by exact discrete closure. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 | dF + d ( mB )| on cubes 1e 14 0 100 200 300 400 500 600 700 count Exact discrete closure after Langevin (max residual 1.22e-14) Figure 12. Closure residual histogram after Langevin sampling. Residuals remain within floating precision, confirming that stochastic dynamics does not spoil exact closure of the coherence complex. 5. Interface-style summaries (diagnostic, parameter-dependent) The interface-enhancement diagnostics (e.g. comparing defect magnitude near the Φwall to bulk baselines) can be useful for parameter exploration, but they are modeland stiffness-dependent and were therefore not used as primary evidence of the capability claim in the main text. For completeness, we include representative diagnostic plots below. 20 0 5 10 15 20 x 1 2 3 4 5 6 7 8 | Ffake xy | y , z Interface enhancement of coherence defects (ensemble averaged) | Ffake xy | ±1 interface (max |dm|) bulk baseline interface window Figure 13. Diagnostic interface plot: ensemble-averaged defect profile with highlighted interface window and bulk baseline. 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