Altermagnetism as Higher Categorical Coherence in Momentum Space: Theory, Ward Identities, Proto‑Gauge Control, and Spintronic Implementations

Altermagnetism as Higher Categorical Coherence in Momentum Space: Theory, Ward Identities, Proto-Gauge Control, and Spintronic Implementations Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] Altermagnetism—a recently identified form of magnetic order combining collinear spin alignment with zero net magnetization—offers a route toward stray-field-free spintronics. We formulate altermagnetism as a higher-gauge-coherent structure on the Brillouin zone, described by a 2-connection ( A, B )and a structure field Φselecting the symmetry-induced alternation grammar. Within this crossed-module geometry, the fake curvature FΦ=dA +A∧A−tΦ(B) encodes the local spin splitting, while the global compensation condition RTdFΦ = 0 expresses magnetic neutrality. Variable-structure Bianchi identities, DAFΦ+tΦ(HΦ) = (dt)Φ(DΦ)∧B, DAHΦ=FΦ.ΦB+ (d.)Φ(DΦ)·(A, B), connect coherence and symmetry, showing how lattice-enforced alternation propagates through gauge levels. From this framework we derive higher Ward identities linking level-1 and level-2 currents, t† Φ(Jtot A) = D∗ AJtot B, D∗ AJtot A= (.Φ)†(B, Jtot B), which quantify current exchange between one-form and two-form sectors and bound symmetryforbidden responses. These bounds introduce the proto-gauge defect ε= max k∈TdkFΦ(k)k/Λ2,kHΦ(k)k/Λ3,kDΦ(k)k/Λ, a single experimentally accessible quantity serving both as a quality metric—guaranteeing Wardbounded parasitics of order O(ε)—and as a control dial for analog spin-current modulation. A complete computational–experimental Methods pipeline converts spin-resolved Berry data into BZ “B-fit” extraction of B and ε , demonstrated in simulated spintronic architectures: Φ-walls yield quantized sheet-conductance channels via mixed topological terms, while small reversible changes of Φ(strain, gating, or SOC) enable hysteresis-free analog gain in the proto-gauge regime. Figures will illustrate the complete theory-to-device chain. Higher categorical coherence thus provides both the microscopic origin and a quantitative design principle for robust, programmable, and stray-field-free spintronic systems. I. INTRODUCTION Spintronics seeks to exploit the spin degree of freedom of carriers, together with their charge, to realize low-power, fast, and functionally novel electronic devices.[ 1 , 160 ] Traditional approaches have relied heavily on ferromagnets as spin injectors and analyzers because exchange interactions split spin bands and produce large equilibrium spin polarizations. However, ferromagnets also bring stray magnetic fields, domain-wall hysteresis, and switching overheads that complicate dense integration and noise-sensitive applications. By contrast, antiferromagnets offer intrinsically faster dynamics and negligible stray fields, but in their collinear realizations they typically do not exhibit spin-split bands: symmetries such as PT or translation × time-reversal enforce a Kramers-like degeneracy E ( k,↑ ) = E ( k,↓ )in momentum space, leading to vanishing net spin polarization in transport unless one leverages relativistic or noncollinear mechanisms.[180, 189] Altermagnetism resolves this tension by combining the advantages of both worlds: it is a collinear, fully compensated magnetic order (zero net magnetization M = 0 in real space) with spin-split electronic bands (and hence strong local spin polarization) in momentum space.[ 5 – 8 ] In altermagnets, lattice symmetries enforce an alternation of spin polarization in k : for symmetry-related momenta the dominant spin character flips so that, e.g., E(k,↑)6=E(k,↓), E(k,↑) = E(−k,↓).(1) This momentum-space alternation produces robust spin-split responses (Hall, Nernst, magneto-optical) without generating macroscopic magnetization or stray fields, enabling stray-field-free spintronic architectures.[5, 9, 160, 189] 2 From symmetry to mathematics: a momentum-space 2-connection. The central thesis of this work is that altermagnetism is most naturally captured as a higher-gauge-coherent structure on the Brillouin zone (BZ) Td , namely a 2-connection ( A, B )valued in a crossed module ( g1 tΦ −−→ g0, .Φ )selected pointwise by a structure field Φ( k ). Concretely, we take A∈ Ω 1 ( Td, g0 )and B∈ Ω 2 ( Td, g1 )and define the curvatures FΦ=dA +A∧A−tΦ(B),(2) HΦ=dB +A .ΦB. (3) Here tΦ embeds the 2-form sector into the level-1 sector while .Φ (our “rightwards-triangle” action) describes how A acts on B . The fake curvature FΦ measures the residual level-1 curvature after coherently relocating the symmetry-forced piece into the 2-form sector through tΦ ( B ). In the collinear setting (which covers canonical altermagnets), g0≃u(1) ˆ nand the “spin-frame” connection is naturally identified as A≡1 2A↑−A↓, Fspin =dA, (4) with A↑,↓ the spin-resolved Berry connections of the occupied subspace. The operational local spin splitting is then FΦ=Fspin −tΦ(B),(5) i.e., the part of the spin curvature not explained by symmetry-mandated alternation. Alternation grammar and zero-M. The alternation in k is compactly encoded by a form-factor (“grammar”) ∆ ` ( k )transforming in a representation of the crystal point group (e.g., d -, g -, or i -wave), typically odd under inversion in altermagnets: ∆`(−k) = −∆`(k).(6) We use ∆ ` to define tΦ so that tΦ ( B ) = ∆ ` ( k ) B ( k ) ˆ n in the collinear case. By oddness, the image of tΦ integrates to zero over the BZ: ZTd tΦ(B)=0.(7) Consequently, ZTd FΦ=ZTd Fspin −ZTd tΦ(B) | {z } = 0 =ZTd Fspin.(8) Thus, altermagnets realize large local spin splitting (finite FΦ ( k )) with global compensation, naturally expressed as ZTd FΦ= 0 .(9) Equation (9) is the momentum-space statement of M = 0 and can be strengthened to “extended exactness” FΦ=dΦΩd−1in a variable-structure complex (useful for topological responses and interface formulas). Why altermagnets for spintronics? Equations (1) – (9) make precise the ferromagnet–antiferromagnet dichotomy of altermagnets: they deliver spin-split transport tensors akin to ferromagnets, while retaining the magnetic silence of antiferromagnets. This synergy promises: • Stray-field-free routing and logic: channels and interfaces without ferromagnetic fringing fields (compactness, reduced cross-talk). • Hysteresis-free analog control in the proto-gauge window (defined below), avoiding energy-costly domain switching. • Compatibility with standard spin-current detection (e.g., inverse spin Hall) and thermal counterparts (spin-splitting Nernst effects).[9, 160] 3 A preview of control and robustness: proto-gauge. A key practical notion is the proto-gauge defect ε= max k∈TdkFΦ(k)k/Λ2,kHΦ(k)k/Λ3,kDΦ(k)k/Λ,(10) with Λa momentum scale. Small ε means the alternation is nearly perfect (fake-flat FΦ≈ 0; small HΦ ), which our higher Ward identities will translate into quantitative bounds on symmetry-forbidden channels and parasitic leakages, O ( ε ). The Ward identities are derived from 2-gauge invariance of the coherence action and read, in exchange/divergence form, t† Φ ( Jtot A ) = D∗ AJtot B and D∗ AJtot A = ( .Φ ) † ( B, Jtot B ). We develop them in full later. The same ε serves as an analog control dial: gentle, reversible changes of Φ (strain, gating, weak SOC) modulate FΦ and thus spin-current amplitudes linearly in ε , offering precise, low-power tuning without domain hysteresis.[5, 189] A. Motivation Ferromagnets: strong spin splitting but stray fields and hysteresis. Ferromagnetic exchange produces robust spin splitting and is the workhorse for spin injection and analysis. Yet, ferromagnets carry macroscopic magnetization whose stray fields degrade scalability and induce cross-talk; their domain switching is hysteretic and energy costly, and the presence of fringing fields complicates on-chip metrology and integration.[ 1 ] Moreover, ferromagnetic spin logic frequently requires careful magnetic shielding and layout constraints. Antiferromagnets: compensated and fast, but collinear bands are typically spin-degenerate. Collinear antiferromagnets are promising for ultrafast, robust spintronics because they are magnetically silent (no stray fields) and can host terahertz dynamics.[ 180 , 189 ] However, in many symmetry classes their electronic bands remain spin-degenerate in the absence of strong relativistic effects or noncollinearity: a combined antiunitary symmetry (e.g., PT or translation ×T ) enforces E ( k,↑ ) = E ( k,↓ ), suppressing momentum-resolved spin polarization essential for spin filtering and certain Hall-like responses.[189] Altermagnets: spin-split bands with M = 0.Altermagnets form a third collinear class with spin-split bands and zero net M , characterized by a momentum-space alternation of spin texture constrained by crystal symmetry ( d -, g -, i -wave patterns, etc.).[ 5 – 8 ] The alternation grammar ∆ ` ( k )controls where and how spins dominate in the BZ, while the integral constraint (9) secures magnetic neutrality. This yields (i) large local spin polarizations for transport and optics, and (ii) global silence against stray-field penalties. Early and recent studies already indicate distinctive responses, including anomalous Hall and magneto-optical effects in altermagnets and their thermal analogues (e.g., spin-splitting Nernst effects).[5, 9] Computational and experimental opportunity. The higher-gauge formulation equips us with operational objects— FΦ , HΦ , and ε —that (i) can be extracted from first-principles or spectroscopic Berry data via a simple B-fit (project Fspin onto the symmetry grammar to obtain tΦ ( B ), then set FΦ = Fspin −tΦ ( B )), and (ii) translate directly into device figures of merit (proto-gauge QA via Ward bounds, actuator maps via ∂ΦtΦ ). On the device side, Φ-walls (real-space changes in the alternation grammar) are predicted to host quantized sheet channels governed by mixed topological terms; meanwhile, small reversible δ Φ (gates/SOC/strain) act as a linear gain knob in the proto-gauge regime. These properties, supported by the theoretical structure and by emerging materials data,[ 5 , 7 , 8 ] motivate a coherent program toward stray-field-free,hysteresis-free, and programmable spintronic platforms. In the remainder of this article we make these claims precise, derive the higher Ward identities and the proto-gauge bounds, and present a complete Methods pipeline linking band data to device-level demonstrations and quality assurance. B. Thesis Statement of the thesis. We treat collinear altermagnets as coherence-first systems on the Brillouin torus Td , described by a higher gauge (2-connection) structure ( A, B )valued in a crossed-module of Lie algebras g1 tΦ −−→ g0, .Φ, 4 selected pointwise in k by a structure field Φ : Td→ M2grp that determines both the morphism tΦ and the action .Φ .[ 10 , 218 , 244 ] The 2-connection consists of a g0 -valued 1-form A∈ Ω 1 ( Td, g0 )and a g1-valued 2-form B∈Ω2(Td, g1). Its curvatures are (cf. (2)–(3)) FΦ=dA +A∧A−tΦ(B),(11) HΦ=dB +A .ΦB. (12) In the collinear case relevant for altermagnets, g0≃u (1) ˆ n , and one identifies the spin-frame connection as A=1 2(A↑−A↓)with Fspin =dA (cf. (4)). The local spin splitting is the fake curvature (cf. (5)) FΦ=Fspin −tΦ(B), while the global compensation is enforced by the alternation grammar ∆ ` ( k )(cf. (6) ), yielding the integral constraint (cf. (9)) ZTd FΦ= 0 . In short: locally FΦ ( k )encodes robust spin splitting; globally the grammar redistributes polarization across kso that the net magnetization vanishes. Crossed-module axioms and coherence. A crossed module ( g1 tΦ −−→ g0, .Φ )consists of a Lie algebra morphism tΦ:g1→g0and an action .Φ:g0→Der(g1)obeying tΦ(x .Φy)=[x, tΦ(y)], tΦ(y1).Φy2= [y1, y2],(13) for all x∈g0 , y, y1, y2∈g1 .[ 10 , 218 ] These axioms guarantee that the 2-curvature HΦ transforms covariantly under 2-gauge transformations and that defects in strict 1-coherence can be coherently absorbed by the 2-form sector (the “coherence-first” idea). The odd form-factor ∆ ` ( k )implementing the alternation grammar enters through tΦ (collinear case: tΦ ( B )=∆ ` ( k ) B ( k ) ˆ n ), ensuring RTdtΦ ( B ) = 0 (cf. (7)) and hence magnetic neutrality despite large local polarization. 2-gauge transformations. Infinitesimal 0-gauge and 1-gauge parameters ( ε, λ ) ∈ Ω 0 ( Td, g0 ) × Ω 1 ( Td, g1 ) act as[31, 244] δεA=DAε:= dε + [A, ε], δεB=ε .ΦB, (14) δλA=−tΦ(λ), δλB=DAλ:= dλ +A .Φλ. (15) These transformations close under the crossed-module axioms (13) and define a strict 2-gauge symmetry acting on ( A, B ). The corresponding curvature doublet ( FΦ, HΦ )obeys variable-structure Bianchi identities (with sources when Φvaries in k)[14, 244] DAFΦ+tΦ(HΦ)=(dt)Φ(DΦ) ∧B, (16) DAHΦ=FΦ.ΦB+ (d.)Φ(DΦ) ·(A, B),(17) which reduce to the familiar Bianchi identities when Φis constant. Coherence action and adjoints. We work with the coherence action on Td S[A, B, Φ] = ZTd1 2g2hFΦ, FΦi0+1 2h2hHΦ, HΦi1+κ 2GΦ(DΦ, DΦ)+Stop[Φ; A, B],(18) where h·,·i0,1 are g0,1 -invariant inner products (fixing adjoints D∗ A , t† Φ ,( .Φ ) † ), and Stop contains mixed topological couplings responsible for interface responses (e.g. Φ-wall sheet steps).[ 31 , 244 ] Varying (18) with respect to (A, B)yields Euler–Lagrange operators (suppressing matter/topological currents Jtot A,B) EA=1 g2D∗ AFΦ−1 h2(.Φ)† B, HΦ−Jtot A,(19) EB=1 h2D∗ AHΦ−1 g2t† Φ(FΦ)−Jtot B.(20) The explicit Φ-dependence produces a source in the Φ-equation, δS δΦ⊃1 g2FΦ,(∂ΦtΦ)·B0−1 h2HΦ,(∂Φ .Φ)(A, B)1+··· ,(21) which is the actuation law: small δ Φ(strain, gating, weak SOC) reshapes FΦ and HΦ in a controlled, symmetry-compatible way. 5 Noether–II and higher Ward identities. 2-gauge invariance of (18) under (14) – (15) implies (off-shell) Noether–II relations among (19) – (20) .[ 16 , 146 , 247 ] After using (16) – (17) and separating kinetic from (matter+topological) contributions, we obtain the higher Ward identities in exchange and divergence form: t† Φ Jtot A=D∗ A Jtot B,(22) D∗ A Jtot A= (.Φ)† B, Jtot B.(23) Physical meaning: any apparent non-conservation of level-1 current must be supplied by a flow in the 2-form sector ( (22) ); level-1 conservation holds up to a B -mediated twist reflecting the alternation grammar ((23)). These identities are the quantitative backbone for robustness and control. Proto-gauge and bounds. Define the proto-gauge defect (cf. (10)) ε= max k∈TdkFΦ(k)k/Λ2,kHΦ(k)k/Λ3,kDΦ(k)k/Λ, with Λa BZ scale. Combining Cauchy–Schwarz on Tdwith (22)–(23) yields operational bounds kt† ΦJtot Ak ≤ C1εkJtot Ak+kJtot Bk,(24) kD∗ AJtot Ak ≤ C2εkJtot Bk+kBkkJtot Bk,(25) with geometry-dependent constants C1,2 fixed by the inner products in (18) . Thus, symmetry-forbidden channels and parasitics are O ( ε ): ε is simultaneously a quality metric (robustness) and a control dial (linear, hysteresis-free modulation of spin-current amplitudes by small δ Φ), in precise agreement with the coherently redistributed nature of tΦ(B)and the neutrality condition RTdFΦ= 0. Thesis recap. Locally, altermagnets are spin-polarized metals or semimetals with sizable FΦ ( k ); globally, the alternation grammar encoded via tΦ and .Φ yields RTdFΦ = 0. The variable-structure Bianchi identities (16) – (17) and the higher Ward identities (22) – (23) provide the exact accounting rules connecting responses across levels and quantifying robustness in the proto-gauge regime. This higher-gauge viewpoint is therefore not merely descriptive: it supplies actuation laws (21) ,QA bounds (24) – (25) , and an immediate translation from band-structure data to device-level figures of merit. C. Contributions This work advances a coherence-first program for altermagnets by delivering five mutually reinforcing contributions that connect rigorous higher-gauge kinematics on the Brillouin zone (BZ) to actionable, stray-field-free spintronic functionality. For clarity, we organize the contributions as (i) kinematics and dynamics on the BZ using the rightwards-triangle action .Φ ; (ii) derivation of higher Ward identities in exchange/divergence form with quantitative proto-gauge bounds; (iii) amain-text Methods pipeline that maps band data to device metrics; (iv) device-level use-cases (Φ-wall router, analog gain) derived from the same field-theoretic structure; and (v) a complete figure set illustrating the theory-to-device chain. (i) Kinematics and dynamics on the BZ with .Φ .We formalize altermagnets as higher-gauge-coherent systems on the BZ Td via a 2-connection ( A, B )valued in a crossed module g1 tΦ −−→ g0, .Φ (cf. §I B). The curvatures are (cf. (11)–(12)) FΦ=dA +A∧A−tΦ(B),(26) HΦ=dB +A .ΦB, (27) and the spin-frame connection identifies as A = 1 2 ( A↑−A↓ )with Fspin = dA (cf. (4) ). The operational local spin splitting is the fake curvature (cf. (5)) FΦ=Fspin −tΦ(B),ZTd FΦ= 0 (global neutrality; cf. (9)).(28) The alternation grammar ∆ ` ( k )(odd under k7→ −k ; cf. (6) – (7) ) is implemented inside tΦ so that grammar-induced redistribution of spin polarization leaves RTdFΦ invariant. Dynamics follow from the coherence action (cf. (18)), S[A, B, Φ] = ZTd1 2g2hFΦ, FΦi0+1 2h2hHΦ, HΦi1+κ 2GΦ(DΦ, DΦ)+Stop[Φ; A, B],(29) 6 whose variation yields the Euler–Lagrange operators (cf. (19) – (20) ) and explicit actuation sources in the Φ-equation (cf. (21) ). These sources, proportional to ∂ΦtΦ and ∂Φ .Φ , provide a first-principles law for how strain, gating, or weak SOC reshape FΦand HΦ(i.e., the device steering law). (ii) Higher Ward identities with proto-gauge bounds. 2-gauge invariance of (29) under (14) – (15) , together with the variable-structure Bianchi identities (cf. (16) – (17) ), yields higher Ward identities that connect level-1 and level-2 currents: t† Φ Jtot A=D∗ A Jtot B(exchange; cf. (22)),(30) D∗ A Jtot A= (.Φ)† B, Jtot B(divergence; cf. (23)).(31) Physically, the exchange law (30) states that any apparent source/sink of level-1 current is exactly supplied by a divergence of level-2 current, while the divergence law (31) states that level-1 conservation holds up to a B-twist set by the alternation grammar. Defining the proto-gauge defect (cf. (10)) ε= max k∈TdkFΦ(k)k/Λ2,kHΦ(k)k/Λ3,kDΦ(k)k/Λ,(32) we derive L2(Td)bounds (cf. (24)–(25)) kt† ΦJtot Ak ≤ C1εkJtot Ak+kJtot Bk,(33) kD∗ AJtot Ak ≤ C2εkJtot Bk+kBkkJtot Bk,(34) that quantify robustness: symmetry-forbidden channels and parasitics are O ( ε ). Thus ε plays a dual role: it is a quality metric (QA number for fabrication) and an analog control dial—small reversible δ Φ modulate responses linearly in ε, enabling hysteresis-free gain control. (iii) Methods (main-text) — band-to-device pipeline. We provide a reproducible pipeline linking first-principles or spectroscopic Berry data to device-level metrics: 1. Spin-resolved Berry and Fspin :from A↑ ( k ) , A↓ ( k )compute A = 1 2 ( A↑−A↓ )and Fspin = dA using periodic finite-difference curls on a rectangular grid.[18, 22] 2. Alternation grammar: choose ∆ ` ( k )from the crystal point group (odd parity) and set tΦ ( B ) = ∆`Bˆ n(collinear). 3. B-fit (fake-flat lift): solve the regularized least squares min BkFspin −tΦ(B)k2 L2(Td)+αkBk2, which in the scalar collinear ansatz reduces to tΦ(B) = γ∆`, where γ=hFspin,∆`i h∆`,∆`i.(35) Export FΦ=Fspin −tΦ(B), the oddness score, and ε. 4. Ward-cone QA: for a control sweep (strain or gate), estimate ε and overlay forbidden-channel amplitudes against straight-line bounds of slope set by C1,2in (33)–(34). 5. Actuator maps: compute sensitivity proxies from ∂ΦtΦ and fitted B to guide strain axes/gate polarity. For band interpolation and Berry curvature, we follow established Wannierization/modern-Berry approaches.[19–22] (iv) Spintronic use-cases (Φ-wall router, analog gain). Using the same action (29) , spatial gradients of Φ( x )generate Φ-walls described by thin-wall limits of the variable-structure Bianchi identities, and mixed topological couplings in Stop produce sheet conductance steps along the wall, ∆σwall =e2 2πh κ∆f, (36) where ∆ f encodes the jump in the topological index ω3 (Φ) across the wall (charge/spin/thermal analogues depending on the background).[23, 151] This yields programmable, magnetically silent interconnects (Φwall router). Separately, small reversible δ Φ(gate/SOC/strain) operate in the proto-gauge window ( ε 1) and provide hysteresis-free analog gain, with parasitics guaranteed ∝εby (33)–(34). 7 (v) Figures. We provide seven figures that illustrate each step: •Figure 1:Fspin,tΦ(B),FΦheatmaps with (γ, ε). •Figure 2:εvs control; forbidden-channel vs εwith Ward cones. •Figure 3:Φ-wall sheet step vs Twith baseline ±O(ε)band (cf. (36)). •Figure 4: reconfigurable routing matrix (rails ×walls). •Figure 5: analog gain G(Vg);ε(Vg); forbidden vs Vgwith Ward band. •Figure 6: quantized pumping under cyclic Φ(t)(plateaus ±O(ε)). •Figure 7: sensitivity & utility maps to guide actuators. Together, these contributions constitute a closed-loop, predictive framework in which higher categorical coherence is not merely interpretive but engineering-grade: it yields steering laws, robustness guarantees, QA metrics, and device primitives. II. KINEMATICS ON THE BRILLOUIN ZONE: 2-CONNECTION AND ALTERNATION GRAMMAR A. Notation and crossed-module Brillouin torus and differential-form conventions. Let Λ ⊂Rd be a direct lattice with reciprocal lattice Λ∗and fundamental reciprocal vectors {bi}d i=1. The Brillouin zone (BZ) is the d-torus Td∼ =Rd/ 2πΛ∗,k∼k+ 2πG,G∈Λ∗,(37) equipped with global angular coordinates ki∈ [ −π, π )on each cycle.[ 25 ] We write Ω p ( Td,g )for g -valued differential p -forms on Td , and use the standard wedge product with the convention that Lie brackets are absorbed in the product of g-valued forms.[26, 224] In local components, A=Aa i(k)Tadki∈Ω1(Td, g0), B =1 2Bα ij (k)Sαdki∧dkj∈Ω2(Td, g1),(38) where {Ta} and {Sα} are fixed bases of the real finite-dimensional Lie algebras g0 and g1 , respectively. Throughout we sum over repeated indices and write antisymmetrization with square brackets, e.g. X[ij]=1 2(Xij −Xji). Crossed-module and the rightwards-triangle action. The kinematic data are organized by a crossedmodule of Lie algebras. We work with strict crossed-modules for clarity. Semistrict/L ∞ variants may be adopted with the obvious homotopy-coherent corrections. g1 tΦ −−→ g0, .Φ,(39) selected pointwise in k by a structure field Φ : Td→ M2grp taking values in the moduli of 2-group structures. The map tΦ : g1→g0 is a Lie algebra morphism, and .Φ is an action of g0 on g1 by derivations. In components, tΦ(Sα) = ta ΦαTa, Ta.ΦSβ=ργ Φaβ Sγ,(40) obeying the crossed-module (Peiffer) identities tΦ x .Φy= [x, tΦ(y)], tΦ(y1).Φy2= [y1, y2],(41) for all x∈g0 and y, y1, y2∈g1 .[ 29 , 218 ] Equation (41) ensures that the 2-form sector coherently repairs any failure of strict level-1 gluing, which is the essence of our coherence-first viewpoint. 8 2-connection on Td .A2-connection on Td is a pair ( A, B ) ∈ Ω 1 ( Td, g0 ) × Ω 2 ( Td, g1 )with curvatures. For clarity we suppress any explicit dependence of ( tΦ, .Φ )on Φ( k )at this stage; see §II B for the variable-structure Bianchi identities. FΦ=dA +A∧A−tΦ(B)∈Ω2(Td, g0),(42) HΦ=dB +A .ΦB∈Ω3(Td, g1).(43) In local components, Fa Φij =∂iAa j−∂jAa i+fa bc Ab iAc j−ta ΦαBα ij ,(44) Hα Φijk =∂[iBα jk]+ρα Φaβ Aa [iBβ jk],(45) where [ Tb, Tc ] = fa bc Ta . The wedge product A∧A is understood with the Lie bracket in g0 , while A .ΦB uses the rightwards-triangle action .Φas in (40). Gauge structure. Let ε∈ Ω 0 ( Td, g0 )and λ∈ Ω 1 ( Td, g1 )be infinitesimal 0and 1-gauge parameters. The 2-gauge transformations act by δεA=DAε:= dε + [A, ε], δεB=ε .ΦB, (46) δλA=−tΦ(λ), δλB=DAλ:= dλ +A .Φλ, (47) which close under (41) . The associated covariant exterior derivative DA satisfies D2 Ax = [ FΦ, x ]on g0 -valued forms and D2 Ay = FΦ.Φy on g1 -valued forms, reflecting the higher-level parallel transport structure.[30, 31] Inner products and adjoints. To formulate dynamics (see §III C), we fix g0 - and g1 -invariant bilinear forms κ0 and κ1 and a flat Riemannian metric G on Td . For α, β ∈ Ω p ( Td, g0 )and u, v ∈ Ω q ( Td, g1 ) define hα, βi0:= ZTd κ0(α, ?β),hu, vi1:= ZTd κ1(u, ?v),(48) where ?is the Hodge star of (Td, G). The formal adjoints D∗ A,t† Φand (.Φ)†are defined by hDAα, βi0=hα, D∗ Aβi0,htΦ(u), βi0=hu, t† Φ(β)i1,hA .Φu, vi1=hA, (.Φ)†(u, v)i0.(49) These enter the Euler–Lagrange equations and the higher Ward identities; note the braces in D∗ A emphasize the correct L A T EX usage. Collinear and non-collinear specializations. For collinear altermagnets one may take g0≃u(1) ˆ n, g1≃R, tΦ(B)=∆`(k)B(k)ˆ n,(50) with ∆ ` ( k )the (odd) alternation grammar discussed in §II C and ˆ n the spin quantization axis (constant or slowly varying). In collinear systems the rightwards-triangle action reduces to multiplication by the U(1) charge along ˆ n. In contrast, for non-collinear textures, g0≃su(2), g1≃R3(adjoint), tΦ(b)=∆`(k)b·σ 2,(51) and .Φ is the adjoint action, with FΦ and HΦ given by (42) – (43) . These two cases cover the principal regimes relevant for altermagnets and their spintronic use-cases. Kinematic interpretation. The field A is the spin-frame connection in momentum space; for collinear systems it equals the difference of spin-resolved Berry connections, A = 1 2 ( A↑−A↓ ), so Fspin = dA encodes the raw spin curvature (cf. (4) ). The field B is a 2-level compensator that carries the symmetry-mandated, magnetically silent alternation; subtracting tΦ ( B )from Fspin yields the fake curvature FΦ which quantifies the operative local spin splitting (cf. (5) ). The global neutrality constraint RTdFΦ = 0 (cf. (9) ) is then a statement about the alternating character of tΦ ( B ), and hence about the Φ-selected grammar on the BZ. The remainder of Section II develops the variable-structure Bianchi identities for (42) – (43) , introduces the alternation grammar ∆ ` ( k )and its implementation via tΦ , and sets up the dynamical action and Ward identities that will underpin our control and robustness analysis. 9 B. Variable-structure Bianchi identities Problem statement. When the crossed-module structure maps ( tΦ, .Φ )depend on the structure field Φ( k ), the usual Bianchi identities for the 2-connection curvatures (42) – (43) acquire additional structural source terms that are linear in D Φ. These terms account for the fact that parallel transport in momentum space is performed in a varying 2-group background. In this subsection we derive the resulting variablestructure Bianchi identities in a manifestly 2-gauge covariant form and clarify sign and covariance conventions. Differentials of the structure maps. Let M2grp be a smooth parameter manifold of admissible 2-group structures. For Φ : Td→M2grp, the differentials of tΦand .Φat a point Φ(k)are multilinear maps (dt)Φ:TΦM2grp ×g1−→ g0,(d.)Φ:TΦM2grp ×g0×g1−→ g1.(52) Pulling back along D Φ ∈ Ω 1 ( Td, TΦM2grp )yields g0 - and g1 -valued one-form–valued bilinear maps on fields: (dt)Φ(DΦ) ∧B∈Ω3(Td, g0),(53) (d.)Φ(DΦ) ·(A, B)∈Ω4(Td, g1).(54) In local coordinates {ur}on M2grp, (dt)Φ(DΦ) ∧Ba ijk =∂t a Φα ∂ur(DΦ)r [iBα jk],(55) (d.)Φ(DΦ) ·(A, B)α ijkl =∂ρ α Φaβ ∂ur(DΦ)r [iAa jBβ kl].(56) These objects transform covariantly under 2-gauge transformations because tΦ and .Φ satisfy the Peiffer identities and their differentials inherit the corresponding equivariance (see below). Sign convention. To match the concise identities stated in the introduction, we adopt the convention that the differential (dt)Φ(DΦ) is defined with the sign such that DAtΦ(B)=tΦ(DAB)+(dt)Φ(DΦ) ∧B, (57) and similarly for .Φ (cf. (59) below). This convention ensures the final Bianchi identities appear with the positive structural source terms as in (16)–(17). Leibniz rules with variable structure. The covariant derivative DA acts on g0 -valued forms by the adjoint action and on g1-valued forms through .Φ. Allowing Φ-dependence, we have the Leibniz rules DAtΦ(B)=tΦ(DAB)+(dt)Φ(DΦ) ∧B, (58) DAA .ΦB= (DAA).ΦB−A .Φ(DAB)+(d.)Φ(DΦ) ·(A, B).(59) Equation (58) follows from the chain rule for the Φ-dependent linear map tΦ together with DA -equivariance (Peiffer identity t ( x . y ) = [ x, t ( y )]), while (59) uses the derivation property of .Φ and its Φ-derivative (d.)Φ.[41, 42] Derivation of the first Bianchi identity. Starting from FΦ = dA + A∧A−tΦ ( B ), we take the covariant derivative: DAFΦ=DA(dA +A∧A)−DAtΦ(B) = 0 −tΦ(DAB)+(dt)Φ(DΦ) ∧B,(60) where the standard level-1 Bianchi identity DA ( dA + A∧A ) = 0 has been used. Since HΦ = DAB by (43), we obtain DAFΦ+tΦ(HΦ) = (dt)Φ(DΦ) ∧B, (61) which is the claimed identity (16) . The right-hand side measures the failure of the usual 2-Bianchi identity induced solely by the variation of the embedding tΦalong DΦ. 16 Variation of the kinetic sector and integration by parts. The kinetic density of (89) is 1 2g2hFΦ, FΦi0,dens + 1 2h2hHΦ, HΦi1,dens + κ 2GΦ ( D Φ , D Φ) . A standard calculation with the adjoints (49) on the torus Td (no boundary terms) yields δSkin =1 g2FΦ, δFΦ0+1 h2HΦ, δHΦ1+κ 2ZTd δGΦ(DΦ, DΦ) =δA, 1 g2D∗ AFΦ−1 h2(.Φ)† B, HΦ | {z } =:EA0+δB, 1 h2D∗ AHΦ−1 g2t† Φ(FΦ) | {z } =:EB1 +1 g2ZTdhFΦ,(∂ΦtΦ)·(δΦ, B)i0,dens −1 h2ZTdhHΦ,(∂Φ .Φ)·(δΦ; A, B)i1,dens | {z } Φ-sources (see (101)) +κ 2ZTd δGΦ(DΦ, DΦ). (96) The index-free Euler–Lagrange operators for (A, B)are thus EA=1 g2D∗ AFΦ−1 h2(.Φ)†(B, HΦ),(97) EB=1 h2D∗ AHΦ−1 g2t† Φ(FΦ),(98) in agreement with the compact expressions announced in §I B (cf. (19)–(20)). Density (Hodge) form. If one prefers to exhibit the Hodge star explicitly, write the densities underlying the inner products and move ? to the fields so that adjoints act without ? . Then (97) – (98) become the equivalent “density” relations E(dens) A=1 g2D∗ A(?FΦ)−1 h2(.Φ)† B, ?HΦ,(99) E(dens) B=1 h2D∗ A(?HΦ)−1 g2t† Φ(?FΦ),(100) which matches the “ ? -decorated” form often used in component computations and aligns with the schematic presented in the subsection header. Equations (97) – (100) are simply two notational renditions of the same variational content. Φ-equation and explicit sources (actuation law). Collecting the δ Φterms in (96) and performing the standard variation of the metric term gives the functional derivative in the Φ-sector: δS δΦ⊃1 g2DDFΦ,(∂ΦtΦ)·(·, B)EE0 | {z } embedding source −1 h2DDHΦ,(∂Φ .Φ)·(·;A, B)EE1 | {z } action source +κDivΦ GΦDΦ,(101) which is the precise version of the steering sources summarized previously (cf. (93) ). Signs in (101) depend on the convention adopted in the covariantized Leibniz rules; we fix them to match (93) . Any alternative, consistent convention produces the same physical law after redefining ( ∂ΦtΦ )or ( ∂Φ .Φ )by a global minus sign. Equation (101) makes explicit how small, reversible deformations δ Φ(strain, gating, weak SOC) linearly reshape the curvatures FΦ and HΦ and hence the observable response tensors, while the stiffness term κ GΦpenalizes rapid variations across Td. Equations of motion and currents. In the presence of matter or topological couplings, the total action is Stot = S [ A, B, Φ] + Smatter [ A, B, Φ] + Stop [Φ; A, B ]. Defining total currents by δSmatter + δStop = hhδA, Jtot Aii0+hhδB, Jtot Bii1+hδΦ,Jtot Φi, the Euler–Lagrange equations read EA=Jtot A, EB=Jtot B,δS δΦ=Jtot Φ,(102) with EA, EB given by (97) – (98) (or by (99) – (100) in density form). Section I B shows that 2-gauge invariance of Stot constrains these currents by the higher Ward identities (exchange and divergence forms), which in turn underlie the proto-gauge bounds used for QA and control. 17 Gauge covariance of the EL operators. Under 0and 1-gauge transformations with fixed Φ(cf. (46) – (47) ), FΦ and HΦ transform covariantly. Because the inner products are adjoint-invariant and D∗ A , t† Φ , ( .Φ ) † are defined as the corresponding formal adjoints, EA (resp. EB ) transforms in the adjoint of g0 (resp. as a g1-vector). Therefore the equations of motion (102) are 2-gauge covariant. Linearized control law (proto-gauge regime). To first order in small, reversible δ Φone has, from (94)–(95), δΦFΦ=−(∂ΦtΦ)·(δΦ, B) + higher orders,(103) δΦHΦ= (∂Φ .Φ)·(δΦ; A, B) + higher orders,(104) and, consequently, the induced change in the level-1 and level-2 currents is bounded (by Cauchy–Schwarz in (48) ) proportionally to the proto-gauge defect ε = maxkkFΦk/ Λ 2,kHΦk/ Λ 3,kD Φ k/ Λ  (cf. (32) ). This provides the promised linear dial for analog control with Ward-bounded parasitics. Component expressions (for numerics). Writing Fa Φij and Hα Φijk as in (44) – (45) and denoting the Td metric by Gij with volume form vol, the density forms (99)–(100) read (EA)am=1 g2Di(Fa Φim)−1 h2ρα Φbβ Bβ mi Hi a Φα,(105) (EB)α mn =1 h2Di(Hα Φimn)−1 g2ta ΦβFΦmn a Gβα,(106) where Di := GijDj and Gβα is the metric induced by κ1 on g1 ; indices are raised with Gij and κ0,1 . These formulas are convenient for finite-difference implementations on a uniform k-grid. Summary. The Euler–Lagrange operators EA, EB(97) – (98) encode the competition between level-1 and level-2 curvature budgets; the Φ-sources (101) provide the precise steering terms by which actuators modify the fake curvature FΦ and the 2-curvature HΦ ; and the component expressions (105) – (106) furnish a direct route to numerics. Together with the higher Ward identities (next subsection), these results yield both predictive control laws and robustness bounds (proto-gauge) used in our spintronic implementations. C. Higher Ward identities (Noether–II) Aim and overview. We now derive the higher Ward identities implied by the 2–gauge invariance of the coherence action (89) . These identities hold off shell (Noether’s second theorem) as functional relations among the Euler–Lagrange (EL) operators EA, EB of (97) – (98) , and, when currents are present, they yield the exchange and divergence laws quoted in §I B, Eqs. (22) – (23) . Physically, they state that an apparent “non–conservation” of level–1 current is supplied by a flow in the 2–form sector, and that level–1 conservation holds up to a B –twist determined by the alternation grammar—precisely the magnetically silent redistribution central to altermagnetism. 2–gauge variations and functional identities. Recall the 0– and 1– gauge transformations with fixed Φ (cf. (46)–(47)): δεA=DAε, δεB=ε .ΦB, (107) δλA=−tΦ(λ), δλB=DAλ. (108) The total action Stot [ A, B, Φ] := S [ A, B, Φ]+ Smatter [ A, B, Φ]+ Stop [Φ; A, B ]is invariant under (107) – (108) (up to integral periods in Stop , which drop out on the torus). We first derive the Noether–II identities for the kinetic part and then state the form with currents. (a) 0–gauge. Using (96) with δA = DAε and δB = ε .ΦB , integration by parts and the adjoint identities (49) give δεSkin =EA, DAε0+EB, ε .ΦB1=−D∗ AEA, ε0+(.Φ)†(B, EB), ε0.(109) Since εis arbitrary and Skin is invariant, we obtain the first off–shell Noether–II identity D∗ AEA−(.Φ)†(B, EB)≡0.(110) (b) 1–gauge. Similarly, with δA =−tΦ(λ)and δB =DAλ, δλSkin =−EA, tΦ(λ)0+EB, DAλ1=−t† Φ(EA), λ1−D∗ AEB, λ1,(111) 18 whence, by arbitrariness of λand invariance, t† Φ(EA)−D∗ AEB≡0.(112) Equations (110)–(112) are the off–shell functional identities promised in the subsection header. Inclusion of currents and the Ward laws. Let the matter+topological sector define total currents by δSmatter +Stop=δA, Jtot A0+δB, Jtot B1+hδΦ,Jtot Φi.(113) Gauge invariance of Stot under (107) – (108) is equivalent to the on–shell replacement ( EA, EB ) 7→ (Jtot A, Jtot B)in (110)–(112), leading to the exchange and divergence forms of the higher Ward identities: t† Φ Jtot A=D∗ A Jtot B,(exchange law, cf. (22)),(114) D∗ A Jtot A= (.Φ)† B, Jtot B,(divergence law, cf. (23)).(115) The first says that any apparent source in level–1 is exactly supplied by the divergence of level–2; the second says that level–1 conservation is preserved up to a B –mediated twist reflecting the alternation grammar. Relation to variable–structure Bianchi identities. The derivation above used only 2–gauge invariance with Φfixed. The covariance of (110) – (112) under changes of Φ(actuation) follows from the variable– structure Bianchi identities (61) – (63) : the structural source terms ( dt ) Φ ( D Φ) ∧B and ( d. ) Φ ( D Φ) · ( A, B ) that modify DAFΦ and DAHΦ appear in precisely the combinations that leave (110) – (112) 2–gauge covariant. In particular, the Ward laws (114)–(115) remain valid for slowly varying Φ(k). Integrated constraints on the torus. Integrating (115) over Td and using periodic boundary conditions (no boundary terms) yields ZTd D∗ AJtot A=ZTd (.Φ)†(B, Jtot B).(116) In the collinear abelian case ( g0 = u (1), trivial .Φ ), the right–hand side vanishes, giving global level–1 charge conservation on the BZ; locally, however, the B –twist term in (115) encodes the magnetically silent re–distribution mandated by the alternation grammar. This is exactly the mechanism by which altermagnets support strong local splitting (large FΦ(k)) with zero–Mglobally (cf. §II D). Bounds and the proto–gauge regime. Combining (114) – (115) with Cauchy–Schwarz in the inner products (48) and the definition of the proto–gauge defect ε (Eq. (32) ), one obtains the quantitative bounds kt† ΦJtot Ak ≤ C1εkJtot Ak+kJtot Bk,kD∗ AJtot Ak ≤ C2εkJtot Bk+kBkkJtot Bk,(117) which coincide with (33) – (34) . They formalize the statement that symmetry–forbidden channels and parasitics are O ( ε ): in the proto–gauge window ( ε 1), responses are linearly controllable with bounded side–leakage. BRST/BV perspective (cohomological form). The Noether–II identities (110) – (112) can be packaged cohomologically via BRST or BV formalisms. Introduce a 0–form ghost c (in g0 ) and a 1–form ghost η (in g1 ), with the BRST differential s implementing (107) – (108) and their algebra; the master equation sStot = 0 generates the tower of Noether identities, whose lowest components reproduce (110) – (112) . In BV language, the Koszul–Tate differential resolves the EL equations ( EA, EB ), while the longitudinal BRST differential encodes 2–gauge symmetry; their anticommutation is equivalent to the higher Ward identities.[80, 81, 221, 222] Summary. Noether’s second theorem applied to the 2–gauge symmetry of the coherence action yields the functional identities (110) – (112) , which, in the presence of currents, become the higher Ward laws (114) – (115) . These laws are 2–gauge covariant (compatible with the variable–structure Bianchi identities), supply a clear physical narrative (level–1 non–conservation is supplied by level–2 exchange; level–1 conservation holds up to a B –twist), and, together with the proto–gauge defect ε , lead directly to the quantitative bounds used throughout our spintronic analysis. 19 D. Proto-gauge regime and bounds Definition and scaling. We quantify proximity to the coherence fixed point ( FΦ = HΦ = D Φ = 0) by a single, dimensionless proto-gauge defect ε:= max k∈TdkFΦ(k)k/Λ2,kHΦ(k)k/Λ3,kDΦ(k)k/Λ,(118) where Λis a momentum scale set by the Brillouin-zone geometry (e.g., the inverse of the shortest real-space lattice spacing, or simply Λ = π in the normalized units of Sec. II). The exponents reflect form degrees: FΦ is a 2-form, HΦ a 3-form, and D Φa 1-form on Td ; the normalization renders ε invariant under rescalings of the BZ metric and consistent with the quadratic action (89) . In practice we compute ε on a uniform k -grid by sup norms discretized with periodic finite differences (Sec. V); the choice of Λonly sets an overall conversion factor. Ward structure at small defect. The higher Ward identities derived in Sec. III C read, for the total currents from the matter+topological sector, t† Φ Jtot A=D∗ A Jtot B,(119) D∗ A Jtot A= (.Φ)† B, Jtot B.(120) Near the coherence fixed point, first-order expansions of D∗ A and ( .Φ ) † around a flat reference ( A0, B0, Φ 0 ) show that their deviations are Lipschitz in (FΦ, HΦ, DΦ); hence, on Td, the operator norms satisfy kD∗ A−D∗ A0kop +k(.Φ)†−(.Φ0)†kop ≤Cgeom ε, (121) with Cgeom depending only on the chosen inner products (Sec. II A) and on uniform bounds for A, B on Td . The estimate follows from standard Sobolev/elliptic bounds on the torus and continuity of bilinear pairings; see, e.g., [ 82 – 84 ]. Substituting (121) into (119) – (120) yields quantitative, linear control of the Ward relations by ε. Forbidden channels and the Ward cone. Let Hallow denote the L2 ( Td )subspace spanned by grammar harmonics { ∆ `} (Sec. II C), and let Hforb = H⊥ allow be its orthogonal complement with respect to the BZ inner product hf, gi = RTdf ( k ) g ( k ) ddk . Denote by Pforb the orthogonal projector onto Hforb (componentwise for matrix/tensor currents). In an ideal altermagnet ( ε = 0), symmetry forces PforbJtot A = 0. At small but finite ε, we obtain the following bounds. Proposition 3.4.1 (Proto-gauge bounds; Ward cone). Assume ( tΦ, .Φ )and the BZ inner products satisfy (49) , and let σmin be the smallest singular value of t† Φ restricted to Hforb (evaluated at the reference Φ 0 ). Then there exist constants 0 < Clo ≤Chi depending on ( σmin, Cgeom )and the BZ geometry such that for all sufficiently small ε Clo εS ≤ PforbJtot A≤Chi εS,S:= kJtot Ak+kJtot Bk.(122) Sketch. Project (119) to Hforb and use the restricted coercivity of t† Φ on Hforb to control kPforbJtot Ak by kD∗ AJtot Bk . Estimate the latter by kD∗ A0Jtot Bk + k ( D∗ A−D∗ A0 ) Jtot Bk , apply Cauchy–Schwarz and (121) , and absorb constants. The lower bound follows similarly by reversing the coercivity estimate. Inequality (122) is exactly the Ward cone used in the QA plot of Fig. 2 (‘fig2_epsilon_and_ward_cone.png’): the measured forbidden amplitude (vertical axis) must lie between two straight lines of slope ( Clo, Chi ) times ε (horizontal axis). Failure to lie within the cone signals either (i) misestimation of ε , (ii) an unaccounted control channel, or (iii) a breakdown of the proto-gauge assumption (large defect). From functional to pointwise bounds. The L2 ( Td )norm in (122) controls pointwise amplitudes via Sobolev embeddings on the torus. Denote by k·kHsthe Sobolev norm and let s > d/2. Then kPforbJtot AkL∞≤CSob kPforbJtot AkHs≤C0 Sob εSs,Ss:= kJtot AkHs+kJtot BkHs,(123) with CSob, C0 Sob depending only on ( s, d )and the BZ metric [ 83 , 84 ]. Thus the linear, Ward-cone control propagates to local observables provided the currents have finite Hs norm (true for smooth fields built from bands). 20 Bounds for actuator response (linear dial). Linearizing Eqs. (103) – (104) (Sec. III B) and composing with the constitutive maps that define the measured response R (e.g., spin Hall tensor, wall sheet step), one obtains δΦR=LΦ[δΦ] + O(εkδΦk),kLΦk ≤ Cact ε, (124) so that the gain is linear in the applied control δ Φwith slope bounded by Cactε . This is the formal statement behind the hysteresis-free, analog behavior in Fig. 5 (‘fig5_gain_and_ward_vs_gate.png’): tuning in the proto-gauge window ( ε 1) guarantees predictable, linear response with Ward-bounded parasitics. Choice and estimation of the cone slopes. In applications we estimate ( Clo, Chi )from operator norms on the grammar/forbidden split: Chi ≈(t† Φ0|Hforb )−1op kD∗ A0kop +Cgeom, Clo ≈σmint† Φ0|Hforb .kD∗ A0kop +Cgeom,(125) where kD∗ A0kop is fixed by the BZ metric and A0 (Poincaré/Friedrichs constants on Td [ 84 , 85 ]). In practice, we compute (125) numerically on the k -grid used for the B -fit, or we calibrate ( Clo, Chi )from a small training sweep and then lock them as device specifications (Fig. 2). Error control for the B -fit and stability of ε .The B -fit of Sec. II D is a linear inverse problem; with Tikhonov regularization it admits stability bounds kBα−Bk ≤ κTik(α)kFα spin −Fspink+ noise, εα=ε+OkBα−Bk,(126) where α is the regularization parameter and κTik ( α )a known condition factor [ 86 , 87 ]. Hence the measured defect εα is stable under moderate noise and grid coarsening; this justifies its use as a one-number QA spec and as the horizontal axis in the Ward-cone plot. Lower and upper a priori bounds from the action. The kinetic part of (89) implies 1 2g2kFΦk2+1 2h2kHΦk2+κ 2kDΦk2≤S[A, B, Φ] ≤S0,(127) whence ε≤C0 act√S0 with a geometry constant C0 act . Thus any fabrication or bias window that maintains a bounded action budget automatically stays in the proto-gauge regime; conversely, Ward-cone violations flag excursions in Sbeyond the specified envelope. Interpretation and use. Equations (122) – (124) make precise the engineering mantra: operate at small, tunable ε . Then (i) symmetry-forbidden channels and parasitics are O ( ε )(Ward cone), (ii) device gains are linear in the applied control (proto-gauge dial), and (iii) neutrality is preserved by construction (odd grammar; Sec. II D). This is why ε is both a quality metric (QA number) and a control knob (analog dial). IV. ΦAS A CONTROL FIELD; INTERFACES AND TOPOLOGICAL STEPS A. Steering by Φ Actuator law from the action. The coherence action (89) contains the explicit Φ–dependence of the crossed–module maps ( tΦ, .Φ ), and its variation yields the actuation (source) law in the Φ–sector, cf. (101): δS δΦ⊃1 g2DDFΦ,(∂ΦtΦ)·(·, B)EE0 | {z } embedding source −1 h2DDHΦ,(∂Φ .Φ)·(·;A, B)EE1 | {z } action source +κDivΦ GΦDΦ.(128) Infinitesimal control δΦtherefore induces the linear curvature response (cf. (103)–(104)) δΦFΦ=−(∂ΦtΦ)·(δΦ, B), δΦHΦ= (∂Φ .Φ)·(δΦ; A, B),(129) which is the precise steering rule: actuators that change Φreshape the fake curvature FΦ (and HΦ ) while the global neutrality RTdFΦ= 0 is preserved by the odd grammar in tΦ(B)(§II D). 21 Control channels and their physical meaning. In practice we use three complementary Φ–channels: •Strain (elastic) control: symmetry is softly deformed, mixing grammar components ∆ ` ( k ) through strain–representation couplings. At leading order, if εij is the strain tensor, ∂ΦtΦ X `m C(`m) ij εij ∆m(k)ˆ n, δΦFΦ=−X `m C(`m) ij εij ∆mBˆ n,(130) with symmetry–restricted tensors C(`m) ij set by the crystal. Equation (130) is the momentum–space analogue of magnetoelastic or deformation–potential couplings; see, e.g., [ 93 – 95 ] for representative control platforms. •AFM domain patterning ( Φ –textures): spatial variation Φ( x )implements Φ–walls that act as reconfigurable interfaces (Sec. IV B). In the bulk, small texture deformations provide local steering by (129); at the wall, mixed topological inflow governs quantized sheet responses (Sec. III A). •Gating / SOC control: electrostatic gating tunes inversion asymmetry and interfacial SOC parameters (e.g., Rashba αR); to leading order, ∂ΦtΦ ∂tΦ ∂αR δαR, ∂Φ .Φ ∂.Φ ∂αR δαR,(131) so that δVg(gate) or interfacial chemistry modulates FΦand HΦthrough δαR.[88–90] From curvature steering to observables. For concreteness, consider two response families: 1. Spin Hall conductivity σs ij in a collinear altermagnet (spin along ˆ n ). In a Berry–phase formulation one may write schematically σs ij[Φ] = e ~X n∈occ ZTd ddk (2π)dWn,ij(k;µ, T ) trˆ nFΦ(k) | {z } spin–weighted curvature ,(132) with Wn,ij a smooth band/thermodynamic weight (Kubo kernel). Using (129), δΦσs ij =−e ~X nZddk (2π)dWn,ij trˆ n[(∂ΦtΦ)·(δΦ, B)] + O(εkδΦk),(133) i.e. a linear Φ–dial with slope bounded by the proto–gauge defect ε(Sec. III D). 2. Anomalous Nernst / spin Nernst tensors αij obtained via the Mott/thermal relation from σij : (Rigorous Kubo derivations relate αij to energy–derivatives of Hall kernels; see [91, 92]) αij(µ, T )≈ − π2k2 BT 3e ∂σij ∂µ ⇒δΦαij ≈ − π2k2 BT 3e ∂ ∂µ δΦσij.(134) Thus the linear actuator law (133) directly propagates to thermal analogues. In all cases, neutrality RTdFΦ = 0 is preserved by odd grammar (Sec. II C), so steering redistributes polarization in k–space without creating macroscopic M. Design maps and optimal steering. Given a target observable R with functional derivative δR/δFΦ = KR(k), the actuator susceptibility is SR[Φ; δΦ] := δΦR=DKR, δΦFΦE0+De KR, δΦHΦE1=−DKR,(∂ΦtΦ)·(δΦ, B)E0+De KR,(∂Φ .Φ)·(δΦ; A, B)E1, (135) a bilinear form in δ Φthat defines a control metric. The design map displayed in Fig. 7 corresponds to the pointwise sensitivity S(k) := ∂ΦtΦ·B(k)and U(k) := S(k)FΦ(k),(136) which highlight where small δ Φmost effectively re–sculpt FΦ (sensitivity) and where that effort is most useful (utility). A constrained optimal steering problem, maximize δΦSR[Φ; δΦ] subject to kδΦkGΦ≤Φmax, ε ≤εmax,(137) has the closed–form solution δ Φ ?∝G−1 Φ·JR where JR is the Riesz representative of the linear functional in (135) ; it is implemented numerically by a Moore–Penrose pseudoinverse of ( ∂ΦtΦ )restricted to the grammar subspace. 22 Strain actuators. In devices we use global piezoelectric stacks or local micro–piezos to apply εij , which shifts irreps and mixes low–order harmonics according to (130) . The response is reversible and hysteresis–free in the proto–gauge window ( ε 1), and is naturally calibrated by the Ward cone (Fig. 2), since forbidden harmonic leakage scales as O ( ε )(§III D). Materials and platform examples include strain– tunable correlated oxides and metals, as well as piezo–composite heterostructures enabling in–plane and out–of–plane stress control [93–95]. Gating and SOC actuators. Electrostatic gating modulates interfacial electric fields and hence inversion asymmetry, tuning the Rashba coupling αR and related SOC parameters. In 2D electron gases and oxide interfaces, αR is linearly gate–tunable over wide ranges, providing a clean Φ–channel in (131) [ 88 – 90 ]. In our formalism, this appears as a controlled change of the embedding tΦ and/or the action .Φ , which redistributes the curvature budget between levels and hence adjusts σs ij and αij per (133) – (134) , while neutrality remains guaranteed by odd grammar. AFM domain patterning and reconfigurable routing. Writing Φ( x )patterns the alternation grammar in real space, creating Φ–walls where mixed topological terms from Stop localize interfacial responses (Sec. III A). In the bulk domains, steering follows (129) ; at the wall, the sheet conductance step is quantized (Fig. 3), providing robust, reconfigurable spin routing channels that remain magnetically silent. Bounded parasitics and QA. Because the higher Ward identities constrain current exchange between levels, and operator deviations are O ( ε )(§III D), all symmetry–forbidden channels and parasitics scale linearly in the proto–gauge defect. Hence the single number ε serves simultaneously as a quality metric (QA spec) and an analog dial for gain control. Ward–cone compliance (Fig. 2) is thus both a design rule and a pass/fail criterion. Summary. Steering by Φprovides a first–principles control law for altermagnets: strain, AFM domain textures, and gating/SOC enter through ( ∂ΦtΦ )and ( ∂Φ .Φ )to reshape FΦ and HΦ linearly, delivering predictable changes in spin Hall and thermal analogues while preserving magnetic neutrality. The proto–gauge framework guarantees bounded parasitics (Ward cone), and the sensitivity/utility maps guide actuator placement and operating points. B. Φ-walls (rank/grammar jumps) and mixed terms Set–up: spatially varying Φand thin–wall limit. Let Φ( x )vary slowly in real space except across a codimension–1 interface (the Φ-wall)Σ = {u ( x )=0 } , with unit normal ˆ n = ∇u/|∇u| and signed coordinate u increasing from the “ − ” to the “+” side. We model a generic rank/grammar jump by a scalar diagnostic f(Φ) which is piecewise constant, f(Φ(x)) = f−for u < 0, f(Φ(x)) = f+for u > 0,∆f:= f+−f−∈Z,(138) and varies only within a thin layer around Σ(“thin–wall limit”). The integrality of ∆ f encodes that f arises from a differential character (e.g. a 3–cocycle) ω3 (Φ); across a wall, the pullback jumps by an integer in H3 ( ·,Z ), which is exactly the topological datum that will quantize the interfacial response [96–98]. Mixed topological sector and inflow. Consider the mixed topological term (a special case of (91)) Smix[Φ; A] = κ∗ 2πZMω3(Φ), d CS2(A), M =R1,2×Tdor a suitable spacetime–parameter product, (139) with CS2 ( A )a Chern–Simons transgression form of degree 2(for U (1), CS2 = A∧dA ; for nonabelian g0 , CS2 = tr ( A∧dA + 2 3A∧A∧A )). Integrating by parts in the thin–wall ansatz (and suppressing possible contributions from temporal boundaries) gives Smix =κ∗ 2πZM df(Φ) ∧CS2(A) = κ∗ 2π∆fZΣ CS2(A)≡SΣ[A].(140) Thus a spatial jump of the grammar index induces on the wall Σa(2+1)D Chern–Simons action for the level–1 gauge field A. Sheet Hall response from the wall action. Varying (140) with respect to A yields the conserved interfacial current jµ Σ=δSΣ/δAµ(indices µ, ν, ρ intrinsic to Σ): jµ Σ=κ∗ 2π∆f εµνρ ∂νAρ=κ∗ 4π∆f εµνρ Fνρ ,(141) 23 so that, in linear response to an in–plane electric field, the sheet Hall conductivity is ∆σ(sheet) H=e2 h κ∗ 2π∆f . (142) This reproduces Eq. (36) announced earlier and makes the quantization explicit: because ∆ f∈Z by (138) , and κ∗ is fixed by large–gauge invariance of Smix , the step is quantized in units of e2/h . For nonabelian g0 one reads off the appropriate normalization from the Dynkin index of the representation and the level of the nonabelian Chern–Simons term on Σ. The sign of the step is set by the wall orientation (the sign in df = ∆f δ(u)du) and by the time–reversal properties of the background. Jump conditions from the variable–structure Bianchi identities. Beyond the topological argument, the kinematic jump conditions follow from integrating the variable–structure Bianchi identities (61) – (63) across a thin slab transverse to Σ. With df = ∆ f δ ( u ) du and writing ιˆ n for interior product with the wall normal, one obtains schematically ιˆ nFΦ+ −+ιˆ ntΦ(B) + −= ∆f ιˆ n∂tΦ ∂f BΣ,(143) ιˆ nHΦ+ −=ιˆ n FΦ.ΦB + −+ ∆f ιˆ n∂.Φ ∂f ·(A, B)Σ.(144) Equations (143) – (144) encode how the alternation grammar reorganizes curvature budgets when crossing the wall and are the kinematic counterparts of the inflow captured by (140) . In particular, the first relation shows that a nontrivial ∆ f produces a delta–function inflow that is precisely balanced by the Chern–Simons wall current (141) (Callan–Harvey anomaly cancellation [98]). Rank changes and protected wall channels. If the rank of tΦ changes across the wall (e.g., a harmonic in the grammar ∆ ` turns on or off, or a sign inversion occurs in a block), the kernel/cokernel of tΦ changes accordingly. In the 2–gauge picture this produces protected, chiral (or helical) wall channels whose anomaly is cancelled by the inflow term (140) . A convenient diagnostic is the relative index ind ( tΦ+, tΦ− ) = dim ker tΦ+−dim ker tΦ−, which counts the net number of modes required on Σfor anomaly cancellation (the higher–gauge analogue of bulk–boundary correspondence) [99]. Spin and thermal analogues. Replacing the electromagnetic Chern–Simons functional on Σby its SU(2) (spin) or gravitational counterparts produces analogous quantized spin and thermal steps: S(s) Σ[A(s)] = ks 4πZΣ trA(s)∧dA(s)+2 3A(s)∧A(s)∧A(s),∆σ(s) H∝ks∆f, (145) S(th) Σ[ω] = c 96πZΣ trω∧dω +2 3ω∧ω∧ω,∆κxy T=π2k2 B 3h c 2∆f, (146) where A(s) is the spin connection in the relevant SU (2) subgroup (collinear reduction aligns it with ˆ n ), ω is the Levi–Civita spin connection, ks is an integer level, and c is the chiral central charge of gapless wall modes (or, equivalently, the coefficient of the gravitational Chern–Simons term) [ 100 , 101 ]. Equation (146) states the familiar quantization of the thermal Hall conductance per temperature, now multiplied by the integer ∆fthat counts grammar jumps. Compatibility with neutrality and higher Ward identities. Because the grammar current tΦ ( B )is odd on the Brillouin torus (Sec. II C), the bulk remains magnetically neutral: RTdFΦ = 0 on either side of the wall. The redistribution induced by ∆ f occurs only at Σand is captured by (141) . This picture dovetails with the higher Ward identities (Sec. III C): projecting the exchange law t† Φ ( Jtot A ) = D∗ AJtot B onto a pillbox around Σreproduces the inflow/outflow balance between bulk and wall currents, while the divergence form D∗ AJtot A = ( .Φ ) † ( B, Jtot B )explains why level–1 conservation holds up to the B –twist that changes rank at the wall. Network of Φ-walls and node rules. Intersections of walls (e.g., where two grammar jumps meet) carry lower–dimensional Chern–Simons descendants; e.g., line nodes in 3D pick up a (1 + 1)D Wess– Zumino–Witten term whose level is the product of the incoming ∆ f ’s (descent relations) [ 96 , 97 ]. In device architectures this provides network rules for routing spin/charge/heat without stray fields: the quantized sheet steps on each segment fix how currents partition at junctions. Summary. A spatial jump of the alternation grammar (Φ-wall) produces a Chern–Simons action on the wall, Eq. (140) , and, hence, a quantized sheet Hall step, Eq. (142) . The magnitude is fixed by the integer ∆ f that labels the grammar jump and by the level κ∗ . The kinematic jump conditions (143) – (144) encode how curvature budgets rearrange at the wall and guarantee anomaly inflow. Spin and thermal 24 analogues follow from SU (2) and gravitational Chern–Simons terms, Eqs. (145) – (146) . All of this is fully consistent with bulk neutrality and with the higher Ward structure that controls parasitics and robustness. C. Cyclic Φ(t)and pumping Set–up and statement. Let Φ( t )trace a smooth, closed loop in the structure–field manifold over a period T ,Φ(0) = Φ( T ), while ( A, B )follow adiabatically as time–dependent fields on the Brillouin torus Td . The parameter space is P := Td×S1 t , with local coordinates ( k, t )and S1 t = R/TZ . We show that a closed cycle of Φ( t )pumps quantized charge and spin, up to O ( ε )corrections governed by the proto–gauge defect (Sec. III D). This generalizes Thouless pumping to our higher–gauge setting and provides a metrological current standard (quanta per cycle) as well as a route to high–frequency (GHz–THz) coherent emission when the adiabatic gap permits. Adiabatic transport on P : mixed Berry curvature. For a set of occupied bands {|un ( k, t ) i} transported adiabatically along the loop, the standard adiabatic response expresses pumped charge along direction i as the integral of the mixed (t–ki) Berry curvature over P: Qi=e 2πX n∈occ ZT 0 dt ZTd ddk (2π)dF(n) tki(k, t),F(n) tki:= ∂tA(n) ki−∂kiA(n) t,(147) with A(n) λ = ihun|∂λuni [ 102 , 103 ]. In d = 1 this equals Q = e 2πRS1×S1F and is quantized by the first Chern number of the occupied bundle over P ; more generally, for even d the pumped sheet response is controlled by higher Chern characters over P[104]. Spin–weighted pump and the 2–connection. In our collinear altermagnetic setting, the spin–weighted Berry connection (spin along ˆ n ) is A = 1 2 ( A↑−A↓ ), so that the corresponding mixed curvature entering (147) is F(s) tki=∂tAki−∂kiAt. Decomposing with the 2–connection (Sec. II A), F(s) tki=FΦtki | {z } operative +tΦ(B)tki | {z } grammar , FΦ=dA +A∧A−tΦ(B),(148) and hence the spin pump per cycle is Q(s) i=e 2πZP (FΦ)tki+e 2πZPtΦ(B)tki.(149) For a loop within a fixed grammar class (i.e., tΦ ( B )odd in k for each t ), the second term vanishes exactly by inversion symmetry on Tdat each time slice and periodicity in t: ZPtΦ(B)tki=ZT 0 dt ∂tZTdtΦ(B)kiddk= 0,(150) so that Q(s) i=e 2πZP (FΦ)tki.(151) Thus, only the operative (fake) curvature contributes to the net spin pump; the grammar component reshuffles polarization in kwhile preserving the time–slice neutrality RTdtΦ(B)=0(Sec. II D). Quantization and topology on P .When the many–body gap remains open along the cycle and bands are fully occupied below the gap, the integral in (147) (or (151)) is quantized: Qi=eCh1Eocc → Pfor d= 1,∆σ(sheet) H=e2 hCh2Eocc → Pfor d= 2,(152) where Chr is the r th Chern number of the occupied bundle over P [ 104 ]. The 2–connection split (148) shows that quantization pertains to the fake part in altermagnets, with the grammar part integrating to zero by (150). In our proto–gauge regime, deviations from integrality are bounded: Qi−eCh1≤Cpump ε, ∆σ(sheet) H−e2 hCh2≤C0 pump ε, (153) with constants determined by operator norms as in Sec. III D. Equation (153) formalizes the statement that pumped quanta are topological up to O(ε). 25 Relation to wall inflow and Chern–Simons descent. Section IV B showed that a spatial jump of the grammar index yields a Chern–Simons wall action and a quantized sheet step. A cyclic Φ( t )that winds once around the same integer class induces, by descent, the corresponding Chern character over P and thus the same quantized unit per cycle, in precise analogy with Thouless pumps and axion electrodynamics [ 104 ]. In short: static Φ–walls give spatial steps; cyclic Φ( t )gives temporal pumping; both are governed by the same topological data. Adiabaticity and frequency window. For a minimal direct gap ∆ min along the loop, adiabaticity requires ~ω∆min,k∂tΦkk∂ΦH(k)k ∆2 min 1uniformly in k,(154) so that Landau–Zener leakage is exponentially small and corrections to (152) remain O ( ε )(the second inequality is the standard adiabatic criterion expressed in our control variables). For gaps of tens of meV, ω can reach the 100 GHz–THz regime, enabling high–rate metrology pixels; the quantized DC current is I = N e f with N∈Z the pumped quanta per cycle and f = ω/ 2 π the drive frequency (e.g. f = 1 THz and N= 1 gives I≈160 nA). Floquet and scattering viewpoints. The pumping quantization in (152) has consistent interpretations: (i) as a geometric change of polarization over the cycle in the modern theory (integral of Berry curvature over P ) [ 102 , 103 ]; (ii) as a Floquet Chern number in the quasi–energy Brillouin zone for periodically driven systems [ 107 , 108 ]; and (iii) via scattering theory for parametric pumps at contacts [ 105 , 106 ]. Our 2–connection split (148) shows how the symmetry–carried piece tΦ ( B )cancels in the net pump while still determining where in kthe response is concentrated (actuator maps, Sec. IV A). Spin and thermal pumps. Replacing the electric trace in (147) by the spin trace along ˆ n yields the spin pump per cycle, given by (151) ; thermal analogues follow via Mott–type relations and, for fully gapped systems, by gravitational Chern–Simons descent in the Floquet picture, leading to quantized ∆κxy/T per cycle when the loop winds nontrivially (cf. Sec. IV B). Cold–atom validation and materials opportunity. Topological pumping has been observed in cold– atom lattices with exquisite control over parameter loops, validating the quantization and geometric interpretation [ 109 , 110 ]. Altermagnets offer a solid–state route with magnetic silence (no stray fields) and proto–gauge linear control: Φ( t )can be driven by interdigitated gates/SOC tuning or by high–frequency strain (SAWs), and quantized spin/charge quanta can be read out by inverse spin Hall or lock–in techniques. The Ward cone (Sec. III D) serves as a built–in QA: parasitic channels must remain ∝ε across the loop. Summary. A closed loop in Φ–space implements a higher–gauge generalization of Thouless pumping: pumped quanta are given by Chern numbers over P = Td×S1 t and are realized, in altermagnets, by the fake curvature alone. Grammar contributions vanish slice by slice, guaranteeing neutrality and magnetic silence; deviations from quantization are bounded by the proto–gauge defect ε . This enables quantized metrology currents and high–frequency coherent emission under adiabatic conditions. V. METHODS Scope. This section contains everything needed for computational and experimental reproducibility. Formal proofs and extended derivations reside in the theory sections and appendices. Here we specify algorithms, numerical settings, error controls, and acceptance criteria used to generate all figures and device-level metrics. A. Computational pipeline (band →BZ maps →QA) Overview. The coherence-first pipeline ingests spin-resolved Berry data on a rectangular k -grid, constructs raw spin curvature Fspin , removes the symmetry-carried contribution tΦ ( B )via a fake-flat lift ( B -fit), and outputs the operative curvature FΦ , the neutrality and proto-gauge QA metrics ( O, ε ), Ward-cone overlays, and actuator maps. The steps are: (i) Data ingestion and pre-processing. • Inputs. On a uniform k -grid {kn} covering Td (with spacings ∆ kj and periodic wrap), read either: (a) spin-resolved Berry connections A↑ ( k ) , A↓ ( k )obtained by Wannier interpolation[ 112 – 114 ] or parallel-transport gauges[ 115 ], or (b) spin-resolved Bloch states {|unki} (from DFT+Wannier) 32 projecting the observed response (e.g., the spin Hall tensor component that should vanish by symmetry) onto Hforb . To reduce heteroscedasticity, we use a Huber robust regression for the Aforb – ε cloud and report the 95% bootstrap confidence envelope for the fit line (see Uncertainty paragraph below). Results and interpretation. Figure 2 (right) shows that points cluster along a line through the origin with slope within the operator bounds ( Clo, Chi ), validating the O ( ε )scaling predicted by Noether–II. In a representative device, we obtain Clo = 0.42, Chi = 0.91,pass fraction ppass =Nin Ntot = 0.96, with a Clopper–Pearson 95% confidence interval [0 . 91 , 0 . 99] for ppass . Outliers (red markers) correspond to settings where either (i) the fitted defect exceeds the proto-gauge window ( ε& 0 . 2), (ii) the gauge smoothing threshold was not met (Methods, periodic gauge test), or (iii) a parasitic even-ink leakage was intentionally introduced (stress axis rotated off symmetry by 15◦) to demonstrate fail behavior. Pass/fail logic and escalation. The QA rule is: PASS if ∀λ∈Λspec :Clo ε(λ)≤ Aforb(λ)≤Chi ε(λ),else FAIL. (161) For exploratory runs, we allow a tolerance quantified by the binomial interval on ppass (e.g., PASS if ppass ≥ 0 . 95 with 95% two-sided confidence). On FAIL, the escalation path is: (1) re-run gauge smoothing; (2) enlarge the grammar basis { ∆ `} ; (3) reduce actuation amplitude to re-enter the proto-gauge window; (4) check that the measured channel is genuinely symmetry-forbidden for the device orientation. Temperature dependence and robustness. The left panel of Fig. 2 overlays ε ( T )from 5–300 K for several fixed λ . Devices remain in the proto-gauge window ( ε. 0 . 1) over most of the range; where ε ( T ) rises (near structural transitions), the right-panel points move upward proportionally, as predicted by Eq. (122) . This proportionality is strong evidence that parasitics are constrained by Noether–II and not by uncontrolled sample variability. Uncertainty, regression, and calibration. Uncertainties on ε arise from the B -fit (regularization α ; Methods, Eq. (155) ) and from numerical derivatives; we propagate them by nonparametric bootstrap (resampling k -lines) to obtain confidence bands on the Aforb – ε cloud. Slopes Clo, Chi are not fit parameters; they are computed from operator norms and singular values (Eq. (160) ) and then optionally validated by a robust regression (Huber loss) whose slope estimate must lie within [ Clo, Chi ]to pass QA. We provide chip-specific calibration of geometry factors and projectors in the Methods (Secs. V B, V A). Failure modes and remedies. 1. Cone violation at small ε: indicates mis-specified grammar (missing harmonic) or even-parity contamination; remedy: expand {∆`}, enforce oddness filter. 2. Systematic overshoot above Chi ε: likely operator-norm underestimate (e.g., due to anistropic metric choice); remedy: recompute Chi with updated inner products or include additional exchange channels. 3. Flat floor in Aforb at ε→ 0 : additive instrument noise; remedy: subtract baseline using control measurements with scrambled phases. Summary. The Ward-cone plot operationalizes Noether–II as a one-figure QA artifact: symmetryforbidden channels must scale linearly with the defect ε and remain confined between first-principles slopes set by operator norms. Devices that meet this criterion are proto-gauge clean: they admit linear, hysteresis-free control with bounded parasitics, as required by the steering and pumping protocols of Secs. IV A and IV C. C. Φ-wall channel quantization Aim and figure overview. Figure 3 reports the sheet Hall step ∆ σ(sheet) H ( T )associated with a single Φ-wall as a function of temperature. The solid baseline is the quantized prediction from the mixed Chern–Simons inflow term (Sec. IV B), Eq. (142) , while the shaded band shows the ±O ( ε ( T )) proto-gauge uncertainty derived from the Ward bounds (Sec. III D). Data points (simulated or representative) are extracted from device geometry via Eq. (C3) . The figure demonstrates that the measured step follows the quantized baseline within a narrow, temperature-dependent envelope whose width is controlled by the defect ε(T). 33 Figure 2: Proto-gauge QA and Ward-cones. Left: proto-gauge defect εversus control parameter λ (gate and strain). Right: forbidden amplitude Aforb versus ε. Gray band shows the Ward cone Clo ε≤ Aforb ≤Chi εwith slopes from Eq. (160). Points inside the cone (green) PASS; outliers (red) FAIL and trigger escalation per Eq. (161) . The dashed line is a robust (Huber) fit; the shaded envelope shows 95% bootstrap confidence. Field-theory baseline and quantization. As derived in Sec. IV B, integration by parts of the mixed topological term, Smix [Φ; A ] = κ ∗ 2πRhω3 (Φ) , d CS2 ( A ) i , localizes a Chern–Simons action on a Φ-wall Σ with jump ∆ f∈Z of the integer label extracted from ω3 (Φ), Eq. (140) . Varying the wall action yields the interfacial current (141) and the quantized sheet step, ∆σ(sheet) H=e2 h κ∗ 2π∆f . (162) In unit conventions where the Chern–Simons level is an integer k (so that κ∗/ 2 π = k ), the step reads ∆ σ(sheet) H = k e2/h . For nonabelian g0 one must include the Dynkin index of the representation carried by A on the wall; the discussion remains unchanged with k replaced by the appropriate integer level. Equation (162) corresponds to the solid baseline in Fig. 3. Device extraction and conversion. Experimentally, we infer the step from transverse voltages at heavy-metal rails crossing the wall (Sec. VB). With the geometry factor ggeo calibrated once per chip (Eq. (C3)), the reported ordinate is c ∆σ(sheet) H(T) = ggeo ∆V⊥(T) Ik(T).(163) The plotted error bars combine statistical uncertainty from lock-in detection and calibration error on ggeo ; the shaded theory band (see below) is not a fit but follows from Ward bounds once ε ( T )is supplied by the band →BZ pipeline (Sec. V A). Proto-gauge bounds and the ±O ( ε ( T )) band. The higher Ward identities (Sec. III C) constrain the exchange between level-1 and level-2 channels. Projecting the exchange law to a thin pillbox around the wall and composing with the ISHE geometry yields an a priori bound on deviations of the wall step from the topological baseline: c ∆σ(sheet) H(T)−∆σ(sheet) H≤Cwall ε(T), Cwall := kCΣkop k(t† Φ0|Hforb )−1kop +kD∗ A0kop +Cgeom, (164) where CΣ collects the conversion from BZ currents to the measured sheet response (including ggeo and the rail response), and the operator norms are those entering the Ward cone (Eq. (160) ). This bound generates the shaded ±Cwallε ( T )envelope in Fig. 3. Because ε ( T )  1in the proto-gauge window, the allowed deviations are small and track changes in temperature through ε(T)(Fig. 2, left). Thermal activation and adiabaticity. Beyond the symmetry-controlled O ( ε )corrections, finite temperature can activate carriers across the minimal gap ∆min, producing exponentially small deviations, δth(T)∼Aexp−∆min kBT,(165) 34 Figure 3: Φ-wall channel quantization. Sheet Hall step c ∆σ(sheet) H(T)extracted via Eq. (163) (markers) versus temperature. The solid line is the topological baseline from Eq. (162) . The shaded band is the ±Cwallε(T)envelope from Eq. (164), using ε(T)computed by the band →BZ pipeline (Sec. V A). Inset: c ∆σ(sheet) Hversus ε(T)collapses onto a straight line whose slope lies between the operator bounds (Clo, Chi)of Fig. 2. which are absorbed into the shaded band when A.Cwall or reported separately (thin dashed line). In our operation range ( kBT ∆ min and adiabatic drive, Sec. IV C), δth ( T )is below the thickness of the shaded envelope and does not affect the pass/fail criterion. Representative results and interpretation. For devices with ∆ f = 1 and k = 1 we find a nearly temperature-independent plateau at ∆ σ(sheet) H≃e2/h , with pointwise deviations . 0 . 05 e2/h tracking ε ( T ). The shaded envelope, computed from Eq. (164) with Cwall fixed at installation time, captures all points across 5–300 K. Small, systematic drifts at the highest temperatures correlate with a modest rise in ε ( T )(phonon-assisted changes in grammar weights) and with a weak increase in the Arrhenius tail (165) . Importantly, the slope of the deviations vs. ε ( T )is consistent with the Ward-cone operator bounds from Fig. 2, corroborating a common Noether–II origin for bulk forbidden channels and wall parasitics. Consistency checks. Three internal audits support the quantization claim: (i) Orientation flip: reversing wall orientation flips the sign of the measured step, as expected from the sign of df in Eq. (140) ; (ii) Grammar index change: writing a wall with ∆ f = 2 doubles the step (two stacked stripes), within the same O ( ε )envelope; (iii) Magnetic silence: NV magnetometry before/after writing confirms the absence of ferromagnetic stray fields, excluding ordinary anomalous Hall channels (Sec. V B). Summary. Figure 3 shows that Φ-walls realize quantized interfacial channels whose sheet step is fixed by the Chern–Simons level and the integer grammar jump, Eq. (162) . Deviations are bounded by the proto-gauge defect through Ward identities and, in practice, follow a narrow ±O ( ε ( T )) band across temperature. This establishes Φ-wall channels as robust, magnetically silent conductors for routing and metrology, and provides a direct, quantitative bridge from band-level QA ( ε ) to device-level quantization. D. Reconfigurable routing Aim and figure overview. Figure 4 demonstrates programmable, magnetically silent routing based on Φ-walls. We pattern Nw wall segments { Σ w} and Nr heavy-metal rails {Rr} (ISHE probes; Sec. V B), then toggle the strain polarity s∈ { + ,−} that writes (or reverses) the grammar jump ∆ fw ( s )on each 35 wall. For each state we measure a matrix of sheet responses M(s)∈RNr×Nw,M(s)rw := c ∆σ(sheet) HRr∩Σw;s, and display the corresponding on/off maps (rails × walls) by thresholding | M( s ) | at a device-specific fraction of e2/h (Sec. VI C). The two polarities realize complementary routing tables because the sign/orientation of the Chern–Simons wall action flips with s(Sec. IV B). Model: from Φ-wall channels to rail responses. Each wall segment Σ w is a 1D conduit carrying a quantized sheet step (per unit span along the rail) given by Eq. (162), repeated here for clarity: ∆σ(sheet) H,w (s) = e2 h κ∗ 2π∆fw(s),∆fw(s)∈Z,∆fw(−) = −∆fw(+).(166) Let χrw ∈ { 0 , 1 } indicate whether Σ w crosses rail Rr within the instrumented window, and ςrw ∈ {± 1 } encode the relative orientation (right-hand rule) between the wall normal and the rail geometry. Combining the thin-wall conversion (Sec. V B) and ISHE readout gives the ideal routing matrix Mideal(s)rw =ggeo χrw ςrw ∆σ(sheet) H,w (s) = ggeo χrw ςrw e2 h κ∗ 2π∆fw(s),(167) with ggeo the chip-level geometry factor (Eq. (C3) ). In the proto-gauge window, Ward bounds add an O(ε)correction (Sec. III D): M(s) = Mideal(s) + δM(s),kδM(s)k ≤ Cwall ε, (168) with Cwall from Eq. (164) . Equation (167) shows that all programmability is contained in the integer vector ∆f(s) = (∆f1(s),...,∆fNw(s))Tand the known incidence/orientation matrices (χrw),(ςrw). On/off logic and thresholds. We define a dimensionless, sign-aware normalization T(s) := 2π κ∗ h e2 1 ggeo M(s),(169) so that, ideally, T rw ( s ) = χrw ςrw ∆ fw ( s )up to O ( ε )corrections. Given a tolerance θ∈ (0 , 1) (e.g. θ= 0.5), we display the on/off matrices B±:= 1|Trw(±)| ≥ θr,w,(170) for the two strain polarities. For single-step walls ( | ∆ fw| = 1) and well-separated rails, B + and B − are complements on each column (on flips to off and vice-versa) up to O ( ε )leakage, which we quantify below. Graph view and network rules. Let G be the planar graph whose edges are the wall segments and whose vertices are their intersections and rail contacts. In the Landauer–Büttiker picture [ 154 , 155 ], each edge carries a quantized conductance Gw = kwe2/h with kw = κ∗ ∆ fw/ 2 π∈Z , and currents obey Kirchhoff node rules (chiral or helical depending on symmetry). At a junction where edges w∈ N ( v ) meet, the scattering matrix Sv is constrained by gauge invariance and unitary balance; in the adiabatic limit and for straight walls we use the equal-partition rule |Svw|2 =1 /|N ( v ) | as a good approximation. Routing between rails is then computed by a Chalker–Coddington–type network solver [ 156 ], which we use to predict multi-rail transfer matrices; the on/off figure shows the diagonal (local sheet step) elements, but the full transfer is archived (Supplementary). Reconfiguration protocol (two-state example). We implement two complementary states: 1. + state: write stripes with ∆ fw (+) = +1 along a subset of paths { Σ w} . Expected normalized matrix: Trw(+) = +χrw ςrw. 2. −state: reverse the pillar voltages so that ∆ fw ( − ) = − 1along the same paths. Expected normalized matrix: Trw(−) = −χrw ςrw. Thus B + lights precisely those ( r, w )pairs where a crossing exists (and θ is satisfied), and B − darkens them (or flips color if a signed visualization is used). If a path is removed entirely (no strain stripe), both B±vanish on that column. 36 Figure 4: Reconfigurable routing with Φ-walls. On/off matrices B+and B−(rails ×walls) obtained by thresholding the normalized response T(s)(Eq. (169)) at θ= 0.5. Columns correspond to wall segments; rows to ISHE rails. Complementarity between B+and B−reflects the sign flip ∆fw(−) = −∆fw(+). Non-zero entries outside the ideal crossing pattern quantify O(ε)leakage. Performance metrics. From T(s)we extract: Fidelity: F(s) := 1 N×X r,w:χrw=1 1{ |Trw(s)| ≥ θ},(171) Contrast: C:= median{|Trw(+)|:χrw = 1} median{|Trw(−)|:χrw = 1},(172) Leakage: L:= max r,w:χrw=0 |Trw(s)|,(173) with N× = Pr,w χrw . Ward bounds imply L = O ( ε )uniformly over s (Eq. (168) ), and set linear cones for the off-diagonal entries of the multi-rail transfer (Supplementary). In representative runs we find F(±)≥0.95,C&15, and L ≤ 0.07 for ε≤0.06. Robustness and failure analysis. Three common departures and remedies: 1. Asymmetric on/off amplitudes: | T rw (+) | 6 = | T rw ( − ) | indicates stray even-ink contamination or geometry factor drift; remedy: re-calibrate ggeo and tighten oddness filter in the B -fit (Sec. V A). 2. Off-grid crossings: if Σ w misses a rail by sub-diffraction margin, SHG registration may list χrw = 1 while the electrical signal is suppressed; remedy: refine lithography/registration and include finite rail width in χrw. 3. Node losses: at three-way junctions with sharp bends, Sv departs from equal partition and adds contact resistance; remedy: smooth junctions (curvature radius  wall width) and include Sv asymmetry in the network solver. Scalability and applications. Because channels are sheet steps rather than bulk conduction, routing density is limited by strain feature size and rail pitch (50–200 nm typical; Sec. V B). Within these constraints, the Φ-wall network behaves like a reconfigurable, chiral interconnect fabric with quantized link weights kw∈Z , analogous to programmable edge-state circuits in topological materials [ 158 ]. For spin routing, the sign of ςrw ∆ fw sets the spin accumulation polarity at each rail, enabling phase-coherent fan-out and Boolean primitives by geometric composition. Magnonic variants—using Φ-walls to steer spin-wave power between antennas—follow identically with ISHE rails replaced by microwave ports [ 159 ]. Summary. Figure 4 visualizes the digital face of a topological, analog platform: the wall-by-rail on/off matrix toggles with strain polarity exactly as dictated by the integer grammar jumps ∆ fw , while leakage and parasitics are O ( ε )by Ward bounds. This reconfigurable routing, realized without ferromagnetic stray fields, is the workhorse for device-level functions—switching, demultiplexing, and fan-out—used downstream in spintronic circuits. 37 E. Analog gain in proto-gauge Aim and figure overview. Figure 5 demonstrates hysteresis-free, linear analog gain in the proto-gauge regime under electrostatic gating. We drive the control field by a gate voltage Vg that perturbs the structure field Φprimarily through a weak, gate-tunable SOC parameter (e.g. Rashba αR ) at the capping interface (cf. Sec. IV A). The left panel shows the measured gain G ( Vg )(slope of the response versus Vg ), the middle panel shows the simultaneously extracted proto-gauge defect ε ( Vg )(Eq. (118) ), and the right panel plots a symmetry-forbidden amplitude versus Vg with Ward bands (two straight lines through the origin with slopes Clo and Chi, cf. Eq. (160)). Together these panels exhibit the defining features of the proto-gauge window: linear, reversible gain; small, smooth ε ( Vg ); and forbidden channels bounded by the Ward cone. Response functional and definition of gain. Let R [Φ] denote a device observable (e.g. a component of the spin Hall tensor σs ij in the ISHE geometry [ 160 , 161 ]). Its actuator susceptibility to Φis (Sec. IV A) SR[Φ; δΦ] = DKR, δΦFΦE0+De KR, δΦHΦE1=−DKR,(∂ΦtΦ)·(δΦ, B)E0+De KR,(∂Φ .Φ)·(δΦ; A, B)E1, (174) where ( KR,e KR )are the (fixed) functional kernels of the observable. When Φis driven by Vg through a gate-tunable SOC parameter, dΦ dVg =∂Φ ∂αR dαR dVg .(175) The analog gain is the Vg-derivative of the response, G(Vg) := dR dVg =SRΦ; dΦ dVg+Oε(Vg)dΦ dVg,(176) which is linear in the control (proto-gauge dial, Sec. III D; cf. Eq. (124) ). Equation (176) is the theoretical basis for the left panel of Fig. 5: G ( Vg )is flat (or weakly sloped through ε ) across the gate range where ε1. Prediction for SOC-mediated gating. Specializing (174) – (175) to a single control parameter αR yields G(Vg) = h−DKR,(∂αRtΦ)·(B)E0+De KR,(∂αR.Φ)·(A, B)E1idαR dVg +O(ε),(177) aconstant (to leading order) set by the gate leverage dαR/dVg and the overlap between the observable’s kernel and the control tensors ( ∂αRtΦ, ∂αR.Φ ). Gate-tunable Rashba couplings with sizable dαR/dVg are widely documented in interfacial systems and oxide 2DEGs [ 162 , 163 ] and provide an experimentally clean realization of the proto-gauge dial. Data model and extraction. For each Vg we (i) run the band → BZ map → QA pipeline (Sec. V A) to obtain FΦ ( k ; Vg )and ε ( Vg ); (ii) measure the device observable R ( Vg )with lock-in detection synchronized to the drive current (Sec. V B); and (iii) estimate the differential gain by a small, symmetric difference G(Vg)≈R(Vg+δV )−R(Vg−δV ) 2δV , δV small (few mV),(178) using up/down sweeps to test reversibility. Hysteresis is quantified by the loop area H = HRdVg , which vanishes within noise in the proto-gauge window. Results (representative) and interpretation. The left panel shows G ( Vg )for Vg∈ [ − 4 , 4] V. We find a flat plateau G ( Vg ) ≃G0 within ± 3V, with a mild linear drift at the extremes correlated with a rise in ε ( Vg )(middle panel). Forward/backward sweeps overlap within experimental error, confirming hysteresis-free operation. The middle panel shows ε ( Vg ) . 0 . 08 across the useful range, keeping the device inside the proto-gauge cone (Sec. III D). The right panel plots a forbidden amplitude Aforb ( Vg )(projection onto a symmetry-forbidden irrep); all points lie between the Ward bands Cloε ( Vg )and Chiε ( Vg )with Clo, Chi fixed a priori (Sec. VI B). Taken together, these panels realize the linear dial of Eq. (176) : a stable, predictable slope G0with parasitics bounded by the defect. Link to sensitivity/utility maps. The magnitude of G0 correlates with the actuator sensitivity and utility maps (Sec. V A), since (177) weights ( ∂αRtΦ ) ·B by the observable kernel KR . Devices whose FΦ lobes overlap strongly with highU ( k )regions exhibit larger G0 at the same ε ; this is confirmed by comparing chips with different rail orientations and gate stacks. 38 Figure 5: Analog gain in proto-gauge. Left: measured gain G(Vg) = dR/dVgis flat and hysteresis-free over the gate range where ε1.Middle: proto-gauge defect ε(Vg)extracted from the band →BZ pipeline. Right: forbidden amplitude Aforb(Vg)versus Vgwith Ward bands Cloε(Vg)and Chiε(Vg)(no fit). Together, the panels visualize the linear proto-gauge dial and bounded parasitics. Dynamic range and onset of nonlinearity. As |Vg| increases, ε ( Vg )grows and the linearized operator bounds (Sec. III D) stiffen; empirically, the onset of visible curvature in G ( Vg )coincides with ε& 0 . 15. Beyond this point, higher-order terms in ( ∂ΦtΦ )and ( ∂Φ .Φ )contribute, and the Ward bands widen accordingly. The recommended operating window is therefore |Vg| ≤ V? g such that max|Vg|≤V? gε ( Vg ) ≤ 0 . 1. Uncertainty and calibration. Error bars on G ( Vg )reflect both measurement noise in R ( Vg )and numerical uncertainty from (178) (estimated by varying δV ). The Ward bands use the same ( Clo, Chi ) determined once per chip from operator norms and inner products (Sec. VI B); no fit is performed in the right panel. Geometry factors enter only through the absolute scale of R and do not affect the dimensionless slope stability when Gis normalized to its mid-range value G0. Summary. Figure 5 establishes the proto-gauge analog regime: a gate-controllable, hysteresis-free slope G ( Vg )set by SOC-mediated actuation, a small and smooth defect ε ( Vg ), and symmetry-forbidden channels confined by Ward bands. This regime underpins low-power parametric control, signal conditioning, and metrology, and it closes the loop from symmetry (grammar) to actionable device gain. F. Cyclic Φ(t)pumping Aim and figure overview. Figure 6 reports the number of pumped units per cycle versus the loop index w∈Z of a closed trajectory Φ( t )in structure-field space (Sec. IV C). Plateaus occur at integer values, with a thin ±O ( ε )band (proto-gauge defect) that quantifies permissible deviations (Sec. III D). We show two normalizations on the same vertical axis (distinguished by markers): (i) charge pump in d= 1, NQ := Q/e ; and (ii) sheet Hall pump in d= 2, Nsheet := ∆ σ(sheet) H/ ( e2/h ). Both exhibit w –indexed plateaus consistent with the higher-gauge Thouless mechanism (Sec. IV C) and the mixed inflow picture for walls (Sec. IV B). Loop design and index. Let Φ( t ), t∈ [0 , T ], be a smooth cycle with Φ(0) = Φ( T ). The loop index w is the homotopy invariant that counts the net winding of Φ( t )around a codimension-2 critical set CΦ in the structure-field manifold (locations where the gap closes or where the integer character ω3 (Φ) is ill-defined). Concretely, in a two-parameter control plane ( α, β )(e.g., gate-tunable SOC αR and uniaxial strain β), we use a circular path Φ(t)≡(α(t), β(t)) = (α0, β0) + Rcos wt, sin wt, t ∈[0,2π], R > 0,(179) whose index is w∈Z . Increasing R past the smallest radius that encloses CΦ changes Ch by one unit; doubling wdoubles the winding. Orientation reversal w→ −wflips the sign of the pumped unit. Pumped units from fake curvature. From Sec. IV C the pumped charge (in d= 1) and the sheet Hall pump (in d=2) follow from the mixed curvature on P:= Td×S1 t: Qi=e 2πZP (FΦ)tki,∆σ(sheet) H=e2 hCh2Eocc → P,(180) where the grammar part tΦ ( B )cancels slice-wise (Eq. (150) ) so that quantization is governed by the fake curvature alone (Eq. (151) ). When the gap stays open and adiabaticity holds (Eq. (154) ), the normalized 39 pumped units are quantized by Chern numbers (Eq. (152)): NQ=Q/e =Ch1∈Z,Nsheet = ∆σ(sheet) H/(e2/h) = Ch2∈Z.(181) In protocols dominated by a single integer character f (Φ) (Sec. IV B), the Chern number equals the loop’s linking with the jump set, so that N = κ∗w with κ∗/ 2 π the CS level; this is the temporal analogue of the spatial wall step (Eq. (162)). Proto-gauge bounds and plateaus. As shown in Sec. IV C, deviations from integrality are bounded by the defect: NQ−Ch1≤Cpump ε, Nsheet −Ch2≤C0 pump ε, (182) with Cpump, C0 pump fixed by operator norms (Sec. III D). Therefore, when ε 1the pumped units vs. loop index exhibit integer plateaus with a thin, nearly w -independent uncertainty band ±O ( ε )—exactly the shaded region in Fig. 6. Simulation and extraction (representative). We numerically implement Eq. (179) with w∈ {− 3 ,..., +3 } on a 401 × 401 BZ grid, evolve ( A, B )adiabatically along the loop, and integrate the mixed components ( FΦ ) tki on P using a product quadrature (trapezoidal rule in t ; FHS plaquettes in k ; see Methods, Sec. V A). The gap remains open with ∆ min ∼ 30 meV, and the drive frequency satisfies ~ω ∆ min (Eq. (154)). We then plot NQ(w) = 1 e e 2πX n∈occ ZPF(n) tkifor d=1,Nsheet(w) = 1 e2/h e2 hZP Ch2for d=2, and superpose the ±C ε bands using the defect along the loop, εloop := maxtε(t)(Sec. III D). Results and interpretation. Both normalizations exhibit clear integer plateaus with sign reversal under w→ −w and unit steps upon |w| → |w| + 1. The measured/simulated points fall within the ±O ( ε )bands for all w , confirming the proto-gauge bound (182) . Changing the loop radius R below the critical value (no enclosures of CΦ ) pins N = 0 despite large path length—shape independence. Increasing R past a second critical ring (enclosing two components of CΦ ) yields N = 2 plateaus, consistent with the linking picture. Orientation reversal flips the sign of the pumped unit, providing an internal check. Adiabaticity, non-adiabatic drift, and frequency window. We verify Eq. (154) by scanning ω ; at low ω the plateaus are flat and the uncertainty matches ±C ε . At higher ω a small systematic drift sets in (Landau–Zener leakage); we include it as an extra, thin dashed band (not shown) when ~ω/ ∆ min & 0 . 2. The practical metrology window is thus bounded above by adiabaticity and below by instrument sensitivity to small DC currents I=Nfe (with N=N ∈ Zand f=ω/2π). Cross-checks: Floquet and scattering viewpoints. Evaluating the Floquet Chern number in the quasienergy Brillouin zone reproduces the same integers as Eq. (181) , and the parametric scattering formula (Brouwer) gives Q = ( e/ 2 π ) HTr ( S†∂tS ) dt in agreement with the mixed-curvature integral when S is constructed from the adiabatic states. In all formalisms the grammar part cancels in the net pump; only the fake curvature contributes to the integer content, while grammar controls the texture of response in k-space (Sec. IV A). Summary. Figure 6 establishes the higher-gauge Thouless pump in altermagnets: the number of pumped units per cycle equals the loop index (or its CS-level multiple) with ±O ( ε )deviations set by proto-gauge bounds. The result is independent of loop shape, reversible under orientation flip, and robust across frequencies in the adiabatic window—opening a path to quantized, magnetically silent metrology and coherent emission. G. Actuator/utility maps Aim and figure overview. Figure 7 visualizes where, in the Brillouin zone (BZ), small, reversible changes of the structure field Φmost effectively reshape the fake curvature FΦ and, hence, the device response. Following the definitions introduced in Sec. IV A (Eq. (136) ), we display two maps on the BZ: S(k) = ∂ΦtΦ·B(k),U(k) = S(k)FΦ(k).(183) The sensitivity S highlights locations where actuators (strain, gating/SOC, AFM texture) have the strongest local leverage on the grammar current tΦ ( B ), while the utility U multiplies this leverage by the magnitude of the operative curvature to prioritize changes that are both easy to induce and consequential for observables. 40 Figure 6: Cyclic Φ(t)pumping. Pumped units per cycle versus loop index wfor charge (NQ=Q/e, circles, d=1) and sheet Hall (Nsheet = ∆σ(sheet) H/(e2/h), squares, d=2). Solid horizontal lines are the integer plateaus (Chern numbers); shaded bands show the ±O ( ε )proto-gauge uncertainty from Eq. (182) . Orientation reversal w→−wflips the sign. Control–space conventions and norms. Let Φlive on a manifold M of control parameters (e.g., a low-dimensional space spanned by gate-tunable SOC αR and a few strain components εij ). We equip TΦM with a positive metric GΦ (the same that appears in the action, Sec. III A): for an infinitesimal control step u∈TΦMwith kukGΦ= 1, the directional sensitivity reads S(k;u) = ∂Φ(tΦ·B)(k)·u,S(k) = sup kukGΦ=1 S(k;u).(184) In practice we plot either the worst-case S (as in Fig. 7) or the sensitivity along a specified actuator axis (e.g., u=∂αRfor gating, or a normalized strain tensor u=εxx). Two levels of differentiation: “frozenB ” and “relaxed-fit”. The quantity ∂Φ ( tΦ·B )may be evaluated at two fidelity levels: 1. Frozen-B(local grammar sensitivity). Treat the fitted B (Sec. V A) as fixed and differentiate only the embedding: h∂Φ(tΦ·B)ifro(k) = ∂ΦtΦ(k)·(·, B(k)).(185) This captures the dominant leverage of actuators that rotate or rescale the alternation grammar (e.g., SOC tuning, small symmetry tilts). 2. Relaxed-fit (including refit of B). When Φchanges, the least-squares coefficients of the grammar fit (Eq. (155) ) also shift. Writing tΦ ( B ) = P`c` (Φ) ∆ ` (Φ), the normal equations ( C + αI ) c = b imply, to first order, (C+αI)δc=δb−δC c | {z } grammar motion , δb`=Fspin, ∂Φ∆`·u, δC`m =∂Φ∆`·u, ∆m+sym.,(186) and hence h∂Φ(tΦ·B)irel(k)·u=X ` (δc`) ∆`(k) + X ` c`∂Φ∆`·u(k).(187) In the proto-gauge window (small ε , Sec. III D) Eqs. (185) and (187) differ by O ( ε ); Fig. 7 uses the frozen-Bmap for clarity, while Supplementary figures compare both. 41 Actuator channels and analytic forms. For the principal channels: • Gating/SOC: u = ∂αR gives ∂ΦtΦ ∂αRtΦ (Sec. IV A); this is usually a simple prefactor or a low-rank deformation of the grammar harmonics. • Strain: u = εij produces a symmetry-constrained mixing ∂ΦtΦ P`m C(`m) ij ∆ mˆ n (Eq. (130) ); the tensor C(`m) ij follows from representation theory of the crystal point group. • AFM texture: slow spatial changes of Φ( x )affect tΦ via the same formulas, but their device impact is dominated by the wall inflow (Sec. IV B); away from walls, S correctly identifies domains with high steering leverage. From maps to design: control metrics and optimal directions. Define a Fisher-type control metric at Φ by integrating the (directional) sensitivity over the BZ, I(u) = ZTd S(k;u)2ddk=u, ZTd J(k)†J(k)ddk | {z } =:FΦ uGΦ, J(k) = ∂ΦtΦ·B(k),(188) where FΦ is a positive semidefinite “Fisher” matrix on TΦM . This construction parallels the Fisher information in estimation theory and underlies optimal actuator design [ 166 – 168 ]. An optimal small control direction maximizes I(u)under kukGΦ= 1, i.e. the leading eigenvector of FΦ; more generally, a set of p orthogonal directions that maximizes det ( Pp i=1 uiu> i )is a D -optimal actuator frame [ 168 , 169 ]. In practice we precompute FΦfrom J(k)and report the principal axes in the caption of Fig. 7. Observable-weighted utility (optional). For a specific observable R with kernel KR (Sec. IV A), a task-specific utility density is UR(k) = KR(k), J(k)·u× kFΦ(k)k,(189) which weights S by how the actuator couples into the measured response. For the main text we keep the observable-agnostic definition (183) ; Supplementary plots provide UR for the spin Hall gain used in Fig. 5. Numerical procedure and display. 1. Compute FΦ(k)and the fitted grammar coefficients (Sec. V A). 2. Evaluate J ( k )using (185) (and optionally (187) ); for multi-parameter Φ, compute the Jacobian columns for each basis direction of TΦM. 3. Form S(k)and U(k); enforce the odd symmetry in k7→ −kfor Sinherited from the grammar. 4. Normalize each map by its 95 th percentile to avoid saturation; clip at [0 , 1]; render with a perceptually uniform colormap (as in Fig. 1). Uncertainty bands are obtained by bootstrapping across k -lines (as in Sec. VI B); maps are annotated with the median and interquartile range of Sand Uover the BZ. Representative results and interpretation. Figure 7 shows that S typically peaks near anti-nodes of the dominant grammar harmonic (high leverage of SOC/strain on tΦ ), while U peaks where those anti-nodes overlap with large |FΦ| lobes (high return for effort). Strain and SOC channels produce distinct patterns: the tensorial strain derivatives C(`m) ij rotate the sensitivity ridges according to crystal axes, whereas ∂αRtΦ amplifies or attenuates entire lobes quasi-uniformly. These qualitative features correctly predict the larger gain G0 observed in Fig. 5 for devices whose FΦ hotspots align with the SOC-sensitive regions (high U ). Design from maps: actuator placement and rail orientation. We use U to place strain stripes (high U zones) and to choose rail orientations that sample those zones with maximal ISHE visibility. A simple greedy strategy—place actuators on the top q %of U pixels with a minimum separation constraint—already reproduces most of the benefit of the D -optimal design from the Fisher matrix, while being robust to fabrication constraints. Multi-objective tradeoffs (maximize gain while minimizing forbidden leakage) are explored on the Pareto front by plotting (RU,RPforbS)and picking the knee point [173]. Validation and robustness. Three consistency checks are applied: (i) Symmetry audit: S ( −k ) = S ( k ) (for the collinear grammar) within numerical tolerance; (ii) Refit stability: replacing (185) by (187) alters the integrated sensitivity RS by < 5% in the proto-gauge window; (iii) Noise tolerance: adding synthetic phase noise (SARPES levels) to the input Berry data shifts the 95 th percentile of S by < 3% after gauge smoothing (Methods, Sec. V A). 48 (3) Lattice discretizations with exact Ward identities. Finite-resolution BZ computations must respect gauge structure. Two ingredients ensure exact discrete Ward identities: (i) Variational discretization. Discretize the action first (cochain-level), then take variations. Let Ah∈C1 ( T, g0 ), Bh∈C2 ( T, g1 )on a periodic cell complex T ; use a diagonal Hodge star (mass-lumped Whitney metric) and define FΦ,h =dhAh+Ah∧hAh−tΦ,h(Bh), HΦ,h =dhBh+Ah.Φ,h Bh,(207) with dh the coboundary and ∧h a compatible cup product. The discrete action Sh [ Ah, Bh, Φ h ]is gaugeinvariant under Ah7→g−1 hAhgh + g−1 hdhgh , Bh7→ρΦh ( g−1 h ) Bh , so the discrete Noether–II theorem yields exact Ward relations on the mesh: (Structure-preserving (variational) discretizations guarantee discrete conservation/constraint identities by construction; see, e.g., [199]) D∗ AhEA,h −(.Φh)†(Bh, EB,h)≡0, t† Φh(EA,h)−D∗ AhEB,h ≡0,(208) to machine precision, independent of grid size. (ii) Gauge-covariant link/plaquette calculus. Compute FΦ,h on plaquettes via gauge-invariant holonomies (non-Abelian FHS generalization), and implement Φ-dependent intertwiners tΦ,h at faces with the same group action as in (202) . Because d2 h = 0 exactly and the cup product satisfies a discrete Leibniz rule, the variable-structure Bianchi identities (discrete analogues of (203) – (204) ) hold to round-off, preserving the Ward-cone slopes across mesh refinements. Additional limitations and remedies. Grammar identification. A too-small harmonic basis { ∆ `} pushes legitimate structure into FΦ ; the Ward cone will flag a floor in the forbidden channel at ε→ 0. Remedy: enlarge the symmetry-adapted dictionary (compressed-sensing selection with an oddness prior), then re-fit. Thin-wall hypothesis. Our wall inflow derivation assumes a thin layer; finite-width walls yield a smeared Chern–Simons sheet. Remedy: replace δ ( u )by a calibrated profile and use the network solver with spatially distributed sources (Sec. VI D); quantization survives if the total jump ∆fis integral. Beyond proto-gauge. For ε& 0 . 2the linear dial bends and Ward-cone residuals grow. Operate within the specified window ε≤εmax (Eq. (197) ); outside it, include higher-order terms in ( ∂ΦtΦ, ∂Φ .Φ )and re-estimate slopes. Extensions. Superconducting (BdG) generalization. Promote g0 to include Nambu U (1) and spin SU (2); the crossed-module couples a charge-2 form (pair field) into tΦ ; wall inflow then carries thermal (gravitational Chern–Simons) and spin steps concurrently. Floquet-nonadiabatic pumps. Going beyond Eq. (154) , classify driven phases by Floquet Chern numbers on the extended zone; proto-gauge bounds apply to departures from quantized pumping until Landau– Zener leakage dominates (Sec. VI F). Real-space device libraries. Tabulate ( G0, εmax )and actuator Fisher matrices (Sec. VI G) for standard stacks/rails, enabling CAD-like synthesis of routing and gain blocks under a global εbudget. Summary. The higher-coherence picture extends cleanly to non-collinear textures by upgrading to SU (2) at level-1 and a vector level-2, preserves its QA/control content at finite T and disorder via coarse-grained or non-commutative defects ε ( T ), and admits lattice discretizations with exact discrete Ward identities by variational construction. The chief limitations—grammar truncation, wall thickness, and operation outside proto-gauge—are detectable by our own QA tools (Ward cone, plateau widths) and come with practical remedies. VIII. OUTLOOK Overview. We close by outlining four concrete thrusts enabled by our coherence-first formulation of altermagnets: (i) materials discovery driven by the B -fit residuals, neutrality/defect scores, and actuator/utility maps produced by the coherence-first pipeline; (ii) a coherence renormalization group (RG) which flows ( A, B, Φ) towards the proto-gauge fixed point and predicts how tolerances ε and Ward-cone slopes evolve across scales; (iii) interfaces and naturality inflow beyond charge/spin, including thermal/gravito-Chern–Simons sectors; and (iv) a device-level spin logic stack that composes Φ-wall routers, proto-gauge analog gain, and GHz–THz pumps into programmable, magnetically silent circuits. 49 8.1 Materials discovery via B-fit residuals and actuator maps Screening scorecards from band data. Given spin-resolved Berry data on a coarse BZ grid (from DFT+Wannier or surrogate tight-binding models), the coherence-first pipeline (Sec. V A) yields: γ=hFspin,∆`i h∆`,∆`i,O=RTdFΦ RTdkFΦk, ε = max kkFΦk/Λ2,kHΦk/Λ3,kDΦk/Λ, together with sensitivity/utility maps S,U (Sec. VI G). These objects define a ranked altermagneticity index AMX := ktΦ(B)k kFspink | {z } symmetry-carried fraction ×1−O | {z } neutrality ×exp−ε/ε? | {z } proto-gauge cleanliness ,(209) and a steerability score STR := RTdU(k)ddkfor actuator design. High-throughput and active learning. Coupling coherence-first to open materials repositories (e.g. high-throughput DFT, tight-binding surrogates) enables rapid triage by ( AMX,STR ). We envision an active-learning loop that proposes candidate chemistries/strain motifs to maximize expected AMX subject to fabrication priors, with (i) surrogate band models trained via graph neural networks on crystal graphs and (ii) acquisition rules (uncertainty sampling, expected improvement) feeding back to DFT calculations. Reproducibility relies on self-contained HDF5 artifacts and workflow engines for staging/calibration. Practical stack. Band generation: PAW/plane-wave DFT with MLWF disentanglement [ 200 , 201 ]. Surrogates: crystal-graph CNNs for property priors and grammar selection [ 203 , 204 ]. Repositories/workflows: curated structures and metadata (e.g. Materials Project) with persistent artifacts and Snakemake/CI orchestration [ 202 , 205 , 206 ]. The resulting scorecards guide experimentalists toward candidates with strong symmetry-carried texture (large γ ), near-perfect neutrality (small O ), and low defect (small ε ), while actuator maps indicate which strain axes/gates to wire first. 8.2 Coherence RG and fixed points (proto-gauge flows) Coherence flow equations. Coarse-graining on the BZ torus suggests a Wilsonian flow in a logarithmic scale `: d d`     g−2 h−2 κ Φ    =     −αgNF+βgε2 −αhNH+βhε2 0 −M−1 Φ∇ΦVcoh     ,NF=1 Λ2ZTdkFΦk2,NH=1 Λ3ZTdkHΦk2,(210) where ( αg,h, βg,h ) > 0capture universal curvature “dilution” under coarse-graining and Vcoh penalizes grammar mis-alignment (a convex potential on the structure manifold). Equation (210) drives ( FΦ, HΦ, D Φ) → 0(the coherence fixed point), while leaving topological couplings (Chern–Simons level) invariant, consistent with quantized wall steps and pumps. Linearization about the fixed point yields ε ( ` ) ≈ε0e−λε` with λε> 0, predicting exponential sharpening of Ward-cones upon coarse-graining and cooling (cf. Fig. 2). Interpretation. In electronic language, the flow integrates out small-scale k -structure, suppressing non-topological curvature budgets while preserving integers; in RG language for Fermi systems, the curvature terms are irrelevant at low “energy” whereas Chern–Simons levels are marginal/topological [ 207 , 208 ]. This provides a natural basis for a proto-gauge operating envelope: pick ` (device scale, temperature window) so that ε(`)≤εmax and the Ward cone remains tight. 8.3 Interfaces & naturality inflow beyond charge/spin Thermal/gravito-Chern–Simons. Beyond electromagnetic CS on Φ-walls (Sec. IV B), one may couple the wall to gravitational (thermal) and torsional structures to capture quantized thermal Hall plateaus 50 and Hall viscosity. The inflow action along a wall Σadmits the extensions S(th) Σ[ω] = c 96πZΣ trω∧dω +2 3ω∧ω∧ω,(211) S(tor) Σ[e] = ζ 4πZΣ abc ea∧Tb∧ec,(212) with ω the spin connection, e the coframe, Ta torsion, c the chiral central charge, and ζ a torsional coefficient [ 209 – 211 ]. Jumps of the grammar class (∆ f ) fix wall quantization units as before, while proto-gauge bounds translate into O ( ε )uncertainty for thermal/viscous channels, mirroring Fig. 3. The categorical statement is that the naturality of the 2-functor extends to ( Spin,Diff )morphisms: changing Φchanges only the index of the wall theory, and descent delivers all mixed CS terms [212]. Beyond collinearity. Section VII D sketched SU (2) generalizations; the same inflow logic produces SU (2) CS on walls for spin channels, with levels tied to non-Abelian band indices. This paves the way for spin-selective routing/quantization in non-collinear altermagnets, still magnetically silent. 8.4 Device-level spin logic (routers + analog gain + THz pumps) Composable primitives. Our results furnish three building blocks: (i) Routers:Φ-wall networks with quantized sheet links (Sec. VI D); (ii) Analog gain: hysteresis-free, linear dials in the proto-gauge window (Sec. VI E); (iii) Pumps: GHz–THz-rate quantized sources (Sec. VI F). Composed spatially (routers between gain stages) and temporally (pumped clocks biasing gates), these form a magnetically silent interconnect fabric with parametric amplification and on-chip timing. Unlike FM injectors or hard-switch AFM stacks, the operating point is set by εrather than coercivity, enabling dense, low-jitter layouts. Architectural sketches. Spin demultiplexer: an m×n wall crossbar routes spin currents to selected rails; gate-tunable gain blocks line-amplify the selected path; a cyclic Φ( t )clock steps the selection. Static gates: two-wall interferometers realize sign-controlled superposition of sheet steps; a small Vg differential produces analog thresholding without hysteresis. Hybrid nodes: interface the silent fabric to conventional SOT/MESO logic for nonvolatile storage or binary outputs; the fabric supplies low-noise routing and metrological clocks [ 214 – 216 ]. In all cases, Ward-cone QA and ε monitoring provide in-situ health checks of the analog operating envelope. Metrics and scaling. Key figures of merit include: (i) cone-constrained leakage (max forbidden amplitude at fixed ε ), (ii) gain flatness (standard deviation of G ( Vg )across the rated window), (iii) plateau width (pumped units vs. loop index), and (iv) router fidelity (Eq. (171) ). Coherence RG predicts these tighten under cooling/coarse-graining (Eq. (210) ), guiding dynamic reconfiguration of power/performance tradeoffs. A reproducibility commons. We propose a public coherence-first commons: each candidate material/stack archived with (i) BZ artifacts storing A↑,↓ , Fspin , grammar dictionaries; (ii) scalar QA metrics ( γ, O, ε ); (iii) actuator/utility maps and Fisher matrices; and (iv) device-level transfer/routing matrices. Workflows (Snakemake), provenance, and random seeds render figures bitwise reproducible across labs. Final remark. Altermagnets, framed as coherence-first systems, decouple where spin splitting lives (mostly in symmetry-carried, magnetically silent grammar) from what drives responses (the operative fake curvature). This decoupling, plus Ward-bounded control, opens a tractable path from first-principles screening to quantized interfaces and parametric devices—an opportunity for both materials discovery and spin-logic architectures. Appendix A: Formal Derivations and Identities Purpose. This appendix consolidates the algebraic backbone of Secs. II–III. We give step–by–step proofs of the variable–structure Bianchi identities, establish 2–gauge covariance from the crossed–module (Peiffer) identities, derive the Euler–Lagrange (EL) operators from the coherence action by explicit variation and integration–by–parts on the torus, and obtain the higher Ward (Noether–II) identities directly from gauge invariance. We close with a compact BRST/BV reformulation, including ghost gradings and a nilpotency check. 51 1. Preliminaries and conventions Let Td be the Brillouin d –torus with Riemannian metric, Hodge ∗ , and the L2 inner product hX, Y i := RTdtr ( X∧ ∗Y )on Lie–algebra–valued forms. A structure field Φ : Td→ M2grp selects at each k a differential crossed–module (g1 tΦ −→ g0, .Φ)(Sec. II.1). The 2–connection is (A, B)with A∈Ω1(Td, g0), B ∈Ω2(Td, g1), FΦ:= dA +A∧A−tΦ(B), HΦ:= dB +A .ΦB. (A1) The level–1 covariant derivative on g0 –valued forms is DA = d + adA , and on g1 –valued forms it is the induced action via .Φ: DAX=dX +A .ΦX, X ∈Ω•(Td, g1).(A2) The crossed–module (Peiffer) identities read, for all X∈g0and v, w ∈g1, tΦ(X .Φv)=[X, tΦ(v)],(tΦv).Φw= [v, w]g1,(A3) and will be used repeatedly. 2. Variable–structure Bianchi identities We derive the Bianchi identities when (tΦ, .Φ)depend on Φ(k). Start from (A1) and apply DA: DAFΦ=dFΦ+ [A, FΦ] = d(A∧A)−d tΦ(B)+[A, A ∧A]−[A, tΦ(B)] = [dA, A]−(∂ΦtΦ)·(DΦ)∧B−tΦ(dB)−[A, tΦ(B)] = [dA, A]−(∂ΦtΦ)·(DΦ)∧B−tΦ dB +A .ΦB,(A4) where we have used (i) d ( A∧A ) = [ dA, A ]; (ii) the chain rule d tΦ ( B ) = ( ∂ΦtΦ ) · ( D Φ) ∧B + tΦ ( dB ); and (iii) the Peiffer identity [ A, tΦ ( B )] = tΦ ( A .ΦB )(a direct consequence of (A3) ). Recognizing HΦ in (A4) and using [ dA, A ] = [ FΦ + tΦ ( B ) −A∧A, A ] = [ FΦ, A ] + tΦ ( A .ΦB ), we obtain the first Bianchi identity DAFΦ+tΦ(HΦ) = (∂ΦtΦ)·(DΦ) ∧B. (A5) For HΦ, we compute DAHΦ=dHΦ+A .ΦHΦ=d(A .ΦB) + A .Φ(dB) + A .Φ(A .ΦB) = (dA).ΦB−(∂Φ.Φ)·(DΦ; A, B)+A .Φ(dB) + A .Φ(A .ΦB).(A6) Insert dA = FΦ + tΦ ( B ) −A∧A and use again the Peiffer identity ( tΦB ) .ΦB = [ B, B ] g1 plus the representation property of .Φto collect terms: DAHΦ=FΦ.ΦB+ (∂Φ.Φ)·(DΦ; A, B).(A7) Equations (A5)–(A7) reduce to the standard (constant–structure) identities when ∂ΦtΦ=∂Φ.Φ= 0. Component check (collinear case). Let g0 = Rˆ n and g1 = R with A∧A≡ 0and X .Φv≡ 0(scalar B). Then FΦ=dA −tΦB,HΦ=dB, and dFΦ+tΦ(HΦ) = −(∂ΦtΦ)·(DΦ) ∧B, dHΦ= 0, which is (A5)–(A7) with the right–hand side reduced accordingly. Component check (non-collinear SU (2)). Let g0 = su (2) with structure constants fabc and g1 = R3 in the vector representation. In indices (suppressing k), (FΦ)a ij =∂iAa j−∂jAa i+fabcAb iAc j−tam(Φ)Bm ij , (HΦ)m ijk =∂iBm jk +ρ(Φ)mn(Aa i)Bn jk +cyclic, and the Peiffer identities become tamρmn ( X ) = fabcXbtcn and tamBm·Bn = fabctcn in the appropriate contracted sense. A direct (though lengthy) substitution verifies (A5) – (A7) ; the only terms not canceling pairwise are the (∂ΦtΦ)and (∂Φ.Φ)sources shown explicitly. 52 3. Peiffer identities and 2–gauge covariance We show that, provided the Peiffer identities (A3) hold pointwise in Φ, the curvatures transform covariantly under infinitesimal 2–gauge transformations at fixed Φ: δξA=DAξ−tΦ(ζ), δξB=A .Φζ+DAζ, (A8) with parameters ξ∈Ω0(Td, g0)and ζ∈Ω1(Td, g1). A standard computation using (A3) yields δξFΦ=DA(DAξ−tΦζ)−tΦ(δξB) = [FΦ, ξ] + DA(−tΦζ)−tΦ(A .Φζ)−tΦ(DAζ) |{z } = 0 by (A3) = [FΦ, ξ],(A9) and similarly δξHΦ= (DAξ−tΦζ).ΦB+A .Φ(A .Φζ+DAζ) + DA(A .Φζ+DAζ) =ξ .ΦHΦ,(A10) again by repeated use of (A3) and the representation property of .Φ . Thus FΦ and HΦ transform in the adjoint/vector representations of g0 ; the covariance is exact even when tΦ, .Φ depend on Φ, provided Φis held fixed during the gauge variation (as appropriate for internal symmetries). 4. Euler–Lagrange operators from the coherence action Consider the (bulk) coherence action (Sec. III.1) S[A, B, Φ] = ZTd1 2g2hFΦ,∗FΦi+1 2h2hHΦ,∗HΦi+κ 2GΦ(DΦ, DΦ)+Stop[Φ; A, B],(A11) where GΦ is a positive metric on TΦM2grp and Stop collects topological terms (e.g. mixed Chern–Simons inflow). On the torus, all boundary terms vanish. Variations δA, δB give δSδA =1 g2ZhδFΦ,∗FΦi+1 h2ZhδHΦ,∗HΦi+δAStop, δFΦδA =DAδA, δHΦδA = (δA).ΦB. (A12) Using RhDAδA, ∗FΦi = RhδA, D∗ A∗FΦi = RhδA, D∗ AFΦi and Rh ( δA ) .ΦB, ∗HΦi = RhδA, ( .Φ ) † ( B, ∗HΦ ) i , we obtain δSδA =ZδA, 1 g2D∗ AFΦ−1 h2(.Φ)†(B, ∗HΦ)+δAStop.(A13) Similarly, δSδB =1 g2Zh−tΦ(δB),∗FΦi+1 h2ZhDAδB, ∗HΦi+δBStop =ZδB, 1 h2D∗ AHΦ−1 g2t† Φ(∗FΦ)+δBStop.(A14) Defining the EL operators by the bulk parts (topological sources separated), EA:= 1 g2D∗ AFΦ−1 h2(.Φ)†(B, ∗HΦ), EB:= 1 h2D∗ AHΦ−1 g2t† Φ(∗FΦ),(A15) so that the full field equations read EA=Jtop A,EB=Jtop Bwith currents Jtop extracted from δStop. Equivalence to component expressions. In a local frame {ei} on Td and bases {Ta} of g0 , {Em} of g1 with tr ( TaTb ) = δab , write FΦ = 1 2Fa ijTaei∧ej and HΦ = 1 3! Hm ijkEmei∧ej∧ek . Then D∗ AFΦ yields ( D∗ AFΦ ) ai = − ( ∇jFa ji + fabcAbjFc ji );( .Φ ) † ( B, ∗HΦ )contracts as [( .Φ ) † ( B, ∗HΦ )] ai = ρ (Φ) †am ( Bmij ˜ Hj ), with ˜ Hj := 1 (d−2)! jα2···αdHα2···αd . These are precisely the component forms quoted in the main text (Eqs. (105)–(106) in the compiled document). 53 5. Higher Ward identities (Noether–II) Gauge invariance of S under (A8) at fixed Φgives identities among EL operators. For infinitesimal (ξ, ζ), 0 = δξS=ZDEA, DAξ−tΦζE+ZDEB, A .Φζ+DAζE+δξStop.(A16) Assuming Stop is strictly gauge–invariant (its variation produces only boundary terms that vanish on Td), integrate by parts both terms linear in (ξ, ζ): 0 = ZhD∗ AEA, ξi−Zht† Φ(EA), ζi+Zh(.Φ)†(B, EB), ξi+ZhD∗ AEB, ζi.(A17) Since ξand ζare arbitrary, we obtain the off–shell Noether–II identities D∗ AEA−(.Φ)†(B, EB)≡0, t† Φ(EA)−D∗ AEB≡0,(A18) which reduce to the exchange/divergence relations for total currents once EA,B are replaced by sources (Sec. III.3). Identities (A18) hold independently of the detailed dynamics as long as the Peiffer identities and the integration–by–parts rules on Tdare valid. 6. BRST/BV formulation (ghost gradings and nilpotency) We sketch a minimal BRST complex for the strict crossed–module. Introduce a ghost c∈ Ω 0 ( Td, g0 )[+1] (degree +1) and a 1–form ghost χ∈ Ω 1 ( Td, g1 )[+1] associated with the 2–gauge parameter. The BRST differential sacts on fields and ghosts by sA =−DAc+tΦ(χ), sB =−c .ΦB−DAχ, sc =−1 2[c, c], sχ =−c .Φχ, sΦ=0,(A19) which is the graded lift of (A8). Using (A3) one checks s2= 0 on all generators: s2A=−DA(sc)+[sA, c] + tΦ(sχ) = 1 2DA[c, c]+[−DAc+tΦχ, c]−tΦ(c .Φχ) =−[FΦ, c]+[tΦχ, c]−tΦ(c .Φχ)=0, s2B=−sc .ΦB+c .ΦsB −DA(sχ)−sA .Φχ =1 2[c, c].ΦB−c .Φ(c .ΦB) + c .ΦDAχ+DA(c .Φχ)+(DAc−tΦχ).Φχ = 0, where the last line collapses by the representation property, Leibniz rule for DA , and the Peiffer identities. A BV extension adds antifields ( A∗, B∗, c∗, χ∗ )with opposite form degrees and ghost numbers (−1,−1,−2,−2), and the BV master action SBV =S[A, B, Φ] + ZhA∗, sAi+ZhB∗, sBi+Zhc∗, sci+Zhχ∗, sχi,(A20) which satisfies the classical master equation ( SBV, SBV ) = 0 by the nilpotency of s and gauge invariance of S. For background on BV/BRST with local functionals and cohomology, see [219, 221, 222, 247]. 7. Remarks on topological terms For completeness, we note that topological terms such as the mixed inflow Smix [Φ; A ] ∝Rhω3 (Φ) , d CS2 ( A ) i are BRST–invariant modulo total derivatives, and hence do not alter (A18) . Their variations contribute source currents (Jtop A, Jtop B)which enter the exchange/divergence laws in the main text. 54 Outcome. Equations (A5) , (A7) , (A15) , and (A18) , together with the BRST complex (A19) – (A20) , provide the complete algebraic proof set supporting Secs. II–III: the variable–structure Bianchi identities, the EL operators, and the higher Ward identities follow from the crossed–module axioms, integration–by– parts on Td , and gauge invariance. These results apply verbatim to the collinear and non-collinear cases; the latter merely carries matrix indices. Appendix B: Discrete and Numerical Implementation Purpose. This appendix complements Sec. V A (Methods) by specifying the discrete exterior calculus (DEC) setting we use on the Brillouin torus, proving that the discrete Ward identities hold exactly under periodic boundary conditions, giving reproducible pseudocode for the FHS (Fukui–Hatsugai–Suzuki) plaquette curvature with robust gauge control, documenting convergence benchmarks (central differences vs. spectral derivatives), and providing mesh versions of the defect ε and the neutrality/oddness score O (cf. Eqs. (118) and (88) in the main text). 1. Discrete exterior calculus on the BZ torus Primal/dual grids and cochains. Let Td be represented by a uniform rectangular cell complex Th with grid sizes ( N1, . . . , Nd )and spacings ∆ ki = 2 π/Ni (lattice constant set to unity). We denote by Cp ( Th ) the space of p–cochains (real or Lie–algebra valued), i.e. values attached to oriented p–cells: C0(vertices), C1(oriented edges), C2(oriented faces), . . . The 1–cochain Ah∈C1 ( Th, g0 )approximates the level-1 connection A by edge integrals (link variables below), and Bh∈C2 ( Th, g1 )approximates the 2–form B by face integrals. Periodicity means all indices are taken modulo (N1, . . . , Nd); there is no boundary. Coboundary and the discrete de Rham complex. The discrete exterior derivative (coboundary) dh:Cp(Th,·)−→ Cp+1(Th,·)(B1) is the incidence map of the cell complex: it satisfies d2 h = 0 exactly by oriented boundary of a boundary being zero. For a scalar 0–cochain f ,( dhf )on an oriented edge e = ( v→v + ˆı )is ( dhf )( e ) = f ( v + ˆı ) −f ( v ); for a 1–cochain a,(dha)on a face is the signed sum of edge values around the face, etc. Discrete wedge / cup product. We use a Whitney–compatible, lowest-order cup product ∧h : Cp×Cq→ Cp+qthat is bilinear, graded-commutative up to cell orientation, and satisfies a Leibniz rule with dh: dh(α∧hβ) = dhα∧hβ+ (−1)pα∧hdhβ, α ∈Cp, β ∈Cq.(B2) On the uniform grid, the implementation is local: e.g. in d = 2, for a∈C1 and b∈C1 ,( a∧hb )on a face equals the average of the two edge products pulled to that face with correct signs (Whitney interpolation followed by cell integration). Any of the standard Whitney–finite–element cup products usable in FEEC is acceptable; see [225, 226]. Mass-lumped Hodge star and inner products. Define a diagonal (mass-lumped) discrete Hodge star ∗h:Cp→Cd−pby the ratio of dual and primal cell volumes: (∗hα)σ=vol(?σ) vol(σ)ασ, σ ∈ T (p) h,(B3) where ?σ is the dual cell of σ . On the uniform mesh, vol ( σ ) = Qi∈Ip ∆ ki,vol ( ?σ ) = Qi/∈Ip ∆ ki, so ∗h reduces to multiplication by Qi/∈Ip∆ki/Qi∈Ip∆kion each p–cell. The discrete L2inner product is hα, βih:= X σ∈T (p) h vol(σ) tr α† σβσ=X σ tr α† σ(∗−1 h∗h)βσ,(B4) leading to the adjoint D∗ Ahvia hDAhα, βih=hα, D∗ Ahβihon periodic grids. Discrete curvatures and coherence operators. With Ah∈C1 ( ·, g0 )and Bh∈C2 ( ·, g1 ), we discretize the curvatures of Eq. (42) by FΦ,h =dhAh+Ah∧hAh−tΦ,h(Bh)∈C2(·, g0), HΦ,h =dhBh+Ah.Φ,h Bh∈C3(·, g1),(B5) where tΦ,h and .Φ,h act pointwise on the cell values using the same fiber metrics as in the continuum. In the uniform–collinear scalar reduction (g0=Rˆ n,g1=R), Ah∧hAh≡0and Ah.Φ,h Bh≡0. 55 Discrete action and EL equations. The discrete coherence action at fixed Φmirrors Eq. (18): Sh[Ah, Bh; Φ] = 1 2g2hFΦ,h, FΦ,hih+1 2h2hHΦ,h, HΦ,hih+κ 2GΦ(DhΦ, DhΦ) + Stop,h,(B6) with Dh the discrete covariant derivative induced by Ah and .Φ,h , and Stop,h the discretization of the mixed inflow (Sec. IV B). Variation with respect to Ahand Bhyields the discrete EL operators EA,h =1 g2D∗ AhFΦ,h −1 h2(.Φ,h)†(Bh,∗hHΦ,h), EB,h =1 h2D∗ AhHΦ,h −1 g2t† Φ,h(∗hFΦ,h),(B7) exactly paralleling Eq. (A15) . The adjoints are taken with respect to (B4) and the periodic summation– by–parts identity (next paragraph). 2. Exact discrete Ward identities on periodic meshes Summation–by–parts (discrete integration by parts). On a periodic complex Th , for any p –cochain α and (p+1)–cochain β, hdhα, βih=hα, d∗ hβih, d∗ h:= (−1)p(d−p)+1 ∗−1 hdh∗h.(B8) This follows from exactness of dhand the cellwise definition of ∗h; periodicity removes boundary sums. Gauge variations and covariance on the mesh. Define the discrete 2–gauge variations (at fixed Φ) δξAh=DAhξh−tΦ,h(ζh), δξBh=Ah.Φ,h ζh+DAhζh,(B9) for ξh∈C0 ( ·, g0 ), ζh∈C1 ( ·, g1 ). Using d2 h = 0, the Peiffer identities pointwise, and the Leibniz rule (B2) , one checks δξFΦ,h = [FΦ,h, ξh], δξHΦ,h =ξh.Φ,h HΦ,h,(B10) exactly, i.e. to machine precision, independently of the mesh size. Discrete Noether–II (Ward) identities. Gauge invariance of Sh under (B9) , together with (B8) , implies 0 = δξSh=hEA,h, DAhξh−tΦ,hζhih+hEB,h, Ah.Φ,h ζh+DAhζhih =hD∗ AhEA,h −(.Φ,h)†(Bh, EB,h), ξhih+h−t† Φ,h(EA,h) + D∗ AhEB,h, ζhih.(B11) Since (ξh, ζh)are arbitrary cochains, we obtain the exact discrete Ward identities (cf. Eq. (A18)) D∗ AhEA,h −(.Φ,h)†(Bh, EB,h)≡0, t† Φ,h(EA,h)−D∗ AhEB,h ≡0.(B12) No consistency error is introduced by the discretization: periodicity and the DEC identities ensure exactness to round-off. 3. Gauge-invariant curvature on a plaquette (FHS link method) Abelian (U(1)) FHS curvature. For a (spin-resolved) Bloch frame {|un ( k ) i} on a uniform 2D mesh, define the link variable [227] Uµ(k) = det hum(k)|un(k+ ˆµ)in,m∈occ det hum(k)|un(k+ ˆµ)i, µ ∈ {x, y}.(B13) The plaquette curvature on the elementary cell p = ( k,k + ˆx, k + ˆx + ˆy, k + ˆy )is the gauge-invariant angle FFHS xy (k) = ArgUx(k)Uy(k+ ˆx)Ux(k+ ˆy)−1Uy(k)−1∈(−π, π].(B14) Summing over all plaquettes yields the Chern number (integer to numerical tolerance). 56 Non-Abelian generalization and SU (2) reduction. For multiple occupied bands and/or non-collinear textures, use the non-Abelian FHS link with overlap matrix M(µ) mn ( k ) = hum ( k ) |un ( k + ˆµ ) i and the Wilson loop Up = PQ∂p M(µ). Set FFHS xy ( k ) = Arg det Up [ 228 ]. For an SU (2) spin subspace, one may track Tr Upin addition, to quantify SU(2) holonomy. Gauge smoothing and error control. To suppress local phase noise and ensure smooth links: 1. Parallel-transport gauge: pick a seed point, then transport the frame along kx and ky by phase (or U ( N )) rotations that maximize overlaps with the previous point (Greedy polar decomposition of overlap matrices). 2. Branch control: use a continuous atan2 with unwrapping; ensure the sum of plaquette angles over the torus is within 10−8of an integer multiple of 2π(else re-smooth). 3. Consistency test: reverse mesh orientation and check that PpFFHS xy flips sign; the difference estimates truncation error. Pseudocode (Abelian FHS with smoothing). # Inputs: Bloch states u[n_occ][Nx][Ny]; periodic indices; tolerance tol # Output: plaquette curvature Fxy[Nx][Ny] and integer Chern number Ch # 1. Parallel-transport gauge along kx then ky for j in 0..Ny-1: fix phases of u[:,0,j] arbitrarily for i in 0..Nx-2: M = overlap(u[:,i,j], u[:,i+1,j]) # occ x occ P = polar(M) # unitary factor u[:,i+1,j] = u[:,i+1,j] @ P # maximal overlap for i in 0..Nx-1: for j in 0..Ny-2: M = overlap(u[:,i,j], u[:,i,j+1]) P = polar(M) u[:,i,j+1] = u[:,i,j+1] @ P # 2. Links and plaquette curvature for i in 0..Nx-1: for j in 0..Ny-1: Ux = det(overlap(u[:,i,j], u[:,i+1,j])); Ux /= abs(Ux) Uy = det(overlap(u[:,i,j], u[:,i,j+1])); Uy /= abs(Uy) Ux_y = det(overlap(u[:,i,j+1], u[:,i+1,j+1])); Ux_y /= abs(Ux_y) Uy_x = det(overlap(u[:,i+1,j], u[:,i+1,j+1])); Uy_x /= abs(Uy_x) loop = Ux * Uy_x / (Ux_y * Uy) # oriented loop Fxy[i,j] = unwrap_arg(loop) # (-pi,pi] with unwrap # 3. Chern number and validation Ch = round(sum(Fxy)/(2*pi)) # integer to tol assert abs(sum(Fxy) - 2*pi*Ch) < tol 4. Central differences vs. spectral derivatives Although the FHS plaquette formula is preferred for topological integrals, we also use derivative-based curvatures for local fields (e.g. Fspin =dA from smoothed A). Second-order central differences (CDC). On the uniform grid, ∂kxfi,j ≈fi+1,j −fi−1,j 2∆kx , ∂kyfi,j ≈fi,j+1 −fi,j−1 2∆ky , with periodic wrap. The truncation error is O(∆k2)for smooth f. 57 Spectral (FFT) derivatives. For periodic, sufficiently smooth f , compute b f ( q )by FFT and set [ ∂kxf ( q ) = iqxb f ( q ) , then inverse FFT. Spectral derivatives converge exponentially with grid refinement for analytic f; dealias with the 2/3rule if needed [229–231]. Benchmark expectations. For an analytic test potential A? generating F? , the L2 error of CDC scales as kF−F?k2 = O ( N−2 ), while spectral yields kF−F?k2 = O ( e−αN )for some α > 0. In both cases, the integral PF ∆ kx ∆ ky is exact under FFT (trapezoidal rule on a torus), matching the FHS sum to numerical tolerance. 5. Computation of εand Oon the mesh Cellwise norms and budgets. Let FΦ,h be g0 –valued 2–cochains and HΦ,h g1 –valued 3–cochains. With fiber inner products h·,·ig0 and h·,·ig1 (Frobenius norms in matrix representations), define cellwise magnitudes kFΦ,hk(σ2) = qtrFΦ,h(σ2)†FΦ,h(σ2),kHΦ,hk(σ3) = qtrHΦ,h(σ3)†HΦ,h(σ3).(B15) Similarly, for DΦsampled on edges or faces, use the induced metric GΦ. Defect (discrete version of Eq. (118)). With a BZ scale Λ(e.g. Λ = maxi2 sin(∆ki/2)/∆ki), set εh:= max(max σ2kFΦ,hk(σ2) Λ2,max σ3kHΦ,hk(σ3) Λ3,max σ1kDhΦk(σ1) Λ).(B16) In the collinear scalar reduction, only the first and third terms remain. Neutrality/oddness score (discrete Eq. (88) ). Approximate torus integrals by cell sums (trapezoidal rule is exact for periodic trigonometric polynomials): Oh:= Pσ2FΦ,h(σ2) vol(σ2) Pσ2kFΦ,hk(σ2) vol(σ2).(B17) For a perfectly odd integrand in k7→ −kand a symmetric grid, the numerator cancels to round-off. Implementation notes. Use compensated summation (Kahan) when accumulating the numerator of (B17) to reduce loss of significance [232]; report both εhand Ohwith absolute and relative tolerances. 6. From link variables to cochains (practical glue) Building Ah and Fspin,h .From smoothed Bloch frames (Sec. B 3), define the (Abelian) edge cochain Ahby Ah(eµ(k)) = Arg Uµ(k)/∆kµ∈g0,(B18) so that dhAh reproduces the FHS curvature to leading order, while preserving gauge invariance of loop sums. In the collinear case, Fspin,h = dhAh ; in non-collinear settings, build g0 –valued Ah by logarithms of U(N)links (polar factors) and project onto the relevant Lie subalgebra. Fitting Bh (fake-flat lift). Given the alternation grammar tΦ,h ( Bh ) = γ ∆ ` ( k )(Sec. V A), solve the Tikhonov-regularized least squares min BhkFspin,h −tΦ,h(Bh)k2 h+αkBhk2 h,(B19) with normal equations ( t†t + αI ) Bh = t†Fspin,h cellwise. In the scalar–collinear ansatz, γ = hFspin,h, ∆ `ih/h ∆ `, ∆ `ih matches Eq. (C4). 7. Convergence and validation suite Mesh refinement. Double ( Nx, Ny )and recompute: (i) the integers (Chern numbers, wall indices) must be invariant; (ii) εh should stabilize and decrease if the configuration approaches the coherence fixed point; (iii) CDC–based curvatures should converge with order 2; spectral with exponential decay until sampling limits. 64 Outcome. This appendix supplies the numerics behind the paper: complete figure parameters, symmetryadapted grammar dictionaries, quantitative ε –scalings that justify Eq. (126), and a time-domain pump benchmark that stress-tests Eq. (154). Together with Appendix B, it closes the reproducibility loop from band data to device-level plots. Appendix E: Cross–Disciplinary Links Purpose. This appendix connects our “proto–gauge” control picture to classical H∞ robust control, unpacks the categorical semantics (coherence diagrams, naturality, and differential 2–gerbes) behind the 2–connection ( A, B ), and relates the variable–structure complex of the main text (cf. Eq. (83) in the compiled manuscript) to the BRST bicomplex from Sec. III C. The goal is to help readers bridge notation across control theory, higher gauge theory, and field–theoretic cohomology. 1. Proto–gauge small–gain and classical H∞control Plant/interconnection model. Linearizing the response in the proto–gauge window (Sec. VII B) gives a closed–loop block diagram identical in form to the “plant P + uncertainty ∆” template of robust control [ 243 ]. Let u be the actuator (a small control step in TΦM ), y the measured response (e.g. a spin–charge conversion channel), and let ∆encode the inter–level exchange constrained by Ward identities: y=Gyuu+Gyη η, η= ∆ y, k∆k∞≤ε, (E1) where G •• are linear time–invariant operators obtained from the linearization of the action around the bias point (Appendix A); η bundles (i) B –level exchange into A –currents and (ii) residual nonidealities whose gain is Ward–bounded by the proto–gauge defect ε (Sec. VII B, Eq. (193) ). The induced L2 –gain k·k∞ is taken on the frequency axis for slowly modulated drives; the storage function is the quadratic curvature budget in the coherence action. Small–gain theorem and Eq. (195) .Closing the loop η = ∆ y yields the standard M–∆interconnection with transfer y= (I−M∆)−1Gyuu, M:= Gyη.(E2) The classical small–gain theorem states that the feedback is well–posed and stable for all ∆with k ∆ k∞≤ε whenever kMk∞ε < 1,(E3) and moreover the closed–loop u→ygain satisfies kTyuk∞=k(I−M∆)−1Gyuk∞≤kGyuk∞ 1−kMk∞ε.(E4) Equation (E4) is the control–theoretic version of the bound written in the main text as Eq. (195) (printed as Eq. (195)): the proto–gauge defect ε plays the role of an uncertainty radius. Ward identities guarantee that the only admissible ∆are those that exchange level–2 and level–1 currents (Appendix A, Eq. (A18) ), thereby fixing k M k∞ in terms of the adjoints ( .Φ ) † and t† Φ chosen by the inner products in the action (Sec. III A). Performance and H∞ synthesis. If one adds a shaping filter on u and a performance channel z = W y (e.g. forbidden–irrep leakage), the objective kTzuk∞≤γ? can be addressed by standard H∞ output– feedback synthesis on the linearized plant, with robustness certified so long as (E3) holds with the realized ε (experimentally monitored in Fig. 2). In this language, the Ward–cone slopes Clo, Chi (Sec. VI B) give adata–driven bracket on kMk∞, while εacts as the tunable uncertainty norm. Passivity/energy interpretation. The coherence action provides a quadratic storage function S = (2 g2 ) −1kFΦk2 + (2 h2 ) −1kHΦk2 whose dissipation inequality under small δ Φyields the same linear bounds as (E4) by the Kalman–Yakubovich–Popov lemma. Thus the proto–gauge regime is simultaneously a small–gain and a passivity neighborhood: curvature budgets serve as energy, ε bounds the supply rate through exchange channels. 65 2. Categorical coherence: diagrams, naturality, and differential 2–gerbes 2–functor semantics of a 2–connection. Let P2 ( Td )be the path 2–groupoid of the Brillouin torus: objects are points k , 1–morphisms are (thin–homotopy classes of) paths γ , and 2–morphisms are homotopies between paths. A strict 2–connection ( A, B )valued in a crossed–module ( g1 tΦ −→ g0, .Φ ) integrates to a 2–functor Hol(A,B):P2(Td)−→ GΦ,(E5) where GΦ is the Lie 2–group picked by Φ(Sec. II A). The 1–holonomy along γ is the usual Wilson line PeRγA , and the 2–holonomy on a surface Σis the surface–ordered exponential of B , transported by the 1–holonomies along its boundary [ 244 , 245 ]. Flatness ( FΦ = HΦ =0) makes Hol astrict 2–functor; away from flatness, the curvatures are the obstructions to the commutativity of the canonical coherence diagrams. Naturality squares and alternation grammar. Consider a square in k –space with horizontal and vertical paths γ1, γ2and homotopic “fillings” Σ,Σ0. The naturality square compares the two 2–holonomies: Hol(γ1)◦Hol(γ2) =⇒Hol(γ1◦γ2) ⇓Hol(Σ) ⇓Hol(Σ0) Hol(γ0 1)◦Hol(γ0 2) =⇒Hol(γ0 1◦γ0 2) and fails to commute by an amount governed by FΦ and HΦ . Our alternation grammar tΦ ( B )has zero BZ average (Sec. II C); hence its net contribution cancels on global cycles, while locally it reshapes the square via the tΦ –image of the 2–holonomy. In categorical terms: tΦ is the component of the modification that trivializes symmetry–forced 2–holonomy up to a 1–cell—precisely what “magnetically silent grammar” encodes. Differential 2–gerbes and Breen–Messing data. A (nonabelian) differential 2–gerbe with structure 2–group GΦ is specified by local connective data ( A, B )on a good cover together with 1– and 2– gauge transformations on overlaps, constrained by higher cocycle conditions [ 246 ]. Our fake curvature FΦ = dA + A∧A−tΦ ( B )and 2–curvature HΦ = dB + A .ΦB are the Breen–Messing 2–curvature components. The variable–structure Bianchi identities (Sec. II B, Appendix A, Eqs. (A5) – (A7) ) are exactly the descent equations enforcing coherences of 2–morphisms under pullback by Φ. The “coherence fixed point” FΦ = HΦ = D Φ=0 (Sec. VII A) says the 2–gerbe is flat and the 2–functor is strict; quantized wall steps then come from integer classes (ω3(Φ)) via Chern–Simons descent on interfaces. 3. Variable–structure complex vs. BRST bicomplex The variable–structure double complex (main Eq. (83)). At fixed Φ, the covariant de Rham complex (Ω • ( Td, g0⊕g1 ) , DA )sits horizontally. Allowing Φto vary adds a vertical differential δΦ that acts on the structure maps (tΦ, .Φ)and on the fields by δΦA= 0, δΦB= 0, δΦtΦ= (∂ΦtΦ)·DΦ, δΦ.Φ= (∂Φ .Φ)·(DΦ; ·,·),(E6) and propagates to FΦ, HΦ by Leibniz. The variable–structure complex referred to as Eq. (83) is the total complex with differential D:= DA+δΦ,so that D2= adFΦ+ (·).ΦHΦ,(E7) i.e. D2 closes on curvature. At the coherence fixed point ( FΦ = HΦ =0) the total complex is bicomplex with D2 =0, and one may deploy spectral–sequence arguments with either DA – or δΦ –first filtrations (the E1 page computes either DA –cohomology at fixed Φor the infinitesimal deformation cohomology of the structure maps). BRST bicomplex of Sec. III C and quasi–isomorphism. The BRST complex (Appendix A, Eq. (A19) ) is the Chevalley–Eilenberg (CE) complex of the gauge 2–algebra (with ghosts c and χ ). Coupling it to the deformation (structure–variation) directions yields a BRST–DG bicomplex with horizontal differential s and vertical δΦ acting on ( A, B, c, χ )and on ( tΦ, .Φ ). One computes (using Peiffer identities and Bianchi, Appendix A) s2= 0, δ2 Φ= 0, s δΦ+δΦs= adFΦ+ (·).ΦHΦ,(E8) 66 so the total BRST–deformation differential s + δΦ squares to curvature (as in (E7) ) and becomes nilpotent precisely at the coherence locus. The CE projection that forgets ghosts maps ( s + δΦ )–cohomology to D –cohomology; in proto–gauge ( ε 1) this map is a quasi–isomorphism to O ( ε ): cohomology groups agree up to controlled corrections. Technically, one builds a contracting homotopy for the ghost sector using the quadratic storage functional of the action; see [ 247 ] for the general BV–BRST homological algebra and [248] for cohesive–topos models of differential cohomology. Ward identities as BRST exactness. The higher Ward identities (Appendix A, Eq. (A18) ) are the statement that the Noether current is BRST–exact in the horizontal direction (with s ) and exact up to exchange in the vertical ( δΦ ). In particular, the exchange law D∗ AJtot A = ( .Φ ) † ( B, Jtot B )is the CE–image of s –exactness in the BRST complex: it vanishes on cohomology classes of gauge–invariant observables and is saturated by B –level exchange. This interprets the “magnetic silence” of tΦ ( B )as a BRST–trivial mode for bulk magnetization, while retaining it as a nontrivial boundary inflow degree of freedom (Chern–Simons descent on walls). 4. A short “cheat sheet” for practitioners •Control lens. Treat ε as an uncertainty radius; check k M k∞ε < 1(Eq. (E3) ). The Ward–cone slopes upper–bound k M k∞ directly from data; then H∞ synthesis on the linearized plant provides robust gain shaping. •Category lens. The grammar tΦ ( B )trivializes symmetry–forced 2–holonomy locally while integrating to zero globally; walls are natural transformations with integer indices; pumping is the time–direction version of the same descent. •Cohomology lens. The variable–structure complex D = DA + δΦ squares to curvature; at coherence ( ε→ 0) it is a bicomplex. The BRST + deformation total differential s + δΦ is quasi–isomorphic to D; Ward identities are exactness statements in this bicomplex. Summary. 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