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1 \section{Links compare neighboring fibers}\label{g:links}
2 
3 A field assigns a vector to each site, but a local change of frame can be chosen independently at every site. A difference such as $\psi_{x+\widehat\mu}-\psi_x$ therefore needs a rule for comparing its two entries. A link supplies that rule. Its orientation is part of its definition, rather than a sign convention to be inferred after a calculation.
4 
5 Let $V$ be a finite periodic lattice and let every fiber be a copy of $\C^r$. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below,
6 \begin{equation}\label{g:backward-link}
7  U_{x,\mu}:\HH_{x+\widehat\mu}\longrightarrow\HH_x,
8  \qquad U_{x,\mu}^*U_{x,\mu}=U_{x,\mu}U_{x,\mu}^*=I.
9 \end{equation}
10 The gauge $g=(g_x)_{x\in V}$ changes coordinates by $\psi_x^g=g_x\psi_x$. Requiring the transported vector to have the frame of its destination gives
11 \begin{equation}\label{g:link-transform}
12  U_{x,\mu}^g=g_xU_{x,\mu}g_{x+\widehat\mu}^{-1},
13  \qquad U_{x,\mu}^g\psi_{x+\widehat\mu}^g
14  =g_xU_{x,\mu}\psi_{x+\widehat\mu}.
15 \end{equation}
16 Products act on column vectors from right to left. More generally, a transporter $T_{y\leftarrow x}$ has the transformation $g_yT_{y\leftarrow x}g_x^{-1}$. This notation checks both the input and output of every product.
17 
18 The forward link is
19 \begin{equation}\label{g:forward-link}
20  u_{x,\mu}=U_{x,\mu}^{-1}:\HH_x\longrightarrow\HH_{x+\widehat\mu},
21  \qquad u_{x,\mu}^g=g_{x+\widehat\mu}u_{x,\mu}g_x^{-1}.
22 \end{equation}
23 The older time-dependent Abelian walk writes this forward link as $L_j$. The later geometric Lean sources use $u$. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward $U$. These formulas describe the same edge in opposite directions. In particular, reversing an edge changes its transporter to the inverse, not to the same matrix in a different notation.
24 
25 \begin{proposition}[Covariance of linked differences]\label{g:stencil-covariance}
26 For spacing $h>0$, define
27 \begin{align}
28  (D^+_{\mu,U}\psi)_x
29  &=\frac{U_{x,\mu}\psi_{x+\widehat\mu}-\psi_x}{h},\label{g:forward-difference}\\
30  (\Delta_U\psi)_x
31  &=\frac1{h^2}\sum_\mu\left(
32  U_{x,\mu}\psi_{x+\widehat\mu}
33  +U_{x-\widehat\mu,\mu}^{-1}\psi_{x-\widehat\mu}-2\psi_x\right).
34  \label{g:laplacian}
35 \end{align}
36 Then $D^+_{\mu,U^g}(g\psi)=gD^+_{\mu,U}\psi$ and
37 $\Delta_{U^g}(g\psi)=g\Delta_U\psi$.
38 \end{proposition}
39 
40 \begin{proof}
41 For the positive edge, substitute \eqref{g:link-transform}. The two gauge factors at $x+\widehat\mu$ cancel, leaving $g_x$ on the left. For the negative edge,
42 \[
43  (U_{x-\widehat\mu,\mu}^g)^{-1}g_{x-\widehat\mu}
44  =g_xU_{x-\widehat\mu,\mu}^{-1}.
45 \]
46 Every term in either difference therefore has the common factor $g_x$.
47 \end{proof}
48 
49 The lattice inner product is $\inner{\phi}{\psi}_V=\sum_x\phi_x^*\psi_x$, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity
50 \begin{equation}\label{g:laplace-form}
51  -\inner{\psi}{\Delta_U\psi}_V
52  =\sum_{x,\mu}\left\lVert(D^+_{\mu,U}\psi)_x\right\rVert^2.
53 \end{equation}
54 Indeed, expanding one squared difference gives
55 $h^{-2}(\norm{\psi_{x+\widehat\mu}}^2+\norm{\psi_x}^2
56 -2\operatorname{Re}(\psi_x^*U_{x,\mu}\psi_{x+\widehat\mu}))$.
57 Summing and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so $\Delta_U$ is self-adjoint and nonpositive. This finite identity uses the periodic boundary condition; an open domain needs its boundary terms or a specified boundary condition.
58 
59 \gaugefigure{01_link_covariance}{A backward link $U$ and its forward inverse $u$ compare the same two fibers. Gauge factors cancel at the input and remain at the destination. The scalar example illustrates the exact transformation law.}
60 
61 
62 
63 \section{A linked walk and its actual charge flux}\label{g:linked-walk}
64 
65 Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let $N\ge1$ be an integer, let $j\in\Z/N\Z$, let $\psi_j^n\in\C^2$, and let $U_j^n\in U(1)$ be backward spatial links. Let $T_j^n\in U(1)$ transport from $(j,n)$ to $(j,n+1)$. At each step choose two on-site unitary coins $C_{1,j}^n,C_{2,j}^n\in U(2)$ and define
66 \begin{align}
67  \eta_j&=C_{1,j}^n\psi_j^n,\label{g:walk-coin}\\
68  \xi_j^+&=(U_{j-1}^n)^{-1}\eta_{j-1}^+,
69  &\xi_j^-&=U_j^n\eta_{j+1}^-,\label{g:walk-shift}\\
70  \psi_j^{n+1}&=T_j^n C_{2,j}^n\xi_j.\label{g:walk-step}
71 \end{align}
72 Here the $+$ channel moves toward increasing $j$, while the $-$ channel moves toward decreasing $j$. The legacy implementation takes the two coins to be equal half mass rotations. The argument needs only their unitarity and the fact that scalar phases commute with them.
73 
74 For a time-dependent scalar gauge $g_j^n$, use
75 \begin{equation}\label{g:temporal-link}
76  (U_j^n)^g=g_j^nU_j^n(g_{j+1}^n)^{-1},
77  \qquad (T_j^n)^g=g_j^{n+1}T_j^n(g_j^n)^{-1}.
78 \end{equation}
79 Spatial transport first brings the gauge factor to site $j$ on the old slice. Temporal transport then brings it to the new slice. Thus the step operator $\mathcal W_n$ satisfies
80 \begin{equation}\label{g:walk-covariance}
81  \mathcal W_n[U^g,T^g]G^n=G^{n+1}\mathcal W_n[U,T],
82  \qquad (G^n\psi)_j=g_j^n\psi_j.
83 \end{equation}
84 Every factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channels followed by unit-modulus multiplication.
85 
86 \begin{proposition}[Exact one-step continuity]\label{g:walk-continuity}
87 Set $\rho_j^n=\norm{\psi_j^n}^2$ and
88 \begin{equation}\label{g:walk-current}
89  J_{j+1/2}^n=|\eta_j^+|^2-|\eta_{j+1}^-|^2.
90 \end{equation}
91 The linked walk obeys
92 \begin{equation}\label{g:walk-balance}
93  \rho_j^{n+1}-\rho_j^n=J_{j-1/2}^n-J_{j+1/2}^n.
94 \end{equation}
95 Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc $A$ with specified left and right boundary edges,
96 $Q_A^{n+1}-Q_A^n=J_{\partial_-A}^n-J_{\partial_+A}^n$.
97 \end{proposition}
98 
99 \begin{proof}
100 The first coin preserves the local norm, so
101 $\rho_j^n=|\eta_j^+|^2+|\eta_j^-|^2$. The second coin and temporal phase preserve the norm after the shift, giving
102 $\rho_j^{n+1}=|\eta_{j-1}^+|^2+|\eta_{j+1}^-|^2$.
103 Subtract these expressions and group the two terms associated with each boundary edge. This is \eqref{g:walk-balance}. The transformed $\eta_j$ is $g_j^n\eta_j$, so its channel norms are unchanged. Summing the balance over the full cycle or the chosen arc cancels its internal currents.
104 \end{proof}
105 
106 This current measures charge transferred during one discrete step. Dividing it by a chosen time step would give a rate. It is not the same expression as the instantaneous Hamiltonian current introduced below. The two expressions belong to different evolution laws, even though both produce a local balance by cancellation.
107 
108 \gaugefigure{06_walk_continuity}{The explicit finite coined walk supplies a current before a continuum interpretation is attempted. The plotted computation checks one-step local balance and total population for its declared finite data.}
109 
110 
111 
112 \section{Unit-circle Clifford channels in the formal transport layer}\label{g:circle-transport}
113 
114 The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let $A=A^*$ and $A^2=I$ on a finite spinor space. Then
115 \begin{equation}\label{g:clifford-projections}
116  P^+_A=\tfrac12(I+A),\qquad P^-_A=\tfrac12(I-A)
117 \end{equation}
118 are self-adjoint complementary orthogonal projections. Idempotence and $P_A^+P_A^-=0$ follow by expanding the products and using $A^2=I$; self-adjointness follows from $A=A^*$. Different spatial Clifford matrices generally have different spectral projections. A permanent assignment of the four spinor coordinates to two Boolean channels cannot replace this construction in all three directions.
119 
120 For a bijection $\tau:V\to V$ and a forward scalar link $u:V\to U(1)$, define
121 \begin{equation}\label{g:clifford-shift}
122  (S_{A,\tau,u}\psi)(x)
123  =u(\tau^{-1}x)P_A^+\psi(\tau^{-1}x)
124  +u(x)^{-1}P_A^-\psi(\tau x).
125 \end{equation}
126 This agrees with the orientation dictionary \eqref{g:forward-link}. Its inverse is
127 \begin{equation}\label{g:clifford-inverse}
128  (S_{A,\tau,u}^{-1}\phi)(x)
129  =u(x)^{-1}P_A^+\phi(\tau x)
130  +u(\tau^{-1}x)P_A^-\phi(\tau^{-1}x).
131 \end{equation}
132 Substitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\tau$ in each channel, gives
133 \begin{equation}\label{g:clifford-inner}
134  \sum_x\inner{S\psi(x)}{S\phi(x)}
135  =\sum_x\inner{\psi(x)}{\phi(x)}.
136 \end{equation}
137 For $u^g(x)=g(\tau x)u(x)g(x)^{-1}$, direct endpoint cancellation gives $S_{A,\tau,u^g}G_g=G_gS_{A,\tau,u}$. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.
138 
139 The genuine unit-circle hypothesis is essential. A nonzero complex scalar is invertible, but multiplication by $2$ multiplies squared norm by $4$. The phase must therefore lie in the unit circle, not merely in $\C^\times$. Norm preservation depends on that mathematical hypothesis. The formalization companion identifies its precise type and transport conventions.
140 
141 The checked four-spinor construction in this layer uses scalar $U(1)$ phases. The $SU(2)$ matrices of the legacy non-Abelian chapter act on an internal two-dimensional fiber. To combine that color action with a Dirac spinor, one would specify a space such as $\C^4\otimes\C^2$, with spin projectors on the first factor and gauge transport on the second. This guarantees their commutation. No assertion here identifies the two-dimensional color fiber with the four-dimensional Clifford spinor or declares the resulting non-Abelian Dirac extension checked.
142 
143 
144 
145 \section{Plaquettes, invariant observables, and curvature phases}\label{g:plaquettes}
146 
147 On a two-dimensional coordinate plane, the backward links define the based plaquette
148 \begin{equation}\label{g:plaquette}
149  W_{x,\mu\nu}
150  =U_{x,\mu}U_{x+\widehat\mu,\nu}
151  U_{x+\widehat\nu,\mu}^{-1}U_{x,\nu}^{-1}.
152 \end{equation}
153 Starting with a vector at $x$, the rightmost factor moves it to $x+\widehat\nu$; the next moves it to the far corner; the last two bring it back to $x$. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving
154 \begin{equation}\label{g:plaquette-covariance}
155  W_{x,\mu\nu}^g=g_xW_{x,\mu\nu}g_x^{-1}.
156 \end{equation}
157 For $U(1)$ the plaquette itself is invariant. For a noncommutative group it is its conjugacy class that is invariant; the matrix entries depend on the frame at $x$.
158 
159 For $SU(2)$ links, the Wilson action of the source is the finite function
160 \begin{equation}\label{g:wilson-action}
161  S_W[U]=\frac1{g_{\mathrm{YM}}^2}
162  \sum_{x,\mu<\nu}\left(1-\tfrac12\operatorname{Re}\operatorname{Tr}W_{x,\mu\nu}\right),
163  \qquad g_{\mathrm{YM}}\in\R\setminus\{0\}.
164 \end{equation}
165 The subscript distinguishes the coupling from the site gauge $g_x$. Cyclicity of trace proves $S_W[U^g]=S_W[U]$. Since an $SU(2)$ matrix has eigenvalues $e^{i\theta},e^{-i\theta}$, a plaquette term is $1-\cos\theta\ge0$. In the Abelian case the analogous term is $1-\operatorname{Re}W$. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interacting field theory.
166 
167 \subsection{An Abelian surface product}
168 
169 For scalar links, multiply the oriented plaquettes on the six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so
170 \begin{equation}\label{g:abelian-cube}
171  \prod_{f\subset\partial c}W_f^{\operatorname{or}(f,c)}=1.
172 \end{equation}
173 The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If $W_f=e^{i\theta_f}$ with each $\theta_f$ chosen as a principal argument, it implies that the oriented phase sum lies in $2\pi\Z$. It does not force that sum to be zero as a real number. An unwrapped small-phase regime or a specified branch is needed for that stronger conclusion.
174 
175 \gaugefigure{02_abelian_plaquette}{For Abelian links, the oriented edge phases cancel around adjacent faces. A holonomy is branch independent; a displayed real phase requires a branch choice.}
176 
177 \subsection{Small smooth plaquettes}
178 
179 Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\theta}$, where the small real phase is $\theta=-q\int_{\square}F_{\mu\nu}\,dS$. Here $q$ is the charge and $F_{\mu\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\mu,a_\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:
180 \begin{equation}\label{g:centered-plaquette}
181  \theta=-qa_\mu a_\nu F_{\mu\nu}(x_c)+O(a^4).
182 \end{equation}
183 Consequently,
184 \begin{equation}\label{g:wilson-expansion}
185  1-\operatorname{Re}W
186  =\tfrac12q^2a_\mu^2a_\nu^2 F_{\mu\nu}(x_c)^2+O(a^6).
187 \end{equation}
188 To see the remainder, use $1-\cos\theta=\theta^2/2+O(\theta^4)$ and $\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.
189 
190 
191 
192 \section{A finite Maxwell--Dirac model and its conserved constraint}\label{g:maxwell-dirac}
193 
194 The finite lattice can also carry dynamical link variables. The following is the particular semidiscrete model in the Abelian source, with time continuous and space a periodic cycle of $N\ge3$ sites. It is useful because its matter current, electric work, and Gauss propagation can all be checked in the same equations.
195 
196 Fix $h>0$, $m\in\R$, and Hermitian spin matrices $\alpha=\alpha^*$ and $\beta=\beta^*$ on $\C^s$. Let $\psi_j\in\C^s$ and $A_j,E_j\in\R$, and set $U_j=e^{-ihA_j}$. The source absorbs the charge into $A$. Its real Hamiltonian is
197 \begin{align}
198  H_h&=H_{\mathrm m}+H_E,\label{g:hamiltonian}\\
199  H_{\mathrm m}
200  &=hm\sum_j\psi_j^*\beta\psi_j
201    +\sum_j\operatorname{Im}(\psi_j^*\alpha U_j\psi_{j+1}),
202  &H_E&=\frac h2\sum_jE_j^2.\label{g:hamiltonian-parts}
203 \end{align}
204 Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions:
205 \begin{equation}\label{g:action}
206  \mathcal S_h=\int\left[
207  \frac{ih}{2}\sum_j(\psi_j^*\dot\psi_j-\dot\psi_j^*\psi_j)
208  +h\sum_jE_j\dot A_j-H_h\right]dt.
209 \end{equation}
210 Variations with fixed endpoint data yield $ih\dot\psi_j=\partial H_h/\partial\overline\psi_j$, $h\dot A_j=\partial H_h/\partial E_j$, and $h\dot E_j=-\partial H_h/\partial A_j$. Written out, these are
211 \begin{align}
212  \dot\psi_j&=-im\beta\psi_j
213  -\frac1{2h}\alpha U_j\psi_{j+1}
214  +\frac1{2h}U_{j-1}^*\alpha\psi_{j-1},\label{g:matter-ode}\\
215  \dot A_j&=E_j,
216  &\dot E_j&=J_j,
217  &J_j&=\operatorname{Re}(\psi_j^*\alpha U_j\psi_{j+1}).
218  \label{g:field-ode}
219 \end{align}
220 For example, if $z_j=\psi_j^*\alpha U_j\psi_{j+1}$, then
221 $\partial_{A_j}\operatorname{Im}z_j
222 =\operatorname{Im}(-ihz_j)=-h\operatorname{Re}z_j$.
223 This fixes the positive sign of $\dot E_j=J_j$ for the conventions in the action.
224 
225 \begin{theorem}[Charge balance and Gauss propagation]\label{g:gauss}
226 For every differentiable solution of \eqref{g:matter-ode}--\eqref{g:field-ode}, define
227 \begin{equation}\label{g:gauss-def}
228  \rho_j=\psi_j^*\psi_j,\qquad
229  \bar\rho=\frac1N\sum_j\rho_j,\qquad
230  \mathcal G_j=\frac{E_j-E_{j-1}}h+\rho_j-\bar\rho.
231 \end{equation}
232 Then
233 \begin{equation}\label{g:gauss-balance}
234  \dot\rho_j+\frac{J_j-J_{j-1}}h=0,
235  \qquad\dot{\bar\rho}=0,
236  \qquad\dot{\mathcal G}_j=0.
237 \end{equation}
238 In particular, initial data satisfying $\mathcal G_j=0$ retain that constraint on their interval of existence.
239 \end{theorem}
240 
241 \begin{proof}
242 Differentiate $\rho_j$ and use $\dot\rho_j=2\operatorname{Re}(\psi_j^*\dot\psi_j)$. The mass term contributes zero, since $\psi_j^*\beta\psi_j$ is real. The forward hopping term contributes $-J_j/h$. The backward scalar is the conjugate of $\psi_{j-1}^*\alpha U_{j-1}\psi_j$, using $\alpha=\alpha^*$ and the scalar nature of $U$. Its contribution is $J_{j-1}/h$. This proves continuity. Periodic summation gives $\dot{\bar\rho}=0$. Finally,
243 \[
244  \dot{\mathcal G}_j
245  =\frac{\dot E_j-\dot E_{j-1}}h+\dot\rho_j-\dot{\bar\rho}
246  =\frac{J_j-J_{j-1}}h-\frac{J_j-J_{j-1}}h=0.
247 \]
248 \end{proof}
249 
250 The subtraction of $\bar\rho$ is structural on a periodic cycle: the discrete divergence sums to zero, so a constraint containing a nonzero total charge without compensating background would be inconsistent. Equation \eqref{g:gauss-def} supplies that neutralizing mean. Conservation of $\mathcal G_j$ is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over $a\le j\le b$ gives the physical-length-weighted charge balance
251 \[
252  \frac{d}{dt}\left(h\sum_{j=a}^{b}\rho_j\right)=J_{a-1}-J_b.
253 \]
254 
255 \begin{proposition}[Electric work and total energy]\label{g:work}
256 Along the same solutions,
257 \begin{equation}\label{g:work-balance}
258  \dot H_E=h\sum_jE_jJ_j,\qquad
259  \dot H_{\mathrm m}=-h\sum_jE_jJ_j,\qquad
260  \dot H_h=0.
261 \end{equation}
262 \end{proposition}
263 
264 \begin{proof}
265 The first formula follows from $\dot E_j=J_j$. For the matter contribution, let $q_j=\partial H_{\mathrm m}/\partial\overline\psi_j=ih\dot\psi_j$. The state-dependent part of the real derivative is
266 $2\operatorname{Re}\sum_jq_j^*\dot\psi_j=0$, because $q_j^*\dot\psi_j=-ih\norm{\dot\psi_j}^2$ is purely imaginary. The remaining explicit link derivative is
267 \[
268  \sum_j\frac{\partial H_{\mathrm m}}{\partial A_j}\dot A_j
269  =-h\sum_jJ_jE_j.
270 \]
271 Adding the two sectors proves total conservation.
272 \end{proof}
273 
274 \subsection{Which gauge transformations remain in this action?}
275 
276 For time-independent real $\chi_j$, the transformation
277 \begin{equation}\label{g:residual-gauge}
278  \psi_j\mapsto e^{i\chi_j}\psi_j,
279  \qquad A_j\mapsto A_j+\frac{\chi_{j+1}-\chi_j}{h},
280  \qquad E_j\mapsto E_j
281 \end{equation}
282 leaves the action and equations invariant. In fact $U_j$ then has precisely the backward-link law \eqref{g:link-transform}, every hopping scalar $z_j$ is unchanged, and the time derivatives of the gauge parameters vanish.
283 
284 This is the residual symmetry of the stated temporal-gauge model. If $\chi_j$ depends on time, the matter time derivative acquires $i\dot\chi_j\psi_j$, while $\dot A_j$ acquires $(\dot\chi_{j+1}-\dot\chi_j)/h$. With $E_j$ fixed and no scalar potential in \eqref{g:action}, these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete linked walk uses its temporal link $T$; it must not be transferred to this action after that degree of freedom has been removed.
285 
286 The statements in this section are ordinary proofs for a declared finite-dimensional classical field model. They establish genuine interaction between its matter and electric variables. They do not constitute a Lean construction of a coupled gauge Hamiltonian, a quantized gauge field, or a global continuum Maxwell--Dirac solution.
287 
288 
289 
290 \section{What a second-order covariant stencil requires}\label{g:consistency}
291 
292 The exact finite identities did not assume a smooth field in the background. A continuum comparison does. Let $A$ be a real $C^2$ function near $x$, let $\psi$ be $C^3$ there, and set
293 \begin{equation}\label{g:integrated-link}
294  U_h(x)=\exp\left(-i\int_x^{x+h}A(s)\,ds\right),\qquad
295  (D_h^A\psi)(x)
296  =\frac{U_h(x)\psi(x+h)-U_h(x-h)^{-1}\psi(x-h)}{2h}.
297 \end{equation}
298 The smooth $A$ in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking $A_j$ to be its edge average $h^{-1}\int_{x_j}^{x_j+h}A(s)\,ds$. This is an additional sampling prescription.
299 
300 \begin{proposition}[Local second-order consistency]\label{g:second-order}
301 With uniform bounds on $A$ and its derivatives through order two and on $\psi$ and its derivatives through order three on a fixed neighborhood,
302 \begin{equation}\label{g:stencil-limit}
303  (D_h^A\psi)(x)=(\partial_x-iA(x))\psi(x)+O(h^2)
304  \qquad(h\to0).
305 \end{equation}
306 The remainder is uniform over points whose neighborhoods have the same derivative bounds.
307 \end{proposition}
308 
309 \begin{proof}
310 For fixed $x$, put
311 $f_x(s)=\exp(-i\int_x^{x+s}A(r)\,dr)\psi(x+s)$.
312 Then the two numerator terms in \eqref{g:integrated-link} are $f_x(h)$ and $f_x(-h)$. The derivative at zero is
313 $f_x'(0)=\psi'(x)-iA(x)\psi(x)$.
314 Taylor expansion of $f_x$ at zero to second order with bounded third derivative gives
315 $f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3)$.
316 Division by $2h$ proves the assertion. The derivative bounds for $A$ and $\psi$ give the required uniform bound on $f_x'''$.
317 \end{proof}
318 
319 A midpoint link $\widetilde U_h(x)=e^{-ihA(x+h/2)}$ has the same conclusion under these bounds. The midpoint quadrature error in the integral is $O(h^3)$, and $|e^{ia}-e^{ib}|\le|a-b|$ for real $a,b$ transfers that error to the link. The two numerator perturbations are each $O(h^3)$ for bounded $\psi$, hence the stencil perturbation is $O(h^2)$.
320 
321 \begin{example}[Left-endpoint sampling can lose an order]\label{g:endpoint-counterexample}
322 Take $A(x)=x$ near the origin and $\psi(x)=1$. With the left-endpoint prescription $\widehat U_h(x)=e^{-ihA(x)}$, the stencil at zero is
323 \[
324  \frac{\widehat U_h(0)-\widehat U_h(-h)^{-1}}{2h}
325  =\frac{1-e^{-ih^2}}{2h}=\frac{i}{2}h+O(h^3).
326 \]
327 The continuum value $(\partial_x-iA(0))\psi(0)$ is zero. Thus this perfectly smooth input has a first-order error. The formula $U_j=e^{-ihA_j}$ by itself does not identify a second-order sampling scheme.
328 \end{example}
329 
330 This qualification does not alter the finite Hamiltonian or its conservation laws. It isolates what is needed to interpret its link values as samples of a continuum connection. Nor does consistency of one stencil prove convergence of the coupled time evolution: stability, compatible initial data, regularity over the comparison interval, and a choice of reconstruction still have to enter that argument.
331 
332 
333 
334 \section{Non-Abelian currents and their frame dependence}\label{g:color}
335 
336 Now take $\psi_x\in\C^2$, $U_{x,\mu}\in SU(2)$, and local gauges $g_x\in SU(2)$, as in the legacy non-Abelian chapter. The linked scalar
337 \begin{equation}\label{g:linked-scalar}
338  z_{x,\mu}=\psi_x^*U_{x,\mu}\psi_{x+\widehat\mu}
339 \end{equation}
340 is gauge invariant: the adjoint factor $g_x^*$ cancels $g_x$, and the factors at $x+\widehat\mu$ also cancel. It follows that the source's scalar link observable
341 \begin{equation}\label{g:su2-current}
342  j_{x,\mu}=2\operatorname{Im}z_{x,\mu}
343 \end{equation}
344 is invariant. To obtain a continuity equation with its particular sign and normalization one must specify the hopping generator. Invariance of the observable alone is not an equation of motion for color charge.
345 
346 Let $\sigma^1,\sigma^2,\sigma^3$ be the Pauli matrices and define the real components $q_x^a=\psi_x^*\sigma^a\psi_x$. Expand the conjugated Pauli basis as
347 \begin{equation}\label{g:adjoint-color}
348  g_x^*\sigma^a g_x=\sum_b R_x{}^a{}_b\sigma^b,
349  \qquad q_x^{g,a}=\sum_bR_x{}^a{}_bq_x^b.
350 \end{equation}
351 The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product $\tfrac12\operatorname{Tr}(AB)$ is preserved by conjugation, and the Pauli basis is orthonormal for it; hence $R_xR_x^{\mathsf T}=I$. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge $\psi_x^*\psi_x$ is separately invariant. A color component relative to a selected Pauli axis is not generally invariant under all local frames.
352 
353 The same distinction applies to a plaquette: $W$ changes by conjugation, while its trace does not. Keeping the full matrix retains transport information that a single trace can lose. Even for $SU(2)$, identical traces at separate plaquettes do not determine the relative frames needed to combine them into a larger loop.
354 
355 \gaugefigure{03_nonabelian_order}{Noncommuting $SU(2)$ factors retain their order. A based holonomy changes by conjugation, while the normalized Wilson trace is unchanged. The color fiber here is separate from the four-component Clifford spinor.}
356 
357 
358 
359 \section{Exact ordered refinement of a two-by-two patch}\label{g:ordered-refinement}
360 
361 The Abelian rule ``multiply the fine plaquettes'' hides two choices in a noncommutative group: where each plaquette is based, and in which order the transported matrices are multiplied. A $2\times2$ patch makes both choices visible without an asymptotic expansion.
362 
363 Write $X_{ij}$ for the backward horizontal link from $(i+1,j)$ to $(i,j)$, where $i=0,1$ and $j=0,1,2$. Write $Y_{ij}$ for the backward vertical link from $(i,j+1)$ to $(i,j)$, where $i=0,1,2$ and $j=0,1$. Define the four fine plaquettes by
364 \begin{equation}\label{g:fine-plaquette}
365  W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},
366  \qquad i,j\in\{0,1\}.
367 \end{equation}
368 Each $W_{ij}$ acts at its own lower-left basepoint $(i,j)$. The two row holonomies, based at $(0,j)$, are
369 \begin{equation}\label{g:row-holonomies}
370  R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},
371  \qquad j=0,1.
372 \end{equation}
373 The conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.
374 
375 \gaugefigure[H]{04_ordered_refinement}{The four fine plaquettes are brought to common basepoints in a specified order. The exact cancellation leaves the eight outer links; a naive product omits the transport information.}
376 
377 \begin{theorem}[Ordered patch identity]\label{g:patch-theorem}
378 For invertible links in any matrix group, the coarse boundary holonomy satisfies
379 \begin{align}
380  W^{(2)}
381  &:=X_{00}X_{10}Y_{20}Y_{21}
382  X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}
383  \label{g:coarse-boundary}\\
384  &=R_0Y_{00}R_1Y_{00}^{-1}.\label{g:patch-identity}
385 \end{align}
386 This identity requires neither small links nor commuting links.
387 \end{theorem}
388 
389 \Needspace{10\baselineskip}
390 \begin{proof}
391 First expand one row while retaining the product order:
392 \begin{align*}
393  R_j
394  &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}
395  X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\
396  &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.
397 \end{align*}
398 The adjacent pairs $X_{0j}^{-1}X_{0j}$ and $Y_{1j}^{-1}Y_{1j}$ cancel. Substituting these row expressions gives
399 \begin{align*}
400  R_0Y_{00}R_1Y_{00}^{-1}
401  ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}
402  \underbrace{Y_{00}^{-1}Y_{00}}_{I}\\
403  &\quad\cdot X_{01}X_{11}Y_{21}X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}.
404 \end{align*}
405 Now $X_{01}^{-1}X_{01}$ cancels, followed by $X_{11}^{-1}X_{11}$. The remaining factors are exactly \eqref{g:coarse-boundary}.
406 \end{proof}
407 
408 All terms in \eqref{g:patch-identity} transform by conjugation at $(0,0)$ once their displayed transports are included. This checks the basepoint typing as well as the algebra. In the Abelian case conjugations disappear and the fine plaquettes can be reordered, recovering the familiar surface product. In the non-Abelian case a naive product of the four untransported matrices can be formed after identifying their coordinate spaces, but it has no reason to obey the required local gauge transformation or to equal the boundary holonomy.
409 
410 This refinement is exact because it refers to one fixed set of fine links and its outer boundary. It does not assert that independently assigned coarse links agree with products of fine links. That separate compatibility question is the starting point of the geometric refinement estimates in the next part of the monograph.
411 
412 
413 
414 \section{Extracting a local curvature and cancelling its first error}\label{g:richardson}
415 
416 Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed $A,B\in\mathfrak{su}(2)$ define the commutator loop
417 \begin{equation}\label{g:commutator-loop}
418  W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.
419 \end{equation}
420 Here $A$ and $B$ are constant skew-Hermitian traceless matrices, and $h$ is a real scalar near zero. Because $W_0=I$, a single analytic matrix logarithm near the identity can be used for all sufficiently small $|h|$. Define
421 \begin{equation}\label{g:log-curvature}
422  F_h=h^{-2}\log W_h,\qquad h\ne0.
423 \end{equation}
424 This construction describes a local constant-matrix model of noncommuting transport. A varying connection has additional derivative terms and must be analyzed with those terms present.
425 
426 \gaugefigure[H]{05_curvature_richardson}{For the specified constant generators, the finite calculation illustrates first-order raw curvature error and cancellation of its leading term. The analytic statement is Proposition~\ref{g:curvature-order}; a fitted slope over finitely many scales is a diagnostic of this example.}
427 
428 \begin{proposition}[Local logarithmic refinement]\label{g:curvature-order}
429 On the common analytic logarithm branch, in any fixed matrix norm,
430 \begin{align}
431  F_h&=[A,B]+hC_3(A,B)+O(h^2),\label{g:log-series}\\
432  C_3(A,B)&=\tfrac12[A+B,[A,B]],\label{g:cubic-coefficient}\\
433  F_h^{[1]}&:=2F_{h/2}-F_h=[A,B]+O(h^2).
434  \label{g:richardson-formula}
435 \end{align}
436 For constant $G\in SU(2)$, simultaneous conjugation of $A,B$ conjugates both $F_h$ and $F_h^{[1]}$ by $G$.
437 \end{proposition}
438 
439 \begin{proof}
440 Multiplying the four exponential power series through degree three gives
441 \[
442  W_h=I+h^2[A,B]+\frac{h^3}{2}[A+B,[A,B]]+O(h^4).
443 \]
444 The analytic logarithm has $\log(I+Z)=Z+O(\norm Z^2)$ near zero. Since $Z=W_h-I=O(h^2)$, its correction begins at order $h^4$. Dividing by $h^2$ proves \eqref{g:log-series}. At half spacing the linear term is $hC_3/2$, so $2F_{h/2}-F_h$ removes it exactly and leaves an $O(h^2)$ remainder. Finally, the power series for both the exponential and the local logarithm commute with conjugation. Applying them term by term proves the covariance statements.
445 \end{proof}
446 
447 The common logarithm branch is essential: unrelated choices can introduce terms of size $h^{-2}$ after rescaling and invalidate the cancellation. The Richardson value is a linear combination of Lie-algebra estimates; it is not asserted to be the logarithm obtained by the exact patch rule.
448 
449 

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