Gauge Symmetry and Conservation-Gauge Unification¶

The complex geometric field $\Psi = K_\phi + i \cdot J_\phi$ admits a nodewise U(1) field-coordinate action in the auxiliary model. This document derives its algebraic invariants and records diagnostic comparisons with conservation, grammar and spectral projections.

Status: HISTORICAL RESEARCH MODEL — algebraic U(1) identities retained; dynamical claims scoped below.

Current scope and supersession (2026-09). The rotation identities, gauge-invariant norms, Parseval identity and the exact harmonic substrate flow are valid within their stated auxiliary models. They do not derive the full nodal equation from the displayed action, make U1–U6 symmetries of that action, prove that engine operators are symplectomorphisms, or establish a conservation/Lyapunov theorem for arbitrary operator trajectories. The implemented connection is the exact one-form $A_{ij}=d(\arg\Psi)_{ij}$ modulo angle wrapping. Its U(1) cycle holonomy is the identity analytically, so the current implementation has no independent gauge curvature, vortices, flux, confinement sector or non-trivial Yang-Mills dynamics. The corresponding public names are retained as compatibility diagnostics. The structural energy is a non-negative finite-snapshot diagnostic and Lyapunov candidate. The separate exact decrease theorem concerns the Dirichlet energy of restricted pure-EPI diffusion; see STRUCTURAL_CONSERVATION_THEOREM.md, TNFR_VARIATIONAL_PRINCIPLE.md, and TNFR_DIFFUSION_STABILITY_THEOREM.md.

1. Nodewise U(1) field-coordinate action¶

1.1 Gauge Transformation¶

The auxiliary transformation rotates the geometric-transport coordinates $(K_\phi,J_\phi)$ at each node while holding the other snapshot fields fixed:

\[ \Psi(i) \;\to\; e^{i\alpha(i)}\,\Psi(i) \]

In component form:

\[ K_\phi'(i) = K_\phi(i)\cos\alpha(i) - J_\phi(i)\sin\alpha(i) \]

\[ J_\phi'(i) = K_\phi(i)\sin\alpha(i) + J_\phi(i)\cos\alpha(i) \]

The fields $\Phi_s$, $|\nabla\phi|$, $J_{\Delta\mathrm{NFR}}$, and $\xi_C$ are singlets by definition of this analysis transform. This transformation is not an engine operator and is not shown to map one reachable TNFR state or trajectory to another.

1.2 Gauge-Invariant Observables¶

The following quantities are invariant under this auxiliary rotation:

Quantity	Definition	Established statement
Energy density $\mathcal{E}(i)$	$\Phi_s^2 + \vert \nabla\phi\vert ^2 + \vert \Psi\vert ^2 + J_{\Delta\mathrm{NFR}}^2$	The $K_\phi,J_\phi$ quadratic sum is rotation invariant
Field magnitude $\vert \Psi(i)\vert ^2$	$K_\phi^2 + J_\phi^2$	Euclidean norm identity
Coherence $C(t)$	Depends on the unchanged graph fields	Unchanged because the analysis transform does not mutate the graph
Legacy topological norm $\vert \mathcal{T}\vert ^2$	$\mathcal{Q}^2 + \tilde{\mathcal{Q}}^2$	Norm of an algebraic doublet
Legacy chirality norm $\vert \mathcal{X}\vert ^2$	$\chi^2 + \tilde{\chi}^2$	Norm of an algebraic doublet

where: - $\tilde{\mathcal{Q}} = K_\phi \cdot |\nabla\phi| + J_\phi \cdot J_{\Delta\mathrm{NFR}}$ (dual topological charge) - $\tilde{\chi} = |\nabla\phi| \cdot J_\phi + K_\phi \cdot J_{\Delta\mathrm{NFR}}$ (dual chirality)

1.3 Rotation doublets and variant quantities¶

The following pairs transform as 2D rotation doublets under $\alpha$:

$(\mathcal{Q},\;\tilde{\mathcal{Q}})$ — topological charge doublet
$(\chi,\;\tilde{\chi})$ — chirality doublet

Two further historical quantities are gauge-frame dependent but are not members of those doublets:

Legacy quantity named Noether charge $Q = \sum(\Phi_s + K_\phi)$ — not invariant
Symmetry breaking $\mathcal{S} = (|\nabla\phi|^2 - K_\phi^2) + (J_\phi^2 - J_{\Delta\mathrm{NFR}}^2)$ — NOT invariant

The norms of the two doublets, $|\mathcal{T}|^2$ and $|\mathcal{X}|^2$, are invariant because their quadratic sums are preserved under rotation. No such norm-invariance statement is made for the separate $Q$ or $\mathcal S$ above.

1.4 Derivation Outline¶

The proof is the Euclidean norm identity for a two-dimensional rotation:

Step 1: Rotate only $(K_\phi,J_\phi)$ and hold four fields fixed.
Step 2: Component rotation formulas (§1.1).
Step 3: Bilinear forms involving one $\Psi$-component and one singlet transform as 2D doublets; their quadratic norms are invariant.

Implementation: src/tnfr/physics/gauge.py — verify_gauge_invariance(), GaugeInvarianceResult dataclass.

2. Exact connection and cycle closure¶

2.1 Gauge Connection¶

The implementation defines the edge connection from a vertex scalar:

\[ A_{ij} = \arg(\Psi_j) - \arg(\Psi_i) \quad\text{(wrapped to } [-\pi,\pi]\text{)} \]

Let $\theta_i=\arg\Psi_i$, with the implementation's deterministic zero phase only when the represented $\Psi_i$ is exactly zero. Every represented nonzero field retains its argument, including arbitrarily small magnitudes; an amplitude threshold would break the nonzero-field covariance identity below. The forward edge is wrapped to $[-\pi,\pi)$ and an undirected reverse is stored as its negative. At a half-turn this gives $-\pi$ and $+\pi$, respectively: distinct representatives of the same circular difference. For an integer $k_{ij}$,

\[ A_{ij}=\theta_j-\theta_i-2\pi k_{ij}. \]

Thus the exponentiated links satisfy $e^{iA_{ij}}=e^{i\theta_j}e^{-i\theta_i}$: they are pure gauge, not independent edge variables. The unwrapped vertex difference is an exact real one-form. The wrapped real edge values can instead sum to a nonzero integer multiple of $2\pi$ around a cycle.

2.2 Covariant Derivative¶

The discrete covariant derivative along edge $(i,j)$:

\[ D_{ij}\Psi = \Psi(j) - e^{iA_{ij}}\,\Psi(i) \]

At nonzero field endpoints, under gauge transformation $\Psi \to e^{i\alpha}\Psi$ and reconstruction of the link:

\[ D_{ij}\Psi \;\to\; e^{i\alpha(j)}\,D_{ij}\Psi \quad\text{(covariant)} \]

Therefore $|D_{ij}\Psi|$ is gauge-invariant. At a zero of $\Psi$, its phase is undefined and the implementation selects phase zero; the complex covariance formula then has no intrinsic meaning there, while the magnitude identity below remains valid.

More strongly, writing $\Psi_i=r_i e^{i\theta_i}$ gives

\[ D_{ij}\Psi=e^{i\theta_j}(r_j-r_i),\qquad |D_{ij}\Psi|=|r_j-r_i|. \]

The implemented covariant difference therefore measures endpoint amplitude contrast after the derived phase alignment. It does not introduce independent parallel-transport dynamics.

2.3 Cycle-closure residual¶

The legacy-named curvature function sums the exact connection around a cycle:

\[ F_C = \sum_{(i,j) \in C} A_{ij} \quad\text{(wrapped to } [-\pi,\pi]\text{)} \]

For $C=(v_0,\ldots,v_{m-1},v_0)$ the vertex-phase terms telescope:

\[ F_C = \sum_{r=0}^{m-1}(\theta_{v_{r+1}}-\theta_{v_r}-2\pi k_r) = -2\pi\sum_r k_r \equiv 0 \pmod {2\pi}. \]

Accordingly, any nonzero returned value is a floating-point wrapping or accumulation residual. It is not evidence of an independent field strength, vortex, topological defect, magnetic flux or confinement. Such phenomena would require independent edge curvature or an additional specified defect model, which this API does not expose. This does not rule out integer winding of a supplied nodal phase field: the final wrap discards that integer. The separate declared-cycle winding interface retains branch and orientation evidence and is not this gauge-curvature diagnostic.

2.4 Yang-Mills Action¶

The compatibility diagnostic

\[ S_{\mathrm{YM}} = \frac{1}{2}\sum_C F_C^2 \]

is therefore zero analytically and reports squared numerical closure error in floating arithmetic. Deriving field equations with a matter current would require an independently varied connection, a coupled matter action and boundary conditions; those are not supplied by this implementation. The compute_yang_mills_equations result is a legacy-named snapshot consistency evaluation, not an equation of motion for TNFR.

Implementation: src/tnfr/physics/gauge.py — compute_gauge_curvature(), compute_covariant_derivative_magnitude(), compute_yang_mills_action().

3. Historical four-label heuristic¶

The API retains four historical labels. They summarize selected coordinates; they are not derived fundamental interactions:

Legacy label	Input coordinate	Valid scope
em_like	$\arg(\Psi) \approx 0$	Projection on the selected $K_\phi$ axis; gauge-frame dependent
weak_like	$\arg(\Psi) \approx \pi/2$	Projection on the selected $J_\phi$ axis; gauge-frame dependent
strong_like	numerical $\vert F_C\vert $	Pure-gauge closure-error slot; zero within tolerance in a valid computation
gravity_like	$\vert \Phi_s\vert \gg \vert \Psi\vert $	Gauge-invariant potential-to-field magnitude comparison

The four scores share a unit reporting budget. The API marks a label active when its normalized score exceeds the equal-share reference $1/N_\text{labels}=1/4$. This is a compatibility convention, not a derived TNFR threshold. In particular, numerical closure noise is suppressed so the strong_like slot cannot be promoted into a physical claim.

Implementation: src/tnfr/physics/gauge.py — classify_interaction_regime(), classify_interaction_regime_formal(), GaugeSnapshot dataclass.

4. Operator and grammar boundary¶

Earlier versions assigned gauge roles to four canonical operators. Those roles are not derived by the implementation:

Historical claim	Current status
UM creates $A_{ij}$	Unsupported: $A$ is computed after the fact from $\Psi$ on every existing graph edge
IL minimizes $\vert D\Psi\vert $	Unsupported without before/after operator telemetry or a theorem
OZ sources $F_C$	Incompatible with a nonzero analytic final wrapped closure residual; the underlying wrapped-edge sum can be a nonzero integer multiple of $2\pi$
RA transports gauge invariants	Unsupported without an induced-map covariance check

Grammar rule U3 constrains the node phase $\phi$ before coupling via $|\operatorname{wrap}(\phi_i-\phi_j)|\le\Delta\phi_{\max}$. It is not a gauge-fixing rule for this auxiliary construction. Since $\Psi$ itself is extracted from graph fields, $\arg\Psi$ has not been established as an independent dynamical degree of freedom.

5. Auxiliary Conservation-Gauge Comparison¶

The variational owner defines the shared quadratic action/energy readouts and harmonic coordinate model. Agreement of the five squared-field sums is a snapshot algebraic identity. Auxiliary Hamiltonian conservation applies to its declared autonomous flow, not automatically to engine trajectories. The conservation owner defines the measured balance residual, including its normalization and units; grammar validity does not force that residual to vanish.

5.4 Availability of the grammar diagnostic¶

compute_grammar_symmetry_mapping in conservation_gauge_unification.py reports the evidence it can assess, rather than deriving conservation from historical rule labels:

Rules	Actual assessment boundary
U1, U2, U4	Always `not_assessed` by this snapshot API; validation needs ordered operator/trigger/debt history and execution context
U5	Always `not_assessed` here; validation needs declared parent/child hierarchy and normalization
U3	Current-edge phase compatibility requires explicit finite endpoint phases and the shared U3 gate
U6	Mean absolute potential drift requires a declared earlier graph or captured potential field; missing reference remains unavailable

These observations do not execute a grammar word, prove a Noether law or supply an all-time storage bound. Grammar owns sequence admission. The different analyze_grammar_stationarity snapshot heuristics retain their separate variational scope.

5.5 Constant oscillator rotation¶

verify_symplectic_gauge_compatibility checks one constant global SO(2) rotation of the auxiliary (K_phi,J_phi) pair. It computes the 2x2 pullback R.T @ omega @ R and determinant residual, with transformation_scope="global_constant_oscillator_rotation" and local_gauge_assessed=False. The legacy product/volume fields are snapshot statistics, not geometric certificates. This check does not supply local gauge dynamics or the derivative of any of the 13 engine operators. A proposed operator map needs the separate Jacobian-level test.

6. Spectral Balance Diagnostics¶

6.1 Graph Fourier Transform¶

The measured structural-balance residual can be represented in the spectral domain using the Graph Fourier Transform (GFT) in the Laplacian eigenbasis $\{\psi_k\}$.

Given the discrete structural continuity equation:

\[ \frac{\Delta\rho(i)}{\Delta t} + \operatorname{div}\mathbf{J}(i) = S_{\text{grammar}}(i) \]

Projecting the measured balance onto eigenvectors $\psi_k$:

\[ \frac{d\hat{\rho}_k}{dt} + \widehat{\operatorname{div}\mathbf J}_k = \hat{S}_k \]

where: - $\hat{\rho}_k = \langle\psi_k | \rho\rangle$ — charge density in mode $k$ - $\widehat{\operatorname{div}\mathbf J}_k = \langle\psi_k | \operatorname{div}\mathbf{J}\rangle$ — current divergence in mode $k$ - $\hat{S}_k = \langle\psi_k | S\rangle$ — source term in mode $k$ - $\lambda_k$ — graph Laplacian eigenvalue; a temporal frequency requires a separate evolution law and clock

A factor $\lambda_k$ appears only after separately representing the divergence as a Laplacian acting on a declared scalar current potential; it must not be multiplied into a quantity already defined as the projected divergence.

6.2 Physical Interpretation¶

Mode regime	Condition	Behaviour
Small $\lambda_k$	Slowly varying graph modes	Measure $\hat S_k$; neither a small residual nor a U5 interpretation follows from the eigenvalue alone
Large $\lambda_k$	Rapidly varying graph modes	Measure redistribution and residuals; rapid temporal dissipation requires a specified dissipative evolution

6.3 Parseval Conservation¶

Energy in the spatial domain equals energy in the spectral domain:

\[ \|\rho\|^2 = \sum_k |\hat{\rho}_k|^2 \]

For one field and one fixed orthonormal basis this is an exact coordinate identity. Apparent drift signals numerical error, a changed graph/basis, or a mismatched normalization; it is not by itself a grammar violation.

6.4 Spectral Energy Decomposition¶

The non-negative structural energy diagnostic decomposes mode-by-mode:

\[ E_k = \frac{1}{2}\left(|\hat{\Phi}_{s,k}|^2 + |\widehat{\nabla\phi}_k|^2 + |\hat{K}_{\phi,k}|^2 + |\hat{J}_{\phi,k}|^2 + |\hat{J}_{\Delta\mathrm{NFR},k}|^2\right) \]

For two supplied snapshots, the implementation reports finite differences $\Delta E_k/\Delta t$. Their sign is empirical. U2 grammar compliance alone does not imply $dE_k/dt\le0$, even for a word containing stabilizers.

Implementation: src/tnfr/physics/spectral_conservation.py — SpectralConservationBalance, SpectralWardIdentity, SpectralLyapunovResult dataclasses.

7. Spectral claims require a separate model¶

U(1) coordinate rotations and Parseval identities do not select a critical parameter, an analytic zeta representation or a nodal pulse law. The Riemann scope retains the inserted-threshold counterexample and finite arithmetic comparisons. Those independent models cannot acquire a dynamical bridge from the gauge diagnostics defined here.

Implementation Reference¶

Module	Content
`src/tnfr/physics/gauge.py`	Auxiliary U(1) rotations, exact connection, closure residuals and legacy four-label heuristic
`src/tnfr/physics/conservation_gauge_unification.py`	Auxiliary action and finite-snapshot compatibility diagnostics
`src/tnfr/physics/spectral_conservation.py`	Spectral residual, Parseval and finite-difference energy diagnostics

Tests: test_gauge.py, test_conservation_gauge_unification.py, test_spectral_conservation.py

Example: examples/02_physics_regimes/26_gauge_structure_demo.py

Implementation & Examples¶

Executable Demonstrations¶

Example	Concept from this document
26_gauge_structure_demo.py	U(1) rotation identities, exact connection and numerical cycle closure

Key Source Modules¶

src/tnfr/physics/gauge.py — Auxiliary U(1) field-coordinate diagnostics
src/tnfr/physics/fields.py — Complex geometric field Ψ = K_φ + i·J_φ

Cross-References¶

Nodal equation: FUNDAMENTAL_THEORY.md §1
Conservation laws: STRUCTURAL_CONSERVATION_THEOREM.md
Variational principle: TNFR_VARIATIONAL_PRINCIPLE.md
Grammar rules: UNIFIED_GRAMMAR_RULES.md
Extended fields ($J_\phi$, $J_{\Delta\mathrm{NFR}}$): EXTENDED_FIELDS_AND_DERIVED_QUANTITIES.md

Quantity	Definition	Established statement
Energy density \(\mathcal{E}(i)\)	\(\Phi_s^2 + \vert \nabla\phi\vert ^2 + \vert \Psi\vert ^2 + J_{\Delta\mathrm{NFR}}^2\)	The \(K_\phi,J_\phi\) quadratic sum is rotation invariant
Field magnitude \(\vert \Psi(i)\vert ^2\)	\(K_\phi^2 + J_\phi^2\)	Euclidean norm identity
Coherence \(C(t)\)	Depends on the unchanged graph fields	Unchanged because the analysis transform does not mutate the graph
Legacy topological norm \(\vert \mathcal{T}\vert ^2\)	\(\mathcal{Q}^2 + \tilde{\mathcal{Q}}^2\)	Norm of an algebraic doublet
Legacy chirality norm \(\vert \mathcal{X}\vert ^2\)	\(\chi^2 + \tilde{\chi}^2\)	Norm of an algebraic doublet

Legacy label	Input coordinate	Valid scope
em_like	\(\arg(\Psi) \approx 0\)	Projection on the selected \(K_\phi\) axis; gauge-frame dependent
weak_like	\(\arg(\Psi) \approx \pi/2\)	Projection on the selected \(J_\phi\) axis; gauge-frame dependent
strong_like	numerical $\vert F_C\vert $	Pure-gauge closure-error slot; zero within tolerance in a valid computation
gravity_like	$\vert \Phi_s\vert \gg \vert \Psi\vert $	Gauge-invariant potential-to-field magnitude comparison

Historical claim	Current status
UM creates \(A_{ij}\)	Unsupported: \(A\) is computed after the fact from \(\Psi\) on every existing graph edge
IL minimizes $\vert D\Psi\vert $	Unsupported without before/after operator telemetry or a theorem
OZ sources \(F_C\)	Incompatible with a nonzero analytic final wrapped closure residual; the underlying wrapped-edge sum can be a nonzero integer multiple of \(2\pi\)
RA transports gauge invariants	Unsupported without an induced-map covariance check

Mode regime	Condition	Behaviour
Small \(\lambda_k\)	Slowly varying graph modes	Measure \(\hat S_k\); neither a small residual nor a U5 interpretation follows from the eigenvalue alone
Large \(\lambda_k\)	Rapidly varying graph modes	Measure redistribution and residuals; rapid temporal dissipation requires a specified dissipative evolution