RTG L0 Master Definition

v1.2: Layer-Consistent Kinematics, SU(2) Lift, and Topology-Safe Computation Order

This publication document records the current RTG L0 master definition as a self-contained, kinematical foundation. L0 is strictly configurational: it operates on a single snapshot of phases and fixed edge data, without importing any dynamical stability concepts (which belong to L1+). The core L0 atom is \(n_i=(\mu_i,\hat{J}_i)\), where \(\mu_i\) is a local frustration magnitude and \(\hat{J}_i\) is a local \(\mathfrak{su}(2)\) frame generator produced by a canonical SU(2) lift specification.

1. Layer-Consistency Rule (Hard Constraint)

L0 objects are functions of a static snapshot \(\{\theta_i\}\in\mathbb{T}^N\) and fixed edge data. L0 must not reference \(\gamma\), \(dt\), integrators, Jacobians of update maps, Neimark–Sacker thresholds, attractors, or equilibrium assumptions. “Equilibrium” can be studied at L1+, but L0 definitions are valid for any snapshot, including transient and far-from-equilibrium states.

2. Configuration Space and Gauge

Configuration space is the \(N\)-torus \(\mathbb{T}^N\). L0 uses only phase differences. A global U(1) shift \(\theta_i\to\theta_i+\phi\) leaves all L0 observables invariant. For reporting, an operational gauge fix may be applied by setting one reference angle to zero (e.g., \(\theta_{\mathrm{ref}}=0\)) or projecting the phase vector onto the orthogonal complement of the all-ones vector.

3. L0 Node Atom: \(n_i=(\mu_i,\hat{J}_i)\)

3.1 \(\mu_i\): Local Frustration Magnitude (Canonical Total Load)

\(\mu_i\) is a nonnegative, purely configurational scalar computed locally from incident edges. The canonical definition is total load (not degree-normalized) to preserve bottleneck and bridge stress signals.

\[\mu_i := \sum_{j\sim i} w_{ij}\,|\sin((\theta_j-\theta_i)-\alpha_{ij})|\]

Canonical weight is fixed to prevent implementation drift: \(w_{ij}:=|\hat{\kappa}_{ij}|\). If no edge gauge is used, set \(\alpha_{ij}=0\). A degree-normalized diagnostic \(\bar{\mu}_i:=\mu_i/\deg(i)\) is permitted for ablations but is non-canonical.

3.2 \(\hat{J}_i\): Local Frame Generator in \(\mathfrak{su}(2)\)

\(\hat{J}_i\) is unit-normalized in \(\mathfrak{su}(2)\) and is constructed by the canonical SU(2) lift specification. No alternative lift is compliant with this master definition unless declared as a new variant identifier.

4. Canonical SU(2) Lift Specification (Fork-Preventer)

The SU(2) lift defines a canonical map from snapshot data \(\{\theta_i\}\), adjacency, and optional \(\alpha_{ij}\) to \(\hat{J}_i\) and edge transport \(U_{ij}\).

4.1 Frame Generator

\[\hat{J}_i := \exp(-i\theta_i\sigma_3/2)\,\sigma_1\,\exp(+i\theta_i\sigma_3/2)\]

Equivalently, \(\hat{J}_i=\cos(\theta_i)\sigma_1+\sin(\theta_i)\sigma_2\).

4.2 Base Transport (Canonical Channel)

\[U^{\mathrm{base}}_{ij} := \exp(-i(((\theta_j-\theta_i)-\alpha_{ij})\sigma_3/2))\]

Alignment check: \(U^{\mathrm{base}}_{ij}\hat{J}_i(U^{\mathrm{base}}_{ij})^\dagger=\hat{J}_j\) if and only if \(\alpha_{ij}=0\). If \(\alpha_{ij}\neq 0\), the residual misalignment is an \(\alpha_{ij}/2\) rotation about \(\sigma_3\); this residual is a direct algebraic seed for nonzero loop closure failure on frustrated faces.

4.3 Probe Transport (Diagnostic Only)

\[U^{\mathrm{opp}}_{ij} := U^{\mathrm{base}}_{ij}\,\exp(+i(\pi/2)\sigma_1)\]

The probe channel is diagnostic-only and is not canonical for loop observables unless explicitly declared in experiment metadata.

5. Resonance Eligibility Filter (Optional L0 Topology Refinement)

L0 may refine the substrate adjacency by an eligibility predicate based purely on configurational similarity:

\[\mathrm{EDGE\_ELIGIBLE}(i,j)\iff|\mu_i-\mu_j|<\varepsilon_\mu\]

\(\varepsilon_\mu\) is an L0 tolerance that controls sparsity and topology. L0 does not derive \(\varepsilon_\mu\) from dynamical parameters. A basic L0-safe nontriviality bound is \(0<\varepsilon_\mu<\mathrm{spread}(\{\mu_i\})\); if \(\varepsilon_\mu\) exceeds the spread of the \(\mu\) field, the filter collapses toward a complete graph and loses structural information.

6. Loop Observable: Pure Trace-Gap (Background-Independent)

For an oriented triangle face \(f=(i,j,k)\), define holonomy \(H_f:=U_{ij}U_{jk}U_{ki}\). The canonical L0 defect seed is pure closure failure:

\[\Delta_f^{(0)} := 1-\tfrac12\operatorname{Tr}(H_f)\]

For \(H_f\in SU(2)\), \((1/2)\operatorname{Tr}(H_f)=\cos(\varphi/2)\) for rotation angle \(\varphi\), hence \(\Delta_f^{(0)}=1-\cos(\varphi/2)\in[0,2]\). \(\Delta_f^{(0)}=0\) corresponds to trivial holonomy \(H_f=\mathbb{1}\), while \(\Delta_f^{(0)}=2\) corresponds to maximal holonomy \(H_f=-\mathbb{1}\).

6.1 Trace-Gap Revision Note

Earlier target-curvature forms such as \(\Delta_f=(1/2)\operatorname{Tr}(H_f)-\cos(F_\triangle/2)\) are deprecated at L0 because they require a reference curvature \(F_\triangle\). L0 uses \(\Delta_f^{(0)}\) only.

7. Topology Loop Hazard and the Required Computation Order

A subtle but fatal implementation hazard exists if \(\mu_i\) is computed only on edges that were first filtered by \(\mu\)-based eligibility. This creates a circular dependency between topology and frustration. The master definition therefore distinguishes substrate adjacency from emergent active adjacency and fixes the execution order.

Object	Meaning	Status
\(A_{\mathrm{base}}\)	Substrate adjacency (loaded input graph)	Given
\(A_{\mathrm{active}}\)	Active adjacency after L0 filtering	Derived

Execution protocol:

• Step 1: Load substrate inputs \((A_{\mathrm{base}},\{\theta_i\},\alpha_{ij},w_{ij}=|\hat{\kappa}_{ij}|)\).
• Step 2: Compute \(\mu_i\) for all nodes using neighbors in \(A_{\mathrm{base}}\) only.
• Step 3: Apply resonance filter on \(A_{\mathrm{base}}\) to produce \(A_{\mathrm{active}}:=\{(i,j)\in A_{\mathrm{base}}:|\mu_i-\mu_j|<\varepsilon_\mu\}\).
• Step 4: Compute SU(2) geometry on \(A_{\mathrm{active}}\) (canonical \(U^{\mathrm{base}}_{ij}\), holonomies \(H_f\), trace-gaps \(\Delta_f^{(0)}\) on faces in \(A_{\mathrm{active}}\)).

This linear order prevents circular “topology loops” and ensures consistent emergence of geometry from configuration via filtration.

8. Vacuum / Silent Regime Protocol

The ordered-phase degeneracy is resolved by a vacuum protocol. If both frustration and loop closure failure are numerically absent, L0 returns an empty structure as a valid outcome (not a failure), while allowing a base adjacency to exist as bookkeeping.

Condition: \(\max_i\mu_i<\varepsilon_{\mathrm{machine}}\) and \(\max_f\Delta_f^{(0)}<\varepsilon_{\mathrm{machine}}\). Interpretation: μ/Δ-derived structure is empty (silent regime). Base adjacency may still exist but carries no active structure under L0 observables.

9. Open Problems and Blocking Dependencies

• \(\varepsilon_\mu\) policy (topology-severe): \(\varepsilon_\mu\) controls sparsity and topology; an explicit calibration protocol is required. A recommended L0-safe path is empirical calibration by matching \(A_{\mathrm{active}}(\varepsilon_\mu)\) to a validated Genesis reference graph under known conditions.
• Gauge-invariant defect content \(Q\) (blocking for compact fork): for \(\partial G=\varnothing\), define a gauge-irreducible defect charge (e.g., Hodge/cohomology projection) to distinguish gauge redistribution from physical annihilation; linked to the prior Hodge-irreducibility findings.
• Phase-4 directionality blocked on L1: L0 can track pattern translation of \(\mu\) and \(\Delta^{(0)}\), but migration direction requires a signed L1 stability/margin layer.
• Phase-3 trace-gap audit: prior results computed under target-curvature trace-gap must be labeled as L1/L2-conditional or reinterpreted under \(\Delta_f^{(0)}\).
• Relational primacy extension (optional): a future edge-primary ontology may recast nodes as derived endpoints of relations; current master is operationally local and lift-defined, but this extension is not yet propagated.

DSN (Dense Summary Note)

L0 MASTER v1.2 (Layer-Consistent, snapshot-only): input snapshot {θ_i}∈T^N plus fixed edge data; forbid γ,dt, integrators, NS/stability, attractors, equilibrium. Gauge: global θ_i→θ_i+φ redundant; fix by θ_ref=0 or projection ⟂(1,…,1). Substrate vs active topology: A_base given; compute μ on A_base; derive A_active by filter; compute SU(2) geometry on A_active. Node atom: n_i=(μ_i,Ĵ_i). Canonical μ (total load, local): μ_i:=Σ_{j~i in A_base}|κ̂_ij|·|sin((θ_j−θ_i)−α_ij)| with α_ij=0 if unused; μ̄_i=μ_i/deg(i) diagnostic only. Resonance filter (optional refinement): EDGE_ELIGIBLE(i,j)⇔|μ_i−μ_j|<ε_μ; ε_μ topology-severe; L0-safe bound 0<ε_μ