The κ₃ Sector in Relational Time Geometry (RTG)

Title: Non-Abelian Geometric Stiffness and Protected Plasticity

Subtitle: The \(\kappa_3\) Sector as a Stratified Active-Matter Phase in RTG/ESM

Status: R2 Validated Program — Layer L2 (Continuum / GR mapping excluded)

Scope: This document defines, measures, and validates the \(\kappa_3\) sector at the Effective Geometry layer (L2) using motifs and grown simplicial clusters with an explicit SU(2) lift and active patch probes.

Reproducibility Status:
• Core mechanisms (K3 unit, K4 barrier, Diamond percolation, quartic shear-response mapping) validated at motif level (✓).
• Cluster Phase-3 results validated on R2_ENSEMBLE (N=10) seeds (✓), with fixed face-coverage per seed (typically 120 faces; seed 25 matched to 120 after redo).
• Relaxation stability under extended “relax longer” confirmed (✓) on seed 19 (Type-I count and key medians invariant across relaxation checkpoints).
• Constitutive law (symmetric \(\pm\alpha\) quartic fits) indicates universal yielding across the Type-I sector (✓); quartic truncation confirmed sufficient by sextic stabilizer check (✓).
• Topological characterization: κ₃ support is anomalously loop-rich (Betti-1 z ~ 8–11 vs null) and structurally redundant (critical edge fraction below null baseline), consistent with loop-stabilized defect trapping (✓).

0. Executive Definition

The \(\kappa_3\) sector is a non-Abelian geometric sector of emergent simplicial manifolds in which SU(2) holonomy produces:

• Anisotropic stiffness under relative shear (OPP shear vs SAME shear response), and
• Constitutive yielding under finite deformation, captured by a negative quartic coefficient in the symmetric shear expansion of the deformation energy.

Empirically, the sector behaves as Stratified Active Matter:

1. Stratification (where the linear stress lives): stress density concentrates on a Singular Stratum of embedded bottleneck hinges, defined operationally as faces with \(n_{\mathrm{active}}=2\) that lie inside the relational backbone (k-core \(\ge 2\)).
2. Plasticity (how the sector responds): yielding (\(Q_{\mathrm{opp}}<0\)) is Type-I universal across probed faces; bottlenecks bear the load, but the entire Type-I sector is plastic.
3. Protection (stability under relaxation): the defect network persists under extended relaxation (seed 19). Topological analysis reveals the protection mechanism: the κ₃ support forms an anomalously loop-rich mesh (Betti-1 z ~ 8–11 vs null) with built-in structural redundancy, consistent with non-Abelian frustration collectively trapped by loop topology.

1. Terminology and Conventions

1.1 SU(2) Lift

Each edge \(e\) carries an SU(2) transport \(U_e\) obtained from an algebra vector \(X_e \in \mathbb{R}^3\):
\[
U_e = \exp_{\mathrm{SU(2)}}(X_e).
\]

Face holonomy around face \(f\) is the ordered product
\[
H_f = \prod_{e \in \partial f} U_e^{\pm 1},
\]
with orientation determining inversion.

1.2 Holonomy Trace and Gap

The gauge-invariant observable is the half-trace
\[
\mathrm{tr}_f := \mathrm{Re}(\mathrm{Tr}(H_f))/2 \in [-1,1],
\qquad
\theta_f = 2\arccos(\mathrm{tr}_f).
\]

Given a prescribed Abelian face flux \(F_f\), the target is \(\cos(F_f/2)\). The per-face holonomy gap is:
\[
\mathrm{gap}_f := \left| \mathrm{tr}_f – \cos(F_f/2) \right|.
\]

1.3 Energy Conventions

Edge variables are often represented in half-angle form \(A_e\). Full SU(2) transport uses \(2A_e\), giving:
\[
E_{\mathrm{full}} = 4 \sum_e \|A_e\|^2 = 4 E_{\mathrm{half}}.
\]

1.4 SU(2) Lift Construction (Operational)

The SU(2) lift is constructed by constrained optimization:

1. Specify target face angles \(F_f\) (manual or autoflux-generated).
2. Initialize edge algebra vectors \(A_e \in \mathbb{R}^3\).
3. Minimize:
   \[
   L(A) = \sum_f \left\|\log\!\left(H_{t,f}^\dagger H_f(A)\right)\right\|_F^2
   + \lambda_{\mathrm{energy}} \sum_e \|A_e\|^2.
   \]
4. Refine/continue to expose stiffness barriers.

2. The K3 Curvature Unit (Calibrated Baseline)

Status: ✓

The triangle \(K_3\) is the calibrated curvature atom. For a single face with flux \(F\), equal splitting gives:
\[
C_{\mathrm{half}} = \frac{F^2}{12},
\qquad
C_{\mathrm{full}} = \frac{F^2}{3}.
\]

Holonomy is exact (\(\mathrm{gap}\approx 0\)) and invariant under global axis rotation.

Consequence: \(K_3\) carries curvature but exhibits no stiffness barrier. \(\kappa_3\) phenomena require coupled non-Abelian loops embedded in a larger constraint network.

3. \(\kappa_3\) Mechanism (Non-Abelian Origin)

Status: ✓

3.1 BCH / Commutator Origin

In SU(2),
\[
\exp(X_1)\exp(X_2)\exp(X_3) \neq \exp(X_1+X_2+X_3).
\]

The Baker–Campbell–Hausdorff expansion implies that even when \(\sum_i X_i \approx 0\), commutator terms generate a finite holonomy deficit:
\[
W = \exp\!\left(\sum_i X_i + \frac{1}{2}\sum_{i 0.
\]

Key insight: \(\kappa_3\) “darkness” arises from commutator curvature invisible to linear (Abelian) closure probes. Importantly, this manifests as a stiffness response only when embedded in coupled loop networks; isolated non-Abelian faces do not exhibit a stable stiffness barrier.

4. Motif Evidence

Status: ✓

4.1 K4 Motif

Opposite shear requires significantly higher energy than same shear at low gap, demonstrating a non-Abelian obstruction.

4.2 Diamond Motif

Stiffness percolates across a shared face: global rotation is soft, while relative shear is energetically costly.

5. Constitutive Plasticity Under Shear

Status: ✓ (R2 Result)

5.1 Quartic Constitutive Law (Opposite-Shear Channel)

We fit the deformation energy under symmetric shear \(\alpha \in [-\alpha_{\max}, +\alpha_{\max}]\) using:
\[
E_{\mathrm{opp}}(\alpha) = E_0 + \frac{1}{2}K\,\alpha^2 + \frac{1}{24}Q\,\alpha^4.
\]

R2 protocol summary:

• Seeds: \(N=10\) (17–26)
• Targets: 10 Type-I faces per seed (total 100 faces)
• Shear grid: 14-point symmetric scan from \(-0.15\) to \(+0.15\) (excluding 0)
• Fit gate: \(R^2_{\mathrm{opp}} \ge 0.95\)

R2 quartic outcomes:

• Fit success: 100/100 (100%)
• Softening fraction: \(P(Q_{\mathrm{opp}}<0)=0.94\) (94%)
• Median \(Q_{\mathrm{opp}}\) among fit-OK faces: \(\mathrm{median}(Q_{\mathrm{opp}}) \approx -22.76\)

5.2 Universality of Yielding

Yielding is Type-I universal within the probed set: both bottleneck hinges and non-bottleneck Type-I faces exhibit high softening fractions (difference not statistically significant in the bottleneck split test).

Interpretation (safe, operational): \(Q_{\mathrm{opp}}<0\) indicates an energetically yielding/buckling response under finite shear. This supports (but does not by itself prove) defect mobility; dynamical mobility requires time-evolution tests (Phase 4).

5.3 Sextic Stabilizer Check

To verify that the negative quartic coefficient is not a truncation artifact, a sextic extension was tested:
\[
E_{\mathrm{opp}}(\alpha) = E_0 + \frac{1}{2}K\,\alpha^2 + \frac{1}{24}Q\,\alpha^4 + \frac{1}{720}S\,\alpha^6.
\]

Results (N=100 faces, OPP channel):

• Fit improvement: \(\Delta R^2\) median \(\approx 9.8 \times 10^{-12}\) (negligible)
• Quartic sign stability: 0/100 faces change sign of \(Q\) between quartic and sextic fits
• Quartic correlation: \(r(Q_{\mathrm{quartic}}, Q_{\mathrm{sextic}}) = 0.9995\)
• Sextic coefficient: 95% of faces have \(S < 0\) (median \(S \approx -307\)); no finite-amplitude restabilization in the probed window

Conclusion: The quartic truncation is sufficient for \(|\alpha| \le 0.15\). Softening is robust to model order: the energy landscape is monotonically softening through both quartic and sextic orders within the probed window, consistent with the onset of a yielding instability.

6. Cluster Spectroscopy and Patch Probes

Status: ✓

6.1 Passive MRI

Faces can be screened via Abelian flux, holonomy gap, and local SU(2) energy to identify candidate regions for active probing.

6.2 Active Shear Probe (Type-I Gate)

Operational Type-I identification (π/2 gate, opposite-shear channel) requires:

• \(n_{\mathrm{active\_faces}} \ge 2\)
• \(\mathrm{ratio}_E := E_{\mathrm{opp}}/E_{\mathrm{same}} \ge 2.6\)
• \(\mathrm{gap}_{\mathrm{opp}} \le 10^{-2}\)

This defines a reproducible “matter-like” subset: low gap under OPP with large anisotropic stiffness.

7. Stratified Anatomy (Structure vs Mechanics)

Status: ✓ R2_ENSEMBLE (N=10)

The \(\kappa_3\) sector is best summarized as a two-stratum system:

• Singular Stratum (hinge skeleton): embedded bottleneck hinges, defined as \(n_{\mathrm{active}}=2\) within the relational backbone (k-core \(\ge 2\)). These carry elevated stress density but are not structural keystones (see §7.4).
• Regular Stratum (yielding Type-I medium): remaining Type-I faces (including k-core bulk and shell) which are constitutively yielding but carry lower linear stress density.

7.1 Stress Anatomy (R2 Aggregate, N=10)

Define stress density proxy:
\[
\kappa_3^{\mathrm{dens}} := \frac{E_{\mathrm{opp}} – E_{\mathrm{same}}}{n_{\mathrm{active\_faces}}}.
\]

Boundary-vs-core (patch-local split) results across all Type-I faces:

• Boundary group (\(n_{\mathrm{active}} \le 2\)): median \(\kappa_3^{\mathrm{dens}} \approx 2.366\) (N=169)
• Core group (\(n_{\mathrm{active}} \ge 3\)): median \(\kappa_3^{\mathrm{dens}} \approx 1.796\) (N=114)
• Mann–Whitney test: \(p \approx 1.10\times 10^{-19}\); Cliff’s \(\delta \approx 0.636\); Cohen’s \(d \approx 1.076\)

This establishes a robust “hinge load” effect: low-constraint faces (especially \(n_{\mathrm{active}}=2\)) carry higher stress density.

7.2 Relational Backbone and Bottleneck Hinges

Relational backbone is defined by k-core decomposition of the Type-I adjacency graph (k-core \(\ge 2\)). Bottleneck hinges are the intersection:
\[
\text{bottleneck}(f) := (\text{k-core}\ge 2)\ \wedge\ (n_{\mathrm{active}}(f)=2).
\]

Empirically, bottlenecks are embedded (interior-biased by distance-to-core diagnostics) and are the primary carriers of linear stress. The bottleneck distinction is mechanical (stress concentration), not structural (core-criticality is uniformly distributed; see §7.4).

7.3 The Adjacency Lift (Connectivity Proof)

Connectivity is adjacency-sensitive:

• Edge-sharing adjacency: Type-I fragments into multiple components (mean \(n_{\mathrm{components}} \approx 6.5\); mean largest component \(\approx 12.1\) faces).
• Vertex-sharing adjacency: the same set merges more strongly (mean \(n_{\mathrm{components}} \approx 3.9\); mean largest component \(\approx 16.3\) faces).

This “adjacency lift” provides the structural mechanism for the backbone: vertex-sharing reveals a more unified skeleton from locally fragmented edge-sharing chains.

7.4 Core-Criticality and Structural Redundancy

Core-critical edges are defined as edges whose removal reduces the k-core number of at least one node. The critical edge fraction of the κ₃ backbone was tested against a null model (500 random face subsets of equal size per seed).

Results:

• Edge adjacency: real critical fraction ≈ 0.79, null baseline ≈ 0.96. The κ₃ backbone is significantly more redundant than random (z = −2.67).
• Vertex adjacency: real critical fraction ≈ 0.62, null baseline ≈ 0.79. Consistent with random (z = −1.35).
• Bottleneck enrichment ≈ 1.0 in both modes: edges incident to bottleneck hinges are not disproportionately represented among critical edges. Criticality is uniformly distributed across the backbone.

Interpretation: The anomalous loop-richness of the κ₃ support (§8) provides structural redundancy — alternative paths that protect the backbone from single-edge failures. The bottleneck distinction is mechanical (stress concentration), not structural (no keystone edges). The backbone’s integrity comes from the collective loop structure, not from specific critical elements.

8. Topological Characterization (Betti Curves and Null Model)

Status: ✓ (Phase-3.5 Result)

8.1 Betti Curves Under κ₃-Density Filtration

Betti-0 (connected components) and Betti-1 (cycle rank) of the Type-I adjacency graph were computed as a function of κ₃-density threshold (high stress enters first). The filtration reveals a percolation-like assembly: high-stress hinges form an initially sparse, acyclic skeleton that becomes loop-rich as lower-stress Type-I faces fill in the surrounding mesh.

At full inclusion (lowest threshold):

• Edge adjacency: mean Betti-0 ≈ 8, max mean Betti-1 ≈ 19
• Vertex adjacency: mean Betti-0 ≈ 3, max mean Betti-1 ≈ 110

8.2 Null Model: Topological Exceptionality

To test whether the loop-richness is a genuine property of the κ₃ support or an artifact of complex geometry, Betti numbers were compared against 500 random face subsets of equal size per seed (drawn from all 120 faces, not just Type-I).

Results:

• Vertex adjacency: Betti-1 z-score mean = 8.54 (range 5.37–12.05); all 10 seeds at 100th percentile. Betti-0 consistently lower than null (fewer components).
• Edge adjacency: Betti-1 z-score mean = 11.30 (range 8.78–14.72); all 10 seeds at 100th percentile. Betti-0 strongly lower than null (real ≈ 6 vs null ≈ 20).

Not a single random draw out of 5,000 per adjacency mode matched or exceeded the real Type-I Betti-1.

Conclusion: The κ₃ defect support is topologically exceptional. It forms a coherent, loop-rich mesh rather than a random scattering of isolated patches. The κ₃ faces preferentially occupy regions of the simplicial complex where they share vertices and edges with each other, forming a densely interwoven defect mesh with significantly more independent cycles and fewer connected components than size-matched random subsets.

9. Relaxation Stability (Extended Relaxation)

Status: ✓ (Seed 19)

Extended relaxation (“relax longer”) preserves the sector as a stable configuration on seed 19 across multiple checkpoints (steps 0–7):

• Type-I count: constant (39/120) and prevalence constant (0.325)
• Bottleneck count: constant (21)
• Median bottleneck stress density: constant (\(\mathrm{median} \approx 2.594\))
• Median non-bottleneck stress density: constant (\(\mathrm{median} \approx 1.873\))

This stability is consistent with the topological protection mechanism identified in §8: non-Abelian closure defects are collectively trapped by the loop topology of the κ₃ mesh. Local relaxation cannot unwind the frustration because the loops provide redundant frustration paths (§7.4).

10. Interpretation: Loop-Stabilized Defect Mesh

At L2, the validated interpretation is:

• \(\kappa_3\) identifies a topologically coherent, loop-stabilized defect mesh within emergent simplicial geometry.
• Topology governs localization: embedded bottleneck hinges inside the relational backbone carry elevated stress density (a mechanical distinction, not a structural keystone role).
• Constitutive response is plastic: quartic softening is broadly present across Type-I faces, not confined to bottlenecks. The softening is robust to model order (sextic check).
• Protection is topological: the defect network persists under relaxation because its anomalous loop-richness collectively traps non-Abelian frustration and provides structural redundancy against single-edge perturbations.

The following table summarizes the three-phase decomposition that constitutes the Stratified Active Matter ontology at L2:

Component	L2 Definition	Mechanical Phase	Topological Role
Vacuum	Type-II / gauge-screened regions	Elastic / weak anisotropy	Background medium
Matter (κ₃ sector)	Type-I(A) active population	Plastic (yielding: \(Q_{\mathrm{opp}} < 0\))	Loop-rich defect mesh (Betti-1 z ~ 8–11)
Hinges	Bottleneck subset: k-core≥2 ∧ \(n_{\mathrm{active}}=2\)	Load-bearing (high \(\kappa_3^{dens}\))	Stress concentrators (not structural keystones)

11. Limits and Non-Claims

• No claim of Einstein equations, continuum metric, or GR limit.
• No claim that “Ricci” terminology implies continuum Ricci tensor; here it is an operational stiffness functional derived from shear response.
• No claim of defect conservation/charge, annihilation, or a completed field theory.
• Yielding (\(Q_{\mathrm{opp}}<0\)) does not by itself prove mobility; mobility requires explicit dynamical tracking.
• Global landscape characterization (unique attractor vs many metastable states) remains open; local softening does not rule out glassy global behavior.
• Core-criticality is uniformly distributed across the backbone; bottleneck hinges are stress-dense but not structural keystones. The “two-stratum” label refers to a mechanical distinction (stress concentration), not a structural fragility distinction.
• Topological results (Betti curves, null models) are computed on the 1-skeleton (face adjacency graph); higher-dimensional persistent homology on the full simplicial complex may reveal additional structure.

12. Minimal Reproducibility Checklist

• K3 calibration verified (\(C_{\mathrm{half}}=F^2/12\), \(C_{\mathrm{full}}=F^2/3\)).
• K4 and Diamond stiffness confirmed (motif-level).
• Type-I gate runs on seeds 17–26 with fixed coverage per seed (typically 120 faces; seed 25 aligned to 120 after redo).
• Stress anatomy computed (boundary vs core split) with nonparametric test + effect sizes.
• Quartic softening fits run on 10 faces per seed with symmetric \(\pm\alpha\) scan and \(R^2\) gating.
• Sextic stabilizer: \(\Delta R^2 \approx 10^{-12}\); 0/100 sign flips; \(r(Q_4, Q_6) = 0.9995\).
• Betti null model: 500 random draws per seed, both adjacency modes; all seeds 100th percentile.
• SCS null model: 500 random draws per seed, both adjacency modes; κ₃ backbone more redundant than random.
• Relaxation stability verified for seed 19 using checkpoint invariance.

13. Evidence Inventory

Data artifacts supporting all validated results:

• Phase-3 core: k3_scan_seed{17..26}.csv, patch_autoflux_shear_results.csv, sensitivity_analysis_N10.csv
• Relational anatomy: k3_relational_faces.csv, k3_relational_summary.csv, k3_relational_edges.csv
• Quartic fits: quartic_softening_fits.csv, quartic_softening_samples.csv, quartic_softening_summary.json
• Sextic check: sextic_fits_opp.csv, sextic_summary_opp.json, sextic_dR2_hist_opp.png, sextic_S_hist_opp.png
• Betti curves: betti_curves_by_seed.csv, betti_curves_mean.csv, betti_curves.png (edge + vertex)
• Betti null model: betti_null_by_seed.csv, betti_null_summary.json, betti_null_distributions.png (edge + vertex)
• SCS null model: scs_null_by_seed.csv, scs_null_bottleneck_decomp.csv, scs_null_summary.json, scs_null_distributions.png (edge + vertex)
• Relaxation: relax_longer_timeseries.csv, relax_longer_seed19.zip
• Stress anatomy: stress_anatomy_stats.json, stress_anatomy_samples.csv

14. Next Steps (Phase 4)

1. Defect mobility: dynamical tracking under controlled perturbation (phase kick / \(\delta\omega\) noise) to distinguish pinned yielding defects from mobile defect fields. The loop-redundancy result (§7.4) predicts that perturbation will be absorbed locally via collective rearrangement rather than propagated along specific channels.
2. Interaction: pairwise defect interaction / screening tests on adjacent bottlenecks.
3. Representation dependence: spin-1 (adjoint) probe to test whether κ₃ is center-sensitive (fundamental-only) or a genuine non-Abelian obstruction. This discriminates between center-vortex-like and non-Abelian-monopole-like defect mechanisms.
4. Micro-bridge: connect defect dynamics to microstate evolution (frequency/phase dynamics) and earlier emergent dimensionality diagnostics.
5. Higher-dimensional TDA: full persistent homology on the simplicial complex (beyond 1-skeleton cycle rank) using GUDHI, to capture topology invisible to graph-based Betti analysis.

End of Document.