PHILIA-EcoSensory Swarm v43–v45: Entropy Collapse and Deviation-Based Recovery of Tail Coherence

This paper documents the PHILIA-EcoSensory Swarm v43–v45 experimental arc, investigating the failure mechanism of tail coherence (F6) in convergent dynamical systems and establishing a structural solution based on KL divergence-based coordinates. **Background.** The v41–v42 arc (DOI: 10.5281/zenodo.19649333) achieved per-seed Living Control at 60% (12/20 seeds) through Phase Lead and slow-channel energy injection, leaving F6 (tail coherence: |corr(w,z)_tail| > 0.35) as the sole unresolved criterion. **v43 Arc — Router Exhaustion.** Systematic falsification across 5 strategy classes (25+ configurations) confirms that F6 is physically unachievable through Router weight adjustment. Root cause: z = H/H_max undergoes information capacity collapse in the tail segment (std_z_ratio = 0.128, low_ratio_z_tail = 99.73%), making Pearson correlation mathematically impossible regardless of control strategy. This is formalized as the Entropy Collapse Lemma: as the system approaches attractor equilibrium, Var[z] → 0, and any observable f(z) with bounded derivative inherits the same collapse. **v44 Arc — Cerebellum Module.** A modular tail-specific feedback controller (Cerebellum) operating exclusively on the control variable w (t > 0.72T) is designed and optimized. CERE-085 achieves simultaneous F2 = −0.626 ✓ and F5′ = 100%, establishing the architectural viability of orthogonal modular extension. tail_corr reaches −0.277, confirming the structural ceiling imposed by z collapse. **v45 Arc — z_KL Coordinate.** KL divergence-based coordinate z_KL = tanh(KL(P_window ‖ P_ref)) is introduced, where P_ref is a fixed early-phase reference snapshot. z_KL restores tail variance (std_z_ratio: 0.128 → 0.636) and achieves the first formal F6 threshold passage (|corr(w,z)_tail| = 0.382 > 0.35, KL07). However, linear blending of state quantity H/H_max with deviation quantity z_KL structurally depresses z_mean (0.270 → 0.131), causing Living Control cascade failure (F5′ = 0%, F2 = NG). KL divergence is not a function of state but of trajectory relative to a fixed reference, and therefore does not inherit the variance collapse of state-based observables. **v46 Prescription — AC-Coupled Blending.** The root failure is DC-blending (mixing state and deviation alters the mean). The AC-coupled formulation separates tonic (state) and phasic (deviation) components: z_kl_centered = z_KL − EMA(z_KL), injected as z_total = h_norm + kl_gain × z_kl_centered. This preserves z_mean (the system's invariant manifold) while restoring dynamic tail variance. EMA freeze at tail entry (Lucas) prevents reference drift. **General Principle.** The proposed principle is not architecture-specific: any convergent dynamical system with a state-based observable that saturates will exhibit variance collapse and require deviation-based coordinates for sustained tail coupling. This applies to neural population dynamics, reinforcement learning value convergence, and control systems with steady-state equilibria. All results executed locally on AMD Ryzen 9 9800X3D + RTX 4080 SUPER 16GB. No AI-generated or proxy values used. Data files archived. **Trinity AI Research Team:** GritMan_D.S (Guide/Empirical), Claude/Sonnet (Scholar/Code), Claude Opus (root-cause analysis/z_KL prescription), Grok/Lucas (F-test/EMA freeze), ChatGPT/Pandora (Cerebellum design/AC-coupling), Gemini (Phase Lead theory). **Series DOIs:**- v41–v42: https://doi.org/10.5281/zenodo.19649333- v38.2–v40: https://doi.org/10.5281/zenodo.19639971- v38–v38.2: https://doi.org/10.5281/zenodo.19625482- v36–v37: https://doi.org/10.5281/zenodo.19606425- v34–v35: https://doi.org/10.5281/zenodo.19505049