Temporal Binding on the Lorentz Manifold: Geodesic Kuramoto Oscillators, Topological Regularisation, and Closed-Loop Geometric Control for Hyperbolic Language Model Distillation

HyDRA v9.x — Causal Diagnostic Protocol for a Hyperbolic Gating Mechanism: Instrumentation Correction and Negative Result Summary This preprint documents a rigorous causal diagnostic study of an angular coupling gate proposed in the HyDRA v7 architecture for hyperbolic language model distillation. The gate couples per-token attention alignment with a Lorentz-manifold kernel through a scalar sigmoid g = σ(W · cos α + b). We report an instrumentation correction, the retraction of a prior interpretive claim, and the systematic execution of four pre-declared causal ablations, from which the gate mechanism as specified does not appear to be load-bearing for validation perplexity under the tested conditions. Contributions Instrumentation correction. The original v9 telemetry contained a validation-path gradient-logging artefact that caused W-gradient columns to appear identically zero during evaluation. The defect is documented, the fix is specified, and the downstream interpretive impact on prior single-seed observations is quantified. Retraction of the moving-optimum hypothesis. A preliminary single-seed trajectory suggesting a non-monotonic "U-turn" in the gate weight W did not reproduce under three-seed replication in the Stage-2 configuration. The claim is retracted in-text; W descends monotonically toward a near-constant asymptote with decreasing |dW/step|. Pre-declared causal diagnostic protocol. A four-diagnostic protocol was declared in advance (D1–D4) with verdict thresholds fixed before any run: D1: no-distillation (teacher removed). D2: gate frozen at near-identity (b = 6, W = 1). D3: gate frozen at initialisation (b = 0, W = 1), reading cos α without learning. D4: cos α replaced by i.i.d. U(−1, 1) noise, gate W, b trainable. Joint result. At seed 42, 5000 steps, cross-entropy-only regime, the four orthogonal gate configurations yield terminal validation perplexity within an 18-PPL band, single-seed (≤ 4.5% of the smallest value). Under the pre-committed single-seed ±5% criterion, the gate mechanism as specified does not appear to be load-bearing for validation perplexity in this regime. Mechanistic observation. The cos α warm-up peak documented in Stage-2 dynamics reproduces identically in D1 and D3 but not in D2, consistent with coupling to the sigmoid's linear operating point rather than to gate learning. Under D4, the optimiser attenuates W approximately twice as deep as under D1, consistent with suppression of a non-informative input. Scope and Limitations Claims are limited to a single seed per diagnostic, 5000 training steps, and the CE-only regime on a 64M-parameter hyperbolic student (12 layers, n_embd = 256, 8 heads). Multi-seed replication (≥ 3 seeds per diagnostic), longer training horizons (≥ 20,000 steps), and a Euclidean-backbone baseline at matched parameter count are pre-publication requirements before any generalisation. Hyperbolic-geometry-specific claims are explicitly not advanced in this work. Context The HyDRA v7 architecture (Zenodo DOI 10.5281/zenodo.19583481) introduced the angular gate alongside other components (SublatticeLMHead, H-AKORN, topological regularisation). This work concerns only the gate mechanism and does not ablate the remaining v7 components. Licence CC BY-NC-SA 4.0.