Application of the Spectral Derivative Variance Functional to Biological Collective Motion: Pre-Registration and Simulation Validation in Starling Murmurations

This repository contains the pre-registration protocol, simulation validation code, and publication figures for the application of the Spectral Derivative Variance (SDV) functional to starling murmuration dynamics — testing the substrate-independence claim of the Universal Boundary Collapse (UBC) framework against biological collective motion.

The Central Question: If the SDV functional describes a universal law of structural boundary collapse, the exact same mathematical formulation used to detect power grid failure, epileptic seizure onset, tumor metastasis, and financial market collapse must also capture the death of a murmuration's unified boundary when a predator strikes. No modification. Same equations. Same signatures.

The Pre-Registration: Four falsifiable predictions are formally stated before any simulation or empirical analysis:

Prediction 1 — Baseline Flocking: Normal murmuration produces stable, elevated SDV. The flock's boundary is intact. Spectral energy flow is steady.

Prediction 2 — Over-Integration (Density Collapse): Predator-induced compression forces birds into hyper-dense formation. Interaction ranges overlap excessively — velocity alignment becomes chaotic. SDV collapses via The Flood trajectory, despite the flock remaining spatially connected.

Prediction 3 — Under-Integration (Fragmentation): Explosive predator-evasion expansion causes the flock to split. The Fiedler value (λ₂) of the spatial Graph Laplacian collapses toward zero. SDV collapses via The Shatter trajectory as structural severing destroys spectral flow.

Prediction 4 — Negative Control (Routine Turn): A non-threat coordinated direction change leaves the topology intact. Spectral flow remains smooth. SDV remains stable. This is the critical control: SDV does not collapse during any collective motion — only during topological failure.

The Simulation Validation: A Vicsek-style coupled oscillator simulation of N=80 birds across 400 timesteps confirms all four pre-registered predictions. Both Over-Integration and Under-Integration produce SDV collapse via opposing Integration trajectories — one from excess coupling, one from structural severing — both reaching the same Boundary Stability → 0 regime. The routine turn leaves SDV untouched.

Empirical Validation — The Next Step: The simulation confirms the mathematical prediction. Empirical validation requires real 3D murmuration tracking data. The STARFLAG dataset (Cavagna et al., 2010, PNAS) — continuous 3D reconstructed trajectories of thousands of individual starlings at ~10 Hz, including predator evasion events — is identified as the target dataset. Researchers with access to STARFLAG or equivalent murmuration tracking data are invited to run the provided code and test the prediction directly.

Related works:

• Universal Boundary Collapse framework: https://doi.org/10.5281/zenodo.20075816
• SDV functional definition: https://doi.org/10.5281/zenodo.20075399
• SDV applied to epileptic brain dynamics: https://doi.org/10.5281/zenodo.20159267
• SDV applied to power grid topologies: https://doi.org/10.5281/zenodo.20323420
• SDV applied to oncological topologies: https://doi.org/10.5281/zenodo.20316524
• SDV applied to financial markets: https://doi.org/10.5281/zenodo.20323041

Data: No external data required. Simulation is fully self-contained and reproducible (seed=42). Empirical validation pending STARFLAG data access.

Requirements: Python 3.x, numpy, scipy, matplotlib

AUTHOR'S NOTE & INTENT

There is a moment — if you have ever stood under a murmuration — when ten thousand birds move as a single mind and you stop breathing.

It is not metaphor. It is not poetry. It is physics. The flock is a network. The boundary is real. The stability is measurable. And when a peregrine falcon cuts through the center of that boundary, the mathematics of its collapse is identical to the mathematics of a power grid failing, a brain seizing, a tumor metastasizing, a market crashing.

That is what the Universal Boundary Collapse framework says. That is what this paper tests.

This pre-registration is a public commitment. These are the predictions. The code is the test. The data — when it arrives — will either confirm or falsify the framework. 

I formally dedicate this work to the open study of collective intelligence in biological systems. To the physicists, ethologists, and computational biologists who have spent careers watching starlings in Rome. To Andrea Cavagna and Irene Giardina, whose extraordinary experimental work made this test possible to even formulate. To every researcher who has ever looked at a murmuration and wondered how.

If you have STARFLAG data — or equivalent 3D flock tracking with predator events — please run the code. The prediction is specific. The test is simple. The implications, if confirmed, are universal.

Math is universal, but its architecture has a conscience. This work is released openly, without restriction, for the advancement of our understanding of how collective systems live, fail, and sometimes save themselves.

— Angel Edgardo Robles Quiroz, 2026