Part IV: The Next Paradigm

Chapter 12 — Mesoscopic Tests of Feedback Geometry

Detecting informational curvature through pattern formation, synchronization, and adaptive networks

At the mesoscopic scale, feedback becomes visible as form. From chemical waves to neural oscillations, systems organize themselves through the continuous negotiation between entropy and structure. IPSC predicts that such self-organization is not merely dynamical but geometric: the Fisher information metric that quantifies the system’s internal correlations should act as an order parameter for coherence. When curvature rises, order stabilizes; when it falls, disorder ensues. The following three proposals describe experiments capable of testing this prediction across biological, chemical, and artificial domains.

Proposal 1: Informational Curvature in Reaction–Diffusion Pattern Formation

Rationale: Reaction–diffusion systems such as the Belousov–Zhabotinsky (BZ) reaction or photosensitive activator–inhibitor media spontaneously form waves and spirals. IPSC interprets these not as purely chemical instabilities but as geometric attractors of information flow. The system’s Fisher curvature, computed from spatial concentration distributions, should increase as patterns form and remain elevated in coherent states.

Experimental Design: Conduct a BZ reaction in a thin film under variable feedback control. Optical sensors record concentration fields at millisecond resolution. External feedback (light or temperature modulation) adjusts the inhibitor–activator ratio dynamically based on pattern stability metrics. Compute instantaneous Fisher information (FI) from normalized concentration histograms:

F = ∫ [ (∂_xp(x,t))² / p(x,t) ] dx

IPSC predicts that FI will peak and stabilize at a fixed range during pattern coherence. Under excessive feedback (or noise), FI should collapse, indicating loss of curvature. By comparing curvature trajectories to pattern persistence, IPSC’s claim that geometry governs feedback stability can be quantitatively assessed.

Expected Signature: Sustained, non-chaotic oscillations in FI preceding or co-occurring with pattern emergence, independent of purely chemical rate constants.

Proposal 2: Neural Synchronization and Fisher–Phase Coupling

Rationale: Neural assemblies form transiently synchronized networks whose coherence depends on reciprocal information flow. IPSC predicts that brain dynamics near criticality maximize informational curvature, measurable via mutual information and Fisher metrics of local field potentials (LFPs) or magnetoencephalography (MEG) data. At such points, systems should display enhanced adaptability and stable global feedback.

Experimental Design: Record high-density LFP or MEG during sensory stimulation or learning tasks. Compute the Fisher information matrix of multivariate time-series distributions and compare its curvature to phase synchrony metrics (e.g., phase-locking value). IPSC predicts that:

Periods of optimal task performance or insight correspond to maximal Fisher curvature.
Curvature peaks precede long-range synchronization bursts, serving as predictors of neural integration events.

Controlled perturbations (transcranial stimulation, pharmacological modulation) can test causality. Reduced curvature should correlate with decreased synchrony or cognitive performance, providing a measurable neuro-informational signature of IPSC dynamics.

Expected Signature: Positive correlation between Fisher curvature and neural coherence; predictive curvature rise prior to synchronization events.

Proposal 3: Adaptive Network Learning as Informational Phase Transition

Rationale: Artificial neural networks, when trained under varying feedback constraints, provide a controllable analog for IPSC’s informational geometry. IPSC predicts that learning corresponds to a geometric phase transition in information space: curvature rises sharply at the onset of generalization — when the network transitions from memorization to abstraction.

Experimental Design: Train deep networks (e.g., convolutional or recurrent architectures) while computing Fisher information of weight distributions per epoch:

F = E[(∂L/∂θ)²]

Plot curvature against validation accuracy. IPSC predicts a non-linear inflection point corresponding to the system’s attainment of self-referential coherence (generalization). This is analogous to critical points in physical systems and neural dynamics, linking learning theory to cosmological feedback geometry.

Expected Signature: Critical transition in Fisher curvature aligning with maximal learning generalization, independent of network size or task complexity.

Cross-Domain Implications

Across chemistry, biology, and computation, the predicted behavior is consistent: feedback-driven systems exhibit quantifiable increases in Fisher information curvature at the onset of ordered dynamics. Whether in a chemical spiral, a neural circuit, or a digital learner, order appears when information organizes itself geometrically.

These mesoscopic tests thus extend IPSC beyond metaphor into measurable physics. The curvature of information, once an abstract mathematical object, becomes experimentally visible as the geometry underlying feedback stability. If verified, it will demonstrate that organization — from cells to cognition — is not an emergent epiphenomenon but a direct expression of informational geometry.

The next chapter moves to the largest scales, where feedback and curvature shape the cosmos itself. There, informational geometry predicts measurable anisotropies, rotational signatures, and fractal clustering that could confirm the universe’s self-referential memory. Chapter 13 — Cosmic Tests of Informational Holonomy develops these proposals.