4.3 Inner Critical Band: Watershed Between the Particle Phase and the Filament-Sea Phase

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The inner critical is not a knife-edge but a thicker, graded band. As one moves inward into this region, stable wraps that constitute particles lose stability in stages. The system transitions from a particle-dominated structure to a boiling state dominated by a dense filament sea.

I. Definition and Why a “Band” Is Inevitable

1. Definition: The inner critical band is the spatial interval over which wrapped, particle-forming states transition continuously into a high-density, filament-sea-dominated state.
2. Why a band:
   • Different thresholds: Distinct particles and composite wraps fail at different stability thresholds, exiting from weaker to stronger.
   • Different timescales: Unbinding, reconnection, and re-nucleation each carry delays, so spatial gradients create temporal tails.
   • Environmental texture: Local tension and shear show organized fine structure rather than a single uniform value.
   • The outcome is a phase-transition zone with clear layering in both composition and response time.

II. Why Stability Fails: Three Interlocking Chains

• Rising external tension and pressure: Inward, tension increases and shear strengthens. Wraps must maintain curvature and twist at smaller radii; costs rise quickly and disassembly follows once thresholds are exceeded.
• Slower internal cadence: Higher tension depresses a wrap’s internal beat. Slower cadence weakens coherence; disturbances are harder to self-heal, so effective stability drops.
• Relentless disturbance packets: Deeper inside, perturbations are more frequent. Their phases and amplitudes scour boundaries, triggering micro-reconnections and breaks. Small failures cascade and push entire classes past instability.

These chains reinforce each other: stronger external tension slows the internal cadence and makes boundary pushing easier, yielding cross-scale, cascading instability.

III. Layering Inside the Band (Outer → Inner)

• Re-nucleation fringe: Short-lived re-nucleation and dense stacking remain possible at the outer edge; composites devolve to simpler wraps and then weaken.
• Weak-wrap exit layer: Lower-stability wraps fail collectively. Short-lived particles and irregular wave-packets increase; background noise rises.
• Strong-wrap exit layer: Higher-stability wraps are broken by shear and reconnection; the particulate state nearly disappears.
• Filament-sea dominance: Entry into a dense, boiling filament sea with frequent shear streaks, reconnection flashes, and multiscale cascades; the whole resembles a thick “soup.”

These strata are statistical. They may interleave, and their boundaries are not straight, consistent with a banded, textured structure.

IV. Two Sides, Clear Contrast

• Outside the band: Particles can still self-sustain. Re-nucleation occurs and dense stacking persists. Responses are slower, and order can be restored after disturbance.
• Inside the band: Filament-sea turbulence dominates. Shear, reconnection, and cascades recur; disturbances tend to spread rather than be locally absorbed. Responses are faster and distinctly chained.

V. Dynamics: Position and Thickness Adjust

• Event breathing: Strong events push segments slightly outward; after they subside, the band contracts.
• Budget constrained: When the overall tension budget rises, the band thickens and moves outward; when it falls, the band thins and retracts.
• Directional bias: Along the spin axis and major orientation ridges, the band’s shape differs from other azimuths—an imprint of intrinsic dynamical alignment, not random noise.

VI. Classify Without a Single Number: Watch Three Things

• Self-sustain ability: Outside, most wraps survive perturbations; inside, most disintegrate into filament-sea components.
• Statistical composition: Outside, long-lived particles dominate and short-lived components are sparse; inside, short-lived particles and irregular wave-packets rise sharply and form contiguous patches.
• Temporal response: Outside, responses are slow and local; inside, they are fast and chained, with unmistakable cascade signatures.

When all three indicators point from self-sustaining to non-self-sustaining behavior, treat that interval as an active segment of the inner critical band.

VII. Summary

The inner critical band is a graded transition zone. Increasing external tension, slowing internal cadence, and repeated disturbance impacts destabilize wraps in stages, shifting the system from particle-led to filament-sea-led behavior. The band has thickness, breathes, and shows directional bias. Identification relies not on a single number but on self-sustain tests, changes in statistical composition, and the character of time responses.