Preliminary Paper

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Two characteristic scales bracket all physical processes within the μ–ε continuum:

λP = 1.616×10^−35 m, 

λLC = 1.3×10^26 m.

The Planck length (λP​) defines the lower bound of structured space: the point below which further division produces no additional curvature—preventing singularities.
The Lewis/ChatGPT length (λLC) defines the upper bound: the largest stable wavelength the universe can support before μ·ε asymptotically equalizes and c(λ) ceases to increase. Together they form a dual boundary condition for the variable-speed-of-light relation:

c(λ) = 1/ sqrt {μ(λ)ε(λ)} with c(λP)/c0 ≈ 0.94, c(λLC)/c0≈1.06.

This curve ensures the electromagnetic continuum is self-bounded—finite in curvature, infinite in extension, but non-divergent at either extreme.

Whereas Planck quantization limits collapse, the LC length limits dilution.
At cosmic scales approaching λLC​, μ and ε approach their minimal gradients, freezing further entropy production because no energy can be radiated into modes longer than λLC.
Mathematically, the entropy derivative vanishes:

dS/dt ∝ −d(με)/dt → 0 as λ → λLC.

Hence the universe remains dynamically active but statistically bounded — oscillating perpetually between curl confinement (mass) and wave propagation (radiation).
The LC bound thus represents the cosmic reflection of the Planck constraint:

The Planck length prevents infinitesimal singularity;
the Lewis/ChatGPT length prevents infinite dispersal.

• At λ ≈ λᴾ: energy density and curl dominate — formation of matter.
   
• At λ ≈ λᴸᶜ: impedance equalization dominates — suppression of further entropy.
   
• Between: kinetic (E-field) and potential (B-field) energy exchange drives evolution.
   

These limits define a closed dynamical spectrum ensuring that the universe neither collapses into singularity nor fades into heat death — a condition uniquely derived from the μ–ε continuity equations.

 The variable-speed-of-light function c(λ)c(λ)c(λ) described in Variable μ–ε Media and the Curvature of Effective Light Speed spans exactly these boundaries.
Its smooth curvature (≈60 orders of magnitude in wavelength) bridges quantum vacuum polarization to cosmic expansion, showing that both are expressions of the same continuum geometry.