Step 5: The Variational Principle - Why R Matters

The redundancy functional becomes powerful when we combine it with energy to form a constrained objective functional:

\[F[\rho, \Phi, \Xi, \Lambda] = E[\rho, \Phi] - \Theta R[\rho, \Xi] + \int d^3x\, \Lambda(x)[\Xi(x) - C[\Phi](x)]\]

where \(E\) is the energy functional, \(\Theta\) is a dimensionless parameter, and \(\Lambda(x)\) is a Lagrange multiplier field enforcing the constraint \(\Xi = C[\Phi]\).

The key principle:

Nature selects configurations that extremize \(F\) - finding a balance between minimizing energy and maximizing redundancy, while respecting the coherence-time-potential constraint. This avoids circular logic by treating \(\Xi\) as independent during variation.

This leads to four fundamental equations from the constrained variational principle:

1. Varying with respect to \(\rho\) (\(\delta F/\delta\rho = 0\)):

\[\frac{\delta E}{\delta \rho} = \Theta \frac{\delta R}{\delta \rho} = \Theta \frac{\kappa T}{2E_c} \omega^2 \Xi\]

Matter distribution equilibrates where energy cost balances redundancy benefit.

2. Varying with respect to \(\Phi\) (\(\delta F/\delta\Phi = 0\)):

\[\frac{\delta E}{\delta \Phi} = \int d^3x'\, \Lambda(x') \frac{\delta C[\Phi](x')}{\delta \Phi(x)}\]

The energy variation balances the constraint forces from the CTP relation.

3. Varying with respect to \(\Xi\) (\(\delta F/\delta\Xi = 0\)):

\[\Lambda(x) = \Theta \frac{\delta R}{\delta \Xi} = \Theta \frac{\kappa T}{2E_c} \rho \omega^2\]

4. Varying with respect to \(\Lambda\) (\(\delta F/\delta\Lambda = 0\)):

\[\Xi(x) = C[\Phi](x) = \exp\left(-\frac{|\nabla\Phi|^2}{\xi^2}\right)\]

This enforces the coherence-time-potential relation.

The remarkable result:

When we work out the energy variation and constraint equations together, this reproduces Poisson's equation:

\[\nabla^2\Phi = \frac{8\pi G}{c^4}\rho\]

This means gravity emerges from constrained redundancy optimization. The gravitational constant \(G\) appears naturally from the redundancy parameters, while the constrained approach avoids circular logic.

Mini-Glossary

Symbol	Name	Meaning	Value / Units	Metaphor
\(F[\rho, \Phi, \Xi, \Lambda]\)	constrained functional	energy minus redundancy plus constraints	energy units	"Nature's constrained cost function"
\(E[\rho, \Phi]\)	energy functional	total energy of configuration	energy	"Energy bill"
\(\Theta\)	redundancy weight	dimensionless parameter	dimensionless	"Information importance"
\(\Lambda(x)\)	Lagrange multiplier	constraint force field	various	"Constraint enforcement"
\(\frac{\delta F}{\delta \rho}\)	matter variation	how F changes with density	volume/mass	"Matter's equilibrium condition"
\(\frac{\delta F}{\delta \Phi}\)	field variation	how F changes with lapse	energy/volume	"Field's equilibrium condition"
\(G\)	gravitational constant	emerges from \(\kappa^2/\xi^2\)	\(m^3/(kg \cdot s^2)\)	"Gravity's strength"
\(\nabla^2\Phi\)	Laplacian of lapse	curvature of time potential	1/length²	"Time curvature"