Step 3: The Coherence Factor - How Lapse Gradients Destroy Records

The third ingredient is crucial: how do spatial variations in \(\Phi\) affect the ability to maintain coherent records?

Consider two nearby quantum systems separated by distance \(\delta x\). They experience slightly different lapses:

\[\Phi(x) \text{ vs } \Phi(x + \delta x) \approx \Phi(x) + \nabla\Phi \cdot \delta x\]

This means they accumulate phase at different rates, leading to dephasing. The phase difference grows as:

\[\Delta\theta \propto \omega (\nabla\Phi \cdot \delta x) \Delta t\]

The coherence time and CTP relation

The coherence time - how long the systems remain correlated - is inversely related to \(|\nabla\Phi|\). We define a coherence factor \(\Xi(x)\) as an independent field, with the coherence-time-potential (CTP) relation connecting it to \(\Phi\):

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

The constraint \(\Xi(x) = C[\Phi](x)\) will be imposed separately via a Lagrange multiplier to avoid circular logic. The CTP relation has key properties:

• \(C[\Phi] = 1\) when \(\nabla\Phi = 0\) (uniform lapse, perfect coherence)
• \(C[\Phi] \to 0\) when \(|\nabla\Phi| \gg \xi\) (steep gradients, rapid decoherence)
• Smooth transition over scale \(\xi\)

Physical interpretation:

Regions with large lapse gradients cannot maintain coherent temporal records. During variational optimization, we treat \(\Xi\) as independent from \(\Phi\), then impose the CTP relation \(\Xi = C[\Phi]\) as a separate constraint. This avoids the circular logic of having \(R\) depend on \(\Phi\) while claiming \(\Phi = \delta R/\delta \rho\).

Mini-Glossary

Symbol	Name	Meaning	Value / Units	Metaphor
\(\Xi(x)\)	coherence factor	independent field during variation	0 to 1 (dimensionless)	"Temporal record quality"
\(C[\Phi](x)\)	CTP relation	coherence-time-potential mapping	0 to 1 (dimensionless)	"Environment-coherence converter"
\(\nabla\Phi\)	lapse gradient	spatial variation of time flow	1/length	"Steepness of time hill"
\(\xi\)	coherence length	scale over which coherence decays	length	"Range of synchronized clocks"
\(\Delta\theta\)	phase difference	relative phase between systems	radians	"Clock desynchronization"
\(\delta x\)	separation	distance between quantum systems	length	"Clock spacing"