Section 18: Relational Measurement and Informational Bootstrap in Temporal Flow Physics
Section 18: Relational Measurement and Informational Bootstrap in Temporal Flow Physics
Core Concept
In Temporal Flow Physics (TFP), measurement is an active, relational process rather than passive observation. It involves causal phase recording and synchronization within a discrete, causally bounded temporal flow network. Observable physical quantities such as spatial separation, energy, and momentum arise from phase relationships and their evolution in a causally synchronized recording system. This framework tightly couples TFP’s fundamental principles to the very act of observation, establishing physical reality as bootstrapped from coherent, consistent information.
Characteristic Units Recap
| Quantity | Physical Dimension |
|---|---|
| Characteristic Length \( L_c \) | [L] |
| Characteristic Time \( T_c \) | [T] |
| Characteristic Energy \( E_c \) | [M L^2 T^{-2}] |
| Characteristic Speed \( c_\text{char} = \frac{L_c}{T_c} \) | [L T⁻¹] |
| Characteristic Mass \( M_c = \frac{E_c}{c_\text{char}^2} = E_c \cdot \frac{T_c^2}{L_c^2} \) | [M] |
| Characteristic Action \( \hbar_c = E_c \cdot T_c \) | [M L² T⁻¹] |
18.1 Temporal Phase as Informational State and Dynamics
Each node \( i \) in the discrete temporal flow network carries a phase \( \theta_i(t) \), defined modulo \( 2\pi \):
Dimension: dimensionless [1] (radians)
Update Rule:
\[
\theta_i(t + \Delta t) = \theta_i(t) + \omega_i \Delta t + \sum_j \left[ R_{ij} \sin(\theta_j(t) - \theta_i(t)) \right] + \delta_\text{asym} \cdot A_i(t)
\]
Where:
- \( \omega_i \): intrinsic angular frequency of node \( i \), dimensionless [1], physically \( \omega_i / T_c \) with dimension [T⁻¹]
- \( \Delta t \): discrete time step, dimensionless [1]
- \( R_{ij} \): coupling strength from node \( j \) to \( i \), dimensionless [1]
- \( \delta_\text{asym} \): asymmetry parameter encoding CPT violation effects (Sections 9.1.2, 12.3), dimensionless
- \( A_i(t) \): local temporal asymmetry biasing phase flow direction, dimensionless
18.2 Phase Differences and Physical Relationality
The fundamental measurable quantity is the phase difference:
\[
\Delta_{ij}(t) = (\theta_i(t) - \theta_j(t)) \bmod 2\pi
\]
Dimension: dimensionless [1]
Interpretation: Values restricted to \( (-\pi, \pi] \). Phase differences acquire physical meaning only when recorded relationally and compared within a closed loop.
18.3 Minimal Recording System and Informational Bootstrap
Measurement consistency requires closure in phase recording across minimal relational systems. For three causally connected nodes \( \{i, j, k\} \), the consistency condition is: \[ \Delta_{ij} + \Delta_{jk} + \Delta_{ki} = 0 \bmod 2\pi \] This closure enforces a coherent causal fabric and constitutes the "informational bootstrap" underpinning emergent reality in TFP.
18.4 Spatial Metric from Phase Distance
Emergent spatial geometry is reconstructed from phase-based relational data.
Define the angular phase distance:
\[
d_{ij} = \min\left( |\Delta_{ij}|, 2\pi - |\Delta_{ij}| \right)
\]
Dimension: dimensionless [1]
Properties:
- Non-negativity
- Symmetry: \( d_{ij} = d_{ji} \)
- Triangle inequality: \( d_{ij} + d_{jk} \geq d_{ik} \)
18.5 Measurement as Synchronization Event
Measurement corresponds to phase synchronization between observer and system nodes.
Before synchronization, the emergent wavefunction \( \Psi(x, t) \) (see Section 10.3.2) encodes probability over phase relations.
Synchronization triggers rapid phase alignment, collapsing differences into recorded values in observer memory.
This is implemented by:
\[
\theta_k(t + \Delta t) = \theta_k(t) + G(\Delta_{ij}(t - \tau))
\]
with typical update form:
\[
G(\Delta) = \alpha \sin(\Delta) + \beta \cos(\Delta)
\]
for dimensionless parameters \( \alpha, \beta \).
18.6 Planck’s Constant and Phase Resolution
Discrete causality imposes a minimal resolvable phase increment:
\[
\Delta \theta_\text{min} = \frac{2\pi}{N_\text{max}}
\]
Where \( N_\text{max} \) is the number of distinguishable phase divisions (dimensionless).
Planck’s constant \( \hbar \) emerges as a conversion factor:
\[
\hbar \sim \Delta \theta_\text{min} \cdot S_\text{unit}
\]
with \( S_\text{unit} = \hbar_c = E_c \cdot T_c \) from Section 1.
This links quantum discreteness to the network’s informational resolution capacity.
Summary
Measurement in TFP is a relational synchronization event. Observable quantities arise from coherent phase relations across a discrete causal network. Spatial geometry emerges from phase distances, while quantum collapse reflects causal memory updates. Planck’s constant links to the network’s minimal phase resolution, rooting quantum physics in informational flow.
Comments
Post a Comment