Section 8: Mapping Emergent Field Parameters to Fundamental Flow Quantities

-

Section 8: Mapping Emergent Field Parameters to Fundamental Flow Quantities

8.1 Overview and Motivation

This section establishes explicit functional relationships between emergent field parameters and the discrete flow dynamics of the Temporal Flow Physics (TFP) framework. These parameters are not inputs but arise as coarse-grained outputs of local flow interactions.

Fundamental Inputs:

  • Fi(n): Flow at node i at discrete time n (dimensionless)
  • α(i,j,n): Exchange strength between nodes i and j (dimensionless)
  • Ci: Processing capacity at node i (dimensionless)
  • Vinternal(F): Local internal potential function (dimensionless)

Emergent Field Parameters:

  • vcond: Vacuum expectation amplitude (dimensionless)
  • λpot: Potential curvature (dimensionless)
  • g: Coupling constant (dimensionless)

8.2 Vacuum Expectation Value: vcond from Internal Potential Structure

8.2.1 Origin from Vinternal(F)

The internal potential function has a double-well form:
Vinternal(F) = (F - F0)² × (F - F1)² (dimensionless)
Here, F0 and F1 represent attractor flow values.

8.2.2 Link Amplitude from Flow Differences

The local emergent amplitude between connected nodes i and j:
Aij(n) = |Fj(n) − Fi(n)| × Cij(n) (dimensionless)
In stable regimes with node flows near attractors:
Aij(n) → |F1 − F0| × Cij(n) (dimensionless)

8.2.3 Coarse-Grained Order Parameter

Define regional order parameter Φa(n) as:
Φa(n) = Σ(i,j ∈ region a) [wij × Aij(n) × exp(i θij(n))] (complex, dimensionless)
Time-averaged amplitude magnitude approximates:
⟨|Φa(n)|⟩ ≈ ⟨wtotal⟩ × ⟨Aij(n)⟩ (dimensionless)

8.2.4 Explicit Formula for vcond

First-order relation:
vcond = ηgeom × |F1 − F0| × ⟨Cij⟩ (dimensionless)
More detailed form:
vcond = |F1 − F0| × sqrt(⟨α⟩ / αcritical) × ⟨coherence_efficiency⟩ (dimensionless)
Where:
ηgeom: geometric averaging factor (dimensionless)
αcritical: threshold for sustained coherence (dimensionless)

8.3 Potential Curvature: λpot from Network Stiffness

8.3.1 Macroscopic Potential Form

The emergent field potential takes the form:
V(Φ) = λpot × (|Φ|² − vcond²)² (dimensionless)
λpot quantifies resistance to deviation from vacuum.

8.3.2 Microscopic Curvature at Minima

Internal potential curvature at minima is:
κinternal = second derivative of Vinternal with respect to F at F = F0 or F1 = 2 × (F1 − F0)² (dimensionless)

8.3.3 Amplification from Collective Stiffness

Flow resistance is scaled by condensed node density and link connectivity:
λpot = κinternal × ρeffective × connectivity_factor (dimensionless)
Where:
ρeffective = number of condensed nodes / condensed volume (dimensionless node density)
connectivity_factor = average node degree / max degree (dimensionless)

8.3.4 Explicit Formula for λpot

λpot = 2 × (F1 − F0)² × (Ncondensed / Vcondensed) × (⟨Ci⟩ / Cmax) (dimensionless)
Alternative expression:
λpot = (2 / vcond²) × (F1 − F0)⁴ × ⟨processing_density⟩ × ⟨capacity_utilization⟩ (dimensionless)

8.4 Coupling Constant: g from Exchange Dynamics

8.4.1 Misalignment Resistance

From Section 6, coupling action Scoupling depends on phase misalignment:
Scoupling = −(2 / g²) × Σneighbors ⟨a,b⟩, n |Φa(n) − Rab(Φb(n))|² (dimensionless)
g controls resistance to phase misalignment between regions.

8.4.2 Dependence on Exchange Strength

Stronger exchange α and tighter capacity utilization Ci increase coupling rigidity.

8.4.3 Processing Load Ratio

Define processing stress as:
processing_stress = ⟨Σj α(i,j,n)⟩ / ⟨Ci⟩ (dimensionless)
Approaching 1 signals nonlinear response increase.

8.4.4 Explicit Formula for g

g² = (⟨Ci⟩ / ⟨Σj α(i,j,n)⟩) × βcoupling × topology_factor (dimensionless)
Simplified proportionality:
g² ∝ ⟨Ci⟩ / ⟨α_total(i,n)⟩ (dimensionless)
Refined formula incorporating fluctuations:
g² = (Cchar / αchar) × (1 + δ_fluctuations) (dimensionless)
Where:
δ_fluctuations = temporal variance in exchange rates (dimensionless)

8.5 Cross-Parameter Relationships

8.5.1 Internal Consistency Constraints

vcond² ∝ (⟨Ci⟩ / ⟨α_total⟩) × |F1 − F0|² (dimensionless)
Product of coupling and potential curvature:
g² × λpot = 4 × (F1 − F0)² × network_efficiency (dimensionless)
Emergent mass squared:
m² = λpot × vcond² = 2 × (F1 − F0)⁴ × (ρnodes × ⟨Cij⟩²) (dimensionless)

8.5.2 Scaling Laws

vcond scales linearly: vcond ∝ |F1 − F0|
λpot scales quadratically: λpot ∝ (F1 − F0)²
Coupling inversely related: g⁻² ∝ ⟨Ci⟩ / ⟨α⟩

8.6 Computational Implementation

8.6.1 Extraction Procedure

Measure from simulations or data:
F0, F1 from internal potential Vinternal(F)
⟨Ci⟩: time-averaged processing capacity
⟨α(i,j,n)⟩: network-wide exchange strengths
Compute parameters:
vcond = |F1 − F0| × sqrt(⟨α⟩ / αcritical) × coherence_factor
λpot = 2 × (F1 − F0)² × ρeffective × (⟨Ci⟩ / Cmax)
g² = (⟨Ci⟩ / ⟨α_total⟩) × βcoupling × topology_factor
Validate consistency:
Check g² × λpot ≈ 4 × (F1 − F0)² × network_efficiency
Confirm m² = λpot × vcond² scales as expected

8.6.2 Time Evolution

Parameter temporal derivatives approximate as:
∂vcond/∂t ∝ ∂(F1 − F0)/∂t + ∂⟨coherence⟩/∂t
∂λpot/∂t ∝ ∂ρnodes/∂t + ∂⟨capacity_utilization⟩/∂t
∂g/∂t ∝ ∂(C/α ratio)/∂t

8.7 Testable Predictions

8.7.1 Parameter Regimes

  • Strong coupling: g² small → λpot large → sharp phase transitions
  • Weak coupling: g² large → λpot small → smooth transitions
  • Low capacity: ⟨Ci⟩ small → g² small → high sensitivity to interactions

8.7.2 Network Size Scaling

  • vcond is approximately constant (intensive quantity)
  • λpot scales with node density (N / V)
  • g² depends on topology, largely independent of node count

8.7.3 Criticality Indicators

  • As ⟨α_total⟩ approaches ⟨Ci⟩:
  • g → ∞ (coupling diverges)
  • λpot → 0 (potential flattens)
  • vcond → 0 (condensate dissolves)

8.8 Summary Table

Parameter Depends On Interpretation
vcond F1 − F0 Vacuum expectation amplitude
λpot (F1 − F0)² × ρ_nodes × ⟨Ci⟩ Field stiffness / potential curvature
g² ⟨Ci⟩ / ⟨α_total⟩ × topology_factor Inter-region coupling strength

Comments

Post a Comment

Popular posts from this blog

The Ethics of two

-
Ethics as an Emergent Property of Interaction The Foundation: Five Core Behaviors Ethics begins with specific, observable interactions between people: Recognition: I see you as an autonomous person with your own needs, boundaries, and goals - not as an object or resource to be used. Attention: I listen to what you're saying and pay attention to the signals you're sending about your state and needs. Respect: I refrain from harming you and acknowledge your right to make your own choices about your life. Trust: We build reliability through consistent actions - when you say something, you follow through, and so do I. Care: I invest energy in your wellbeing because I value your existence and flourishing. These aren't rules imposed from outside. They're what naturally develops when two people encounter each other and choose to interact constructively rather than destructively. The Process: How Ethics Emerges The First Encounter If I were the only person in the...
Read more

Temporal Physics: A New Framework

-
Temporal Physics: A New Framework Table of Contents 1. [Fundamental Framework](#1-fundamental-framework) 2. [Emergence of Space](#2-emergence-of-space) 3. [Fields and Force Dynamics](#3-fields-and-force-dynamics) 4. [Particle Properties](#4-particle-properties) 5. [Quantum Entanglement](#5-quantum-entanglement) 6. [Quantum-Classical Transition](#6-quantum-classical-transition) 7. [Observable Phenomena](#7-observable-phenomena) 8. [Experimental Implications](#8-experimental-implications)  1. Fundamental Framework Time as the Fundamental Entity - Traditional physics treats space as primary; we invert this paradigm. - Core Assertion : Time flows generate spatial dimensions, implying that space is a byproduct of temporal interactions. - The interplay of temporal flows shapes all physical phenomena, creating a dynamic framework that transcends classical interpretations.  Core Principles 1. Time is Primary, Space is Emergent : Space does not exist independently but arises from the c...
Read more

Thinking Through Tools: AI, Cognition, and Human Adaptation

-
I see intelligence not as an absolute measure of “smarter” or “dumber,” but as something fundamentally shaped by the tools and environments we operate within. Consider a carpenter and a caveman: each has developed cognitive architectures optimized for their respective toolsets. The carpenter excels with saws, levels, and electrical tools, while the caveman masters rocks, fire, and natural materials. Neither is inherently more intelligent—their minds have simply organized around different problem-solving contexts. This tool-dependent view of intelligence becomes crucial when examining AI adoption. People using AI may lose proficiency at certain tasks when performed without assistance—just as the carpenter might struggle to build shelter using only natural materials. But this is not cognitive decline; it’s cognitive reallocation. The brain shifts resources from one set of skills to optimize for new challenges and opportunities. When someone transitions from traditional cognitive work to ...
Read more