Coherence Map — Thermodynamics
TriadicFrameworks /docs/theories/thermodynamics/coherence_map.md#
This map defines how coherence behaves across temperature, entropy, free energy, flows, equilibrium, and RTT regimes.
1. Coherence Dimensions#
Thermodynamic coherence is evaluated across five substrate‑level dimensions:
1.1 Constraint Coherence#
- validity of state variables
- consistency of constraints (T, S, F, U, V, P)
- non‑negativity of entropy
- ensemble‑consistent definitions
1.2 Potential Coherence#
- convexity of free‑energy surfaces
- stability of minima
- well‑defined gradients
- ensemble‑appropriate potentials (F, G, Ω)
1.3 Gradient Coherence#
- flows follow gradients
- directionality preserved
- no oscillatory or mechanical drift
- monotonic relaxation
1.4 Entropy Coherence#
- monotonicity (dS/dt ≥ 0)
- valid regime boundaries
- correct open‑system behavior
- irreversibility structure
1.5 Equilibrium Coherence#
- fixed‑point structure
- ∇F = 0
- dS/dt = 0
- stability via second‑derivative tests
2. Coherence Levels (C0–C4)#
C0 — Incoherent#
- constraints violated
- entropy negative or undefined
- free‑energy surfaces non‑convex
- flows not gradient‑aligned
C1 — Weak Coherence#
- constraints partially valid
- entropy monotonicity fragile
- gradients noisy or inconsistent
- equilibrium unstable
C2 — Moderate Coherence#
- constraints valid
- free‑energy surfaces mostly convex
- flows gradient‑aligned
- equilibrium stable but sensitive
C3 — Strong Coherence#
- full constraint integrity
- convex potentials
- monotonic flows
- stable equilibrium fixed‑points
C4 — Perfect Coherence#
- idealized constraint geometry
- perfectly convex potentials
- exact monotonicity
- globally stable equilibrium
C4 is theoretical; real systems approach C3.
3. Coherence Field#
The coherence field is a gradient over:
- constraint validity
- potential convexity
- gradient alignment
- entropy monotonicity
- equilibrium stability
High gradients indicate coherence instability, typically near:
- phase transitions
- constraint changes
- ensemble switches
- environment coupling
4. Collapse Modes#
Thermodynamic coherence fails through four canonical collapse modes:
M1 — Constraint Collapse#
- invalid state variables
- negative entropy
- inconsistent ensembles
M2 — Potential Collapse#
- non‑convex free‑energy surfaces
- unstable minima
- undefined gradients
M3 — Gradient Collapse#
- flows not aligned with −∇F or −∇T
- oscillatory or mechanical drift
- loss of directionality
M4 — Entropy Collapse#
- dS/dt < 0
- irreversibility violated
- open‑system inconsistency
5. RTT Regime Coherence#
R1 — Constraint Substrate Regime#
Coherence strongest.
- constraints fundamental
- entropy monotonic
- free‑energy convex
- flows gradient‑aligned
R2 — Statistical Mechanics Regime#
Coherence refined.
- microstates explicit
- partition functions define potentials
- fluctuations appear
R3 — Field‑Theoretic Regime#
Coherence embedded.
- free energy field‑dependent
- phase transitions field‑level
- vacuum structure influences stability
R4 — Cosmological Regime#
Coherence geometric.
- temperature geometric
- entropy horizon‑scale
- equilibrium cosmological
6. Diagnostics#
A thermodynamic system is coherent when:
- S ≥ 0
- dS/dt ≥ 0
- free‑energy surfaces convex
- flows follow gradients
- equilibrium is a fixed‑point
A system is incoherent when:
- constraints violated
- entropy decreases
- potentials non‑convex
- flows misaligned
- equilibrium unstable
Summary#
Thermodynamic coherence is:
- constraint‑first
- potential‑structured
- gradient‑aligned
- entropy‑monotonic
- equilibrium‑fixed‑point
- RTT‑dependent
Coherence is strongest in R1, refined in R2, embedded in R3, and geometric in R4.