Coherence Map — Thermodynamics
TriadicFrameworks /docs/theories/thermodynamics/coherence_map.md#
Thermodynamics is the R1 constraint‑first substrate grammar of the RTT stack. Coherence in Thermodynamics refers to the structural integrity of constraint geometry, potential surfaces, gradient flows, and monotonicity. It does not refer to mechanical stability, particle motion, or kinetic behavior.
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.