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.