ENTROPY (RDG)
Entropy = lost information when a system changes irreversibly.
All entropy formulas are projections of one operator:
R_discard(f) = S(X) − S(Y)
A process f : X → Y discards microstate detail → entropy increases.
SID : Microstate Geometry (Boltzmann)
S_SID ≈ k_B ln Ω
ASCII:
Before: [ o o o o o o o o ]
After : [ o o o o ]
Loss = ln(volume_before / volume_after)
Interpretation:
• shrinkage of admissible microstate volume
• geometric loss under coarse-graining
PED : Probability Flattening (Shannon)
S_PED = −∑ p_i log p_i
[0.8, 0.1, 0.1] → [0.33, 0.33, 0.33]
Loss = flattening cost
Interpretation:
• diffusion of probability mass
• irreversible spreading
Q : Thermodynamic Cost (Heat / Temperature)
ΔS = ∫ (δQ_rev / T)
[tight] → [loose] → [cannot return]
Interpretation:
• irreversible heat cost
• boundary constraints on allowed transformations
UNIFIED RDG TRIANGLE
( SID )
microstate geometry
▲
|
( Q ) ◄-- R_discard --► ( PED )
thermodynamic probability
cost flux
MASTER OPERATOR
R_discard : State → ℝ₊
R_discard(X → Y) = information lost under f
Projections:
S_SID = π_SID ∘ R_discard
S_PED = π_PED ∘ R_discard
S_Q = π_Q ∘ R_discard
One operator.
Three coordinate systems.
Same invariant everywhere.
R_discard is the one invariant behind entropy. SID sees lost microstate volume, PED sees distribution flattening, Q sees irreversible cost. Three projections, one operator.