Chapter 3

Entropy as surviving history space

In RSG, entropy is the Shannon readout of the normalised survival measure over generated histories. It measures how much unresolved, survival-compatible history space remains represented after filtering.

Where Entropy Enters

Entropy is not added as primitive disorder. It is applied after the survival measure has already been defined.

generated histories
  -> survival weights
  -> normalised representation measure
  -> Shannon scalar

Shannon Readout

For represented history i, the Shannon information content is -log_b p_i. The average information content of the represented measure is:

I_b(i) = -log_b p_i
H_b(p) = -sum_i p_i log_b p_i

Because the RSG survival weights use exp(-A_i), the natural-logarithm form is the default in the entropy note.

H_surv(t) = -sum_i p_i(t) ln p_i(t)

RSG Survival Form

Substituting the survival-normalised weights gives a compact RSG form that connects entropy directly to accumulated loss and the survival normaliser.

p_i(t) = exp(-A_i(t)) / Z(t)
Z(t) = sum_j exp(-A_j(t))

H_surv(t) = _p + ln Z
_p = sum_i p_i A_i

Entropy Flow

A history gains represented weight when its current effective loss is below the represented average. This turns survival filtering into an explicit concentration rule.

lambda_i(t) = Gamma_i(sigma_i(t)) W_i(phi_i(t))
dot p_i = p_i(_p - lambda_i)

lambda_i < _p  =>  dot p_i > 0

The concise entropy-flow identity is:

dot H_surv = -Cov_p(A, lambda)
           = -Cov_p(A, Gamma W)

Effective Live Histories

The visualisation's Nlive readout is the intuitive count-like form of entropy.

N_eff(t) = exp(H_surv(t))

High H_surv

Many survival-compatible histories remain live and unresolved.

Low H_surv

The survival filter has concentrated representation into fewer dominant basins.

Openness And Closure Diagnostics

For a finite active family of N histories, entropy can be normalised into an openness index and a complementary survival-closure index.

H_max = ln N
O_surv = H_surv / H_max
C_surv = 1 - H_surv / H_max

This does not make entropy identical with closure. It makes entropy the scalar diagnostic from which a closure index can be derived after the active family or coarse-graining has been fixed.

Topological And Continuous Coarse-Graining

Histories can be grouped by closure class, macroscopic bin, or continuous base measure. The safe statement is always relative to the chosen history family, base measure, and coarse-graining.

p_C(t) = sum_{i : class_D(X_i) = C} p_i(t)
H_class(t) = -sum_C p_C(t) ln p_C(t)

dP(gamma) = (exp(-A[gamma]) / Z) dmu(gamma)
H_surv^mu = -integral p(gamma) ln p(gamma) dmu(gamma)